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 GIFT TO THE LIBRARY 
 
 Civil Engineering Department 
 
 UNIVERSITY OF CALIFORNIA 
 
 BY 
 
 PROFESSOR FRANK SOULE 
 1912 
 
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UNIVERSITY OF CALITOrr 
 
 DEPARTMENT OF CIVIC ENGINEER! NO 
 Owfi.;. :<.£•, UA IFOR, 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 Microsoft Corporation 
 
 http://www.archive.org/details/elementsoflogariOOpeirrich 
 
THE ELEMENTS 
 
 OF 
 
 LOGARITHMS 
 
 WITH AN EXPLANATION OF THE 
 
 THREE AND FOUE PLACE TABLES 
 
 OF LOGARITHMIC AND TRIGONOMETRIC 
 FUNCTIONS 
 
 BY 
 
 JAMES MILLS PEIRCE 
 
 UNIVERSITY PROFESSOR OP MATHEMATICS IN HARVARD UNIVERSITY 
 
 BOSTON 
 
 PUBLISHED BY GINN BROTHERS 
 
 1873 
 
>4 
 
 Engineering 
 Library 
 
 :ering 
 library 
 
 Entered according to Act of Congress, in the year 1873, by 
 
 JAMES MILLS REIRCE, 
 
 In the Office of the Librarian of Congress, at Washington. 
 
 camhridp.e: 
 press of john wilson and son. 
 
PREFACE. 
 
 r I ^HIS little book is meant chiefly for beginners ; 
 but I have also had in view the wants of 
 more advanced students who are seeking to refresh 
 and deepen their knowledge. The sections included 
 in brackets are inserted for their benefit; and it is 
 my purpose to add, in a future edition, chapters on 
 the natural system, series, and the errors incident to 
 simple interpolation, so as to make the book a com- 
 plete manual of all that can be considered as belong- 
 ing to the elements of the subject. 
 
 Logarithms ought not to be comprised, as they 
 often are, in the midst of a treatise on algebra. For, 
 in the first place, they are not algebraic functions; 
 and, besides this, the student is unlikely to form an 
 adequate comprehension of their purpose or to appre- 
 ciate the importance of acquiring skill in the use of 
 his tables, if he takes them up in the course of a 
 study to which they have no application. If loga- 
 rithms must needs be combined with any other branch 
 of mathematics, their true alliance, on grounds both 
 
 789544 
 
IV PREFACE. 
 
 theoretical and practical, is not with algebra, but 
 with trigonometry. But it seems to me desirable 
 that so important a subject should be studied by 
 itself, and studied fully and thoroughly; and the 
 aim of the teacher should be the double one of ex- 
 pounding the doctrine of logarithms in a concise, 
 accurate, and clear form, and of inculcating and 
 enforcing good practical principles in the use of 
 tables. It has been my endeavor, accordingly, in 
 writing these pages, not, of course, to convert the 
 learner into an accomplished computer, but to set 
 him on the right road towards becoming such if he 
 pleases, and, at all events, to lead him to form the 
 sound habits of work without which little beneficial 
 discipline is derivable from this study. 
 
 I hope that teachers who are preparing pupils for 
 Harvard College, where logarithms are now required 
 for admission, will find that this volume furnishes 
 them with a satisfactory text-book. The course I 
 would recommend for such students comprises all that 
 precedes the Appendix, except the bracketed portions. 
 Those teachers who have not time to go over so much 
 ground will find it possible to omit also §§ 7, 8, 13, 
 15, 17, 18, and 43, and perhaps some other sections. 
 Those, on the other hand, who are able to exceed 
 the course described above are advised to take up 
 the table of Logarithms of Sums and Differences. 
 
 Only the rudiments of the theory of logarithms 
 
PREFACE. V 
 
 are required at the examination, but candidates are 
 expected to be thoroughly drilled in practical work. 
 Applicants in Course II., however, should be well 
 prepared on all the unbracketed parts of the book, 
 including the Appendix. The use of the trigono- 
 metric tables is not required for admission in 
 Course I. 
 
 I would further respectfully ask the attention of 
 professors of mathematics in other colleges to the 
 desirableness of putting into the hands of students 
 tables of three or four places instead of the bulky 
 volumes which even practical computers need for 
 only the more delicate kinds of work, and especially 
 instead of the inconvenient and poorly constructed 
 tables often found bound up in the same covers with 
 a work on trigonometry. This has been pointed out 
 by several of the best authorities abroad, and is, no 
 doubt, widely acknowledged here as well. I venture 
 to hope, therefore, that the publication of this little 
 treatise will induce a more general employment of 
 the tables to which it is designed as a companion. 
 
 J. M. Peiece. 
 
 Cambridge. October 1873. 
 
CONTENTS. 
 
 Page 
 
 CHAPTER I. — General Principles 1 
 
 CHAPTER H. — The Denary System 17 
 
 CHAPTER HI. — Computation by Logarithms . . 26 
 
 Table of Proportional Parts 37 
 
 Three-Place Table of Logarithms of Numbers ... 38 
 
 Four-Place Table of Logarithms of Numbers ... 39 
 
 Arithmetical Complement. Reciprocal 43 
 
 Multiplication by Logarithms 44 
 
 Division by Logarithms 45 
 
 The Rule of Three by Logarithms 46 
 
 Involution by Logarithms 46 
 
 Evolution by Logarithms 47 
 
 Compound Operations 48 
 
 Exponential Equations 48 
 
 Logarithms of Sums and Differences 49 
 
 APPENDIX. — Explanation of the Trigonometric 
 
 Tables 55 
 
 Three-Place Table of Trigonometric Functions ... 56 
 Four-Place Table of Logarithmic Trigonometric Func- 
 tions 63 
 
 Inverse Trigonometric Functions 69 
 
 Examples in Trigonometry adapted to the Four-Place 
 
 Tables 76 
 
THE 
 
 ELEMENTS OF LOGARITHMS. 
 
 CHAPTER I. 
 
 GENERAL PRINCIPLES. 
 
 § 1* Logarithms are numbers that are substituted in 
 computation for other numbers, to which they bear such 
 a relation that the operations to be performed on the 
 latter are represented by simpler operations performed on 
 the former. The method of logarithms was first pro- 
 pounded in 1614, in a book entitled Mirifici Logarith- 
 morum Canonis Description by John Napier (latinized 
 Neperus), Baron of Merchiston in Scotland, who was 
 born about 1550, and died in 1618, four years after the 
 publication of his memorable invention. This method 
 has contributed very powerfully to the modern advance 
 of science, and especially of astronomy, by facilitating 
 the laborious calculations without which that advance 
 could not have been made. It is also constantly employed 
 in surveying, navigation, and other branches of practical 
 mathematics ; and it may be used with much advantage 
 in all multiplications and divisions of numbers of three 
 or more figures. 
 
 Besides their usefulness in computation, logarithms 
 fill an important place in the higher theoretical mathe- 
 matics ; but, in this book, we shall consider them almost 
 wholly in their practical aspiect. 
 
1 THE ELEMENTS OF LOGARITHMS. 
 
 The word logarithm, which is due to Napier himself, 
 is formed from Xo'/o^, ratio, and df/idfiog, number, and 
 means a number that indicates a ratio. It refers to the 
 proposition, which will be proved in its proper place, and 
 which was made by Napier his fundamental theorem, 
 that the difference of two logarithms determines the 
 ratio of the numbers for which they stand, so that an 
 arithmetical series of logarithms corresponds to a geo- 
 metric series of numbers. 
 
 § So The base of a system of logarithms is a fixed 
 number to which all numbers are referred in that system. 
 It may have any positive value except unity. 
 
 The logarithm of a number in any system is the 
 exponent of the power to which the base of the system 
 must be raised to produce that number. 
 
 The antilogarithm of a number is that number of 
 which the given number is the logarithm; in other 
 words, it is that power of the base of which the given 
 number is the exponent. 
 
 Thus, in a system of logarithms of which 8 is the 
 base, 
 
 log 8 = 1, antilog 1 = 8, 
 
 log G4 = 2, antilog 2 = G4, 
 
 log 512 = 3, antilog 3 = 512, 
 
 log 4096 = 4, &c, antilog 4 = 4096, &c. 
 
 In general, if x and u have any such values as to 
 satisfy the equation 
 
 we have, in the system of which a is the base, 
 
 u = log x, x = antilog u ; 
 
 and either of the two latter equations may be regarded as 
 equivalent to the former. 
 
GENERAL PRINCIPLES. 3 
 
 Napier at first called logarithms artificial numbers, 
 and antilogarithms natural numbers ; and the latter desig- 
 nation is still often used. The word antilogarithm is 
 modern ; and, though a convenient term, is used by a few 
 writers only. 
 
 §3. The following formulas are proved in algebra, 
 and the proofs, which are readily derived from the prin- 
 ciple that the exponent of a quantity denotes the 
 number of times the quantity is taken as a factor in 
 the term into which it enters, should here be given by 
 the student: — 
 
 a u a v a w = a u + v + w + , (1) 
 
 a w 
 
 — = a u ~ v , (2) 
 
 aP v } 
 
 (a u ) v = a™, (3) 
 
 „ u 
 
 Va u = a». (4) 
 
 § 4-. The following principles are less familiar : — 
 If, in (2), u = v, 
 a u 
 a v ' 
 
 /. rf = l; (5) 
 
 that is, the zero power of any quantity is unity. This 
 result conforms to the principle cited in § 3. For to 
 introduce any quantity into a term zero times as a factor 
 is equivalent to introducing unity as a factor, since either 
 of these operations would leave the term unchanged. 
 That is, a°b = 5, since b is multiplied by a zero times, 
 and also lb=b; so that a°b = lb, or a = 1. 
 If, in (2), u = 0, then, by (5) and (2), 
 
 a u 1 
 
 — = — , a u ~ v = a~ v : 
 
 /. a-v = — ; (6) 
 
4 THE ELEMENTS OF LOGAEITIIMS. 
 
 that is, a quantity with a negative exponent denotes 
 the reciprocal* of the same quantity with the corre- 
 sponding positive exponent. Thus, 8~ 2 = j%. This re- 
 sult may also be brought into conformity with the prin- 
 ciple of § 3. For to divide by a v , or to multiply by — 
 is to take away a as a factor v times; that is, to take it v 
 times less than before ; and this is precisely what must be 
 meant by taking it ( — v) times more than before. Hence 
 — is the same thins? as a~ v . 
 
 If, in (4), u is not divisible by v, the fractional ex- 
 ponent may still be used to denote the operation of 
 successively raising to the power denoted by the numer- 
 ator and extracting the root denoted by the denominator. 
 
 Thus, 8^ = V8 2 = 4, 8* = J. This result, again, is con- 
 formable to our general principle. For Va" is that 
 quantity which, taken v times as a factor, gives a taken 
 u times as a factor. We may then regard it as a taken 
 
 7 times as a factor, and hence write Wa u = a v . 
 
 Exponents may then be either positive, negative, or 
 null, and may be either integral or fractional. They may 
 even be incommensurable^ for the value of a power hav- 
 ing an incommensurable exponent can be found, within 
 any required limits of error, by taking a sufficiently ap- 
 proximate fractional value of the exponent. 
 
 * The reciprocal of a quantity is the quotient obtained by dividing 
 unity by that quantity; e.g., the reciprocal of x is — . 
 
 f Two quantities are said to be incommensurable to each other, when 
 they have no common measure. A number is called incomnu araradfa, when 
 it is incommensurable to unity, and therefore cannot be exactly expressed 
 by an integer, or by the ratio of two integers; i.e., by a fraction. Circulating 
 decimals must not be confounded with incommensurable numbers. Thus, 
 4/2 is incommensurable, but O.G is not incommensurable. The ratio of two 
 quantities incommensurable to each other is an incommensurable number. 
 An incommensurable number can be expressed fractionally within, any re- 
 quired limit of accuracy. Thus, L4142 expresses y2 within ±0.00005. In- 
 commensurable numbers are also called surds. 
 
GENERAL PRINCIPLES. D 
 
 It is easily proved that formulas (1-4) hold for any 
 values of the exponents. For we have shown that the 
 general principle from which those formulas are derived 
 may be extended to all exponents. For example, we 
 have 
 
 VaVa Va 
 
 § 5. Since any number may be an exponent, any num- 
 ber may be a logarithm. 
 Thus, if 8 is the base, 
 
 8-2 = &, 8* = 4, 8"* = J. 
 
 .«.— 2 = ]og& f = log4, -§=logi. 
 
 § 6. Since a = 1, therefore, 
 
 in any system, log 1 = 0, antilog = 1. (7) 
 Since a 1 = a, therefore, 
 
 in any system, log base = 1, antilog 1 = base. (8) 
 By (6), if a is the base, 
 
 or the logarithm of the reciprocal of any number in any 
 system is equal to the negative of the logarithm of the 
 number in the same system. 
 
 If a > 1, the result obtained by taking it as a factor 
 an infinite number of times is itself infinite ; so that 
 
 a+« =oo,«-° =^=4- =0.* 
 
 * If x increases indefinitely, - decreases indefinitely ; and if x decreases 
 indefinitely, — increases indefinitely. Hence 
 
 1=0, L=oo. 
 co 
 
 1* 
 
Ill 
 
 THE ELEMENTS OP LOGARITHMS. 
 
 A log oo = + oo , antilog (+ oo ) = oo , \ ~q. 
 
 log = — oo , antilog ( — oo ) = 0. ) 
 
 If a < 1, a— 1 > 1, and hence 
 (a-i ) +• =a-» =oo, a+- =-_ =4 = 0. 
 
 A log oo = — oo , antilog (+ oo ) = 0, \ ^. 
 
 log = -j- oo , antilog ( — oo ) = oo . ) 
 
 § 7. If the base is greater than unity (as is the case 
 „ the common systems), its positive integral j)owers are 
 all greater than unity, and their reciprocals, the negative 
 integral powers, are, consequently, all less than unity. 
 For example, 5 3 > 1, 5~ 3 < 1, &c. 
 
 771 
 
 Any fractional power of the base, a 11 , is, in like manner, 
 greater or less than unity, according as its nth power, a m , 
 is greater or less than unity (provided n is positive) ; and 
 a m j s greater or less than unity, according as m is positive 
 or negative ; that is, according as the original exponent 
 — is positive or negative. Hence, all the positive powers 
 of the base (both integral and fractional) are greater than 
 unity, and all the negative powers are less than unity ; so 
 that, if the base is greater than unity, the logarithms of 
 all numbers greater than unity are positive, and the loga- 
 rithms of all numbers less than unity are negative. 
 
 It is shown in the same way that, if the base is less 
 than unity, the logarithms of all numbers less than unity 
 are positive, and the logarithms of all numbers greater 
 than unity are negative. 
 
 Thus, if 8 is the base, log 4 = + h log J = — § ; but 
 if £ is the base, log 4 = — §, log \ = + }. 
 
 § 8. If x and (x -\- h) are two numbers, and u and 
 (u -\- k) their logarithms, then 
 x = a u , x + h = a u + k = a u a k by (1), h = a u (a k — 1). 
 
GENERAL PRINCIPLES. 7 
 
 If then x -f- h > x ; that is. if h is positive ; a k must be 
 greater than 1 ; so that, by § 7, if a > 1, Jc is positive, and 
 w -J- k > w, but if a <[ 1, & is negative, and u -f- k <^ u. 
 
 Hence, the greater of two numbers has the greater 
 logarithm in a system of which the base is greater than 
 witty, and the less logarithm in a system of which the 
 base is less than unity. 
 
 Thus, if 8 is the base, log 8 > log 4 ; but if \ is the 
 base, log 8 < log 4. 
 
 [§ O.* We have seen that rt» = l, if w = 0, and that 
 a u differs from 1, if u differs from 0. But it is possible to 
 make the difference between a u and 1 less than any 
 quantity that may be assigned by making u sufficiently 
 small. 
 
 For if m is any value of u (positive, if a > 1 ; negative, 
 if a < 1), and if a m =/>> 1, then if we reduce the value 
 of u to Jm, x = ah m =pi = V p. Now, by the principle 
 a* — V* = (a + b) (a — b), 
 
 But since p > 1, \fp -f- 1 > 2. 
 
 .-. v!p-i<J(i>-1). 
 
 Hence, by halving the value of u, the difference (x — 1) 
 is reduced to an amount less than half its former value. 
 Now we can successively halve the value of u as many 
 times as we please, and if (a; — 1) were halved at the 
 same time, we could, by carrying the process far enough, 
 make that difference less than any assigned quantity; 
 then we can certainly accomplish this, when (x — 1) is 
 reduced more rapidly than by successive halving. 
 
 We have seen that if x ^> 1, we can find a value of u 
 that will make (x — 1) as small as we please. Then the 
 
 * Sections included in brackets may be omitted by the beginner. 
 
8 THE ELEMENTS OF LOGARITHMS. 
 
 reciprocal of x, which is less than 1, can be made to differ 
 from 1 as little as we please, since 
 
 1 — 1 = *=*<> — L 
 
 X X ^ 
 
 But the value of u for — is the negative of the value of 
 u for x. See (9). 
 
 Hence, by making the logarithm sufficiently near 0, and 
 either positive or negative, we can bring (he antilogarithm 
 as near 1 as ice please, making it at the same time either 
 greater or less than 1 at our discretion.^ 
 
 [§ 10. If x and (x -\-h) are two numbers, and u and 
 (u -\- k) their logarithms, then, as in § 8, 
 
 h — a u («* — 1) =x (a* — 1). 
 
 But, by making k sufficiently small, we can, by § 9, 
 make (a h — 1) as small as we please, and therefore small 
 enough to render h less than any quantity that may be 
 assigned. Hence, by making a sufficiently small change 
 in the value of a logarithm, we can make the correspond- 
 ing change in the antilogarithm as small as we please; 
 and if the logarithm increases continuously (that is, by 
 insensible gradations) from — go to -)- go , the antiloga- 
 rithm will also change its value continuously ; increasing 
 from to -f-oo, if the base is greater than unity ; and 
 decreasing from -f-oo to 0, if the base is less than unity ; 
 and, in either case, passing through every intermediate 
 value. 
 
 We see then that, in any system of logarithms, every 
 positive number has one and only one real logarithm, and 
 conversely that every real number (positive or negative ) 
 has one and only one antilogarithm, which is always 
 p>ositive. 
 
 Negative numbers have no real logarithms, and when 
 they occur in logarithmic calculations, the arithmetical 
 
GENERAL PRINCIPLES. 9 
 
 value of the number and its sign must be considered se- 
 parately.] 
 
 [§11. The word real is used in the last article in a 
 technical sense, which is likely to be new to the student. 
 Heal quantities, or reals, are whole numbers, ratios be- 
 tween whole numbers (that is, fractions), and incommen- 
 surable numbers (which can be approximately represented 
 by fractions), and are either positive or negative ; that is 
 to say, they are the ordinary quantities of arithmetic and 
 elementary algebra. Distinguished from them are imagi- 
 nary quantities or imaginaries. Of these algebra affords 
 examples in the even roots of negative quantities (such 
 as \/— 1 ), which cannot be real, since no real quan- 
 tity produces a negative result when raised to an even 
 power. In the elementary mathematics, imaginaries ap- 
 pear merely as absurd expressions, to which little atten- 
 tion is paid ; but in the higher study of the relations of 
 quantity, they assume great importance. In the higher 
 theory of logarithms, it is found that every number 
 (whether positive, negative, or imaginary) has, in any 
 system, an infinity of logarithms, which are, in general, 
 all imaginary ; but that if the number is real and positive, 
 one and only one of its logarithms is real, and this is the 
 logarithm ordinarily considered. 
 
 Imaginary logarithms cannot be made use of in prac- 
 tical computation, and we shall not have occasion to 
 consider them further. We only refer to them here in 
 order to explain why we have so carefully guarded the 
 language of the foregoing article.] 
 
 [§ 12. We have said that a real logarithm has but 
 one antilogarithm, which is always positive. Other values 
 of the antilogarithm might also be admitted. For exam- 
 ple, since ah = s/a has two values, we might consider 
 
10 THE ELEMENTS OF LOGARITHMS. 
 
 antilog 0.5 as having two values, one positive and the 
 other negative. But mathematicians have found it best 
 to admit only one value of the antilogarithm, even in that 
 higher theory of logarithms to which allusion has just 
 been made ; and in the exponential equation 
 
 x is regarded as having but one value for each value of u, 
 and as being real and positive whenever u is real, assum- 
 ing a to be real and positive.] 
 
 § 13* If several logarithms form an arithmetical pro- 
 gression, their antilogarithms form a geometric progres- 
 sion. 
 
 For if the logarithms are 
 
 u> u -f- k, u -f- 2£, u -\- 3&, 
 
 and if we denote a u by cc, and a k by A, the antilogarithms 
 of the given logarithms will be, by (1) and (3), 
 
 x, hx, h 2 x, h z X) 
 
 It will be observed that £, the constant difference in the 
 series of logarithms, is the logarithm of h, the constant 
 ratio in the series of antilogarithms. 
 
 Example. In the system of which 8 is the base, 
 
 
 antilog 4 
 
 
 = 
 
 4096, 
 
 
 antilog 31 
 
 
 = 
 
 1024, 
 
 
 antilog 2§ 
 
 
 r=: 
 
 256, 
 
 
 antilog 2 
 
 
 = 
 
 64, 
 
 
 antilog \\ 
 
 
 = 
 
 16, 
 
 
 antilog f 
 
 
 = 
 
 4, 
 
 
 antilog 
 
 
 rz= 
 
 1, 
 
 
 antilog (- 
 
 i) 
 
 = 
 
 b 
 
 
 antilog (— 
 
 -14) 
 
 = 
 
 tV, 
 
 
 antilog (— 
 
 -2) 
 
 = 
 
 T&, &C 
 
 Here fc= 
 
 _f,A=i = 
 
 = antilog 
 
 (-§)• 
 
GENERAL PRINCIPLES. 11 
 
 [§ 14L The terms of any arithmetical progression of 
 which one term is may be considered as logarithms 
 of the corresponding terms of any geometric progression 
 of which one term is 1 ; those terms being regarded as 
 corresponding which are similarly situated relatively to 
 the terms and 1 respectively, and the base of the system 
 being that root of the constant ratio of the latter series of 
 which the exponent is the constant difference of the former 
 series. 
 
 Let 7c ±= the constant difference of the arithmetical pro- 
 gression, 
 h ±± the constant ratio of the geometric progres- 
 sion. 
 Then mk and h m will denote any two corresponding 
 terms of the two series, and if we take «, the base of the 
 system, such that 
 
 k i_ 
 
 a = wh = h k , or a k = A, or a"* = h m t 
 we have 
 
 Jc = logh, m7c = \og h m ; 
 which proves the proposition, and shows that the constant 
 difference of the arithmetical series is the logarithm of 
 the constant ratio of the geometric series.] 
 
 § 15. Let w, u\ and u" be three successive logarithms 
 in the series of § 13, and x, x', and x" their antilogarithms. 
 Then 
 
 u" — u'=.u' — u = &, x' — x = (h — l)x, 
 x"—x'= (A — l) hx = h (x' — x). 
 
 If now Jc be taken very near 0, h will be very near 1, since 
 h = a k ; and consequently 
 
 (x" — x 1 ) = {x' — x) approximately ; 
 that is, if the successive differences of three logarithms are 
 equal and very small, the differences of their antiloga- 
 rithms are approximately equal. 
 
12 TIIE ELEMENTS OF LOGARITHMS. 
 
 Hence it follows that if the successive differences of 
 three logarithms are very small and not equal, the differ- 
 ences of their antilogarithms are approximately propor- 
 tional to those of the logarithms. For if the successive 
 differences of the logarithms be divided into equal parts, 
 those of the antilogarithms will be divided into the same 
 number of parts, which will be approximately equal. 
 
 We do not here stop to inquire hoic small the differ- 
 ences must be in order that the error resulting from the 
 application of this theorem may remain within definite 
 limits. That is a question which requires for its answer 
 higher principles of logarithms than those explained in 
 this chapter. The theorem itself is, however, necessary in 
 the practical use of logarithms. 
 
 § 1G. We have now considered the general nature of 
 logarithms. Their utility in computation is derived from 
 the following theorems ; — 
 
 I. The logarithm of the continued product of several 
 quantities is the sum of their logarithms. 
 
 II. The logarithm of the quotient of two quantities is 
 the excess of the logarithm of the dividend over that of 
 the divisor. 
 
 III. The logarithm of any power of a quantity is the 
 product obtained by multiplying the logarithm of the 
 quantity by the exponent of the power. 
 
 IV. The logarithm of any root of a quantity is the 
 quotient obtained by dividing the logarithm of the 
 quantity by the exponent of the root. 
 
 These theorems are also conveniently stated in the 
 following formulas: — 
 
 I. log (xyz ) = log*+logy+logz + , (12) 
 
 which may be written, if Z is used for the word sum, and 
 II for the word product. 
 
GENERAL PRINCIPLES. 13 
 
 I. \ogIIx = Z\ogx. (13) 
 
 X 
 
 II. log — = log x — log y. (14) 
 
 III. log x n = n log x. (15) 
 
 n. 1 
 
 IY. log Vx = — log x. (16) 
 
 The proofs of these theorems are easily derived from 
 formulas (1-4). 
 
 I. Let u = log x, v = log y, w = log 2, &c. 
 Then, if a is the base of the system, 
 
 x = a u i y = a v , z = a w t &c. 
 
 Then, by (1), 
 
 xyz =za u + v + w + , 
 
 or IIx=z a2« . 
 
 .-. by § 2, log II x = Zu z= 2 log x. 
 
 II. With the same notation, by (2), 
 
 2/ 
 .-. by § 2, log — = u — v = \ogx — logy. 
 
 if 
 
 III. With the same notation, by (3), 
 
 x n _ a nu t 
 
 ,\ by § 2, log x n = nu = n log a; , 
 
 where it should be observed that n may be positive or 
 negative, integral or fractional. 
 
 IV. With the same notation, by (4), 
 
 u 
 
 n _ 
 
 Vx = a n . 
 
 n ifi 
 
 .\ by § 2, logV x = — = — log x. 
 n n ° 
 
14 THE ELEMENTS OF LOGARITHMS. 
 
 These theorems show that the use of logarithms en- 
 ables us to replace the operations of multiplication, divi- 
 sion, involution, and evolution, by operations respectively 
 simpler. 
 
 Multiplication is replaced by Addition. 
 Division is replaced by Subtraction. 
 Involution is replaced by Multiplication. 
 Evolution is replaced by Division. 
 
 § 17. The logarithms of proportional numbers are 
 equidifferent. 
 
 For the proportion 
 
 m p 
 n q 
 
 gives, by (14), 
 
 log m — log n = \ogp — log q ; (17) 
 
 whence is derived the following formula for the logarithm 
 of the fourth term of a proportion, 
 
 log q = log p -f- log n — log m. (18) 
 
 This proposition is given by Napier as the First 
 Proposition of the theory of logarithms. 
 
 § 18. An exponential equation is easily solved by 
 means of logarithms. 
 
 For let the equation be 
 
 b x = m y 
 
 where x is the unknown quantity, and b and m are posi- 
 tive. Theu, by (15), 
 
 log m = log b x = x log b. 
 
GENERAL PRINCIPLES. 15 
 
 [§ 19. If b be made the base of a system of logarithms, 
 then, by § 2, x = log m in the system of which b is the 
 base. We obtain, then, from (19) the following theo- 
 rem: — 
 
 If the logarithm of a number in any system be divided 
 by the logarithm of a second number in the same system, 
 the quotient is the logarithm of the first number in a new 
 system, of which the second number is the base. 
 
 When logarithms belonging to different systems are 
 brought into the same formula, it is convenient to distin- 
 guish them by writing the symbol of the base as a suffix 
 to the symbol " log." Thus, 
 
 log a m = log m in the system of which a is the base, 
 
 log& m = log m b ; 
 
 and we have 
 
 '•-"Sr-] (20) 
 
 [§ 30. If m = a, log a m = 1, by (8), and (20) becomes 
 
 log & a s= j^-p or loga b. log 6 a = 1 ; (21) 
 
 so that, when we change from one system of logarithms to 
 another, the logarithm of the old base in the new system 
 is the reciprocal of the logarithm of the new base in the 
 old system.'] 
 
 [§ 21. Combining (20) and (21), we have 
 
 log & m = log & a. loga m.] (22) 
 
 [§ 22. We see, by (20) and (22), that, if the logarithms 
 of a series of numbers are computed in any system, we 
 can convert them to any other system, by dividing all the 
 computed logarithms by one fixed quantity, namely the 
 logarithm of the new base in old system, or by multiply- 
 
16 THE ELEMENTS OF LOGARITHMS. 
 
 ing all the computed logarithms by one fixed quantity, 
 namely the logarithm of the old base in the new system. 
 Thus, a system of logarithms founded on the base 8 
 can be converted into a system of which £ is the base, by 
 multiplying the logarithms of the former system by — 3, 
 which equals log 8 in the latter system (since (£)~ 3 = 8) ; 
 as will be seen by the following table : 
 
 Nos. 
 
 logs 
 
 logs 
 
 
 (base 8) 
 
 (base h) 
 
 4096, 
 
 4, 
 
 -12, 
 
 1024, 
 
 3J, 
 
 -10, 
 
 256, 
 
 2§, 
 
 - 8, 
 
 64, 
 
 2, 
 
 - 6, 
 
 16, 
 
 H, 
 
 - 4, 
 
 8, 
 
 1, 
 
 — 3, 
 
 4, 
 
 h 
 
 - 2, 
 
 2, 
 
 h 
 
 - 1, 
 
 1. 
 
 0, 
 
 0, 
 
 h 
 
 -h 
 
 1, 
 
 h 
 
 — h 
 
 2, 
 
 h 
 
 -1, 
 
 3, 
 
 A, 
 
 -1J> 
 
 4-] 
 
THE DENARY SYSTEM. 17 
 
 CHAPTER n. 
 
 THE DENARY SYSTEM. 
 
 § 33. There is an infinite variety of possible systems 
 of logarithms ; but only three have been actually used by 
 mathematicians. These are the system of Napier, the 
 natural system, and the denary system. 
 
 The denary, or decimal, system is that which has the 
 number 10 for its base. In practical computation, it is 
 far more convenient than any other, and is now used ex- 
 clusively, on account of its relation to the ordinary nu- 
 meral system of arithmetic. It was proposed in 1617 by 
 Henry Briggs, then Professor of Geometry in Gresham 
 College, London, and afterwards Savilian Professor of 
 Geometry at Oxford. This mathematician was among 
 the first to recognize the importance of the invention of 
 logarithms, and he made two journeys to Scotland for 
 the purpose of visiting Napier, in consultation with whom 
 he formed his new system, — a system distinguished from 
 the original one not only in being founded on the number 
 10, but also in some important simplifications of theory. 
 
 Denary logarithms are sometimes called JBriggsian, 
 and sometimes common, or vulgar. 
 
 The natural system, though not so well adapted to 
 use in computation as the denary, is the most important 
 of all systems in the higher theory of logarithms. It may 
 be regarded as a modification of the system of Napier, 
 with which it is often erroneously confounded. But the 
 original Neperian system was immediately superseded by 
 2* 
 
18 THE ELEMENTS OF LOGARITHMS. 
 
 that of Briggs, with the approval of Napier himself, and 
 now has only an historical interest. 
 
 [§ 24. It is a principle of the Theory of Numbers that 
 a composite number can be resolved into prime factors in 
 only one way. If then m is any positive integer, 10 m = 
 2 m 5 m can only be formed of rn 2's and m 5's. If, again, n 
 
 is any positive integer not a divisor of m, V10 OT = 10 n 
 cannot be an integer. For the prime factors of such an 
 integer must be divisors of 10 m and must therefore be a 
 certain number of 2's and 5's, which repented n times 
 would give m 2's and m 5's ; but this is impossible, since 
 
 n is not a divisor of m. Again, VlO m cannot be a com- 
 mensurable fraction irreducible to an integer. For the 
 ?ith power of such a fraction would itself be a fraction 
 
 in 
 
 irreducible to an integer. Moreover, 10 ~~ n can be neither 
 an integer nor a commensurable fraction ; for, if it were, 
 
 m 
 
 its reciprocal, 10* would be commensurable. Hence, if u 
 is a commensurable irreducible fraction (positive or nega- 
 tive), 10" is incommensurable; and hence, conversely if 
 10" is commensurable, u is incommensurable, if it is not 
 an integer. In the denary system, then, the logarithms 
 of all commensurable numbers, except the integral powers 
 of 10, are incommensurable, and the antilogarithms of 
 all commensurable fractional logarithms are incommen- 
 surable^ 
 
 § 25. The denary logarithm of any number is either 
 an integer, or else consists of an integer (which may be 0) 
 plus a fraction, which is customarily expressed decimally 
 to a greater or less number of places. The integral part 
 of the logarithm is called its characteristic, and the deci- 
 mal part its mantissa. Thus the characteristic of the 
 logarithm 1.6875 . . is 1, and its mantissa is .6875 . . ; 
 
THE DENARY SYSTEM. 19 
 
 the characteristic of 0.3010 . . is 0, and its mantissa is 
 .3010 . . . 
 
 § S6. The significant figures of any number are those 
 figures that remain when all zeros at the beginning and 
 end of the number are omitted. Thus, the significant 
 figures of the numbers 81004000, 81004, 810.04, and 
 0.081004, are 81004. 
 
 It will be seen that zero is a significant figure, when it 
 occurs between other significant figures, so that it cannot 
 be omitted without changing their relative numeral posi- 
 tion. 
 
 § 27. If two numbers consist of the same series of 
 significant figures, their logarithms, in the denary system, 
 have the same mantissa, and the difference of their char- 
 acteristics is equal to the number of numeral places by 
 which the units' place of one number is removed from the 
 units' place of the other. 
 
 For it is a principle of arithmetical numeration that 
 the operation of moving the units' place in any number n 
 places is equivalent to multiplying the number by 10 ± w , 
 the exponent being positive if the units' place is moved to 
 the right, negative if it is moved to the left. But to mul- 
 tiply a number by 10 ±» is, by (13), equivalent to adding 
 the integer ± n to its logarithm ; and this addition does 
 not affect the mantissa of the logarithm, but increases or 
 diminishes its characteristic by n, the number of places by 
 which the units' place has been removed. 
 Thus, if we know that 
 
 log 48.7 = 1.6875 . . , 
 it follows that 
 
 log 48700 = 4.6875 . . , 
 
 log 4.87 = 0.6875 . . , 
 
 log 0.00487 = — 3 + .6875 . . 
 = — 2.3125 . . . 
 
20 THE ELEMENTS OF LOGAEITHMS. 
 
 The above theorem expresses the fundamental property 
 of the denary system, on which the peculiar utility of that 
 system depends. 
 
 § 28. In any system of logarithms, log 1 = 0, by (7). 
 We have, then, in the denary system, by § 27, 
 
 log 10 = 1, log 0.1 = — 1 
 
 log 100 = 2, log 0.01 = — 2 
 
 log 1000 = 3, log 0.001 = — 3(* ( 23 ) 
 
 log 10000 = 4, log 0.0001 = — 4. 
 
 &c, &c., 
 
 that is, the logarithm of a number which co?isists of a 
 
 single figure 1 and cyphers is an integer ', and is equal to 
 
 the number of places by which the units' place is removed 
 
 from the figure 1, being positive or negative, according as 
 
 the units' place is on the right or left of the figure 1, and 
 
 null if 1 is in the units' place. 
 
 The principle of numeration referred to in § 27 shows 
 that the numbers here spoken of are the integral powers 
 of 10 ; that is, that 
 
 10i= io, 10-1 = 0.1, 
 
 102=100, 10- 2 = 0.01, 
 
 and (23) may also be obtained directly from these equa- 
 tions, by the definition of a logarithm, § 2. 
 
 § 29. Any positive number is either an integral 
 power of 10, or else it lies between two successive in- 
 tegral powers, of which the lower contains the figure 1 
 in the same numeral place as the first significant figure of 
 the given number, and the higher contains the figure 1 
 in the next place to the left. For example, 
 
 10 < 48.7 < 100, 
 0.001 < 0.00487 <0.01, 
 &c. 
 
THE DENARY SYSTEM. 21 
 
 Then, by § 8, the logarithm of a number which is not 
 an integral power of 10 lies between the two successive 
 integers which are the logarithms of the integral powers 
 determined as above ; and it is therefore equal to the less 
 of those two integers plus a fractional quantity less than 
 1. The latter quantity is the mantissa of the logarithm; 
 the integer is the characteristic of the logarithm. 
 
 Hence, the characteristic of the logarithm of any num- 
 ber, in the denary system, is equal to the number of places 
 by which the units' place is removed from the first signifi- 
 cant figure, being positive if the units' place is on the 
 right of the first significant figure (that is, if the number 
 > 10), negative if it is on the left (that is, if the number 
 < 1), and null if the first significant figure is in the 
 units' place (that is, if the number is between 1 and 
 10). 
 Thus, 
 
 characteristic of log 
 
 characteristic of log 
 
 characteristic of log 
 
 characteristic of log 
 
 characteristic of log 
 
 characteristic of log 
 
 characteristic of log 
 
 characteristic of log 0.00487 
 &c, &c. 
 
 § 30, The above rule for detennining the characteristic 
 makes it necessary that that which is added to the charac- 
 teristic to produce the logarithm, that is, the mantissa, 
 shall be positive in all cases, even when the logarithm, as 
 a whole, is negative. Thus, log 0.00487 is between — 3 
 and — 2, and we have seen (in § 27) that it is either 
 
 — 3 + . 6875 . . or — 2 — . 3125 . . 
 
 But our rule makes the characteristic — 3, and therefore 
 
 48700 = 
 
 4, 
 
 4870 = 
 
 3, 
 
 487 = 
 
 2, 
 
 48.7 = 
 
 1, 
 
 4.87 = 
 
 o, 
 
 0.487 = - 
 
 -1, 
 
 0.0487 = - 
 
 -2, 
 
 1.00487 = - 
 
 -3, 
 
22 THE ELEMENTS OF LOGARITHMS. 
 
 necessitates the adoption of the first of these forms. In 
 order to avoid writing the sign -f~ between the character- 
 istic and mantissa, we write the sign — over the charac- 
 teristic, as a way of indicating that it affects that figure 
 only. For example, we write 
 
 log 0.00487 ="3.6875... 
 But as negative characteristics are often a source of con- 
 fusion in computation, it is customary to add 10 to every 
 negative characteristic, and to write — 10 after the loga- 
 rithm. Thus we have 
 
 log 0.00487 = 7.6875 . . — 10. 
 
 The — 10 is often omitted in writing, and left to be 
 remembered. 
 
 There is sometimes no advantage in adding 10 to the 
 negative characteristic ; but the student is advised to use 
 this method in all cases, for the sake of having a fixed 
 rule. 
 
 § 31. The characteristic of a logarithm depends, as we 
 see, wholly on the position of the units' place in the 
 antilogarithm, and not at all on the significant figures 
 which compose it. This part of the logarithm is then 
 readily found by inspection of the number. The man- 
 tissa, on the other hand, can only be obtained by laborious 
 calculations, and the practical computer must consult his 
 table for it. But this part of the logarithm depends 
 entirely on the series of significant figures that compose 
 the antilogarithm, without regard to the position of the 
 units' place. Hence, if a table is so made as to give the 
 logarithm of any number between tico successive integral 
 powers of 10, as 100 and 1000, it will give the logarithm 
 of any number whatever, since all possible series of sig- 
 nificant figures can be found within the chosen limits. 
 This is a great advantage which the denary system of 
 logarithms has over any other. 
 
THE DENARY SYSTEM. 23 
 
 But we can suppose an infinity of numbers between 
 any two given numbers, as 100 and 1000, and it is impos- 
 sible that the logarithms of all these numbers should be 
 explicitly contained in any table. They can only be given 
 at certain intervals ; but if these intervals are made suf- 
 ficiently small, the logarithms of intermediate numbers 
 can be found from the table by a process called inter- 
 polation, which is founded on the principle of § 15, and 
 will be fully explained in the next chapter. The greater 
 the degree of accuracy with which the logarithm is re- 
 quired, the smaller should be the ratio of the intervals, 
 in the table, to the numbers which are separated by those 
 intervals. Thus, a table of four-place logarithms may be 
 limited to the logarithms of all integers from 100 to 1000, 
 where the above-named ratio has for its greatest value 
 T -J o ; while a seven-place table should include the loga- 
 rithm of every integer from 10,000 to 100,000, where the 
 greatest value of the ratio is T q (jo^« 
 
 [§ 32. TVe are indebted to Briggs, not only as the 
 author of the denary system, but also as the founder, 
 and, in great part, the computer of the tables now actually 
 used. In 1617, he published the first instalment of his 
 own table, containing the logarithms of all integers below 
 1000 to eight places of decimals; and this he followed, in 
 1624, by his Arithmetica logarithm tea, containing the 
 logarithms of all integers from 1 to 20,000 and from 90,000 
 100,000 to fourteen places of decimals, together with a 
 learned introduction, in which the theory and use of log- 
 arithms are fully developed. The interval from 20,000 
 to 90,000 was filled up by Adrian Ylacq, a Dutch com- 
 puter; but in his table, which appeared in 1628, the loga- 
 rithms were given to only ten places of decimals. 
 
 " The total number of errors found in Vlacq," says Mr. 
 Glaisher, "amounts to 603, which probably includes very 
 nearly all that exist ; this cannot be regarded as a great 
 
24 THE ELEMENTS OF LOGARITHMS. 
 
 number, when it is considered that the table was the 
 result of an original calculation, and that more than 
 2,100,000 printed figures are liable to error." {Athenaeum, 
 15 June 1872. See also the Monthly Notices of the Royal 
 Astronomical Society for May, 1872.) All the modern 
 published tables are founded on that of Vlacq, of which 
 an edition, containing many corrections, was issued at 
 Leipzig in 1794 under the title Thesaurus Logarith- 
 morum Completus by George Vega. Concerning Vega's 
 table Professor De Morgan writes, " This is, no doubt, up 
 to this time, the table of logarithms ; the one of all others 
 to which ultimate reference should be made in questions 
 of accuracy." (English Cyclopaedia, Arts and Sciences, 
 Vol. VII., article " Table," London, 18G1.) 
 
 Callet's seven-place table (Paris, 1795), instead of 
 stopping at 100,000, gives the eight-place logarithms of 
 the numbers between 100,000 and 108,000, in order to 
 diminish the errors of interpolation, which are greatest 
 in the early part of the table ; and this addition has been 
 since generally included in seven-place tables. But the 
 only important published extension of Vlacq's table has 
 been made by Mr. Sang (1871), whose table contains the 
 seven-place logarithms of all numbers below 200,000. 
 
 Briggs and Vlacq also published original tables of the 
 logarithms of the trigonometric functions. 
 
 Besides the tables we have named, we ought to men- 
 tion the great collection, called Tables du Cadastre, which 
 was constructed under the direction of Prony, by an 
 original computation, under the auspices of the French 
 republican government of the last century. This work, 
 which contains, besides other tables, the logarithms of all 
 numbers up to 100,000 to nineteen places, and of the num- 
 bers between 100,000 and 200,000 to twenty-four places, 
 exists only in manuscript, "in seventeen enormous folios," 
 at the Observatory of Paris. It was begun in 1792 (year 
 ii.) ; and " the whole of the calculations, which to secure 
 
THE DENAEY SYSTEM. 25 
 
 greater accuracy were performed in duplicate, and the 
 two manuscripts subsequently collated with care, were 
 completed in the short space of two years." {English 
 Cyclopcedia, Biography > Vol. IV., article " Prony.") 
 
 For further information on the subject of this section, 
 and for an enumeration and description of tables pub- 
 lished up to 1861, the student is referred to the article 
 " Table " in the English Cyclopcedia already cited.] 
 
26 THE ELEMENTS OF LOGARITHMS. 
 
 CHAPTER III. 
 
 COMPUTATION BY LOGAEITHMS. 
 
 § 33. This chapter on the practical use of logarithms 
 is adapted to the author's Three and Four Place Tables 
 of Logarithmic and Trigonometric Functions. 
 
 § 34. Quantities are divided into two classes : con- 
 stants and variables ; and variables are divided into two 
 classes : independent variables and functions. 
 
 A constant is a quantity which is conceived to be 
 restricted to one or more fixed values; a variable is a 
 quantity which is not so restricted, and (in general) may 
 have any value. 
 
 When the value of a variable is regarded as being as- 
 sumed arbitrarily, the variable is called an independent 
 variable; when it is regarded as being determined by 
 the values of other variables, the variable is called a func- 
 tion of the variables by which it is determined. 
 
 The independent variables which enter into any ques- 
 tion are often called simply the variables. 
 
 If, for instance, in the equation a u = x, a denotes the 
 base of a definite system of logarithms, while x denotes 
 any number, and u its logarithm, a is a constant, and x 
 and u are variables, either of which may be regarded as 
 an independent variable, and the other as a function of it. 
 But if a is variable, as well as x and u, any two of the 
 three quantities may be regarded as independent variables, 
 and the third as a function of those two. Thus, a func- 
 
COMPUTATION BY LOGARITHMS. Zl 
 
 tion may depend on only one independent variable or on 
 any number of independent variables. 
 
 A logarithm in a given system may be regarded as a 
 function of its antilogarithm, or an antilogarithm as a 
 function of its logarithm. But we shall generally con- 
 sider the relation in the former light, and speak of the 
 antilogarithm as the independent variable, and of its loga- 
 rithm as a function of that variable ; and in that case we 
 shall commonly use the equation 
 
 U = lOg Xy 
 
 in which we shall call x the independent variable and u a 
 function of x. When we wish to regard u as the inde- 
 pendent variable and a; as a function of u, we shall prefer 
 to write 
 
 x = a u , or x = antilog u, or x = log -1 u. 
 
 The last form of notation should receive the special 
 attention of the student. The exponent — 1 attached to 
 a symbol of quantity shows that the quantity is to be used 
 as a divisor, instead of a multiplier, in tho term into 
 which it enters, e. g., on— hi = — ; or if n = mx, x = 
 m— x n. See (6). By analogy, then, the exponent — 1 is 
 attached to a symbol of relation to denote the inversion 
 of that relation ; so that, for example, if u = log x, x = 
 log -1 u. 
 
 § 35. A mathematical table is an orderly arrange- 
 ment of the values of some function for certain selected 
 values of its independent variables. Thus, a table of 
 logarithms contains the logarithms of certain selected 
 numbers (for example, of all integers from 100 to 1000) 
 arranged in the order of magnitude and in a form con- 
 venient for reference. 
 
 The independent variables, of which the successive 
 values are assumed arbitrarily, and generally at equal 
 intervals, are called the arguments of the table. A table 
 
28 THE ELEMENTS OF LOGARITHMS. 
 
 of logarithms is a table of one argument, namely the anti- 
 logarithm ; a multiplication table is a table of two argu- 
 ments, namely the two factors. Most of the Three and 
 Four Place Tables are tables of one argument, and their 
 argument is generally distinguished by being printed in 
 full-faced type. 
 
 § 3(5. The tabulated values of either the argument or 
 the function in any table are those that are explicitly con- 
 tained in the table. The difference between two succes- 
 sive tabulated values of either quantity is called a tabular 
 difference. The letter A followed by the letter which de- 
 notes any quantity is used by all writers to denote the 
 difference between two values of that quantity. We 
 shall here employ this notation to designate a tabular 
 difference. Thus, if x denotes the argument of a table, 
 and u the function, and if two successive tabulated values 
 of x are x x and a? 2 , and the corresponding values of u are 
 u x and w 2 , then the corresponding tabular differences are 
 
 Ax = x 2 — x v Au = u 2 — u v 
 
 The tabular difference of the argument generally has 
 & fixed value throughout any given table; but this is not 
 always the case. 
 
 When a value of any quantity is intermediate between 
 two successive tabulated values, such value may be said 
 to lie within the corresponding tabular difference. 
 
 § 37, A table maybe used in two ways: directly and. 
 inversely. The direct use of the table consists in finding 
 the value of the function for an assumed value of the 
 argument; the inverse use, in finding the value of the 
 argument for an assumed value of the function. In either 
 case, if the assumed value is tabulated, the required value 
 is readily found by inspection ; but if the assumed value is 
 not tabulated, we must resort to interpolation. 
 
COMPUTATION BY LOGARITHMS. 29 
 
 § 38. We shall not enter on the complete theoiy of 
 interpolation, but shall confine ourselves to simple inter- 
 polation, which is all that is necessary in the tables we 
 have to explain. Simple interpolation rests on the prin- 
 ciple that corresponding values of the argument and the 
 function lie within corresponding tabular differences, and 
 divide those differences in the same ratio. This principle, 
 which is called the principle of proportional parts, is ap- 
 plicable, with approximate correctness, to a table of 
 logarithms, provided the tabular differences are made 
 sufficiently small, as we have proved in § 15 ; and in every 
 properly constructed logarithmic table, the tabular differ- 
 ences are made sufficiently small to permit the application 
 of the principle up to the limit of accuracy belonging to 
 the table. 
 
 If, in any mathematical table to which simple interpo- 
 lation is applicable, x 1 and x 2 are two successive tabulated 
 values of the argument, and u\ and u% the corresponding 
 values of the function, and if x and u are any two cor- 
 responding intermediate values of the argument and 
 function, then the principle of proportional parts 
 gives : — 
 
 u — Xl\ u 2 — u u 2 — u\ Au 
 x Xi X2 — x X2 — x\ Ax ' 
 
 whence we derive the following formulas : — 
 
 (24) 
 
 u = tn + ^-p-/lu, (25) 
 
 ^2 — u A /oe\ 
 
 x = x 2 -^T Ax ' < 28 ) 
 
30 THE ELEMENTS OF LOGAEITHMS. 
 
 If, then, x is given, u may be found by either (25) or 
 (26) ; and if u is given, x may be found by either (27) or 
 (28). 
 
 In these formulas, the first term of the second member 
 may be regarded as an approximate value of the required 
 quantity, and the second term as a correction to be applied 
 to that approximate value. The approximate value of 
 the required quantity is that tabulated value which cor- 
 responds to either of the two tabulated values of the 
 assumed quantity between which the given value lies. 
 Whichever of these two values be taken, the same result 
 will be obtained by the above formulas ; but it is best to 
 form the habit of taking that tabulated value which is 
 nearest to the given value, whether above it or below it, 
 because that course is preferable in some abbreviated 
 methods of interpolation which we shall presently have 
 to explain. 
 
 § 39. If, as it often happens, zfa = l, (25-28) become 
 
 u = «i -j- (as — x\ )du> (29) 
 
 u = u 2 — (#2 — x)/Ju, (30) 
 
 *=*> *T' (32 > 
 
 § 40. The figures of the correction found are gener- 
 ally to be carried out only to a certain numeral place deter- 
 mined by the nature of the table in which the correction 
 is applied. If, in this case, the figures rejected amount to 
 more than half a unit in the place of the last figure re- 
 tained ; that is, to more than five units in the place of the 
 first figure rejected ; the last figure retained is to be in- 
 creased by 1 ; for the figures retained will thus most accu- 
 
c>£Fa.F. l l^CNT OF CIVIL C»NG«N££RI 
 
 u^ii.. '-u.1, _ j -or- 
 
 COMPUTATION BY LOGARITHMS. 31 
 
 rately represent the true value of the correction. For 
 example, 18.723 lies between 18 and 19, but is nearer to 
 the latter than the former, since it exceeds 18.5. Hence 
 if the decimal is rejected, the number should be repre- 
 sented by 19, rather than by 18. On the other hand, 
 672.18 should be represented by 672, not by 673, if the 
 decimal is rejected. 
 
 If that which is rejected is precisely 5 in the first 
 rejected place, it is equally correct to increase the last 
 figure retained and to leave it unchanged. Some com- 
 puters adopt the rule for this case of increasing the last 
 figure retained if it is odd, and not if it is even, so as to 
 increase it in half the cases that occur in the long run, 
 and to leave it unchanged in the other half. Accord- 
 ing to this rule, we should convert 3.5, 106.5, 24.5, 11.5 
 into 4, 106, 24, 12. But sometimes a reason for adopt- 
 ing some different proceeding may suggest itself. 
 
 The precepts of this section should be applied to all 
 cases in computation in which the figures of any number 
 are rejected beyond a certain numeral place. 
 
 § 41. We shall illustrate what has been said in the 
 foregoing sections by reference to the following fragment 
 of a four-place table of denary logarithms : — 
 
 x\ 100 101 102 103 104 105 106 107 108 109 
 
 X 
 
 u 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 
 
 u 
 
 A« 1 43 43 42 42 42 41 41 40 40 
 
 An 
 
 The argument of this table is the antilogarithm, desig- 
 nated by jc, and it is tabulated from 100 to 109, its tabular 
 difference being constant and equal to 1. The function 
 is the mantissa of the logarithm, which is printed without 
 the decimal-point. Its tabular difference varies from 43 
 to 40 (or, more properly, from .0043 to .0040). The char- 
 
32 THE ELEMENTS OF LOGARITHMS. 
 
 acteristic of a logarithm can readily be found by § 29 ; and 
 it is therefore omitted in the modern tables. 
 
 The number 108.2 and its logarithm may be said to 
 lie within the last tabular differences of this table, accord- 
 ing to the phraseology of § 36. 
 
 § 42o The following examples of interpolation are 
 adapted to the little table of logarithms given in the last 
 section: — 
 
 Ex. 1. Find log 10.537. Here, by § 29, the charac- 
 teristic is 1, and we have only to find the mantissa, which 
 by § 27, is the same as that of 105.37. The nearest tabu- 
 lated value of the antilogarithm is here 105. "We have 
 then 
 
 xi = 105, u t = 0212, Ax = l,Au = 41, x — x 1 = 0.37. 
 
 The correction to be added to u\ is then, by (29), 
 
 0.37 X 41 = 15.17, 
 
 of which the unit-figure is to be added to the fourth figure 
 of u\ . The decimal is, then, to be added to unknown 
 figures of \i\ , and it is, therefore, useless, and should be 
 rejected. We have then 
 
 mantlogl05 =0212 
 0.37 X 41 = 15 
 
 mant log 10537 = 0227 .-. log 10.537 = 1.0227. 
 
 Ex. 2. Find log 1017.95. Here we adopt 102, rather 
 than 101, as the nearest tabulated value ; and we have, by 
 (30), since 102 — 101.795 = 0.205, 
 
 mant log 102 = 0086 
 0.205 X 43 = 9 
 mant log 101795 = 0077 .-. log 1017.95 = 3.0077. 
 
 Here 0.205 X 43 = 8.815, which is represented by 9, 
 according to the precept of § 40. 
 
COMPUTATION BY LOGARITHMS. 33 
 
 Ex. 3. Find log 0.01068. 
 
 mantlog 107 = 0294 
 
 0.2. X 41= 8 
 
 mantlog 1068 = 0286 /. log 0.01068 = 8.0286 — 10. 
 
 Ex. £. Find log- 1 0.0157. The nearest tabulated man- 
 tissa is 0170. Hence we have, by (32), 
 
 0170 = mant log 104 
 
 i§= _03 
 
 0157 = mant log 1037 /. log- 1 0.0157 = 1.037. 
 
 The antilogarithm should generally be computed, as 
 in this instance, to only one place beyond the tabulated 
 figures. 
 
 Ex.5. Find log- 1 (7.0102 — 10). 
 
 0086 = mant log 102 
 
 0102 = mant log 1024 .-. log- 1 (7.0102 — 10) =0.001024. 
 
 § 43. If we suppose the argument of a table to 
 undergo a gradual and uniform change of value, the 
 function will, at the same time, pass through a gradual, 
 but not generally uniform, change. Thus, if x in the 
 table of § 41 increases at a uniform rate from 100 to 109, 
 log x will increase continuously, though more and more 
 slowly. Now the derivative of a function is another 
 function which measures the rate of change of the original 
 function as compared with its variable, at the moment 
 that the variable passes through any given value ; and 
 this secondary function can always be found by methods 
 of the higher mathematics. In some of the Three and 
 Four Place Tables, the value of the derivative is given 
 for each tabulated value of the function, and can be used 
 as a multiplier, instead of -—-, in the formulas of interpo- 
 lation, through half the tabular difference before and 
 
34 THE ELEMENTS OF LOGAEITHMS. 
 
 after the tabulated value against which it stands. This 
 depends on the principle already cited in § 38; for it 
 merely assumes that the function varies at a uniform 
 rate, as compared with the variable, through one tabular 
 difference. 
 
 § 44. The results obtained by logarithms are liable 
 to errors, which arise from the use of only a certain num- 
 ber of decimal-places of the logarithms. The greater this 
 number of places, the greater will be the degree of ac- 
 curacy of the results, supposing the data to be accurately 
 known. But it must be remembered that, in any practical 
 computation, the data themselves are subject to errors of a 
 certain magnitude, resulting from the imperfection of the 
 means by which they are obtained ; and it is a mere waste 
 of labor to employ logarithms which will enable us to 
 carry the results to a greater number of places than the 
 accuracy of the data warrants. " Of the misuse of tables," 
 says De Morgan, " no instance is more common than that 
 which consists in taking tables of too many places of 
 figures." In astronomy and geodesy, the precision of in- 
 struments and methods justifies the use of seven-place 
 tables. But in ordinary work, such as that of navigation 
 or common surveying, the accuracy of the results furnished 
 by a four-place table is as great as we need or can expect 
 to attain ; and as four-place logarithms take, in the long 
 run, only half the time required by those of five places, 
 and one-fourth of that required by those of seven places, 
 it is desirable to employ them for all purposes which they 
 will serve. A four-place table is also well adapted to be 
 used in instruction, where the leading object is the incul- 
 cation of principles. But the student who wishes to 
 become a skilful computer must, after acquiring the com- 
 mand of such a table, familiarize himself with those of five, 
 six, and seven places as well. 
 
 Three-place logarithms may be used advantageously 
 
COMPUTATION BY LOGARITHMS. 35 
 
 in rapid work preliminary to a detailed computation, and 
 wherever only a low degree of accuracy is required. 
 
 § 4o. Before entering on the explanation of particu- 
 lar tables, we would call the attention of the student to 
 a few general principles of work. 
 
 First, the beginner in the use of tables should take 
 pains to learn the best methods of dealing with them, 
 the most convenient way of holding them, the simplest 
 order to be followed in consulting them, and the proper 
 employment of the helps to interpolation with which 
 they are provided ; and to learn these things he must 
 begin by making a careful study of their plan. Secondly, 
 he must form the habit of using them with strict atten- 
 tion to rule, and especially of interpolating with accuracy, 
 when the best possible results are sought. But, thirdly, 
 it must be remembered that the object of logarithms is 
 to shorten numerical work. This object is only fully 
 accomplished, when the computer can use his table 
 rapidly, and interpolate, both directly and inversely, at 
 a glance. A great deal of this facility can be gained in a 
 short time by practice ; and the learner should do his best 
 to acquire it, so far as he can do so without losing his habit 
 of correct work. We may add that the computer ought 
 to keep his eyes open, and avoid falling into mistakes 
 from which a little common sense would be enough to 
 save him. 
 
 Among points which belong under the head of accu- 
 racy, we would refer again to the precept of § 40 ; that, 
 when figures are rejected at the end of a number, amount- 
 ing to more than half a unit in the place of the last figure 
 retained, the last figure retained should he increased hy 1. 
 Another rule in which the student requires instruction is 
 that zeros at the end of a decimal should not be omitted ; 
 for although they do not affect the value of the number, 
 they show that its value is known to a certain number 
 
36 THE ELEMENTS OF LOGARITHMS. 
 
 of places. For a similar reason, when zero is the char- 
 acteristic of a logarithm, it is always written, and it is a 
 good habit to write it in the case of any decimal num- 
 ber between and 1. Thus, log 1.04, if we are using 
 four-place logarithms, is written 0.0170, not simply .017, 
 for the last form would leave the fourth decimal figure 
 altogether in doubt, and would suggest a question whether 
 the characteristic had been thought of. Lastly, the stu- 
 dent must observe that it is as great an offence against 
 the spirit of accuracy to carry out a result beyond the 
 number of places which its logarithm justifies as it is to 
 fall short of that number. Thus, we have found (§ 42, 
 Ex. 4), log- 1 0.0157 = 1.037. We might carry out the 
 correction of the tabulated number indefinitely, obtaining 
 
 if = 0.309524.., 
 log-* 0.0157 == 1.03690476 . . . 
 
 But this would be to restrict too closely the number deter- 
 mined by a four-place logarithm. In fact, the antiloga- 
 rithm of 0.0157 . . may lie anywhere between 
 
 1.03669 . . = log- 1 0.0156500 . . 
 and 1.03693 . . = log- 1 0.0157500 . . 
 
 If we assume that the value of the given logarithm is 
 0-0157000 . . , its antilogarithm is 1.036812 ... In work- 
 ing with four-place tables, we commonly carry out our 
 antilogarithms to four (or sometimes, in the early part 
 of the table, to five) significant figures ; but the last 
 figure must be regarded as probable, rather than abso- 
 lutely certain, owing to the accumulation of small errors 
 in the various logarithms employed to obtain the result. 
 Thus, slightly different answers are sometimes found by 
 different methods of performing one example, or even in 
 looking out a single logarithm or antilogarithm by differ- 
 ent processes of interpolation. This ought not to pro- 
 duce any distrust of the method of logarithms, but rather 
 
COMPUTATION BY LOGARITHMS. 37 
 
 to impress on the student the fact that results derived 
 from imperfect data must be subject to corresponding 
 imperfections. In general, it may be said that, in all 
 computations, whether logarithms are used or not, the 
 results should not be liable to such errors as would be 
 perceptible in observation ; nor is it a sound method of 
 working to attempt to carry them decidedly beyond the 
 point of accuracy which observation could detect. 
 
 All that has been said in this section must be modified 
 by the remark that the experienced computer may and 
 should permit himself a freedom in the use of his tables 
 which would be unsafe for a beginner. But good habits 
 of approximate work are only acquired by those who 
 have first learned to be exact. 
 
 Table of Proportional Parts. 
 
 §46. This table is designed to be used in connex- 
 ion with other tables, as an aid in interpolation. It con- 
 tains the product of every integer from 1 to 100 by every 
 tenth from 0.1 to 0.9 ; and is readily used for multiplying 
 any number of two figures by any decimal. The table is 
 arranged in five rows of twenty columns each. The value 
 of the multiplicand is printed in full-faced type over every 
 fifth column, and the values of the multiplier are printed, 
 without the decimal-point, on the right and left of each 
 row of columns. The products are given as if the mul- 
 tipliers were tenths. Thus 0.7 X 28 = 19.6, and this 
 result is found in the column of 28 and the line of 7. 
 If the multiplier is in the place of hundredths, thou- 
 sandths, or any other numeral place, it is only necessary to 
 make a change in the place of the decimal-point in the 
 product. Thus 0.03 X 62 = 1.86. To multiply a number 
 of two figures by any decimal, we must find the products 
 which correspond to the successive figures of the multi- 
 plier and add them together. The decimal part of the 
 4 
 
38 THE ELEMENTS OF LOGARITHMS. 
 
 result is generally to be discarded, and in that ease the 
 rule of § 40 must be observed. Thus, let it be required 
 to find 0.619 X 37. We have 
 
 0.6 X 37=22.2 
 0.01 X 37= 0.37 
 0.009 X 37= 0.333 
 .-. 0.619 X 37 = 23. 
 
 In like manner, we find 0.27 X 15 = 4, 0.59 X 73 = 
 43, 0.78 X 69 = 54, 0.96 X 84 = 81, 0.36 X 57 = 21, 
 0.289 X 51 = 15, 0.483 X 93 = 45, 0.374 X 82 = 31. 
 
 The table can also be used inversely. Thus, let it be 
 required to find, to two places of decimals, what part 36 
 is of 79. Looking in the column of 79, we find 31.6 = 
 0.4 X 79 ; 36 — 31.6 = 4.4, and the tabulated product 
 nearest to this is 4.74 = 0.06 X 79. 
 
 .-.ft = 0.46. 
 
 In like manner, we find ff = 0.43, |f = 0.81, §£ = 
 0.32, f f = 0.37, ff = 0.79, 11 = 0.30. 
 
 The student should practise himself in the use of this 
 table till he can obtain such results as those found in this 
 section mentally and by a rapid glance. 
 
 Three-Place Table of Logarithms of Numbers. 
 
 § 47. This table contains, to three places of deci- 
 mals, the mantissa of the denary logarithm of every num- 
 ber of one or two significant figures. The antilogarithm 
 is the argument of the table, and is printed in full-faced 
 type, the first significant figure being given in the left- 
 hand column, and the second significant figure (for which 
 is to be substituted, if there is but one significant figure) 
 at the top of the table. The mantissa of the logarithm is 
 given, in common type, in the line and column deter- 
 mined by the two figures of the antilogarithm. Thus, 
 
COMPUTATION BY LOGARITHMS. 39 
 
 we find log 8.7 = 0.940, log 7 = 0.845, log 0.61 = 9.785 
 — 10. 
 
 The last line of the table contains the logarithms of 
 numbers of three figures, beginning with 10. Thus, we 
 find log 1070 = 3.029. 
 
 The three-place logarithm of any number whatever 
 and the antilogarithm of any three-place logarithm can 
 be found from this table by the method explained and 
 exemplified in §§ 38-42, and the process may be shortened 
 by the use of the table of Proportional Parts. The anti- 
 logarithm should be computed to one figure only beyond 
 those tabulated, as it cannot be determined more accu- 
 rately by its three-place logarithm. It must be observed 
 that, in their order of succession, the logarithms run 
 across each line from left to right, and then begin again 
 at the left of the next line. 
 
 Let the student find the following logarithms and anti- 
 logarithms by this table : — 
 
 437.2 =2.640, 
 
 log 0.694 = 9.841 - 
 
 -10, 
 
 2.78 = 0.444, 
 
 log 17.92 = 1.253, 
 
 
 32910. = 4.518, 
 
 log 0.00217 = 7.336- 
 
 -10; 
 
 log- 1 2.335 = 216, log- 1 0.794 = 6.23, 
 
 log- 1 1.872 = 74.5, log- 1 (9.647 — 10) = 0.444, 
 
 log- * (8.427 — 10) = 0.0268, log- * 2.533 = 342. 
 
 Four-Place Table of Logarithms of Numbers. 
 
 § 4§. The arrangement of this table is similar to that 
 of the three-place table. The antilogarithm is the argu- 
 ment, and is given to three significant figures, for the 
 last one or two of which is to be substituted in the case 
 of a number of less than three figures. The first two 
 significant figures must be sought at the left of the 
 table, in the column headed "Natural Numbers," and 
 
40 THE ELEMENTS OF LOGARITHMS. 
 
 the third at the top of the table. The second, third, and 
 fourth decimal-figures of the logarithm will be found in 
 the line and column thus determined, and the first deci- 
 mal-figure in the column, and either on the same line 
 with the first two significant figures of the antilogarithm, 
 or, if no figure stands there in the first decimal place, 
 then on the nearest line above which contains a figure in 
 that place. Thus, log 3.23 = 0.5092, log 48.6 = 1.6866, 
 log 7480=3.8739, log 0.285=9.4548 — 10, log 921 = 
 2.9643, log 0.0052 = 7.7160 — 10. 
 
 If the second decimal-figure is printed in small type, 
 the first decimal-figure is to be sought at the beginning 
 of the next following line. Thus, log 2.59 = 0.4133, log 
 126 = 2.1004, log 0.0797 = 8.9015 — 10, log 63.2 = 1.8007, 
 log 50400 = 4.7024. 
 
 As in the three-place table, the order of succession of 
 the logarithms is from left to right through the succes- 
 sive lines ; but there are no repetitions in this table, and 
 the last logarithm in each line is followed by the first in 
 the next line. 
 
 The four-place logarithm of any number or the anti- 
 logarithm of any four-place logarithm can be found from 
 this table by the method of §§ 38-42, by the aid of the 
 table of Proportional Parts, already explained. The anti- 
 logarithm should generally be computed to four figures 
 only, that is, to one figure beyond the tabulated figures ; 
 but in the early part of the table it may be computed to 
 five figures. Thus, we have 
 
 log 59.43 = 1.7740, log 0.008147 = 7.9110 — 10, 
 
 log 284.8 =2.4545, log 572610. =5.7579, 
 
 log 0.073748 = 8.8678 — 10, log 0.5017 = 9.7004 — 10, 
 log 3.1607 = 0.4998, log 99.968 = 1.9999 ; 
 
 log- 1 1.9155 = 82.32, log- 1 (5.8760 — 10) = 0.00007517, 
 log- 1 0.0547 = 1.1342, log- 1 (8.1577 — 10) = 0.014377, 
 log- 1 3.8291 = 6747, log- 1 (9.5757 — 10) = 0.3765. 
 
COMPUTATION BY LOGARITHMS. 41 
 
 § 49. Interpolation in this table is facilitated by the 
 use of the right-hand division of the table, headed " Pro- 
 portional Parts." If we follow out the line which 
 contains the tabulated logarithm corresponding to the 
 first three significant figures of the given number, we 
 shall find, under the fourth significant figure, the correc- 
 tion which corresponds to that figure, pointed off as if 
 the mantissas in the table were whole numbers, and, in 
 like manner, under the fifth significant figure of the anti- 
 logarithm, we shall find the correction for that figure, pro- 
 vided we move the point in that correction one place to 
 the left, and so on. Thus, suppose it is required to find 
 log 1273.64. We find 
 
 mant log 127 = 1038, correction for .3 == 10.4 
 
 „ „ .06 = 2.09 
 „ „ .004= 0.139 
 
 „ .364 = 13 
 /.log 1273.64 = 3.1051. 
 
 In like manner, we find 
 
 log 0.073748 = 8.8678 — 10, log 0.287428 = 9.4586 —10, 
 log 3.1607 = 0.4998, log 10.0086 = 1.0004, 
 
 log 908.4 =2.9583, log 0.09994 =8.9998—10. 
 
 In using this table of Proportional Parts, we should 
 work from the nearest tabulated value. For each line of 
 corrections is computed from the average value of the 
 tabular difference for that line, and the smaller the pro- 
 portional part taken, the less is the probability of error. 
 Hence, if the fourth significant figure is as great as 5, we 
 should increase the third significant figure by 1, and from 
 the tabulated logarithm thus obtained subtract a suitable 
 correction. For example, suppose it is required to find 
 log 1.2084. We have 120.84 = 121 — 0.16. Then 
 
42 THE ELEMENTS OF LOGABITHMS. 
 
 mant log 121 = 0828, correction for 0.1 = 3d 
 
 „ 0.06 = 2.09 
 * 0.16 = 6 
 .-. log 1.2084 = 0.0822 
 In like manner, we find 
 
 log 0.3077 = 9.4882 — 10, log 63.071 == 1.7998, 
 log 5518.3 se 3.7418, log 9.426 = 0.9743. 
 
 § 50 o This sub-table of Proportional Parts can also 
 be used inversely. Thus, suppose it is required to find 
 log- 1 1.9155. We have 
 
 9154 == mant log 823, 9155 = 9154 + 1, 
 1.1 = correction for 0.2 ; 
 .-. log- 1 1.9155 = 82.32. 
 
 Again, if log -1 3.8291 is required, we have 
 
 8293 =j mant log 675, 8291 = 8293 — 2, 
 
 1.9 = correction for 0.3, 675 — 0.3 = 674.7 
 
 .-. log-i 3.8291 = 6747. 
 
 In like manner, we find 
 
 log- 1 0.4438 == 2.779, log- 1 (5.8760 — 10) = 0.00007516, 
 log- 1 0.0547 = 1.1342, log- 1 (8.1577 — 10) = 0.014377, 
 log- 1 2.8272 = 671.7, log- 1 (9.5757 — 10) == 0.3764. 
 
 § 51. The extreme left-hand column of the table of 
 logarithms, headed "Angles," is introduced to facilitate 
 finding the logarithm of the number of minutes in an 
 angle expressed in degrees and minutes or hours and 
 minutes, or of seconds in an angle expressed in minutes 
 and seconds. The unit-figure is to be taken at the top of 
 the table, and the characteristic is always 2. Thus we 
 obtain log (6° 27') = 2.5877, log (10* 12») = 2.7868, 
 log (4' 08") = 2.3945, log (13° 15'.8) == 2.9008. 
 
 The small table under the table of logarithms is to be 
 
COMPUTATION BY LOGARITHMS. 43 
 
 used for trigonometric work, and will be explained in 
 another place. 
 
 Arithmetical Complement. Reciprocal. 
 
 § 452. The arithmetical complement of the logarithm 
 of a number, which may also be called the cologarithm 
 of the number, is the remainder obtained by subtracting 
 that logarithm from 10. It is commonly denoted by 
 "arco log" or by " colog." The above definition is then 
 expressed by the formula, 
 
 colog x =z 10 — log x. (33) 
 
 Since 
 
 10 = 9.99 9 -f- (1 in the last place), 
 
 the arithmetical complement may be found by subtracting 
 each figure of the logarithm from 9, and then, adding 1 to 
 the last figure of the remainder, or, in other words, by 
 beginning with the characteristic and subtracting each suc- 
 cessive figure of the logarithm from 9, down to the last sig- 
 nificant figure, and subtracting that figure from 10. By 
 this method the cologarithm of a number can be written 
 down from inspection of the table almost as rapidly as 
 the logarithm itself. Thus we have 
 
 '&■ 
 
 colog 59.43 = 8.2260, colog 0.008147 = 12.0890, 
 
 colog 284.8 = 7.5455, colog 572610. = 4.2421, 
 
 colog 0.073748 = 11.1322, colog 0.5017 = 10.2996, 
 
 colog 3.1607 = 9.5002, colog 99.968 = 8.0001, 
 
 colog 1273.64 = 6.8949, colog 908.4 = 7.0417, 
 
 colog 0.287428 = 10.5414, colog 10.0086 = 8.9996, 
 
 colog 1.2084 = 9.9178, colog 0.3077 =10.5118. 
 
 § 53. By (33) and (9), 
 
 colog x — 10 = — log x = log - ; (34) 
 
44 
 
 THE ELEMENTS OP LOGARITHMS. 
 
 that is, if 10 be subtracted from the cologarithm of any 
 number, the remainder is the logarithm of the reciprocal 
 of that number. Thus we have 
 
 z^s 
 
 = 0.01683, 
 
 tt.ooVttt 
 
 = 122.74, 
 
 irsVs 
 
 = 0.003512, 
 
 Z72^TJ) 
 
 = 0.000001746, 
 
 t.tttW** 
 
 = 13.560, 
 
 ^■sVtT 
 
 = 1.993, 
 
 ^•tV(J7 
 
 = 0.3164, 
 
 ■^•^.V^-g" 
 
 = 0.010002, 
 
 t^tVs? 
 
 = 0.0007850, 
 
 ¥oW 
 
 = 0.0011008, 
 
 tf.TsV 4 2""S 
 
 = 3.478, 
 
 TU.fos'S 
 
 = 0.09990, 
 
 T.2oS¥ 
 
 = 0.8276, 
 
 TJ.^OTT 
 
 = 3.249. 
 
 Multiplication by Logarithms. 
 
 § 5J:« Multiplication, division, involution, and evo- 
 lution are easily performed by logarithms by the aid of 
 the theorems of § 16. The theorem for multiplication is 
 that the logarithm of the continued product of several 
 quantities is the sum of their logarithms ; or 
 
 log TIx = Zlog x. 
 
 If any of the factors are less than 1, their logarithms 
 should be increased by 10, as in § 30, and the resulting 
 logarithm diminished by 10 times the number of such 
 factors. 
 
 If any of the factors are negative, their logarithms are 
 to be taken without regard to their signs, and the sign of 
 the result determined by the rules of algebra. It is well 
 to write the letter n after any logarithm which corresponds 
 to a negative number. Then if the number of logarithms 
 so marked is odd, the product is negative ; if even, the 
 product is positive. This indicates the usual method of 
 dealing with negative numbers in logarithmic compu- 
 tation. 
 
 Ex. 1. Find 908.4 X (—0.79) X (—101.92) X 3.1607 
 x (—0.0004). 
 
COMPUTATION BY LOGARITHMS. 45 
 
 10 
 
 Am 
 
 908.4 
 
 log 
 
 2.9583 
 
 — 0.79 
 
 log 
 
 9.8976?* 
 
 — 101.92 
 
 log 
 
 2.0083;*, 
 
 3.1607 
 
 log 
 
 0.4998 
 
 — 0.0004 
 
 log 
 log 
 
 6.6021n 
 
 — 92.50 
 
 1.9661^ 
 
 10 
 
 Ex. 2. Find 0.05392 X 0.9 X 1534.1 X 2.117. 
 
 Arts. 157.6. 
 
 Ex. S. Find (—7.1309) X 0.004006 x 0.7172 x 
 (—0.015) X 21.362. A)is. 0.006564. 
 
 Division by Logarithms. 
 
 § 55, The logarithm of a quotient is found by sub- 
 tracting the logarithm of the divisor from that of the 
 dividend. By this principle and by (34), 
 
 x 
 log — = log x — log y = log x -\- colog y — 10. (35) 
 
 if 
 
 Hence the logarithm of a quotient may be found by add- 
 ing together the logarithm of the dividend and the cologa- 
 rithm of the divisor and subtracting 10 from the residt. 
 The student should habituate himself to working by this 
 latter rule, as it is more convenient than the former, espe- 
 cially where compound operations are to be performed, as 
 in §§ 56 and 59. 
 
 If the characteristic of the dividend is increased by 10, A 
 that of the quotient must be diminished by 20. 
 Ex. 1. Divide 0.9316 by 15.42. 
 
 0.9316 log 9.9692 — 10 
 
 15.42 colog 8.8119 
 Ans. 0.06041 log 8.7811 — 10 
 
 Ex. 2. Divide 0.039008 by — 0.0097. Ans. — 4.022. 
 
 Ex. 3. Divide — 67.18 by — 0.4573. Ans. 146.93. 
 
46 THE ELEMENTS OF LOGARITHMS. 
 
 The Rule of Three by Logarithms. 
 
 § 56. If any three terms of a proportion are known, 
 the fourth can be found by the Rule of Three ; and the 
 operations required by this rule may be performed, in a 
 single process, by logarithms. See § 17. 
 
 Ex. 1. 0.5334 : x — —0.001907 : —2.008. Find x, 
 
 rru j> i * rru • °- 5334 X (—2.008) 
 The Rule of Three gives x = _ 0t001907 — - 
 
 0.5334 log 9.7270 — 10 
 
 — 2.008 log 0.3028» 
 
 — 0.001907 colog 12.7197^ 
 x = 561.7 log 2.7495 
 
 Ex. 2. —37.82 : 503.7 = 0.09852 : x. Find x. 
 
 Ans. x = — 1.3119. 
 
 Ex. S. 0.71972 : 3.156 = x : 0.04983. Find as. 
 
 Ana. x = 0.011363. 
 
 Involution by Logarithms. 
 
 § 57, The logarithm of any power of a quantity is 
 found by multiplying the logarithm of the quantity by 
 the exponent of the power / or 
 
 log x n = n log x. 
 
 If the logarithm of the given quantity is increased by 
 10, the product must be diminished by 10 times the expo- 
 nent of the required power, since 
 
 n (log as-J— 10) =n log x -f- lOn. 
 
 The results found by this method are hardly to be 
 relied on beyond the third significant figure, since the 
 error in the logarithm of the given quantity is multiplied 
 by the exponent. 
 
COMPUTATION BY LOGARITHMS. 47 
 
 Ex. 1. Find the cube of 0.8724 
 
 0.8724 log 9.9407 — 10 
 3 
 
 Ans. 0.664 log 9.8221 — 10 
 
 Ex. 2. Find the 7th power of 3.051. Ans. 2400. 
 Ex. 3. Find the square of 21.27. Ans. 452.5. 
 
 Evolution by Logarithms. 
 
 § 58. The logarithm of any root of a quantity is 
 found by dividing the logarithm of the quantity by the 
 exponent of the root ; or 
 
 n I 
 
 log \/x = log X. 
 
 n 
 
 If the characteristic of the logarithm of the given 
 quantity is negative, it should be increased before division, 
 not by 10, but by 10 times the exponent of the required 
 root, and then the characteristic of the quotient should be 
 decreased by 10; for 
 
 log x 4- lOrc log x . . A , » . % . 
 
 — = — \- 10 = log \fx -j- 10. 
 
 n n 
 
 Ex. 1. Find the 4th root of 0.005627. 
 
 0.005627 4 )37.7503 — 40 
 
 Ans. 0.2739 log 9.4376 — 10 
 
 Ex. 2. Find the 9th root of 0.0007808. 
 
 Ans. 0.4516. 
 Ex. 3. Find the 5th root of 370.42. 
 
 Ans. 3.264. 
 Ex. ^. Find the cube of the 11th root of 61.98. 
 
 Ans. 3.081. 
 Ex. 5. Find the 10th power of the 7th root of 0.8213. 
 
 Ans. 0.7550. 
 
48 THE ELEMENTS OF LOGARITHMS. 
 
 Compound Operations, 
 
 § 50. Examples involving the combination of multi- 
 plications, divisions, involutions, and evolutions are espe- 
 cially suited to be solved by logarithms. 
 
 Let the student work out the following results: — 
 
 Ex. 1. 
 
 a 
 
 V 
 
 0.05639 X 1.0728 3 X 4^21.3 X 10.00 
 
 : ■ ■ = 4.068. 
 
 V0.007623 10 X 52.46 X 0.7000 
 Ex. 2. 
 
 /V7394. xM2998y / 532.4 X 0.290 _ 
 
 V 1.0063 2 X 57.00 / X V V0.0695 X 9.995 _ u - uuu ^ uy - 
 
 Ex. 3. 
 
 3 U \/Q.019Q7 3 X 3.796 y = g574 
 
 V \0.05545* X 1000.0 X 0.0007241/ 
 
 Ex. 4- 
 
 4 / \^43.04 X 809.8 X 0.2516 3 X 0.09093 _ 2386> 
 
 V 2139. X 3.007 X ^0.008188 X 0.7433 4 
 
 Ex. 5. 
 "l.0007 5 X 0.3710 . ^45.56x0.2203 
 
 V \A).005278 * 7.197 X 0.6411 
 
 Ex. 6. 
 
 /0.7628 X 79328. X 0.01333% {%$ 
 
 8.140. 
 
 40.07 X Vl5.21 
 
 -) =o. 
 
 2034. 
 
 Exponential Equations. 
 [§ 60. We have seen, in § 18, that if 5* = m y 
 
 log rn 
 
 ~ log b * 
 
COMPUTATION BY LOGARITHMS. 49 
 
 or log x = log log m — log log b \ ,^q\ 
 
 = log log m -j- colog log b — 10. > 
 
 Ex. 1. 21* = 15.75. Find x. 
 
 m = 15.75 log 1.1973 log 0.0783 
 b =21 log 1.3222 colog 9.8787 
 
 x =0.9058 log 9.9570 — 10 
 
 Ex. 2. 40.56*= 8276. x = 2.437.] 
 
 [§ 61. If either b or m is less than 1, log b or log m is 
 negative, and we proceed as in § 54. 
 
 Ex. 1. Find x from the equation 0.5175* = 6.208. 
 
 m = 6.208 log 0.7930 log 9.8993 — 10 
 
 b = 0.5175 logT.7139 = — 0.2861 colog 10.5434 ;* 
 x = — 2.771. log 0.4427/1 
 
 Ex. 2. Find x from the equation 0.027907* = 0.8213. 
 
 Ans. x = 0.05501.] 
 
 [§ OS. It has already been remarked in § 19 that solv- 
 ing an exponential equation by logarithms is equivalent 
 to converting a logarithm from one system to another. 
 Thus, in the above examples, we have found the following 
 logarithms from their values in the denary system: — 
 
 log 21 15.75 = 0.9058, log . i5175 6.208 = — 2.771, 
 log4o.56 8276 = 2.437, log/usm 0.8213 = 0.05501.] 
 
 Logarithms of Sums and Differences. 
 
 [§ 63. Addition and Subtraction cannot be performed 
 by logarithms. Hence arise inconvenience and liability 
 to error, when these operations occur in the midst of 
 others which can be performed by logarithms. The table 
 of Logarithms of Sums and Differences is one form of 
 a table proposed by Gauss to obviate this difficulty. 
 
50 THE ELEMENTS OF LOGARITHMS. 
 
 The argument of this table, designated by the letter 
 A, is any logarithm, which we may call log x. Its char- 
 acteristic (increased by 10, if negative) is at the top of 
 the table, and the first two figures of the mantissa are 
 in the left-hand column. The remaining figures of the 
 mantissa are for the tabulated values of the argument. 
 The function given in the table, designated by B, is log 
 (x -\- 1). Its characteristic, accompanied sometimes by 
 one or two figures of the mantissa, is printed at the top 
 and bottom of each column. The remaining figures are 
 to be found in the body of the table ; the first of them 
 being printed only in the first line in which it has a new 
 value and also in the first line of each division of five 
 lines. When this first figure is in small type, the earlier 
 figures of B are to be taken from the foot of the column ; 
 otherwise they are to be taken from the top of the 
 column. 
 
 We shall call B, or log (x-\- 1), the Gaussian of A, or 
 log a:, and A the anti- Gaussian of B; and Ave shall use 
 the symbols (s) and ©— 1 to designate these relations. 
 Thus we write 
 
 B = ©A, or log (* + 1) == ® log x, ) 
 
 A = ©-IB, or log (x — 1) = ®-l log x\ J K } 
 
 where the exponent — 1 is used according to the prin- 
 ciple laid down in § 34. 
 We have then 
 
 © (9.2100 — 10) = 0.0653, © 0.9800 = 1.0232, 
 
 ® 0.6300 = 0.7215, © (8.4800 — 10) = 0.0129. 
 
 These results may be verified by means of the table 
 of Logarithms of Numbers ; which gives, for instance, 
 
 log- 1 (9.2100 — 10) = 0.1622 = a-, 
 log (x -f 1) = log 1.1622 = 0.0653.] 
 
 [§ 64, The Gaussian or anti-Gaussian of any four-place 
 
COMPUTATION BY LOGARITHMS. 51 
 
 logarithm may be found to four decimal-places from this 
 table by simple interpolation. In the principal columns 
 of the table, a multiplier (derivative), to be used instead 
 of the tabular difference, is printed in small type after 
 each value of the function. This multiplier should prop- 
 erly be used through half the tabular interval on each 
 side of the value to which it is attached ; or, in other 
 words, we should apply our correction to the nearest 
 tabulated value. The general table of Proportional Parts 
 may be used in computing corrections. 
 
 The multiplier is omitted in the earlier columns of the 
 table, where the tabular differences are so small that cor- 
 rections are easily computed, and also in the later columns, 
 after the occurrence of the letter c, where its value is 100, 
 and A and B vary by equal amounts. 
 
 The table extends only from A = 6.0000 — 10 to A = 
 4.0000. The reason of this limitation is that 
 
 if A < 6.0000 — 10, B = 0.0000 to four places, 
 and if A > 40000, B = A to four places. 
 
 Let the student find the following values from this 
 table : — 
 
 © 1.0960 ==1.1295, © 3.8129 =3.8130, 
 
 © (7.5265 — 10) = 0.0015, © (9.6431 — 10) == 0.1582. 
 
 ©-11.0960 = 1.0597, ©-13.8129 = 3.8128, 
 
 ©-10.1051 = 9.4373 — 10, ©-11.0216 = 0.9782.] 
 
 [§ G«>o If m and n are any two numbers, we have 
 
 m-\-n=zn (— + l)» m — n=n( 1), 
 
 log (m + n) = log n -f- log (^ + 1 V 
 
 log (m — n) = log n + log C^- — 1\ ; 
 
52 TIIE ELEMENTS OF LOGARITHMS. 
 
 or, by (37), 
 
 771 
 
 log O |«)=logw + @log- 
 
 = log n-\-($ (log m + colog ??)> 
 log (>?i — n) = log » -f- ©- 1 log — 
 
 ±= log n -f- G)- 1 (log «i + colog n) ; 
 
 (38) 
 
 and by these formulas we can obtain log (m-f-n) and log 
 ( m — w ) when log m and log w are known, without having 
 to find in and ?i themselves. 
 
 Putting in (38) ?w = l and ?i = x, we have, by (7), 
 
 log (1 + sb) = log x + ® colog aj, j 
 
 log (1 — #) = log x -f- ©- 1 (colog x) ) ^ ' 
 
 Formulas (37-39) will be found at the bottom of the 
 table.] 
 
 [§ 66. The use of this table is illustrated by the fol- 
 lowing examples : — 
 
 Ex. 1. Find the hypotenuse h of a right triangle of 
 which the legs are a = 43.76 and b = 71.02. 
 
 The Pythagorean Proposition gives h 2 = a 2 -f- b 2 , or, 
 h = \/(a 2 -f- b 2 ). Working without the Gaussian table, 
 we have 
 
 a = 43.76 log 1.6411 b = 71.02 log 1.8514 
 2 2 
 
 «2 = 1915 log 3.2822 log 3.7028 
 
 b 2 = 5044 
 
 a? +0*= 6959 2)3.8425 
 h =83.40 log 1.0212 
 
 The Gaussian method saves two references to the 
 table. By that method we have 
 
COMPUTATION BY LOGARITHMS. 53 
 
 log a =1.6411 
 
 colog b = 8.1486 
 
 log^ =9.7897 — 10 
 b 
 
 log ^- = 9.5794 — 10 © 0.1397 = log C * + ** 
 
 I © 0.0698 = log 1 — 
 
 log b 1.8514 
 h = 83.40 log 1.9212 
 -Eb. 2. Compute 
 
 (0.2194)2 x V^915.7 + 4.4 X V'O/Tx 32.03 
 (571.6 X 0.0235 — 0.04004 X 6.72 X 10.97)3 
 
 Am. 0.1143.] 
 
APPENDIX. 
 
 EXPLANATION OF THE TRIGONOMETRIC TABLES. 
 
 (§ 1.) The following formulas, in which qp denotes any 
 angle, are proved in Trigonometry : — 
 
 log sin cp == colog esc qp — 10, 
 log esc qp = colog sin qp — 10, 
 log tan qp = colog ctn qp — 10, 
 log ctn qp = colog tan qp — 10, 
 log sec qp = colog cos qp — 10, 
 log cos cp = colog sec qp — 10. t 
 
 (A) 
 
 See Seaver's Formulas, 6 ; Peirce's Trigonometry, § 10 ; 
 Chauvenet's Trigonometry, § 19 ; Elements of Logarithms, 
 §53. 
 
 sin (90° — qp) = cos qp, , 
 esc (90° — qp) = sec qp, 
 tan (90° — g>) = ctn qp, 
 ctn (90° — cp) = tan qp, 
 sec (90° — qp) = esc qp, 
 cos (90° — qp) = sin cp. 
 
 (B) 
 
 See Seaver, 35 ; Peirce, § 7 ; Chauvenet, §§ 18, 38. 
 
56 THE ELEMENTS OF LOGARITHMS. 
 
 sin (90° -f- q>) = cos <jp, 
 cos (90° -|- op) = — sin r/>, 
 tan (90° -f- <p) = — ctn cp ; 
 sin cp = (— )* sin (2k 90° + q>) 
 
 = —(—)* cos ([2k -f 1] 90° + cp), ) (C) 
 cos 9 = (— )* cos (2£ 90° + ?) 
 
 = (— )* sm (PA+l]90o+qp), 
 tan g> = tan (2k 90° + q) 
 
 = — ctn ([2& + 1] 90° + cp). 
 
 See Seaver, 35, 28, 29, 30 ; Peirce, §§ 63, 65, 68 ; Chau- 
 venet, §§ 38, 41, 43, 45, 51, 52. 
 
 <P) 
 
 1st Qu. 2nd Qu. 3rd Qu. 4th Qu. 
 
 sin + + — — 
 
 cos -f- — — + 
 
 tan + — + — 
 
 See Seaver, 4, 5 ; Peirce, §§ 62, 66 ; Chauvenet, § 47. 
 
 Three-Place Table of Trigonometric Functions. 
 
 (§ 2.) This table contains the values of the six simple 
 trigonometric functions, generally to three significant 
 figures, and their logarithms to three places of decimals, 
 for every degree of the quadrant. The logarithms of the 
 functions occupy the left-hand division of the table, and 
 the natural values the right-hand division. The degrees 
 run down the left-hand column of the table to 45°, and 
 then up the right-hand column. The names of the func- 
 tions are to be taken from the top of the page when the 
 degrees are taken from the left-hand column, marked ° at 
 the top, and from the bottom of the page when the de- 
 grees are taken from the right-hand column marked ° at 
 the bottom. The integral part of the value of the func- 
 tion, whether natural or logarithmic, is generally printed 
 only at the top and bottom of the column, unless it 
 
APPENDIX. 57 
 
 changes its value in the column. The characteristics, 
 when negative, are increased by 10, so as to become posi- 
 tive. This is the case with the log sin and log cos of all 
 angles and with the log tan of angles less than 45° and 
 the log ctn of angles greater than 45°. We shall find it 
 convenient to follow this usage in citing the logarithmic 
 functions, in our illustrations, and to leave the term — 10 
 to be added (or remembered) by the student when he em- 
 ploys the functions in examples. We have then 
 
 sin 17° = 0.292, tan 17° = 0.306, sec 17° = 1.046, 
 esc 17° = 3.42 ctnl7° = 3.27, cosl7° = 0.956; 
 
 Iogsinl7 = 9.466,logtanl7° = 9.485,logsecl7° = 0.019, 
 Iogcscl7° = 0.534,logctnl7 = 0.515,logcosl7° = 9.981; 
 
 sin 52° = 0.788, tan 52° = 1.28, sec 52° = 1.62, 
 esc 52° = 1.269, ctn 52° = 0.781, cos 52° = 0.616; 
 
 Iogsin52° = 9.897,logtan52 = 0.107,logsec52° = 0.211, 
 log esc 52° = 0.103, log ctn 52° = 9.893, log cos 52° = 9.789. 
 
 The method of simple interpolation explained in the 
 treatise on Logarithms, §§ 38-42, may be applied to this 
 table both directly and inversely. The tabular differences 
 are printed between the lines in small type, except for the 
 last two columns of logarithmic functions. The table of 
 Proportional Parts may be used in computing the correc- 
 tions. The student must be careful to obseiwe whether 
 the correction, in any given case, should be applied to 
 the tabulated value by addition or subtraction. This can 
 be determined by the theorem that the sine, tangent, and 
 secant of an acute angle increase with the increase of the 
 angle, while the cosine, cotangent, and cosecant decrease; 
 or, more simply, by the principle that corresponding 
 values of the argument and the function must lie within 
 corresponding tabular differences. For example, to find 
 the functions of 71°. 8, we apply to those of 72° two- 
 tenths, or to those of 71° eight-tenths, of the tabular dif- 
 
58 THE ELEMENTS OF LOGARITHMS. 
 
 ferences for the interval between 71° and 72°, adding or 
 subtracting in each case so as to bring the result between 
 the tabulated values for 71° and 72°. Thus we obtain 
 
 sin 71°.8 = 0.94G -f 0.004 = 0.951 — 0.001 = 0.950, 
 
 esc 71°.8 = 1.058 — 0.006 = 1.051 -f- 0.001 = 1.052, 
 
 tan 71°.8 = 2.90 + 0.14 = 3.08 — 0.04 = 3.04, 
 
 ctn 71°.8 = 0.344 — 0.015 = 0.325 + 0.004 = 0.329, 
 
 sec 71°.8 = 3.07 + 0.14 = 3.24 — 0.03 = 3.21, 
 
 cos 71°.8 = 0.326 — 0.014 = 0.309 -f 0.003 = 0.312 ; 
 
 log sin 71°.8 ss 9.976 -f 0.002 = 9.978 — 0.000 = 9.978, 
 
 log esc 71°.8 = 0.024 — 0.002 = 0.022 + 0.000 = 0.022, 
 
 log tan 71°.8 = 0.463 -f 0.020 = 0.488 — 0.005 = 0.483, 
 
 log ctn 71°.8 = 9.537 — 0.020 = 9.512 + 0.005 = 9.517, 
 
 log sec 71°.8 = 0.487 + 0.018 = 0.510 — 0.005 = 0.505, 
 
 log cos 71°.8 = 9.513 — 0.018 = 9.490 + 0.005 = 9.495. 
 
 The two methods of interpolation here indicated lead to 
 the same result. But the best habit, as we have remarked 
 in the book on Logarithms, is that of working from the 
 nearest tabulated value, which is, in this case, 72°. 
 
 In the inverse use of this table, where an angle is found 
 from the logarithm of one of its trigonometric functions, 
 the result should be expressed to the nearest tenth of a 
 degree. E.g., if 
 
 log tan y = 0.367, 
 
 the nearest tabulated value of the log tan is 0.372 = log 
 tan 67° ; then, using formula (32) of the book on Loga- 
 rithms, we have 
 
 Au = 21, y 2 - u = 5, ^=p = A = 0.2, 
 y = 67° — 0.2 = 66°.8. 
 
 In like manner, let the student find the following 
 values : — 
 
 If log sin q. = 9.325, cp = 12°.2; if cos <y = 0.494, <p = 60°.4, 
 if logctn(p = 9.960,9 = 47 o .6; if ctn 9=2.05, cp = 20°.7. 
 
APPENDIX. 
 
 59 
 
 (§3.) It is to be observed that the two angles, taken 
 from opposite sides of the page, which correspond to any 
 one number, found or interpolated in a given column of 
 the table, are complements of each other, and also that the 
 two names of any column, given at the top and bottom of 
 the page, and corresponding to the two angles, are com- 
 plementary. Thus, 
 
 0.292 = sin 17° = cos 73°, 0.494 = cos 60°.4 == sin 29°.6, 
 0.107 = log tan 52° 9.633 = log tan 23°.2 
 
 = log ctn 38°, = log ctn 66°.8. 
 
 This relation is in accordance with formula (B) of (§ 1).* 
 
 Again, it will be seen that the log sin and log esc, the 
 log tan and log ctn, the log sec and log cos are respectively 
 arithmetical complements. Thus, 
 
 log tan 17° =9.485, logctnl7° = 0.515 = co 9.485; 
 log sec 52° =0.211, log cos 52° = 9.789 = co 0.211; 
 log sin 71°.8 = 9.978, log esc 71°.8 = 0.022 = co 9.978. 
 
 This relation agrees with (§ 1) (A), since 10 should be sub- 
 tracted from one of the logarithms of each pair, as taken 
 from the table, in order to reduce it to true value. It is a 
 consequence of this relation that the tabular differences 
 are alike in the columns of log sin and log esc, also in the 
 columns of log tan and log ctn, and lastly in the columns 
 of log sec and log cos. The pairs of columns thus 
 related are bracketed together in the logarithmic part 
 of the table, so that the six columns are printed as 
 three double columns. 
 
 Other formulas of trigonometry, such as 
 
 log tan qp = log sin qp -\- log sec qp, 
 
 log cos qp = log sin qp -|- log ctn qp, 
 
 sin 2 qp -|- cos 2 qp = 1, sec 2 cp — tan 2 qp = 1, &c, 
 
 * The numbers of the sections of the Appendix are distinguished from 
 those of the book by being included in parentheses. 
 
60 THE ELEMENTS OF LOGARITHMS. 
 
 are readily verified by this table. Thus we have 
 
 log sin 78° = 9.990 
 log sec 78° = 0.682 
 
 log sin 78° -f log sec 78° = 0.672 
 log tan 78° = 0.673; 
 
 where a slight discrepancy arises from the accumulation 
 
 of errors in the addition of the logarithms. 
 
 Again, 
 
 log tan 22° = 9.606 
 
 log tan 2 22° = 9.212 
 © log tan 2 22° = log (1 -f tan 2 22°) = 0.066 
 .-. log sec 22^ = i log (1 -J- tan 2 22°) =0.033. 
 
 (§ 4.) Where the printed tabular differences are omit- 
 ted near the top of a column, being printed in the lower 
 part of the same column, the principle of interpolation 
 cannot properly be applied to the table. In this case, we 
 may employ the small table in the upper right-hand corner 
 of the page. The direct use of this table is to find the log 
 sin or log tan of a small angle, by finding the logarithm 
 of the angle itself, expressed in degrees, and adding to 
 that logarithm a logarithm (S for the log sin or 7 T for the 
 log tan) of which the value is given in the small table. 
 Thus, we have 
 
 log sin 5°.23 = 0.718 -f 8.241 = 8.959, 
 log tan 5°.23 = 0.718 + 8.243 = 8.961. 
 
 The log esc and log ctn, as we have seen in (§ 3), are 
 the arithmetical complements of the log sin and log tan. 
 
 log esc 5°.23 = 1.041, log ctn 5°.23 = 1.039. 
 
 The natural esc and ctn are the cuitilogarithms of the 
 log esc and log ctn. E.g., 
 
 esc 5°.23 = 11.00, ctn 5°.23 = 10.95. 
 
APPENDIX. 61 
 
 The log cos, log ctn, log sec, log tan, and natural sec 
 and tan of an angle near 90° are, by (B), found by finding 
 the complementary functions of the complement of that 
 angle. E.g., 
 
 log cos 83°.81 = logsin 6°.19 = 0.791 +8.241 = 9.032, 
 log ctn 83°.81 = 9.035, log sec 83°.81 = 0.968, 
 log tan 83°.81 = 0.965, 
 sec 83°.81= 9.30, tan 83°. 81 = 9.22. 
 
 This table can also be used inversely. Thus, to find 
 the angle whose log sin is 9.123. The large table shows 
 that it is between 7° and 8°. .-. S= 8.241 or 8.240. 
 9.123 — 8.240 = 0.883 = log (7.64 < 7.83). .-. 8= 8.241. 
 9.123 — 8.241 = 0.882 = log 7.62, /. 9.123 = log sin 7°.62. 
 In like manner, we find 
 
 9.055 = log sin 6 Q .51, 9.120 = log tan 7°.52, 
 1.470 = log ctn 1°.94, 0.962 = log sec 83°.73, 
 1.245 = log esc 3°.26, 8.127 = log ctn 89°.232, 
 15.75= ctn 3°.64, 100.0= esc 0°.573, 
 10.00 = sec 84°.26, 37.71 = tan 88°.48. 
 
 This small table may be explained by reference to the 
 principle that the sine and tangent of a small angle are 
 nearly equal to the arc which is subtended by the angle in 
 the unit-circle. But the circumference of the unit-circle 
 is 3.14159 . . . Hence, the arc of 1° in that circle is 
 3 - 1416 - = 0.01745 . . , of which the three-place logarithm 
 is 18 8.242. Then 
 
 arc cp in unit-circle = (cp in degrees) X 0.01745 . . , 
 log (arc cp in unit-circle) = log (cp in degrees) -f- 8.242. 
 
 This formula is exact for any angle cp, and if qp is small, it 
 gives an approximate value of log sin cp and log tan cp, 
 the accurate values to three places being found by sub- 
 stituting for 8.242 the values of S and T given in the 
 table. 
 6 
 
62 THE ELEMENTS OF LOGARITHMS. 
 
 (§5.) This table is directly applicable to acute angles 
 only. If we have to find a function of an angle 
 in any other quadrant than the first, we must subtract 
 from the angle the greatest multiple of 90° which it 
 contains, and then find the same function of the re- 
 mainder if we have subtracted an even multiple, or the 
 complementary function if we have subtracted an odd 
 multiple. See formula (C). The sign of the result is 
 most readily determined by observing the quadrant in 
 which the given angle lies. See (D). The sign affecting 
 the angle may be disregarded in the first part of the work, 
 but must be considered in determining the quadrant. 
 Thus, if log sin 302°.7 is required, we subtract 270^ = 3 
 X 90°, obtaining 32°.7 as the remainder. Then, as we 
 have subtracted an odd multiple, 
 
 log sin 302°.7 = log cos 32°.7 = 9.925, (neglecting signs). 
 
 But 302°. 7 being in the fourth quadrant, where the sine is 
 negative, 
 
 log sin 302°.7 = 9.925?i. 
 
 In like manner we have 
 
 cos 143°.4 — — 0.803, log tan 205°.6 =9.680, 
 sin (— 1055°.7) = 0.412, log sec (— 49P.8) = 0.176ft, 
 ctn (— 16°.1)= — 3.47, log esc 120°.2 =0.063. 
 
 In using the table inversely for angles in any quadrant, 
 we should observe that any one value of a trigonometric 
 function corresponds to two angles between 0° and 360°, or 
 between — 180° and -f- 180°, or in any continuous angular 
 space of four right angles. Thus, 
 
 if log ctn gp = 9.663n, g>=114°.7 or = 294°.7; 
 if log sin y = 9.886, y = 50°.3 or = 129°.7 ; 
 if cos cp = 0.810, cp = 3&°.9 or = 324°.l ; 
 
 if log sec cp = 0.30b*, q = 244°.2 or = 115°.8. 
 
APPENDIX. 63 
 
 Four-Place Table of Logarithmic Trigonometric 
 Functions. 
 
 6.) This table occupies five pages, the last four of 
 which we shall consider first. These pages contain the 
 logarithms of the sine, tangent, cotangent, and cosine, at 
 intervals of 10', from 5° to 85° ; the characteristics, when 
 negative, being increased by 10. The log esc and log sec 
 are, by (A) and (§ 3), the arithmetical complements of the 
 log sin and log cos ; while the log tan and log ctn are 
 arithmetical complements to each other. To find the 
 logarithm of any function of any tabulated angle, we seek 
 tbe degrees, if below 45°, in the left-hand column, and if 
 equal to or above 45°, in the right-hand column, and take 
 the name of the function on the same side of the table 
 as the degrees, and immediately against the degrees. In 
 the line determined by the name of the function, and in 
 the column determined by the minutes, taken at the top 
 of the table if the degrees are found at the left, and at 
 the bottom of the table if the degrees are found at the 
 right, will be found the second, third, and fourth decimal 
 figures of the function. The characteristic and first deci- 
 mal-figure will be found in the left-hand column of the 
 values of the function, unless the second decimal-figure is 
 printed in small type, and, in that case, they are in the 
 right-hand column of the values of the function. E.g., 
 
 log tan 8 P 40' = 9.1831, log sec 38° 00' = 0.1035, 
 
 log sin 18° 10' = 9.4939, log esc 52° 00' = 0.1035, 
 
 log sin 18° 30' = 9.5015, log sec 59° 50' = 0.2988, 
 
 log ctn 32° 20' = 0.1986, log ctn 68° 20' = 9.5991. 
 
 (§ 7.) The functions of angles not tabulated, and lying 
 between 5° and 85°, can be found from this table by inter- 
 polation, care being taken to apply the correction so as to 
 bring the result within the proper tabular difference. To 
 
64 THE ELEMENTS OF LOGARITHMS. 
 
 facilitate the interpolation, the proportional parts of the 
 tabular differences (except those less than 10) occurring 
 on the second and third pages of the table (pp. 8 and 9) 
 are printed on those pages; while the fourth and fifth 
 pages (pp. 10 and 11) contain the corrections for each 
 line arranged as in the table of Logarithms of Numbers. 
 The latter plan is inapplicable to pp. 8 and 9, because the 
 tabular differences vary so fast that an average value can- 
 not safely be used throughout a line. 
 
 Thus, let the functions of 7° 18'.4 (= 1 Q 20' — 1'.6) be 
 required. We have 
 
 log sin log tan log ctn log cos 
 for 7° 20' 9.1060, 9.1096, 0.8904, 9.9964, 
 tabdiff 99, 101, 101, 2; 
 
 and the proportional parts of the tabular differences are 
 for 1' 9.9 10.1 
 
 for 0'.6 ^94 (M)6 while — ^ = 0.32. 
 
 for 1.6 16. 16. 
 
 Then, applying the corrections so as to bring the results 
 between the values for 7° 10' and 7° 20', we have 
 
 log sin = 9.1044, log tan = 9.1080, log cos = 9.9964, 
 log esc = 0.8956, log ctnr= 0.8920, log sec == 0.0036. 
 
 Again, if the functions of 53° 47'.9 (= 53° 50' — 2'.1) 
 are required, we have 
 
 log sin log tan log ctn log cos 
 for 53° 50' 9.9070 0.1361, 9.8639, 9.7710, 
 
 and the lines of Proportional Parts give 
 
 for log sin for log tan and ctn for log cos 
 
 for 2' 1.9 5.3 3.4 
 
 forOU ^09 0.26 0.17 
 
 for2U 2. 6. 4T~ 
 
APPENDIX. C5 
 
 Applying these corrections, we have 
 
 for 53° 47 / .0, 
 
 log sin = 9.9068, log tan = 0.1355, log cos = 9.7714, 
 log csc= 0.0932, log ctn= 9.8645, log sec = 0.2286. 
 
 In like manner, we find : — 
 
 log tan 12° 41'.3 = 9.3525, log cos 37° 28'.7 = 9.8996, 
 log sin 18° 26'.2 = 9.5001, log ctn 51° 43'.7 = 9.8970, 
 log sec 69° 15'.3 = 0.4507, log esc 60° 57'.9 = 0.0583. 
 
 (§ 8.) The first page of this table (p. 7) contains the log 
 sin and log tan for every minute from 0° to 6°, the degrees 
 and names being given at the top, and the minutes in the 
 left-hand column, marked ' at the top, and the log cos and 
 log ctn for every minute from 84° to 90°, the degrees and 
 names being given at the bottom, and the minutes in the 
 right-hand column marked ' at the bottom. Only the third 
 and fourth decimal-figures of the log tan (or log ctn) are 
 printed, the preceding figures being the same as for the 
 log sin (or log cos), except when the third decimal-figure 
 of the log tan (or log ctn) is printed in small type, in 
 which case the second decimal-figure is to be increased by 
 1. Thus: — 
 
 log sin 1° 23'= 8.3828, log cos 86° 37' = 8.7710, 
 
 log tan 1° 23' = 8.3829, log ctn 86° 37' = 8.7717, 
 
 log sin 4° 33' = 8.8994, log cos 87° 40' = 8.6097, 
 
 logtan 4° 33'=8.9008, log ctn 87° 40' = 8.6101. 
 
 "When the angle contains a fraction of a minute, inter- 
 polation may be employed ; and the numbers printed in 
 small type after the values of the functions are multipliers 
 (derivatives), which may be used instead of the tabular 
 differences through half the intervals before and after the 
 line on which they stand, and for both the functions in that 
 line. For example, to find the log sin and log tan of 
 l p 03'.6 (== 1° 04' — 0'.4), we have 
 
66 THE ELEMENTS OF LOGAEITHMS. 
 
 0.4 X 68 = 27.2, log sin 1° 03'.6 = 8.2699 — 27 = 8.2672, 
 log tan 1° 03.6 = 8.2700 — 27 = 8.2673. 
 
 In like manner : — 
 
 log sin 1° 22'.2 = 8.3786, log cos 87° 37'.8 = 8.6165, 
 log tan 1° 22'.2= 8.3787, log eta 87° 37'.8= 8.6169. 
 
 Where the printed multiplier is omitted, as in the 
 upper part of the first column of the table, interpolation 
 cannot properly be applied to the table, and the method 
 which will be explained in (§ 10) must be used for angles 
 not tabulated. 
 
 The log esc and log ctn of angles between 0° and 6^ 
 and the log sec and log tan of angles between 84° and 90° 
 can be found by (A). Thus we have 
 
 log esc 1° 22'.2 = 1.6214, log sec 87° 37'.8 = 1.3835, 
 log ctn 1° 22'.2 = 1.6213, log tan 87° 37'.8 = 1.3831. 
 
 (§ 9.) The right-hand division of page 7 is a table for 
 finding the log sec or log cos of any angle below 9° 51' or 
 the log esc or log sin of any angle above 80° 09'. The 
 first column of this division, marked " Angles " at the top, 
 contains those small angles, expressed to the nearest tenth 
 of a minute, at which the last figure of the log sec and log 
 cos expressed to four decimal places changes its value. 
 The second column, marked " Angles " at the bottom, con- 
 tains the complementary angles of the former. In the 
 intermediate lines, and in the third column, marked "sec" 
 at the top and "esc" at the bottom, are the third and 
 fourth decimal figures of the log sec of the angles which 
 lie between those named in the first column, and of the 
 log esc of the angles between those named in the second 
 column; the preceding figures, which are always 0.00, 
 being given at the top and bottom of the column. Thus, 
 0.0006 is the log sec (to four places) of any angle between 
 2° 53'.0 and 3° 08'.0 and the log esc of any angle between 
 80° 52'.0 and 87 9 07\0. 
 
APPENDIX. G7 
 
 The arithmetical complement of the tabulated loga- 
 rithm is the log cos or log sin of the given angle. Thus, 
 we have 
 
 log sec 1° 22'.2 = 0.0001, log esc 87° 37'.8 = 0.0004, 
 log cos 1° 22fJ2 = 9.9999, log sin 87° 37 / .8 = 9.9996. 
 
 This little table leads us to observe the slow rate of 
 variation of the cos and sec when their angle is near 0°, 
 and of the sin and esc when their angle is near 90°. 
 Hence a small angle cannot be very accurately found from 
 its cos or sec, or an angle near 90° from its sin or esc ; for 
 the function, or its logarithm, expressed to a given number 
 of places, may correspond to any angle included icithin a 
 considerable range. See also (§ 17). 
 
 10.) The log sin, log tan, log esc, and log ctn of 
 
 small angles and the log cos, log ctn, log sec, and log tan 
 of angles near 90° can also be found by the sin ail tables 
 at the bottom of the table of Logarithms of Numbers, 
 which correspond to that explained in (§ 4). We must find 
 the logarithm of the given angle (or of its complement in 
 the case of an angle near 90°) in minutes, and for this 
 purpose we may generally use the extreme left-hand 
 column of the table of logarithms, headed "Angles," 
 without reducing the angle to minutes. Thus: — 
 
 log sin 4° 23'.6 = 2.4209 + 6.4633 = 8.8842, 
 log tan 4° 23'.6 = 2.4209 + 6.4646 = 8.8855, 
 log sin 1° 03'.6 = 1.8035 + 6.4637 = 8.2672, 
 log tan 1° 03'.6 = 1.8035 + 6.4638 = 8.2673, 
 log cos 87° 37'.8 = log sin 2° 22'.2 
 
 = 2.1529 + 6.4636 == 8.6165, 
 log ctn 87° 37'.8 = 2.1529 + 6.4640 = 8.6169, 
 log ctn 89° 59'.90127 = log tan 0° 00'.09873 
 
 = 8.9944 + 6.4637 = 5.4581. 
 
OS THE ELEMENTS OF LOGARITHMS. 
 
 (§ 11.) The functions of angles in all quadrants may- 
 be found by the method which lias already been explained 
 in (§ 5). The table at the top of page 10, the arrangement 
 of which will be readily understood, may be used in this 
 connexion. E.g. : — 
 
 log sin 97° 18'.4 == 9.9964, 
 
 log ctn 413° 47'.9 = 9.8645, 
 
 log tan 192° 41'.3 == 9.3525«, 
 log cos (—108° 26'.2) = 9.5001>i, 
 
 log esc 339° 15'.3 = 0.4507m, 
 
 log cos (— . 37° 28'.7) = 9.8996, 
 
 log sec 590° 27'.0 = 0.1960;?, 
 
 log sin 178° 56'.4 = 8.2672. 
 
 (§ 12.) In astronomy, angles are sometimes expressed 
 in time, instead of arc, the circle being divided into 24 
 hours. 
 
 The angles in our table are accordingly expressed in 
 this system, as well as in degrees and minutes. This is 
 the case with all the angles in the principal table of p. 7, 
 and with the exact degrees on the other pages of the 
 table, little tables for converting minutes into time and 
 time into minutes being given on each of those pages, in 
 the corners of the main table. Thus, we have 
 log sin h 10™08* = 8.6454, log ctn 2 h 31» l 15* =: 0.1101, 
 log esc b h 33™ 20 s = 0.0029, log sec 4 7t 09» l 53 s = 0.3352. 
 
 (§ 13.) In navigation and surveying, horizontal angles 
 are often expressed in points of the compass, of which 
 eight constitute the quadrant, and one is equal to 11° 15'. 
 At the top of page 11 will be found a table of logarithmic 
 trigonometric functions for every quarter-point. The 
 points at the top are to be used with the names on the 
 left, and the points at the bottom with the names on 
 the right. Thus, we have 
 
APPENDIX. 69 
 
 log sin li pt == 9.3856, log ctn 5J pt = 9.7280, 
 log sec 7 pt = 0.7098, log cos 10$ pt = 9.6734. 
 
 Interpolation sliould not be applied to the compass- 
 table. 
 
 Inverse Trigonometric Functions. 
 
 (§14.) The angle which corresponds to a given value 
 of any logarithmic trigonometric function may be found 
 by the inverse use of the table explained above, in (§§ 6-13). 
 But a more convenient way of arriving at the angle is 
 furnished by the table of Inverse Trigonometric Func- 
 tions. In the headings of this table, A denotes any tri- 
 gonometric function, log A its logarithm, and sin - x A, 
 cos - 1 A, &c, the angle of which A is the sine, cosine, 
 <&c, as the case may be. This is in accordance with cus- 
 tom and with the principle of notation explained in Loga- 
 rithms, § 34. The symbol sin -1 may be read inverse 
 sine, or antisine, or the angle (or arc)of which the sine is. 
 It is sometimes denoted by " arcsin." Thus we have 
 
 log — 1 = antilog, sin "~ 1 = arcsin. 
 
 The inverse of " log sin " may be written in either of the 
 following ways : — 
 
 (log sin) — ! = sin — * log -1 = arcsin antilog. 
 
 The argument of this table is the logarithmic trigono- 
 metric function (the characteristic being increased by 10, 
 when negative), and is denoted by log A. In the first 
 three divisions of the table, occupying page 12 and the 
 first half of page 13, the argument is given, at intervals 
 of 0.01, from 8.5000 (— 10) to 0.0000 in the left-hand 
 column of the table, and from 0.0000 to 1.5000 in the 
 right-hand column of the table. The names at the top 
 are to be taken with the left-hand column, and the names 
 
70 TIIE ELEMENTS OF LOGARITHMS. 
 
 at the bottom with the right-hand column. Thus, under 
 the name sin- 1 A, cos — 1 A, tan -1 A, or ctn- 1 A, and 
 in the same line with a value of log A, found at the left, 
 is the angle of which that value of log A is the log sin, 
 log cos, log tan, or log ctn ; and over the name tan -1 A, 
 ctn — 1 A, sec -1 A, or esc -1 A, and in the same line with 
 log A at the right, is the angle of which log A is the log 
 tan, log ctn, log sec, or log esc. E.g. : — 
 
 (log sin)- 1 8.8200= 3° 47'.3, 
 (log tan)- 1 9.7400 = 28° 47', 
 (log esc) - 1 0.7800 = 9° 33', 
 (log cos) -i 9.0700 = 83° 15', 
 (log ctn) - 1 0.1000 = 38° 28', 
 (log sec) - 1 1.4200 = 87° 49/3. 
 
 The angle corresponding to a non-tabulated value of 
 log^l may be found by direct interpolation; and the num- 
 ber in small type on the right or left of any tabulated angle 
 is a multiplier {derivative) which may be used, instead of 
 the tabular difference, in this interpolation, through half 
 the interval before and after the angle against which it 
 stands, being so pointed off as to express the number of 
 minutes by which the angle is varied in that part of the 
 table, by a change of one unit in the third decimal-place 
 of log A. The general table of Proportional Parts (p. 2) 
 may be used in finding the correction. 
 
 For instance, suppose we have to find the angle of 
 which the log sin is 8.9023(— 10). 
 
 We have, if log A = 8.9000, sin- 1 A = 4° 33'.4 
 
 by table of Proportional Parts, 0'.63 X 2.3= 1'.4 
 whence, if log A = 8.9023, sin - 1 A = 4° 34'.8 
 
 adding so as to approach the value for 8.9100. 
 
 Next, to find (log ctn)- 1 9.7984 (—10) : — 
 
APPENDIX. 71 
 
 We have, if log A = 9.8000 
 
 (== 9.7984 + 0.0016), ctn- 1 ^1 = 57° 45'; 
 
 by table of Proportional Parts, 3'.6 X 1-6 = 6' ; 
 
 whence, if log A = 9.7984, ctn - 1 A = 57° 51' ; 
 
 again adding, so as to approach the value for 9.7900. 
 
 Again, to find (log sec)— 1 0.5157: — 
 
 We have, if log A = 0.5200 
 
 (=0.5157 + 0.0043), sec-* .4 = 72° 25'; 
 
 by table of Proportional Parts, 2'.5 X 4.3 = 11'; 
 
 whence, if log A = 0.5157, sec- 1 .4 = 72° 14'; 
 
 subtracting so as to approach the value for 0.5100. 
 
 In like manner, let the student find the following 
 results : — 
 
 (log tan)- 1 0.6931 = 78° 33', 
 (log esc)- 1 1.3574= 2° 31'.0, 
 (log sin)- 1 9.0590= 6° 35', 
 (log cos)- 1 9.6062=66° 11', 
 (log tan)- 1 8.9507= 5° 06'.1, 
 (log ctn)- 1 0.5555 = 15° 33'. 
 
 TJie angle is, in every case, to be found icith the same 
 degree of accuracy as that of the tabulated angles between 
 which it is interpolated ; that is, to the nearest tenth of a 
 minute if found from the first division of the table, and 
 otherwise to the nearest minute. The reason of the dif- 
 ference is that the angle varies less rapidly at the begin- 
 ning of the table than at the end for a given change in 
 log A, and hence is more accurately determined by a value 
 of log A given to four places. Towards the end of the 
 table, it is sometimes not even determined to the nearest 
 minute, as we shall presently explain more particularly. 
 See (§ 17). 
 
 (_§ 15.) The student should remark that the angles in 
 the columns of sin ~ x A and cos ~ x A are complements of 
 
72 THE ELEMENTS OP LOGARITHMS. 
 
 each other; also in those of tan -1 A and ctn - 1 A; 
 and again in those of sec -1 A and esc -1 A. This is in 
 accordance with (B). 
 
 Again, the values of the argument on opposite sides of 
 the page are arithmetical complements, and therefore cor- 
 respond to reciprocal values of A ; while the angles bear 
 to these values of A reciprocal relations, as the upper and 
 lower names of the same column (sin -1 A and esc -1 A ; 
 cos -1 A and sec -1 A ; &c.) indicate. See (A). 
 
 (§ 1G.) The multipliers for the first and second columns 
 are omitted in the lower half of the third division of the 
 table; because in that part of the table they vary too 
 rapidly to admit of accurate interpolation through so wide 
 an interval as 0.01. Hence, the last three divisions of 
 the table are given, containing the argument at smaller 
 intervals, from 9.7500 to 0.0000, the multipliers being still 
 so expressed as to measure the variation in minutes for a 
 change of 0.001 in the argument. Thus, we find : — 
 
 If log sin cp = 9.8236, g>=41° 56' — 1.4 X 7M =41° 46' ; 
 if log sin cp = 9.8812, cp = 49° 39' —0.8 X 9.'3=49° 32'; 
 if log sin 9 = 9.9173, <? = 55° 42' + 0.3 X 12' =55* 46'. 
 
 The values of cos -1 A are the complements of those 
 of sin - 1 A; and the arithmetical complements of the 
 tabulated values of log A are the values of the log esc for 
 the angles in the column sin — 1 A, and of the log sec for 
 the complements of those angles. Thus, we have : — 
 
 If log cob <p=z 9.9267, sin- 1 A = 57°38',g> = 32 22 ; 
 if log esc cp = 0.0265, log sin cp = 9.9735, q, = 70° 11' ; 
 if log sec cp = 0.1089, log cos cp = 9.8911, cp = 38° 54' ; 
 
 if log cos cp = 9.8183, cp = 48° 51' ; 
 
 if log sec cp = 0.1252, cp =41° 27'; 
 
 if log esc cp = 0.1879, cp = 40° 27' ; 
 
 if log sec cp = 0.1999, cp = 50° 52'. 
 
APPENDIX. 73 
 
 (§ B7.) When a change of 0.0001 in any logarithmic 
 trigonometric function produces a change of more than 1' 
 in the corresponding angle, the four-place logarithm does 
 not determine the logarithm to minutes, since the uncer- 
 tainty of ±0.00005 in the logarithm will leave an un- 
 certainty of ±0'.5 in the angle. This is never the case with 
 the log tan or log ctn ; but an angle can always be found 
 to minutes from either of these functions and by means 
 of the first three divisions of the table. But with the 
 other functions this uncertainty may occur ; and the point 
 at which it begins (at log A= 9.8940 or = 0.1060) is indi- 
 cated by a note in the table. The angles are, however, 
 still tabulated to minutes, and the given values may be 
 regarded as the probable values for the assumed values of 
 the functions. Thus, if log sin cp = 9.9900, of which the 
 true value may lie anywhere between 9.98995 and 9.99005, 
 (jp may have any value from 77° 43' to 77° 47'. But 
 if 9.9900 expresses the value of the logarithm exactly, 
 cp = 77° 45', which is therefore adopted as the probable 
 value. 
 
 At the bottom of the last column of the table, interpo- 
 lation becomes inaccurate. But the uncertainty just 
 spoken of is here so great that the angle cannot be deter- 
 mined with nicety from the given function ; and the inac- 
 curacy of interpolation is, therefore, of little consequence. 
 
 It is to be observed that, with four-place logarithms, 
 the angle can be determined to minutes from its log tan 
 or log ctn in all parts of the quadrant ; from its log sin 
 or log esc in the first half of the quadrant ; and from its 
 log cos or log sec in the last half This observation will 
 often direct the computer in the choice of his method, 
 where several are open to him. 
 
 (§ 18.) This table is only applicable to values of the 
 logarithm between 8.5000 (— 10) and 1.5000. If the 
 logarithm lies outside of these limits, the angle may be 
 
 7 
 
74 TIIE ELEMENTS OF LOGARITHMS. 
 
 found by the table of Logarithms of Numbers, by the 
 formula printed between the two divisions of page 12, 
 which is obtained by reversing those at the bottom of pp. 
 4 and 5» For those formulas give 
 
 log (cp in minutes) = log sin cp — S=. log tan cp — T. 
 
 Hence, if £= T= 6.4637, we have 
 
 log (cp in minutes) = log A + 3.5363 — 10, 
 
 whether A denotes sin cp or tan cp. S has always the 
 above value, when log A < 8.5000, but T may reach the 
 value 6.4639, and thus give a (generally unimportant) 
 correction of our formula. 
 
 If the given logarithm is the log cos or log ctn of the 
 required angle, that angle will be the complement of cp 
 determined as above. If log A > 1.5000, we must take 
 its arithmetical complement, which will be the logarithm 
 of the reciprocal function, and proceed as above. 
 
 Thus: — 
 
 If log sin cp = 7.5192, log <r = 1.0555, cp = 11 ; .3G ; 
 
 if log cos c f == 8.0079, log cog>= 1.5442, co qp= 35'.01, 
 
 9 = 89° 24'.99; 
 if 1. >g tan c f = 6.8197, log cp = 0.3560, cp = 2'.270 ; 
 
 if log ctn c; = 1.9981, log tan cp = 8.0019, cp — 34'.53 ; 
 
 if log sec c f — 1.5019, log cos cp = 8.4981, cp = 8$° 1T.7; 
 
 if log esc qp = 4.1254, log sin cp = 5.8746, cp = 0'.2576. 
 
 (§ 10.") Acute angles only nre directly given by this 
 table. When angles in any quadrant are admissible, the 
 signs of the given trigonometric functions must be noted, 
 and it must be remembered that to every value of any 
 function two angles correspond, in any four consecutive 
 right angles ; for example, between 0° and 360°, or be- 
 
ArPEXDix. 75 
 
 tween — 180° and -f- 180°. The values of the angles 
 may be found from the table by the aid of (C). Thus : — 
 
 (log sin)- 1 8.9023/1 = 184° 34'.8 
 
 or = 355°25'.2; 
 
 (log ctn)- 1 9.7984 = 57° 51 
 
 or = 237° 51' ; 
 
 (log sec)- 1 0.5157/1 = 107° 46' 
 
 or= 252° 14'; 
 
 (log tan)- 1 0.6931w = 101° 27' 
 
 or= 281° 27'; 
 
 (log cos)- 1 9.6062 = 66° 11' 
 
 or = — 66° 11' ; 
 
 (log sin)- 1 9.0590 = 6 Q 35' 
 
 or = 173° 25'. 
 
 In many questions, the required angle can only be 
 acute or obtuse. It is then fully determined by the sign 
 of the given function, unless that function is a sine or co- 
 secant ; and, in that case, two values of the angle are 
 admissible if the function is positive, none if the function 
 is negative. Thus, in such a question : — 
 
 (log ctn)- 1 0.5555/1 = 164° 27'; 
 
 (log sin)- 1 9.8236 = 41° 46' or = 138° 14' ; 
 
 (log cos)- 1 9.9267 =32° 22'' ; 
 
 (log sin)- 1 9.8812/1 is an inadmissible angle. 
 
 (§ SO. - ) This table can also be used inversely to find a 
 logarithmic trigonometric function of a given angle ; and 
 if we were restricted to one trigonometric table, it would 
 probably be found more convenient to use this table both 
 directly and inversely than that which precedes it. 
 
 Let the student find from this table : — 
 
 logtan57°46' =0.2003, log sec 50° 27' =0.1961, 
 log cos 63° 56' = 9.6429, log sin 4° 23'.6 = 8.8843, 
 log ctn 4° 23'.6 = 1.1144, log ctn 87° 37'.8 = 8.6170. 
 
 (§ SI.") If the natural value to four figures of any tri- 
 gonometric function of a given angle, or the angle which 
 corresponds to such a value, is required, we must use the 
 
76 THE ELEMENTS OF LOGARITHMS. 
 
 table of Logarithms of Numbers, in connexion with one 
 of the four-place trigonometric tables. Thus : — 
 
 tan 57° 46' = 1.586, sin 4° 23'.6 = 0.0766, 
 sec- 1 26.30 = 87° 49 / .3, ctn-i 0.8217 = 50° 36'. 
 
 Examples in Trigonometry for the Four-Place Tables. 
 
 Seaver's Formulas of Trigonometry (Boston, John Allyn) will be found 
 a convenient book of reference in these examples. 
 
 (§ 22.) Plane Bight Triangles. 
 Ex. 1. Given : h = 0.4958, A = 54° 44'. 
 
 A =0.4958 log 9.6953 9.6953 
 
 A = 54° 44' log sin 9.9120 log cos 9.7615 
 
 log a 9.6073 
 
 log b 9.4568 
 
 i? = 90°— .4 = 35° 16' a= 0.4048 
 
 b = 0.2863 
 
 Ex. 2. Given: h = 54.57, a = 23.48. 
 
 
 A = 54.57 
 
 colog 8.2630 
 
 a = 23.48 
 
 log 1.3707 
 
 h + a = 78.05 log 1.8924 jg ™ J } 9.6337 
 
 h — a — 31.09 log 1.4927 A = 25° 29' 
 
 2)3.3851 B = 04° 31' 
 
 log b 1.6926 b =49.28 
 Ex. 3. Given : a = 0.04437, b = 0.02216. 
 
 a = 0.04437 log 8.6471 8.6471 
 b =0.02216 colog 1.6544 
 
 A = 63° 28' log tan ) ~~~ log esc ) 
 
 B = 26° 32' log ctn } °- 3015 log sec \ °- 0483 
 
 h = 0.04959 log 8.6954 
 
 We may also find h by the Gaussian table, as in 
 Logarithms, § 66. 
 
APPENDIX. 77 
 
 Ex. 4. Given: h = 0.3736, B= 12° 30'. 
 To be computed : A = 77° 30', a = 0.3648, ft = 0.08086. 
 
 JSfe 5. Given: ft = 14.548,-5= 54° 24'. 
 To be computed : A = 35° 36', A = 17.888, « = 10.415. 
 
 Ex. 6. Given : ft = 11.111, .4 = 11° 11'. 
 To be computed: B= 78° 49', h = 11.324, a = 2.196. 
 
 JSb. 7. Given: 7* = 0.06723, ft = 0.05489. 
 To be computed : a = 0.03882, A = 35° 16, B = 54° 44'. 
 
 £fc 8. Given : a = 8.148, ft = 10.864. 
 To be computed: h = 13.58, ^ = 36° 53', B = 53° 07'. 
 
 (_§ 23.) /Special Cases of Plane Bight Triangles. 
 
 Seaver's Formulas, 130. 
 
 Ex. 1. Given: h = 4602.8360, ft = 4602.2106. 
 
 h—b= 0.6254 log 9.7962 
 
 2h = 9205.6720 colog 6.0359 
 
 
 2)5.8321 
 
 M 
 
 log sin 7.9160 
 
 
 3.5363 
 
 £-4=0° 28'.33 
 
 log 1.4523 
 
 yl = 0° 56'.66 
 
 .5=89* 03'.34 
 
 .2. Given: a =3792.8 
 
 1714, ft = 3791.8692. 
 
 a = 3792.8714 
 
 colog 6.4211 
 
 ft = 3791.8692 
 
 colog 6.4212 
 
 a _ft = 1.0022 
 
 log 0.0009 
 
 a+ ft =7584.7406 
 
 log 3.8800 
 
 2. 
 
 colog 9.6990 
 
 ^ — i? 
 
 log tan 6.4222 
 
 
 3.5363 
 
 A — B = 0° 00'.9088 log 9.9585 
 A + B = 90° OO'.OOOO 
 
 A = 45° 00'.4544 B=U° 59'.5456. 
 
78 THE ELEMENTS OF LOGARITHMS. 
 
 Ex.3. Given: h = 210.54219, a = 210.48823. 
 
 h — a= 0.05396 log 8.7321 
 2 h = 421.08438 colog 7.3756 
 
 2) 6.1077 
 
 45° — \ A log sin 8.0538 
 
 3.5363 
 
 45° 
 
 -M= 
 
 - C 
 
 ' 38'.92 
 
 log 
 
 1.5901 
 
 
 \A: 
 
 = 44 c 
 
 ' 21'.08 
 
 
 
 
 A-- 
 
 = 88° 
 
 42U6 
 
 B-- 
 
 = l°17 / .84 
 
 Ex. 4. 
 
 Given : 
 
 A = 
 
 107.9823, 
 
 B = 
 
 :2°08'.64. 
 
 To be computed : a = 107.8867, b = 4.040, A = 87° 51'.36. 
 
 Ex.5. Given: a = 920139, 5 = 925897. 
 
 To be computed: ^ = 44° 49'.27, J5 = 45° 10'.73. 
 
 Ex. 6. Given: h = 2.096889, ^ = 86° 25'.38. 
 
 To be computed : a = 2.092802, b = 0.1308, B= 3° 34'.62. 
 
 (§ 24.) Plane Oblique Triangles. 
 Ex. 1. Given: a= 0.3578, -Z?=32°41', C= 47° 54'. 
 
 ^1 — 180° — (J5+ C) = 99° 25'. 
 
 a =0.3578 log 9.5537 9.5537 
 
 B = 32° 41' log sin 9.7324 (7 = 47° 54' log sin 9.8704 
 A = 99° 25' W esc 0.0059 0.0059 
 
 b =0.1959 log 9.2920 c = 0.2691 log 9.4300 
 
 Ex. 2. Given : c= 325.06, A = 154° 22', (7=8° 03'. 
 To be computed : B = 17° 35', a = 1004.3, 5 = 701.2. 
 Ex. 3* Given : a = 4236, J = 5124, B = 124° 50'. 
 
 * Each of the examples 3-9 should be illustrated by a figure. 
 7* 
 
APPENDIX. 79 
 
 a = 4*236 log 3.6270 
 
 b =5124 colog 6.2904 log 3.7096 3.7096 
 
 B = 124° 50' log sin 9.9142 log esc 0.0858 0.0858 
 
 A x = 42° 44' log sin 9.8316 
 A 2 =137*16' 
 <7i= 12° 26' log sin 9.3330 
 
 (7 2 =— 82° 06' log sin 9.9959;i 
 
 log 3.1284 log 3.7913^ 
 C! = 1344 c 2 =— 6184 
 
 Ex. 4- Given : a = 4236, b = 5124, A = 124° 50'. 
 
 Compute: B x = 83° 04', <7i= — 27° 54', ci= — 2416. 
 J? 2 = 96° 56', <7 2 = — 41° 46', c 2 = — 3438. 
 
 Ex.5. Given: a = 49.00, c= 67.98, (7=32° 18/ 
 
 Compute : A\ = 22° 39', B x = 125° 03', h = 104.17 ; 
 A 2 = 157° 21', B 2 = — 9° 39', 6 2 = — 21.32. 
 
 JS&. & Given: a = 49.00, c = 49.00, (7=32° 18'. 
 
 Compute : A x = 32° 18', B x = 115° 24', b x = 82.84 ; 
 ^2=147*42^,^2= 0*00', 5 2 = 0. 
 
 -Sis. 7. Given: a = 49.00, c = 31.24, (7=32° 18'. 
 
 Compute : A x = 56* 57', B x = 90° 45', h = 58.46 ; 
 ^ 2 = 123* 03', B 2 = 24* 39', h = 24.38. 
 
 ^c.5. Given: a =49.00, c= 26.18, <7<=32* 18'. 
 
 To be computed :A 1 =A 2 = 90* 00', i?i = B 2 =57° 42', 
 ^ = h = 41.42. 
 
 jEk. 9. Given: a =49.00, c= 12.50, (7=32* 18'. 
 
 We find log sin .4 = 0.3211. .-. sin A > 1, which is 
 impossible, and there is no triangle which satisfies the 
 conditions. 
 
 Ex. 10. Given : a = 6.239, b = 2.348, C= 110* 32'. 
 
80 THE ELEMENTS OF LOGARITHMS. 
 
 a + b = 8.587 colog 9.0662 
 
 a — £ = 3.891 log 0.5900 
 
 90° — L(7 = L(^_L-i?)— 34° 44' log tan 9.8409 
 
 \ (A — B) = 17° 26' log tan 9.4971 
 A = 52° 10' 
 
 B = 17° 18' log esc 0.5267 
 
 £7=110° 32' log sin 9.9715 
 
 b = 2.348 log 0.3707 
 
 c= 7.395 log 0.8689 
 
 The side c can also be found in the following ways : — 
 
 (a + b) log 0.9338 (a — b) log 0.5900 
 i(A + B) log cos 9.9147 log sin 9.7557 
 
 x \a — B) log sec 0.0204 log esc 0.5235 
 
 c= 7.395 log 0.8689 c = 7.400 log 0.8692 
 i (^ — B) is so small that the slight error in its com- 
 puted value gives rise to a perceptible error in its log esc, 
 and therefore in c as computed by the last formula. 
 
 Ex. 11. Given : a = 0.01511, c = 0.03100, B = 
 169° 45'. 
 To be computed : A = 3° 21'.3, (7 = 6° 53'.7, b = 0.2541. 
 
 Ex. 12. Given : a = 0.13561, b = 0.12348, c = 
 0.14091. 
 
 s = 0.20000 colog 0.6990 
 
 5 — a — 0.06439 log 8.8088 co 1.1912 
 5 — 5 = 0.07652 log 8.8838 co 1.1162 
 5— c = 0.05909 log 8.7715 co 1.2285 
 
 2) 17.1631 
 
 log r 8.5816 8.5816 
 
 A = 61° 18' \A — 30° 39' log tan 9.7728 
 B= 53° 00' \B = 26° 30' log tan 9.6978 
 
 6 — 65° 42 ' \C= 32° 51' log tan 9.8101 
 
 2^=180° 00' 
 
APPENDIX. 81 
 
 Ex.13. Given: a =17.856, 5 = 13.349, c = 11.111. 
 To be computed : A = 93° 20', B = 48° 16', C= 38° 24'. 
 
 Ex. U. Given: a = 32.571, A =29° 47'.61, J? = 
 33° 10'.37. 
 
 Compute by Seaver, 131, Third Method: 5 = 35.868. 
 
 Ex. 15. Given : C= 121* 42', log a = 1.8968, log 6 
 = 2.0014. 
 Compute by Seaver, 133, Second Method : ^ = 25° 20', 
 ^=32° 58'. 
 
 Ex.16. Given: a = 0.062387, b = 0.023475, C = 
 110° 32'. 
 Compute by Seaver, 133, Third Method : c = 0.07395. 
 
 (§ 25.) Heights and Distances. 
 Ex. 1. At the distance of 277.3 feet from a tower, which 
 stands on a horizontal plane, the angle of elevation of the 
 tower is observed to be 37° 28'. Find its height. 
 
 Ans. 217.5 feet. 
 
 Ex. 2. From the top of a tower 367.6 feet high, an 
 object is observed in the horizontal plane on which the 
 tower stands, with an angle of depression of 49° 43'. 
 Find the distance of the object from the foot of the 
 tower. Ans. 311.4 feet. 
 
 Ex. 3. Two observers, stationed on opposite sides of 
 a cloud, observe the angles of elevation to be 51° 23' and 
 30° 18', their distance apart being 1000 feet. Find the 
 height of the cloud and its distances from the two ob- 
 servers. Ans. 2712 feet ; 3472 feet ; 5401 feet. 
 
 Ex. 4. The angle of elevation of the top of a tower, 
 which stands on a horizontal plane, is observed at one 
 station to be 68° 19', and at another station, 546 feet farther 
 from the tower, to be 32° 34'. Find the height of the 
 
82 THE ELEMENTS OF LOGARITHMS. 
 
 tower, and its distances from the two points of observa- 
 tion. Ans. 467.6 feet ; 185.9 feet ; 731.9 feet. 
 
 Ex. 5. An observer on shipboard sees a cape bearing 
 N.N.E. ; and after sailing 30 miles N.W. by N., he sees the 
 same cape bearing E. by S. Find the distances of the cape 
 from the two points of observation. 
 
 Ans. 21.65 miles ; 25.43 miles. This example is most 
 easily solved by finding the angles in points (namely, 5 
 pts, 4 pts, and 7 pts), and then using the table at the top 
 of p. 11 of the Tables. 
 
 Ex. 6. Coasting along shore, I saw a cape of land 
 bearing N.N.E. ; and after sailing W.N.W. 20 miles, I 
 saw it bearing N.E. by E. Find the distances of the cape 
 from the ship at both stations. 
 
 Ans. 29.93 miles ; 36.00 miles. 
 
 Ex. 7. Being at sea, we saw two headlands, of which 
 one bore S.W. by W. and the other W. byN. The chart 
 showed that the first headland bore S.E. from the second, 
 and was distant from it 23.25 miles. Find our distances 
 from both headlands. Ans. 18.26 miles ; 32.25 miles. 
 
 Ex. 8. Two ships sail from the same port, the one S.W. 
 30 miles, and the other S.E. by S. 40 miles. Find the bear- 
 ing and distance of the second ship from the first. 
 
 Ans. S. 74° 30' W. ; 45.08 miles. 
 
 Ex. 9. An observer from a ship saw two headlands ; 
 the first bearing N.E. by E., and the second N.W. After 
 he had sailed N.N.W. 10.25 miles, the first headland bore 
 E. by N., and the second W.N.W. Find the bearing and 
 distance of the first headland from the second. 
 
 Ans. N. 88° 02' E., or nearly due east ; 35.25 miles. 
 
 Ex 10. An observer saw two headlands; the first 
 bearing S.E., the second E.S.E. After sailing E. by N. 
 10 miles, he saw the first bearing S. by E., and the second 
 
ArrENDix. 83 
 
 S.E. by S. Find the bearing and distance of the first 
 headland from the second. Arts. S.S.W. 6.89 miles. 
 
 Ex. 11. At one station, the bearing of a cloud is 
 N.N.W., and its angle of elevation 50° 35'. At a second 
 station, bearing from the first N. by E. and distant 1 mile, 
 the bearing of the cloud is W. by N. Find the height of 
 the cloud and its distance from each station. 
 
 Am. 7727 feet; 10002 feet; 8838 feet. 
 
 Ex. 12. In the midst of a level plain, which is crossed 
 by a straight road, stands a tower, 250 feet high. An ob- 
 server at the top of the tower sees an object which moves 
 on the road. At first, it bears N.N.W., and its angle of 
 depression is 16° 08' ; five minutes later, it bears E. by S., 
 and its angle of depression is 32° 18'. Find the direction 
 of the road, its distance from the tower, and the rate at 
 which the object is moving. 
 
 Am. S.E. i S.; 250.9 feet ; 2.575 miles per hour. 
 
 (§ 20.) Areas of Triangles. 
 
 Ex. 1. Given : a = 12.34 chains, b= 17.97 chains, (7= 
 135° 04. 
 
 To be computed : Area = 7.832 acres = 7 A 3R 13r. 
 Ex. 2. Given : a— 17.95 chains, _Z? = 100°, (7=70°. 
 To be computed : Area = 85.90 acres = 85A 3R 24r. 
 
 Ex. 3. Given: a = 45.56 chains, 5 = 52.98 chains, c = 
 61.22 chains. 
 
 To be computed : Area = 117.3 acres = 117A 1.2R. 
 
 Ex. 4- Given: a= 32.56 chains, 5 = 57.84 chains, c = 
 44.44 chains. 
 
 To be computed: Area= 71.93 acres = 71 A 3R 29r. 
 
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