to w 11601 to Z Cross hedge 890 Cross hedge 70 From n tv- 1032 Cross hedge H Cross hedge 570 Cross hedge40376G 1676) 256150 89624 (to corner From W 144 130 632 Cross bedge 50 180 From • 14 588 T 580 to •15 towards 15 40 500 76 300 From T to 1576 100 Lines 0. let Off-sets left. Station Prob. 6. To survey a town or city.* Here it will be proper to have an instrument for taking angles, as a theodolite or a plain table : the latter is a very convenient instrument, because the minute parts may be drawn upon it on the spot. A chain of 50 feet long, divided into 50 links of one foot each, will be more convenient than the common surveying chain, and an off-set staff of ten feet long will be very useful. Begin at the meeting of two, or more, of the principal streets, through which you can have the most extensive view, and conse quently the longest station-lines. Having fixed your instrument at this point, draw lines of direction along the streets, using two men as marks, or use poles fixed on wooden pedestals, or such objects as may conveniently present themselves, viz. windows, doors, corners of houses, &c. Measure the station-lines with the chain, and take off-sets with the staff to all remarkable places on the right and left, such as churches, public buildings, markets, halls, colleges, remarkable houses, &c. and where necessary take their dimensions. Remove the instrument to another station in one of the lines already measured, then proceed as before, and so on till the whole is finished. The principal streets being measured, proceed to the smaller and intermediate streets; and, lastly, to the lanes, alleys, courts, yards, and every part requisite to be represented in your plan. Thus, let B, fig. 10, be the first station, where a street runs off to the right, take the angle CBF, and measure the length of the street BC ; when you come to the street O, on the left hand, note its dise tance froin B, and measure the width of it. Remove the instrumen to C, the mark which is at F to B, and that which is at C to D, and measure the angle BCD, noting the street on the right hand of C. Measure CD, and be careful not to omit taking the dimensions of the streets m and n on the left and right, and also their positions. Remove the instrument to D, the mark which is at B to C, and that which is at D to E, then take the angle CDE. Bring the mark C to D, and E to F; then fis the instrument at E, and take the angle DEF; measure DE and EF: in measuring along EF, note the two streets on the right and left at V. Again, Place the instrument at F, and take the angle EFE ; at F, a street goes forwards to G, and another turns to the right: measure FB, and in the way take the width and situation of the two streets at Z; having arrived at B, a part of your survey is completed, and may be laid down on paper. In a similar manner, you must proceed till the whole is finished. Prob. 7. To cut off from a plan a given number of acres, &c. by a line drawn from any point on the side of it. Let A, fig. 11, be the given point, from which a line is to be drawn, towards B, so as to cut off five acres 2 roods 14 perches. Draw AB, so as to cut off a quantity ABC, as near the quantity proposed as you cau judge; and suppose the true quantity of ABC, when calculated, to be only 4 A. 3 R. 20 P., which is 2 R. 34 P.=114 perches=71250 square links too little. Then measure AB, which suppose equal to 1234 links, by the half of which, viz. 617 links, let 71250 links be divided ; the quotient, 115 links, will be the altitude of the triangle to be added, whose base is AB, therefore make BD=115 links, and draw AD, which will cut off the quantity required. OF GAUGING. The business of cask gauging is commonly performed by two instruments, namely, the gauging or sliding Tule, and the gauging or diagonal rod. 1. Of the Gauging Rule. This instrument serves to compute the contents of casks, &c. after the dimensions have been taken. It is a square rule, baving various logarithmic lines on its four sides or faces ; and three sliding pieces, running in grooves in three of them. On the first face are three lines, namely, two marked A, B, for multiplying and dividing; and the third, MD, for malt depth, because it serves to gauge malt. The middle one B is on the slider, and is a kind of double line, being marked at both the edges of the slider, for applying it to both the lines A and MD. These three lines are all of the same radius, or distance, from 1 to 10, each containing twice the length of the radius. A and B are placed and numbered exactly alike, each beginning at 1, which may either be 1, or 10, or 100, &c. or 'l, or 01, or '001, &c. but whatever it .s, the middle division, 10, wi'l be ten times as much, and the last division 100 times as much. But i on the line MD is opposite 215, or more exactly 2150-4 on the other lines, which number 2150°4 denotes the cubic inches in a malt bushel ; and its divisions numbered retrogade to those of A and B. On these two lines are also several other marks and letters : thus, on the line A are MB, for malt bushel, at the number 2150-4; and A for ale, at 282, the cubic inches in an ale gallon; and on the line B, is W, for wine, at 231, the cubic inches in a wine gallon : also si, for square inscribed, at •707, the side of a square inscribed in a circle whose diameter is 1 ; se, for square equal at :886, the side of a square which is equal to the same circle ; and c, for circumference, at 3•1416, the circumference of the same circle. On the second face, or that opposite the first, are a slider and four lines, marked D, C, D, E, at one end, and root, square root, cube, at the other ; the lines C and D containing respectively the squares and cubes of the opposite numbers on the lines D, D; the radius of D being double to that of A, B, C, and triple to that of E; so that whatever the first I on D denotes, the first on C is the square of it, and the first on E the cube of it; so that if D begin with 1, C and E will begin with 1 ; but if D begin with 10, C will begin with 100, and E with 1000; and so on. On the line C are marked oc at 0796 for the area of the circle whose circumference is 1 ; and od at •7854, for the area of the circle whose diameter is 1. Also on the line D, are WG, for wine gauge, at 17:15; and AG for ale gauge, at 18.95; and MR, for malt round, at 52:32 ; these three being the gauge points for round and circular ineasure, and are found by dividing the square roots of 231,282, and 21504 by the square root of 7854 : also MS, for malt square, are marked at 46 37, the malt-gauge point for square measure, being the square root of 2150 4. On the third face are three lives, one on a slider marked N; anc two on the stock, marked SS and SL, for segment starding and seg. went lying, which serve for ullaging standing and lying casks. And on the fourth, or opposite face, are a scale of inches, and three other scales, marked spheroid, or 1st variety, 2d variety, 3d variety; the scale for the 4th, or conic variety, being on the inside of the slider in the third face. The use of these lines is to find the mean diameters of casks. Besides all those lines, there are two others on the insides of the two first sliders, being continued froni the one slider to the other The one of these is a scale of inches, from 12 to 36, and the other is a scale of ale gallons, between the corresponding numbers 435 and 3.61; which form a table to show, in ale gallons, the contents of all cylinders whose diameters are from 12 to 36 inches, their commod altitude being 1 inch, The Use of the Gauging Rule. Prob. 1. To mul.iply two numbers as 12 and 25. Set 1 on B, to either of the given numbers, as 12, on A ; then against 25 on B, stands 300 on A ; which is the product. Prob. 2. To divide one number by another, as 300 by 25. Set 1 on B, to 25 ou A ; then against 300 on A, stands 12 ou B, for the quotient. Prob. 3. To find a fourth proportional, as to 8, 24, and 96. Set 8 on B, to 24 on A ; then against 96 on B, is 288 on A, the 4th proportional to 8, 24, 96, required. Prob. 4. To extract the square root, as of 225. The first I on C standing opposite the one on D, on the stock ; then against 225 on C, stands its square root 15 on D. Pro). 5. To extract the cube root, as of 3375. The line D on the slide being set straight with E; then opposite 3375 on E, stands its cube root 15 on D. Prob. 6. To find a mean proportional, as between 4 and 9. Set 4 on C, to the same 4 on D; then against 9 on C, stands the mean proportional 6 on D. Prob.7. To find numbers in duplicate proportion. As to find a Number which shall be to 120, as the Square of 3 to the Square of 2. Set 2 on D, to 120 on C; then against 3 on D, stands 270 on C, for the answer. Prob. 8. To find numbers in subduplicate proportion. As to find a Number which shall be to 2 as the Root of 270 to the Root of 120. Set 2 on D, to 120 on C; then against 270 on C, stards 3 on D, for the answer. Prob. 9. To find numbers in triplicate proportion. As, to find a Numler which shall be to 100, as the Cube of 36 is to the Cube of 40. Set 40 on D, to 100 on E; then against 36 on D, stands 72.9 on E, for the answer. Prob. 10. To find numbers in subtriplicate proportion. As, to find a Number which shall be to 40, us the Cube Root of 729 is to the Cube Root of 100. Set 40 on D, to 100 on E; then against 72:9 or E, stands 36 on D, for the answer. Prob. 11. To compute malt bushels by the line MD. As, to find the Malt Bushels in the Couch, Floor, or Cistern, whose Length is 230, Breadth 562, and Depth 5'4 Inches. jet 230 on B, to 5'4 on MD; then against 582 on A stands 33:6 pushels op B, for ihe answer. Note.-—The uses of the other marks on the rule will appear in the examples farther on. of the Gauging, or Diagonal Rod. The diagonal rod is a square rule, having four faces ; being commonly four feet long, and folding together by joints. This instrument is used both for gauging or measuring casks, and computing their contents, and that from one dimension only, namely, the diagonal of the cask, or the length from the middle of the bung-hole to the meeting of the head of the cask with the stave opposite to the bung ; being the longest line that can be drawn within the cask from the middle of the bung. And, accordingly, on one face of the rule is a scale of inches for measuring this diagonal ; to which are placed the areas, in ale gallons, of circles to the corresponding diameters, in like manner as the lines on the under sides of the three slides in the sliding rule. On the opposite face are two scales of ale and wine gallons expressing the contents of casks having the corresponding diagonals. And these are the lines which chiefly form the difference between this instrument and the sliding rule ; for all their other lines are the same, and are to be used in the same manner. Er. Tlie rod being applied within the cask at the bang-hole, the diagonal was found to be 34:4 inches ; required the content in gallons. Now to 34:4 inches corrrespond, on the rod, 90$ ale gallons, or 111 wine gallons, the content required. Note. The contents exhibited by the rod, answer to the most common form of casks, and fall in between the ed and 3d varieties following. of Casks as divided into Varieties. 1. The middle frustum of a spheroid, And if the content of any of these be computed in inches, by their proper rules, and this be divided by 282, or 231, or 2150°4, the quo tient will be the content in ale gallons, or wine gallops, or malt busbels, respectively. Because 282 cubic inches make 1 ale gallon. I wine gallon. I malt bushel. To the square of the head diameter, add double the square of the bung diameter; and multiply the sum by the length of the cask. Then let the product be multiplied by •00091, or divided by 1077, for ale gallons : |