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In practice, however, it is usual to make all the indices positive. This is done by adding 10 to each negative index ; observing to reject an equal number from the final result. Thus, for negative-1, we may put down positive 9.

for negative-2, we may put down positive 8.

for negative-3, we may put down positive 7, &c. Because, minus 1, plus 10, equals 9.

minus 2, plus 10, equals 8.

minus 3, plus 10, equals 7, &c. 5. Repeating the third example, we have Numbers. Logarithms.

Logarithms.
3.902
0.591287

0.591287
597.160
2.776091 =

2.776091
.0314728
2.497935 or

8.497935

Prod. 73.3333

1.865313

1.865313 Here the sum of the indices is 11: from which reject 10, and the result is the same as before.

COMPOUND INTEREST BY LOGARITHMS.

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RULE. Find the amount of 1 dollar for 1 year ; multiply its logarithms by the number of years; and to the product, add the logarithm of the principal. The sum will be the logarithm of the amount for the given time.

From the amount substract the principal, and the remainder will be the interest.

EXAMPLE. The last example in Daboll's Arithmetic, under Compound Interest, is as follows :-“ What will 50 dollars amount to in 20 years at 6 per cent."

This question wrought out in the most expeditious manner by common arithmetic, would take the student scarcely less than three hours, and the final result would be, if done correctly,

$160.3567736106422365941496492288974572748800. But, by logarithms, this sum may be done in as many minutes, and the correct answer, in dollars and cents, obtained with very few figures. Thus: Amount of 1 dollar for 1 year is 1.06 logarithm 0.0253059 Multiply by the time

20

0.5061180 Add log. of principal 50 1.6989700

Amount as above=$160.35,7 2.2050880 From the foregoing general principles of the nature and application of logarithms, are derived an infinite number of specific rules, adapted to particular cases.

OF THE TABLE OF NATURAL SINES AND TANGENTS.

To those who are unacquainted with logarithms, it will be interesting to know, that all the cases in right and oblique angled trigonome

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surveying, may, by the help of natural sines and tangents, be solved exactly in the same way, and with the same facility, as he would solve a simple question in the Rule of Three. Natural sines are merely decimals, bearing the same proportion to unity, or 1, that the corresponding number of degrees and minutes bears to radius, or 90°. Naiural tangents bear the same proportion to unity, or 1, that the corresponding number of degrees and minutes bears to 45°, because it is a ivell known principle, that the sine of 90°, and the tangent of 45°, are each equal to radius. That is, 1 is assumed as the natural sine of 90° in the table of natural sines, and as the tangent of 45° in the table of tangents, and every other number in each of these tables, is calculated accordingly.

GENERAL RULE. 1. State the question in every case, as already taught: 2. Multiply the second and third terms together, and divide the product by the first.

The manner of taking natural sines and tangents from the tables, is the same as for logarithmic sines and tangents; only that there is in the tables, no column of differences as in the latter, for the more readily finding the odd seconds, when required. But these may be found by making a proportion for the aliquot parts.

There are some problems to which natural tangents afford a much more simple and ready solution, than any process by logarithms. The following one, in heights and distances, will illustrate this.

EXAMPLE. The allitude of an inaccessible object taken at an unknown disiance from its base, is 55° 54'; and when taken again at the distance of 93 feet from the place of the first observation, in a direct line with it, the altitude is 33° 20': Required, the height of the object.

RULE. Divide the difference of the natural co-tangents of the angles of elevation, by the distance between the stations. ThusCo-tangent of 33° 20' is 1.52043 of 55° 54' is .67705

feet.
Divide by the diff. = .84338)93.0000(110.27 Ans.

Note. This is the shortest solution possible, and perfectly easy.

Again : Given the latitude and departure, in transverse sailing or surveying; to find the course.

RULE. Divide the departure by the latitude, the quotient will be the natural tangent of the course: or, divide the latitude by the departure, and the quotient will be the co-tangent of the course. Universally, If in any right angled triangle, the perpendicular be divided by the base, the quotient will be the tangent of the angle at the base ; and if the base be divided by the perpendicular, the quotient will be the tangent of the angle at the vertex of the perpendicular.

OF THE TRAVERSE TABLE, OR TABLE OF LATITUDE AND DEPARTURE.

This is calculated for degrees and quarters of degrees, and for any distance up to 100 rods, chains, &c.; by which the northings and

PROBLEM XII.- To find the latitude and departure, or northing, doc. for any course and distance.

If the course be less than 45°, look for it at the top, but if more than 45°, at the bottom of the page, and look for the distance in the right or left hand column; against the distance, and directly under or over the course, stand the northing, &c. in whole numbers and decimals.

If the course be less than 15°, the northing or southing will be greater than the easting or westing; but if more than 45°, the easting or westing will be the greatest.

When the distance exceeds 100, take any two or more numbers, which, added together, will equal the distance, and find the latitude and departure for each of these numbers; add the several latitudes together, and the sum will be the whole latitude; and so for the departure. And when the distance is in chaius and links, or whole numbers and decimals, find the latitude, &c. for the chains or whole numbers, and then for the links and decimals, remembering to remove The decimal point in the table further to the left, according to the given decimal.

1. Required the latitude and departure for 45 rods, on a course N. 15° 15' W.

Under 15° 15', and against 45, is 43.42 for the northing, and 11.84 for the westing

2. Required the latitude and departure for 120 rods, on a course S. 59° 30'E.

Take one third of 120, which is 40; against this number, over 580 30', is 20.90 for the latitude, and 34.11 for the departure. These multiplied by 3 give 62.70 for the southing, and 102.33 for the easting.

3. Required the latitude and departure for 37.36 rods, or 37 chains and 36 links, on a course N. 26° 45' E. For 37. Lat. 33.04

Dep. 16.65 0.36

.16

.32

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NOTE. When the minutes are not 15, 30, or 45, the northings, &c. may be had by proportion, or they may be calculated by natural sines, or by trigonometry.

PROBLEM XIII.—To calculate the northing or southing, foc. for any course and distance, by natural sincs.

Find the nat. sine and co-sine of the course, and into each of these multiply the distance ; the products will be the latitude and departure

Required the latitude and departure for 6 chains and 22 links on a course N. 38° 27', W. Nat. sine of 38° 27', 0.62183

Nat. co-sine, 0.78315 6.22

6.22

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3.8677826

4.8711930

USEFUL NUMBERS, AND THEIR LOGARITHMS.

Numbers. Logarith. Circumference of a circle to diameter, 1 Area of a circle to radius,

3.14159265359 0.4971499 Surface of a sphere to diameter, Circumference of a circle to radius, 1 6.28318530718 0.7980799 Solid contents of a sphere to diameter, 1 0.52359877560 -1.8950899 Solid contents of a sphere to radius, 1 4.18879020479 0.6220886 Square of

3.14159265359 9.86960440109 0.9942997 Square root of

3.14159265359 1.77245385091 0.2485750 1-360 of

3.14159265359 0.00872664626 -3.9408474 1-24 of

3.14159265359 0.13089969390 -1.1169387 1-4

of do. or area of cir. to diam. 1 0.78539816340 -1.8950899 365th root of $1.05, or amount of $1. for 1 day, 1.00013368072 0.00005805 365th root of $1.06, or amount of $1. for 1 day, 1.00015965359 0.00006933 12th root of $1.05, or amount of $1. for 1 mo. 1.00407412 0.00176577 12th root of $1.06, or amount of $1. for 1 mo. 1.0048675505 0.00210882 360 degrees expressed in seconds,

1296000 6.1126050 Arc, equal to radius, in degrees,

57.295780 1.7581226 in minutes,

3437.74677 3.5362739 in seconds,

206264.8 5.3144251 Length of an arc of 1"=sine of 1"

0.000004848 -6.6855749 of 21= sine of 2

0.000009696 -6.9866049 of 3' =sine of 3

0.000014544 -5.1626961 of 1' = sine of 1'

0.000290888 -4.4637261 of 1°

0017453293 -2.2418774 Sine of 1°

0.017452406 -2.2418553 Mile, reduced to rods,

320 2 5051500 yards,

1760 3.2455127 feet,

5280 3.7226340 inches,

63360 4.8018152 Square mile, in acres,

6401 2.8061800 square rods,

102400 5.0103000 square yards,

3097600 6.4910254 square feet,

27.178400 7 4452679 square inches,

4014489600 9.6036304 Equatorial diameter of the earth, in miles,

7924 3.8989445 in rods,

2535680 6.4040945 in yards,

13946240 7.1444572 in feet,

41838720 7 6215785

in inches 502064640 8.7007597 Circumference of the Equator, in miles,

24893.98 4.3960944 in feet,

131440217 8.1187283

in inches, 1577282608 9.1979095 Radius of Earth's orbit, in miles,

95273869 7.9789738 Sun's horizontal parallax,

89.57760 09333658

A TABLE

OR

LOGARITHMS OF NUMBERS

FROM 1 to 10,000.

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Log. 0.000000 0.301030 0.477121 0.602060 0.698970 0.778151 0.845098 0.903090 0.954243 1.000000 1.011393 1.079181 1.113943 1.146128 1.176091 1.204120 1.230449 1.255273 1.278754 1.301030 1.322219 1.342423 1.361728 1.380211 1.397940

N, 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

li 12 13 14 15 16 17 18 19 20 21 22 23 24 25

N, long.
26

1.414973
27 1.431364
28

1.447158 29 1.462398 30 1.477121 31 1.491362 32 1.505150 33 1.518514 34 1.531479 35 1.544068 36 1.556303 37 1.568202 38 1.579784 39 1.591065 40 1.602060 41 1.612784 42 1.623249 43 1.633468 44 1.643453 45 1.653213 46 1.662758 47 1.672098 48 1.681241 49 1.690196 50 1 1.698970

Log. 1.707570 1.716003 1.724276 1.732394 1.740363 1.748188 1.755875 1.763428 1.770852 1.778151 1.785330 1.792392 1.799341 1.806180 1.812913 1.819544 1.826075 1.832509 1.838849 1.845098 1.851258 1.857333 1.863323 1.869232 1.875061

N. 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Log. 1.880814 1.886491 1.892095 1.897627 1.903090 1.908485 1.913814 1.919078 1.924279 1.929419 1.934498 1.939519 1.944483 1.949390 1.954243 1.959041 1.963788 1.968483 1.973128 1.977724 1.982271 1.986772 1.991226 1.995635 2.000000

69 70 71 72 73 74 75

N. B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thence the annexed first two figures of the Logarithm in the

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