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The first column contains the measured distances from the starting point to every forward station expressed in links of Gunter's chain. The two central columns, headed "Back Sight" and "Fore Sight," contain the readings of the two staves at the back and fore observations respectively. The difference of such readings is placed in one of the two side columns headed "Rise" or "Fall," according as the ground at the forward station is higher or lower than that at the back station. If it be highest (or the ground rises, as it is called), then the forward reading will be the smaller of the two; but if it be the lowest (or the ground falls), then the forward reading will be the greater of the two; thus, in our first reading, the back observation is 13.71, and the forward observation 7.88, their difference= 5.83 feet, which is the difference of level between the two points; and as the forward rea ing was the smaller of the two, it is clear that the ground was rising at that place, and therefore the difference of the readings, viz. 5.83, is placed in the column of rises. In the next three successive pair of sights, the forward readings are the greatest, indicating a continued descent of the surface line, and the differences of those readings are inserted in the column of falls, viz. 6.90, 7.84, and 9.78. At the next following sight, the forward reading is again the smallest, therefore the difference 13.67 is placed in the column headed "Rise," and so on of the rest. No mistake can arise by placing the subtraction in the wrong column, as in every instance it must be

placed in the column adjoining the larger quantity; thus if the fore sight is greater than the back sight, the resulting quantity must be placed in the column of falls, which is adjoining to that containing the reading of the fore sight, and vice versa.

The adjoining column, headed "Reduced Levels," contains the absolute heights of each forward station above the datum line of the section, or a horizontal line passing through the starting point or bench mark A ; these quantities, which are technically called the reduced levels, are obtained by the constant addition and subtraction of the numbers contained in the columns of "Rise," and "Fall," the former being considered as positive, and the latter as negative quantities; thus, assuming the level of the starting point A as the datum, we have the first forward station 5.83 feet higher than the datum, therefore in the column of reduced levels it is marked + (plus); next we have a fall or negative quantity of 6.90 feet, which must be subtracted; but as it is greater than 5.83, it shows that this station is below the datum line, by the difference between 5.83 and 6.90=1.07 feet, which is the depth of the second forward station below the datum line, and therefore is marked (minus); the next is likewise a fall of 7.84, and as our last result was below the datum line, this additional negative quantity will take us still lower by its whole amount; it must, therefore, be added to 1.07, giving 8.91 feet for the depth of our third forward station below our datum, and it is therefore entered in

the column of reduced levels with a minus sign. The next is also a fall of 9.78, which, applied as the last, gives 18.69 for the depth of the fourth forward station below the datum. The ground then rises again, and we have an ascent of 13.67 feet, which will bring us nearer to our datum; and as it diminishes our depth below the datum line it must be subtracted from the last result; thus, 18.69 13.675.02 feet for the depth of the fifth forward station below the datum; we have then a rise of 15.55, which will carry us above the datum by the amount of difference between it and 5.02, leaving 10.53 feet for the height of the sixth forward station above the datum line; the next is a fall of 4.74, which diminishes our height by that quantity, and therefore must be subtracted from 10.53, leaving 5.79 as the height of the seventh forward station above the datum.

In like manner every other pair of sights in our example was reduced, applying each difference of the back and forward readings with their proper signs, until, at the close of the work, the point B (the last forward station) was found to be 49.33 feet above the datum line, or level of the starting point A.

The reduction of levels becomes a simpler operation when the height of the bench mark (used as a starting point) above the intended datum line is known; thus (in our example), suppose the height of the bench mark A was 100 feet above the level of high-water Trinity mark at London Bridge, and that it was intended to assume the level of that mark as the datum line of our


section; then 5.83 feet, the rise to the first forward station, must be added to 100, giving 105.83 for the height of the ground at the point a above datum; next from 105.83 subtract the fall 6.90, which gives 98.93 for the height of the point b above datum; then from 98.93 subtract 7.84, which gives 91.09 for the height of c above datum; and in like manner, by adding the quantities of rise, and subtracting those of the falls, the whole line of levels may be reduced to the line assumed as the datum.

As a proof of the accuracy of the arithmetical operation, the columns of back and fore sights should be added up, and the lesser sum subtracted from the former; the result of the agreement with that by the reduced levels is a proof of accuracy. Likewise another proof may be obtained by adding up the contents of the column of rise and fall; and if upon taking the lesser sum from the greater, the remainder represents the same quantity as obtained by both the other operations, there can be no doubt of the correctness of the reductions of the levels, as in our example. By the reduced levels, the height of B above A is 49.33 feet. The sum of the back readings is 87.95, and that of the forward readings 38.62; their difference also gives 49.33 for the height of B above A; and, lastly, the sum of the rises is 54.88, and that of the falls is 5.55, the difference giving, as before, 49.33 feet.

It is, perhaps, to be recommended that the observer should reduce his levels as he proceeds in the field, as

it will occupy but very little time, and can be frequently done while the staff-man is taking a new position; besides, the observer will frequently be able to detect by the eye if he is committing any glaring error, as, for instance, inserting a number in the column of rises, when it ought to occupy a place in that of the falls, the surface of the ground at once reminding him that he is going down hill instead of ascending.

If the foregoing method of reducing levels be found difficult or troublesome, on account of the introduction of plus and minus signs, they can be dispensed with as well as the columns of "Rise" and "Fall" by proceeding in the following manner. Assuming the starting point to be any even number of feet high; or, what is the same thing, assume a datum line any even number of feet below the starting point, as 100 or 1000, taking care that your choice falls upon a number greater than the number of the whole fall you are likely to experience in the operation; then from this assumed height subtract the reading of the forward staff, and to the remainder add the reading of the back staff; the result will be the height of the first forward station above the assumed datum line; then from this height subtract the next forward reading, and to the remainder add the reading of the back staff; the result will be the height of the second forward station above the assumed datum, and so on throughout the whole levelling operation. The difference between any two of the readings will be the difference of level between the corresponding points on the ground.

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