MR. BIRAM'S IMPROVEMENTS The accompanying figures show the system as applied according to the first arrangement, to the propelling of a railway locomotive carriage. Fig. 1 is a side elevation, partly in section, of the engines; fig. 2, an end elevation; and fig. 3, a plan. The engines are proposed to be placed fore and aft, and worked in connection by means of double cranks. Mr. Biram illustrates the advantages of his system by several other exemplifi IN OSCILLATING ENGINES. cations. In one it is applied to the water supply pump of a locomotive engine, worked by an eccentric on the shaft of the driving-wheels. In another it is adapted to a fire-engine, whereby "the same work would be performed with one cylinder, which is now done by two of the same diameter, the friction would be reduced one-half, and a rotary mode of working be substituted for the reciprocating." Another arrangement for adoptFig. 2. ing the system to a fire-engine is shown, in which an upright lever is introduced and the piston worked by a rowing action, which is allowed on all hands to be the best of any. Mr. Biram observes farther of this last arrangement, "It is also well-adapted for a ship's pumps, in which case the pipe above the valve-box should be conveyed through the ship's side, as near as conveniently may be above the load water line; it will be also evident that, by attaching a horizontal rod to the end of the upright lever, and passing handles through it at convenient distances (say about a yard asunder), a great number of men might be conveniently employed at the same pump; and further, that by attaching an air vessel, it would at once become a powerful fire-engine, by which the water could be sent to any part of the vessel." Mr. Biram's last exemplification is an air-pump, "by turning the handle of which in one direction, it is a condensing pump; and by reversing the motion, it becomes an exhausting one." Mr. Biram's claim is-1st, to "the con 4 METHOD OF COMPUTING THE DIRECT DISTANCE BETWEEN structing of oscillating engines, whether worked by steam, water, or any other fluid, with passages placed as before described for the entrance and exit of the steam, water, or other fluid, to and from the cylinders, and opened or closed, wholly or partially in manner before described, according as the progress of the cranks, and the state of the steam or the cylinders may require." And, 2nd, "the application of the same peculiar system of action to water-pumps, air-pumps, fire-engines, and other machines and instruments employed in the raising or propelling of fluids as before respectively exemplified and described." METHOD OF COMPUTING THE DIRECT DISTANCE BETWEEN ANY TWO PLACES ON THE SURFACE OF THE EARTH, THE CORRECT GEOGRAPHICAL POSITIONS, OR THE LATITUDES AND LONGITUDES OF THOSE PLACES BEING KNOWN. -APPLICABLE MORE PARTICULARLY TO THE RUN OF STEAM SHIPS. In reading the published accounts of the performances of some of the ocean steamers of the larger class that go on lengthened voyages, the distances they are said to run between one observation and another, when compared with the times that are stated to be occupied in accomplishing those distances, are sometimes very astonishing and eminently calculated to excite our surprise; and when placed in juxta-position with the performances of the swiftest sailing vessels, we are apt to imagine that the officers on board have fallen into some error in registering their remarks. It ought however to be considered, that steamers possess an immeasurable advantage over sailing vessels in running a direct course, being comparatively but little influenced either by wind or tide; whereas, a sailing vessel when working up against a head wind, being obliged to traverse on oblique courses, has both the distance and the time of accomplishing it greatly lengthened. In fact, the method of conducting a steam ship from one place to another, under any circumstances, is altogether a different thing from that of conducting a vessel under sails; and in this particular case, as well as in many others when the elements happen to be adverse, the practice of navigation has undergone a very complete and remarkable change; so much so indeed, that the difficult and intricate manoeuvring of fifty years ago on a stormy and turbulent ocean is now in a great measure assimilated to the mimic operations of school-boys on the bosom of a placid lake or a gently flowing river. The problem which enables us to test the correctness of steam-boat logs, in so far at least as the distances are concerned, is that which assigns the direct distance between any two places on the surface of the globe, when their true geographical positions, or their latitudes and longitudes, are known. There is nothing particular in the problem itself, either as regards the difficulty of its solution, or the interest that attaches to it, being in reality well known to all the readers of general geography; but from its peculiar applicability to the subject we are now considering, we have been induced to propose it in an insulated form, independent of all other problems of a kindred nature, thinking that by so doing it will be more likely to meet the eye and arrest the attention of our readers. It is generally understood, that the nearest direct distance between any two places on the surface of the earth, is an arc of a great circle passing through those places, and such, that if its plane were produced, it would pass through the centre of the globe and cut it into two equal and similar parts. And in like manner, if the planes of the meridians passing through the poles and each of the two places were produced, they would respectively pass through the centre of the globe and divide it into two equal and similar parts. But supposing that each of the planes above mentioned should extend no further than to the centre, they would cut out a solid gore or portion of the sphere, separated into two triangular pyramids supplemental to each other, and whose bases are portions of the spheric surface. It is to these pyramids so constituted, that we must apply for the solution of the present problem. Since the latitudes and longitudes of the two places are given, the compliments of the latitudes and the sum or difference of the longitudes, according as the one is east and the other west, or both east, or both west; are also given, the former constituting two sides of a spherical triangle, and the latter the angle at the pole comprehended between those sides, or be 1378 D ANY TWO PLACES ON THE SURFACE OF THE EARTH, ETC. tween the meridians passing through the places; the third side of the triangle being that which the problem requires us to determine-namely, the direct distance between the places proposed, and subtending the angle at the pole which is formed by the sum or difference of longitudes. In all cases where the spheric projections are not clearly understood, the development of the above-mentioned pyramids on a plane is the most readily comprehended, and from such a development the rules of calculation for the several cases into which the problem divides itself are deduced with the greatest ease and facility; this, therefore, is the method which we intend to employ in the present instance, and for this purpose, let the straight line C A be drawn in any direction and to any distance at pleasure; and in it assume the point C as the centre of the sphere or vertex of the triangular pyramid; then A fan a B tan c tan a at the point C, make the angles I C B and B C G respectively equal to the given co-latitudes. From the point A, any how assumed in CA, let fall the perpendicular A B, which produce to meet the extension of C G in F. At the point B, make the angle F B H equal to the given sum or difference of longitude, and on B as a centre, with the distance B A as radius, describe the circular arc A PB meeting B H in the point H, and draw HF. Then upon the straight line C F, 5 which is given, construct the triangle F C D, of which the sides C D and FD are respectively equal to C A and F H; then is D C F the angle at the centre of the sphere, which is measured by the arc GK, the required distance between the places. That the preceding is the true development of the triangular pyramid abstracted from the sphere as already described, will be most clearly understood from its reconstruction. Thus, let the figure A CDF HB, be drawn on pasteboard, and correctly cut out from the card along the several bounding_lines A C, CD, D F, FH, H B and B A; then let the figure thus separated be folded up upon the lines C B, C F, and F B, and it will be found that the points A, D, and H coincide, as also do the lines A C, D C, and A B, H B; thus constituting the triangular pyramid developed by the preceding process; the circular arcs IB, B G and G K being the co-latitudes and distances between the places respectively. Let C B, the common intersection of the planes A C B and F C B, be made the radius; then will C A and C F be the secants of the angles B C A and BCF to that radius, and A B, B F the tangents; and if the radius C B be assumed equal to unity, the several lines just referred to will be represented simply by the trigonometrical quantities or angular functions named on the respective lines; the angles B C A, B C F and DCF being indicated by the small Italian letters a, c, b, placed therein, while the angle FB H is denoted by the Roman capital B. Now, the solution of the problem requires the determination of the angle F C D=b, in the plane triangle CFD; but by examining the conditions, we find that the two sides CD and C F, containing the required angle, only are given, and these of themselves are insufficient to determine the angle; but by the principles of construction, the side F D, which subtends the required angle, is equal to FH in the plane triangle F B H, and this we can readily find from the data, which are the two sides B F, B H and the contained angle F B H-B. By the principles of plane trigonometry, it is FH = √BF2 + B H2± 2 B F. B H cos. FB H; that is, FH √tan.2a+tan.2 c +2 tan. a tan. c. cos. B. 6 METHOD OF COMPUTING THE DIRECT DISTANCE BETWEEN Here, then, we have the side F H expressed in known terms, but F D is equal to FH; hence it follows, that in the plane triangle F C D, all the sides. are given to determine the angle subsec. a + sec.2c-tan.2 acos. b. = tended by the side F D, and the writers on trigonometry have shown that the formula for that purpose is, cos. F C D= CF2+ CD2F D2 but by the property of the circle in reference to the secant and tangent of an arc or angle, we have sec.2 a -tan.2 a = 1, and sec.2c-tan.2c=1; therefore by substitution, ; ; that is, cos. bcos. B sin. a sin. c cos. a cos. c. This, then, is the final form of the equation, for calculating the direct distance between any two places on the surface of the earth, where it must be observed, that a is the complement of one latitude, and c that of the other, and that the minus or plus sign is to be employed, according as the angle B, or the side a or c is greater or less than ninety degrees, the cosines of angles between 90° and 180° being negative. The same formula, it is manifest, will apply to the determination of the distance between the sun and moon, or between the moon and a fixed star, when their right ascensions and declinations are known; so that this problem, when taken in connexion with that at page 410, vol. xlii., constitutes one of the best methods of finding the longitude of a ship at sea, and that which is generally known to seamen by the name of Lunar Observations. Our present object, however, is only to determine the distance between places on the earth, and for this purpose, we must consider the problem under the various cases into which it naturally divides itself, and these are, 1. When the latitudes are both north or both south, and the sum or difference of the longitudes less than ninety degrees or a right angle. 2. When the latitudes are both north or both south, and the sum or difference Longitude of Halifax 63° 37' 30" West. Longitude of Liverpool 2 59 30 West. of the longitudes greater than ninety degrees or a right angle. 3. When the latitudes are, the one north and the other south, and the sum or difference of the longitudes less than ninety degrees or a right angle. 4. When the latitudes are, the one north and the other south, and the sum or differences of the longitudes greater than ninety degrees or a right angle. These are the four distinct cases of the problem, and the method of applying the above general formula to the resolution of each case, will become manifest from what follows. EXAMPLE 1. What is the direct distance in English miles, between St. Paul's Church at Liverpool, and the pillar in the Dockyard at Halifax, North America, the latitudes being respectively 53° 24′36′′ and 44° 39′ 24′′ north, and the longitudes 2° 59' 30" and 63° 37' 30" west of Greenwich? Here the latitudes are both north and the longitudes both west, which corresponds to the first of the preceding cases, and also to that represented by the construction; for since the longitudes are both of one name, the angle contained between the meridians of the two places is equal to the difference of the longitudes, and being less than ninety degrees, the conditions of the first case are completely satisfied. The operation is therefore as follows: Diff. of longitudes...... 60 38 0 Log. cos. 9.690548 Nat. num.................. + 0.20793 Log. +0.56434 add Nat. cos. b+ 0.77227; therefore we have b=50° 33′ 30′′=3033.5 geographical miles; but the geogra |