110 



THE POPULAR EDUCATOR. 



nounces upon actions before and without any consideration of 

 their utility or effects, say, first of all, that our judgments of 

 right and wrong are instantaneous. In most cases, they say, 

 we pronounce our judgment as to the morality of an action at 

 once ; and that if there are certain cases in which we are unable 

 to do this, this arises from the difficulty of having all the facts 

 present to our mind, and not from any doubt we have as to the 

 decision when once the facts are clearly ascertained. An act of 

 murder or sacrilege we decide to be criminal and vicious the 

 moment it is mentioned, without waiting to balance any reasons 

 or arguments against one another ; simply because we know 

 that nothing could lead us to arrive at a different conclusion. 

 We have no more doubt or difficulty in such a case than in 

 judging that it is dark at midnight when once we open our 

 eyes. 



On the other side it is urged, that the fact that a judgment is 

 instantaneous or immediate is no proof that it is innate in our 

 minds. Long practice and habit may render the operation so 

 rapid, that we are unable to perceive the process by which it is 

 arrived at. The perception of distance by the eye, for instance, 

 was from its apparent instantaneousness supposed to be per- 

 ceived directly and immediately by the sight ; though it is now 

 clearly ascertained that it involves a comparison by the mind 

 between different objects, and a judgment founded thereon. So 

 also a person accustomed to any particular science will perform 

 operations in a moment, which to a learner will cause consider- 

 able delay and difficulty ; and, it is urged, the judgments of 

 conscience are not more rapid than some of these. Indeed, it 

 may be also said, it is only in the simpler moral questions which 

 arise every day, that there can be fairly said to be any such 

 rapidity at all. 



LESSONS IN LOGARITHMS. II. 



NATURE AND USE. 



15. Hitherto we have dealt with numbers and their powers, 

 and have illustrated the use of logarithms by the manipulation 

 of indices, whether whole or fractional numbers. We proceed 

 now to a further definition of logarithms. 



16. Given a fixed number, called a base. The logarithm of a 

 number with regard to that base is the index of the power to 

 which the base must be raised in order to produce the number. 



17. If 2 be assumed as a base, then the powers of 2 will be 

 the natural numbers, and the indices of those powers will be 

 the logarithms of the natural numbers ; thus 



TABLE OF LOGARITHMS TO BASK 2. 



Natural Numbers. Logarithms. Natural Numbers. Logarithms. 



1 ... 123 ... 7 



2 ... 1 256 ... 8 

 4 ... 2 512 ... 9 

 8 ... 3 1024 ... 10 



16 ... 4 2048 ... 11 



32 ... 5 4096 ... 12 



64 ... 6 



18. By means of this table, logarithmic calculations may be 

 exemplified on a small scale, in the following manner : 



19. (a) To Multiply two or more Numbers together. If the 

 logarithms of the factors be added together, the sum is the 

 logarithm of the product. Thus, to multiply 128 by 8, add 7 

 and 3 together, the logarithms of the factors ; the sum 10 is 

 the logarithm of the product 1024. Again, to multiply 4, 8, and 

 16 continuously together, add 2, 3, and 4 together, the loga- 

 rithms of the factors ; the sum 9 is the logarithm of the 

 product 512. 



20. (b) To Divide one Number by another. If the logarithm 

 of the divisor be subtracted from the logarithm of the dividend, 

 the remainder is the logarithm of the quotient. Thus, to divide 

 256 by 64, subtract 6, the logarithm of the divisor, from 8, 

 the logarithm of the dividend ; the remainder 2 is the logarithm 

 of the quotient 4. 



21. (c) To find a fourth Proportional to three given Terms. 

 If the logarithms of the second and third terms be added 

 together, and from the sum the logarithm of the first term be 

 subtracted, the remainder is the logarithm of the fourth term. 

 For example, to find a fourth proportional to 8, 32, and 64 : 

 If 8 : 32 : : 64 : the fourth term ; then add 5 and 6 together, the 

 logarithms of the second and third terms, and from the sum 11 



subtract 3, the logarithm of the first term ; the remainder 8 is 

 the logarithm of the fourth term, 256. 



22. (d) To find any Power of a Number. If the logarithm 

 of the number be multiplied by the index of the required power, 

 the product is the logarithm of that power. Thus, to find the 

 square of 16, multiply 4, the logarithm of the number, by 2, the 

 index of the square ; the product 8 is the logarithm of the 

 square 256. 



23. (e) To find any Root of a Number. If the logarithm 01 

 the number be multiplied by the index of the required root, or 

 be divided by its denominator, the quotient is the logarithm of 

 that root. Thus, to find the cube root of 64, divide 6, the 

 logarithm of the number, by 3, the denominator of the index of 

 the cube root ; the quotient 2 is the logarithm of the cube root 4. 



24. The nature and use of logarithms having been thus illus- 

 trated and exemplified in the system of which the base is 2, we 

 shall now give a full explanation of the system in common use. 



COMMON SYSTEM OF LOGARITHMS. 



25. The number 10 has been assumed as the base of the 

 common system of logarithms, because it is the root of the 

 decimal scale of notation, and on this account possesses certain 

 advantages which have led to its universal adoption by mathe- 

 maticians. 



26. The powers of the number 10 being respectively unity 

 with as many ciphers annexed as are denoted by the indices of 

 the different powers, the construction of the following table is 

 sufficiently evident to the student : 



TABLE OF POWERS. 



10' = 10000000 . 7th power. 

 10 s = 100000000 . 8th power. 

 10 9 = 1000000000 . 9th power. 

 10 10 = 10000000000 . 10th power. 

 10" = 100000000000 llth power. 

 10"= = 1000000000000 12th power, 

 etc. 



27. These powers of 10 being the natural numbers, and their 

 indices the logarithms of those numbers, the construction of 

 the following table is rendered evident by the table in the pre- 

 ceding article : 



FIRST SKELETON TABLE OF LOGARITHMS TO BASE 10. 

 Natural Numbers. 

 1 



10 

 100 

 1000 



10000 . 

 100000 . 

 1000000 . 



28. If unity, the first natural number, be divided by the suc- 

 cessive natural numbers in the preceding table, the quotients 

 will be a series of decimal fractions viz., '1, '01, '001, etc. 

 The logarithms of these quotients will be found by subtracting 

 the logarithms of the natural numbers from 0, the logarithm of 

 unity. Now though it be impossible, arithmetically, to sub- 

 tract the logarithms 1, 2, 3, etc., from the logarithm 0, yet the 

 operation that should be performed is indicated by placing the 

 sign of subtraction before each of these logarithms ; thus, -1, 

 2, 3, etc. Hence, the construction of the following table 

 of decimal fractions, with their logarithms, is evident to the 

 student : 



SECOND SKELETON TABLE OF LOGARITHMS TO BASE 10. 



Natural Numbers. 

 1 



01 . 

 001 . 

 0001 . 

 00001 . 

 000001 



Logarithms. 

 1 



. 2 



. 3 



. 4 



. 5 



. 6 



29. These logarithms, being of an opposite character to the 

 former, are called negative, while the former are denominated 

 positive. From the remarks in the preceding article, it is evi- 

 dent that the logarithm of every proper fraction is essentially 

 negative, and that the logarithms of such fractions numerically 

 increase in proportion as the fractions themselves decrease in 

 value, compared with unity. Hence, when the vahie of a frac- 

 tion is indefinitely small, its logarithm, numerically considered, 

 must be indefinitely great ; and when the value of a fraction is 



