larly on level ground. At any convenient distance, as at C place your semicircle or such other instrument as you judge best for taking an angle of altitude, as a quadrant or the like; observe that your semicircle be placed horizontally, by making a plummet line fixed to the centre fall just upon 90°. Then move the index up and down, till you espy the top of the tree or building, thro' the sights, at A; see then what degree upon the limb is cut by the index, as suppose 55°. Then measure the distance C B, between your instrument and the object, which let be 30 feet. 55°. 30 feet. A B Draw a line C B at pleasure; at B erect a perpendicular. From B set off 30 feet towards C, taken from any scale. Lay your protractor on the point at C, and against 55° make a mark, through which, and the point C, draw a line intersecting the perpendicular at A; then take the line AB in your compasses and applying them to the same line on the scale, you will find the height of the object A B to be 45 feet, from the level of your instrument; to which add the height of your instrument from the ground, and you will have the height of the object. To measure heights by a staff or rod. 1. By the aid of a quadrant, a carpenter's, or mason's plumb, erect perpendicularly a staff of a given length, above the sur face. 2 Having erected your staff, in a fair day, measure the length of its shadow, and also the length of the shadow of the object, whose height you wish to ascertain; and then you will Have all the requisites necessary for operation: Suppose The length of the shadow of the staff, 15 ft. The length of the staff, 10 ft. The length of the shadow of the object, 135 ft. By the Rule of Three, thus: As 15: 10 :: 135: 90 the height of the object. N. B. In all operations of this description, the person is supposed to stand on an horizontal plane; otherwise his work will be erroneous. On such a plane, the perpendicular height of any object, may be taken, in a clear day. To measure distances by the chain only; or by a cord equally divided into feet and inches, By either of these instruments, may be measured the distance of an object on the opposite side of a river, pond, lake; or, the distance of any visible, inaccessible object, on the earth. 18,76, Ans. ARITHMETIC. The breadth of a river pond, or lake, is required. A Erect a staff, or any object, at the ch. L. E. F. 2 30 The 5,20 223 8.30 ch. L. D As 2,30: 5,20 :: 8,30: 18 76 the breadth of the river. N. B. This measurement is founded on the geometrical theorem, that, if in two triangles, all the angles of one be each respect. ively equal to all the angles of the other; then the legs opposite to the equal angles will be proportional. It is plain, that the sum of all the angles, in the less triangle, D E, F, is equal to the sum of all the angles in the greater, A, B, F. And of course there must be a proportion between their respective legs. And, consequently, by ascertaining the length of the base and perpendicular of the less triangle, and also the base of the greater triangle, you may easily find the perpendicular of the greater triangle, which is the breadth of the river. This is done by the Rule of Three Direct. The above principles account for the measurement of heights, by a staff and shadows. Of the line of numbers on Gunter's Scale. On Gunter's scale there is a line, marked at the right hand, num. which is divided. The marks of the large division being numbered from left to right, the marks of the smaller division immediately below the preceding, not numbered. To prove the line by compasses, observe that the distance from 1 to 2, is equal to the distance from 2 to 4; and from 5 to 20 is equal to the distance from 4 to 8; and from 4 to 8 is equal to the distance from 3 to 6. To find a number on the line, as suppose 134. For the 1 hundred, account one on the line; and for 3, take 3 of the large divisions; and for 4, take 4 of the smaller divisions, and that is the point. To find 750, for 7 take 7 on the line; for 50, take 5 of the great divisions; and that is the point. To find a small number, as 12; for 10 take one as before, and for 2, take 2 of the large divisions, and that is the point. Multiplication by Gunter's line. To multiply 3 by 7, set one foot of the compasses on 1, in the left hand line, and extend the other to 3, towards the right hand, and with the same extent place one foot in 7, and the other foot will fall on 21 in the right hand line, which is the answer. Division by Gunter's line. To divide 63 by 3, extend from 3 to 1, towards the left hand, and that extent will reach the same way from 63 to 21, the an swer. Note. Totiply you must extend your compasses from 1, towards the right band; and to divide you must extend them to the left hand. 2. Divide 350 by 25; extend from 25 to 1, towards the left hand; and that extent will reach the same way from 350 to 14, the quotient. Rule of Three Direct. 1. If 4 bushels of oats cost 9 shillings, What will 36 bushels cost?-Extend from 4 to 9, towards the right hand, and that extent will reach the same way from 36 to 81, the answer in shillings. 2. If 5 yards of cloth cost 10s. 6d. What will 30 yards cost? Extend from 5 to 10, towards the right hand, and that extent the same way will reach from 30 to 63, the answer. To Measure Boards. 1. If a board be 9 inches wide, and 18 feet long, What is the content in superficial square feet ?-Extend from 12, (the cen tre of foot measure) to 9, towards the left hand, and that extent the same way will reach from 18 to 134, the answer. To Measure Timber by Gunter's Line. A piece of timber 20 inches square and 9 feet long, What is the content in solid feet-Extend from 12, the centre, to 20, towards the right hand, and that extent twice the same way will reach from 9 to 25 feet, the content. TO MEASURE A SPHERE OR GLOBE. A Sphere or Globe, is a round solid body, in the middle of which is a point, from which all lines drawn from the surface are equal. RULE. Multiply the cube of the diameter by ,5236, and the product will be the solid content. Note. When the solidity of a Globe is give may be found by dividing the solidity by ,5236 the cube root of the quotient. the diameter nd extracting To measure the solidity of a Frustrum or Segment of a Globe. The frustrum of a Globe, is any part cut off by a plane. RULE. To three times the square or the semidiameter, of the base, add the square of the height; then multiply that sum by the height, and the product by,236, you will have the solid content. Let A D C represent a coal-pit; the height at the chimney, B D, 9 feet; the diameter, A C, 34 feet; to find how many cords of wood it contains, making no allowance for the chimney. feet in a cord128)2417,4612(18,886 cord, Ane, |