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MEASURING OF BOARDS AND TIMBER.
1st. OF BOARD MEASURE. To measure a board is no other than to measure a long square.
1. If a board be 16 inches broad, and 13 feet long, how many feet are contained therein ? Ans. 17 ft. 4 inc.
2od. OF SQUARED TIMBER. By squared timber is here meant such as have equal bases, and the sides straight and parallel. The rules for measuring all such solids are shewn in the measuring of a parallelopipedon, page 174.
2. If a piece of timber be 15 inches square, and 18 feet long, how many solid feet are contained therein?
Ans. 28 ft. 1 inc. 6 p. 3. If a piece of timber be 2 feet 9 inches deep, 1 foot 7 inches broad, and 16 feet 9 inches long, how many feet of timber are in that piece ?
Ans. 72 ft. il inc. Multiply the depth, breadth and length together, and the product will be the content.
Error.-A common error is committed, for want of art, in measuring these last sorts of solids. Vany add the depth and breadth together, and take half of that sum for the side of a mean square. This error, though it is but small, when the depth and breadth are pretty near equal, yet if the difference between the depth and breadth be great, the error is very consider. able; for the piece of timber thus measured will be more than the truth, by a piece whose length is equal to the length of the piece of timber to be mea. sured, and the square equal to half the difference of the breadth and depth.
The answer to the above qnestion, in the false way, is 78,62 feet.
I will shew both ways in the Key.
Always divide 1728, (the solid inches in a foot,) by the area of the base, and the quotient is the length of a foot.
4. If a piece of timber be 18 inches square, how much in length will make a foot solid ?
Ans. 5 inches. 5. If a piece of timber be 22 inches deep, and 15 inches broad, how much in length will make a foot ?
Ans. 5,23 incheş.
3d. OF UNEQUAL SQUARED TIMBER. By unequal squared timber is meant all such as have unequal bases, that is, such as is thicker at one end than at the other, as most timber trees are when they are hewn, and brought to their squares. The usual way to measure such timber, is to take a square about the middle of the piece, which is taken for a mean square. This way, when the piece is pretty near as thick at one end as at the other, is something near the truth; but when there is a great disproportion between the ends of the piece, the error is considerable. All such solids being the frustums of pyramids, the true way of measuring them is by the rules laid down for frustums of pyramids. I will give the answer both by the true and false way.
6. If a piece of timber be 25 inches square at the greater end, and 9 inches square at the lesser end, and 20 feet long, how many feet of timber are in that tree
Ans. 40,13 feet by the false way.
43,101 feet by the true way. 7. If a piece of timber be 32 inches broad, and 20 inches deep at the greater end, and 10 inches broad and 6 deep at the lesser end, and 18 feet long, how many feet of timber are in that piece ?
Ans.. Content the true way, 37,33 feet.
Content the false way, 34,12 feet. 4th. OF ROUND TIMBER, WHOSE BASES ARE EQUAL.
The usual way to measure round timber trees, is to girt them about the middle with a string, and take the fourth part of that girth for the side of a square, by which they measure the piece of timber, as if it was square.
That this method is an error, I shall make appear in the following demonstration: If the circumference of a circle be 1, the area will be ,07958; then the fourth part of 1 is
25 which squared is ,0625. This they take for a inean area, instead of ,07958: therefore the true content always bears such proportion to the content found by the above customary false way, as ,07958 is to ,0625 which is nearly as 23 to 18; so that in measuring by the customary false way, there is above the one-fifth part lost of what the true content ought to be. This error, though it has been in practice for a series of years, ought to be omitted, if they follow the correct method, as is here laid down.
8. If a piece of timber be 96 inches in circumference, or girth, and 18 feet long, how many feet of timber are con tained therein ?
Ans. Content the false way, 72 feet.
Content the true way, 91,67 feet. 9. If a piece of timber be 86 inches girth, and 20 feet long, how many feet are contained therein?
Ans. Content the false way, 64,2 feet.
Content the true way, 81,74 feet.
5th. OF ROUND TIMBER, WHOSE BASES ARE UNEQUAL
The usual way to measure round timber I have shewn, and its error in timber that is all the way of an equal thickness; and it must be much more so in timber that is tapering; and the more tapering it is, the greater is the error. Therefore, to measure all such timber according to art and truth, such a piece ought to be considered as a frustam of a cone, and should be measured by the rule for the frustum of a cone.
10. If a piece of timber be 9 inches in diameter at the lesser end, and 36 inches at the greater end, and 24 feet long, how
Ans. 74,22 feet. 11. If a piece of timber be 136 inches in circumference at one end, 32 inches in circumference at the other end, and 21 feet long, how many feet of timber are contained in that piece?
Ans. 92,34 feet, 6th. OF THE FIVE REGULAR BODIES. These bodies may all be measured by the rule laid down for measuring a pyramid, except it be a cube, or hexaedron, which is already measured.
1st, OF A TETRAEDRON. A tetraedron is a solid contained under four equal and equilateral triangles.
Required the solidity of a tetraedron, whose side is 12 inches, and the perpendicular height 9,798 inches.
Ans. 203,641632 inches.
2nd. OF AN OOTAEDRON. The octaedron is a body contained under eight equal and equilateral triangles.
· RULE. An octaedron is composed of two quadrangular pyramids, joined together by their bases; therefore, if the area of the base be multiplied into a third part of the length of both pyramids, the product will be the solid content.
Required the solid content of an octaedron, whose side is 12 inches?
Aps. 814,5792 the 'solidity.
3d. OF A DODECAEDRON. A dodecaedron is a solid body, contained under twelve pentangular planes, whose vertexes all meet in the centre.
RULE. Find the content of one of the pyramids, and multiply that by 12, (the number of pyramids in the dodecaedron, and that product will be the solidity of the dudecaedron.
Let each side of a dodecaedron be 12 inches ; required the solidity thereot ?
Ans. 13241,85392 inches, 4th. OF AN 100SAEDRON, The icosaedron is a solid body, contained under twenty equal and equilateral triangles. The icosaedron is composed of twenty triangular pyramids, with their vertexes all joined in the centre.
Rule. Find the solid content of one pyramid, multiply that by twenty, and the product is the solid content of the icosaedrop.
Lét each side of an icosaedron be 12 inches; required the solidity thereof?
Ans. 3769,945840 inches. NOTE. The cube, or hexaedroa, is already measured. By the following table, the contents, either superficial or solid, of any of these bodies may very readily be found; for all like superficial figures are in proportion one to another, as are the squares of their like sides: therefore it will be, as the square of 1, (which is 1,) is to the superficialcon. tent in the table, so is the square of the side of the like budy to the superficial content of that body. Therefore, if the number in the table be muitiplied by the square of the side given, the product is the superficial content required.
Again, ail like solids are in such proportion to each other, as are the cubes of their like sides: therefore it will be, as 1, (which is the cube of 1,) is to the solid content in the table, so is the cube of the side given to the solid content required. Therefore, if the number in the table be multiplied by the cube of the given side, the product will be the solid content of the same body.
A Table showing the solidity and superficial content of any of the regulast
bodies, the side being 1, or unity, s.
I will give two examples; according to the tabular numbers.
i. Let the side of a dodecaedron be 12 inches, (as before;) what is the content, solid and superficial?
Ans. 13241,869632 the solid content.
2972,981976 the superficial content. 2. Let the side of an octaedron be 20 inches; what is the content, solid and superficial?
Ans. 3771,2860000 the solid content.
1385,640800 the superficial content. 7th. HOW TO MEASURE ANY IRREGULAR SOLID.
have any piece of wood or stone that is craggy or uneven, and you desire to find the solidity of it, put the body into any regular vessel, as a tub, a cistern, or the like, and pour in as much water as will just cover it; then take out the solid, and measure how much the fall of the water is, and so find the solidity of that part of the vessel.
Suppose a piece of wood or stone to be measured; and suppose a tub 32 inches in diameter, into which is set the stone or wood, and covered with water. When the solid is taken out, the fall of the water is 14 inches. I demand the solidity of that body?
Ans. 6,51 feet,
PRACTICAL QUESTIONS. 1. If a pavement be 47 feet 9 inches long, and 18 feet 6 inches broad, I demand how many yards are contained therein ?
Ang. 98 yards 1 foot. 2. There is a room whose length is 21,5 feet, and the breadth 17,5 feet, which is to be paved with stoues, each