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racy of their positions. Now the measurement of the length of such a straight fence as co, which is nearly perpendicular to the two chain lines that cross it, gives a very imperfect proof of the accuracy of its position, especially if it be a long one. In this latter case the fence co might be a full chain from its true position at one end, and its length, as shown by the plan, would be so near the measured length that the error would not be detected; whereas the tie line m n, shown on the plan, would detect the error at once, by its increased or diminished length. However, when two chain lines cross a straight fence obliquely, the length of the fence, from crossing to crossing, evidently gives a sufficient proof of the accuracy of its position; yet to measure a short tie line, at one of the crossings, is a shorter method of proof, in all cases where the length of the fence is considerable. The method of proof recommended by the Commissioners in question, ought, therefore, never to be adopted, except where one or both of the chain lines cross the straight fence obliquely, and at a short distance. It seems hardly necessary to remark that the positions of the straight fences G B, C F are determined by the lines measured close to them. Thus the survey of six fields may be made, and its accuracy proved, by five lines, with the two short tie lines mn, no, which may be regarded as mere offsets. Besides, had the fences GB, C F been crooked, the same lines would have effected the survey by offsets thereon. Moreover, the survey of these six fields, all the internal fences being as shown in the plan, may be accurately effected by four lines, in the following manner. The triangle ADE remaining as the foundation of the survey, let a station, or rather direction point, p be entered in the field notes, in the direction of the straight fence a b, and another similar point at q, in the direction of the straight fence o c; leave also a proper station mark, or pole, at r; on arriving at s in E A, leave another station, in such a position, that a line from s to r will cross all the four straight fences. This last line will prove the fundamental triangle AD E and the positions of the four straight fences, at the same time, without measuring the lines by the fences GB, CF; which can have now three crossings by chain lines, and the other two fences a b, o c have each two crossings by chain lines, and each one direction point, viz: Р and q; through which points these fences must respectively pass, after they have been drawn through their crossings on DE, and the other line from s to r, not shewn on the plan, This method of determining the position of straight fences, though theoretically elegant, cannot always be easily
practised, especially where the fences are high, or the ground hilly, thus preventing the directions of the straight fences being seen to distant chain lines, as in the cases of the fences a b, o c, with respect to the chain line AD.
THE METHOD OF MEASURING HILLY GROUND.
When the ground, over which lines are measured, rises or falls, or both alternately, the horizontal distances are what are required in plotting the survey, as well as for finding the content thereof, and not the actual distances measured along the surface of the ground.
For many ordinary purposes the horizontal measurement may be obtained by holding the end of the chain up so as to keep it horizontal, as nearly as can be judged by the eye, the arrow being placed vertical under the end so held up: but when a large and accurate survey is required, the distances must be measured along the line of ground, and the angles of elevation and of depression of the several inclined parts of the line must be taken, either with a common quadrant, or afterwards with the theodolite (to be hereafter described), and the lengths of the several rises and falls must be noted; from which the corresponding correct horizontal distances may be readily computed. The following table shews the number of links to be subtracted from every chain, or 100 links, for the angles there set down.
TABLE shewing the reduction in links and decimals of a link upon 100 links for every half degree of inclination from 3. to 20° 30'.
By this table the trouble of computation is avoided, only the
distance, measured on each rise or fall, requiring to be multiplied by the reduction in chains corresponding to the angle of each rise or fall, and the product, subtracted from that distance, will give the correct distance, as in the following
1. A line was measured 12.43 chains, on ground naving a rise of 8 degrees, required the horizontal length of the line. Here to 810, or 8° 30′ corresponds the reduction 1.10 links, whence
12.29.327 horizontal distance,
in which the decimal, being less than half a link, is rejected; thus making the correct horizontal distance 12.29 chains, or 1229 links.
2. The acclivity of a hill rises 20°, and measures 16·14 chains, its declivity falls 1140 and measures 32.28 chains, required the horizontal distance between the extremities of the line thus measured over the hill, it being level at the top 2.80 chains,
Here 16.14 × 6·08 98.1312
32.28 × 2.01 = 64.8828
163-0140 or 163 links, or 1·63 chains.
49.59 chains, the horizontal distance required.
NOTE. When fences are crossed, stations made, &c., on the acclivity or declivity of a hill, the horizontal distance up to such points must be found. Some surveyors place the arrow forward a distance equal to reduction, due to the angle of acclivity or declivity, at the end of every chain measured, and thus obviate the necessity of reducing the line afterwards, having for the purpose a small pocket quadrant, so graduated that the plumb-line thereof shews, on observing the angle of elevation or depression, the reduction required for each chain. great deal of trouble is thus saved, as the theodolite cannot be conveniently carried about for this purpose. Such pocket quadrants are not made by the mathematical instrument makers, being the productions of clock makers or other mechanics, according to the various designs of surveyors.
THE USE OF THE PARALLEL RULER,
IN REDUCING CROOKED FENCES TO STRAIGHT ONES, TO FACILITATE THE COMPUTATION OF THE CONTENTS OF FIELDS.
As some surveyors prefer the parallel ruler to the method already given, for reducing crooked fences to straight ones, the method of using that instrument for this purpose is here given. This method is founded on a well-known proposition of Euclid, in which it is shewn that triangles on the same base, and between the same parallels, are equal.
Let ABC, ABD be triangles on the same base A B, and between the same parallels AB, CD; then the triangle ABC is equal to the triangle AB D. And, if the triangle A EB, which is common to the other two triangles, be taken away, the remaining triangles A E C, BED will also be equal; whence equal areas may be transferred from one side of a line to the other, which is the principle, on which, as already said, the following Problems are founded.
IT IS REQUIRED TO REDUCE THE OFFSET-PIECE ABCDE TO A RIGHT ANGLED TRIANGLE AEC, BY AN EQUALIZING LINE Ec, WITH THE PARALLEL RULER.
to B, which will cut A c at a, where a mark must be made. Lay the ruler from a to D, and the further side thereof being now held fast, bring the near side to C, marking Ac at b. Lay the ruler from 6 to E, move it parallel to D, marking A c Join Ec; then A Ec is a right angled triangle required, and its area may be found by taking half the product of AE and A c.
THE FOLLOWING IS A GENERAL RULE FOR SOLVING PROBLEMS OF THIS KIND.
Draw a temporary line, as A c at right angles, or at any other angle to the chain line, as A E, of the offsets.
1. Lay the ruler from the first to the third angle, and move it parallel to the second angle; then make the first mark on the temporary line.
2. Lay the ruler from the first mark on the temporary line to the fourth angle, and move it parallel to the third angle; then make the second mark on the temporary line.
3. Lay the ruler from the last named mark to the fifth angle, and move it parallel to the fourth angle; then make the third mark on the temporary line.
4. Lay the ruler from the last named third mark on the temporary line to the sixth angle, and move it parallel to the fifth angle; then make the fourth mark on the temporary line.
In this manner the work of casting by the parallel ruler may be conducted to any number of angles. Great care must be taken, during the operation, to prevent the ruler slipping, as such an accident will derange the whole of the work, if not discovered and immediately corrected.
TO REDUCE A CURVED OFFSET-PIECE TO A RIGHT-ANGLED TRIANGLE.
Let A abcde B be the 5 curved offset-piece. Divide the curve by points a, b, &c., so that the parts A a, a b, &c., may be straight, or nearly so; and draw A 5 perpendicular to A A B. Lay the ruler from A
to b; move it parallel to a, and mark A 5 at 1. Lay the ruler from 1 to c; move it parallel to b, and mark A 5 at 2. Lay the ruler from 2 to d; move it parallel to c, and mark A 5 at 3. Lay the ruler from 3 to e; move it parallel to d, and mark A 5 at 4. Lay the ruler from 4 to B; move it parallel to e, and mark A 5 at 5. Draw the line B 5; then will A B 5 be a right angled triangle equal in area to the offset-piece A adc de B, as required.
EXAMPLES FOR PRACTICE ON THE TWO PRECEDING PROBLEMS.
1. Lay down a right-lined offset-piece, from the following notes; reduce it to a triangle by the parallel ruler; and find its content, both by calculation from the offsets and the casting of the ruler.