UC-NRLF 
 
 lllillll lull Hill lllil'l"' 
 
 $B 532 =i^3 
 
 '•-.....l-'ilJ '!.>,.„ !'■' .il- 
 
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LIBRARY OF 
 
 ALLEN KNIGHT 
 
 CERTIFIED PUBLIC ACCOUNTANT 
 
 502 California Street 
 
 SAN FRANCISCO. CALIFORNIA 
 
GIFT OF 
 
 OsilSliw- 
 
 N^WvMD^t^ 
 
LOGARITHMS 
 
 TO 12 PLACES 
 
 AND THEIR USE IN 
 
 INTEREST CALCULATIONS 
 
 By CHARLES E. SPRAGUE 
 Author of "Thb Phii^osophy of Accounts" 
 '•Text-Book of the Accountancy of Investment 
 and "Extended Bond Tabi^es" 
 
 New York, 1910 
 publisht by the author 
 
s^ 
 
 ^-7 
 
 Copyright, 1910, by Charles E. Sprague. 
 
 ^ Oid^ \t^^sA3r 
 
 TRUNK BROS. 
 
 18 FRANKFORT ST. 
 
 NEW YORK 
 
PREFACE. 
 
 The need of a logarithmic table for special cases, where 
 the usual five-figure and seven-place results are insufiBcient, is 
 often felt by the accountant and the actuary. Rough results 
 will answer for approximativ purposes ; but where it is desir- 
 able, for instance, to construct a table of amortization, sinking 
 fund or valuation of a lease at an unusual rate, for a large 
 amount and for a great many years, exactness is desirable and 
 becomes self-proving at the end. 
 
 It is, of course, a slower process than that for a few places, 
 but as the figures from which all results are obtainable are 
 containd in two pages instead of 200, there is, on the other 
 hand, a great saving in the mechanical labor of turning leaves. 
 
 It also contains a thoro analysis of the entire doctrin of 
 interest, explaining every process by the use of logarithms, as 
 well as arithmetically and algebraically. 
 
 CHARLES E. SPRAGUE. 
 
 Union Dime Savings Bank, 
 New York, January, 1910. 
 
 380313 
 
TABLE OF CONTENTS. 
 
 PART I. — Thk Properties of Logarithms. 
 
 Page 
 
 The Nature of Logarithms 1 
 
 Multiplication by IvOgarithms 3 
 
 Division of Logarithms 5 
 
 Tables of Logarithms 6 
 
 To Find the Number 8 
 
 To Form the Logarithm 16 
 
 Less than 12 Places 21 
 
 Multiplying Up 22 
 
 Signs of the Characteristics 27 
 
 Different Bases 28 
 
 PART II. — Tables for Obtaining Logarithms and 
 Antilogarithms to 12 Places of Decimals. 
 
 Table of Factors 30 
 
 Table of Interest Ratios 32 
 
 Table of Sub-Reciprocals 33 
 
 Table of Multiples 34 
 
 Logarithmic Paper 36 
 
 PART III. — The Doctrin of Interest. 
 
 Definitions 39 
 
 The Amount 41 
 
 The Present Worth 43 
 
 The Compound Interest and Discount 45 
 
 Finding Time or Rate 46 
 
 The Annuity 47 
 
 Amount of Annuity 48 
 
 Present Worth of Annuity 50 
 
 Amortization 52 
 
 Special Forms of Annuity 53 
 
 The Unit of Time 55 
 
 Frequency of Payment 57 
 
 Coefficients of Frequency 58 
 
 Fractional Periods 63 
 
 Sinking Funds 65 
 
 Interest-Bearing Securities 67 
 
 Multiplying Down 72 
 
 Computing Amortizations 73 
 
 Discounting 75 
 
 Intermediate Purchases 76 
 
 Intermediate Balances > 77 
 
 Short Periods 80 
 
 Finding the Income Rate ... 83 
 
 Interest Formulas 86 
 
PART I 
 
 The Properties of Logarithms 
 
PART I. 
 THE PROPERTIES OF LOGARITHMS. 
 
 1. — If we multiply 5 lO's together, 10 x 10 x 10 x 10 x 10, 
 we may write the result as 
 
 100000 
 or 10^ 
 
 or the fifth power of ten. 
 The little " ^ " is the exponent of the power. We may 
 form a series of the powers of 10 : 
 
 00000 or 10^ 
 
 10000 
 
 10* 
 
 1000 
 
 10« 
 
 100 
 
 10« 
 
 10 
 
 10^ 
 
 1 
 
 10° 
 
 2. — The following observations may then be made : 
 
 1. The number of the zeroes in the first colum is 
 the exponent in the second. 
 
 2. Each term in the first colum is one-tenth of the 
 one above it, while in the second colum each exponent is 
 one less than the exponent above it. This leads to the 
 result that 10° = 1, which at first seems paradoxical. 
 
 3. If we multiply together any two terms in the first 
 colum, we add the exponents in the second. 
 
 3. — Logarithms are auxiliary numbers having relation 
 to a base. When the base is once fixt, every possible number 
 has its logarithm. The customary and most convenient base 
 is 10, because our whole system of numeration is based upon 
 ten. The logarithms are simply exponents and we re-write 
 the above series thus : 
 
Thk Properties of Logarithms. 
 
 The base being 10, 
 
 100000 is the number whose logarithm is 5 
 
 or contracted, 
 
 100000. 
 
 nl 
 
 5 
 
 10000. 
 
 nl 
 
 4 
 
 1000. 
 
 nl 
 
 3 
 
 100. 
 
 nl 
 
 2 
 
 10. 
 
 nl 
 
 1 
 
 1. 
 
 nl 
 
 
 
 .1 
 
 nl 
 
 — 1 
 
 .01 
 
 7ll 
 
 -2 
 
 .001 
 
 nl 
 
 -3 
 
 .0001 
 
 nl 
 
 -4 
 
 .00001 
 
 nl 
 
 -5 
 
 The copula {nl) means "is the number whose logarithm is ;" 
 
 while {In) means "is the logarithm of the number ." 
 
 4. — We have here logarithms of a few numbers, but we 
 
 need the logarithms of a great many others. All possible 
 
 numbers must lie between some of the logarithms now ascer- 
 
 taind. The numbers between 1 and 10 must have their 
 
 logarithms between and 1; that is, the logarithms must be 
 
 fractions, and these are exprest decimally to as many places 
 
 as desired, the difficulty in calculation greatly increasing as 
 
 the number of places is increast. Similarly, as the numbers 
 
 of two figures lie between 10 and 100, their logarithms must 
 
 lie between 1 and 2; that is, they must be 1 -f a decimal 
 
 « 
 fraction. 
 
 5. — We will now illustrate the properties of logarithms, 
 confining our attention to the single-figure numbers 2, 3, 4, 5, 
 6, 7, 8 and 9, which are as follows, rounded at 12 places : 
 
 .301029 995 664 
 
 In 
 
 2 
 
 .477 121254 720 
 
 In 
 
 3 
 
 .602 059 991328 
 
 In 
 
 4 
 
 .698 970 004 336 
 
 In 
 
 5 
 
 .778 151250 384 
 
 In 
 
 6 
 
 .845 098 040 014 
 
 In 
 
 7 
 
 .903 089 986 992 
 
 In 
 
 8 
 
 .954 242 509 439 
 
 In 
 
 9 
 
MuivTlPLICATION BY LOGARITHMS. 
 
 6. — The third observation in Art. 1 leads to the following 
 rule : 
 
 The sum of the logarithms of several numbers is the logarithm 
 of their product. 
 
 2 111 .301029 995 664 
 
 3 7il .477 121254 720 
 2x3= (o nl .778 151250 384 
 
 2 X 5=~ 
 
 2 X 10=^ 
 7.— In the 
 
 2 
 
 5 
 10 
 
 2 
 
 20 
 logj 
 
 nl 
 nl 
 nl 
 nl 
 
 7ll 
 
 irith 
 
 .301029 995 664 
 .698 970 004 336 
 
 1.000 000 000 000 (See Art. 3) 
 
 .301029 995 664 
 1.301029 995 664 
 ms of 20, 200, 2000, 20.000, 200,000 
 
 2,000,000, etc., we shall find the same decimal part 
 .301029 995 664, {_ln 2) preceded by the figures 1, 2, 3, 
 4, 5, 6, etc., indicating the distance from the units place of 
 their left-hand figure, or the number of zeroes interpolated to 
 hold that position. This is also true of any combination of 
 figures ; the decimal part of the logarithm is the same what- 
 ever their place-value, while the whole number prefixt indi- 
 cates the place-value, being the number of places to the left 
 of units. 
 
 Thus, if the logarithm of 2 . 378 is 
 
 .376 211850 283, then 
 
 1.376 211850 283 
 
 2.376 211850 283 
 
 3.376 211850 283 
 
 4.376 211850 283 
 
 5.376 211850 283 
 etc. 
 
 8. — Where the number is less than unity (a decimal frac- 
 tion) the characteristic or index (the prefixt figure) is 
 negativ, altho the decimal (or mantissa) remains positiv. It 
 is usual to put the minus sign over the characteristic: 
 
 .2378 {7il) 1.376 211850 283 
 
 .02378 {7il) 2.376 211 850 283 
 
 .002378 («/) 3.376 211850 283 • 
 
 etc. etc. 
 
 23.78 
 
 7ll 
 
 237.8 
 
 7ll 
 
 2,378 
 
 nl 
 
 23,780 
 
 nl 
 
 237,800 
 
 nl 
 
 
 etc. 
 
4 The Properties of Logarithms. 
 
 Here the position of the left-hand figure of the number again 
 determins the characteristic. 1 indicates that the left-hand 
 figure, 2, is in the Jirsi place to the right of the unit place ; 2 
 indicates that this figure is in the second place, and so on. 
 The following list of characteristics will show that the left- 
 hand figure of the combination, by its location to the right 
 and left of the unit figure, determins the characteristic. 
 
 Unit 
 
 Places 000 000 00 0- 000 000 000 
 
 Characteristics 9 876 543 21 1 2 3 45 6 78 9 
 
 This principle saves a vast amount of time in the compu- 
 tation of logarithms, and also in their application. 
 
 9. — Since division is the converse of multiplication, it may 
 be performd by subtraction as that is by addition. 
 
 The difference of the logarithms of two numbers is the 
 logarithm of their quotient. 
 
 Required the quotient of 6 -H 2. 
 
 6 nl .778 151250 384 
 
 2 '' .301029 995 664 
 
 6/2 nl .477 121254 720 In 3 
 Required the value of >^ or 1 -f- 2 
 
 1 nl .000 000 000 000 
 
 2 '• .301029 995 664 
 
 1/2 ;2/ 1.698 970 004 336 In .^ 
 Required the value of Yi 
 
 1 nl .000 000 000 000 
 3 '' .477 121254 720 
 
 1/3 w/ 1.522 878 745 280 /;^ .3333 
 
 .522 878 745 280 is called the cologarithm of 3 or the 
 logarithm of the reciprocal of 3. 
 
 10. — Powers of numbers are found by multiplication. 
 Let it be required to find the third power of 2, which may be 
 written 2^ or 2x2x2. By the process first shown 
 2 nl .301029 995 664 
 2 " .301029 995 664 
 
 2_ " .301029 995 664 
 
 2 X 2 X 2 = 2^ nl .903 089 986 992 In 8 
 
Roots and Powers. 
 
 Required the square (2d power) of 3 
 
 3 nl .477 121254 720 
 3 " .477 121254 720 
 
 3' nl .954 242 509 440 In 9 
 In each of the above examples it would have been simpler 
 to multiply the logarithm by the exponent. 
 2^ nl (.301 029 995 664) x 3 = .903 089 986 992 In 8. 
 3^ nl (All 121 254 720) x 2 = .954 242 509 440 In 9. 
 Therefore, to "raise" a number to a certain power, we 
 multiply its logarithm by the exponent and then find the 
 number corresponding to the product-logarithm. 
 
 11. — The second power is usually called the square, and 
 the third power the cube. 
 
 12. — If a certain number is a power of another, we call 
 the latter a root of the former. Thus if 2^ = 32, we may say 
 that the 5th root of 32 is 2. The usual way of expressing 
 
 this is c- / 
 
 Y 32 = 2, or 
 
 32^ = 2. 
 Using the latter form gives a symmetrical list of ex- 
 ponents and their meanings: 
 a" A positiv exponent denotes a power 
 a~** A negativ exponent denotes the reciprocal of a power; 
 ajr A fractional exponent denotes a root, or the root of a power; 
 a* The exponent ^ denotes the number itself; 
 a° The exponent ^ denotes unity. ^ 
 
 13. — As roots are powers with fractional exponents, there- 
 fore roots are found (or extracted) by dividing logarithms 
 insted of multiplying. Thus if it be required to find the 6th 
 root of 64, we take (from Colum A of the Table of Factors) 
 the logarithm of 64, and divide it by 6. 
 64^ nl (1.806 179 973 984 /6) == .301 029 995 664 In 2. 
 
 Therefore 2 is the 6th root of the number 64. 
 
 14. — Such an exponent as f may require explanation. It 
 signifies the third power of the fourth root or the fourth root 
 of the third power. 
 
 15. — Fractional exponents may be represented as decimal, 
 insted of vulgar fractions. Thus we may write 2*^^ insted of 
 2* or 3* for 3 . In fact, that is what most logarithms are: 
 fractional exponents of 10, exprest decimally. 
 
The; Properties of Logarithms. 
 
 Tabi^es of Logarithms. 
 
 16. — The decimal fractions j^ch constitute that part of 
 the logarithm requiring' tU^>ilation are interminate ; their 
 values may be computed to any number of decimal places. If 
 all the logarithms in a certain table are carried to 5 deci- 
 mal places, it is called a 5-place table, and so on. Thus the 
 logarithm of 2 has been computed, with great labor, to 20 
 places and even further. 
 
 2 nl .301029 995 663 981195 21 + 
 In a 4-place table this would be rounded off to 
 
 .3010; 
 in a 7-place, .3010300; 
 inalO-place, .30102 99957; 
 
 in a 12-place, .301029 995 664. The terminal decimal is 
 never quite accurate, but is nearer than either the next 
 greater or the next less. 
 
 17. — The number of figures in the numbers for which the 
 logarithms are given must also be considered. The tables 
 most in use, like those of Vega, Chambers and Babbage, are 
 of five figures and seven places. A six-figure table would 
 have to contain ten times as many logarithms and occupy ten 
 times the space. A sixth and a seventh figure may be obtaind 
 from them by interpolation. The United States Coast Survey 
 tables (now out of print) are five-figure ten-places. Nine 
 figures may be obtained by simple proportion, but the tenth 
 is, for the most of the work, unreliable. Both of the foregoing 
 systems give auxiliary tables of proportionate parts, or 
 differences. 
 
 18. — Peter Gray and Anton Steinhauser have 
 publisht tables of 24 and 20 places respectivly, but the plan 
 for extending the numbers of figures is quite different from 
 the simple interpolation above referd to. They both procede 
 by subdividing the number into factors, and adding together 
 the logarithms of those factors. 
 
 19. — All logarithmic calculations end with the ascertain- 
 ment of a number which the problem calld for. The more 
 
Tables of Logarithms. 7 
 
 decimal places the tables give, the more exact the resulting 
 number, or answer, will be, and the number of figures in the 
 answer can never be more than the number of places in the 
 final logarithm. 
 
 20. — I have selected twelve figures as the most useful 
 limit for the accurate computation of interest problems, that 
 being the kind for which the work is specially designd. The 
 logarithms are given to two figures and thirteen places, the 
 extra place insuring the accuracy of the 12th, which would 
 otherwise sometimes be 1, 2 or even 3 units in error, thru the 
 roundings being preponderant in one direction or the other. 
 
 21. — The method used is that of factoring, it being pos- 
 sible to construct the logarithm of any number of twelve 
 figures or less (900,000,000,000 in all) by some combina- 
 tion of the 584 logarithms given on the two pages of the 
 Table of Factors. 
 
 Colum A contains numbers of two figures, 11 to 99, and 
 their logarithms to thirteen places. 
 
 Colum B contains the logarithms of four-figure numbers 
 1.001 to 1.099, each beginning with 1.0. . 
 
 Colum C contains the logarithms of six-figure numbers 
 1.00001 to 1.00099, each beginning with 1.000. . . 
 
 Colum D, 1.0000001 to 1.0000099, beginning with one 
 and five zeroes. 
 
 Colum E, 1.000000001 to 1.000000099, beginning with 
 one and seven zeroes. 
 
 Colum F, 1.00000000001 to 1.00000000099, beginning 
 with one and nine zeroes. 
 
 For example, opposit 34 in the table we find : 
 
 A .531478 917 042,3 In 3.4 
 
 B .014 520 538 757,9 In 1.034 
 
 C .000147 635 027,3 In 1.00034 
 
 D .000 001476 598,7 In 1.0000034 
 
 E .000 000 014 766,0 In 1.000000034 
 
 F . 000 000 000 147,7 In 1 . 00000000034 
 
 By omitting all the prefixt zeroes, the printed table is made 
 very compact, each line containing only 53 figures insted of 
 78. It will be understood hereafter that C 34, for example, 
 means the number 1.00034, and F 34 means 1.00000000034. 
 
The Properties of I^ogarithms. 
 
 To Find the Number when the Logarithm is Given. 
 
 22. — In this process there are two stages : first, to divide 
 the logarithm into a number of partial logarithms among those 
 containd in the T F (Table of Factors) ; second, to multiply 
 together the numbers corresponding to these logarithms. Of 
 course the decimal part only of the logarithm is used and the 
 number has the position of its units figure determind from the 
 characteristic. 
 
 23.— Let the logarithm .753 797 472 366,5 be one which 
 has been obtaind as the result of an operation, and the 
 corresponding number be required. Search in Coluni A for 
 the highest logarithm which does not exceed the given one. 
 This is found to be .748 188 027 006,2, which stands op- 
 posit 56. 
 
 Subtracting from 753 797 472 366,5 
 
 A 56 .748 188 027 006,2 
 
 we have the remainder 5 609 445 360,3 
 
 This is smaller than any logarithm 
 in Colum A. We search for it in 
 Colum B and find opposit 13 pre- 
 cisely the same figures 5 609 445 360.3 
 
 These two logarithms added together make the given loga- 
 rithm ; hence the product of their numbers gives the number 
 required. 
 
 To multiply 56 by 1.013 
 
 56 ^ 
 
 
 
 1013 
 
 1013 
 
 
 5 
 
 5065 
 
 56 
 
 
 e 
 
 6078 
 
 56 
 168 
 
 r °^ 
 
 
 56728 
 
 56728 
 
 24. — This process may be greatly simplified as follows, 
 placing the figures of the multiplier in vertical order at the 
 
 side: 
 
 56 56 
 
 66 13 X 5 065 
 
 168 or 13 X 6 078 
 
 56728 56728 , 
 
 Notice that the first product is moved two colums to the right 
 of the multiplicand. 
 
To Find the Number when the Logarithm is Given. 9 
 
 25. — We will now take a little 
 
 larger logarithm 
 
 and continue the subtraction A 56 
 
 B13 
 
 C26 
 
 D29 
 
 E58 
 
 G65 + 
 
 753 911 659 107,4 
 
 748 188 027 006,2 
 
 5 723 632 101,2 
 
 5 609 445 360,3 
 
 114 186 740,9 
 
 112 901 888,7 
 
 1 284 852,2 
 
 1 259 452,2 
 
 25 400,0 
 25 189,1 
 
 210,9 
 
 208.5 
 2,4 
 2,4 
 
 There is no colum G; but it is found by simply taking the 
 first two figures from E. It may be either 55 or 56, which 
 may make the thirteenth figure of the result doutful, but 
 probably not the twelfth. 
 
 See Note 1. 
 See Note 2. 
 
 See Note 3. 
 
 See Note 4. 
 
 5600 
 56 
 
 168 
 
 567280000 
 113456 
 3 4 3 6 8 
 
 4 2 74928000 
 * 
 11.348550 
 5106847 
 
 507 
 
 429 
 
 97 
 1.5. 
 
 94 
 70 
 54 
 28 
 3 
 
 5 6 7 4 2 9 17 15 2 6, 
 
lo The PropertiEvS op Logarithms. 
 
 Note 1. — The second multiplication jumps its right-hand 
 figure (6) y^z^r places to the right, which may be markt off 
 by four zeroes, or four dots. 
 
 Note 2. — Having extended the product to include the 
 13th figure, contraction begins in this multiplicand ; its first 
 figure used being the 7th (markt *) allowing for the carrying 
 from the 8th. Thus the starting point for this multiplication 
 is moved six places back. 
 
 Note 3. — The multiplicand need no longer be extended, 
 as has been done at successiv stages above, but remains the 
 same to the end. For convenience, dots may be placed in 
 advance under the first figure to be used in multiplication in 
 each line. j 
 
 Note 4. — The thirteenth figures are added, but only used 
 for carrying to the twelfth. In this example the total of the 
 last colum is 31, but it does not appear, except as contribut- 
 ing 3 to the next colum. 
 
 The dot below a figure indicates where the contracted 
 multiplication begins, all the figures to the right being 
 ignored, except as to their carrying power. 
 
 25. — Another example in which there is no suitable loga- 
 rithm in A and w^e must begin with B. 
 
 Required the number for log. Oil 253 170 227 
 
To Kind the Number when the Logarithm is Given, ii 
 
 Formation of Number from Logarithm. 
 
 Logarithm 
 
 
 
 1 
 
 1 
 
 2 
 
 5 
 
 3 
 
 1 
 
 7 
 
 
 
 1 
 
 2 
 
 7 
 
 
 
 A — 
 
 B 26 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 1 
 
 4 
 
 7 
 
 3 
 
 6 
 
 
 
 7 
 
 7 
 
 5 
 
 8 
 
 
 
 
 1 
 
 
 
 5 
 
 8 
 
 
 
 9 
 
 3 
 
 5 
 
 1 
 
 2 
 
 C 24 
 
 
 
 
 1 
 
 
 
 4 
 
 2 
 
 1 
 
 8 
 
 1 
 
 7 
 
 
 
 
 
 
 
 
 
 
 1 
 
 5 
 
 9 
 
 1 
 
 1 
 
 8 
 
 1 
 
 2 
 
 D 36 
 
 
 
 
 
 
 1 
 
 5 
 
 6 
 
 3 
 
 4 
 
 5 
 
 7 
 
 3 
 
 
 
 
 
 
 
 
 2 
 
 7 
 
 7 
 
 2 
 
 3 
 
 9 
 
 E 63 
 
 — 
 
 — 
 
 
 
 
 
 
 2 
 
 7 
 
 3 
 
 6 
 
 
 
 5 
 
 
 
 
 
 
 
 
 3 
 
 6 
 
 3 
 
 4 
 
 F 83 
 
 
 
 
 
 
 
 
 
 
 3 
 
 6 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 9 
 
 G 67 ^^^'^ 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 9 
 
 A — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 B 
 26 
 
 1 
 
 
 
 2 
 
 6 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 2 
 
 6 
 
 
 
 
 • 
 
 
 
 
 
 
 C 2 
 
 
 
 
 
 2 
 
 
 
 5 
 
 2 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 4 
 
 1 
 
 
 
 4 
 
 
 
 
 
 1 
 
 
 
 2 
 
 6 
 
 2 
 
 4 
 
 6 
 
 2 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 
 D 3 
 
 
 
 
 
 
 
 3 
 
 
 
 7 
 
 8 
 
 7 
 
 3 
 
 9 
 
 6 
 
 
 
 
 
 
 • 
 
 
 6 
 
 1 
 
 5 
 
 7 
 
 4 
 
 8 
 
 1 
 
 
 
 2 
 
 6 
 
 2 
 
 4 
 
 9 
 
 9 
 
 3 
 
 4 
 
 4 
 
 8 
 
 7 
 
 E 6 
 
 
 
 
 
 • 
 
 
 
 
 6 
 
 1 
 
 5 
 
 7 
 
 5 
 
 3 
 
 
 
 
 • 
 
 
 
 
 
 
 3 
 
 
 
 7 
 
 9 
 
 F 8 
 
 
 
 • 
 
 
 
 
 
 
 
 
 8 
 
 2 
 
 1 
 
 3 
 
 
 • 
 
 
 
 
 
 
 
 
 
 
 3 
 
 1 
 
 G 6 
 
 • 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 2 
 
 6^ 
 
 2_ 
 
 5 
 
 0^ 
 
 01 
 
 
 
 0^ 
 
 ^ 
 
 0^ 
 
 
 In this example we illustrate the procedure when B 
 furnishes the first logarithm. It also shows the convenience 
 of using paper ruled for the purpose. 
 
12 The Properties of Logarithms. 
 
 26. — In order to set down the partial products without 
 hesitation, remember the numbers 2, 4, 6. 
 In multiplying by B 
 
 the first figure of the product moves two places to the right. 
 In multiplying by C 
 
 the first figure of the product movers/our places to the right. 
 In multiplying by D 
 
 the first figure of the 7nultiplica?td moves six places to the left. 
 
 27. — The following rule may now be formulated for this 
 process: 
 
 Rule. — 1. By successiv subtractions separate the given 
 logarithm into a series of partial logarithms found in the 
 colums of the T F, setting opposit each its letter and number. 
 
 2. By successiv multiplications find the product of all the 
 numbers thus found, allowing, in the placing of the partial 
 products, for the prefixt 1 and zeroes. 
 
 28. The work may be made to occupy fewer lines by 
 setting down the factors E, F and G as one number at the top, 
 multiplying it by A and incorporating it thereafter as one 
 multiplicand with the preceding figures. The result will not 
 be affected. Let the factors be, as before: A 56 B 13 C 26 
 D29 E58 F48 G 55. 
 
 E F G 
 
 584855 
 
 ^^^ 2924275 
 
 350913 
 
 5600000327519 
 
 B13 56000003275 
 
 16800000 9 82 
 
 5672800331776 
 
 C26 1134560066 
 
 340368029 
 
 567427525987 1 
 
 D29 11348551 
 
 5106848 
 
 567429171 5 afiXL 
 
To Find the Number when the Logarithm is Given. 13 
 
 29. — Required the number whose logarithm is .5 or >^. 
 
 .500 000 000 000.0 
 A 31 491 361 693 834,3 
 
 B 20 
 
 8 638 306 165,7 
 8 600 171 761,9 
 
 C 08 
 
 38 134 403,8 
 34 742 168,9 
 
 D 78 
 
 3 392 234,9 
 3 387 483,7 
 
 E 10 
 
 4 751,2 
 4 342,9 
 
 F 94 
 G 03 
 
 408,3 
 408,2 
 
 0,1 
 
 The resulting factors 
 
 A 31 B 20 C 08 D 78 E 10 F 94 G 03 
 when combined produce the result 3.16227766017. 
 
 30. — The multiplication illustrates how zeroes are treated 
 when they occur in the multipliers. 
 
 31. — The result is the square root of 10, to 12 places, as may- 
 be demonstrated by multiplying 3.16227766017 by itself. 
 Method by Multipi^es. 
 32. — In order to facilitate the multiplication of the factors, 
 A, B, C, etc., Mr. A. S. Little, of St. Louis, has devised a Table 
 of Multiples, giving the product of each number from 1 to 9 
 by every number from 2 to 99. (See page 35.) Thus the 
 multiples of 89 read in one line as follows : 
 
 123456789 
 089 178 267 356 445 534 623 712 801 
 Then, if it be desired, for example, to multiply 68792341 
 by 89, we would select from the above table 
 under 6 5 3 4 
 
 8 712 
 
 7 623 
 
 9 801 
 
 2 178 
 
 3 267 
 
 4 356 
 
 1 089 
 
 6122518349 
 
14 .Thk Properties of I^ogarithms. 
 
 We have thus multiplied each figure of the multiplicand 
 by both figures of the multiplier, setting down each partial 
 product unhesitatingly. 
 
 33. — The work may be made more compact by piling the 
 partial products like bricks, using only three lines: 
 5 3 4.801,356, 
 7 12,178,089 
 6 2 3,2 6 7, 
 
 6122518349 
 34. — Three figures must be set down for each partial 
 product, even if the first be a zero. 
 
 35. — To use this method in combining the factors of a 
 number, the letters A, B, C, etc., are written above alternate 
 figure spaces, which is facilitated by the use of paper properly 
 ruled. Then the first partial product under each letter is 
 placed with its middle figure under that letter at the top. 
 
 36. — The following is an example of a combination al 
 ready performd in another form : 
 
 A B C D E F G 
 A 56 1 584855 
 
 2 8 0,448 
 4 4 8,280 
 2 2 4,2 8 
 
 56 327519 
 
 B 13 6 5 3 9,0 6 
 
 078 2 6,2 
 
 091 
 
 5672800331778 
 C26 13 0.052,000, 
 15 6,208,078 
 18 2.0 0,0 8 
 
 5674275259864 
 
 D 29 14 5.116,15 
 
 17 405 8,1 
 
 2 3.203 
 
 5 67429171526 
 37. — Mr. Little has also suggested a process for verifying 
 a numerical result by using a different set of factors in a 
 second operation. 
 
 38. — Required the number corresponding to 
 .305 773 384 163.0 
 
To Find the Number when the Logarithm is Given. 15 
 
 The factors are A 20 B 10 C 97 D 21 E 94 F 94 G 33. 
 The number is 2.02195383809. 
 
 In order to check the result and make sure of perfect ac- 
 curacy, we may solve the problem a second time, using two 
 subtrahends from A. The first subtraction may be of any 
 suitable number ; 11 is found to give the greatest facility. 
 
 305 773 384 163,0 
 
 A 11 041 392 685 158.2 
 
 264 380 699 004,8 
 
 A 18 255 272 505 103,3 
 
 9 108 193 901,5 
 
 B 21 9 025 742 086,9 
 
 82 451 814,6 
 
 C 18 78 165 972,0 
 
 4 285 842,6 
 D 98 4 256 065,1 
 
 29 777,5 
 E 68 29 532.0 
 
 245,5 
 F 56 243.2 
 
 2.3 
 G 53 2.3 
 
 39. — The remainder of the operation may be by either method: 
 A B C D E F G A B C D E F G 
 
 A 18 685653A18 685653 
 
 548522 548522 
 
 18 1234175 18 1234175 
 A 11 18 123418 A 11 1^ 123418 
 
 19 8 135759 198 135759 
 B21 396 2715 B21 021 02 1,15 
 
 198 136 18 9 6 3,1 
 
 2021580138610 1 68 105 
 
 C18 202158014 202 1580138610 
 
 16172 6 411 C 18 3 6,0 1 8,0 0,1 
 
 2021944023035 0.090.018 
 
 D98 18197496 3 6.14 4.0 5 
 
 1617555 2021944023034 
 
 202196383809 D 98 19 6 9 8,39 
 
 0.8 8 2.0 
 
 ♦ 196392 
 
 202196383809 
 In this example the first method appears to be preferable, 
 especially in the earlier part. 
 
1 6 The Properties op Logarithms. 
 
 To Form the Logarithm of a Number. 
 
 40. — This consists in two processes : first, the number is 
 separated into a series of factors corresponding to the six 
 colums of the thirteen-place table ; second, the logarithms of 
 these factors are copied from the table and added together. 
 
 41. — The factoring is effected by a progressiv division, 
 the divivSor receiving successivly more and more of the figures 
 of the number. 
 
 42. — To illustrate this division we will assume a number 
 in which the division will be soon completed. 
 
 To find the logarithmic factors. A, B, C, etc., of 5.6728. 
 First extend the number to 12 places, 567 280 000 000. The 
 first factor A is always the first two figures of the number 
 itself. A 56)56 7 2 80 000 000 (1 . 013 B 
 
 56 
 72 
 56 
 
 168 
 168 
 
 It will readily be seen that one 56 might have been omitted. 
 A 56)7 280 000 000 (B 13 
 56 
 
 168 
 168 
 
 Turning then to the Table we have only to set down the 
 logarithms of these two factors : 
 
 A 56 nl 748 188 027 006,2 
 B 13 nl 5 609 445 360,3 
 
 56728 nl 753 797 472 366 5 
 B 13 may be regarded as an abbreviation of L.013. 
 
 43.— We will now give an example where a second divisor, 
 at least, is required. 
 
 A 56) 7 4 2 9 1 7 1 5 2 6 (B 13 
 56 
 
 182 
 168 
 
 A B 56 728 ) 1 4 
 
To Form the Logarithm op a Number. 17 
 
 The second divisor is the product of A and B. It might 
 be obtaind in either of three ways. 
 
 1.013 = 56728 
 
 multiplication 
 
 56 X 
 
 addition 
 
 5Q 
 
 
 + 56 
 
 
 + 168 
 
 56728 
 But the easiest way is 
 
 by subtraction 56742 five figures of the number 
 
 — 14 the remainder 
 56728 
 This is the proper method for forming all divisors after 
 the first ; subtract the remainder from the original number so 
 far as used. 
 
 44. — We resume the division, bringing down /our more 
 figures, to the ninth inclusiv. 
 
 AB)56728 )14 9 171526(C26 
 113456 
 357155 
 340368 
 
 ABC)56742 7,5) *1678726(D29 
 
 1134855 
 
 543871 
 
 510685 
 
 5 67 42 9,174- 3 3 1 8 6 (E 58 
 
 28371 
 
 4815 
 
 4539 
 
 2 7 6 (F 48,7 
 
 227 
 
 49 
 
 45 
 
 4 
 
 The third divisor A B C is also formd by subtracting from 
 
 the number 5674291715 
 
 * the remainder 16 7 8 7 
 
 5674274928 
 
1 8 Thb Properties of Logarithms. 
 
 As only six figures are needed for the divisor and one for 
 carrying, this is rounded up to 5 6 7 4 2 7,5 
 
 The fourth divisor is practically the number itself so far 
 as needed, and this lasts to the end. 
 
 45. — The entire process is now repeated, but for greater 
 accuracy in the twelfth figure we will divide out to the 
 thirteenth. . 
 
 A56) 7 429 17 15 26,0 (B 13 
 56 
 
 A B 56 728) 
 
 182 
 168 
 
 149171 
 113 4 5 6 
 
 (C26 
 
 357155 
 340368 
 
 A B C 56 742 749 28) 
 [Contracted division begins here] 
 
 56 742 92) 
 
 1 6 7 8 7 2 6,0 (D 29 
 113485 5,0 
 
 5 4 3 8 7 1,0 
 5 106 84,7 
 
 33 186,3 (E58 
 2837 1,4 
 
 4 8 1 4,9 
 45 3 9,4 
 
 2 7 5,5 (F48 
 2 2 7.0 
 
 485 
 454 
 
 (G55 
 
 31 
 
 28 
 
 It remains only to add together the logarithms : 
 A 56 {nl) 748 188 027 006,2 
 
 B 13 " 
 
 5 609 445 360,3 
 
 C 26 •' 
 
 112 901 888,7 
 
 D29 " 
 
 1 259 452,2 
 
 E58 " 
 
 25 189,1 
 
 F 48 " 
 
 208,5 
 
 G55 '* 
 
 2.4 
 
 567 429 171 526 («/) 
 
 753 911 659,107 
 
To Form the Logarithm of a Number. 19 
 
 46. — The figures in the last colum are only usg^ for car- 
 rying to the twelfth, which otherwise would give Stnsted of 7. 
 
 47. — We may now formulate the following rule for finding 
 the logarithm: 
 
 Rule.— 1. Make the number to 13 figures, by adding 
 cifers or cutting off decimals. 
 
 2. Cut off the two left-hand figures by a curve, giving A. 
 
 3. Divide the next three figures by A, giving the two 
 figures of B, and a remainder. 
 
 4. Form the second divisor A B, by subtracting the re- 
 mainder from the first five figures of the number. 
 
 6. Bring down four more figures to the remainder and 
 divide by A B, giving the two figures of C and a remainder. 
 
 6. Form the third (and last) divisor A B C by subtracting 
 the remainder from ten figures of the number. 
 
 7. Divide the remaining figures by the third divisor. As 
 there are ten figures in the divisor and only eight in the divi- 
 dend, contraction begins immediately. Having obtaind the 
 figures of D, the divisor for E, F and G is simply the number 
 itself contracted. 
 
 8. Write down the logarithms of A, B, C, D, E and F, 
 obtaind from the several colums of T F; also that of G, being 
 the first two figures of E. The sum will be the logarithm, the 
 thirteenth figure being used for carrying only. 
 
 48. — It is advisable to make all logarithmic computations 
 on paper ruled with thirteen down-lines, every third being 
 darker. A specimen is given on page 36. 
 
 49. A few examples for practis are given below with the 
 factors and the solution: 
 
 5674 = A 56 B 13 C 21 D 15 E 35 F 42 G 70 
 log. 5674 = 3.753 889 331 458 
 38.8586468578 = A 38 B 22 C 58 D 31 E 39 F 02 G 25 
 
 log. do. = 1 .589 487 673 453 
 
 3.1415926535898 + = A 31 B 13 C 41 D 16 E 33 F 11 G 91 
 log. do. =.497 149 872 694 
 
 (This number is the ratio of the circumference of a circle toils diameter.) 
 1.02625 = B 26 C 24 D 36 E 63 F 83 
 log. do. = .011 253 170 127 
 This number begins with an expression of the form B (1.026), hence 
 no division by A occurs. 1026 is the first divisor. 
 
20 The Properties of I^ogarithms. 
 
 B 1026) 2 5 
 2052 
 
 C24 
 
 
 4480 
 
 
 
 4104 
 
 
 B C 102624624) 3 7 6 0.0 
 3 7 8 7 3,9 
 
 D36 
 
 
 6 8 1 2 6,1 
 
 6 1574,8 
 
 
 102625) 
 
 6 55 1,3 
 
 6 1 5 7,5 
 
 3 9 3,8 
 
 3 7,9 
 
 K63 
 
 
 8 5,9 
 
 8 2,1 
 
 F83 
 
 
 3,8 
 3,1 
 
 7 
 
 G70 
 
 B 26 
 C24 
 D36 
 E63 
 F83 
 G70 
 
 Oil 147 360 775,8 
 
 104 218 170,0 
 
 1563 457,3 
 
 27 360,6 
 
 360,5 
 
 3,0 
 
 Oil 253 170 127 
 
 
 This result will be found also in the Table of Interest-Ratios, but 
 even more extended. 
 
21 
 
 Logarithms to Less than 12 Places. 
 
 60. — The T F may be cut down to any lower number of 
 places. In the example in Art. 45 it may be required to give 
 9 places only, the tenth being used for carrying. We cut 
 down the original logarithm to ten figures, with a comma after 
 the ninth and it becomes 
 
 753 911 659,1 
 A 56 748 188 027.0 
 
 
 5 723 632,1 
 
 B 13 
 
 5 609 445,4 
 
 
 114 186,7 
 
 C 26 
 
 112 901.9 
 
 
 1 284,8 
 
 D 29 
 
 1 259,5 
 
 
 25,3 
 
 B58 
 
 25,2 
 
 F 24 
 
 1 
 
 A 
 
 56 
 
 B 1 
 
 5 6 
 
 3 
 
 168 
 
 
 567280000 
 
 C 2 
 
 113456 
 
 6 
 
 3 4036,8 
 
 
 56742749 2,8 
 
 D 2 
 
 1 1 3 4,9 
 
 9 
 
 5 10,7 
 
 E 5 
 
 2 8,4 
 
 8 
 
 4,5 
 
 F 2 
 
 1 
 
 56742917 1,4 
 
 The number is slightly in error in its tenth place, but 
 correct to the ninth. 
 
 51. — If a table of factors for 18 or some other number of 
 places should hereafter be prepared, the methods which have 
 been explaind would be applicable. 
 
22 The Properties of Logarithms. 
 
 Multiplying Up. 
 
 52. — Mr. Edward S. Thomas, of Cincinnati, has suggested 
 another method for obtaining the factors of the number in 
 forming its logarithm. 
 
 53. — It procedes by multiplication insted of division, the 
 latter operation being notabh^- the more laborious. The num- 
 ber, at first taken as a decimal less than 1, is successivly mul- 
 tiplied up to produce 1.000,000,000,0 and these multipliers 
 are the A, B, C, D, E, F and G, whose logarithms added to- 
 gether make the cologarithm, from which the logarithm is 
 easily obtaind. 
 
 54. — A is a number of two figures, a little less than the 
 reciprocal of the number, which will be calld the sub-recipro- 
 cal of its two initial figures. A Table of Sub- Reciprocals is 
 given on page 33. The number multiplied by A will always 
 give a product beginning with 9. B is always the arithmetical 
 complement of the two figures following tlve nine, or the re- 
 mainder obtaind by subtracting those two figures from 99. 
 Multiplication by B will usually give a result beginning with 
 999. C is the next complement and gives 5 9's, 999,99. D 
 similarly brings 999,999,9 >{^ * ^ * >K H^ ^ * . No further multipli- 
 cation is necessary, when D has been used ; the six figures in 
 the places of the stars are the complements of E, F and G. 
 
 55. — To illustrate, let it be required to obtain the loga- 
 rithm to the 12th place of 3.14 159 265 359 0. The object is 
 to multiply .314 159 265 -859 up to 1 .000 000 000 000 0. The 
 first step is to find the sub-reciprocal of 31, or A. Turning to 
 the Table of Sub-reciprocals, opposit 81 we find 31, by which 
 we multiply. 
 
Multiplying Up. 
 
 23 
 
 A 31 
 
 99 — 73 = 26 
 
 B 26 is therefore 
 the next multipli- 
 er; dropping the 
 last two figures 
 
 (99—21) C 78 
 
 (99—43) D 56 
 
 (99-47) 
 
 (99-03) 
 
 (100-77) 
 
 E 52 
 F 96 
 G23 
 
 .3141592653590 
 
 .9424777960770 
 31415 92653 5 9 
 .9 738937226129 One 9 has been secared 
 
 194778744523 
 58433623357 
 
 999214959400 9 
 
 6994504716 
 
 799 3 719 6 8 
 
 Three nines secured 
 
 99999434706 9 3 Five nines 
 49999718 
 5999966 
 
 .9999999470377 Seven nines 
 52 
 9 6 
 23 
 
 A31;^/ .4 913616938343 
 
 B26 111473607758 
 
 C78 3386176522 
 
 D56 24320423 
 
 E52 225833 
 
 F 96 4 16 9 
 
 G 23 _j 1^ 
 
 colog. 0.5 02850127306 
 
 log. 1.4 97149872694 
 
 56. — It may happen, in the course of multiplication, that 
 the complement of the figures following the 9 does not suffice 
 to secure two nines more. In this case, another supplementary 
 multiplication must take place. This occurs in the following 
 example, which has alredy been solvd. 
 
24 The Propkrties of I,ogarithms. 
 
 57. — Required the logarithm of the number 
 567 429 171 526. 
 
 In this example the C multiplication also requires an 
 additional figure. This seldom occurs. 
 
 .567 429171526 
 A 17 .397 200 420 068 2 
 
 .964 629 591594 2 
 
 B 35 28 938 887 747 8 
 
 4 823 147 958 
 
 .998 391627 300 
 " 01 998 391 627 3 
 
 .999 390 018 927 3 
 C 60 599 634 0114 
 
 .999 989 652 938 7 
 " 01 9 999 896 5 
 
 .999 999 652 835 2 
 
 D 03 299 999 9 
 
 .999 999 952 835 1 
 47 164 9 
 
 { 
 
 A 17 
 
 230 448 921 378 3 
 
 B 35 
 
 14 940 349 792 9 
 
 01 
 
 434 077 479 3 
 
 C 60 
 
 260 498 547 4 
 
 01 
 
 4 342 923 1 
 
 D 03 
 
 130 288 3 
 
 B 47 
 
 20 411 8 
 
 F 16 
 
 69 5 
 
 G 49 
 
 21 
 
 
 .246 088 340 892 7 
 
 
 .753 911659 107 3 
 
 As the multiplication by 35 brings only 998 insted of 999, 
 we multiply again by B 01, which brings it up. 
 
MuivTiPLYiNG Up. 25 
 
 58. — In the next example there is a large defect in B, 
 which requires an additional multiplication by 7. 
 
 110 175 
 
 A 83 881 400 (83, subreciprocal of 11) 
 33 052 5 
 
 914 452 5 
 B 85 73156 200 
 4 572 262 5 
 
 B 07 
 C 87 
 
 999 995 469 056,9 
 
 D 45 3 999 981,9 
 
 499 997,7 
 
 992 180 962 5 
 6 945 266 737,5 
 
 999 126 229 237.5 
 
 799 300 983,4 
 
 69 938 836,0 
 
 999 999 969 036,5 
 30 963,5 
 
 A 83 
 
 919 078 092 376,1 
 
 B 85 
 
 35 429 738 184,5 
 
 B 07 
 
 3 029 4*70 553,6 
 
 C 87 
 
 377 670 935,8 
 
 D 45 
 
 1 954 320^8 
 
 K 30 
 
 13 028,8 
 
 F 96 
 
 416,9 
 
 G 35 
 
 1,5 
 
 957 916 940 818 
 042 083 059 182 
 
 The number 11075 was purposely selected, very slightly 
 in excess of the highest number in colum B, so as to produce 
 the shortage of 7. 
 
26 The Properties of I^ogarithms. 
 
 59. — Little's Table of Multipliers may be used in the 
 multiplication, as in the following example. It will be found 
 •that the logarithm when computed has the same figures as the 
 number itself; a remarkable peculiarity which no other com- 
 bination of figures can possess. 
 
 . 137 128 857 423 9 
 
 A 
 
 71 
 
 710 715 682 846 4 
 
 213 142 355 142 
 
 49 756 849 721 3 
 
 
 . 973 614 887 709 7 
 
 B 
 
 26 
 
 23 415 620 818 2 
 
 1 820 262 080 
 
 078 104 182 2 
 
 
 . 998 928 874 790 1 
 
 tt 
 
 01 
 
 998 928 874 8 
 
 
 . 999 927 803 664 9 
 
 C 
 
 07 
 
 69 994 946 3 
 
 
 . 999 997 798 611 2 
 
 D 
 
 22 
 
 1 981 981 8 
 
 198 198 
 
 19 815 4 
 
 
 . 999 999 998 606 4 
 
 
 
 01 393 6 
 
 
 
 E F G 
 
 A 
 
 71 
 
 851 258 348 719 1 
 
 B 
 
 26 
 
 11 147 360 775 8 
 
 B 
 
 01 
 
 434 077 479 3 
 
 C 
 
 07 
 
 30 399 549 8 
 
 D 
 
 22 
 
 955 446 8 
 
 E 
 
 01 
 
 434 3 
 
 F 
 
 39 
 
 169 4 
 
 G 
 
 36 
 
 16 
 
 
 862 871 142 576 1 
 
 .137 128 857 423 9 
 which is the log. of 1.371 288 574 239 
 
27 
 
 Signs of the Characteristic. 
 
 60 — We have seen (Art. 8) that while the decimal part 
 of the logarithm is always positiv, the characteristic is often 
 negativ and has the minus sign above it. 
 
 61. — In adding together several logarithms with different 
 signs, the positivs and the negativs must be added separately; 
 the less sum must be subtracted from the greater, and the 
 remainder has the sign of the greater sum. The carrying 
 from the decimal part counts with the positivs. 
 
 34 7il. 
 
 1.531478 917 042,3 
 
 2900 " 
 
 3.462 397 997 899,0 
 
 .73 '' 
 
 1.863 322 860120,5 
 
 .056 ♦' 
 
 2.748 188 027 006.2 
 
 The sum of the 
 
 
 decimals is. . . . 
 
 2.605 387 802 068,0 
 
 The positivs are 
 
 1. 
 
 and 
 
 3 
 
 Total 
 
 + 6 
 
 The negativs are 1 
 
 , 
 
 2 
 
 — 3 
 
 Sum of the logarithms +3.605 387 802 068,0 
 
 The decimal point in the result must follow the fourth 
 figure, as indicated by the characteristic 3. 
 
 62. — In subtracting one logarithm from another, when 
 the decimal of the subtrahend is the greater, and a unit^ 
 " borrowed," the unit is considered as one more negativ: '^ut ^CT^'^'y'' * 
 the total characteristic changes its signs from plus to minus or /Tv^^^ 
 from minus to plus. . 
 
 290 2 462 397 997 899,0 <^^^^=-^ ^ 
 
 .0058 3.763 427 993 562,9 
 Operate on the deci- 
 mals only . 462 397 997 899,0 
 
 0.763 427 993 562,9 
 
 1.698 970 004 336,1 
 Negativ from subtrahend 3 
 Total negativ 4 
 
 Sign changed +4 Y^^^"^^^ 
 
 From minuend ... -h 2 ) 
 
 6 . 698 970 004 336,1 In 5 000 000^ 
 
 y 
 
28 The PrOPERTIBS of lyOGARlTHMS. 
 
 63. — To multiply a logarithm having a negativ character- 
 istic (in order to obtain a power of a decimal) , multiply the 
 decimal part and the characteristic separately and add the two 
 together: 
 
 2.301029 995 664.0 x 5 
 
 Decimal part _ 1 . 505 149 978 320,0 
 
 Characteristic 10. 
 
 9.505 149 978 320,0 
 Therefore the 5th power of .02 is .000 000 003 2. 
 
 64. — To divide a logarithm having a negativ characteristic, 
 (for the extraction of a root;) if the characteristic is exactly 
 divisible, divide the decimal part and the characteristic 
 separately: 
 
 12.690 196 080 028,5 -f- 6 
 2.115 032 680 004,7 
 
 But if the characteristic be not so divisible, add to it a 
 negativ quantity, which will make it divisible, and prefix to the 
 decimal part in compensation an equal quantity positiv. 
 
 12.690 196 080 028,5 -f- 5 
 Add 3 15. 
 
 Add 3 3.690196 080 028.5 
 
 Quotient 3 . 738 039 216 005,7 
 
 Different Bases. 
 
 65. — Ten is the base of the logarithmic system which we 
 have been explaining; it is the most useful of all systems, 
 because ten is also the base of our numerical system. These 
 are usually calld common or vulgar, or Briggsian logarithms, 
 but decimal logarithms would seem more appropriate. 
 
 66. — Any number might form the base of a system of lo- 
 garithms, but the only other in actual use is one known as 
 the "natural" system, having for its base the number 
 2. 718281828459 -h which is the sum of the series 
 
 1 J_ 1 1 1 
 
 ■^ "^ i "^ 1x2 "^ 1x2x3 "^ 1x2x3x4 "^ 1x2x3x4x5^^^' 
 
 This is only used in theoretical inquiries, and is seldom of 
 utility to the accountant. 
 
PART II. 
 
 TABLES 
 
 FOR OBTAINING 
 
 Logarithms and Antilogarithms 
 
 TO 12 PLACES OF decimals 
 
30 
 
 TABLE OF FACTORS 
 
 (V 
 
 a 
 
 01 
 02 
 03 
 04 
 
 05 
 06 
 07 
 
 08 
 09 
 
 10 
 11 
 12 
 13 
 14 
 
 15 
 16 
 17 
 
 18 
 19 
 
 20 
 21 
 22 
 23 
 24 
 
 25 
 ^26 
 
 27 
 28 
 29 
 
 30 
 31 
 32 
 33 
 34 
 
 35 
 36 
 37 
 38 
 39 
 
 40 
 41 
 42 
 43 
 44 
 
 45 
 46 
 47 
 48 
 49 
 
 A 
 
 * . * 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 041 392 685 158,2 
 079 181 246 047,6 
 113 943 352 306,8 
 146 128 035 678,2 
 
 176 091 259 055,7 
 204 119 982 655,9 
 230 448 921 378,3 
 255 272 505 103,3 
 278 753 600 952,8 
 
 301 029 995 664,0 
 322 219 294 733,9 
 342 422 680 822,2 
 361 727 836 017,6 
 380 211 241 711,6 
 
 397 940 008 672,0 
 414 973 347 970,8 
 431 363 764 159,0 
 447 158 031 342,2 
 462 397 997 899,0 
 
 477 121 254 719,7 
 491 361 693 834,3 
 505 149 978 319,9 
 518 513 939 877,9 
 531 478 917 042,3 
 
 544 068 044 350,3 
 556 302 500 767,3 
 568 201 724 067,0 
 579 783 596 616,8 
 591 064 607 026,5 
 
 602 059 991 328,0 
 612 783 856 719,7 
 623 249 290 397,9 
 633 468 455 579,6 
 643 452 676 486,2 
 
 653 212 513 775,3 
 662 757 831 681,6 
 672 097 857 935,7 
 681 241 237 375,6 
 690 196 080 028,5 
 
 B 
 
 1.0* H« 
 
 434 077 479,3 
 
 867 721 531,2 
 
 1 300 933 020,4 
 
 1 733 712 809,0 
 
 2 166 061 756,5 
 
 2 597 980 719,9 
 
 3 029 470 553,6 
 3 460 532 109,5 
 3 891 166 236,9 
 
 4 321 373 782,6 
 
 4 751 155 591,0 
 
 5 180 512 503,8 
 
 5 609 445 360,3 
 
 6 037 954 997,3 
 
 6 466 042 249,2 
 
 6 893 707 947,9 
 
 7 320 952 922,7 
 
 7 747 778 000,7 
 
 8 174 184 006,4 
 
 8 600 171 761,9 
 
 9 025 742 086,9 
 9 450 895 798,7 
 9 875 633 712,2 
 
 10 299 956 639,8 
 
 10 723 865 391,8 
 
 11 147 360 775,8 
 11 570 443 597,3 
 
 11 993 114 659,3 
 
 12 415 374 762,4 
 
 12 837 224 705,2 
 
 13 258 665 283,5 
 
 13 679 697 291,2 
 
 14 100 321 519,6 
 14 520 538 757,9 
 
 14 940 349 792,9 
 
 15 359 755 409,2 
 
 15 778 756 389,0 
 16*197 353 512,4 
 
 16 615 547 557,2 
 
 17 033 339 298,8 
 17 450 729 510,5 
 
 17 867 718 963,5 
 
 18 284 308 426,5 
 
 18 700 498 666,2 
 
 19 116 290 447,1 
 19 531 684 531,3 
 
 19 946 681 678,8 
 
 20 361 282 647,7 
 20 775 488 193,6 
 
 C 
 1.000+* 
 
 4 342 923,1 
 
 8 685 802.8 
 
 13 028 639,0 
 
 17 371 431,8 
 
 21 714 181,2 
 
 26 056 887,2 
 
 30 399 549,8 
 
 34 742 168,9 
 
 39 084 744,6 
 
 43 427 276,9 
 
 47 769 765,7 
 
 52 112 211,2 
 
 56 454 613,2 
 
 60 796 971,8 
 
 65 139 287,0 
 
 69 481 558,7 
 
 73 823 7^7,1 
 
 78 165 972,0 
 
 82 508 113,5 
 
 86 850 211,6 
 
 91 192 266,3 
 
 95 534 277,6 
 
 99 876 245,5 
 
 104 218 170,0 
 
 108 560 051,0 
 
 112 901 888,7 
 
 117 243 682,9 
 
 121 585 433,8 
 
 125 927 141,2 
 
 130 268 805,2 
 
 134 610 425,9 
 
 138 952 003,1 
 
 143 293 536,9 
 
 147 635 027,3 
 
 151 976 474,3 
 
 156 317 878,0 
 
 160 659 238,2 
 
 165 000 555,0 
 
 169 341 828,4 
 
 173 683 058,5 
 
 178 024 245,1 
 
 182 365 388,3 
 
 186 706 488,2 
 
 191 047 544,7 
 
 195 388 557,7 
 
 199 729 527,4 
 
 204 070 453,7 
 
 208 411 336,6 
 
 212 752 176,1 
 
 D 
 
 1.00000** 
 
 43 429,4 
 
 86 858,9 
 
 130 288,3 
 
 173 717,8 
 
 217 147,2 
 
 260 576,6 
 
 304 006,0 
 
 347 435,4 
 
 390 864,9 
 
 434 294,3 
 
 477 723,7 
 
 521 153,1 
 
 564 582,5 
 
 608 011,8 
 
 651 441,2 
 694 870,6 
 738 300,0 
 729.4 
 158,7 
 
 781 
 825 
 
 868 588,1 
 912 017,5 
 955 446,8 
 998 876,2 
 1 042 305,5 
 
 1 085 734,8 
 
 1 129 164,2 
 
 1 172 593,5 
 
 1 216 022,8 
 
 1 259 452,2 
 
 302 
 346 
 389 
 433 
 476 
 
 881,5 
 310,8 
 740,1 
 169,4 
 598,7 
 
 520 028,0 
 563 457,3 
 606 886,6 
 650 315,9 
 693 745,2 
 
 737 174,5 
 780 603,7 
 824 033,0 
 867 462,3 
 910 891,5 
 
 1 954 320,8 
 
 1 997 750,0 
 
 2 041 179,3 
 2 084 608,5 
 2 128 037,7 
 
 
 
 ^H 
 
 
 
 o 
 
 E 
 
 F 
 
 a • 
 
 3 
 
 1.0'** 
 
 1.0%* 
 
 
 
 ^ 
 
 434,3 
 
 004,3 
 
 01 
 
 868,6 
 
 008,7 
 
 02 
 
 1 302,9 
 
 013,0 
 
 03 
 
 1 737,2 
 
 017,4 
 
 04 
 
 2 171,5 
 
 021,7 
 
 05 
 
 2 605,8 
 
 026,1 
 
 06 
 
 3 040,1 
 
 030,4 
 
 07 
 
 3 474,4 
 
 034,7 
 
 08 
 
 3-908,7 
 
 039,1 
 
 09 
 
 4 342,9 
 
 043,4 
 
 10 
 
 4 777,2 
 
 047,8 
 
 11 
 
 5 211,5 
 
 052,1 
 
 12 
 
 5 645,8 
 
 056,5 
 
 13 
 
 6 080,1 
 
 060,8 
 
 14 
 
 6 514,4 
 
 065,1 
 
 15 
 
 6 948,7 
 
 069,5 
 
 16 
 
 7 383,0 
 
 073,8 
 
 17 
 
 7 817,3 
 
 078,2 
 
 18 
 
 8 251,6 
 
 082,5 
 086.9 
 
 19 
 
 20 
 
 8 685,9 
 
 9 120,2 
 
 091,2 
 
 21 
 
 9 554,5 
 
 095,5 
 
 22 
 
 9 988,8 
 
 099,9 
 
 23 
 
 10 423,1 
 
 104,2 
 
 24 
 
 10 857,4 
 
 108,6 
 
 25 
 
 11 291,7 
 
 112,9 
 
 26 
 
 11 726,0 
 
 117,3 
 
 27 
 
 12 160,2 
 
 121,6 
 
 28 
 
 12 594,5 
 
 125,9 
 130,3 
 
 29 
 30 
 
 13 028,8 
 
 13 463,1 
 
 134,6 
 
 31 
 
 13 897,4 
 
 139,0 
 
 32 
 
 14 331,7 
 
 143,3 
 
 33 
 
 14 766,0 
 
 147,7 
 
 34 
 
 15 200,3 
 
 152,0 
 
 35 
 
 15 634,6 
 
 156,3 
 
 36 
 
 16 068,9 
 
 160,7 
 
 37 
 
 16 503,2 
 
 165,0 
 
 38 
 
 16 937,5 
 
 169,4 
 173,7 
 
 39 
 40 
 
 17 371,8 
 
 17 806,1 
 
 178,1 
 
 41 
 
 18 240,4 
 
 182,4 
 
 42 
 
 18 674,7 
 
 186,7 
 
 43 
 
 19 109,0 
 
 191,1 
 
 44 
 
 19 543,3 
 
 195,4 
 
 45 
 
 19 977,5 
 
 199,8 
 
 46 
 
 20 411,8 
 
 204,1 
 
 47 
 
 20 846,1 
 
 208,5 
 
 48 
 
 21 280,4 
 
 212,8 
 
 49 
 

 
 TABLE OF 
 
 FACTORS— Continued 
 
 
 
 31 
 
 HI 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 F 
 
 
 s 
 
 50 
 
 * • * 
 
 1.0** 
 
 1.000** 
 
 1.00000** 
 
 1.0%* 
 
 1.0%* 
 217,1 
 
 S 
 50 
 
 698 970 004 336,0 
 
 21 189 299 069,9 
 
 217 092 972,2 
 
 2 171 467,0 
 
 21 714,7 
 
 51 
 
 707 570 176 097,9 
 
 21 602 716 028,2 
 
 221 433 725,0 
 
 2 214 896,2 
 
 22 149,0 
 
 221,5 
 
 51 
 
 52 
 
 716 003 343 634,8 
 
 22 015 739 817,7 
 
 225 774 434,3 
 
 2 258 325,4 
 
 22 583,3 
 
 225,8 
 
 52 
 
 53 
 
 724 275 869 600,8 
 
 22 428 371 185,5 
 
 230 115 100,3 
 
 2 301 754,7 
 
 23 017,6 
 
 230,2 
 
 53 
 
 54 
 
 732 393 759 823,0 
 
 22 840 610 876,5 
 
 234 455 722,9 
 
 2 345 183,9 
 
 23 451,9 
 
 234,5 
 
 54 
 
 55 
 
 740 362 689 494,2 
 
 23 252 459 633,7 
 
 238 796 302,1 
 
 2 388 613,1 
 
 23 886,2 
 
 238,9 
 
 55 
 
 56 
 
 748. 188 027 006,2 
 
 23 663 918 197,8 
 
 243 136 837,9 
 
 2 432 042,3 
 
 24 320,5 
 
 243,2 
 
 56 
 
 57 
 
 755 874 855 672,5 
 
 24 074 987 307,4 
 
 247 477 330,3 
 
 2 475 471,5 
 
 24 754,8 
 
 247,5 
 
 57 
 
 58 
 
 763 427 993 562,9 
 
 24 485 667 699,2 
 
 251 817 779,4 
 
 2 518 900,7 
 
 25 189,1 
 
 251,9 
 
 58 
 
 59 
 60 
 
 770 852 Oil 642,1 
 
 24 895 960 107,5 
 
 256 158 185,1 
 
 2 562 329,9 
 2 605 759,1 
 
 25 623,4 
 
 256,2 
 260,6 
 
 59 
 60 
 
 778 151 250 383,6 
 
 25 305 865 264,8 
 
 260 498 547,4 
 
 26 057,7 
 
 61 
 
 785 329 835 010,8 
 
 25 715 383 901,3 
 
 264 838 866,3 
 
 2 649 188,3 
 
 26 492,0 
 
 264,9 
 
 61 
 
 62 
 
 792 391 689 498,3 
 
 26 124 516 745,5 
 
 269 179 141,9 
 
 2 692 617,4 
 
 26 926,3 
 
 269,3 
 
 62 
 
 63 
 
 799 340 549 453,6 
 
 26 533 264 523,3 
 
 273 519 374.0 
 
 2 736 046,6 
 
 27 360,6 
 
 273,6 
 
 63 
 
 64 
 
 806 179 973 983,9 
 
 26 941 627 959,0 
 
 277 859 562,8 
 
 2 779 475,8 
 
 27 794,8 
 
 277,9 
 
 64 
 
 65 
 
 812 913 356 642,9 
 
 27 349 607 774,8 
 
 282 199 708,3 
 
 2 822 905,0 
 
 28 229,1 
 
 282,3 
 
 65 
 
 66 
 
 819 543 935 541,9 
 
 27 757 204 690,6 
 
 286 539 810,3 
 
 2 866 334,1 
 
 28 663,4 
 
 286,6 
 
 66 
 
 67 
 
 826 074 802 700,8 
 
 28 164 419 424,5 
 
 290 879 869,0 
 
 2 909 763,3 
 
 29 097,7 
 
 291,0 
 
 67 
 
 68 
 
 832 508 912 706,2 
 
 28 571 252 692,5 
 
 295 219 884,3 
 
 2 953 192,4 
 
 29 532,0 
 
 295,3 
 
 68 
 
 69 
 
 70 
 
 838 849 090 737,3 
 
 28 977 705 208,8 
 
 299 559 856,2 
 
 2 ^996 621,6 
 
 29 906,3 
 
 299,7 
 304,0 
 
 69 
 70 
 
 ^45 098 040 014,3 
 
 29 383 777 685,2 
 
 303 899 784,8 
 
 3 040 050,7 
 
 30 400,6 
 
 71 
 
 851 258 348 719,1 
 
 29 789 470 831,9 
 
 308 239 670,0 
 
 3 083 479,9 
 
 30 834,9 
 
 308,3 
 
 71 
 
 72 
 
 857 332 496 431,3 
 
 30 194 785 356,8 
 
 312 579 511,8 
 
 3 126 909,0 
 
 31 269,2 
 
 312,7 
 
 72 
 
 73 
 
 863 322 860 120,5 
 
 30 599 721 966,0 
 
 316 919 310,3 
 
 3 170 338,1 
 
 31 703,5 
 
 317,0 
 
 73 
 
 74 
 
 869 231 719 731,0 
 
 31 004 281 363,5 
 
 321 259 065,4 
 
 3 213 767,3 
 
 32 137,8 
 
 321,4 
 
 74 
 
 75 
 
 875 061 263 391,7 
 
 31 408 464 251,6 
 
 325 598 777,1 
 
 3 257 196,4 
 
 32 572,1 
 
 325,7 
 
 75 
 
 76 
 
 880 813 592 280,8 
 
 31 812 271 330,4 
 
 329 938 445,5 
 
 3 300 625,5 
 
 33 006,4 
 
 330,1 
 
 76' 
 
 77 
 
 886 490 725 172,5 
 
 32 215 703 298,0 
 
 334 278 070,5 
 
 3 344 054,6 
 
 33 440,7 
 
 334,4 
 
 77 
 
 78 
 
 892 094 602 690,5 
 
 32 618 760 850,7 
 
 338 617 652,2 
 
 3 387 483,7 
 
 33 875,0 
 
 338,7 
 
 78 
 
 79 
 80 
 
 897 627 091 290,4 
 
 33 021 444 682,9 
 
 342 957 190,4 
 
 3 430 912,9 
 
 34 309,3 
 
 343,1 
 347,4 
 
 79 
 80 
 
 903 089 986 991,9 
 
 33 423 755 486,9 
 
 347 296 685,4 
 
 3 474 342,0 
 
 34 743,6 
 
 81 
 
 908 485 018 878,6 
 
 33 825 693 953,3 
 
 351 636 136,9 
 
 3 517 771,1 
 
 35 177,9 
 
 351,8 
 
 81 
 
 82 
 
 913 813 852 383,7 
 
 34 227 260 770,6 
 
 355 975 545,1 
 
 3 561 200,2 
 
 35 612,1 
 
 356,1 
 
 82 
 
 83 
 
 919 078 092 376,1 
 
 34 628 456 625,3 
 
 360 314 910,0 
 
 3 604 629,2 
 
 36 046,4 
 
 360,5 
 
 83 
 
 84 
 
 9^4 279 286 061,9 
 
 35 029 282 202,4 
 
 364 654 231,5 
 
 3 648 058,3 
 
 36 480,7 
 
 364,8 
 
 84 
 
 85 
 
 929 418 925 714,3 
 
 35 429 738 184,5 
 
 368 993 509,6 
 
 3 691 487,4 
 
 36 915,0 
 
 369,2 
 
 85 
 
 86 
 
 934 498 451 243,6 
 
 35 829 825 252,8 
 
 373 332 744,4 
 
 3 734 916,5 
 
 37 349,3 
 
 373,5 
 
 86 
 
 87 
 
 939 519 252 618,6 
 
 36 229 544 086,3 
 
 377 671 935,8 
 
 3 778.345,6 
 
 37 783,6 
 
 377,8 
 
 87 
 
 88 
 
 944 482 672 150,2 
 
 36 628 895 362,2 
 
 382 Oil 083,8 
 
 3 821 774,6 
 
 38 217,9 
 
 382,2 
 
 88 
 
 89 
 90 
 
 949 390 006 644,9 
 
 37 027 879 755,8 
 
 386 350 188,6 
 
 3 865 203,7 
 
 38 652,2 
 
 386,5 
 390,9 
 
 89 
 90 
 
 954 242 509 439,3 
 
 37 426 497 940,6 
 
 390 689 249,9 
 
 3 908 632,7 
 
 39 086,5 
 
 91 
 
 959 041 392 321,1 
 
 37 824 750 588,3 
 
 395 028 267,9 
 
 3 952 061,8 
 
 39 520,8 
 
 395,2 
 
 91 
 
 92 
 
 963 787 827 345,6 
 
 38 222 638 368,7 
 
 399 367 242,6 
 
 3 995 490,9 
 
 39 955,1 
 
 399,6 
 
 92 
 
 93 
 
 968 482 948 553,9 
 
 38 620 161 949,7 
 
 403 706 173,9 
 
 4 038 919,9 
 
 40 389,4 
 
 403,9 
 
 93 
 
 94 
 
 973 127 853 599,7 
 
 39 017 321 997,4 
 
 408 045 061,8 
 
 4 082 348,9 
 
 40 823,7 
 
 408,2 
 
 94 
 
 95 
 
 977 723 605 288,8 
 
 39 414 119 176,1 
 
 412 383 906,5 
 
 4 125 778,0 
 
 41 258,0 
 
 412,6 
 
 95 
 
 96 
 
 982 271 233 039,6 
 
 39 810 554 148,4 
 
 416 722 707,7 
 
 4 169 207,0 
 
 41 692,3 
 
 416,9 
 
 96 
 
 97 
 
 986 771 734 266,2 
 
 40 206 627 574,7 
 
 421 061 465,6 
 
 4 212 636,0 
 
 42 126,6 
 
 421,3 
 
 97 
 
 98 
 
 991 226 075 692,5 
 
 40 602 340 114,1 
 
 425 400 180,2 
 
 4 256 065,1 
 
 42 560,9 
 
 425,6 
 
 98 
 
 99 
 
 995 635 194 597,5 
 
 40 997 692 423,5 
 
 429 738 851,4 
 
 4 299 494,1 
 
 42 995,2 
 
 430,0 
 
 99 
 
32 
 
 TABLE OF INTEREST RATIOS 
 
 1 + i 
 
 Logarithm 
 
 1 + i 
 
 Logarithm 
 
 1.00125 
 
 1.0015 
 
 1.00175 
 
 1.002 
 
 1.00225 
 
 000 542 529 092 294 
 000 650 953 629 595 
 000 759 351 104 737 
 000 867 721 531 227 
 000 976 064 922 559 
 
 1.01375 
 
 1.014 
 
 1.01425 
 
 1.0145 
 
 1.01475 
 
 005 930 867 219 212 
 
 006 037 954 997 317 
 006 145 016 376 364 
 006 252 051 369 365 
 006 359 059 989 323 
 
 1.0025 
 1.00275 
 1.003 
 1.00325 
 1 . 0035 
 
 001 084 381 292 220 
 001 192 670-653 684 
 001 300 933 020 418 
 001 409 168 405 876 
 001 517 376 823 504 
 
 1.015 
 
 1.01525 
 
 1.0155 
 
 1.01575 
 
 1.016 
 
 006 466 042 249 232 
 006 572 998 162 075 
 006 679 927 740 826 
 006 786 830 998 449 
 006 893 707 947 900 
 
 1.00375 
 1.004 
 1 . 00425 
 1 . 0045 
 1 . 00475 
 
 001 625 558 286 737 
 001 733 712 809 001 
 001 841 840 403 709 
 
 001 949 941 084 268 
 
 002 058 014 864 072 
 
 1.01625 
 
 1.0165 
 
 1.01675 
 
 1.017 
 
 1.01725 
 
 007 000 558 602 125 
 007 107 382 974 057 
 007 214 181 076 625 
 007 320 952 922 745 
 007 427 698 525 323 
 
 1.005 
 
 1.00525 
 
 1.0055 
 
 1.00575 
 
 1.006 
 
 002 166 061 756 508 
 002 274 081 774 949 
 002 382 074 932 761 
 002 490 041 243 299 
 002 597 980 719 909 
 
 1.0175 
 
 1.01775 
 
 1.018 
 
 1.01825 
 
 1.0185 
 
 007 534 417 897 258 
 007 641 111 051 437 
 007 747 778 000 740 
 007 854 418 758 035 
 007 961 033 336 183 
 
 1.00625 
 
 1.0065 
 
 1.00675 
 
 1.007 
 
 1.00725 
 
 002 705 893 375 925 
 002 813 779 224 673 
 
 002 921 638 279 469 
 
 003 029 470 553 618 
 003 137 276 060 415 
 
 1.01875 
 
 1.019 
 
 1.01925 
 
 1.0195 
 
 1.01975 
 
 008 067 621 748 033 . 
 008 174 184 006 426 
 008 280 720 124 194 
 008 387 230 114 159" 
 008 493 713 989 132 
 
 1.0075 
 1.00775 
 1.008 
 1 . 00825 
 1.0085 
 
 003 245 054 813 147 
 003 352 806 825 089 
 003 460 532 109 506 
 003 568 230 679 656 
 003 675 902 548 784 
 
 1.02 
 
 1.02025 
 
 1.0205 
 
 1.02075 
 
 1.021 
 
 008 600 171 761 918 
 008 706 603 445 309 
 008 813 009 052 089 
 
 008 919 388 595 035 
 
 009 025 742 086 910 
 
 1.00875 
 
 1.009 
 
 1.00925 
 
 1.0095 
 
 1.00975 
 
 003 783 547 730 127 
 003 891 166 236 911 
 
 003 998 758 082 352 
 
 004 106 323 279 658 
 004 213 861 842 026 
 
 1.02125 
 
 1.0215 
 
 1.02175 
 
 1.022 
 
 1.02225 
 
 009 132 069 540 472 
 009 238 370 968 466 
 009 344 646 383 631 
 009 450 895 798 694 
 009 557 119 226 374 
 
 1.01 
 
 1.01025 
 
 1.0105 
 
 1.01075 
 
 1.011 
 
 004 321 373 782 643 
 004 428 859 114 686 
 004 536 317 851 323 
 004 643 750 005 712 
 004 751 155 591 001 
 
 1.0225 
 
 1.02275 
 
 1.023 
 
 1.02325 
 
 1.0235 
 
 009 663 316 679 379 
 009 769 488 170 411 
 009 875 633 712 160 ' 
 
 009 981 753 317 307 
 
 010 087 846 998 524 
 
 1.01125 
 
 1.0115 
 
 1.01175 
 
 1.012 
 
 1.01225 
 
 004 858 534 620 329 
 
 004 965 887 106 823 
 
 005 073 213 063 604 
 005 180 512 503 780 
 005 287 785 440 451 
 
 1.02375 
 
 1.024 
 
 1.02425 
 
 1.0245 
 
 1.02475 
 
 010 193 914 768 475 
 010 299 956 639 812 
 010 405 972 625 180 
 010 511 962 737 214 
 010 617 926 988 539 
 
 1.0125 
 
 1.01275 
 
 1.013 
 
 1.01325 
 
 1.0135 
 
 005 395 031 886 706 
 005 502 251 855 626 
 005 609 445 360 280 
 005 716 612 413 731 
 005 823 753 029 028 
 
 1.025 
 1 . 02525 
 1.0255 
 1.02575 
 1.026 
 
 010 723 865 391 773 
 010 829 777 959 522 
 010 «35 664 704 385 • 
 Oil 041 525 638 950 
 Oil 147 360 775 797 
 
TABLE OF INTEREST RATIOS— 
 Continued 
 
 TABLE OF SUB-RECIPROCALS 
 
 (Art. 51) 33 
 
 1 + i 
 
 Logarithm 
 
 Initial Figures 
 
 Sub-reciprocal 
 
 1.02625 
 
 1.0265 
 
 1.02675 
 
 1.027 
 
 1.02725 
 
 Oil 253 170 127 497 
 Oil 358 953 706 611 
 on 464 711 525 690 
 Oil 570 443 597 278 
 Oil 676 149 933 909 
 
 10 
 11 
 12 
 13 
 14 
 
 90 
 83 
 76 
 71 
 
 66 
 
 1.0275 
 
 1.02775 
 
 1.028 
 
 1.02825 
 
 1.0285 
 
 Oil 781 830 548 107 
 Oil 887 485 452 387 
 Oil 993 114 659 257 
 012 098 718 181 213 
 012 204 296 030 743 
 
 15 
 
 16 ' 
 17 
 18 
 19 
 
 62 
 58 
 55 
 52 
 50 
 
 1 . 02875 
 1.029 
 1 . 02925 
 1.0295 
 1.02975 
 
 012 309 848 220 326 
 012 415 374 762 433 
 012 520 875 669 524 
 012 626 350 954 050 
 012 731 800 628 455 
 
 20 
 21 
 22 
 23 
 24 
 
 47 
 45 
 43 
 41 
 40 
 
 1.03 
 
 1.0305 
 
 1.031 
 
 1.0315 
 
 1.032 
 
 012 837 224 705 172 
 
 013 047 996 115 232 
 013 258 665 283 517 
 013 469 232 309 170 
 
 • 013 679 697 291 193 
 
 25 
 26 
 
 27 
 28 
 29 
 
 38 
 37 
 35 
 34 
 33 
 
 1.0325 
 1.033 
 1 . 0335 
 1.034 
 1.0345 
 
 013 890 060 328 439 
 
 014 100 321 519 621 
 014 310 480 963 307 
 014 520 538 757 924 
 014 730 495 001 753 
 
 30 
 31 
 32 
 33 
 34 
 
 32 
 31 
 30 
 29 
 
 28 
 
 1.035 
 1 . 0355 
 1.036 
 1.0375 
 1.038 
 
 014 940 349 792 937 
 
 015 150 103 229 471 
 015 359 755 409 214 
 
 015 988 105 384 130 
 
 016 197 353 512 439 
 
 35-36 
 
 37 
 38-39 
 
 40 
 41-42 
 
 27 
 26 
 25 
 24 
 23 
 
 1.039 
 1.04 
 1.041 
 1 . 0425 
 1 . 043 
 
 016 615 547 557 177 
 
 017 033 339 298 780 
 
 017 450 729 510 536 
 
 018 076 063 645 795 
 018 284 308 426 531 
 
 43-44 
 45-46 
 47-49 
 50-51 
 52-54 
 
 22 
 21 
 20 
 19 
 18 
 
 1.044 
 
 1.045 
 
 1.046 
 
 1.0475 
 
 1.048 
 
 018 700 498 666 243 
 
 019 116 290 447 073 
 
 019 531 684 531 255 
 
 020 154 031 638 333 
 020 361 282 647 708 
 
 55-57 
 58-61 
 62-65 
 66-70 
 71-75 
 
 17 
 16 
 15 
 14 
 13 
 
 1.049 
 
 1.05 
 
 1.055 
 
 1.06 
 
 1.065 
 
 020 775 488 193 558 
 
 021 189 299 069 938 
 023 252 459 633 711 
 025 305 865 264 770 
 027 349 607 774 757 
 
 76-82 
 
 83-89 
 
 90 
 
 12 
 
 11 
 
 1 
 
 1.07 
 
 1.075 
 
 1.08 
 
 1.09 
 
 1.10 
 
 029 383 777 685 210 
 031 408 464 251 624 
 033 423 755 486 950 
 037 426 497 940 624 
 041 392 685 158 225 
 
 * 
 
 •• 
 
34 
 
 TABLE OF MULTIPLES 
 
 1 
 
 2 
 
 3 
 
 
 4 
 
 5 
 
 6 
 
 
 7 
 
 8 
 
 9 
 
 001 
 
 002 
 
 003 
 
 
 004 
 
 005 
 
 006 
 
 
 007 
 
 008 
 
 009 
 
 002 
 
 004 
 
 006 
 
 
 008 
 
 010 
 
 012 
 
 
 014 
 
 016 
 
 018 
 
 003 
 
 006 
 
 009 
 
 
 012 
 
 015 
 
 018 
 
 
 021 
 
 024 
 
 027 
 
 004 
 
 008 
 
 012 
 
 
 016 
 
 020 
 
 024 
 
 
 028 
 
 032 
 
 036 
 
 005 
 
 010 
 
 015 
 
 
 020 
 
 025 
 
 030 
 
 
 035 
 
 040 
 
 045 
 
 006 
 
 012 
 
 018 
 
 
 024 
 
 030 
 
 036 
 
 
 042 
 
 048 
 
 054 
 
 007 
 
 014 
 
 021 
 
 
 028 
 
 035 
 
 042 
 
 
 049 
 
 056 
 
 003 
 
 008 
 
 016 
 
 024 
 
 
 032 
 
 040 
 
 048 
 
 
 056 
 
 064 
 
 072 
 
 009 
 
 018 
 
 027 
 
 
 036 
 
 045 
 
 054 
 
 
 063 
 
 072 
 
 081 
 
 010 
 
 020 
 
 030 
 
 
 040 
 
 050 
 
 060 
 
 
 070 
 
 080 
 
 090 
 
 Oil 
 
 022 
 
 033 
 
 
 044 
 
 055 
 
 066 
 
 
 077 
 
 088 
 
 099 
 
 012 
 
 024 
 
 036 
 
 
 048 
 
 060 
 
 072 
 
 
 084 
 
 096 
 
 108 
 
 013 
 
 026 
 
 039 
 
 
 052 
 
 065 
 
 078 
 
 
 091 
 
 104 
 
 117 
 
 014 
 
 028 
 
 042 
 
 
 056 
 
 070 
 
 084 
 
 
 098 
 
 112 
 
 126 
 
 015 
 
 030 
 
 045 
 
 
 060 
 
 075 
 
 090 
 
 
 105 
 
 120 
 
 135 
 
 016 
 
 032 
 
 048 
 
 
 064 
 
 080 
 
 096 
 
 
 112 
 
 128 
 
 144 
 
 017 
 
 034 
 
 051 
 
 
 068 
 
 085 
 
 102 
 
 
 119 
 
 136 
 
 153 
 
 018 
 
 036 
 
 054 
 
 
 072 
 
 090 
 
 108 
 
 
 126 
 
 144 
 
 162 
 
 019 
 
 038 
 
 057 
 
 
 076 
 
 095 
 
 114 
 
 
 133 
 
 152 
 
 171 
 
 020 
 
 040 
 
 060 
 
 
 080 
 
 100 
 
 120 
 
 
 140 
 
 160 
 
 180 
 
 021 
 
 042 
 
 063 
 
 
 084 
 
 105 
 
 126 
 
 
 147 
 
 168 
 
 189 
 
 022 
 
 044 
 
 066* 
 
 
 088 
 
 110 
 
 132 
 
 
 154 
 
 176 
 
 198 
 
 023 
 
 046 
 
 069 
 
 
 092 
 
 115 
 
 138 
 
 
 161 
 
 184 
 
 207 
 
 024 
 
 048 
 
 072 
 
 
 096 
 
 120 
 
 144 
 
 
 168 
 
 192 
 
 210 
 
 025 
 
 050 
 
 075 
 
 
 100 
 
 125 
 
 150 
 
 
 175 
 
 200 
 
 225 
 
 026 
 
 052 
 
 078 
 
 
 104 
 
 130 
 
 156 
 
 
 182 
 
 208 
 
 234 
 
 027 
 
 054 
 
 081 
 
 
 108 
 
 135 
 
 162 
 
 
 189 
 
 216 
 
 243 
 
 028 
 
 056 
 
 084 
 
 
 112 
 
 140 
 
 168 
 
 
 196 
 
 224 
 
 252 
 
 029 
 
 058 
 
 087 
 
 
 116 
 
 145 
 
 174 
 
 
 203 
 
 232 
 
 261 
 
 030 
 
 060 
 
 090 
 
 
 120 
 
 150 
 
 180 
 
 
 210 
 
 240 
 
 270 
 
 031 
 
 062 
 
 093 
 
 
 124 
 
 155 
 
 186 
 
 
 217 
 
 248 
 
 279 
 
 032 
 
 064 
 
 096 
 
 
 128 
 
 160 
 
 192 
 
 
 224 
 
 256 
 
 288 
 
 033 
 
 066 
 
 099 
 
 
 132 
 
 165 
 
 198 
 
 
 231 
 
 264 
 
 297 
 
 034 
 
 068 
 
 102 
 
 
 136 
 
 170 
 
 204 
 
 
 238 
 
 272 
 
 306 
 
 035 
 
 070 
 
 105 
 
 
 140 
 
 175 
 
 210 
 
 
 245 
 
 280 
 
 315 
 
 036 
 
 072 
 
 108 
 
 
 ■ 144 
 
 180 
 
 216 
 
 
 252 
 
 288 
 
 324 
 
 037 
 
 074 
 
 111 
 
 
 148 
 
 185 
 
 222 
 
 
 259 
 
 296 
 
 333 
 
 038 
 
 076 
 
 114 
 
 
 152 
 
 190 
 
 228 
 
 
 266 
 
 304 
 
 342 
 
 039 
 
 078 
 
 117 
 
 
 156 
 
 195 
 
 234 
 
 
 273 
 
 312 
 
 351 
 
 040 
 
 080 
 
 120 
 
 
 160 
 
 200 
 
 240 
 
 
 280 
 
 320 
 
 360 
 
 041 
 
 082 
 
 123 
 
 
 164 
 
 205 
 
 246 
 
 
 287 
 
 328 
 
 369 
 
 042 
 
 084 
 
 126 
 
 
 168 
 
 210 
 
 252 
 
 * 
 
 294 
 
 336 
 
 378 
 
 043 
 
 086 
 
 129 
 
 
 172 
 
 215 
 
 258 
 
 
 301 
 
 344 
 
 387 
 
 044 
 
 088 
 
 132 
 
 
 176 
 
 220 
 
 264 
 
 
 308 
 
 352 
 
 396 
 
 045 
 
 090 
 
 135 
 
 
 180 
 
 225 
 
 270 
 
 
 315 
 
 360 
 
 405 
 
 046 
 
 092 
 
 138 
 
 
 184 
 
 230 
 
 276 
 
 
 322 
 
 368 
 
 414 
 
 047 
 
 094 
 
 141 
 
 
 188 
 
 235 
 
 282 
 
 
 329 
 
 376 
 
 423 
 
 048 
 
 096 
 
 144 
 
 
 192 
 
 240 
 
 288 
 
 
 336 
 
 384 
 
 432 
 
 049 
 
 098 
 
 147 
 
 
 196 
 
 245 
 
 294 
 
 
 343 
 
 392 
 
 441 
 
TABLE OF MULTIPLES— Continued 
 
 35 
 
 1 
 
 2 
 
 3 
 
 
 4 
 
 5 
 
 6 
 
 
 7 
 
 8 
 
 9 
 
 050 
 
 100 
 
 150 
 
 
 200 
 
 250 
 
 300 
 
 
 350 
 
 400 
 
 450 
 
 051 
 
 102 
 
 153 
 
 
 204 
 
 255 
 
 306 
 
 
 357 
 
 408 
 
 459 
 
 052 
 
 104 
 
 156 
 
 
 208 
 
 260 
 
 312 
 
 
 364 
 
 416 
 
 468 
 
 053 
 
 106 
 
 150 
 
 
 212 
 
 265 
 
 318 
 
 
 371 
 
 424 
 
 477 
 
 054 
 
 108 
 
 162 
 
 
 216 
 
 270 
 
 324 
 
 
 378 
 
 432 
 
 486 
 
 055 
 
 110 
 
 165 
 
 
 220 
 
 275 
 
 330 
 
 
 385 
 
 440 
 
 495 
 
 056 
 
 112 
 
 168 
 
 
 224 
 
 280 
 
 336 
 
 
 392 
 
 448 
 
 504 
 
 057 
 
 114 
 
 171 
 
 
 228 
 
 285 
 
 342 
 
 
 399 
 
 456 
 
 513 
 
 058 
 
 116 
 
 174 
 
 
 232 
 
 290 
 
 348 
 
 
 406 
 
 464 
 
 522 1 
 
 059 
 
 118 
 
 177 
 
 
 236 
 
 295 
 
 354 
 
 
 413 
 
 472 
 
 531 
 
 060 
 
 120 
 
 180 
 
 
 240 
 
 300 
 
 360 
 
 
 420 
 
 480 
 
 540 
 
 061 
 
 122 
 
 183 
 
 
 244 
 
 305 
 
 306 
 
 
 427 
 
 488 
 
 549 
 
 002 
 
 124 
 
 186 
 
 
 248 
 
 310 
 
 372 
 
 
 434 
 
 496 
 
 558 
 
 063 
 
 126 
 
 189 
 
 
 252 
 
 315 
 
 378 
 
 
 441 
 
 504 
 
 567 i 
 
 064 
 
 128 
 
 192 
 
 
 256 
 
 320 
 
 384 
 
 
 448 
 
 512 
 
 576 
 
 065 
 
 130 
 
 195 
 
 
 260 
 
 325 
 
 390 
 
 
 455 
 
 ^20 
 
 585 
 
 066 
 
 132 
 
 198 
 
 
 264 
 
 330 
 
 396 
 
 
 462 
 
 528 
 
 594 
 
 067 
 
 134 
 
 201 
 
 
 268 
 
 335 
 
 402 
 
 
 469 
 
 536 
 
 603 i 
 
 068 
 
 136 
 
 204 
 
 
 272 
 
 340 
 
 408 
 
 
 476 
 
 544 
 
 612 •! 
 
 069 
 
 138 
 
 207 
 
 
 276 
 
 345 
 
 414 
 
 
 483 
 
 552 
 
 621 ' 
 
 070 
 
 140 
 
 210 
 
 
 280 
 
 350 
 
 420 
 
 
 490 
 
 560 
 
 630 
 
 071 
 
 142 
 
 213 
 
 
 284 
 
 355 
 
 426" 
 
 
 497 
 
 568 
 
 639 1 
 
 072 
 
 144 
 
 216 
 
 
 288 
 
 360 
 
 432 
 
 
 504^ 
 
 576 
 
 648 1 
 
 073 
 
 146 
 
 219 
 
 
 292 
 
 365 
 
 438 
 
 
 511 
 
 584 
 
 657 
 
 074 
 
 148 
 
 222 
 
 
 296 
 
 370 
 
 444 
 
 
 518 
 
 592 
 
 666 
 
 075 
 
 150 
 
 225 
 
 
 300 
 
 375 
 
 450 
 
 
 525 
 
 600 
 
 675 
 
 076 
 
 152 
 
 228 
 
 
 304 
 
 380 
 
 456 
 
 
 532 
 
 608 
 
 684 
 
 077 
 
 154 
 
 231 
 
 
 308 
 
 385 
 
 462 
 
 
 539 
 
 616 
 
 693 
 
 078 
 
 156 
 
 234 
 
 
 312 
 
 390 
 
 468 
 
 
 546 
 
 624 
 
 702 
 
 079 
 
 158 
 
 237 
 
 
 316 
 
 395 
 
 474 
 
 
 553 
 
 632 
 
 711 
 
 080 
 
 160 
 
 240 
 
 
 320 
 
 400 
 
 480 
 
 
 500 
 
 640 
 
 720 
 
 081 
 
 162 
 
 243 
 
 
 324 
 
 405 
 
 486 
 
 
 567 
 
 048 
 
 729 
 
 082 
 
 164 
 
 246 
 
 
 328 
 
 410 
 
 492 
 
 
 574 
 
 656 
 
 738 
 
 083 
 
 166 
 
 249 
 
 
 332 
 
 415 
 
 498 
 
 
 581 
 
 664 
 
 747 
 
 084 
 
 168 
 
 252 
 
 
 336 
 
 420 
 
 504 
 
 
 588 
 
 672 
 
 756 
 
 085 
 
 170 
 
 255 
 
 
 340 
 
 425 
 
 510 
 
 
 595 
 
 680 
 
 765 
 
 086 
 
 172 
 
 258 
 
 
 344 
 
 430 
 
 516 
 
 
 602 
 
 688 
 
 774 
 
 087 
 
 174 
 
 261 
 
 
 348 
 
 435 
 
 522 
 
 
 609 
 
 696 
 
 783 
 
 088 
 
 176 
 
 264 
 
 
 352 
 
 440 
 
 528 
 
 
 616 
 
 704 
 
 792 
 
 089 
 
 178 
 
 267 
 
 
 356 
 
 445 
 
 534 
 
 
 623 
 
 712 
 
 801 
 
 090 
 
 180 
 
 270 
 
 
 360 
 
 450 
 
 540 
 
 
 630 
 
 720 
 
 810 
 
 091 
 
 182 
 
 273 
 
 
 364 
 
 455 
 
 546 
 
 
 637 
 
 728 
 
 819 
 
 092 
 
 184 
 
 276 
 
 
 368 
 
 460 
 
 552 
 
 
 644 
 
 736 
 
 828 
 
 093 
 
 186 
 
 279 
 
 
 372 
 
 465 
 
 558 
 
 
 651 
 
 744 
 
 837 
 
 094 
 
 188 
 
 282 
 
 
 376 
 
 470 
 
 564 
 
 
 658 
 
 752 
 
 846 
 
 095 
 
 190 
 
 285 
 
 
 380 
 
 475 
 
 570 
 
 
 665 
 
 760 
 
 855 
 
 096 
 
 192 
 
 288 
 
 
 384 
 
 480 
 
 576 
 
 
 672 
 
 768 
 
 864 
 
 097 
 
 194 
 
 291 
 
 
 388 
 
 485 
 
 582 
 
 
 679 
 
 776 
 
 873 
 
 098 
 
 196 
 
 294 
 
 
 392 
 
 490 
 
 588 
 
 
 686 
 
 784 
 
 882 
 
 099 
 
 198 
 
 297 
 
 
 396 
 
 495 
 
 594 
 
 
 693 
 
 792 
 
 891 
 
36 
 
 SPECIMEN OF RULED PAPER 
 
 RECOMMENDED FOR USE WITH THE FOREGOING TABIvES. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 1 1 
 
 i 1 
 
 
 
 
 
 . i 
 
 i 1^ 
 
 ! 1 
 
 1 
 
 
 
 
 
 ■ 
 
 
 
 
 
 j. i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 - 
 
 
 
 
 — 
 
 — 
 
 — 
 
 __ 
 
 
 ! 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 ' 
 
 
 
 
 
 ! 
 
 
 1 
 
 i 
 
 
 
 
 i 
 
 
 
 
 T 
 
 
 
 
 
 
 
 i 
 
 ! 
 1— 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 1 i 
 
 
 
 
 
 
 
 i 
 
 
 
 
 r •: 
 
 i 
 
 
 
 
 
 
 
 
 
 j 
 
 i 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 T - 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 1 ! 
 
 i 
 
 
 7"! ■ 
 1 ■ 
 
 
 
 
 
 
 
 
 
 
 1 
 t 
 
 
 ! 1 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 1 : 
 i i 
 i i 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 .,J_„_ 
 
 
 
 
 
 
 
 
 
 
 I 
 ! 
 
 
 
 
 
 - 
 
 
 
 
 i 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 .-,_,.. j.._.,^-_ 
 
 
 ■ r -! 
 
 
 
 
 
 
 
 
 
 
 
PART III. 
 
 The Doctrin of Interest 
 
PART III. 
 
 THE DOCTRIN OF INTEREST. 
 
 Interest. 
 
 67. — Interest, mathematically considerd, is the increase 
 of an indettedness by lapse of time. The rate of such increase 
 varies with circumstances, * and is subject to bargaining ; the 
 resulting contract, exprest or implied, must embody the fol- 
 lowing terms: 
 
 Principal. The number of units of value (dollars, pounds, 
 francs, marks, etc.,) originally loand or invested. 
 
 Interest Rate. The fraction which is added to each 
 unit by the lapse of one unit of time ; usually a small decimal. 
 
 Frequency. The length of the unit of time, measured 
 in years, months or days. 
 
 Time. The number of units of time during which the 
 indettedness is to continue. 
 
 68. — As each dollar increases just as much as every other 
 dollar, it is best at first to consider the principal as one dollar 
 and when the proper function thereof has been calculated, 
 to multiply it by the number of dollars. 
 
 69. — The interest rate is usually spoken of as so much 
 percent per period or term. "6%' per annum" means an 
 increase of . 06 for each term of a year. We will designate 
 the interest rate by the letter i ; as, ? = .06. At the end of 
 one term the increast indettedness is 1 -f /, (1.06), a very 
 important quantity in computation. 
 
 * For discussion of the causes for higher or lower interest rates, see 
 The Rate of Interest, by Prof. Irving Fisher. 
 
40 The Doctrin op Interest. 
 
 70. — Punctual Interest. The usual contract is that the 
 increase shall be paid off in cash at the end of each period, 
 restoring the principal to its original quantity. Let c denote 
 the cash payment ; then 1 + z — ^r = 1 ; and the second term 
 would repeat the same process. The payment of cash for 
 interest must not be regarded as the interest : it is a cancel- 
 lation of part of the increast principal. Many persons, and 
 even courts, have been misled by the old definition of interest, 
 "money paid for the use of money," into treating uncollected 
 or unmatured interest as a nullity, tho secured precisely in the 
 same way as the principal. 
 
 71. — But the interest money may not be paid exactly at 
 the end of each term, either in violation of the contract or by 
 a special clause permitting it to run on, or by the det being 
 assigned to a third party at a price which modifies the true 
 interest rate. In this case the question arises : how shall the 
 interest be computed for the following periods ? This gives 
 rise to a distinction between simple and compound interest. 
 
 72. — Simple Interest. During the second period, altho 
 the borrower has in his hands an increast principal, 1 + i, he 
 is at simple interest only charged with interest on 1, and has 
 the free use of /, which tho small has an earning power pro- 
 portionate to that of 1. His indettedness at the end of the 
 second term is 1 + 2/, and thereafter 1 + 3/, 1 -f- 4/, etc. 
 After the first period he is not charged with the agreed per- 
 centage of the sum actually employed by him, and this to the 
 detriment of the creditor. For any scientific calculation, 
 simple interest is impossible of application. 
 
 73.— Compound Interest. The indettedness at the end 
 of the first period is 1 H- /, and up to this point punctual, simple 
 and compound interest coincide. But in compound interest the 
 fact is recognized that the increast principal, 1 -f z, is all sub- 
 ject to interest during the next period, and that the det 
 increases by geometrical progression, not arithmetical. The 
 increase from 1 to 1 + z is regarded, not as an addition of i to 
 1, but as a multiplication of 1 by the ratio of increase 
 (1 -f z) . We shall designate the ratio of increase by r when 
 convenient, altho this is merely an abbreviation of 1 H- z, and 
 the two expressions are at all times interchangeable. 
 
Thk Amount. 41 
 
 74. — For the second period, 1 + / is the actual and 
 equitable principal, and it should be again increast in the ratio 
 1 -j- i. The total indettedness at the end of the second period 
 is therefore 1 x (1 + x (1 + z) = (1 + 0' = r\ At 
 the end of the third period it will have become r^, and at the 
 end of term No. /, r^. 
 
 Thb Amount. 
 
 75. — The sum to which $1 will have increast at compound 
 interest at ^ (or 100/ per cent.) in / periods, is called the 
 Amount, and will be designated as s. We then have the 
 following equation: 
 
 5 = rt= (1 + iy 
 
 76. — To find the amount of one dollar, raise the 
 ratio to a power whose exponent is the number of 
 periods. 
 
 77. — The logarithm of the ratio of increase is the most 
 important logarithm for interest calculations. If the interest 
 rate does not exceed two figures, the logarithm will be found 
 in full in col. B, TF. For convenience we will designate it by 
 a capital letter L. Thus, if i = .065, L will be found opposit 
 Q^ in B. If i = .065 ; log. r = I, = .027 349 607 774,8. 
 
 78. — As powers are found by multiplying the logarithm, 
 L must be multiplied by i. 
 
 r^ nl tl, 
 
 79. — To find the amount, multiply the logarithm of 
 the ratio by the number of periods, and the correspond- 
 ing number will be the amount of $1. 
 
 80. — Let the interest rate be 3.5%' per annum, payable 
 annually, what will be the amount of $1 at the end of 100 
 years? Turning to col. B, TF, we find opposit B 35, (or 
 1.035) the logarithm .014 940 349 792,9. 
 
 1.= .014 940 349 792,9 
 
 / = 100 /L = 1 . 494 034 979 29 
 
 From the characteristic 1, it appears that the amount will be 
 in the tens of dollars ; and as the decimal part of the logarithm 
 is a little more than that which is opposit 31 we know that the 
 amount is $31 and some cents. Thus a rough idea of the 
 amount may be gaind almost instantly. 
 
42 The Doctrin of Interest. 
 
 81. — To obtain a more accurate value and one which will 
 be sufi&ciently near for a large principal, we proceed as follows: 
 
 82. — In the first place we can only obtain ten correct 
 figures from lOOL. The final figure 9 is never perfect ; it may 
 be 8 . 51 or 9 . 49 or anywhere between. We must, therefore, use 
 only eleven in the logarithm and finally get ten in the number. 
 
 (?) nl 494 034 979 29 
 
 A 31 
 
 491 361 693 83 
 
 
 2 673 285 46 
 
 B 06 
 
 2 597 980 72 
 
 
 75 304 74 
 
 C 17 
 
 73 823 79 
 
 
 1 480 95 
 
 D 34 
 
 1 476 60 
 
 
 4 35 
 
 E 10 
 
 4 34 
 
 F 2 
 
 1 
 
 
 31 
 
 B 06 
 
 186 
 
 
 31186 .... 
 
 C 1 
 
 31186 
 
 7 
 
 2 18 3 2 
 
 
 3119130162 
 
 D 3 
 
 9357 
 
 4 
 
 1248 
 
 
 3119140767 
 
 E 10 
 
 3 1 
 
 
 3 119 140 798 = 5 (/ 100, z .03) 
 
 83. — In order to give accurate results up to twelve figures 
 for one hundred interest terms, we have provided on page 32 
 a special table of the logarithms of the 150 interest ratios (1 + i) 
 which most frequently occur, calculated to 15 places, which 
 allows two places for loss in multiplication. 
 
43 
 The Present Worth. 
 
 84. — The sum which if now invested at i will in / periods 
 amount to $1 is evidently less than $1. It is in the same pro- 
 portion to 1 as 1 is to s. Designating the present worth by /, 
 we have 
 
 p \\ \\\\ s 
 
 ox p =— = 5~^ 
 s 
 
 or the amount and the present worth are reciprocals of each 
 
 other. 
 
 A series of amounts reads 
 
 1, r\ r% r% r*, r", etc. 
 A series of present worths reads 
 
 1, r"\ r~% r % r"*, r~% etc. 
 Reversing the latter series and connecting it with the 
 former we have a continuous series in geometrical progression: 
 r"% r"*, r~^, r"% r"\ 1, r\ r^ r^ r"*, r^ . 
 Using 1.03 as the ratio, the series becomes 
 
 f' .86260878 
 
 r-^ .88848705 
 
 r-« .91514166 
 
 r"' .94259591 
 
 r' .97087379 
 
 r° 1. 
 
 r* 1.03 
 
 r' 1.0609 
 
 r^ 1.092727 
 
 r* 1.12550881 
 
 r"^ 1.15927407 
 
 In this series, which might be extended indefinitly upward 
 and downward, every term is a present worth of any which 
 follows it and an amount of each which precedes it. .86260878 
 is the present worth at 10 interest periods of 1 . 15927407 ; 
 1.12550881 is the amount at eight periods of .88848705. 
 
 85. — If any term be multiplied by 1.03, the product will 
 be the next following term ; if it be divided by 1.03 or (which 
 is the same thing) be multiplied by .97087379, the product 
 will be the next preceding term. 
 
44 I^HB DocTRiN OF Interest. 
 
 86. — To find the logarithm of the present worth, subtract 
 the logarithm of the amount (for the same time) from zero. 
 
 In the preceding example, but using L from the 15 place 
 table 
 
 s nl /fL = 1.494 034 979 293,7 
 
 p=^\ls nl —/L = 2.505 965 020 706,3 
 
 A 32 505 149 978 319,9 
 
 815 042 386,4 
 
 B 01 434 077 479,3 
 
 380,964 907,1 
 
 C 87 377 671935.8 
 
 3 292 971,3 
 
 D 75 3 257 196,4 
 
 35 774,9 
 
 K 82 35 612.1 
 
 162,8 
 F 37 160,7 
 
 G 49 2,1 
 
 32 
 3 01 032 
 
 3 2 032 . . . . 
 C 8 256256 
 
 7 224224 
 
 32 0-5 986784. . . 
 D 7 22441907 
 
 5 1602993 
 
 3206010828900 
 
 E 8 256481 
 
 2 6412 
 
 F 3 962 
 
 7 22 
 
 G 4 2 
 
 9 
 
 3206011092779 
 ,0 32060110928=;^ (1.035)^°° to 12 places. 
 
The Compound Interest and Discount. 45 
 
 87. — That the amount and the present worth are correct 
 reciprocals may be tested by multiplying them together. 
 Taking a few figures of each we have 
 
 31 ■ 19 14 
 .03 
 
 2 
 
 06 
 01 
 
 935742 
 
 62383 
 
 1871 
 
 1 
 
 1 . 00000 
 Every pair of reciprocals gives a product of 1. 
 
 The Compound Interest and Discount. 
 
 88. — We have hitherto used the word "interest" 
 abstractly as denoting that force or principle which effects the 
 increase of the amount of an indettedness as time goes on. The 
 interest-increment which is thus added is also frequently called 
 " the Interest," which may be v/ritten with a capital letter. 
 
 89. — If we take the original principal away from the 
 amount, we evidently have the Interest. For a single period 
 
 2 = 1 + z _ 1 = r — 1. 
 
 When there are more than one period it is the compound 
 Interest, obtaind in the same way and represented by a 
 capital I = (1 + z)^ — 1 = ?-* — 1 == .S — 1. 
 
 Thus the compound Interest of $1 at 3% per period for 
 100 periods is $31.19 — 1.00 = $30.19. For two periods it 
 is 1.0609 — 1 = 0.0609. 
 
 90. — In the opposit case of a present worth there is a 
 diminution of the principal. The present worth of $1 at 3 per 
 cent, one period, is .97087379; the Discount is not .03, but 
 .02912621, the true principal being not $1, but .97087379, 
 which X . 03 = . 02912621. Representing the simple Discount 
 by d, we have d —\ — p = i y, p = i/s. 
 
 91. — If there be more than one term involvd, it is com- 
 pound Discount, which will be represented by D. Thus, at 
 3 per cent for 5 periods D = 1 — .86261 = .13739. D is 
 also the present worth of the compound Interest for the same 
 time. .15927 x .86261 = .13739. 
 
46 The Doctrin of Interest. 
 
 In general D = l — p — \p = 1/s, 
 
 92.— Thus we see that the variance from par (|1) is 
 called compound Interest or compound Discount, according 
 as regarded from the past or the future point of view and that 
 their properties are as follows: 
 
 D ^ 1 — ^t 
 and their relation is D = /I ; or I = ^D. 
 
 Finding Time or Rate. 
 
 93.— 'By time we mean the number of periods, terms or 
 intervals, and by this number the logarithm of the interest- 
 ratio is multiplied to produce the logarithm of the amount. 
 
 (/ X L) /« 5 
 / X log (1 + i) = log s 
 
 94. — If the amount is known and the rate, but the num- 
 ber of periods unknown, we can transform the above equation 
 into this: 
 
 log s 
 
 h 
 
 95. — At .03 interest, in how many periods will $1 amount 
 to $2, or how long will it take a sum to double itself ? 
 logs = log2 = .3010299956640 
 I, = log 1.03= .0128372247052 
 
 Using only seven places 
 
 . 0128372) . 3010300 (23.47 
 
 2 5 6 7 4 4 
 
 442860 
 
 385116 
 
 57744 
 
 49349 
 
 8395 
 
 The money will double in 24 periods, as it is not quite 
 doubled at 23. 
 
 96.' — How many periods must a det of $1 be deferd to be 
 worth now 30 cents, at 3>^% ? 
 
Finding Time or Rate. 47 
 
 Lo^ 1.035 = .01494035 
 Log 1.035-' = 1.98505965 
 Log- .30 = 1.47712125 
 
 98 5 05965) 1.47712125( 
 
 — .01494035)— .52287875(34.9997 
 
 4 4 8 2 10 5 
 7466825 
 5976140 
 1490685 
 1344631 
 146054 
 134463 
 11591 
 Practically 35 periods. 
 For convenience in division, the minus sign is made to 
 extend over the entire logarithms. Then, as both divisor and 
 dividend are of the same sign, the quotient is positiv. 
 
 97. — If the rate be unknown, the equation/ x log. (1 -|- z) 
 = log. s may again be transformed to 
 
 log.(l + 0='-^. 
 
 98. — 20 periods having elapst and the amount of $1 being 
 now $3.20713547, what is the rate? 
 
 log. 3.20713547 = .506117303 
 .506117303/20 = .025305865 In 1.06 
 1 + 1 = 1.06.-. z= .06 
 
 The Annuity. 
 99. — We have now investigated the two fundamental 
 problems in compound interest : viz. , to find the amount of a 
 present worth, and to find the present worth of an amount. 
 The next question is a more complex one : to find the amount 
 and the present worth of a series of payments. If these pay- 
 ments are irregular as to time, amount and rate of interest, the 
 only way is to make as many separate computations as there 
 are sums and then add them together. But if the sums, times 
 and rate are uniform, we can devise a method for finding the 
 amount or present worth at one operation. 
 
48 The Doctrin of Interest. 
 
 100. — Annuity. A series of payments of like amount, 
 made at regular periods, is called an annuity, even though the 
 period be not annual, but a half year, a quarter or any other 
 length of time. Thus, if an agreement is made for the follow- 
 ing payments: 
 
 On Sept. 9 1904 |5100. 
 
 On March 9 1905 100. 
 
 On Sept. 9 1905 100. 
 
 and on March 9 1906 100. 
 
 this would be an annuity of $100 per period, terminating after 
 4 periods. It is required to find on March 9, 1904, assuming 
 the rate of interest as 3% per period : First, what will be the 
 total amount to which the annuity will have accumulated on 
 March 9, 1906 ; second, what is now, on March 9, 1904, the 
 present worth of this series of future sums ? It is evident that 
 the answer to the first question will be greater than ;^400, and 
 that the answer to the second question will be less than $400. 
 
 Amount of an Annuity. 
 
 101. — It is easy, in this case, to find the separate amounts 
 of the payments, for the number of terms is very small, and 
 we have already computed the corresponding values of $1.00. 
 The last $100 will have no accumulation, and will 
 
 be merely $100. 
 
 The third $100 will have earned in one period |3.00, 
 
 and will amount to 103 . 
 
 The second $100 will amount to 106.09 
 
 The first $100 (rounded off at cents) will amount to 109 . 27 
 
 and the total amount will be $418.36 
 
 102. — If, however, there were 50 terms instead of 4, the 
 work of computing these 50 separate amounts, even by the use 
 of logarithms, would be very tedious. 
 
 103. — Let us write down the successiv amounts of $1.00 
 
 under one another: 
 
 a 
 
 Amounts of $1. 
 1.00 
 1.03 
 1.0609 
 1.092727 
 
Amoun'T of an Annuity. 49 
 
 104. — Now, as we have the right to take any principal we 
 choose and multiply it by the number indicating the value of 
 $1.00, let us assume one dollar and three cents, and multiply 
 each of the above figures by 1.03, setting the products in a 
 second colum: 
 
 a. b. c. 
 
 Amounts of %\ . 00 Amounts of %\ . 03 Amounts of |0 . 03 
 
 1.00 1.03 
 
 1.03 1.0609 
 
 1.0609 1.092727 
 
 1.092727 1.12550881 
 
 105. — Our object in doing this was by subtracting colum 
 a from b to find the amount of an annuity of three cents. 
 Before subtracting, we have the right to throw out any num- 
 bers which are identical in the two colums. Expunging these 
 like quantities, we have left only the following: 
 
 a. b. c. 
 
 Annuity of $1 . 00 Annuity of $1 . 03 Annuity of |0 . 03 
 
 1.00 1.12550881 
 
 less 1.00 
 
 1.12550881 Amount 0.12550881 
 
 That is, an annuity of three cents will amount, under the 
 conditions assumed, to twelve cents and the decimal 550881. 
 Therefore, an annuity of one cent will amount to one-third of 
 .12550881 or .04183627. An annuity of ^1.00 will amount 
 to 100 times as much, or ;^4. 183627, which agrees exactly 
 with the result obtained by addition, in Article 45. 
 
 106.— The number .12550881 ( obtaind by subtracting 
 1 .00 from 1 . 12550881) is actually the compound Interest for the 
 given rate and time, and the number .03 is the single Interest) 
 the amount of the annuity of ^1.00 is .12550881 -f- .03 = 
 4.183627. This suggests another way of looking at it. The 
 compound Interest up to any time is really the amount of a 
 smaller annuity, one of three cents instead of a dollar, con- 
 structed on exactly the same plan, and used as a model. 
 
 107.— Rule. To find the amount of an annuity of ^1.00 
 for a given time and rate, divide the compound Interest by a 
 single Interest, both exprest decimally . 
 
50 The Doctrin of Interest. 
 
 108. — Let S and P represent the amount and the present 
 worth, not of a single $1.00, but of an annuity of $1, then 
 S = I -f- /. 
 
 Exprest in symbols the reasoning would be this: 
 
 Amount of annuity of 1 = r*-^ -f r^-{- r^ + r + 1 (a) 
 
 Multiplying by r. 
 
 Amount of annuity oi 1 -\- i = r^ -\- r^'^ . .r* + r^-\-r^ ■{• r {b) 
 
 Subtracting (a) r^-'^ r^ + ^M- r + 1 
 
 Amount of annuity of / — r^ — 1 (^:) 
 
 = r^ — 1 = I 
 Amount of annuity of 1 = S = 1/i 
 
 109. — If the number of periods were 50, insted of 4, the 
 advantage of this process, with the use of logarithms, will be 
 very evident. 
 
 The rate being .03, the logarithm of the ratio, or 
 L = .012 837 224 705 172 
 50L == .641861235 258,6 
 Factors, A 43 B 19 C 50 D 34 E 60 F 10 G 14 
 5= 4 . 38390601876 
 
 - 1 
 
 I = 3 . 38390601876 
 
 I->. 03 = 112. 796867292 = 8 
 Compare this with the diflSculty of finding the result by 
 arithmetic for even ten periods. 
 
 Present Worth of an Annuity. 
 110. — To find the present worth of an annuity, we can, of 
 course, find the present worth of each payment and add them 
 together ; but it will evidently save a great deal of labor if we 
 can derive the present worth immediately, as we have learnd 
 to do with the amount. 
 
 111. — The like course of reasoning will give us the result. 
 Take the four numbers representing the present worths of $1.00 
 at 4, 3, 2 and 1 periods respectivly, and multiply each by 1.03. 
 a. b. 
 
 Present Worth of Present Worth of 
 
 Annuity of $1 . 00 Annuity of $\ . 03 
 
 .888487 .915142 • 
 
 .915142 .942596 
 
 .942596 .970874 
 
 .970874 1.000000 
 
Present Worth of an Annuity. 51 
 
 Canceling all equivalents, we have c, 
 
 . 888487 Present Worth of 
 
 Annuity of .03 
 
 1.000000 
 
 1.000000 less .888487 
 
 .111513 
 Annuity of $1 . 00 = . 111513 -H . 03 = 3 . 71710 
 This is the same result (rounded up) as that obtaind by 
 adding column a. 
 
 112.— But .111513 is the compound discount of $100 for 
 four periods, and we therefore construct this rule: 
 
 113. — Rule. To find the present worth of an annuity of 
 $1.00 for a given time and rate, divide the compound Discount 
 for that time and rate by a single interest. Symbolically 
 P = D -f- /. We might give this the form P = S -^ 5, being 
 the present worth of the amount of the annuity. 
 
 114. — It may assist in acquiring a clear idea of the work- 
 ing of an annuity, if we analyse a series of annuity payments 
 from the point of view of the purchaser. 
 
 115.— He who invests $3.7171 at 3%, in an annuity of 4 
 periods, expects to receive at each payment, besides 3% on his 
 principal to date, a portion of that principal, and thus to have 
 
 his entire principal gradually repaid. 
 
 Principal. 
 
 His original principal is 3 . 7171 
 
 At the end of the first period he receives 1 .00, con- 
 sisting of 3% on 3.7171. .1115 
 
 and payment on principal .8885 .8885 
 
 leaving new principal 2 . 8286 
 
 (or present worth at 3 periods). 
 
 In the next instalment 1 00 
 
 there is interest on 2.8286 .0849 
 
 and payment on principal 9151 .9151 
 
 leaving new principal 1 .9135 
 
 Third instalment 1 .00 
 
 Interest .0574 
 
 on principal .9426 .9426 
 
 ^9709 
 
 Last instalment 1 .00 
 
 Interest .0291 
 
 Principal in full 9709 .9709 
 
52 
 
 Thk Doctrin of Interest. 
 
 Thus the annuitant has received interest in full on the 
 principal outstanding, and has also received the entire original 
 principal. The correctness of the basis on which we have been 
 working is corroborated. 
 
 116. — It is usual to form a schedule showing the com- 
 ponents of each instalment in tabular form. 
 
 Total 
 Instalment 
 
 Interest 
 Payments 
 
 Payments 
 
 on 
 Principal 
 
 Principal 
 Outstanding 
 
 1904 Mar. 9 
 
 1904 Sept. 9 
 
 1905 Mar. 9 
 
 1905 Sept. 9 
 
 1906 Mar. 1 
 
 1.00 
 1.00 
 1.00 
 1.00 
 
 .1115 
 .0849 
 .0574 
 .0291 
 
 .8885 
 .9151 
 .9426 
 .9709 
 
 3.7171 
 2.8286 
 1.9135 
 0.9709 
 0.0000 
 
 4.00 
 
 2829 
 
 3.7171 
 
 • 117. — The payments on principal are known as amorti- 
 zation, which may be defined as the gradual repayment of a 
 principal sum thru the operation of compound interest. It 
 differs from the ordinary compound interest in this, that the 
 new principal for each period is less than the previous one. 
 
 118. — As an example of logarithmic evaluation of an 
 annuity, take an annuity of $1, as before, for 50 periods at the 
 rate of .03 per period. At the beginning of the first period, 
 what is its present worth, or what should be paid in one sum 
 for such annuity ? 
 
 i= .03 r=1.0S nl .012 837 224 705 172 (to 15 places) 
 
 60Iy= .641861235 258,6 
 As we are discounting, not 
 accumulating, we must take _ 
 
 the cologarithm — 50 L 1 . 358 138 764 741,4 
 
 and find the number. Factors A22 B36 C82 DOS El 2 F73 G23 
 
 p = 1.03-5 0= .228 107 079 790 
 D = l — p = .771892 920 210 
 D-f-.03 =25.729 764 007 =P 
 
 This may be proved down to maturity by amortization, 
 the schedule beginning thus: 
 
 
 Instalment 
 
 Payments of 
 
 Payments on 
 
 Principal 
 
 
 Interest at 3% 
 
 Principal 
 
 Outstanding 
 
 
 
 
 25.729 764 
 
 1 
 
 1.00 
 
 .771893 
 
 .228107 
 
 25.501 657 
 
 2 
 
 1.00 
 
 .765 050 
 
 .234 950 
 
 25.266 707 
 
 3 
 
 1.00 
 
 .757 901 
 
 .242 099 
 
 25.024 608 
 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 49 
 
 1.00 
 
 .057 404 
 
 .942 596 
 
 .970 874 
 
 50 
 
 1.00 
 
 .029 126 
 
 .970 874 
 
 .000 000 
 
Special Forms of Annuity. 53 
 
 119. — It may be notist that each payment on principal, or 
 amortization for one period, is the present worth of the instal- 
 ment at the beginning of its period. From this the instalment 
 of amortization may be calculated at any point independently 
 of any other figures. Thus the payment on principal in the 
 21st instalment of $1 is the present worth of $1.00 in 30 
 periods, or . 411987 ; because at the beginning of the 21st 
 period there were 30 instalments yet to come. 
 
 120. — It will also be notist that each amortization multi- 
 plied by 1 . 03 becomes the next following, these being a series 
 of present worths ; and that thus they may be derived from 
 one another, upwards or downwards. 
 
 Speciai. Forms of Annuity. 
 
 121. — The annuities heretofore spoken of are payable at 
 the end of each period, and are the kind most frequently 
 occurring. To distinguish them from other varieties they are 
 spoken of as ordinary or immediate annuities. 
 
 122. — When the instalment (or rent) of the annuity is 
 payable at the beginning of the period, it is called an annuity 
 due, altho ''prepaid" would seem more natural. It is evi- 
 dent that this is merely a question of dating. The instalments 
 compared with those in Art. 103 are as follows: 
 
 / 
 
 Immediate 
 Annuity 
 4 Periods 
 
 Annuity 
 
 Due 
 4 Periods 
 
 Immediate 
 Annuity 
 5 Periods 
 
 1 
 
 1.00 
 
 1.03 
 
 1.00 
 
 
 1.03 
 
 1.0609 
 
 1.03 
 
 Amounts of / 
 
 1.0609 
 
 1.0927 
 
 1.0609 
 
 ( 
 
 1.0927 
 
 1.1255 
 
 1.0927 
 1.1255 
 5.3091 
 —1 
 
 4.1836 4.3091 4.3091 
 
 To find the amount of an annuity due, for t periods, 
 find the amount of an immediate annuity for t ■\-\ periods and 
 subtract $1. 
 
54 I'hk Doctrin of Interest. 
 
 123. — In finding the present worth: 
 
 Immediate 
 Annuity 
 4 Periods 
 
 Annuity 
 
 Due 
 4 Periods 
 
 Immediate 
 Annuity 
 3 Periods 
 
 .888487 
 .915142 
 .942596 
 
 .915142 
 .942596 
 .970874 
 
 .915142 
 .942596 
 .970874 
 
 .970874 
 
 1.00 
 
 2.828612 
 +1. 
 
 3.828612 3.828612 
 
 To find the present "worth of an annuity due for t 
 periods find the present worth of an immediate annuity 
 for f — 1 periods and add $1. 
 
 124. — A deferd annuity is one which does not commence 
 to run immediately, but after a certain number of periods, as 
 an annuity of 5 terms, 4 terms deferd, which would begin at 
 the fourth period from now and continue to the ninth inclusiv. 
 
 Its present worth is r^ -\- r-^ -{- r^ -\- r^ -}- r^ 
 An annuity of the entire nine terms would be worth now 
 
 1 + ;-! + r-2 + H + r-* + r^ + r^ + r^ + r^ 
 If from this the value of the four deferd terms be subtracted 
 it will leave the value of the deferd annuity. 
 
 125. — To find the present worth of an annuity for m 
 terms, deferd n terms, subtract from the value of w + « 
 terms that for n. 
 
 126. — A perpetual annuity, or a perpetuity, is one which 
 never terminates. Its amount is infinity, but its present worth 
 can be calculated at any certain rate of interest. If the rent 
 of the annuity is $1 and the rate is . 05, the value of the annuity 
 is such a sum as will produce $1 at that rate or $200, being 
 $1 / .05. The compound discount is the entire $1, being for 
 an infinit number of terms ; therefore the rule still holds : 
 divide the compound discount by the rate of interest. 
 
 127. — Annuities at two successiv rates may occur ; say 5 
 per cent, for 10 years and then 4 per cent, for 10 more. The 
 second part is evidently a deferd annuity, and therefore its 
 present worth is the same as 
 
 20 years at 4% 
 
 less 10 years at 4% 
 
 + 10 years at 5% 
 
The Unit of Time. 55 
 
 128.— In all these examples of annuities it has been 
 assumed that the term or interval between payments is the 
 same length of time as the interest-period. For example, the 
 rate of interest may be so much per year, while the payments 
 are half-yearly or quarterly ; or there may be yearly payments 
 while the desired interest-rate is to be on a half-yearly basis. 
 We shall defer the treatment of these cases until the subject 
 of nominal and effectiv rates has been discust. 
 
 129. — There may also be varying annuities, where the 
 instalment changes by some uniform law. These seldom occur 
 in practice. Where the change is simple, as in arithmetical 
 progression, the annuity may be regarded as the sum of several 
 annuities, otherwise the values must be separately calculated 
 for each term. An annuity running for 5 terms, as follows : 
 13, 18, 23, 28, 33, may be regarded as (1) an annuity of 13 of 
 5 terms ; (2) an annuity of 5, 4 terms ; (3) an annuity of 5, 
 3 terms ; (4) an annuity of 5, 2 terms ; (5) a single amount 
 of 5. 
 
 The Unit of Time. 
 
 130. — It makes no difference in the result whether each 
 term is a year, or a month, or a day, so long as the number of 
 terms (/) and the rate per term {t) are ascertaind. But unfor- 
 tunately the habit has been fixt in common speech of stating 
 the rate, not at so much per term, but so much per annum, 
 even when the interest is payable or chargeable semi-annually 
 (which is the prevalent custom), or quarterly, or monthly, 
 
 131. — When we refer hereafter to a nominal rate per 
 annum, we shall write "per cent." in full, using for actual 
 rates per period the symbol % or the decimal. The letters 
 a, s, q, or m, will stand for "payable annually," "semi- 
 annually," "quarterly," or ** monthly." 
 
 132. — The following phrases need interpretation into 
 more exact language: 
 
 (a) ** Six per cent, per annum, payable annually," means 
 what it says : six per cent, per term, the term being a year. 
 
 {F) "Six per cent, per annum, payable semi-annually," 
 means three per cent, each half year ; which is more than six 
 per cent, per year. 
 
56 Tun DocTRiN OF Interest. 
 
 (c) "Six per cent, per annum, payable quarterly,'' means 
 one-and-one-half per cent, per term of three months. 
 
 (d) ** Six per cent, per annum, payable monthly," means 
 one-half per cent, per month. 
 
 133.— In cases (^), (c) and (d), the ''6" is fictitious. 
 The ratios which must be used are 1.03, 1.015 and 1.005, not 
 1.06 at all. *' Six per cent." is known as the nominal rate, 
 but the effectiv rate for the entire year is different. 
 
 Taking up the above four cases: 
 
 (a) Here the nominal and the effectiv rate are identi- 
 cal; .06. 
 
 (3) Here the effectiv rate is .03 per half year; for the 
 year .0609. 
 
 (c) Here the effectiv rate is . 015 per quarter ; for the 
 year .06136355. 
 
 (d) Here the effectiv rate is .005 per month; for the 
 year .06167781. 
 
 134. — Thus the words ** six per cent, per annum " have 
 four different meanings, according to the qualifying phrase 
 used, or understood. Let / represent the nominal rate "per 
 annum," i being the rate per term, and k the effectiv rate 
 per year. 
 
 Then in (a), where r = 1.06 and / = 1, 
 
 1 + y^ = 1 -l-y= 1.06 
 In (5), where r = 1.03, and / = 2, 
 
 1 + ^ = ^-^^ (1 + y2jy= (1.03)^= 1.0609 
 
 In (c) , where r = 1 . 015 and t = 4, 
 
 l-{.k = r'=(l + }ijy= (1.015)*= 1.06136355 || 
 In (flO, where ?- = 1 . 005 and if == 12, 
 
 l-\-k = r^'= (1 + i^y)^^= 1.005)^^= 1.06167781 1| 
 These values may be ascertained by logarithms or by 
 arithmetic. 
 
 135. — Case (3) furnishes an arithmetical solution which is 
 very convenient. Expanding (1 •\-j/^y by the binomial theorem 
 we have 1 +/ +/ V4. To the nominal rate the quarter of its 
 square is to be added to give the effectiv rate if compounded 
 
The Unit of Tims. 57 
 
 at half periods. Thus at 6% for 7, .06^ = .0036, .0036/4 = 
 .0009; .06 + .0009 = .0609. At 8%, .08^ = .0064; 
 .0064/4=1 .0016. k= .0816. 
 
 136. — The rate k being =/ -{-j^/4, we may factor this, 
 making ity(l +y/4). 1 +y/4 is thus a multiplier, reducing 
 the nominal rate payable semi-annually to an effectiv annual 
 rate. For six per cent, this multiplier would be 1.015, 
 (. 0609 / . 06) ; for five, 1 . 0125 ; for four, 1 . 01 ; for 3%, 1 . 0075 ; 
 for 2%, 1.005. The same reasoning applies to a nominal half- 
 yearly rate, payable quarterly. If 3% is i for the half-year, 
 3 (1.0075) is y for the half year with quarterly payments, 
 or 3.0225. 
 
 137. — But the annual rate given may be the effectiv rate 
 (z) and the question be, what rate (y) will be equivalent for 
 the case of more frequent payments, giving k as the nominal 
 rate per annum for that frequency. 
 
 Case {a) is the same as before. 
 
 Case (3) 1 +y =: (1 + 0^= (1 + y2k) For i = 6%, 
 l-}-J=:(1.06y/-=l. 02956301 ; and /^ = 2j = . 05912602. 
 That is^ to produce 6% payable annually, we must invest at 
 5.912602% per annum, payable semi-annually, or 2.956301% 
 per period of six months. 
 
 (c) 1 + y == (1 + t)H =a + H^) For / = 6%, 
 l^ k = 1.05869538, payable quarterly. 
 
 (d) 1 +y =: (1 + i)^ = (1 + iW For i = 6%; 
 k = .058269, payable monthly. 
 
 138. — In annuity calculations the period or interval 
 between cash payments is to be considerd as well as the fre- 
 quency of compounding the interest. Here, also, the terms 
 are reduced to the **per annum" standard. An annuity of 
 $50 per half year is usually spoken of as an annuity of $100, 
 payable semi-annually. What the actual value of the yearly 
 revenue is, depends upon the rate of interest assumed in the 
 problem. 
 
 139. — If ^a represents the instalment or "rent" of the 
 annuity for each half-year, and i the rate of interest for the 
 half-year, the equivalent of these two cash payments for the 
 
58 Ths Doctrin of Interest. 
 
 year will be Yza ■}- }4a (1 -^ i) = a + }4ai = a (1 -f K^')- 
 If y is the nominal rate per annum or 2i, then the annual 
 effectiv payment is a (1 -\- j/^) and 1 +y/4isa multiplier for 
 transforming a yearly annuity into a half-yearly one. This is 
 the same multiplier which was alredy found to transform a 
 yearly nominal rate of interest, compounded semi-annually 
 into its corresponding ejffectiv rate. This multiplier, 1 +y/4, 
 will be found important in practis. It may be called the 
 co-efl5cient of double frequency, or C^^^ The ^^^ represents the 
 ratio of the frequency of compounding to that of payment. 
 
 140. — If the rate of interest is 3%" per half year (6 per 
 cent. , s) and the annuity payment $1 per annum, to find the 
 amount of the annuity for four years, we may reduce the 
 interest to the annual standard, the cash being alredy there. 
 
 The annual equivalent of the rate is .0609 (6 x 1.015). 
 Twice the logarithm of 1 . 03 . 012837224705 
 
 is log. 1.0609 = .025674449410. 
 The first step is to find the amount, for which purpose the 
 logarithm is multipHed by 4, . 1026977976400. 
 This is also 8 times the logarithm of 1.03, so that we gained 
 nothing by squaring 1.03. From either view the amount is 
 1 .26677008 and the compound interest is .26677008. This is 
 next to be divided by the rate of interest, which is not . 03, 
 nor .06, but .0609. 
 
 .0609). 26677008(4. 3804601, amount of annuity. 
 2436 
 
 2317 
 1827 
 
 4900 
 
 4872 
 
 2808 
 2436 
 
 372 
 365 
 
The Unit of Time. 59 
 
 141. — We may test this result as follows: 
 
 End of first year ; cash 1.00000 
 
 third half year ; interest .03 on 1.00. . .03 
 ♦• second year; " .03 on 1.03.. .0309 
 
 " •• " cash 1 .00000 
 
 Total 2 . 0609 
 
 End of fifth half year ; interest .03 on 2.061 . . .06183 
 
 '• third year; interest .03 on 2.123. . .06369 
 
 cash _1^ 
 
 Total 3.18642 
 
 End of seventh half year ; interest . 03 on 3 . 1 86 .09558 
 
 •• fourth year ; interest .03 on 3.282 .09846 
 
 cash J_^ 
 
 Total 4.38046 
 
 142. — We may simplify this method a little further. Had 
 we made the instalment 60 cents each half year, the compound 
 interest would have been half as much, or . 13338504. This 
 would have been divided by .03, giving 4.446168. It would 
 have been the same had we divided the compound interest of 
 $1 by .06. But we did divide it by .0609, which is 
 .06 X 1.015, the latter being the coeflficient of double fre- 
 quency. We might, therefore, have divided the amount of the 
 annuity when payable semi-annually by the C^^^ 
 4.446168/1.015 = 4.38046 
 
 143. — Therefore, an annuity payable annually is trans- 
 formd as to its amount into one payable half-yearly by multi- 
 plying it by the C^^). 
 
 144. — The present worth of the annuity is subject to the 
 same law ; when the annual payment is divided into two equal 
 sums its present worth is increast in the ratio of 1 +y/4 or 
 1 -j- z/2. In the case given above 
 
 the logarithm . 1026977976400 
 
 would have been changed 
 
 to its cologarithm 1 . 8973022023600 
 
 the number of which would 
 
 be the present worth . 789409234 
 
 The compound Discount would be .210590766 
 and the rate or divisor as before .0609 
 giving the present worth of the 
 
 annuity as 3 . 45797645 
 
6o The Doctrin op Interest. 
 
 145. — The correctness of this may be demonstrated as 
 follows : 
 
 Amount invested in annuity 3.45797645 
 
 Half-year's interest on 3.457976 + .10373929 
 
 3.56171574 
 Half-year's interest on 3.561716— . 10685147 
 
 3.66856721 
 Annual instalment 1.00000000 
 
 2 . 66856721 
 Half -year's interest on 2.668567+ . 08005702 
 
 2 . 74862423 
 Half-year's interest on 2.748624+ .08245873 
 
 2.83108296 
 Annual instalment 1.. 00000000 
 
 1.83108296 
 Half-year's interest on 1.831083— .05493249 
 
 1.88601545 
 Half-year's interest on 1.88 6 015+ . 5 658046 
 
 • 1 . 94259591 
 Annual instalment i 1.00000000 
 
 .94259591 
 Half-year's interest on .942596— .02827788 , 
 
 . 97087379 
 Half-year's interest on '.,9 708738+ .02912621 
 
 1 . 00000000 
 Ivast instalment 1.0000000 
 
 146. — Had the payments been half-yearly, each being 60 
 cents, the compound discount would 
 
 have been 105295383 
 
 and we should have divided by .03, 
 
 giving 3.5098461 
 
 Dividing by the C^^) i oi5, we should 
 
 again have the value 3.457976 + 
 
 147. — The conclusion is that there are two ways of calcu- 
 lating the amount or present worth of an annuity where the 
 interest compounds with twice the frequency of the cash 
 payments. 
 
 (1) Procede as if both were at the greater interval, tak- 
 ing care to use the effectiv rate of interest in dividing. 
 
 (2) Procede as if both were at the smaller interval, the 
 instalment being half as much and divide the result by the C^^). 
 
The Unit of Time. 6i 
 
 148. — Where the interest-period is greater than the pay- 
 ment-period, or the payments are made twice as frequently as 
 the interest is compounded, the solution is less easy because it 
 depends on evolution. 
 
 149. — Half of the instalment is paid when only half the 
 interest-period has elapst. It may be considered as earning 
 interest for the other half -period, but the rate must be taken 
 effectivly. Thus, if the interest for a period is .03, the ratio 
 for the half-period is the square root of 1.03, or 1.01488916. 
 
 The half-instalment paid at the half- period becomes at the 
 
 end of the full period 50 X 1 .01488916 = .50744458 
 
 The other half-instalment is only .50 
 
 and the total of both is 1.00744458 
 
 This is the effectiv instalment, insted of the nominal instal- 
 ment, $1. It is a coejB&cient of frequency as to payments, and 
 may be represented by C^^), meaning that the interest is com- 
 pounded only half as often as a payment is made. 
 
 150. — If the period of compounding is the half year at .03 
 per period (a nominal rate per cent, of .06), the effectiv rate 
 per quarter is 1.01488916 and the G^^) is 1.00744458, being 
 half the square root of 1 + half the nominal rate. If the 
 nominal rate is 3 . 8 per cent, (s), take first the logarithm of 
 
 1.019 008 174 184 006,4 
 
 and divide by 2... 004 087 092 003,2 
 
 The number corresponding to this 
 
 is ^ 1.009 455 299 
 
 halving the decimal part . 1 . 004 727 649 
 
 is the C^^^ for a rate of 3 .8 per annum (.y), payments quarterly, 
 or$l (^). 
 
 151. — The coefficient of frequency (C^^O has to be com- 
 puted at the commencement of each problem by the above 
 method. 
 
 152. — The computation of the amount, or of the present 
 worth, as the case may be, then goes on just as if the payments 
 took place at the same times as the compoundings. When 
 completed, the result is multiplied by O^K 
 
 153. — When the interest-period is semi-annual and the 
 instalments are paid quarterly, it is better to ignore the ' * per 
 
62 The Doctrin of Interest. 
 
 annum" rate and treat of the periods (half years) and half 
 periods (quarters) , after the commencement, 
 
 154.— An annuity of $2 per annum, payable quarterly, 
 interest to be compounded semi-annually, for 2 years at 3i^ 
 per cent, per annum, would be stated as an annuity of $1 per 
 period, payable by half-periods interest at lA^ per period, and 
 continuing for four periods. The present worth of this annuity, 
 omitting the condition ''payable by half -periods," would be 
 3.81698703, which x 1.00472765 is 3.8350324, the present 
 worth when the annuity is paid at the quarters or half-periods. 
 Tested as follows: 
 
 Present worth 3 . 8350324 
 
 Interest at .019 .0728656 
 
 3.9078980 
 First and Second Instalments, with interest 
 
 on the first 1.0047276 -|- 
 
 2 . 9031704 
 Interest at .019 .055 1602 
 
 2.9583306 
 Third and Fourth Instalments, as before. . . 1 .0047277 
 
 1.9536029 
 Interest at .019 .0371185 
 
 1 .9907214 
 
 Fifth and Sixth Instalments 1.0017276 
 
 .9859938 
 Interest at .019 .0187339 
 
 1 . 0047277 
 Seventh and Eighth Instalments 3 .0047277 
 
 The C<^) being almost exactly 1.00472765, it is taken 
 alternately as 1.0047276 and 1.0047277. 
 
 155. — Values of C^^) for all ordinary rates are found by 
 taking half the decimal part of the figures under ' ' Square 
 Root " in Table VI. of the Text Book of the Accountancy of 
 Investment, Part III, the ** 1" remaining where it is. 
 
 156. — To find the amount or the present worth of an 
 annuity where half of each instalment is collected midway of 
 the period, procede as if the entire instalment were collected 
 at the end and then multiply the result by C^^\ being 
 
 1 + ^ (^rr^'-i). 
 
Fractionai. Periods. 63 
 
 157. — In some theoretical computations interest is con- 
 ceived as compounding momently or continuously. Interest 
 at 6% per annum, when compounded momently, gives an equiv- 
 alent effectiv rate of .061837. This is obtained by multiply- 
 ing the rate .06 by the constant quantity .4342944819 (or as 
 many figures as required); considering this as logarithm, 
 its number will be the ratio sought, 1.061836546539. If .06 
 is the effectiv rate and it is desired to find the nominal rate, 
 multiply the logarithm of the ratio (L) by the constant quan- 
 tity 2 . 302585092994, or so much as required and the result will 
 be the nominal rate. 
 
 log. 1.06 = .0253058652648. This X 2.30585092994 
 = .0583689 + . These constants depend on the Naperian 
 logarithms. 
 
 Fractionai. Periods. 
 
 158. — We have hitherto treated only of entire periods, but 
 it is quite usual that the number of periods should be a mixt 
 number, sometimes a fraction only. 
 
 159. — A det is due in one year from now, at six per cent, 
 annually ; but the dettor has the privilege of paying at the 
 half year ; what interest should he then pay ? There are two 
 answers to this question, depending on whether it is to be con- 
 siderd legally or equitably — by simple interest or by compound 
 interest. 
 
 160. — Legally, the rate is .03 per half year, the law not 
 recognizing the justice of compound interest. Equitably, that 
 is not the true proportion in which the interest should be 
 divided. The creditor gets, not six per cent, annually, but 
 six per cent, semi-annually, which we have seen to be more 
 profitable. 
 
 161. — The compound interest for a half term is at the rate 
 of .02956301 only, not .03. Compound interest for several 
 periods is greater than simple interest ; conversely, for part of 
 a period the compound interest is the lesser. 
 
 162.— If the det spoken of is $1,000,000 and is discharged 
 at midyear by a payment of $1,030,000, the creditor has the 
 
64 The Doctrin of Interest. 
 
 use for six months of $30,000, at some rate from which the 
 dettor has no benefit, besides the use of the $1,000,000 to 
 which he is entitled. 
 
 163. — If interest were not a constant force, but a periodi- 
 cal incident, there would be no such thing as interest between 
 the periodical dates ; one would have to pay a full period or 
 nothing. 
 
 164. — The result of this inconsistency is that, conven- 
 tionally, when interest is calculated on a certain number of 
 terms and a fraction of a term, the interest compounds for the 
 integral terms, but remains simple during the fraction of a term. 
 
 165. — For four-and-a-half years on the conventional inter- 
 est plan at six per cent, annually, the compound interest must 
 
 be calculated for four years ; amount 1.26247696 
 
 then this must be multiplied by 1.03 (the con- 
 ventional ratio for the half year) producing 1.30035127 
 This number is exactly midway, arithmetically, 
 
 between the amount at four years 1.26247696 
 
 and that at five years 1.33822558 
 
 This plan of dividing the difference in proportion to the 
 time elapst is generally used where the even periodical values 
 can be obtaind from tables, especially in case of valuation of 
 bonds, as will be shown hereafter. 
 
 166. — In scientific interest, the ^ forms part of the num- 
 ber of terms. The log. 1 . 06 025 305 865 264,8 
 
 being multiplied by 4.5 gives 113 876 389 191,6 
 
 the number for which is 1 .29 979 957 070 
 
 This result might have been obtaind by 
 
 multiplying the 4 year amount 1 . 26 247 696 
 
 by the inconvenient number 1 . 02 956 301 
 
 which is the effectiv ratio. 
 
 167. — When annuities are to be sumd or valued, it is 
 necessary to get the value for the entire terms first and then 
 multiply by the effectiv rate for scientific interest ; for conven- 
 tional interest either multiply by the conventional rate, or 
 "split the difference," according to time elapst. It is impos- 
 sible to value or sum the annuity in one operation by a 
 fractional multiplier, for the reason that these processes depend 
 entirely on a uniform ratio. 
 
Sinking Funds. 65 
 
 168. — It is the universal custom in actual business to treat 
 parts of terms by simple interest, not by compound ; conven- 
 tionally, not scientifically. 
 
 Sinking Funds. 
 
 169. — We have hitherto assumed the periodical instal- 
 ment, or rent of an annuity, to be 1. When this is some other 
 number, the amount or present worth of $1 is multiplied by 
 that other number ; that is, the amounts (or present worths) 
 are directly proportionate to the rent. But sometimes we have 
 given the amount or the present worth as a fixt sum and wish 
 to find an instalment which will produce that amount or ex- 
 tinguish that present worth. 
 
 170. — We have seen that the amount of an annuity of $1 
 at 3% for 50 periods is $112.79687. If the amount were 
 $1000 insted of $112.79687, it is evident that each instalment 
 must be increast as many times as $112 . 79687 is containd in 
 $1000. The quotient is 8.8655. Therefore, under the same 
 conditions where $1 amounts to $112.79687, $8.8655 will 
 amount to $1000. If the growth of the two annuities be com- 
 pared it will be seen that at any point the one which is to 
 accumulate to $1000 is 8 . 8655 times as large as the one which 
 accumulates to $112.79687. 
 
 Instalments Instalments of 
 
 of $1 $8.8655 
 
 8 . 8655 
 
 . 2660 
 
 8 . 8655 
 
 1 
 
 . 0000 
 
 
 . 0300 
 
 1 
 
 
 2 
 
 . 0300 
 
 
 . 0609 
 
 1 
 
 
 3 
 
 . 0909 
 
 
 . 0927H- 
 
 1 
 
 
 17.9970 
 
 . 5399 
 
 8.8655 
 
 27 . 4024 
 
 .8220 
 
 8 . 8656 
 
 4 . 1836 37 . 0899 
 
 etc. etc. 
 
 Therefore, to find the instalment which contributed each 
 period, will amount to a given sum S, divide S by the amount 
 of an annuity of $1. 
 
66 The Doctrin of Interest. 
 
 171. — Where an annuity is so constructed that it shall 
 accumulate to a certain amount at a certain time, it is called a 
 sinking fund. Frequently the uniform periodical contribu- 
 tion is itself calld the sinking fund, and is found in the fore- 
 going manner. 
 
 172. — Where the present worth is the quantity given, the 
 process of finding the uniform contribution which will gradu- 
 ally extinguish or amortize that present worth by the aid of 
 compound interest is similarly performd. The fixt quantity is 
 the present worth of an annuity of x dollars ; the given present 
 worth divided by P, the present worth of $1 gives the instal- 
 ment, x^ necessary to amortize it. 
 
 173. — It is required to find what annual payment will 
 clear off $1000 in 50 periods, allowing .03 interest. We have 
 alredy calculated that a payment of $1 per period will pay off 
 $25 . 729764, with interest. $1000 is 38 . 8655 times $25.729764 ; 
 therefore the contribution must be $38 . 8655 per period, which 
 will, by forming a schedule, be found to amortize the $1000. 
 174. — As a provision for liquidating indettedness, or for 
 replacing vanishing assets, sinking fund and amortization are 
 two different applications of the same principle. Formerly, 
 the terms were used interchangeably, but more recently they 
 are distinguisht as follows: 
 
 175. — The sinking fund permits the det to stand 
 till maturity, but in the meantime provides a fund 
 which at maturity pays off the entire det, the Interest 
 on the original sum being paid separately. 
 
 176. — The amortization plan accumulates nothing, 
 \iU\. gradually reduces the det, applying to this reduc- 
 tion all the excess of the contribution over the Interest. 
 177. — The two operations which we have performd show 
 that the sums necessary to be set aside for a det of $1000 dur- 
 ing 50 periods at . 03 are, 
 
 Sinking Fund $ 8.8655 
 
 Amortization 38.8655 
 
 The difference is the $30 
 
 per period, which on the sinking fund plan is required to pay 
 the current interest, so that actually the two methods of con- 
 tribution come to the same thing. 
 
Interest-bearing Securities. 67 
 
 178. — The number of terms necessary for a certain contri- 
 bution per period to amount to a certain principal may be 
 found, but first the amount of a single dollar must be found. 
 
 The amount of the annuity is I/z, the total compound 
 interest divided by the rate of interest. Multiplying that 
 amount by the rate gives, therefore, the compound interest. 
 Adding to this $1 we have the amount of a single dollar, ^ or 
 S X z -h 1. We then proceed as shown in Art. 93. 
 
 Similarly the present worth of the annuity being D/t, 
 /> = 1 — P X z, and / may be deduced therefrom. 
 
 179. — The rate of interest oi an annuity cannot be ascer- 
 taind by any direct formula, as it involvs the solution of 
 equations of higher degrees. 
 
 180. — A special method for finding the income-rate of 
 securities by gradual approximation will be given hereafter. 
 (Art. 231). 
 
 Interest- BEARING Securities. 
 
 181. — A bond (which is the most usual form of interest- 
 bearing security) is a complex promise to pay: 
 
 1. A certain sum of money at a future time ; this is 
 known as the principal, the par or the capital. 
 
 2. Certain smaller sums, proportionate to the principal, 
 and at various earlier times. These are usually known as the 
 "interest," but as they do not necessarily correspond to the 
 true rate of interest, it will be better to speak of them as the 
 coupons. 
 
 182. — These various sums are never worth their face or 
 par until the stipulated times arrive, but are always at a dis- 
 count. The principal is never worth its face until its maturity; 
 the coupons are never worth their face until the maturity of 
 each. Yet while both principal and coupons are at a discount, 
 the aggregate may easily be worth more than the par, and it 
 is the aggregate, principal and coupons which is the subject of 
 the valuation. 
 
 183. —If the bond is sold at par, the coupon and the in- 
 terest are equivalent. Take a five per cent. (5) bond for 
 $10,000, due in 5 years, at par. Its value consists of 
 
68 Thb Doctrin of Interest. 
 
 EXAMPLE 1 
 
 1. The present worth of $1000 at 10 periods at .025 781 . 1984 
 
 2. The present worth of an annuity of |25 per period, 
 
 10 periods 218.8016 
 
 Aggregate 1000.0000 
 
 EXAMPLE 2 
 
 But if the coupons were $30 each, the bond being ** six per 
 
 cent," the principal would still be valued at : 781 . 1984 
 
 while the coupons would be worth 262 . 5619 
 
 Aggregate 1043.7603 
 
 EXAMPLE 3 
 
 If the bond were a "four per cent." bond, the coupons 
 
 being $20 each, the valuations would be, principal 781 .1984 
 
 coupons 175.0413 
 
 Aggregate 956.2397 
 
 All the above calculations may be made by logarithms, 
 commencing with the logarithm (L) of 1 . 025. 
 
 184. — From these computations we may draw the follow- 
 ing inferences : 
 
 1. If the coupon rate is the same as the income rate, the 
 bond is at par. 
 
 2. If the coupon rate is greater than the income rate, 
 the bond is worth more than par. 
 
 3. If the coupon rate is less than the income rate, the 
 bond is worth less than par. 
 
 185. — Rule I. Any bond may be valued so as to earn 
 a given interest rate by adding together 
 
 1. Present worth of the principal ; 
 
 2. Present worth of the annuity, consisting of all the 
 
 coupons. 
 
 186. — Representing the coupon rate or the proportion 
 which the coupon bears to the principal, by c and the value 
 of the bond for $1 by V 
 
 1 — /-* 
 t 
 r^ is the only quantity which requires logarithms for its com- 
 putation, which always begins with L, the logarithm of r. 
 /L is the logarithm of r^ and subtracted from zero is the loga- 
 rithm of r^. In the above example 
 
Interest-bearing Securities. 
 
 69 
 
 I, or log. (1 + or log. 1 . 025 = . 010 723 865 391,8 
 log. 1.025^° = /L = __.107 238 653 918 
 log. 1025-^ « = 1.892 761 346 082 
 1.892 761 346 082 /« .781198 401727 
 Substituting the above value for r-^ will give the results in 
 Examples 2 and 3. 
 
 187. — In the second and third case the correctness of the 
 figures may be proved by forming a schedule of amortization, 
 which, starting with the present value, will bring the value, up 
 or down, to par at maturity. 
 
 Six Per Cent. Bond, Net Income 025. 
 
 Coupons 
 
 Interest at .025 
 
 Amortization 
 
 1043.7603 
 
 30. 
 
 25.0940 
 
 3.9060 
 
 1039.8543 
 
 30. 
 
 25.9964 
 
 4.0036 
 
 1035.8507 
 
 30. 
 
 25.8902 
 
 4.1038 
 
 1031.7469 
 
 30. 
 
 25.7937 
 
 4.2063 
 
 1027.5406 
 
 30. 
 
 25.6885 
 
 4.3115 
 
 1023.2291 
 
 30. 
 
 25.5808 
 
 4.4192 
 
 1018.8099 
 
 30. 
 
 25.4702 
 
 4.5298 
 
 1014.2801 
 
 30. 
 
 25.3570 
 
 4.6430 
 
 1009.6371 
 
 30. 
 
 25.2410 
 
 4.7590 
 
 1004.8781 
 
 30. 
 
 25.1219 
 
 4.8781 
 
 1000.0000 
 
 300. 
 
 356.2397 
 
 43.7603 
 
 
 Four Per Cent. Bond, Net Income .025. 
 
 Coupons 
 
 Interest at .025 
 
 Amortization 
 
 956.2397 
 
 20. 
 
 23.9060 
 
 3.9060 
 
 960.1457 
 
 20. 
 
 24.0036 
 
 4.0036 
 
 964.1493 
 
 20. 
 
 24.1038 
 
 4.1038 
 
 968.2531 
 
 20. 
 
 24 2063 
 
 4.2063 
 
 972.4594 
 
 20. 
 
 24.3115 
 
 4.3115 
 
 976.7709 
 
 20. 
 
 24.4192 
 
 4.4192 
 
 981.1901 
 
 20. 
 
 24.5298 
 
 4.5298 
 
 985.7199 
 
 20. 
 
 24 6430 
 
 4.6430 
 
 990.3629 
 
 20. 
 
 24.7590 
 
 4.7590 
 
 995.1219 
 
 20. 
 
 24.8781 
 
 4.8781 
 
 1000.0000 
 
 200. 
 
 243.7603 
 
 43.7603 
 
 
70 Thk Doctrin of Interest 
 
 In the six per cent, example the amortization is subtracted ; 
 in the four per cent, example the amortization of discount 
 (called also accumulation or accretion) is added. The figures 
 in the two amortization colums are identical. 
 
 188. — This is the most natural method of valuation, and 
 for one who only occasionally employs it, perhaps the safest. 
 There are other methods which in practis are briefer. 
 
 189. — The excess over par in the second example (six per 
 cent, coupons), $43.7603, is known as premium. 
 
 In the third example (four per cent, coupons), the value 
 is less than par and the difference is known as discount, a 
 word which has several meanings. When I have occasion to 
 speak of both premiums and discounts I shall use the word 
 variance ; that is, variance from par. 
 
 190. — The difference between the coupon-rate, or cash- 
 rate, and the interest-rate, or income-rate, is the sole cause of 
 the variance. This difference will be called the interest- 
 difference. 
 
 . 025 being assumed as the interest-rate, and the coupon- 
 rate .03 or .02, the interest- difference is .005. 
 
 191. — Where the coupon is .03, the .025 may be con- 
 sidered as interest on $1, and each .005 is a future benefit or 
 extra profit, which should be paid for. Reduced to present 
 values, these benefits are the present worth of an annuity of 
 .005 per period. 
 
 192. — The present worth of an annuity of .005 for 10 
 periods is .0437603, or on $1000, $43.7603, the same variance 
 as found by the previous process. 
 
 193. — Rule II. The variance is the present worth of an 
 annuity of the interest-difference. When the coupon-rate is the 
 greater, the variance is added to the par ; when the coupon- 
 rate is the less, the variance is subtracted from par. 
 
 194. — Representing the variance by Q, the second rule 
 
 may be exprest as follows: 
 
 1 — ^' 
 
 or ^^ (1 — r-') 
 
Intkrbst-bearing Securities. 71 
 
 195. — Multiplying both numerator and denominator by 
 
 200 will not alter the value of the fraction, hence it will be 
 
 the same thing if we use the nominal annual rates. Insted 
 
 , .03— .025 6 — 5 , . , . 
 
 of ^r^ we may use — = — which is easier. 
 
 In the above example, No. 2, the variance would be 
 obtaind thus : 
 
 Q = 5-=-^ (1 — .781198401727 
 o 
 
 V = 1 + ?-=^ (1— .78119840172/ 
 5 
 
 = 1+ i (. 218801598273) 
 o 
 
 = 14- .0 43760319655 = 1.0 437603 + 
 
 In the third example, where c = 4 
 
 Q = ^^^ (.2188016) 
 
 = — i (.2188016) 
 o 
 
 V = l— .0 437603=. 9 562397 
 The results may be carried to 11 or 12 decimals if desired. 
 
 196.— A third method (suggested by Mr. Arthur S. Little) 
 is based upon the value of a perpetual bond. This, as there is 
 no redemption, is merely a perpetual annuity, or perpetuity of 
 the ' ' coupon. ' ' The value of such a perpetuity is c/i. A six per 
 cent, (s) bond to pay five per cent. (5) is .03/.025 = 6/5 = 1.20, 
 and this value is perpetual, there being no redemption. But 
 if it is known that the variance (.20) will vanish 10 years from 
 now, the value of the bond is now lessend by the present worth 
 of that variance. 
 
 197. —Rule III. The terminant value is the perpetuity 
 value, minus the present worth of the perpetuity variance. 
 
 v= ^_(£_i)(^.) 
 
 In Example 2, the perpetuity value is 6/5 = 1 . 20 
 The present worth of the vanishing quan- 
 tity .20 is 1562397 
 
 Remainder 1.0437603 
 
72 The Doctrin of Interest. 
 
 Here the first step is to obtain the perpetuity value by 
 simple arithmetic. The variance is, of course, .20. Then 
 r-t==.781 198 401 727 is found as usual and multiplied by .20, 
 giving 156239680+ 
 
 In Example 3, the perpetuity is 80 
 
 but the variance is still .20, and its present 
 
 worth is .15623968 
 
 which is added, making 95623968 
 
 because minus a minus is plus. 
 
 198. — Multiplying down. Whichever of these three 
 methods has been employd for ascertaining the value of the 
 bond at a certain date, if the successiv values for each period 
 are expected to be required (and they usually are) , it is pre- 
 ferable to find them by schedules of amortization rather than 
 resort to independent logarithmic calculation for each. A 
 thoro test of the correctness of all the intermediate values is 
 the fact that the series reduces to par at maturity. In this 
 test, insted of a formal schedule in colums, all the figures may 
 be brought into a single colum, so that no marginal computa- 
 tion may be needed. The amount of each amortization is not 
 exprest, but implied, in the following example of such a single- 
 col um schedule : 
 
 A four per cent, (j) bond for 4 years to net three 96/100 (5). 
 The r is 1.0198: 
 
 1.0198 nl .008 515 007 6315 
 
 X8 _.068 120 061 052 
 
 Subtract from zero : 1 .931 879 938 95 
 
 (A85 B05 C67 D93 E75 F27) 
 
 Present worth of |1 854 830 361 69 
 
 Compound discount 145 169 638 31 
 
 Divide by .0198 7.331 799 914.75 
 
 Multiply by interest difference 000 2 
 
 Premium .001 466 359 98 
 
 Value 1 .001 466 359 98 
 
 .0198 .01 010 014 663 60 
 
 .009 009 013 197 24 
 
 .0008 .000 801 173 09 
 
 1.021 295 393 91 
 
 Coupon 02 
 
 1.001295 393 91 
 
MuivTiPLYiNG Down. 73 
 
 We save a number of figures by adding and subtracting at the same 
 time, putting a circle round the coupon to indicate subtraction : 
 
 Resuming 1 . 001 295 393 91 
 
 .01 10 012 953 94 
 
 .009 9 011658 55 
 
 7)801 036 32 
 
 & 
 
 1.001121042 72 
 
 The operation may be still further abridgd by amortizing the premium 
 only, but subtracting the interest-difference only, not the entire coupon : 
 
 .001 121041 72(1) 
 11 210 43 
 ^0 089 38 
 (2) 896 83 
 
 943 239 36 (2) 
 9 432 39 
 ^^8 489 15 
 (2) 754 59 
 
 761 915 49 (3) 
 7 619 15 
 6 857 24 
 (2) 609 53 
 
 577 001 41 (4) 
 5 770 01 
 5 193 01 
 (V) 46160 
 
 388 426 03 (5) 
 3 884 26 
 ^^3 495 83 
 (2) 310 74 
 
 196 116 86 (6) 
 1 961 17 
 1 765 05 
 (2) 156 89 
 
 Error in decimals 03 
 
 .000 000 000 00(7) 
 The (2) is two places further to the right than in the first procedure. 
 
 199. — Computing Amortizations. It may sometimes be 
 advisable to find and verify at first the instalments of amorti- 
 zation, leaving this series of amounts to stand, not filling in 
 the remaining colums of the schedule, until required. The 
 obtaining of the amortization colum is remarkably easy, as 
 shown in Art. 119. 
 
74 The Doctrin of Interest. 
 
 200. — Starting with the premium or discount at t periods, 
 
 as above explaind, it is next amortized to the extent of p 
 
 that is the present worth of the interest-difference. This mul- 
 tiplied by r gives the next amortization, — ^ and so on down. 
 
 201. — In the last example, the premium at 4 years was 
 
 found to be 001 466 359 98 
 
 The next amortization is simply the present worth at 8 periods 
 of . 0002. The present worth of $1 has been found to be 
 .854 830 36169; this x .0002 = .000170 966 072 238 
 Three figures may be dropped from this with safety, leaving 
 the decimals as much extended as in the previous operation. 
 
 1.000 000 170 966 072 (1) 
 
 .01 1709 661 
 
 .009 1538 695 
 
 .0008 136 773 
 
 174 351 210 (2) 
 
 1 743 512 
 
 1 569 161 
 
 139 481 
 
 177 803 355 (3) 
 
 1 778 034 
 
 1 600 230 
 
 142 243 
 
 181 323 862 (4) 
 1 813 239 
 1 631 915 
 145 059 
 
 184 914 075 (5) 
 1 849 141 
 1 664 227 
 147 931 
 
 188 575 374 (6) 
 1 885 754 
 1 697 178 
 150 860 
 192 309 166 (7) 
 1 923 092 
 1 730 782 
 
 153 847 
 
 . 000 196 116 887 (8) 
 
 These are the eight instalments of amortization, which, added 
 together, should equal the total premium. 
 
Computing Amortization. 75 
 
 .000 170 966 07 
 174 351 20 
 177 803 35 
 181 323 86 
 184 914 07 
 188 575 37 
 192 309 17 
 196 116 89 
 
 .001 466 359 98 
 
 But if this test were not available, the last amortization would have 
 to be multiplied up by 1 .0198, which should produce .000200. 
 
 .000 196 116 887 
 
 1 961 169 
 
 1 765 052 
 
 156 893 
 
 .000 200 000 001 
 
 202. — Small discrepancies in the last figure are to be ex- 
 pected and disregarded ; therefore, the decimals should be 
 carried beyond the figures which are to be utilized. 
 
 203. — Discounting. A series of values in reverse order, 
 beginning from maturity, may be obtaind (without using 
 logarithms) by division, the interest-ratio being the divisor. 
 The entire amount to be receivd on the above bond is: princi- 
 pal $1, coupon .02, total $1.02. This should be divided by 
 
 1.0198. 
 
 1.02 -V- 1.0198 = 1.000 196 116 89 
 
 which is the value 1 period before maturity. The coupon .02 
 must be added before the second discounting process. 
 
 1.020 196 116 89 ^ 1.0198 = 1.000 388 426 03 
 
 .02 
 
 1.020 388 426 03 
 
 1.020 388 426 08 -f- 1.0198 = 1.000 597 001 41 
 
 This is a laborious process, even if, insted of dividing, we mul- 
 tiply, by the reciprocal of 1 . 0198, . 980584428. 
 
76 
 
 The Doctrin of Interest. 
 
 1.02 
 
 -|- coupon 
 
 .980 584 428 
 .019 611 689 
 
 1.000 196 117 
 .02 
 
 1.020 196 117 
 
 .980 584 428 
 
 .019 611 689 
 
 098 058 
 
 88 253 
 
 5 884 
 
 98 
 
 10 
 
 7 
 
 1.000 388 427 
 
 Table of Multiples 
 
 1. 
 
 .980 584 428 
 
 2. 
 
 1.961168 857 
 
 3. 
 
 2.941753 285 
 
 4. 
 
 3.922 337 713 
 
 5. 
 
 4.902 922 142 
 
 6. 
 
 5.883 506 570 
 
 7. 
 
 6.864 090 998 
 
 8. 
 
 7.844 675 427 
 
 9. 
 
 8.825 259 855 
 
 204. — Intermediate Purchases. It happens very often 
 (perhap? in a majority of cases), that bonds are not purchast 
 on the very day when the interest is payable. In the preced- 
 ing examples it was supposed that exactly 8 or 10 periods 
 would elapse from the purchase of the bond till its maturity ; 
 but the purchase may have been a month, or several months, 
 or months and days, after the beginning of the period. 
 
 205. — We saw (Art. 167) that an annuity cannot be valued 
 by the usual formula when the number of terms is a mixt 
 number. We must derive it from the next regular term or 
 interpolate it between the two nearest. It was also explaind 
 that this interpolation maybe done in two ways: one by simple 
 proportion, conventionally, or by compound interest, scientifi- 
 cally. In the present state of knowledge the conventional or 
 non-scientific method is establisht by usage, altho it works 
 injustice to the buyer. The difference is usually not very 
 large. 
 
 206. — Let us suppose that in the above Examples, in Art. 
 182 the purchase had been made 9}^ periods before maturity, 
 that is, 4 years 9 months. 
 
Intermediatf Purchases. 77 
 
 The value at 10 periods (Ex. 2) is. . . 1043.7603 
 
 the value at 9 periods is 1039.8543 
 
 the difference, or amortization, is 3 . 9060 
 
 As half the time has elapst, we assume 
 that half the amortization has taken 
 
 effect 1.9530 
 
 and this we subtract from the 10- 
 
 period value 1043.7603 
 
 making 1041.8073 
 
 which is exactly half way between the 9-period and the 
 10-period values. Besides this, however, the purchaser must 
 pay one-half of the current coupon, or $15.00, as "accrued 
 interest," the entire cost being 1056.8073. This is called the 
 flat price, and formerly this was the form in which securities 
 were quoted at the Exchanges ; now, however, the quotations 
 are understood as so much "and interest," meaning that the 
 accrued interest is made a separate item in the bill. 
 
 207.— Had the 10-year value been multiplied by 1.0125 
 (half the interest rate) the same flat value would have been 
 obtaind. 1043.7603 X 10125 = 1056.8073 
 
 208-— To apply the scientific plan, the 1043.7603 would 
 have been multiplied by 1.01242284, giving 1056.7267 
 
 insted of 1056.8073 
 
 a difference of .0806 
 
 in favor of the purchaser, but he could not claim it under the 
 law and the customs of the market. 
 
 209. — As the usual period is divided into six months, and 
 as the odd days are considerd as thirtieths of a month, the 
 amortization for each day is 1/180 of that for a half year. If 
 2 months, 17 days had past after the interest-date, then 1.9530 
 must be multiplied by 77/180, giving .8359 as the proportion- 
 ate amortization for 77 days. 
 
 210. — Thus the value of a bond at any date from its issue 
 to its maturity, at some given rate of interest, may be calcu- 
 lated by first valuing it at at two consecutiv interest dates, 
 according to the rules given, and then "splitting the differ- 
 ence" by dividing it into 180ths. 
 
78 The Doctrin of Interest. 
 
 211. — Even if the unit of time employed in the coupon- 
 payments and the interest-compoundings be different, the rules 
 given in the section entitled " The Unit of Time " will enable 
 these to be allowed for, if Rule I (Art. 185) is used for valuing. 
 
 212. — Intermediate Balances. — When the regular 
 interest-periods do not coincide with the date of the balance 
 sheet, it becomes necessary to adjust the valuations for that 
 purpose in the manner just described for purchases at odd 
 times. 
 
 213. — If the interest-dates are May 1 and November 1, 
 and the dates for balancing are January 1 and July 1, the bond 
 must, on January 1, have been amortized to the extent of one- 
 third of that from November to May, conventionally. 
 
 214. — A six per cent, (s) bond for $1000, to yield five per 
 cent. (^) due Nov. 1, 1925, is worth on 
 
 November 1, 1920 1043.7603 
 
 on May 1, 1921 1039.8543 
 
 the amortization for 6 months is, 
 
 therefore 3.9060 
 
 For 2 months it is one-third 1 . 3020 
 
 and this subtracted from 1043.7603 
 
 leave the value on Jan. 1, 1921 1042.4583 
 
 Applying the same method between 
 
 May 1 and Nov. 1, 1921 1039.8543 
 
 1035.8507 
 
 3) 4.0036 
 
 1.3345 
 
 1039.8543 
 
 we have the balance, July 1, 1921 . . 1038.5198 
 
 If we take the January value 1042 . 4583 
 
 and multiply down ; x .025 26.0615 
 
 1068.5198 
 
 — Coupon 30. 
 
 we also get the July result 1038.5198 
 
 Thus we have the choice of interpolating each balance- 
 value, or having obtaind one, of multiplying down to maturity, 
 
Intermediate Bai^ances. 
 
 79 
 
 which can be done on the conventional, but not on the scientific 
 plan. The resulting schedule would be as follows: 
 
 Date 
 
 Collected 
 
 Interest 
 at .025 
 
 Amortization 
 
 Value 
 
 Jan. 1,1921 .... 
 
 Value at 
 
 5 per cent. 
 
 basis 
 
 1042.4583 
 
 Julyl, " .... 
 
 30.00 
 
 26.0615 
 
 3.9385 
 
 1038.5198 
 
 Jan. 1, 1922.... 
 
 30.00 
 
 25.9630 
 
 4.0370 
 
 1034.4828 
 
 Julyl. - .... 
 
 30.00 
 
 25 8621 
 
 4.1379 
 
 1030.3449 
 
 Jan. 1, 1923.... 
 
 30.00 
 
 25.7586 
 
 4.2414 
 
 1026.1035 
 
 Julyl, " .... 
 
 30.00 
 
 25.6526 
 
 4.3474 
 
 1021.7561 
 
 Jan. 1, 1924.... 
 
 30.00 
 
 25.5439 
 
 4.4561 
 
 1017.3000 
 
 Julyl, - .... 
 
 30.00 
 
 25.4325 
 
 4.5675 
 
 1012.7325 
 
 Jan. 1, 1925.... 
 
 30.00 
 
 25.3183 
 
 4.6817 
 
 1008.0508 
 
 Julyl, - .... 
 
 30.00 
 
 25.2012 
 
 4.7988 
 
 1003.2520 
 
 Nov. 1, " .... 
 
 20.00 
 
 16.7480 
 
 3.2520 
 
 1000.0000 
 
 The last period is of only 4 months, July 1 to Nov. 1, Yz of 
 the half year. The cash collected is, therefore, considerd as 
 only $20, yz of $30 ; each previous coupon had included Yi of 
 the following half year, and this must now be squared up. In 
 the colum " Interest at .025 " the procedure is peculiar. 
 
 The 16,7480 is composed of two parts : 
 
 1. Yi of .025 on the $1000 par $16.6667 
 
 2. .025 infullon the $3.2520 .0813 
 
 $16.7480 
 215. — To explain why interest in full for the half-year is 
 reckond on the premium, go back to the normal schedule in 
 Art. 187, and it will be seen that the premium on May 1 was 
 4.8781. Now, on the conventional plan, based on simple 
 interest, this 4 . 8781 should not vary during the period ; there- 
 fore the interest ought to be : 
 
 Yz of .025 of 4 8781 = .0813 
 
 which is the same as .025 of ^ of 4.8781 (3.520) = .0813 
 Hence in the last or broken period the variance from par 
 must be treated as having earnd interest during the entire 
 period, while the par itself has only earnd interest for the 
 actual time, as four months. 
 
 216.— It will also be notist that 3.2520 is ^ of 4.8781, so 
 that if we had calculated 4.8781 by discounting, it would have 
 been sufi&cient confirmation of the preceding values to take Yz 
 of 4.8781 and compare it with 3.2520. 
 
8o The Doctrin of Interest. 
 
 217. — It must be rememberd that the periods introduced 
 for balancing purposes are artificial, and that, strictly speaking, 
 amortization takes place only at the dates when interest 
 becomes due. The charging of part of the coupon, tho not 
 yet collected, is fictitious, but in each period until the last, this 
 borrowing is compensated for by a fresh loan. 
 
 218.— Short periods, terminal or initial. — It happens 
 sometimes (altho it should be avoided) that the bond does not 
 mature at the end of an interest period, but at some previous 
 date. This gives rise to a fractional period, not an artificial 
 one like those establisht for balancing purposes (Art. 212), 
 but an actual one, which must be taken into account in the 
 valuation. 
 
 219. — We will take the case of a six per cent. (5) bond for 
 $1000. issued Jan. 1, 1921, and payable Nov. 1, 1925, interest 
 payments January and July 1, and valued to pay five per 
 cent. (5). There are 9 full periods and a short period of 4 
 months, or ^ of a period. The coupon for this short period 
 would be $20 insted of $30, as in such cases the last coupon is 
 always proportional to its time. The interest ratio is also 
 reduced for that period to 1 + ^ of .025, or 1.016^ by the 
 conventional plan. 
 
 220. — Using the first method of evaluation, the following 
 are the components of the value : 
 
 An annuity of 9 terms of |30 each at . 025 $239 . 1260 
 
 The present worth of $1020 at .025 for 9 terms 
 and .016^ for 1 term. 
 
 $1 for 9 terms .8007284 
 
 Divided by 1 . 016% .7876017 
 
 Multiply by 1020 803.3537 v 
 
 1042.4797 
 It will be observd that this differs slightly 
 from the value obtaind in Art. 209 for the 
 same length of time, but with interest May 
 and November 1042 . 4583 
 
 The $1020 referd to is composed of the principal and the 
 last or partial coupon. 
 
 221. — To divide by a number like 1 .016^, it is easier first 
 to multiply both divisor and dividend by 3, converting each 
 into a whole number. 
 
 1.0116^3 X3 -=3.05 
 
 .8007284 X3 =2.40218508 
 2 . 40218508 -- 3 . 05 = . 78760167 
 
Short Periods, TerminaIv or Initiai. 8i 
 
 222. — To illustrate the case of an initial short period, 
 suppose that the above bond had been issued on October 1, 
 1920, 3 months earlier than has been assumed ; issued Oct. 1, 
 1920, interest January and July, principal maturing Nov. 1, 
 1925. There is then a preliminary coupon for 3 months, $15, 
 to be discounted at 1 . 0125 ; 9 coupons of $30 each forming an 
 annuity ; one coupon of $20 and the principal, $1000, dis- 
 counted for 9 periods at 1.025, and one period at 1.016^. 
 The value on Jan. 1, 1921, obtaind as before, is 1042.4797. 
 The simplest way will probably be to add to this the initial 
 coupon $20, and discount back the entire 1062.4797 by divid- 
 ing by 1.015, giving 1046.7780. 
 
 223. — We may have two other complications : the bond 
 may be purchast within one of the odd periods ; the balancing 
 period may be at still another date. 
 
 224 — If the above bond were bought on Dec. 1, 1920, the 
 price would be between 1046.7780, the October value, and 
 1042.4797, the January value, a three months' interval. The 
 difference is 4 . 2983 ; and either Yz of this (1.4328) may be 
 added to the January value or ^ (2.8655) subtracted from 
 the initial value. 
 
 1042.4797 + 1.4328 = 1043.9125 
 1046 . 7780 — 2 . 8655 = 1043 . 9125 
 
 225. — The adjustment of values to balancing dates pre- 
 sents no special difi&culty, being performd by simple proportion. 
 
 226. — Cash payments on principal. — Bonds at the same 
 rates of coupon and of interest, tho at different dates of matu- 
 rity, may be combined into one schedule. This may be done 
 even if the interest-dates are different, but it is practically 
 better in that case to keep the schedules distinct. 
 
 227. — We must commence with an aggregate value, made 
 up of the separate values of the groups of bonds maturing on 
 the same day. 
 
 228. — $2000 six per cent. (5) bonds, maturing as follows : 
 $1000 on Nov. 1, 1923, $1000 on Nov. 1, 1925 ; interest .025, 
 interest payments May and November. Required, value on 
 Nov. 1, 1920. 
 
 The value of the first bond is 1018.8099 
 
 The value of the second 1043.7603 
 
 Aggregate value 2062.5702 
 
82 
 
 The Doctrin of Interest. 
 
 This value, multiplied down in the usual manner, gives the 
 following schedule. At the date of the maturity of bond No. 
 1, the cash colum must contain not only the coupon $60, but 
 the $1000 payable on principal. 
 
 Date 
 
 Cash 
 Collections 
 
 Interest 
 at .025 
 
 Payments 
 on Principal 
 
 Investment 
 Value 
 
 1920 Nov. 1 
 
 
 
 
 2062.5702 
 
 1921 May 1 
 
 60.00 
 
 51.5643 
 
 8.4357 
 
 2054.1345 
 
 " Nov. 1 
 
 60.00 
 
 51.3534 
 
 8.6466 
 
 2045.4879 
 
 1922 May 1 
 
 60.00 
 
 51.1372 
 
 8.8628 
 
 2036.6251 
 
 •• Nov. 1 
 
 1060.00 
 
 50.9156 
 
 1009.0844 
 
 1027.5407 
 
 1923 May 1 
 
 30.00 
 
 25.6885 
 
 4.3116 
 
 1023.2291 
 
 •♦ Nov. 1 
 
 30.00 
 
 25.5808 
 
 4.4192 
 
 1018.8099 
 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 229. — The remainder of the schedule continues as in 
 Article 187. 
 
 230. — For intermediate balances (Article 214), the interest 
 requires adjustment at the date of partial payment. We now 
 assume that the above group of bonds is to be valued on 
 January 1 and July 1 of each year. The values of the bonds 
 under consideration for January 1, 1921, would be, by inter- 
 polation 1017.3000 
 
 and 1042.4583 
 
 Afir2're2'ate. 
 
 
 
 
 . 20.^9 7.^ft.^ 
 
 
 Dates 
 
 Cash 
 Collections 
 
 Interest 
 at .025 
 
 Payments 
 on Principal 
 
 Investment 
 Value 
 
 1921 Jan. 1 
 
 
 
 
 2059.7583 
 
 - July 1 
 
 60.00 
 
 51.4940 
 
 8.5060 
 
 2051.2523 
 
 1922 Jan. 1 
 
 60.00 
 
 51.2813 
 
 8.7187 
 
 2042.5336 
 
 '• July 1 
 
 60.00 
 
 51.0633 
 
 8.9367 
 
 2033.5969 
 
 1923 Jan. 1 
 
 1050.00 
 
 42.5066 
 
 1007.4934 
 
 1026.1035 
 
 - July 1 
 
 30.00 
 
 25.6526 
 
 4.3474 
 
 1021.7561 
 
 
 etc. 
 
 etc. 
 
 etc. 
 
 etc. 
 
 ^31. — The entries under January 1, 1923, are peculiar. 
 The $1000 paid off was only in possession for 4 months, Yi 
 of a period; therefore, $20 is the appropriate sum to be con- 
 siderd as paid with it, and if it was kept in a separate account, 
 that is all which would be allocated to it. The other $1000 is 
 on interest during the full period and $30 is charged to it. 
 
Short Periods, Terminai, or Initial. 83 
 
 Cash entries : 
 
 For bond No. 1 par 1000 
 
 Gross interest thereon, .03, 4 months. . 20 
 Gross interest on No. 2, .03, 6 months._^ 30 
 
 1050 
 Interest entries : 
 
 Bond No. 1, Yz of period at .025 16.6667 
 
 Bond No. 2, full period, (Art. 210), at 
 
 .025 on 1033.5969 25.8399 
 
 42.5066 
 Applied on principal : 
 
 1050.00-42.5066== 1007.4934 
 
 232. — While this procedure may be applied where the 
 successiv partial payments on principal are of the most irregu- 
 lar amounts and intervals, their chief utility is in what are 
 known as serial bonds, where regular payments of principal 
 are made, usually annually. Each bond or group of bonds, as 
 it is dropt from the total, carries with it the appropriate cash 
 and interest entries exactly as exemplified above. 
 
 To Find thb Income Rate. 
 
 233. — When the cash-rate, the time and the price of the 
 bond are known, it is very desirable to know what is the 
 income-rate, for, of course, every one wishes to get the highest 
 income, security being equal. 
 
 234. — There is no positiv, direct method of doing this 
 beyond three periods, as an equation of a higher degree is not 
 directly soluble. There are only methods of approximation 
 and trial. 
 
 235. — When printed tables are accessible it is easy to 
 make a rough approximation by observing between what values 
 the given price lies. The smaller the interval between the 
 rates of the table, the closer is the approximation, and an addi- 
 tional decimal may be obtaind by proportion. With an ex- 
 tended table at close intervals, the result is suflSciently accu- 
 rate for commercial purposes. 
 
 236. — There is a method devised by the author which will 
 produce even greater accuracy, up to 12 places, at the expense 
 of considerable labor. It appears in the later editions of his 
 " Text- Book of the Accountancy of Investment," from which 
 it is here quoted. 
 
$4 The Doctrin of Interest. 
 
 To Find the Income Rate. 
 
 1. — Given a bond on which there is a premium or discount 
 Q, cash-rate c payable in n periods, what is the income-rate, /? 
 
 2. — Every premium or discount is the present worth of an 
 annuity of n terms, each instalment of which is the difference 
 of rates ; or it is the difference of rates X such an annuity of 
 $1 (Art. 193). Writing P for the present worth of an annuity 
 of $1 (Art. 108): Q = P X {c — i). If instead of the pre- 
 mium on $1 we use that on $100, we have 100 Q = P X 
 (100^ — 100/). It will not affect the value of the right-hand 
 side if we halve one factor and double the other. 100 Q = 
 ^ P X (200^ — 2000. 200^ is the rate per cent, per annum 
 as conventionally termed. Thus we pay 4 per cent, per 
 annum, meaning .02 per period. So also of 200/ for i. 
 
 3. — It is evident that if we divide lOOQ, the premium on 
 $100, by %P (which we will hereafter call the trial-divisor), 
 we shall find the difference of rates. But as the annuity de- 
 pends on the unknown rate, this does not help us at all. 
 
 4. — Let us assume the rate of income per annum to be any 
 rate whatever, and calculate the trial- divisor at that rate. 
 Then there is this property : If the assumed rate is too large, 
 the quotient or difference of rates will be too small, and yet will 
 be nearer the truth, and vice versa. From this approximate 
 difference of rates we derive a new rate and proceed with this 
 as a trial-rate. 
 
 The result of this trial will give a new rate still nearer 
 and so on. We may slightly modify any rate to make it more 
 easy to work. If we select our first trial-rate near the true 
 rate, fewer successiv approximations will be necessary. 
 
 5. — Fortunately for our purpose, any table of bond values 
 will readily give the trial- divisor, by taking the difference 
 between the values at the same income-rate of two successiv 
 $100 bonds, say a 3% and a 4%, a 5% and a 6%, always 1% 
 apart. 
 
 6. — For example, a 6% bond for $100 (semi-annual) for 
 50 years is sold at 133 . 00, what is the income rate ? 
 
To Find the Income Rate. 85 
 
 7. — With so large a premium as 33, the income-rate is 
 evidently much less than 6. Let us assume 4. Then from 
 any bond table we find on the 4% line the value of a 5% bond 
 
 to be 121.55 
 
 and that of a 4% bond to be 100 00 
 
 The first trial-divisor is therefore 21 . 55 
 
 33.00 -f- 21. 55 = 1.531. the difference of rates. 6 — 1.531 = 
 4.469, the new trial rate. Taking 4.45 as more convenient, 
 the new trial-divisor is 19.98. 33.00-^-19.98 =- 1.651. 
 6 — 1.651 = 4.349. We find that 20.315 is the trial-divisor 
 for 4.35. 33.00 -> 20. 315 = 1.6244. 6 — 1.6244 = 4.3756. 
 Next using 4.37, trial-divisor 20.25 : 33.00 -f- 20.25 = 4.37, 
 almost exactly, so that 4. 37 has reproduced itself. The value 
 of the bond at 4.37, as computed by logarithms, is 133.0069, 
 an error of less than one cent. 
 
 8. — It will be noticed in the foregoing example that the 
 results always swing to the opposit side of the true rate ; that 
 is, if the trial-rate is too large the next rate is too small, and 
 the true rate is between them. The successiv rates were 
 4. . .4.469. . .4.349. . .4.3756. . .4.37. 4.37 lies between any 
 pair of these. This is always the case with bonds above par. 
 With bonds below par it is different. The true rate always 
 lies beyond the approximation. 
 
 9. — As an example of a bond below, take a 3% bond pay- 
 able in 25 years. If purchased at 88. 25, what is the income- 
 rate ? The following may be the steps, the dividend being 
 always 11.75, the discount. 
 
 Trial-rates 3.70 3.725 3.7265 
 
 Trial- Divisor 16 . 2190 16 . 175 16 . 17245 
 
 Result 3.7244 3.7264 3.7265 
 
 As 3 . 7265 reproduces itself, it must be correct to the 4th 
 decimal ; and the value of a 3% bond for 25 years to yield 
 3.7265 is found by logarithms to be 88.25015. 
 
86 The Doctrin of Interest. 
 
 237. — Since the publication of the above method some 
 simplifications have been suggested, referring particularly to 
 the employment of Sprague's Extended Bond Tables. 
 Insted of obtaining a divisor as shown in par. 5, we may first 
 multiply Q itself by the interest-difference and then use the 
 entire variance at the trial-rate (Q'') as the divisor, giving the 
 same result. Thus in par. 7, where the first assumed rate is 
 4% and the interest difference 2, finding from the Table that 
 
 the premium is 430.983.52 
 
 33 X 2 -^43.098352 = 1.531 
 
 insted of 33-^^21.55 = 1.531 
 
 238. — The valuable suggestion has further been made by 
 Mr. E. S. Thomas that by using the two nearest income rates 
 and interpolating, five decimals may be obtaind at once. In 
 the example where Q = 33.00, 200^ = 6, select 4.35 as 200i^ 
 and 4.40 as 200/2. From the Tables we find opposit 4.35, 
 335,201.06 and opposit 4.40, 322,371.36, the interest-differences 
 being 1.65 and 1.60. 
 
 4 . 35% 33 . 00 X 1 . 65 -^ 33 . 520106 = 1 . 624398 
 
 4 . 40% 33 . 00 X 1 . 60 -^- 32 . 237176 = 1 . 637861 
 
 4 . 35 -f 1 . 624398 = 5 . 974398 = 6 . — 025602 
 
 4.40 + 1.637861 = 6.037861 = 6. + .037861 
 Difference in error . 063463 
 
 239. — If the same operation be performd on a third 
 rate, 4.45 : 
 
 4.45% 33.00 X 1.55 -h 30.974371 = 1.651333 
 
 4.40% 33.00 X 1.60 -f- 32.237176 = 1.637861 
 
 4.45+1.651333 = 6.101333 
 
 4.40 + 1.637861 = 6037861 
 
 Difference .063472 
 
 which differs so slightly from .063463 that further differenc- 
 ing may be neglected. 
 
To Find the Income Rate. 87 
 
 240. — By proportion, we may now ascertain at what rate 
 the coupon will become 6. From 4.35 to 4.40, an extent of 
 .05, it varies by .06347. 
 
 Therefore .06347 : .025602 : : .05 : .0201686 
 
 Add 4.35 
 
 Rate required 4.3701686 
 
 or safely 4.37017 
 
 241. — In the other example (par. 9), where Q = 11.75, 
 
 it appears from the Tables, (p. 10) , that the nearest rates are 
 
 3.70% and 3.75%; the corresponding values being 88.646703 
 
 and 87.900397 and the discounts 11.353297 and 12.099603. 
 
 3 . 70% — 11 . 75 X . 70 -f- 11 . 353297 = — . 72445916 
 
 3.75% — 11 75 X .75 -^ 12.099603 = — .72832968 
 
 3 . 80% — 11 . 75 X . 80 -f- 12 . 837828 = — . 73221109 
 
 Diff. 
 3.70 — .72445916 = 2.97554084 
 3.75— .72832968 = 3.02167032 -04612948 
 3 . 80 — . 73221109 -= 3 . 06778891 04611859 
 3 — 2.97554084 -= .02445916 
 
 .04612 : .02446916 : : .05 : .02650169 
 
 3.70 
 
 3.72650169 
 or safely 3.7265 
 
 242. — The process may be still further abridged by mak- 
 ing the interest-difference the standard of comparison insted 
 of the coupon, giving the same result. Thus in 238, insted of 
 4.35 + 1.624398 = 5.974398 = 6. - .025602 
 4.40 + 1.637861 =-6.037861 = 6. + -037861 
 
 .063463 
 might have been written 
 
 1.65 — 1.624398 = + .025602 
 
 1.60 — 1.637861 = — .037861 
 
 And in 241 
 
 .70 — .72445916 = — .02445916 
 
 •75 -- .72832968 = + .02167032 
 
 .80 — -73221109 = + .06778891 
 
 .063463 
 
 .04612948 
 .04611859 
 
88 
 
 INTEREST FORMULAS. 
 
 i = Rate of Interest, or the Interest on Unity for 1 period. 
 
 r~ 1 + /, or the Ratio of Increase. 
 
 / =^ Number of Periods of Time. 
 
 r*^ = (1 + 0* = Amount = s. 
 
 rt^-= (i + /)-t = -^^^^ Present Worth = /. 
 
 r^ — 1 = Compound Interest = I. 
 
 1 — r-^ = Compound Discount = D. 
 
 Amount of Annuity — l-^-r-^-r^-^-r^ + r^-^ = l/t = S. 
 
 Present Worth of Annuity = 1 + r^ + r^ + r^ . . r^-^= B/i = P. 
 
 y = Nominal Rate Per Annum. Coefficient of Double Fre- 
 quency = C(^) = 1 + V*- Coefficient of Half Frequency 
 
 = c(^^) = i + y2 (y"rT7— 1) 
 
 Sinking Fund = 1/S = 2*/I. 
 
 Amortization = 1/ P =:=^ /*/ D = //I + t. 
 
 c = Coupon Rate of Bond, or Cash Rate. 
 
 V • = Value of Bond at c to Earn i — r'^ -\- cV. 
 
 or = 1 + {c—i) P. 
 
 or= f-(f-l)P. 
 
UE ON THE LAST DATE 
 
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