Incumbrances.—An « incumbrance" is the technical name given to any object which takes up a portion of the space included within the boundary of a solid, or the inner surfaces of a vessel, and requires that a certain deduction be made from the gross content, in order to arrive at a correct estimate of the quantity of goods which such solid contains, or is capable of containing, in whole or in part. Incumbrances usually consist of columns or pillars placed in vessels or rooms for purposes of support, or of the machinery used in manufacturing processes, such as the rakes in a brewer's mash-tun, &c. In computing the net area or content of a vessel, &c., affected by incumbrances, the necessary dimensions of each pillar or other substance are taken separately, and the amount of space occupied by it is determined in gallons or bushels. The gross area or content of the vesselthat is, the total space inclusive of that taken up by the incumbrances-being also ascertained, the difference between the results of the two operations gives the net area or content required. As, in some cases, an incumbrance extends only through part of a vessel, the allowances on the areas must, of course, be confined to the sections which are thus affected.

Officers are to enter in their « dimension books," or « table books," an account of the dimensions of every incumbrance existing in a fixed vessel, in which gauges are directed to be taken, and to add a note, descriptive of the nature of the incumbrance, stating its form, the particular sections of the vessel to which it applies, and whether it is placed in a perpendicular, horizontal, or oblique position, so that the correctness of the registered areas may, at any time, be readily checked, and any alteration discovered.

Further observations as to the treatment of incumbrances will be found under each of the articles relating to the classes of vessels employed by different traders.

Standard Tables of Areas.-For the purpose of facilitating the calculation of areas and contents by the pen, two sets of printed tables are issued to officers, under the authority of the Board. The principal of these is called "A Table of the Areas of Circles in Imperial Gallons." The figures contained in it may be regarded as strictly accurate. Opposite to the diameter of every circular body, from one inch to four hundred inches in diameter, is inserted the area in gallons at one inch deep. The construction of the table is as follows:-Each diameter is squared, and the square divided by 353-036, or multiplied by .0028326, on the principle exemplified at page 248. The result shows the cylindrical area proper to the given diameter. In the actual formation of the table, since the areas of different circles are proportional to the squares of their diameters, all that was necessary after having found to several decimal places, the area of a circle whose diameter is 1 inch, was to multiply such area successively by 2a, 32, 42, &c., in order to obtain the areas corresponding respectively to diameters of 2, 3, 4, &c. inches. The other table supplied to officers is that entitled "Table of the Areas of Semi-squares in Imperial Gallons." This table is applicable only to bodies bounded by plane surfaces, the areas of which in gallons, are calculated by multiplying together two dimensions, and dividing the product by 277-274. An addition and a subtraction performed on the proper tabular numbers gives the same result with less labor.

Suppose a square, a side of which is 7 inches; a second square, a side of which is 3 inches; and a third, a side of which is equal to the difference of the sides of the other squares, that is, 4 inches. Now half the area of the first square is 492 24.5 square inches, and of the second, 9 ÷ 2 = 4.5 square inches. Add together these areas, and we have 29 square inches. Subtract from this half the area of the third square, namely 8 square

inches, and the result is 21 square inches. But 21 is the product of 7 x 3, and expresses the area in square inches of any rectangle, triangle, &c., the dimensions of which are, respectively, 7 inches and 3 inches, the values of the sides of the first and second squares above supposed. The same property holds good universally.* Therefore, when two numbers are given, their product may be found by subtracting half the square of their difference from half the sum of their squares. So, when the sides of a rectangle are given, its area may be found by subtracting half the square of the difference of the sides from half the sum of the squares of the sides. If, then, a table be formed, containing a series of the areas of "semi-squares," in gallons, it is plain that the area of any rectangle may be obtained in that denomination, simply by adding the tabular number opposite to the length to that opposite to the breadth, and deducting from the sum the number opposite to the difference between the length and breadth. Triangles and oblique parallelograms may be similarly treated, substituting base and height for length and breadth. This table is almost exclusively used in the determination of the areas of malt utensils, for which purpose, the result, as shown in gallons, must be divided by 8, to reduce it to bushels. But no good computer who will practice the short method of calculating malt-areas exemplified a little further on, will resort to the table of semi-squares, except as a check on the work of the pen.

(2) MALT GAUGING.-The duty on malt made for general purposes is levied wholly by gauge. Instead of the tedious and inconvenient process of ascertaining the quantities of materials in operation by successive fillings of a small standard measure, the system is adopted of requiring the trader to steep and couch the whole of his grain in rectangular vessels or utensils, the area of which, when once computed from the dimensions and recorded in the official books, affords a ready means of arriving at a close approximation to the number of bushels present, simply by multiplying such constant area by the average depth of the goods. Every utensil so constructed and employed is, in effect, a large measure, and yields nearly the same result as would otherwise be attained in a much more laborious manner, by resorting to actual detailed measurement of the ordinary kind. The only inaccuracies to which the method of gauging is liable, are such as proceed from the difficulty of laying considerable masses of grain with a uniformly level surface, and from the unavoidable compression exerted by the weight of these masses upon their lower portions. The former objection may, however, be obviated in a great degree, by causing the grain, as the law directs, to be made as flat and even at the top as is fairly practicable, and then taking a sufficiency of depths, properly distributed, to give the true average depth of the entire bulk. In order to guard against the effects of excessive natural compression, as well as to facilitate the gauging of the grain, the malt acts impose a certain limit on the depth of cisterns, and also on the depth to which corn may be laid in a couch-frame. Any artificial compression of the grain in couch-frame-that being the utensil in which the charge of duty generally arises-is rendered punishable by fine. After removal from the couch-frame to the floor of the malthouse, it is provided that the grain shall be laid level, and in such form as will admit of its being conveniently gauged. In consequence of the weight with which particles of grain press upon each other when placed together in deep layers, the amount of the most exact gauge of the dry barley, &c., contained in a cistern will, if the depth be much greater than 10 inches, prove to be considerably less than the quantity which would be obtained on subjecting the grain to actual admeasurement. But as soon as the mass is covered with water, a portion of its weight becomes sustained by the fluid, and the results of gauging and of measurement will now exhibit no material difference. It should be noted on the other hand, that when the water is put in before the grain, so much buoyancy is for a time induced, as to make the first gauge of the steeping somewhat higher in amount than the quantity ascertainable by measure. Cisterns and Couch-Frames.-On the erection of a new cistern or couchframe in a malthouse, it should first be observed whether the various legal

• A conclusive geometrical proof may be readily derived from Euclid, Book II.

conditions respecting the form, &c. of each such utensil have been practically complied with. The fact of the sides and ends, or in the case of a couch-frame, the sides and the bottom, being straight and at right-angles to each other, is best tested by the application of a T square, or failing that, by one of the methods described at page 169.

It must also be seen that the bottom of a cistern is even-that is, free from any gross inequalities of the surface-and that the inclination for the drip is not more than the law allows. Some limitation, it is obvious, must be set to the amount of fall given to the bottom of a cistern, for otherwise the depths of the grain in different places might vary so greatly as to prevent a correct account being taken, by gauge, of the quantity in steep.

The next step is to ascertain the average length and breadth of each utensil, and, as regards a cistern, its greatest depth. For a couch-frame the average depth should be taken. This dimension is readily found by measuring perpendicularly, in several places, with a graduated dipping-rod. In order to determine the horizontal length and breadth with precision, correct gauging rods, supported, if necessary, in the middle, must be employed; but before applying them, the sides and ends of the vessel should be marked out, according to their length, into two or more equal sections, straight lines being drawn with chalk or a charred stick, from the top to the bottom of each at the points of division, and other straight lines at the allotted distances across these. The gauging rods should then be extended between the centres of the opposite and corresponding pairs of parallelograms that have thus been formed, and the distances read off in inches and tenths at the junction of the sliding pieces.

A

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Example. Let the length and breadth of the upper edges of the rectangular utensil figured in the margin be respectively 260 and 84 inches, as at first roughly measured with a tape, and the depth 32 inches. Also suppose that it appears sufficient to try the interior length between the ends in six places, and between the sides in eight places. Now to ensure the regular and horizontal measurement of these distances, the upper and lower edge of each of the sides should be divided into four equal portions by straight lines. The edges of the ends should be divided into three equal sections, and similarly connected. Then let a straight line at half the depth of the vessel be drawn all round the inner surfaces. Each side will thus be mapped out into 8 equal rectangles, and each end into 6. By stretching the rods between the centres-as nearly as can be judgedof the opposite spaces, writing down the several lengths so measured, and dividing the total by the number of dimensions, the average length and breadth of the utensil will be obtained with sufficient exactness for all purposes.

P

Assuring the true average length to be 260-7 and the breadth 83-8 inches, to calculate the area at one inch deep.

Square factor for bushels, reversed for the performance of contracted multiplication, see p. 86.

But a more simple and expeditious process than any of these is one derived from the following considerations:

To divide by 2218-192 is to multiply by its reciprocal, 000450818. For the three first ciphers in this factor, point off three decimal places from the right of the multiplicand, and the factor then becomes 450818. Now, to multiply by 45 is to multiply successively by 5 and 9, the product of which is 45. But multiplication by 5 is equivalent to division by 2, and multiplication by 9 is the same as the deduction of one-tenth of the number multiplied from itself. Again, instead of multiplying the original number by 000818, and adding the result to the product by 45, we may increase such product by its 550th part, for 45÷000818= (almost exactly) 550; and to divide by 550 is to multiply by 02 and divide by 11, since 1100 = 180 X 11.

550

The general rule for computing reetangular malt areas is, therefore-Take half of either the length or breadth, and multiply it by the other dimension; point off three decimal places and subtract one-tenth of the product from itself. The result thus obtained is a good first approximation, agreeing closely with the area given by the slide-rule. To ensure correctness, it is only necessary to add 1 bushel for every 550 bushels, or '02 bushels for every 11 bushels.

Example. Let the length and breadth of a malt utensil be, as before, 260-7 and 83-8 inches respectively. Required the area in bushels.

As in working out the areas of malt utensils for insertion in the official books, it is needless to proceed beyond the second place of decimals, except where incumbrances have to be deducted, the foregoing operation may be shortened by employing the contracted method of multiplication and applying the final correction no further than it is seen will give the centesimal figure of the result with exactness.

It may be relied upon that the mode of calculation here exemplified-the process being conducted at full length or otherwise according to the magnitude of the dimensions-will furnish, without error, all areas not consisting of more than five figures of whole numbers and decimals-that is, all areas under 1000 bushels; but very few utensils occur in practice which hold so much as 100 bushels to the inch.

When allowance has to be made for incumbrances, and the areas, as carried to four places of decimals agreeably to the Instructions, contain more than one figure of whole bushels, it will be proper to divide the product of the length and breadth by 2218-192, or to employ the table of semi-squares, since we cannot be certain that the result of any other process will be accurate to the extent required in these cases.

For an example of the mode of treating a rectangular incumbrance in a malt utensil, officers are referred to the Malt Instructions.

In addition to the directions therein given, it may be observed, that if an incumbrance be of a cylindrical shape, its area in cubic inches is most readily determined by measuring the circumference in one or more places with a tape, and multiplying the square of the average girth so found by 07958 (see page 211). Otherwise the diameter may be ascertained, if convenient, by means of a pair of large bent compasses: the square of the diameter multiplied by 7854 gives the area in cubic inches. In order to make the table of the areas of circles applicable to such bodies, the diameters must be obtained either from the circumference by calculation, using the factor 3183 (page 194), or directly by the use of calliper compasses. It is seldom that the incumbrances met with in maltsters' utensils are of other than cylindrical or rectangular forms, but where the figure differs materially from either of these, the area must be found agreeably to the rules laid down in the Principles of Mensuration for solids which bear the nearest resemblance to the object in question.

When there are two or more incumbrances in such utensils, a variation sometimes occurs in the third or fourth decimal place of the total area, according to the system of computing that is adopted. Usually, the areas of the several incumbrances are determined separately in bushels, and the results added together before making the deduction. It is also the practice, however, to calculate and set forth the cubic inches in each area, and to reduce the total result into the required denomination. The latter method is approved at the Chief Office on the ground that it allows to the trader the utmost possible abatement from the gross area.

There is thus a difference of 01 bushels against the trader by employing the first method.

Another diversity of practice exists with regard to the casting of areas that are required in bushels, some officers finding the result in gallons from the table of semi-squares or the table of circular areas, and then reducing into bushels, whilst others determine the cubic inches and apply the divisor or factor for bushels. The fourth decimal figure is thus occasionally affected where incumbrances exist; and as the chief object in all these cases must be to secure uniformity of procedure based on a correct principle, there can be no doubt, that as respects maltsters' utensils from which charges of duty may arise, the latter system should always be adopted.

Average Depth."-When the surface of the grain has been made reasonably level, the officer proceeds to obtain a sufficient number of dips, at regular distances apart, by means of a graduated rod or dipping-piece, and a small metal plate fastened to a handle. The rod passes through a hole in the plate, and as soon as it reaches the bottom of the grain, the plate is brought down flat upon the surface, and being retained in this position when the rod is withdrawn, marks against the edge of the latter the number of inches and tenths in each depth. Care should always be taken to insert the rod perpendicularly, to let the plate come into fair and even contact with the grain, but to use no positive pressure; and to hold the plate and the rod firmly together at the time of withdrawing the latter, so as to prevent the plate from slipping upwards and thus indicating an unduly high gauge.

These dips, when formed into an average have the effect of compensating slight irregularities that may exist after the grain has been levelled, and of thus giving, as it were, an uniform depth to the mass. It is essential that the dips should be sufficient in number and be distributed on a correct principle, for if otherwise, the result may often differ considerably from the true average. The only proper method is to divide the surface, or to conceive it to be divided, into a series of equal rectangles, by lines drawn between its opposite sides, and then to dip the grain, as nearly as can be judged, in the centre of each section. In performing this important service, it is advisable, with respect at least to the

The principle of arithmetical means or averages has been fully explained in the chapter on Arithmetic, page 187.