DESCRIPTION OF A LOCOMOTIVE SPACE BOILER, ON THE PATENT PLAN OF MR. JAMES JOHNSTON, OF WILLOW PARK, FIG. 1 is a side elevation of the boiler. A B is the furnace, L is one of Hosking's water gauges. D, the smoke-box, is made of extra length, in order to give greater effect to a practice which has been in use for two or three years in some of the locomotives, both on the English and Scotch lines, viz., the insertion of a plate or damper in the smokebox, in such a manner that the products of combustion are made to pass beneath the lower edge of the plate previous to their getting into the chimney. I first saw this practised in 1842, in the locomotives on the Glasgow and Greenock railway; but it has not yet come into general use, although a saving of fuel is effected by it. Fig. 2 is a front elevation of the boiler. Fig. 3 is a cross section, through the body of the boiler. Fig. 4 is a cross section through the three furnaces. Figs. 5 and 6 are separate sketches of the pieces of metal which close up the ends of the sheet-water spaces between the flues. The body of the boiler is 6 feet in length, and through it there pass twentyseven chambers or flues, as shown in fig. 3. Between each of these flues there is a sheet-water space, in. wide, 2 ft. deep, and 6 ft. in length. The outer flue on each side of the boiler (as indicated by the dark shading) is filled up with soot, or any other good non-conductor of heat. This is done on purpose to prevent the heat acting on the two large water spaces called descending water spaces, which are situated between the soot-filled flues, and the external casing of the boiler. The superiority of this kind of boiler depends on the mass of water within it being divided and kept in two different states or conditions. The joint area of a cross section of the two descending water spaces, is rather greater than the joint area of the cross sections of all the sheet-water spaces between the flues and around the furnaces. There are two divisions of the water, viz., the descending water spaces, and *For some previous remarks on the principle of this boiler, see p. 21 of our present volume. GREENOCK. BY THE INVENTOR.* the ascending water spaces. As to the condition of the water in those two kinds of spaces, I have already stated that no heat is allowed to act on the descending water spaces; therefore, the water in those spaces is in the state technically termed solid; that is, there is no steam amongst it. Now as the sheet, or ascending water spaces, are acted on by the fire, steam will be formed in them. Therefore, in the two kinds of water spaces, we have water in two different states or conditions; in the descending spaces it is solid; in the ascending spaces, it is continually being converted into steam; and as steam is 1728 times lighter than water, it is evident that the water in the descending spaces will rush down, and displace the steam from the ascending spaces, as the two kinds of water spaces have free communication with each other at top and bottom. B B, figs. 1, 2, and 4, are the conductors which convey the descending current of water to the sheet-water spaces of the furnaces. A A, figs. 1, 2, and 4, are the conductors which convey the steam and surplus water from the water spaces of the furnaces into the interior of the boiler. The advantages that are gained by the powerful currents of water that are continually rushing up the sheet-water spaces are as follows: First. The bubbles of steam are removed from the plates of the boiler the instant they are formed, and therefore the plates of the boiler are kept cool, and cannot be injured although the fire be urged to the utmost. Second. The formation of deposit is prevented, owing to the plates of the boiler being kept cool. It is the overheating of the plates of the common kinds of boilers which causes the deposits to form on them, owing to the soluble bicarbonate of lime being converted by the heated plates into the insoluble carbonate. Third. Fuel is saved, owing to the plates being cool, and consequently in a fitter condition to receive heat than when over-heated, as is the case in common boilers. Fourth. In consequence of the currents passing through the sheet-water spaces, these spaces can be made of much smaller dimensions than the water spaces of other boilers; and, consequently, boilers made on this plan are of less bulk and weight in proportion to their power than any boiler hitherto invented. The boiler represented in the accompanying engraving is intended for a locomotive to run on the narrow gauge railways. It has 750 square feet of heating surface in the flues, 135 square feet of heating surface in the furnaces, and 20 square feet of fire-grate; that is, 4 square feet more fire-grate, and 75 square feet more heating surface in the furnaces, than there is in the furnace of the large locomotives that work the incline on the Edinburgh and Glasgow railway. In consequence of the current in these boilers being produced by the difference in weight between the contents of the descending and ascending water spaces, it is evident that anything which would increase this difference would increase the rapidity of the current throughout the boiler. For this purpose I insert in these boilers the feed at the top of the descending water spaces, but under the surface of the water, so that the cold feed cannot affect the steam in the steamchamber. In consequence of this arrangement, the currents are accelerated, owing to the cold feed water increasing the density of the water in the descending water spaces. If the cold feed were inserted at the bottom of the boiler, under the ascending water spaces, or at the lower part of the furnace, the current throughout the boiler would be diminished, as the cold water, if inserted at these places, would tend to equalize the weight of the contents of the ascending and descending water spaces. The following is an extract from the report of a committee of the Franklin Institute, appointed to enquire into the cause of the explosion of the Richmond locomotive in the United States : "The large engines foam so much, especially when new, that every cock in the upper part of the boiler will indicate water, while the water spaces are filled with dry steam. In this case the gaugecocks become worse than useless-they are actually deceptive; while the dangerous state of things around the fire-box requires no comment. Surely our ingenious mechanics will not be at a loss to contrive a remedy for this defect; and in view of this, the Committee would respectfully invite their attention to the new form of boiler devised by Mr. Johnston, which involves a principle, as it appears to the Committee, perfectly fitted to meet the case." J. J. GEOMETRICAL EXTRACTION OF THE Amongst the numerous subjects connected with mechanical enquiries, there is no operation that more frequently requires to be performed than the extraction of the square root; yet, singular as it may appear, there is not a rule in the whole compass of arithmetic, of which the nature and application are less understood by the generality of practical men. This may be attributed to the circumstance, that the rationale of the process is of too abstract a character to come within the comprehension of the mere arithmetician; a position that will readily be admitted, when it is considered that the foundation of the rule cannot be traced without some knowledge of the elementary principles of algebra, the operation itself being nothing more than the analysis of an algebraic binomial product. It is true that tables of the squares and square roots of numbers have been computed to a sufficient extent, and with a degree of precision beyond anything that can ever be required in actual practice; but then such tables are not at all times accessible; and besides, cases may frequently occur where they do not directly apply. On all such occasions it is desirable to possess some rule of easy application, whereby to approximate to the root of any number proposed, and to serve as a check on the result obtained by the regular or common rule, when the operator is not sufficiently conversant with the subject, as to place an implicit reliance on his own performances. It is a rule of this nature that we now propose to supply, requiring no further knowledge of elementary geometry, than how to raise a perpendicular, and describe a circle. It is rather surprising, that in books of arithmetic and mensuration, where the use of the square root is taught, the method of approximating to the numerical value of the root by a geometrical construction should never be adverted to. There are several propositions in Euclid's Elements of Geometry from which the process may be inferred; but we are not aware that they have ever been made available for that purpose, either in text books for schools, or in books that are expressly designed to guide the operations of practice. The roots obtained in this way, when considered merely as geometrical magnitudes, are rigorously accurate; but when referred to numerical values measured from a scale, they are only approximative, and the degree of approximation is more or less accurate, according to the delicacy of the instruments, and the dexterity with which the process is performed. One of Euclid's propositions from which the operation may be inferred is the 14th of the second Book, where it is required "to describe or make a square equal to a given rightlined figure." Now, it requires no great stretch of judgment to perceive that the side of this square is nothing else than the square root of the quantity by which the area of the right-lined figure is expressed; and, consequently, the process of determining an equal square is nothing else than the geometrical extraction of the square root; for it is only the side of the square, and not the square itself that the process determines. When a number is proposed, of which it is required to approximate to the square root, we have only to consider that number as an area equivalent to the rightlined figure in the proposition, and then the side of an equal square will be the root of the number given. The operation is performed in this way:-the proposed number is divided into any two factors, of such convenient magnitudes as are suitable to the size of the paper on which the figure is to be delineated. It is of no consequence, in other respects, what factors be taken, provided that their product is precisely the same as the given number; but the nearer the factors are to an equality, the better; indeed, if they happen to be equal, either of them is the root sought, and in that case no construction is necessary; but when they are unequal, as will generally happen, draw the straight line A B, Fig. 1. SI of any length at pleasure, and in it take any point, P, at which erect the perpendicular P S, produced as far as necessary; then, from the point P, and in the direction towards A, set off P R equal to one of the factors into which the given number has previously been divided, the other being set off from P towards B, and falling in the point Q. Bisect QR in C, and on C as a centre, with the distance C R or C Q as radius, describe the semicircle R D Q, meeting the perpendicular P S in D; then is PD the side of the square, or the root required. And this is rigorously correct, when considered geometrically, but only approximative when applied to a scale for numerical valuation. Now nothing can be more simple than the above process; and every workman is sufficiently acquainted with the use of his instruments to be able to erect GEOMETRICAL EXTRACTION OF THE SQUARE ROOT. the perpendicular P S, and describe the semicircle R D Q; and since this in every case is all that is required after the given number is separated into its factors, practical men will find it to their advantage to render themselves familiar with the process of construction; and if they will only take some pains in taking off the numbers, and use good instruments, the results in general will be more than sufficiently accurate for every practical purpose. We shall illustrate the principle by an example wherein the number whose root is required is separated into several pairs of factors. This is a favourable case for showing the general nature of the construction, as the result will be the same whatever factors be taken; and thus it will be seen that no difficulty will be encountered in separating the number in a proper way. Example.-Let it be required to approximate to the square root of 360, by a 69 construction similar to that above exemplified. Now 360 is divisible into several pairs of factors, some of which are very convenient for the purpose of delineation and others not, in consequence of the space that they would occupy on the paper, if taken from a scale of such dimensions as to render them appreciable; the several pairs of factors are as follows, viz.: 2, 180; 3, 120; 4, 90; 5, 72; 6, 60; 8, 45; 9, 40; 10, 36; 12, 30; 15, 24; and 18, 20-in all eleven pairs. It is, however, evident, that in a limited space, some of the first pairs are not convenient for construction, although in other respects they are equally so with the rest; we will therefore perform the process with the last five pairs 9, 40; 10, 36; 12, 30; 15, 24, and 18, 20; the last pair being nearly equal, shows that the root is somewhere between them, being greater than 18 and less than 20. Here follows the construction. Draw the straight line A B, fig. 2, of any convenient or indefinite length, in which assume any point, P, at pleasure; at the point P thus assumed, erect the perpendicular P S, which can be produced as far as may be required. Then from the point P as an origin, and in the direc tion P A, towards the left hand, set off from a scale of equal parts, the distances, Pa, Pc, Pe, Pg, and P i, respectively, equal to the series of greater factors 40, 35, 30, 24, and 20. And from the same point P, but towards B on the right hand, set off the several distances Pb, Pd, Pf, Ph, and Pk, respectively equal to the series of smaller factors 9, 10, 12, 15 and 18; then bisect the distances a b, c d, ef, gh and ik in the points 1, 2, 3, 4 and 5; 4 5P and on these points as centres, with half the corresponding distances as radii, describe the semicircles as in the figure; these semicircles will all intersect the perpendicular P S in the same point D, so that PD shall be the square root of 360, the number proposed; accurately when viewed in its geometrical character, but approximatively when valued numerically by applying it to the scale of construction; by measurement it will be found as nearly as possible equal to 19, the square of 19 being 361 instead of 360; the calculated root is 18 973, &c., carried to any degree of exactness at pleasure. This solution has been given on the supposition that the position of the point P is fixed, the corresponding factors originating there and extending on each |