REMARKS ON TAKING AND ON BALANCING SURVEYS. The small misclosures in surveys which are judged correct, usually arise from errors in taking courses which are caused by local attractions, and from excess of measure on the distances. There is danger of increasing distances from the irregularities of surface. Inexperienced chainmen should not be permitted to exercise their judgment in making allowance for surface in such cases. In balancing surveys, there is less danger from diminishing distances than there is from increasing them. Where there are no obstructions and the land is level between the terminating points, it may be supposed that distances are correctly measured, which should not be much increased or diminished. When a course is near northeast and southwest, or northwest and southeast, if by diminishing the distance, both the latitude and departure are so diminished as to favour the balancing, it is good evidence of an excess of measure. may occur in which distances may be increased, but the surveyor should be very cautious about it. The method of balancing surveys by subtracting half the sum of the difference from the numbers in the larger column and by adding it to those in the less, is not always correct. Sometimes it is very incorrect. Cases On three sides of a survey there may be no obstructions to taking courses or to measuring distances, when on the other side there may be so many obstructions to both, that a very correct survey of it cannot be taken. In such a case, most of the corrections should be made on the lines where the difficulties are found. In general, corrections should be made on long lines, and it will require time and experience for the learner to see whether they should be made on courses or on distances, or whether on both. DECLINATION OF THE MAGNETIC NEEDle. In the appendix, page 85, is an account of my observations of the declination of the magnetic needle from 1805 to 1825. From 1813 to the latter period, I could not satisfy myself as to the motion of it. Sometimes the declination appeared to increase, at other times to decrease, but the motion either way was so small that the needle appeared about stationary. Since 1825, the needle has declined so far to the west as to complete a full degree since 1805. At Salem in Massachusetts, the observations of Mr. Bowditch are recorded in direct opposition to mine. The distance between this town and Salem cannot exceed 120 miles. The celebrity of that gentleman is too well established to have his correctness questioned, and his observations give currency to an opinion that in the northern states the needle has had no retrograds motion. According to a record kept at Hartford for many years past by a friend of mine, Mr. N. Goodwin, the needle has declined to the west. About the year 1810, the late Mr. Spencer of New Hartford in Litchfield County, published in the Connecticut Courant that for a number of years then past, the west declination had increased. Observations made in the State of New-York, by the Surveyor General, also give ar. increase of west declination. In 1813, an article appeared in a Philadelphia paper from which the following was extracted. At Philadelphia, " 1701, W. declination 8° 30' by Mr. Scull. 1793, W. 1° 30' by R. Brooks. 1794, the needle was observed to recede westward by H. Brooks of Philadelphia, M. Humphrys of Md. and others in Virginia. 1802, more than 10 30 W. by R. Howel. 1804, 20 W. by several men of science. 1813, 2° 27' W. by Th. Whitney." If it is sufficiently proved that in various places the west declination has increased, while at Salem it has decreased; who can account for the anomaly? Is it because Salem is contiguous to the ocean? Can that affect it? What is the strange unknown something which gives polarity to the needle? Where is its residence? Is it in the atmosphere, or is it in the earth a fixed law of nature? If the latter, why is the needle at every place continually changing its position? Why is the line of no declination a perpetual wanderer without a home, intruding on every land and every sea? To the young Surveyor.-In performing practical operations, never call to your assistance Sir Richard Rum. LOGARITHMS. Let there be a series of numbers, increasing by a common difference, as for instance, by 1, viz. 0, 1, 2, 3, 4, 5, 6, &c. and another, viz. 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, &c. increasing by a common multiplier, 10. The former are the LOGARITHMS of the latter. It will be seen that 1 is added in the upper series, as often as 10 is made a multiplier in the lower. If two logarithms, then, be added, and the numbers below them multiplied, the sum of the logarithms will be the logarithm of the product. Thus, if the logs. 1 and 2 be added, and the corresponding numbers, 10 and 100 be multiplied, the sum will be 3, and the product 1,000. These numbers may be seen to correspond to each other, in the two series above; there, 3 stands over 1,000, and is, of course, its logarithm. So, if 2 and 4 be added, and 100 and 10,000 multiplied, the sum will be 6, and the product, 1,000,000, which may be likewise seen above to correspond. Furthermore, if, from a greater logarithm a less be taken, and, at the same time, the number corresponding to the greater be divided by the number corresponding to the less, the remainder will be the logarithm of the quotient. Thus, if from the log. 5, the log. 3 be subtracted, and, at the same time, 100,000, the number corresponding to 5, be divided by 1,000, the number corresponding to 3, the remainder will be 2, and the quotient 100, which may be seen above to correspond to each other. With a set of logarithms, then, calculated, not merely for the numbers 10, 100, 1000, &c., but for all numbers whatever, it is plain that we might perform the operations of multiplication and division, by addition and subtraction only. Such logarithms are calculated and arranged in tables for use. By observing the two ranks of numbers at the head of this article, it will be perceived, that the logarithms increase by a constant addition, (of 1,) and the corresponding numbers by a constant multiplication, (by 10,); and therefore, that when the logarithms are in arithmetical progression, the numbers are in geometrical progression. On this account, logarithms have been defined to be a series of numbers in arithmetical progression, corresponding to another series in geometrical progression. But it will be seen that the numbers, 10, 100, 1,000, 10,000, &c. are the first, second, third, fourth, &c. powers of 10, and that their corresponding logarithms, 1, 2, 3, 4, &c. are the indices of these powers. Hence a better definition is, LOGARITHMS OF NUMBERS ARE THE INDICES, EXPRESSING THE POWERS TO WHICH A GIVEN NUMBER MUST BE RAISED, TO PRODUCE THOSE NUMBERS. This "given number" is called the RADIX, or BASE, of the system, and may be any number whatever. The number 10, however, is most con The logarithm of 1 is 0, and that of 10 is 1. Hence it is evident, that, for numbers between 1 and 10, the logarithm will be between 0 and 1. Of course, it will be a fraction; and, when placed in the tables, it is expressed by a decimal. For numbers between 10 and 100, the logarithm is between 1 and 2; for numbers from 100 to 1,000, between 2 and 3; from 1,000 to 10,000, between 3 and 4, and so on, consisting, of course, in each case, in part of a whole number, and in part of a decimal. From the above, it will be sufficiently evident, that THE INTEGRAL (OR WHOLE NUMBER) PART OF ANY LOGARITHM, IS A UNIT LESS THAN THE NUMBER OF INTEGRAL PLACES IN THE CORRESPONDING NUMBER. OR, IT SHOWS HOW FAR THE HIGHEST FIGUre of THE NUMBER IS DISTANT FROM THE UNITS' PLACE. THE INTEGRAL PART OF A LOGARITHM IS CALLED THE INDEX, or CHARACTERISTIC, OF THAT LOGARITHM. As this characteristic is always a unit less than the number of integral places in the corresponding number, it is evident that when this num ber has but one integral place, the characteristic will be 0; which is the case with the last log. above. If, then, the number be a proper fraction, or a decimal, and have, therefore, no integral place, or places, the characteristic of the log. on this principle, ought to be less than nothing. Now, since we cannot diminish nothing, so as to render it less than nothing, we employ, whenever it is necessary to use logarithms of this kind, a characteristic with a mark over it, thus, 1, 2, 3, 4. &c. The mark shows that the characteristic ought to be less than nothing, and the number, over which it is drawn, informs us, how much it should be less than nothing. In all cases, then, where it is necessary to add the logarithm, we must subtract this characteristic; and in all cases where it is necessary to subtract the logarithm, we must add the characteristic. For, since adding nothing to a number does not alter it, adding less than nothing ought to diminish it. And, since subtracting nothing from a number does not alter it, subtracting less than nothing ought to increase it.* It will be seen above, that for every division of the number by 10, (or, every removal of its decimal point towards the left) the characteristic becomes a unit less. *An apology may, perhaps be considered necessary for the language here employed. The writer would not be thought to advocate the absurdity that a number may actually be less than nothing; but the phrase is so concise and expressive, and notwithstanding its absurdity, conveys to the mind of one unacquainted with the nature of negative quantities, the idea intended, so much more perfectly, than any explanation we should here have room to make, could do, that it has been thought advisable to employ it. This treatise is intended for practical men, and not for metaphysicians nor scholars. To those who have at 4.763353, &c. By observing these logarithms, it will be seen, that THE NEGATIVE CHARACTERISTIC OF A LOGARITHM, SHOWS HOW FAR THE FIRST SIGNIFICANT FIGURE OF ITS CORRESPONDING DECIMAL IS DISTANT FROM THE UNITS' PLACE. EXPLANATION OF THE TABLE. When the logarithms of numbers, from 1 upward to any other given number, are calculated, and arranged in a table, they constitute a table of logarithms. Tables of logarithms of great extent have been calculated, but those in common use extend to numbers no higher than 10,000. This is far enough for the purposes of ordinary calculation. In the first column on the left of each page, are arranged the natural numbers, and to distinguish this column from the others, the letter N is placed over it. Opposite these numbers in the next ten columns, are arranged the logarithms. For the first 100 numbers, however, on the first page of the table, there is but a single row of logarithms, opposite each column of numbers, and there are four rows of numbers on the page. TO FIND THE LOGARITHM OF ANY WHOLE NUMBER. If the number be less than 100, look for it in the column headed N, and directly opposite to it, in the column headed Log. will be found its logarithm. If the number be greater than 100, but less than 1,000, find it as before, in the column headed N, and directly opposite, in the column headed 0, will be found its logarithm. It will be seen, that when the first two figures of several successive logarithms are alike, they are omitted in the table, after having been once inserted, and only four figures are retained. When, therefore, there are but four figures in the logarithm opposite the given number, cast the eye up the blank till you find two more, which prefix to those already found. It will likewise be observed, that for the logarithms of numbers above 100, no characteristic is inserted in the table. But this may easily be supplied, since it is always a unit less than the integral places in the number. For numbers between 100 and 1,000, then, it is 2. Find the logarithm of 868. Opposite this number in the table, are found the figures 8520. Casting the eye up the blank we meet with 93, which we prefix to 8520, making 938520. The characteristic, 2, being then prefixed, we have the complete logarithm, 2.938520. If the number be greater than 1,000, and less than 10,000, find the first three figures in the column, headed N, and the fourth at the head of the page. Then, exactly opposite the first three, and in the column headed by the fourth, will be found the last four figures of the required logarithm. To these must be prefixed two others, found as above, in the column headed 0. contain six figures, the first two of those must be prefixed. If not, cast the eye up the blank, in search of the proper ones, as before. There is one exception to this rule. In some logarithms, will be seen points or periods occupying the places of figures. When this is the case, ciphers must be written in place of the points, and the two figures to be prefixed, must be sought in the next lower line, in the column, headed 0. Thus, Find the log. of 4,177. Opposite 417, and in the column headed 7, are found the figures 0364. The opposite log. in the column headed 0, has six figures, of which the first two, viz. 62, are to be prefixed to 0864, making 620364. The characteristic being joined, the log. is 3.620864. For 4,143, we find opposite 414, and under 3, the figures 7315. Here we must look along up the blank in the column, headed 0, for the two figures to be prefixed, which are 61, making the log. 617315, which, with its characteristic, is 3.617315. For 4,366, we find opposite 436, and under 6, the figures..84. Here, therefore, we must write ciphers for the periods, thus, 0084, and look in the next lower line of the column headed 0, for the two figures to be prefixed; which are 64, making the log. 640084, which, with its characteristic, is 3.640084, If the number exceed 10,000, find the logarithm of the first four figures as above. By the remaining figures, multiply the number opposite, in the column headed D; from the right of the product, reject as many figures as the given number has places more than four, and add what is left to the logarithm previously found. This sum will be the log. required. The column, marked D, is a column of differences: that is, it contains the differences between each two successive logarithms. In many cases a mental estimate may be made, of what should be added to a log. without the use of this column. The logarithms of pure or mixed decimal numbers should be taken out as if for whole numbers, care being taken to prefix the proper characteristic, in each case. Of vulgar fractions the logs. may be found by reducing them first to decimals, or by the rule given below, in division. TO FIND THE NUMBER CORRESPONDING TO ANY GIVEN LOGARITHM. Find the logarithm next less than that given, in the column headed 0; pass the eye to the right, along that horizontal line, into the other columns, and you will find either the log. given, or one very near it; the first three figures of the corresponding number will then be found opposite, in the column headed N, and the fourth figure, directly over the log. at the top of the page. The integral places in this number will be determined by the characteristic of the log. If this characteristic be 3, all the places will be integral; if it be 2, one decimal must be pointed off; if it be 1, two decimals, &c. On the other hand, if it be 4, a cipher must be annexed to the number; if it be 6, two ciphers must be annexed, and so on. When the given log. cannot be found in the table, however, the number may be more accurately obtained by the following process. Find in the table the next less log., and take the difference between it and the given one. Make this difference the numerator of a fraction, and the number opposite in the column headed D, the denominator; reduce this fraction to a decimal, which write immediately after the four figures corresponding to the tabular log. used. Afterwards place the decimal point where the characteristic of the given log. may require. |