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Thus, if the logarithms 2 and 6; are added to. gether, the sum is 8, wbich answers to the logarithm of 256, the number that is obtained by the multiplication of 4 and 64, which are the numbers standing under the logarithms 2 and 6. By adding the logarithms 4 and 5 we have 9, which stands over 512, the number obtained by multiplying together 16 and 32. Hence the addition of logarithms answers to multiplication in common numbers.

2. “ Subtraction in logarithms answers to division of common numbers."

Divide 256 by 8, and you have 32, over which stands 5=8 -3: the logarithms standing above.

3. “Multiplication in logarithms answers to in volution of common numbers."

Ex. The square of 8 is 64 ; now 3 is the logai rithm answering to 8, and 3x2, (because 2 is the index of the square), is equal to 6, which is the logarithm of 64.

-4. Division in logarithms answers to evolution in common arithmetic."

Ex. 1. The square root of 256 is 16, over which stands the logarithm 4; which answers to 82, 8 being the logarithm of 256.

Ex. 2. The cube root of 512 is 8; and 9, which is the logarithm of 512, divided by 3, the sign of the cube, gives 3, which is the logarithm of 8.

The same indices will serve for any geometric series ; but the logarithms generally used are those wbich increase in a tenfold proportion, as 0, 1. 2. 3.

5. 6. &c. 1. 10. 100. 1000. 10000. 100000. 1000000. Here it is evident, that the logarithms of non

are

bers between 1 and 10, are greater than 0, and less than one, thus the logarithms of 2, 6, 8, &c. .3010300, .7781513, .9030900, &c.

The logarithms of the numbers between 10 and 100, are greater than 1, and less than 2; thus the logarithm of 15 is 1.1760913, and the logarithm of 95 is 1.9777236.

The logarithms of numbers between 100 and 1000, are greater than 2, and less than 3; thus the logarithm of 165 is 2.2174839, and of 984 is 2.9929951.

The logarithms between 1000. and 10000, must be somewhere between 3 and 4, and so on.

The logarithms in the above series are called indices, which are frequently neglected, the decimal part only being put down; thus, if it be required to find the logarithm of 248, it will be sufficient to put down.3944517, and the number being between 100 and 1000, I know the index is 2. Therefore the rule for finding the index is this:

The index is always, one less than the number of figures in the whole number : or the figures in the whole number must be always one more than the index." The logarithm of 248 is 2.3944517

2480

3.3944517 24800 4.3944517 24.8

1.3944517 2.48

0.3944517 .248

1.3944517 .0248

2.3944517 .00248

3.3944517 Here the decimal figures remain the same: and the only difference is in the indices, which are in-,

creased or diminished by unit for every ten-fold increase or decrease of the whole number. It will be observed, that where there is but one whole number, the index will be 0; but if the figures be decimals, as .248, the index is one minus, or · 1; by the prefixing o to the decimal figure, the value is diminished in a ten-fold proportion, then the index is 2, or minus two.

We cannot pursue the subject farther for want of tables, which would be incompatible with this small work.

LOGIC, the art of thinking and reasoning justly ; or, it may be defined the science or history of the human mind, inasmuch as it traces the progress of our knowledge from our first and most simple ideas through all their different combinations, conceptions, and all those numerous deductions that result from variously comparing them one with another.

LOG-WOOD, in the arts, is derived from a low: prickly tree, which is found in great plenty at Campeachy, in the bay of Honduras, and is denominated “ hæmatoxylon campechianum.” It copies to Europe in large logs, cleared from the bark, and is very hard, compact, heavy, and of a red colour. It is in high request among dyers, especially in dying black. It gives out the colour both to water and alcohol; the liquor at first assumes a fine red colour with a shade of purple. The infusion be. comes gradually deeper, and at last almost black. To cloth, previously boiled in alum and tartar, it gives a beautiful violet colour, which, however, will not stand.

Alkalies rènder the colour darker, acids change it to yellow. Froin a variety of ex. periments it is found that the colouriog matter of

log-wood bears, in many respects, a strong analogy to tannin, bat in others it differs from it.

LONGITUDE, in geography, the distance of any given point from another, in the direction of east of west; as latitude is that distance, in the direction of north or south. Latitude is reckoned in degrees from the equator; longitude, from a meridian (one of the perpendicular lines, on maps or globes, or a Jine parallel to these), which is fixed upoñ at plea. sure: thus the meridian that passes over Greenwich is the meridian of Greenwich ; and it is from this point that the English reckon the distance of places.

As perpendicular lines, drawn from the opposite poles of a globe, are necessarily wider apart at its greatest circumference, than at any other point between that and those poles, it follows that the width of a degree of longitude, which is determined by those lines, increases, either in a southward or northward direction, in the ratio that it approaches the equator. When, therefore, a degree of longi: tude is mentioned, it is impossible to know what number of miles it contains, unless the degree of latitude be also ascertained. The following table shows how many miles answer to a degree of longitude, at every degree of latitade. Lat. Miles. Lat. Miles. Lat. Milet.

Miles. 1 59.99 10 59.08 19 56.73

28 52.97 2 59.97 11 58.89 20 56.38, 29 52.47 3 59.92

12 58.68 21 56.01 30 51.96 :: 4 59.86 13 58.40 22 55.63 91 51.43

5 59.77 14 58.22 23 55.23 32 50.88 6 59.67 15 57.05 34 54.81 33 50.32 7 59.56 16 57.67 25 54.38 34 49.74 8 59.42 17 57.37 126 53.93 35 49.15 9 59.26 18 57.06 +27 53.46 36 48.54

Lat.

Lat. Milcs.

87

bat Miles.

'Lat. Miles. Lat. Miles. 37 47.92

51 37.76 65 25.36 78 12.48 38 47.28

52 36.94 66 24.41 79 11.45 39 46.62

53 36.11 67 23.4+ 80 10.42 40 45.95 54 35.27 68 22.48 81 9.38 41 45.28 55 34.41 69 21.50 82 8.35 42 44.59 56 33.55 70 20.52

83 7.32 43 43.88 57 32.68 71 19.54 84

6.28 44 43.16 58 31.79 72 18.55 85 5.23 45 42,43 59 30.90 79 17.54 86 4.18 46 41.68 60 30.00 74 16.55

3.14 47 40.99 61 29.09 75 15.52 88 2.09 48 40.15 62 28.17 76 14.51 89 1.05 49 39.36 63 27:24 77 13.50 90 0.00 50 38.57 64 26 30

The use of this table will be readily understood. From it, we learn that 10° of longitude in 80° latitude, amount to a hundred and four miles, and two bundredth-parts; while 10° of longitude, in 40 of latitude contain five hundred and ninety-eight miles, and six hundredth-parts.

Longitude, in navigation, is of so much importance to safety and expedition, that the following rewards have been offered by an act of the British parliament as an encouragement to any person who shall discover a proper method for finding it out: the author or authors of any such method, shall be entitled to the sum of 10,0001., if it determines the longitude to one degree of a great circle; to 15,0001., if it determines the same to twothirds of that distance, and to 20,0001., if it determines the same to one half of the same distance ; and half of the reward shall be due and paid when the commissioners of the navy, or the major part of them, agree that any such method extends to

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