ART. 47. The side of the Tetraedron being 3, to find the solid and superficial content. Cube=3x3x3=27, and 27×·11785=3-18195=solidity. OF THE OCTAEDRON. This solid is contained under eight equal and equilateral triangles, which may be conceived to consist of two quadrangular pyramids of equal bases joined together, the sides of whose bases are equal to the given sides of the triangles, under which it is contained. ART. 48. The side of an Octaedron being 3, to find the solid and superficial content. Cube=3x3x3=27, and 27x-4714-12-7278=solidity. OF THE DODEcaedron. This solid is contained under 12 equilateral pentagons, and may be conceived to consist of twelve pentagonal pyramids, of equal bases and altitude, whose vertices meet in the centre of the dodecaedron. ART. 49. The side of a Dodecaedron being 3, to find the solid and superficial content. Cube=3x3x3=27, and 27×7·663119=206·904. OF THE EICOSIEDRON. This solid is contained under twenty equal and equilateral triangles, and may be conceived to consist of twenty equal triangular pyramids, whose vertices all meet in the centre. ART. 50. The side of an Eicosiedron being 3, to find the solid and superficial content. Cube=3x3x3=27, and 27×2·18169=58.90563=solidity, As the figures of some of these bodies would give but a confused idea of them, I have omitted them; but the following figures, cut out in pasteboard, and the lines cut half through, will fold up inte the several bodies. Tetraedron. A Hexaedron. Octaedron. Dodecaedron. Eicosiedron. OF CASK GAUGING. Among the many different canons drawn from Stereometry, for Gauging casks, the following is as exact as any. Take the dimensions of the cask in inches, viz. the diameter at the bung and head, and length of the cask; subtract the head diameter from the bung diameter, and note the difference. If the staves of the cask be much curved or bulging between the bung and the head, multiply the difference by 7; if not quite so curve, by 65; if they bulge yet less, by 6; and if they are almost or quite straight, by 55, and add the product to the head diameter; the sum will be a mean diameter, by which the cask is reduced to a cylinder. Square the mean diameter, thus found, then multiply it by the length; divide the product by 359 for ale or beer gallons, and by 294 for wine gallons. Note 1. The length is most conveniently taken by callipers, allowing, for the thickness of both heads, 1 inch, 1 inch, or 2 inches, according to the size of the cask; but if you have no callipers, do thus; measure the length of the stave, then take the depth of the chimes, which with the thickness of the head, being subtracted from the length of the stave, leaves the length within. Note 2: You must take the head diameter, close to its outside, and, for small casks, add three tenths of an inch: for casks of 30, 40, or 50 gallons, 4 tenths, and for larger casks, 5 or 6 tenths, and the sum will be very nearly the head diameter within. In taking the bung diameter, observe, by moving the rod backward and forward, whether the stave, opposite the bung, be thicker or thinner than the rest, and if it be, make allowance accordingly. By the Sliding Rule. On D is 18-94, the gauge point for ale or beer gallons, marked AG, and 17:14, the gauge point for wine gallons, marked WG: set the gauge point to the length of the cask on C, and against the mean diameter, on D, you will have the answer in ale or wine gallons accordingly as which gauge point you make use of. By the Scale. Take the extent from the guage point to the mean diameter, set one foot of the dividers in the length, and turning them twice over, they will point out the content. ART. 51. Required the content in ale and wine gallons, of a cask, whose bung diameter is 35 inches, head diameter 27 inches, and length 45 inches? Bung diameter=35 Square of the diameter=1062.76 Head diameter=27 Length= 45 Squared 1062.76 ART. 52. A round mash tub is 42 inches diameter at the top, within, and 36 inches at the bottom, and the perpendicular height 48 inches; required the content in beer and wine gallons? 183 This being the lower frustum of a cone, to the product of the di ameters add of the square of their difference; multiply this sum by the length, and it will give the solidity in such parts as the dimensions are taken in. If they be taken in inches, divide by 359 for beer, and 294 for wine gallons. 42x36+ 42-36X42-36 3 X48 (359=2031 ale gallons. ART. 53. Let the difference of diameters of this tub be 6 inches, the height 48 inches, and the content 2033 gallons, to find the diameters? Multiply the content, if beer measure, by 359; if wine measure, by 294, and divide the product by the length: from the quotient subtract of the square of the difference of the diameters; to this remainder add the square of the difference of the diameters, and extract the square root of the sum; from the square root subtract the difference of the diameters, and it will give the least diameter to great exactness, to which add the difference of the diameters, and the sum is the greatest diameter. 203-75X359 48 6×6 3 —+3×3—3=36, and 36+6=42. The diameters are 36 and 42. The content of any vessel in gallons, &c. may be thus found: measure the inside of the vessel, according to the rule of the figure, and find the content in cubick inches; then, ART. 54. To ullage a Cask, lying on one side, by the Gauging Rod, when the Bung Diameter, and the Content, one, or both are greater or less than the Table on the Rod is made for. RULE. As the bung diameter of the cask to be measured, is to the bung diameter that the table is made for; so are the dry inches of the cask, to a fourth number, which find in the table on the rod, and note the number of gallons answering to it. Then as the content of the cask that the table is made for, is to the content of the cask to be measured; so is the number of gallons answering to the aforesaid fourth number, to the number of gallons your cask wants of being full. ART. 55. To find a Ship's Burthen, or to Gauge a Ship. There is such a diversity in the forms of ships, that no general rule can be applied to answer all varieties; however, the following rules are practised. RULE 1. Multiply the breadth at the main beam, half the breadth, and length together; divide the product by 94, and the quotient is the tons. RULE 2. Divide the continued product of the length, breadth, and depth, in feet, by 100, for ships of war, and 95 for merchant ships, in which nothing is allowed for guns, &c. and the quotient is the tons. RULE 3. Take the length from the stern post to the upper part of the stem; subtract two thirds of her breadth from that length; multiply the remainder by the whole breadth, and that product by half the breadth, in feet, and divide by 100 for war, and 94 for merchant tonnage. RULE 4. The weight of a ship's burthen is half the weight of water she can hold. What is the tonnage of a ship, whose length is 97 feet, breadth 31 feet, and depth 15 feet. By Rule 1st. breadth 15.5 Breadth 31 155 By Rule 2d. 465 480-5 Length 97 33635 43245 94)46603-5495 03 tone. Carried over. |