5. Again, the numbers 3, 6, 12, 24, &c., are in geometrical progression, for each number is formed from the one immediately preceding by multiplying by 2. If we take the following series of powers, 31, 32, 33, 3, 3, &c., we find that the exponents proceed in arithmetical progression, and the quantities themselves in geometrical progression. 6. Def.-Logarithms are a series of numbers in arithmetical progression answering to another series in geometrical progression, so taken that o in the former corresponds with 1 in the latter. I Thus, 0, 1, 2, 3, 4, 5, 6, &c., are the logarithms or arithmetical series, and 1, 2, 4, 8, 16, 32, 64, &c., are the numbers or geometrical series, answering thereto-the latter being called the natural number. Or, 0, 1, 2, 3, 4, 5, the logarithms, and 1, 5, 25, 125, 425, 5125, the corresponding numbers. Or, o, 1, 2, 3, 4, 5, the logarithms, and I, 10, 100, 1000, 10000, 100000, the corresponding numbers. In which it will be seen, that by altering the common ratio of the geometrical series, the same arithmetical series may be made to serve as logarithms of any series of numbers. As above, when the common ratio of the geometrical series are 2, 5, and 10 respectively. I 7. The common ratio in the geometrical series corresponding to the common difference of in the arithmetical series is called the base of the system. Thus, the base of the first specimen exhibited is 2, the base of the second is 5, and the base of the third is 10. In the specimens just exhibited we have, in each, taken two ascending progressions, but they might equally well have been two descending progressions, or the one descending and the other ascending. Logarithms, however, as now used in practice, are limited to the case of two progressions, either both ascending or both descending the former giving the logarithms of integers, the latter of fractional numbers. But a better way of considering logarithms is as follows: 8. Def.-The logarithm of a number to a given base is the index of the power to which the base must be raised to give the number. For instance, if the base of a system of logarithms be 2, 3 is the logarithm of 8, because 8 = 23 = 2 X 2 X 2. And if the base be 5, then 3 is the logarithm of the number 125, because There may be thus as many different systems of logs. as we please; but, for practical use, it is necessary to select and adhere to one. employed now is called Briggs' system. That usually 9. We now proceed to describe what is called the common system of logarithms. In the common system of logarithms unity is assumed to be the logarithm of 10; that is, 10 is the constant base. All the logarithms registered in the Tables commonly used, are indices of the radix or base 10; a Table of logarithms of numbers is in fact nothing more than a Table of the exponents of 10 placed against the several numbers themselves. Accordingly I Now, if the above Tables were amplified by the insertion of the logarithms of all the numbers between 1 and 10, between 10 and 100, &c., we should have a Table of logarithms of all numbers from 1 to 10000; and whatever may be the difficulty of determining the intermediate logarithms, it is at once easily seen that the logarithms of all numbers between 1 and 10, i.e., between 10° and 101 must lie between o and 1, and will be o + a fraction, that is, a decimal less than 1; of all numbers between 10 and 100, i.e., between 101 and 102 must lie between 1 and 2, and will be 1 + a fraction, or a decimal between 1 and 2; of all between 100 and 1000 will be 2 + a fraction, and so forth; or the integral part of each intermediate logarithm will be one less than the number of integral figures in the quantity of which it is the logarithm. Thus, the logarithms of 2, 3, 4, &c., to 9, have o as the integral part; those of 10, 11, 12, &c., to 99 have I as the integer; those of 100, 101, 102, &c., to 999, have 2 as the integer; and so forth. Hence Tables of logarithms usually supply only the fractional or decimal part; the integral part is always known from the number of integers in the value whose logarithm is wanted. Very few logs. can be expressed in terminating decimals, but this causes little inconvenience, since a log. carried to six or seven decimal places is sufficiently exact for all common purposes. 10. The integers 1, 2, 3, 4, &c., which are the logarithms of 10 and its powers (see 9), are chief indices, and the logarithms intermediate to these, as for instance 1778151 (which is the logarithm of 60) consisting of an integer and a decimal fraction, though they are also indices, are usually referred to as consisting of an index* and mantissaf, the integral part being specially termed the index or characteristic, because it indicates, by being one less, how many integral places are in the corresponding natural number, and the annexed decimal being called the mantissa. EXAMPLE.-In the log. 4'616339, the figure (4) standing to the left of the decimal point is the characteristic or index, and the remaining portion (616339) is the mantissa or decimal part. In order to avoid confusion from the use of the word "index" to signify two things, we shall throughout this work employ the term characteristic when speaking of logarithms, and index when speaking of roots or powers. Mantissa, a Latin word signifying an additional handful; something over and above an exact quantity. 11. To find the characteristic of the logarithm of any number greater than unity we have, therefore, the following rule: RULE I. The characteristic of the logarithm of a number greater than unity, i.e., of a whole or mixed number, is one less than the number of the digits of its integer part Thus: the characteristic of the logarithm of 849 is 2; for the number 849 is an integer consisting of three digits (that is the number between 100 and 1000) and 1 less than 3 is 2. Also, the index of the log. of 264'96 (which is a mixed number) is 2, since the integral part of the number, namely 264, is a number between 100 and 1000, or consists of 3 digits, and one less than 4 is 3. Again, 3 is the characteristic of the logarithm of 3847 216, since this number has 4 integral digits; while o is the characteristic of the logarithm of 3.847216, since this number has one integral digit. Again, the characteristic of the log. of a number of one place of integers (such as 5 or 5'08, or 5'0801) is 0. Again, every number with two places of integers (such as 50, or 50·8, or 50.813) is 1. Again, every number with three places of integers (such as 508, or 508'2, or 508-25) has for its characteristic 2, and so on. EXAMPLES FOR PRACTICE. Write down the characteristics of the logarithm of the following numbers: 12. 16. 473'908 17. 54793000 18. 21256.8 19. 2*14006 20. 50'7406 I It has been shown that in the common system of logarithms (Briggs') the log. of is o; consequently, if we wish to extend the application of logs. to fractions, we must establish a convention by which the logs. of numbers wholly decimal, i.e., less than unity, may be represented by numbers less than zero, i.e., by negative numbers. Extending, therefore, the above principles to negative exponents, since *1, or I and, its log. is negative, yet not so small It follows from this, that when the number is a decimal with all its digits significant, in value between as the log. of, which is 1. Its log. therefore will be something between o and I with some positive decimal added. Hence I is its characteristic. When the number is a decimal with zero as its first digit, in value therefore below but not so low as To, its log. is less than 1, but not so small as -2, and so will be -2 with some positive decimal attached. Thus is the characteristic. The log. of a decimal between or and 'oo is some number between -2 and -3, and its characteristic is -3; of a number between oo and 0001 its log. is between -3 and -4, and its characteristic 4; and generally, following this reasoning, it will appear that the characteristic of a decimal fraction is negative, and may be known from its denoting the place of the first significant figure of the decimal, as being the 1st, 2nd, 3rd, &c., place after the point; hence, 13. To find the characteristic of any number less than unity, i.e., of a decimal, we have the following RULE II. The characteristic of the logarithm of a number less than unity, and reduced to the decimal form, is negative and one more than the number of cyphers following the decimal point. A negative characteristic is denoted by writing over it the negative sign (—), thus ī, 2, 3, &c.* Thus the chatacteristic of the logarithm of 00521 is 3, since the number of cyphers following the decimal point increased by 1 is 3. 14 But in order to avoid the confusion that might arise by the addition and subtraction of negative indices, the following rule is frequently used. RULE III. Add 1 to the number of cyphers between the decimal point and the first significant I figure, and subtract from 10; the remainder is the index required. Thus the characteristic of the log. of '04 is 2, or 8, since I adled to the number of cyphers following the decimal point is 2, then 2 from 10 is 8. (a) If the characteristic of a vulgar fraction is required, it must first be reduced to an equivalent decimal fraction, and then the index is found by the rule. Write down the characteristics of the logarithms of the following decimal fractions: The negative sign (—) is written above the characteristic, thus 2, instead of before it, to show that it affects only the characteristic and not the mantissa, which remains positive. If it were written in front of the complete logarithm it would signify that the entire logarithm was negative, but such logarithms are never employed in the operations connected with navigation. 15. The characteristic may also be found as follows: RULE IV. Place your pen between the first and second figure, (not cypher), and count one for each figure or cypher, until you come to the decimal point; the number thus given will be the characteristic: but observe that if you count to the left you must subtract the number found from 10, and consider the remainder as the characteristic. Thus, in finding the log. of 4.6017, if you place the pen between the first figure (4) and second (6), it falls on the decimal point; in this case the characteristic is o. 46017 1123 Next, in the case of log. of 46017, place your pen between 4 and 6, and count ; the characteristic is 3. Next, in the case 4601700, here the decimal point falls behind the last cypher (No. 5). Hence, counting as before, we have Again, in the case of log. 00046017 the first significant figure is 4. Hence, counting, *00046017 we have 4321 4601700 and the characteristic is 6. but here we count to the left, so that the characteristic is negative, or 4, which taken from 10 is 6. Again, in the case of log. of 46017, we have 46017, and the characteristic is ī, or 9. 13 16. The mantissa of the logarithm depends entirely on the relative value of the figures composing the quantity whose logarithm it is, and not at all upou the numerical value of that quantity: thus, the mantissa of the log. of is 1 113943, which is also the mantissa of 1.3, or 130, or 1300, for in each case the and the 3 have the same relative value. So the mantissa of a logarithm is always the same, if the significant figures remain the same, and is not altered by the addition of cyphers to the right or left of these figures, or what is equivalent, by the multiplication or division of the quantity by 10, or any power of 10; it is only the characteristic which alters its value by an alteration in the position of the decimal point, I being added to the characteristic for every place the decimal point is removed to the right, that is, for every 10 by which the quantity is multiplied; or, 1 is subtracted from the characteristic for every place the decimal point is removed to the left, that is, divided by 10. |