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QUESTION v. In a field I took two stations P and e, at the distance of 10 chains from each other : OPA = 21° 2017

(PQD =

10° 40'
APB
10

DRC =
BPC = 57 12

COB = 42 OO
CPD = 29 40

BQA =

67 05 DPE 64 25

AQF = 137

00
EPF
16

FQE =
Required the figure and area.

= 49

18 30

= 79

62 52

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From a station within a field of five sides I meafured the distances to the several corners, beginning with that on the west, and measuring round towards the north, viz. Ist distance 7345 lin''s, the 2d = 5980, the 3d = 6495, the 4th = 6015, and the 5th = 7050; also the angles formed by these distances in the same order were 711, 557,494, and 81, degrees. What is the area? Anf. 979ac. 2r. 35-921p.

VII.

QUEST'ION From a place o, near the middle of a field ABCDEF, from which I could see all the angles, I measured the distances to the several corners, and observed the quantities of the angles formed at o by those diltances, as below.

Distances

Distances
OA = 4315 links
OB = 2982
oc = 3561
OD = 5010
OE = 4618

OF = 3606
What is the area?

Angles
AOB = 60° 30'
BOC 47 40
COD 49 50
DOEN 57 10
EOF = 64 15

FOA = 80 35
Ans. 412ac. ir. 1783224p.

QUESTION

VIII. Having made choice of a station within a piece of land, I measured from thence the several bearings and distances of the corners as below : Required the

area.

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Bearings Distances
NE

3540
NE

4785 Nwbw

3915

4125 SSE 7°E

2030 EbS3E

2945 Anf. 319ac. 3r. I'193P.

swbs

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SECTION II.

OF CASK GAUGING.

TH

HE meaning of the word Gauging is restricted

to the measuring of casks, and other things falling under the cognizance of the officers of the

excife;

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excise ; and it has received its name from a gauge rod used by the practitioners of the art.

The business being performed, or the calculations made, commonly by means of the instruments, called the gauging or diagonal rod, and the sliding rule or gauging rule, it will be necessary to treat of these instruments, which I shall do as below.

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The Defcription and Use of the Sliding Rule. This is a square rule, having consequently four fides or faces, three of which are furnished with sliding picces running in grcores. The lines upon them are mostly logarithmic ones, or distances which are proportional to the logarithms of the numbers placed at the ends of them; which kind of lines were placed upon rules, by Mr. Edmund Gunter, for expeditiously performing arithmetical operations ; in which business he used a pair of compailes for taking the several logarithmic distances : but instead of the compaíses, siiding pieces were added, by Mr. Thomas Everard, as being more convenient and certain in practice.

Upon 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 is upon 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 be either i, or 10, or 100, &c, or •1, or 'on, or .001, &c; but whatever it is, the middle division 10, will be 10 times as

much,

much, and the last division 100 times as much. But I on the line mo is opposite 215, or more exactly 2150°4 on the other lines, which number 215004 denotes the cubic inches in a malt bushel, and its divisions numbered retrograde to those of A and B. Upon these two lines are also several other marks and letters: thus, on the line A are MB, for malt bulhel, at the number 2150:4; and A for ale, at 282, the cubic inches in an ale gallon; and upon the line e 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.

Upon the second face, or that opposite the first, are a flider and four lines, marked D, C, D, E, at one end, and root, square, root, cube, at the other; the lines c and e 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 rond denotes, the first on c is the square of it, and the first on E the cube of it, ; so if d begin with 1, c and e will begin with '1; but if p begin with 10, c will begin with 100, · and E with 1000; and so on. Upon 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 whole diameter is 1. Also, upon 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 or circular measure, and are found by dividing the square roots of 231, 282, and 2150:4 by the square root of 7854. allo ms, for malt square, are marked at 46.37, the malt gauge point for square measure being the {quare root of 2 1 504. 003

Upon

Upon the third face are three lines, one upon a slider inarked N; and two on the stock, marked ss and sl, for segment standing and segment lying, which serve for ullaging standing and lying casks.

And upon the fourth, or opposite face, are a scale of inches, and three other scales, marked spheroid or ist variety, 2d variety, 3d variety; the scale for the 4th, or conic variety, being on the inside of the flider 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 niders, being continued from 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 num. bers •435 and 3:61 ; which form a table to sew, in ale gallons, the contents of all cylinders whose diameters are from 12 to 36 inches, their common altitude being i inch.

As the sliding rule is for performing, very expeditiously, any operations of multiplication, divifion, and extraction of roats, which may be required by any precept proposed in words, &c; so the manner of making these operations will appear in the following problems.

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To find the ProduƐt of Two Given Numbers,

by the Sliding Rule. RULE. To either of the given numbers on a set I on B, then against the other number on is the product on A.

EXAMPLE 1. Required the product of 12 and 25.

By placing i on B. under 12 on A, above 25 on B stands 300 on A; which is the product required. NOTE. When the I on B has been set to the one

factor

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