and the adverfaries of revelation, will feize with avidity, as arguments in favour of their avowed principles. The interpretations of Scripture which are agreeable to reafon and common fenfe, or are clearly pointed out in other parts of the facred writings, a mind truly difinterested and candid will readily admit; but the fables of fuperftition adopted by the Jewish Rabbies, and the dreams of enthufiafm which fome of the myfterious and allegorical Writers have advanced, as the genuine fenfe of the word of God, ought to be uniformly and univerfally exploded as injurious to the effential interests of unmingled Chriftianity.

An Account of the Mathematical and Phyfico-mathematical papers, in the fecond part of the fiftieth volume of the Tranfactions of the Royal Society, just published, for the year 1758. 4to. 12s. 6d. Davis and Reymers.

Article 63. Concerning the Fall of Water under Bridges. By J. Robertfon, F. R. S.

HE motion of Fluids in general is one of the moft

THU ufeful, though most intricate, branches of the Ma

thematics. It is therefore no wonder, that many celebrated Authors have made it their chief ftudy.-Neverthelefs, this fubject has not, as yet, been freed from all its difficulties. Sir Ifaac Newton, in the fecond book of his Principia, has given feveral propofitions for determining the quantity of water difcharged out of a veffel in a given time, through an orifice in the bottom; which is objected against by Dan. Bernouilli, as being only true in particular cafes. But though this Author has wrote very largely upon the fubject, and made a great number of experiments, to confirm his theory, he is, nevertheless, fubject to many objections;-indeed, the fame thing may be faid of almoft all other Writers..

When water is to be conveyed from one place to another, through pipes, it is highly neceflary to know the quantity that will keep them full, and no more, for fear of bursting them by an over-charge; and how much they will difcharge in a given time; which, though many experiments have been made at Paris and Verfailles, as may be feen in Mr. Belidor's Architecture Hydraulique; yet no general theory has hitherto been established, from which the quantity of water difcharged in different circumftances can be determined. The reafon is, that the various forms of pipes, their finuofities, and their rife

13

or

or fall, will ever prevent the discovery of univerfal rules applicable to every occafion.

When water is to be conveyed to a town, by rivers or canals, it is of the utmost importance to give the stream such a descent, as not to be too rapid, nor yet too flow; for a rapid course will deftroy the banks, and make it too expensive to keep them in conftant repair; and it may also happen that the fource of the ftream will not be fufficient for a conftant fupply. It is imagined, that if the fall be three inches in a mile, it is fufficient; which is faid to be the fall of the New River at Iflington.

When rivers are to be made navigable, or two rivers are joined by a canal, the defcent must be regulated in such a manner, as to be every where alike; and when this cannot be done, fluices are made in proper places, to fupport the water where the defcent is too great in refpect to the reft: and when bridges are to be built over large rivers, it is neceffary that the piers fhould not take up fo much space as to impede the current, and confine the water above, which might occafion its overflowing the adjacent country in rainy feafons, and cause fo great a fall below, as to endanger the bridge, and render the navigation hazardous.

In order to compute this fall, when the velocity of the ftream, the breadth of the river, and the water-way, are given, Mr. Robertfon thought it would not be unacceptable to the curious to give this problem, efpecially as the folution is not generally known. He lays down five propofitions by way of principles, upon which his folution depends; the third of which is, That water forced out of a larger channel through one or more fmaller paffages, will have the ftreams through thofe paffages contracted in the ratio of 25 to 21. This principle he has borrowed from Mr. Jones, and the demonftration is referred to Sir Ifaac Newton, Princip. B. ii. prop. 36.

As this propofition relates to water running through an orifice in the bottom of a veffel, it does not appear that its application to the fall of water in a river contracted by the piers of a bridge, is juft; becaufe the water is forced through the orifice, not only by the weight of that above it, but likewife by the weight of that contained in a hyperbolic folid, generated about its aflymptote, which is much greater than the column above the orifice. This does not happen in a river confined to a lefs breadth.-But not to contend upon principles which may be difputed and applied according to an Author's fancy, the truth of our objection will best appear from his own rule

for

for finding the height of the fall. He calls the breadth of the river in feet b; the mean velocity of the water in feet per fec. v; the breadth of the water-way between the obstacles c; and a the height of the fall in a fecond: from his principles

he finds

2561

21 C

2

ขบ
IX
4 a

for the height of the fall. Now as this rule is to be general, whatever b and c may be, it is evident that when bc, this expreffion fhould become nothing, fince it expreffes the difference between the fall of the water, when the breadth of the river is reduced by the piers, and that of the ftream, when it runs through its natural channel;

but in that cafe we get

25 21

2

1: which is a plain contradiction.

If the ratio of 21 to 25 be neglected, the rule thus cor

bb

บข

4 a

rected, viz. to -IX > comes nearer to the truth than the Author's; as will appear from the example of Westminsterbridge, where b 994, c=820, v2, and a = 16.1: Thefe values fubftituted into the expreffion above, gives .43 of an inch, which Mr. Labellye obferved to be the real fall; where as Mr. Robertfon's rule gives an inch and a tenth, and therefore above double to what it should be.

The example given of the fall under London-bridge is, we conceive, out of the question; for the velocity found above bridge is occafioned by the water-way, as it is now contract ed; whereas the velocity of the water before it is contracted fhould be known, according to the Author's rule: moreover it is poffible, that the fame velocity may be found above a cataract of any height; and therefore the velocity found above, when the bridge is built, cannot, we imagine, ferve to find the height of the fall under the bridge. That the example which the Author gives of London-bridge, agrees nearly with the real fall, is owing to two fuppofitions he makes, and which are not demonftrable; the one we have confidered already, and the other that the water-way is reduced from 236 feet to 196}, by the piles drove round the piers, remains to be proved; as the ftarlings were never exactly measured, and befides thefe piles reach very little above low-water mark.

Article 70. Trigonometry abridged, by Patrick Murdoch, A. M.

F. R. S.

The feveral branches of the mathematics have, within this century, been fo much cultivated and improved, that there

1 4

is

is fcarce any room left for new inventions; a fubject can hardly now be thought on, which Sir Ifaac Newton, his cótemporaries and followers, have not already treated of. What remains therefore for the prefent mathematicians, is to reduce their thoughts to a lefs compafs, and to render their demonftrations plainer and easier. In which application of their abilities, though an Author cannot difplay his genius in fo confpicuous a point of view, yet he may fhew his fagacity and judgment, by felecting and improving what is of real fervice to mankind, preferably to what is merely fpeculative; which valuable end this learned Author has chiefly in view, throughout all his performances.

Moft Writers upon spherical trigonometry have reduced the cafes of right-angled triangles to fixteen, and thofe of oblique triangles to twelve: but they never conceived, that the rules of plain trigonometry were applicable to the fpheric! for want of which their trigonometrical works were extended to lengths which the subject by no means required; rendering that science very tedious, and much more difficult than it would have been, had their principles been fewer, and no more than were neceflary.

Those who are converfant with this fubject, will be furprized to fee that our Author has given all the cafes, both of plain and fpheric trigonometry, in four quarto pages and a half; the whole being reduced to three theorems only, whose demonftrations are very fhort and clear: efpecially if the Reader makes ufe of figures cut out of card-paper, fo as to raise fuch parts as fall above the plane, and are marked with the lines which are to be confidered.

As the practical part of aftronomy chiefly depends on the computation of fpheric triangles, what the Author has given in this fhort paper, will be of the utmost confequence, by leffening the labour of calculation.

Article 73. Of the best form of Geographical Maps. By the fame Gentleman.

In 1746, Mr. Murdoch publifhed a fmall octavo, entitled, The Elements of univerfal Perspective; wherein he has shewn, that all kind of projections may be reduced to one common principle, which he has illuftrated by feveral examples. He likewife publifhed a thin quarto, containing tables of meridional parts, adapted to the true figure of the earth, and not to the fpheric form, as has been the cuftom: and from these tables, navigation and geography would have received confiderable 'improvements, had the menfurations in Peru been agreeable to what was expected from those made in France, and at the arctic circle.

In

In the present paper, he introduces a new construction of maps, by representing a part of the globe upon a conic furface, flattened into a plane, which he conceives will reduce linear and fuperficial measures, nearer to that on the globe, than any other projections whatsoever; the reafons will be best understood by the Author's own words.

When any portion of the earth's furface is projected on a plane, or transferred to it by whatever method of description, the real dimenfions, and very often the figure and pofition of countries, are much altered and mifreprefented. In the common projection of the two hemifpheres, the meridians and parallels of latitude do, indeed, interfect at right angles, as on the globe; but the linear diftances are every where diminished, excepting only at the extremity of the ⚫ projection: at the center they are but half their juft quantity, and thence the fuperficial dimenfions but one fourth part: and in lefs general maps this inconveniency will always, in fome degree, attend the Stereographic projection.

The orthographic, by parallel lines, would be ftill lefs exact, thofe lines falling altogether oblique on the extreme C parts of the hemisphere. It is ufeful, however, in defcribing the circum-polar regions: and the rules of both projections, for their elegance, as well as for their ufes in aftronomy, ought to be retained, and carefully ftudied. As to Wright's or Mercator's nautical chart, it does not here fall under our confideration: it is perfect in its kind.

After this the Author obferves, that the particular methods of projection propofed or used by geographers, are so various, that we might, on that very account, fufpect them to be faulty; and proceeding to fhew, upon what foundation his conftruction is to be made, he mentions the following properties.

1. The interfections of the meridians and parallels will be • rectangular.

< 2. The diftances north and fouth will be exact; and any ⚫ meridian will ferve as a fcale.

< 3. The parallels, where the line which generates the conic furface, interfects the quadrant, or any fmall distances of places that lie in thofe parallels, will be of their just quantity. At the extreme latitudes they will exceed, and in the mean latitudes, between the two foregoing interfections, they will fall fhort of it. But unless the zone is very broad, neither the excefs nor the defect will be any where ⚫ confiderable.

< 4. The