(7.) What must be the depth of spirits in a vessel, so that if this depth be called one-tenth of an inch more than it really is, and multiplied by the area of the vessel, the result, diminished by one-half per cent. of itself, shall be exactly equal to the quantity of spirits present ? Answer. 19.9 inches.* (8.) If, by selling spirits at 12s. 4d. a gallon, 3 per cent. be lost, at what price must it be sold to gain 15 per cent. ? Answer. 14s. 74d. (nearly). (9.) The prime cost of 50 gallons of spirits is £25; 10 gallons are lost by leakage. At what price per gallon must the remainder be sold so as to gain 10 per cent. on the original cost ? Answer. 138. 9d. (10.) Divide 560 into two parts, such that of one part added to y of the other shall amount to 297. Answer. 190 and 370. (11.) Spirits were pilfered from a cask containing 128 gallons, at 24.6 per cent. O.P. The deficiency was made up with water, in consequence of which the strength fell to 8.4 O. P. What quantity of spirits was abstracted ? Answer. 16-6 gallons. (12.) A sample of cut tobacco on being dried loses 35 of its original weight; a sample of the leaf from which this tobacco was prepared, loses in drying 12 of its weight. How much water per cent. has been added by the manufacturer ? Answer. 35.39. (13.) The produce of a back of wash of the original gravity 52°, attenuated to 0., amounts to 11.6 proof gallons per cent. over the attenuation charge. Required the produce per cent. for every 5o attenuated. Answer. 1.12 gallons. (14.) Assuming that a back of wash produces 1 gallon of proof spirits for every 5 per cent. attenuated, and having a certain back of wash declared at 48° and attenuated to 3°, the produce of which is 10.4 per cent. of proof spirit, what was the true original gravity ? Answer. 550. (15.) How much rum, at 35 per cent, O.P., is needed to raise the strength of 12° gallons at 23 per cent. U. P. to 10 per cent. U. P. ? Answer. 35.24 gallons. (16.) A round's charge of malt duty amounts to £89 10s. 3d., of which 1 arises from net floor bushels, and the remainder from couch. Required the gross quantity of each. Answer. 165 gross floor bushels. ) 708.6 „ couch (17.) A quantity of spirit at a certain temperature weighs 8.849 lbs. per gallon. At a higher temperature a gallon of this spirit weighs 8.806 lbs. Calculato the increase of bulk per cent. Answer. 0.49. * This question, which is founded on the instructions relative to the modo of taking charges of spirits in certain receivers, may be solved without reference to the area of the vessel, and amounts simply to asking. What number is that which, if increasod by one-tenth, and the result dimi. pished by its own 200th part, will exactly reproduce itself? (18.) From a cask containing 86 gallons of spirits at 54 per cent. O. P., 20 gallons are drawn off, and the deficiency made up with water. Find the strength of the spirits remaining in the cask when this has been repeated 4 times. Answer. 466 U. P. (19.) By how much does the weight of 137 gallons of wort, at the gravity of 64°, exceed the weight of an equal quantity of water ? * Answer. 87.68 lbs. (20.) The standard silver of this realm contains 37 parts in 40 of fine silver and i lb. troy of standard silver is coined into 66s. What is the amount of money that can be coined from 100lbs. avoirdupois of fine silver, a pound troy being to a pound avoirdupois as 144 : 175 ? Answer. £433 11s. 2d. (21.) If the rents of a parish amount to £2340 178. 6d., and a rate of £137 10s. 8d. be levied, what portion of it must be paid by an estate of which the rental is £143 9s. 10d. ? Answer. £8 88.74d. (22.) The population of five parishes being 1,236, 452, 364, 516, and 3,430 respectively, find what the population of a sixth parish must be in order that the average population of the six may be 1256-5. Answer. 1,541. (23.) By selling malt at 8s. 4d, a bushel, a malt-factor clears 1th of the prime cost. He then raises the price to 88. 10d. How much does he clear per cent, at the latter price ? Answer. 177. * For the data necessary to the solution of this question, see page 105. CHAPTER III. PRINCIPLES OF MENSURATION; LOGARITHMS ; DESCRIPTION OF THE SLIDE RULE. Elementary Definitions.-Practical Geometry.--Methods of finding the Areas of Surfaces, Contents of Solid Bodies, and the Capacities of Vessels.-Nature and use of Logarithms.- Origin of the Slide Rule : construction and properties of its various scales. Mensuration, in its literal and more obvious sense, is simply the act or art of measuring; that is, the ascertaining by actual trial, how often one magnitude of known dimensions, such as an inch, a yard, a cubic foot, &c., is contained in any other magnitude of the same kind. But by a natural enlargement of meaning, the term Mensuration is also employed to designate a collection of rules and processes founded on the relation of algebra and arithmetic to geometry, by which, when certain parts or dimensions of a figure are given, it is possible without further measurement to infer or calculate the remaining parts or dimensions of that figure. Of the results thus deducible, the most important, as applied to the purposes of common life, are numerical expressions of the areas of planes or surfaces, the contents of solid bodies, and the capacities of vessels. The theory of Mensuration in general is a very extensive branch of mathematical science, and may be said properly to comprise the whole of Trigonometry, or the doctrine of the properties of triangles and angular magnitudes, as well as the construction of fixed or standard measures. It is, however, usual and convenient to treat trigonometry as a separate study, and to introduce into works specially devoted to Mensuration, only such problems relating to triangles as admit of being solved without reference to angular values. In the following pages it will not be necessary to explain more than the first principles of ordinary Mensuration, or to consider any portion of the subject at greater length than will be sufficient to afford the young officer an intelligent insight into the reasons of the methods adopted in practical gauging, so far as that art applies to the securing of the few existent duties of Excise. For this purpose all that is essential is an acqnaintance with decimal arithmetic, and a clear perception of the leading truths of geometry, such as are laid down in every edition of the Elements of Euclid. DEFs., &c. (1.) A figure is the general name given to any portion of space bounded by distinct outlines, regular or irregular, and having the form of a surface, a solid, or a hollow body. The drawing or representation of a magnitude is also called a figure. (2.) By a dimension of a figure is usually meant the measure of its lineal extension in any particular direction, as, for instance, its length, breadth or width, depth or height. The term may be applied to any distance measured in a straight line along, across, or through the space occupied by a figure, and, as a necessary condition, terminating in opposite points of its boundary, that is, completely traversing the figure. But in practice, the phrase, dimensions, is employed chiefly to denote those lines or distances which constitute the ordinary elements of the computation of areas or solid contents. Thus, the dimensions of a rectangle are commonly understood to be its length and breadth ; of a triangle, its base and altitude; because these distances furnish, respectively, the most simple and convenient data, from which the total magnitude of the figure can be inferred. By some writers, the entire boundary, or the sum of the separate sections of the boundary of a figure, is regarded as forming one of its dimensions; and occasionally, also, the idea of square or cubic measure is included under the same title, as when it is said that a certain object has the “ dimensions of so many square or cubic feet, &c.” It is desirable, however, for the sake of precision, that the term should be used in the sense of lineal extension only. (3.) The perimeter of a figure signifies the entire length of its boundaries, as the sum of the several sides of a rectangle, a triangle, &c. This expression is seldom employed, except with reference to plane rectilineal figures,—those described on flat surfaces and enclosed by straight lines. (4.) A point, mathematically considered, is merely an indication of position in space, and must be supposed to have, of itself, no magnitude or degree of extension whatever,--to be, in fact, infinitely small and indivisible. It is evi. dent that a point answering to this definition cannot exist in nature or be produced by art, as the extremity of the finest needle possesses certain dimensions of its own, minute though these must be, and if impressed on paper or some other surface, will leave a mark, which, even though it cannot be perceived without the aid of a microscope, will be found to occupy an appreciable portion of that surface, and must be allowed, therefore, to have, at least, the positive attributes of length and breadth. The necessity of imagining such a point as that contemplated in geometry arises from the fact, that were it granted that a point might have the least conceivable magnitude, it would, in that case, form a part of the object in which it is placed, and, consequently, would interfere with the rigour, or absolute exactness, of demonstrations respecting the space included within the external limits of the object containing it. Practi. cally, we are obliged to make the best representation of a geometrical point, which the instruments at our command will permit, and then, by the faculty of abstraction, to divest the visible mark or dot so produced, of all idea of bulk or size. (5.) Lineal measurement,—that is, the art of drawing straight lines between any system of points, and ascertaining their length in some known denomination of measure,-is the groundwork of every process in practical geometry and Mensuration. A mathematical line must be conceived to have no other dimension than that of length; it must not possess in any degree the attributes of breadth or thickness. If, for instance, the boundary lines of a surface were regarded as having the slightest amount of breadth, they would to that extent compose a part of the surface which they enclose, and would be of themselves surfaces and not simple lines or marks of the termination of a surface. Perhaps the clearest notion of the formation of a mathematical line will be obtained by supposing a mathematical point (see def. 4) to move in any direction, through space, and to indicate its course by leaving behind it a trace or track. Now, as a mathematical point is assumed to have no dimensions whatever, its traco or track can have no dimension but that of length, for to contend that such a track may have positive breadth or depth would involve the inference that the point which generated it had dimensions corresponding to that breadth or depth, and this is opposed to the definition of a mathematical point, as above given. Similarly, a physical line--one such as it is possible to draw or represent by the aid of any material appliances—may be aptly conceived to originate in the motion of a physical point, and therefore to have a degree of breadth and thickness equivalent to the magnitude of the physical point by the movement of which it is supposed to be produced. It will thus be seen that the extremities of a line-geometrical or physical must be points, and that these points must partake of the character of the lines which they terminate. A geometrical line cannot end in a physical point, or a physical line in a geometrical point. It will also be apparent that if one geometrical line be drawn across another, the place at which they intersect, or cut one another, must be a geometrical point. There are two general classes of lines : straight lines, and curved lines or curves. In geometry, the word right is employed with regard to lines in the same sense as the more popular term, straight. A straight line is defined to be one which lies erenly, that is, in one continuous direction, between its extreme points. It follows as an obvious and necessary consequence of this definition, that a straight line is the shortest distance between any two points. A curved line is one which varies in direction at every point of its course. It is a property of a curve as distinguished from a straight line, that if a curved line of any shape be conceived to turn round its own extremities, as fixed points or pivots, and as it turns to leave behind it a trace or track, that track would enclose a certain portion of space, or form the outline of a solid figure. The revolution of a straight line, on the contrary, can never be attended with this effect; it will not enclose space when it is caused to turn round its own extremities. There are many kinds of curves of regular and definite contour. Of these the circumference of the circle supplies the most familiar example, and is the most easily described. All straight lines are measured by comparing their lengths respectively with some smaller line of known length taken as a measuring unit. Any appropriate instrument bearing a scale of equal parts, each representing the length of the measuring unit, and graduated if requisite into equal sub-divisions of the unit, as for instance, into inches and tenths of an inch, may be used to effect the Mensu. ration of lines or distances. The number of times and parts of a time which any given line contains the unit of measure applied to it, constitutes the relative numerical value of the line so measured with reference to that particular line or unit. If a unit of different length be employed, as a foot instead of an inch, a yard in place of a foot, &c., the numerical expression of the value of the line will, of course, undergo a proportional alteration. There is no standard unit of measure for curved lines. As curves are of such various shapes, and continually change their direction, it would be impossible to dovise any general or standard curved measure, which on being repeated a certain |