L OGA RITH MIC A ND
TRIGONOMETRIC TA BLES
RE VISED EDITION
PREPA RED UNDER THE DIRECTION OF
EARLE RAYMO ND HEDRICK
ENTIRELY RE-SET IN TYPE FAC E
NEW YORK
THE MACMILLAN COMPANY C O YRIGH T 19 13 A ND 1920 P , ,
B r T H E MA CMIL L A N COMPA NY.
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e U and electrot d . Revis edition ublish A s 192 . Re rinte S t p ype ed p ed ugut , 0 p d A ril 1924 Ma Se tember D ecember 1925 A ril June 1926 January p , ; y , p , , ; p , , ; , l 2 mbe r 1 m m Juy, 19 7 ; March , D ece 928 ; Nove ber, 1929 October Nove ber 1930 ; , ; , , A ril October 19 1 F r ar 19 2 A r 3 a m r 1933 p , , 3 ; eb u y , 3 ; p il, 193 ; M y , 1933 ; Nove be , ;
July, 1935.
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The present edition of this book contains all of the tables in the r i u i i ns All have been reset in a new and ver re l ev o s e o . a p d t y dab e type . Great care has been exercised to preserve and to increase the great e r r li ili ha xiste in th revio e i i n d g ee Of e ab ty t t e d e p us d t o . For careful rea in Of the roo s either in the first roo s ma e romt or i h d g p f , p f d f ype n t e roo s m romcast lates I amin ebte to m u h r h p f ade f p , d d y da g te Elisabet n i h r L Mill r e r l a d h r h n Mr R c a . e t s ve a Of m e usba d, . d , o y own students, and to h llowin rien s in other institutions some ims wi h h t e fo g f d , t e t t e i r a h en s : Pro ess r . urrier B own ni rsi d Of t eir stu t o H . ve Pr d f C C , U ty ; o ess r i ni ersit f In iana Pro s r o H T . Dav s v O e o H . B ht f . , U y d ; f s . Dwig , M hue s Insti ute of Technolo Pr ess r B F ni assac s tt t o o . r gy ; f W. o d , U
versi Of Mi hi an Pro essor A . . ar in ni r i ty c g ; f M H d g, U ve s ty Of Arkansas ; Pr r Pomona olle Pr s r L h o ssor G . Jae e e o es o S n ni fe C . g , C g ; f . . Jo ston, U r s r A K n ver i i Pr s s . J m s t o et o t o e o . e n r a A hin n y f D ; f p e d C . . Hutc so , ni rsi f olora o Pro essor G Mullin r r ll U ve ty o C d ; f . W . s, Ba na d Co ege ol mi i r it Pr ess r L M P n h i u a n ve s o o . . assa o M sa nst (C b U y) ; f , as c usetts I u h l r r t O T no o P o esso s H . L . Rietz Ros n F te f ec gy ; f , coe Woods, a d J. . i w Pr ss r rs Reill ni t Of o a o e o E . E . s n I h y, U vers y I ; f Wat o , owa State Teac e
ll r Falls r E . ilson mri n r o e e at e a D . . a b e ass a d P o C g C d ; W W , C dg , M . ; fess r a hr n ant Athens olle e A hen Al am h Of o K t y Wy , C g , t s , ab a. Eac h s ns or rou s has rea th omlet r i h lin t e e perso g p d e c p e p oof . W t deep fee g, I may record also that the late Professor Louis Ingold of the University Of Miss uri rea the roo s u a 54 n h n m h l of o d p f p to p ge , a d ad se t e t e ast
h within a wee Of his en ath n nu r 5 1 5. t ese pages k sudd de o Ja a y 2 , 93 ’ These careful readings render the possibility Of printers errors ex r ml m il h l l i n h r h an t e e y re ote . Wh e t e ca cuat o of t e p obability t at un iscovere err r exists is not simle a s ri t a un has en e t Of d d o p , t c cco t be k p each error found and of the total number not found by any one group Of rea ers so that a basis for a statistical calculation is nown : the result d , k ’ ing probability that even one undiscovered printers error exists is not mre h n one in man th usan o t a y o ds . I esire to ex ress here m than s ll tho ar ic larl those d p y k to a se , p t u y mentione abo e who hav assis e in h eff r t ma h s tables d v , e t d t e o t o ke t e e so ree romerrors and there ore s relia l now of no comarable f f f o b e . I k p me h f r rin hi li t od o secu g t s qua ty in a set Of tables . I repeat also my acknowledgment made in the original edition to man reviousl existin tabl s r i l rl h f an those y p y g e , pa t cua y t ose O Vega d of Houl urin th ro - rea in h h ha have e . D e o o w o a si te g p f d g, t se ve s s d comare these tables wi h a rea ari in a l in lu in p d t g t v ety Of exist g t b es, c d g several hi h- lace tables and h val h n r al la an g p , t e ues ave bee ec cu ted d chec e whenever a isa r mn h n o r k d d g ee e t as bee disc ve ed . Finall I i h mn n y, w s to e tio the excellent cooperation Of the editorial stafi Of the Macmillan oman un r h l ir i n r F C p y de t e ab e d ect o Of M . . T . h Sutp en.
E . R. HEDRICK CONTENTS
EX PLANATION or THE TABLES
TABLES PRINCIPALLY TO FIVE PLACES
T LE MM N LO RITHMS or NUM ERS AB I. CO O GA B
T LE AB Ia.
T LE II A r THE TRI ONOMETRIC FUNCTIONS 21—44 AB . CT UAL VALUES o G
T LE III or S AND T roa INTE RrOLATION AB a. VALUES
T LE O L or TRI NOMETRIC F NC AB III. COMM N OGARrrHMS THE GO U
TABLE REDUc rION or DE GREES TO RADIANS
T L TRI N METRIC FUN TI NS r AN LES IN RADI NS 2- 3 AB E V. GO O C O o G A 9 9
T LE I r E REE S AB Va. REDUC T ON o RADIANS To D G TABLE 94- 111
T LE I 112- 114 AB VI .
L r r l M TABLE VIII. MU TIPLES o M AND o /
T LE IX ND L T M r HrrERBOLIC FUNCTI NS 116- 122 AB . VALUE S A OGARI H S o O
T LE L ND R T MS r VE R NES 123—125 AB X . VA UES A LOGA I H o HA SI
T LE F R LE— r PRIMES 120 - 127 AB X I. ACTO TAB LOGARITHMS o
T LE X IIa AB .
LE X IIb C M D TAB . O POUN DISCOUNT
T LE X II AMOUNT or ANNUIT r AB c. AN PRE SENT VALUE or AN ANNUIT r
rOR I CO P TIONS TABLE X IIe. LOGARITHMS NTEREM M UTA AME RICAN Ee R-NOR MORTALIT r TABLE
LE II IMPORT T C NST NTS TAB X I. AN O A
BRIEF TABLES— PRINCIPAL LY TO FOUR PLACES
LE a MM N L RITHMS 134—135 TAB X IV . CO O OGA
LE ANTH OGARITHMS 136—137 TAB X IVb.
E AND LO RITHMS or TRI N METRIC FUNC TABLE X IVc. VALU S GA GO O TIONS 138- 142 EX PLANATION OF THE T ABLES
—PLACE COMMON LOGARITHMS OF TABLE I. FIVE NUMBERS FROM 1 TO 10 000
h m a n Lo arithms. The wer to whic 10 ust be r ised to roduce 1. Commo o g p m p an number n is called the common lo arith of n . Thus lo 10 1 y g g , — — 2 1 3 etc. lo 1 0 lo 1 lo 2 log 100 , log 000 , ; g , g ; g , ‘ - e In eneral if 10 n l is called the common lo arithm log 3, tc . g , , g
f n and i denote b lo n . O , s d y g
l r m a - 2. Fundamental Princip es. Loga ith s constitute great labor saving de m The rin f n a vice in arithmetical co putations . p ciples O their applicatio re stated as follows : r is e ual to the mo he a o ors: I. The logarithmof a p oduct q su f t log rithms f the fact ” ll ws r mthe a h t if 1 ‘ a 10 b l ab l a l b. T o o o ct t a 0 and og og og his f f f , ‘+L m r n ri ulti add a ithm. 0 b. e : to l 1 w I b f p y, log s ' I T h o a ion is ua the di erence b ined in I . he b garit mf fract eq l to fl o ta by subtract g the b garithmof the denominator fromthe logarithmof the numerator: log (a/b) if ‘ a an 5 h "L l a b. F r 10 d b t en 1 a . rie : og log o , 10 , O b In b f
divide ract o arithms . to , subt l g Th a mo a w r e ual to the a mo m ied III. e b g rith f po e is q log rith f the base ultipl “ he ex n n o he w r: l a b lo a T ll w r m e a t h t by t po e t f t po e og g . his fo o s f o th f c t a ‘ “ “ if 10 a, then 10 a .
the n m r index o he roo : T ll mthe u be by the f t t og (log a)/b. his fo ows fro a h if 10‘ a then 10t m f ct t at , a f II The r hmo h r r o r ne ive Corollary o . b ga it f t e ecip ocal f a numbe is the gat o h arithmo the number: lo l a — lo a ince 1 0 f t e log f g ( / ) g , s log .
n v 3. Characteristic a d Mantissa. E ery real positive number has a real If a and b are an w that common logarithm. y t o real positive numbers such e t er ze n a m a b, then log a log b. N i h ro or ny negative nu ber has a real
Inspection of the preceding table Shows that the lo arithmof ever number between 1 and 10 i a er n g y s prop fractio , the lo arithmof ever number between 10 and 1 1 a ra n g y 00 is f ctio , the logarithmof every number between 100 and 1000 is 2 a fraction ;
Common logarithms are exponents of the base 10 ; other systems of logarithms have hm
' Na i rian lo T 12 e n m1 e arithm able II . 1 v difl re t fro 0 ; p g s (see V , p ) ha e a base denoted by c, an irrational numbe r whose value is approximately When it is necessary to call attention to the base the ex res ion lo a will mean common lo arithmof n l a m n h , p s gxo g ; og, will ea t e viii EXPLANATION OF THE TABLES [53
vi n that the lo arithmOf ever number not an exact and so on . It is e de t g y ( power Of 10) consists Of a whole number a fraction (usually written as a e number is called the characteristic the decimal is decimal) . The whol greater than 1 may be determined as follows :
e r thmOf ever number between and 1 is — 1 a raction th loga i y f , h mof ever numbe between and i — 2 a tion t e logarith y r s frac , — the logarithmof every number between and is 3 a fraction;
and so on .
m r there is a reat racti a advanta e however in comutin to nu be ; g p c l g , , p g, racte ti w — 1 0 - 2 0 — write these cha ris cs as follo s : 9 1 , 8 1 , 3 — 10 etc. Thu the lo arithmof is 1 but thi houl 7 , s , g s s d writ n 10 an rl for all num r le than 1 be te ; d simila y be s ss . l RULE II. The d aractevi stic of a number less than I is found by subtracting rom9 the numbero ci her e w h decimal oint and the rst si ni a d f f p s b t een t e p fi g fic nt igit,
i Thus, the characteristic of log 645 s 2 by Rule I; the characteristic of log is 1 by (I) ; of log is 0 by (I) ; of log is 9 10 by (II) ; of i log s 8 10 by (II) .
n t han e h man a ut nl the ha does o c g t e tiss b o y c racteristic. Thus 5345 534500 all have the same man , , tissa.
4. Use of the Table. To use logarithms in computation we need a table arran e so as to ble us to find with as li tl fio an im g d ena , t e e rt d t e as possible,
ules I an I R d I . viii onl manti are This ne Ta , p , y ssas given. is do in ble I.
‘ h r r Anot e ule for finding the chn scte risfia based on this property , is often useful: if the ecimal oint were ust af r the fir i ni ure th h r r ic would be zero star d p j te st s g ficant fig , e c a acte ist ; t at this point and count the digits passed over to the left or right to the actual decimal point; so] COMMON LOGARITHMS ix
r m a um Fi e the LE M To nd the lo a ith o iven n ber. rst determin PROB 1. fi g f g ,
i h l k in h b e f th man i a. characterist c, t en oo t e ta l or e t ss To find the mantissa in the table when the given number (neglecting the decimal int consists of our or less di its exclusive Of ci hers at the po ) f , , g ( p innin or end look in the column marked N for the rst three di it and beg g ) , fi g s select the column headed by the fourth digit : the mantissa will be found at h i f w Th he a hm f t e intersect on o this ro and this column . us to find t log rit o
2 Ob rve rst e I ha th ha teri tic is 4. To find the manti a 7 050, se fi (Rul ) t t e c rac s ss , a n n h 2 2 in lumn a o i it in fix ttentio o t e digits 7 05; find 7 0 co N, nd opp s te column 5 is the desire mantissa hence lo 72050 The d , g mantissa Of is found opposite 782 in column 6 and is hence 1 log 0.
ti i n in h 5. Interpola on. If there are more than four Sign fica t figures t e iven number its man rinted in the table but it n be ound g , tissa is not p ; ca f approximately by assuming that the mantissa varies as the number varies in the small interval not tabulated; while this assumption is not strictly cor
ct it i f cientl accura r use with t table . re , s sufi y te fo his Thus find he l thm f 20 4 w rv tha o 2 5 , to t ogari o 7 5 e Obse e t l g 7 0 0 and that log 72060 Hence a change Of 10 in the number causes a change of in the mantissa; we assume therefore that a change of 4 in the number will caue a r ximatel a chan e of s , pp o y, g X (dropping the sixth place) in the mantissa; and we write log 72054
' The difierence between two successive values printed in the table is called bular difier nce v rti nal ar f thi difierence a ta e abo e) . The propo o p t o s to be added to one of the tabular values is called the correction ve and i und m ti h the r riate abo ) , s fo by ul plying t e tabular difierence by app op a w h h fraction bove) . These proportional parts are usually written it out t e zeros and are rinte at he ri t-han e each e be u when , p d t gh d sid of pag , to sed
m l Bra ple 1. Find the ogarithmof Opposite 126 in column 4 find ‘ the tabulardifierence is 34 (zeros dropped) ; X 34 is give n in the margin as 24 ; this correction d iv th ad ed g es as e mantissa Of hence log 10 . m l Bru pts 2 . Find the ogarithmof Opposite 185 in column 6 find tabular difierence 23 ; X 23 is given in the margin as 10 ; this correction added gives as the mantissa Of hence log
6. Reverse Readin of the T R LE 2 To nd the number wh n g able. P OB M . fi e ‘ m ‘ Fi nl find m lo arith known. rst xi at n i n on he man i o r its g is , fi ng te t o t t ssa y, f o means of the two following rules : i L h h m v in wh a e h RU E III. If t e characteristic of t e logarith is positi e ( ich c s t e man n o b n h n d n more than the tissa is otj llc wed by egi at t e left, cou t igits o e
Th n r h l al e umbe W ose ogarithmis h is Often c led the antllogarlthmOf k. ' An nv ni m f follo r zero the 1 other co e ent for o these rules is as ws : if the cha acteristic we re , decimal point would fall just after the first significant figure ; mov e the decimnl point one place f to the right or each positive unit in the characteristic. one place to the left for each negative x EXPLANATION OF THE TABLES [56
m r e 9 n ci preceded by a nu be n and followed by pr fix phers, and place he e h her the decimal point to t l ft of t ese cip s.
Example 1. Given log a: to find 2 . Since the mantissa is 22737 we look for 22 in the first column and to the right and below for
l Th mbe r is therefore 1688 . in 37 h we mn 8 o o i 168 . e nu S 7 , whic find in co u pp s te ce the l d n characteristic is +1. we begin at the left , count 2 p aces, an place the point ; he ce a:
Example 2 . Given log a: to find 3 . This mantissa is not found in the table ; in such cases we interpolate as follows : select the man i a in the table nex less than the iven mantissa and write d own the or s n n t ss t g , c re po di g ‘ ‘ number here 774 the tabular difierence is 25 the actual difierence foun b s b rac in ; , 1 ; ; ( d y u t t g the mantissa Of 1774 fromthe given mantissa) is 17 ; hence the proportionality factor is 17[25 or e n en h in movin the decimal oin d n afi the man (to th earest t t ) . S ce g p t oes ot ect
foll w tha di i in the r r n mber are 17747 to five laces . The hara tissa, it o s t the g ts equi ed u ( p ) c c teristic 2 directs to count 3 places fromthe left ; hence a:
ULE n era w he v ma a no in R . I gen l, hen t gi en ntiss is t found the table, write down four digits of the number corresponding to the mantissa in the table next less
' than the iven man a de ermine a h ue dividin he a ua i erence g tiss , t fift fig r by g t ct l d fi ' b h r i a he m h h r e d er nd deci nt mean o t e ar c e stic. y t tabula fi ence, locate t al poi by s f c a t i
l arit 7. Co og hms. We might add the logarithms of the factors in the nu merator and fromthis sumsubtract the logarithmof the denominator; but we can shorten the operation by adding the negative of the logarithmof the denominator f h The n ative of the instead o subtracting t e logarithmitself . eg logarithmof a number (when Written in convenient formfor computation) is called the l ari of the n r We ma fin the n a ive of an co og thm umbe . y d eg t y number b subtractin i r mzero and t i nvenient in lo arithmic comu y g t f o , i s co g p tion wri rm the n ative of ta to te zero in the fo 10. Thus eg is 10; the negative of 10 is Remembering that the cologarithmof a number is its negative we have the following rule : To find the cob garithmof a number begin at the left of its logarithm(including the characteristic and subtract each di it rom9 exce t h wh trac ) g f , p t e ich sub t rom10 i the lo arithmhas not — 10 a h ma i a w 0 a h f ; f g fter t e nt ss , rite 1 fter t e result; i the lo ari thmhas — 10 a er the mantis a no — 1 a f g ft s , do t write 0 fter the result. By this rule the cologarithmOf a number can be read directly out of the table Without takin the trouble to write d wn the g o logarithm. Attention must be iven not to or et hara g f g the c cteristic. The use of the cologarithmis governed by the principle :
a CON E E TH I . D NS D LOGARI MS AND ANTILOGARITHMS
8. Me thod of mui ari Co p t ng Log thms. This table is a rearrangement Of the
‘ conden iv ' ber sed table en b HoiIel. Fromi the lo arithmof an num g y 1 t, g y whatever may be obtained to within 5 in the fifteenth place ; or to any desired e ree of accurac l n h d g y ess tha t is. To illutra kin s te the r we hall mu lo r nine la . Ta p ocess, s co p te g to p ces g
n' 26535 7 we divide it b the rst i ni cant di it Obtainin 89 9, y 3, fi s g fi g , g
If the lo men s in one or more ci hers the last si ni cant di it is to be un ersd garith d p , g fi g d
' n mr - F mulas et de Tables u é i u 3d cd . P 1901. 1 HOOE L, Recueil dc or q a , , aris, Gauthier Villars, 510] TRIGONOMETRIC FUNCTIONS xi
4 1 7 We then divide this uotient b e 1 0 7 9 55 q y tc . , obtaining
r 52172
n Obtain the lo arithmof each Of the rst our actors romthis bl We ca g fi f f f ta e . The logarithmof the last factor can be obtained by multiplying its decimal part by M 44819; for the error made in writing log (1 + 2 ) M1:
n M ’ e find Ma: either b in th act h is less tha z l2. W y us g e f t at the last column in h table ives multi les Of M or referabl b Table II t is g p , (p y) y V I, page 115. A din the five lo arithms ust mentioned we find d g g j ,
log r 98727 4, which is surel rre t within 1 in the tenth lace Th y co c to p . e comet value is 98726 9
Th ro ma a li an ther number n anal e p cess y be pp ed to y o in a ogous manner. Such high-place logarithms are occasionally needed in statistical work and in the preparation of tables .
thod of mutin Antilo arithms. T n 9. Me Co p g g he co densed table of anti l r ms iv eleven i ni t ure ten de i la o a ith es s can c mal ces . Fromi g g g fi fig s ( p ) t, the antilogarithmOf any number can be computed to within 5 in the tenth t i it Significan d g . m f Thu to co ute the antilo arithmo .4342944819 to 8 si ni ant s, p g g fic figures, we may write
«010 am ) .
T e t five a rs ma ta e dir r mthe able The las t a r h firs f cto y be Ob in d ectly f o t . f cto may be calculated fromthe formula 10’ 1 The error in this formula is less than 3 in the (2k)th decimal place if a: is less than where
k 1.
H wever a m h mre i de n s on the f Table and X 1 o , uc o rap d process pe d use O s I t le Thu T l I 10° nearl B Table X I with his tab . s, by ab e , y. y , “3“ m” log 94524 Hence ) ° a (10 Obtaining the second factor fromthis ble an he last ac r r m e orml 10’ 1 b Table III ta , d t f to f o th f ua y V “ 3" we find 1828 ; the correct value is 27 18281828459 Thi r re nl o n m i s p ocess quires o y tw lo g ultiplicat ons.
II F - PLACE TABLE OF THE ACT AL AL ES OF THE . IVE U V U TRIGONOMETRIC FUNCTIONS OF ANGLES
n n n and 10. Direct Readi gs. This table gives the Sines, cosines, ta ge ts, ° co n n of the n r m0 and a imle device indicate b ta ge ts a gles f o to by s p , d y ° ° he rint n he a f r an l r m4 to 90 ma be t p i g, t v lues O these functions fo g es f o 5 y ° rea ire l r mth me le For an le l than 4 read down the d d ct y f o e sa tab . g s ess 5 e he d r bei a h and the minu on the le t for an les pag , t eg ees ng found t t e top tes f ; g ° eater than 4 d e r es bei und at the bottomand gr 5 rea up th page, the deg e ng fo h min th h t e utes on e rig t . ° f r x m e which oes an l 1 . To find a function Of an g e (such as 5 27 6, o e a pl ) d t ce an in n m f min e we emlo he ro of inter no redu to tegral u ber o ut s, p y t p cess xii EXPLANATION OF THE TABLES [510
° la T ill e d 1 po tion. o ustrate, l t us fin tan 5 In the table we find ° ’ ° ’ ° tan 15 27 and tan 15 28 we know that tan 15 n n Th t lies betwee these two umbers . e process of in erpolation depends on the ° ’ ° ’ assumption that between 15 27 and 15 28 the tangent Of the angle varies irectl as the an le while this assumtion is not ri tl t d y g ; p st c y rue, it gives an
° assume that tan15 is halfway between and We may
mn that e tan en a e as h assu i g th g t v ri s t e angle, an increase Of in the angle will increase the tangent by X (retaining only five places) ; hence
° tan 15
' Table I the abular i er nce a ve i , t d fi e bo ) . The proport onal part of the tabular difierence which is use is called the correction d above) , and is found by multiplying the tabular difierence by the appropriate fraction of
m ° Exa ple 1 . Find sin 63 ° ’ sin 63 52
rr n co ectio X (to be added) . ° Hence sin 63
° Emmplc 2 . Find cos 65 ° ’ cos 65 24 tabular difierence 26; X 26 21
° H ence we 65
subtract it.
R n lati n a o in n h an 11. Reverse eadings. I terpo o is ls used fi ding t e gle when one of its functions is given.
Emmple 1. Given sin a: to find 3 .
Looking in the table we find the sine which is next less than the given sine to be 3 2832, ° and this belongs to 19 Subtract the value of the Sine selected fromthe given sine to obtain the actual difiere nce note that the tabular difierence The actual dif ference divided by the tabular difierence gives the correction as the decimal of a ° minute (to be added) . Hence a: 19
fin as. Emmpls 2 . Given cos a: to d
H n ference is 28 ; the actual difierence is 3; correction 3128 (to be subtracted) . e ce ° a 73
' E To nd an e when n o its tri onometnc unctions is iven : select RUL . fi an gl o e f g f g 512] TRIGONOMETRIC FUNCTIONS xiii
' it by the tabula r diflerence; this gives the correction which is to be added if the given ne or n n and o be r i function is si ta ge t, t subt acted f the given function is cosine or
F —PL CE COMMO GARITH III. IVE A N LO MS OF THE TRIGONOMETRIC FUNCTIONS
12. Use of the Table . If it is required to find the numerical value Of ° x X sin 51 we may apply logarithms as follows :
° ' log sin 51 27 10 (add) log x 0:
° The only new idea here is the method of finding log sin 51 27 which means ° the lo arithmofthe sine of51 27 The m bvi wa is in Tab e II g ost o ous y to find l , ° ' sin 51 27 and then to fin i T d n able I, log 10, but this involves consultin two tables T av id he ne o in thi g . O o t cessity f do g s, T able III ives the lo arithms of the Sines sines tan n and g g , co , ge ts, cotangents. The arrangement and the principles of interpolation are similar to those given l on . vn for Tab e I. The Sines and of all a a he nts p cosines cute ngles, t tange ° of all acute angles less than 45 and the cotangents Of all acute angles greate r ° than 45 are ro er ractions and their lo arithms end with 10 which i p p f , g , s not rinted in the ta le w h h h e a p b , but hic s ould be written down w enev r such a t is ue log ri hm s d . n h rin b v u r a r I t e p ted ta le, al es a e st ted so that 10 should be s ubtracted in eve y case.
m n ° Exa ple 1 . Find log si 68 ° ’ a i a m i h h fin lo in On the page h v ng 68 t the botto , and in the row hav ng 25 on t e rig t d g s ° ’ 68 25 10 ; the tabular difference is 5 ; X 5 is given in the margin as 2 ; this is ° the correction to be add ed , giving log sin 68 10 . a In case of sine and tangent add the correction . In case of cosine and cotangent . subtr ct
the correction .) l m . iv n o . Exa ple 2 G e g cos x 10, to find x ° ' The logarithmic cosine next less than the given one is 10 and belongs to 57 53 the actual difierence is 19 ; the tabular difie rence is 20 ; hence the correction is 19l20 ° (to the nearest tenth) ; (subtract) ; hence x 57
— din lo ctn a for an an le a note that lo ctn a lo tan a since In fin g g y g , g g , ’ n ar recisel the same ctn a I/tan a . He ce the tabular difierences for log ctn e p y
n hr u he ab k i ev ed order. Likewise as those for log ta t o ghout t t le, but ta en n r ers , — l s — n he v of lo sec at log sec a og co a , log csc a log sin a ; he ce t alues g l are omit and og csc a ted. ° For angles near 0 or near the interpolations are not very accurate if a or an ent near the difierences re large . For the calculation Of sine t g 4 Table IIIa, page 5, gives the values of
' ’ == — a — o A S log sin A log A and T log tan A l g ,
’ A for values Of where A is the given angle and A is the number of minutes in , ° A between 0 and Then
' ’ lo A T log sin A log A S and log tan A g , ° for n l Moreover since we have cos A Sin 90 A) and small a es . g , ( ° ctn A tan (90 A) , xiv EXPLANATION OF THE TABLES [s12
° ° log cos A - log (90 and log ctn A - log (90
when A is near Another method practically equivalent to the preceding is to use the ap
’ ’ log ain A - log ain B - log A - log B
’ ’ l A - o tan B - lo - l og tan l g g A og B ,
where A is the given angle and B is the nearest angle to A that is given in the ° If A 3 and A — B these ormulas table. < l l f give log sin A and log tan A to five decimal places .
— RA IAN EA IV V. D M SURE
mu tions in Radian Measure. The reduc ion 13. Co p ta t of degrees to radians
this table may be regarded as a table of multiples of The values of sin x cos x tan x are state for ever an m , , , d y gle x fro to radians at intervals of radian in Table V— Trigonometric n Meas ure The values of an of t Functions in Radia . y hese functions for larger values of x may be computed by first converting the value of the angle in m de ree measure b Table Va and radian easure to g , y , then finding the value mT I of the function fro able I . The reduction of radians to degrees can be performed directly by Table V; reater accurac b the su lementar Table or, for g y, y pp y Va.
ERS— ROOT — RE IP VI. POW S C ROC ALS
mn T e re cube s uare 14. Arrange e t. his table is arranged so that th squa , , q l n five decimal laces for root, cube root, or re ciproca ca be read directly to p ’ “ fi u T a thi not onl n n ii an number n f three i ni can res . o ttai n s y o s g fi t g , y , , V , i l n u l n l n 1 n are rin n ever a e All values b t a 10 i O 100 o . V , l , so 4 , l , i p ted y p g hav been ull m k e caref y reco puted and chec ed.
Thus to find read ID V; column the result : To find read in the same line in 10 n l mn the r ul : T find 117 r ad 10 times the entr in ; column , 4 co u es t o V , e y V ,
Similar! fromif; column ; fromthe same line in QIO n lumn 1 r mth s me li co ; 1 7 f o e a ne in 0100 n column .
The efiect of a chan e in the decimal oint in n’ n’ and 1 n g p , , I is only to Shift the decimal oint in the result Without alterin the di i p , g g ts printed.
VII NAPIERI N N T . A OR A URAL LOGARITHMS
— 15. The Base e . Natural Logarithms . The number e i alled the natural base of lo arith Th l s c g ms . e ogarithms Of numbers to this base are given in Table VII at intervals of from to and at unit r m1 to 4 Th inte rvals o 0 09 . e undamental relati n l f f o og. n log. 10 X log", n enables us to trans er romthe base 10 to the base e or n e f f , co v rsely; where logg 10 519] OTHER TAB LES xv
M LTIPLES OF M AND OF M VIII. U N
16. Multiples of M and 1/M . This table is convenient whenever a number m i lied M or M T r n v is to be ult p by by H . his occus whe e er it is desired to n e mcommon lo arithms to natural lo arithm or nver in cha g fro g g s, co sely, s ce
M log . e and since we have
x l x lo e M l an l x l logm ( og. ) ( gw ) og. x d og. ( /M) logmx. Other formulas that require these multiples are
’ - fl - logme x logme x M and log. (l o x) log. x
nd he a roximate ormulas see x a t pp f ( 8, 9, pp . , xi)
lo 1 x x' M and 1 gi c ( :l: ) : l: :l: (I/M x.
L ES AND LOGARITHMS OF HYPERB L T IX . VA U O IC FUNC IONS
' " lic unc ons . Th . H erbo F ti i a ive 17 yp s t ble g s the values of e , e , sinh x, cosh x tanh x and the lo arithms of e’ inh x h x at var in in , ; g , s , cos , y g tervals
" n ’ fromx 0 to x 10. It is to be noted that log e log e and log tanh x lo inh x lo h Th le a x g s g cos x. e tab my be e tended indefinitely by means of Table III since lo e” x M for this reason Table II ma be V , gm ; VI y arde a ble of values f l ' reg d as ta o ogi e e .
AL ES AND X. V U LOGARITHMS OF HAVERSINES
v T 18. Ha ersines. his table gives the values and the logarithms of the ° ° haversin of an les r m0 1 at in a f n es g f o to 80 terv ls o The haversi e, which
hav A (it) vers A 00 8 A) ;
n fiv ie m he ce its values to e p ces may be co puted fromthe table of cosines. n el in navi a ion an it ma It is used exte siv y g t , d y be used to advantage in the n solution of ordinary Oblique tria gles .
F R TABLE— L MS OF X I. ACTO OGARITH PRIMES
of omo s . Lo ari of . F o s C site Number thms Prim The 19 act r p g es . uses t bl are evident in ue ions involvin actorin an r n in hi h of his ta e q st g f g, d fo fi d g g l arithms of numbers wh e rime actors are less than 201 place og os p f 8. We shall illustrate the finding of logarithms of other numbers by finding
T a : log 1 . aking r 26536, divide by 3 (the first digit) , obtaining 2 vide thi uotient in neral t nea 7551 Di s q by ( ge , by he rest rst our di its Obtainin 683 B Table III the a roxi fi f g ) , g 8 y V , pp mate formula log (1 E: x) gives log 8683 1944 (Table VIII) 12547 (Table X I) log log 3 log 66817 (Table X I) 9880
while the true value of lo r s 2 at the erroris less than 1 in g i 987 6 9, so th the ei hth la f an number ce . In neral t e lo arithmO g p ge , his process will give th g y to within 6 in the ei ht dec m la a r less than g h i al p ce , and the prob ble erro is in the ei hth lace ill an 10. g p . For st greater accuracy, see Table Ia d 5 xvi EXPLANATION OF THE TABLES [520
X II. INTERE ST TABLES
20. Interest Tables . Tables X II a b c d ive m , , , g co pound interest and f r various er cents u to fif ear annuity data o p p ty y s . Aside fromthe obvious
Table X IIe ives the lo arithms of 1 r to fifteen la e r g g ( ) p c s, fo all ordinary values of r rom to For other values of r lo 1 r ma be f , g ( ) y computed fromTable Ia (see The final result in interest calculations
(See 5 Ta X I is th American Ex eri ce rtalit Ta ble If e p en Mo y ble.
The value of such four-place tables consists in the greater speed with which he in th d r e of a urac the afiord is or t y can be used, case e eg e cc y y sufi cient f e th purpose in hand.
n ur si ni cant ur logarithmof any umber offo g fi fig es can be read directly.
V ithm. T ble be XI b . Autilogar s his ta will found to facilitate approximate T e i calculations to a marked degree . h proport onal parts are stated in the
alues and Lo arithms of Tri onometric Functions . In this table me . V g g ,
a a n ctn an their mmon lo rithm are s ted the values Of sin , cos , ta a , a , d co ga s, ta
Greek Alphabe t
LET TE RS NAMES LET TERS NAME S LETT ERS NAMES
A a Alpha H 11 Eta N v Nu T r Tau B 3 Beta 9 o Theta e X i T v
P 7 Gamma I t Iota O O Omicron 4) Pin A 6 Delta K x Kappa H r Pi X x Chi E e Epsilon A A Lambda P p Rho \I' p Psi Z Zeta M a Mu E a 3 Sigma 0 00 Omega LOGARITHMIC AND
TRIGONOMETRIC TABLES
TABLE I
COMMON LO GA RITHMS OF NUMBERS
FROM
1 TO 10 000
‘ FIVE DECIMAL PLACES
1 — 100 100 Logarithms of Numbers 150
— 200 — Logarithms of Numbers 250 250 Logarithms of Numbers 300
‘ I] 450 Loganthms of Numbers 500
I] 650 — Logarithms of Numba 's 700 13
7 1 0J 0E 2 1A 1E 3 2J 1E 4 2B 2A 5 3fi 3D 6 4E 3B 7 4S 4E 8 5B 4B 9 6B 5A 14 700 Logarithms of Numbers 750 750 Logarithms of Numbers 800 15
I] 850 Logarithms of Numbers 900 17
94 002 18 900 Logarithms of Numba 's 950
20 Table Ia CondensedLogarithms and Antilogarithms [Ia
CONDENSED LOGARITHMS TO FIFTEEN DECM AL PLACES
[The first digits of n are given in the first row at the tOp ° the last digit of n In - l he co m f l i hms are hose of 1 2 3 m. T l n o o ar t t the left hand co u n first u g , , , mi in colum ive lo 1 x where x times 1 2 The re a n g ns g g ( ) , , ,
a - x . 00000001 lo 1 x x M Wi [For , g ( ) , to thin 3 in the 17th place , where M 3B H ence the last column gi ves multiples of M except for the l e column h decImal ace . All th s t at would follow have the same significant digits . l displac each time one p ace .]
CONDENSED ANTILOGARITHMS TO TEN DECIMAL PLACES
T he first di its of n are iven in the first row at the ta n x x [ g g p ; ; 1 . 2, - “ 3 9 are iven In the le t han column . Th , g f d e first digits in 10 are given in the
second row at the top .l
“ orn 10 a 1 n 1 M to within 3 in the 12th d mal l e whe ( / ) eci ac , re A ) 3 302585 Hence the 1 olu i es multiples ( 1/ except for h ca rn 1s ee A ' ia e d 3 . ll the columns mw i p f ofiggOfiow contain ti:0 same sigmficant dI Its dIs laced one lace for each new colum g p p n.] TABLE II
OF THE
TRIGONOMETRIC FUNCTIONS
FROM
0° TO LS OF ONE MINUTE
FIVE DECIMAL PLACES ° 0 Values of Trigonometric Functions 1 [II
3437 7
.00145 572 96
12 312 5 .99999
245 55
180 93
.01018 .01018
.01164 .01164
81 847
.01309 .01309
73 139
. 01454 .01455 67 402
.01600 .01600
59 266 ° ° 11] 2 Values of Trigonometric Functions 3 ° ° 24 4 Values of Trigonometric Functions 5 [11
. 06976 . 06993 .99756
. 07005 . 07022 24 1
. 07121 . 07139 14 008
.07266 .07285 13 727 99736
.07431 13 457 99725
.07556 .07578 13 197 99714
.07724 12 947 99703
.07870 12 706 99692
. 07987 3520
.07991 .08017
.08020 046
.08136 .08163 12 251 99668
.08281 .99657 11 992
.08456 11 826
.08571 .08602 11 625 99632
.08716 .99619 ° ° II] 6 Values of Trigonometric Functions 7 25
. 10453 . 10510 9 5 144 99452
. 10657 9 3831
. 10805
. 10952
. 10981 . 1065
. 11011 0821 10973
. 11031 . 11099 9 0098
. 11176 .11246
. 11320 . 11394
. 11465 . 11541
.5772
. 11609
. 11836 .99307
.3863
. 11898 . 11983
. 12013 3245
11985
.12131
. 12187 . 12278 ° ° 26 8 Values of Trigonometric Functions 9 [II
. 13917 7 1154 99027
13975
. 14061 . 14202
.0264 .99002
. 98998 6 9972
14205 . 1435 1 6 9682
14349 14499
14493
. 14637 6 7584
. 14975 . 6779
. 15005 6646
. 14925
14982
. 15069 . 15243
. 15212 . 15391 .98836
. 15356 . 15540 98814
. 15689 6 3737
. 15838
° ° 28 12 Values of Trigonometric Functions 13 [11 ° ° 11] 14 Values of h igonometric Functions 15 29 ° ° 30 16 Values of Trigonometric Functions 17 [11 ° ° 11] 18 Values of Trigonometric Functions 19 31
.30902 .32492 3 0777 95106
.31012
.31040 . 32653 95061
.31178 32814 95015
.33007 0296
3 0178
.31593
.9916
.31730
.9858
.9772
.31868
. 31979
.9572
. 9487 702
.33978 19431
. 34010 9403
.32282 .34108 2 9319
. 32419
. 32557 .94552 ° ° 32 20 Values of Trigonometric Functions 21 [11
Te n Ctn
9 .9 9 9 . 34202 . 383 7 3 6 430 7450
.36562 2 7351 93919
.34475 .36727 .93869 7204
.34612 .36892 2 7106 93819
. 3699 1 27034
. 37024 . 7009
.37057 .93769 6961
.37223 2 6865 93718
34993 93667
.35157 .93616 6605
.35293 . 37720 . 93565
.37887 2 6395 93514
. 37986 16325
. 38020 . 5302
.35565 . 38053 2 6279 93462
. 35701 . 38220 .934 10 6142
. 35837 . 38386 .93358
Ctn Tan ° ° II] 22 Values of Trigonometric Functions 23 33
. 39073 2 3559 92050
13483 92005
.39207 .42019
. 39341 91936
.91879
.42998 . 3257
.43032 3238
.43136 2 3183 91822
2 3090 91764
. 39875 .43481 2 2998 91706
. 39982 .2925
.91648
.2871
.40141 2 2817 91590
2274 .43950 5 91531
.44175 .91472 2620
2 2549 914 14
.91355 ° ° 34 24 Values of Trigonometric Functions 25 [11
.44523 2460 558 2443 593 2425
.44697
.44872
44977
.45222
.45397
.45573
.45748
.45924
45995
.46101
.46277
.46454
.46631
° ° 36 28 Values of Trigonometri c Functions 29 [11 ° ° II] 30 Values of Trigonometric Functions 31 37 ° ° 38 32 Values of Trigonometric Functions 33 [11 ° ° II] 34 Values of Trigonometric Functions 35 39
.55919 . 67451 1 4826
55992
. 67663 1 4779 82822
.4751
.56160 .67875 1 4733 8274 1
1 . 68002 1705
1 4687 82659
.56401 .68301 1 4641
.56521 . 68514 1 4596 82495
. 68728 .82413 4541
.56760 .82330
68985 .4496
.69157 1 4460 82248
.4433
.56976
.57000 . 69372 .82 165 4406
.57119 .69588 1 4370 82082
34335 .82015
.57238 . 81999 4317
69977 .4290
.81915 ° ° 40 36 Values of Trigonometric Functions 37 [11 ° ° 11] 38 Values of h igonometric Functions 39 41 ° ° 42 40 Values of Trigonometric Functions 41 [ H
.64279 . 83910
. 83960 . 1910
.84009 1903
.64390 . 84158 1 1882 76511
. 64501 . 84407 1 1847 76417
. 64612 1 1812 76323
. 84906 . 76229
. 84956 . 1771 1764 192
.85157 76135
.85408 1 1708 76041
64989 11695 . 76003 1688 75984
.85660 1 1674 75946
.85912 .75851
.85963 . 1633
.86014 1626
. 65276 .86166 .75756 1599
. 86419 1 1571 75661
. 86674 1 1538 75566
0 776 21524
2 827 . 1517
84 878 . 1510
. 65606
COMMON LOGARITHMS
OF TH E
TRIGONOMETRIC FUNCTIONS
FRO M
0° TO 90° AT INTERVALS OF ONE MINUTE
FIVE DECIMAL PLACES
Table Illa— AuxiliaryTable of S and T for A in Minutes
— ' — ' 8 a log sin A log A and T - log tan A log A
’ ' For small angles : lO sin log A S and log log A ° ° es near 90 fog cos A - log (90 ctn A - log ’ ° T Ww ere A - number of in A and 90 Jef - number + , ( ° in 90 A . ° 0 Logarithms of Trigonometric Functions
a3 373 627 . . a3 r 5 524 3 g s o2 915 q.0.579 421 730 1 7 S 5 812 118 4 5 318 1 7 8 203 5 373 g 627 ! 8 0 5 g 488 8 5 g S H 709 ! 4 0 0 “3 e fl 233 I a ~ 8 05 CO 014
. 018 : : a » fl o “ ac 215 wh flO oo 0 ac 582 l - f flw ui 100 b 12 25 2 qo 75
I I a ~ ~c ' ’ I o 1 7 ~ c ' q 60 405 o 4-1 s 8 o 0 0 385 ! o s 5 r 0 b 2 454 0 a8 g 0 2 12 15 606
: : “8 s a I 0 I S ~ 0 ~ 871 129 S 490 I S ~ 91 089 911 I 0 O S ~ 6( fi C 12 07 387 s 205 510 490 w 9Q 8 111 a 08 225 775 s 8 522 478
. 219 0 9 9 06 c8 996 0 0 s- 0 00 I 3 806 0 5 0 9 1 647 519
4 19 f 0 o 9 347 0 9 300 0 9 278 0 0 280
. 304 0 9 349 0 9 4 15 0 9 500 0 0 605
. 727 0 9 867 0 9 024 0 9 196 0 0 384
. 1 587 0 0 1 805 0 0 036 0 0 1 280 0 0 1 538 808 ° III] 1 Logarithms of TIigononietric Functions 47
808 090
688 11 73 004
668 ° 9 014 371 ° 0 .30 263 737 112 0 9 495 9 9 888 0 ° 9 0 0 0 0 o 9 ( 0 8 t 698 114 8 539 8 971 w 410 8 o 0 g 5 H 0 11 63 857
s 771 a 238 711 191
168
170
193 713 238 768 11 56 304
939 493 052 615 183 755 331 911 495 083 675 271 11 49 870
080 690 304 921 541 165 792 422 055 1 692 ° 2 Logarithms of Trigonometric Functions [III
° 87 Logarithms of Trigonometric Functions III] tP - Lqpfimmmc figmmmd fi c mfimul 49
m5 e 060
m. 2 120 g 819 2 9 r 580 » fl o m. 35 u : ! > ] P a 69 ma 0 4 Q a 34 1 a S . c 92 e m 104 a : Na 115 mqt § o 3 . c 138 m 868 mq ' 3 m.q3 30 w I : 0 I 0 m0 C 0 O a 400
’ 0 ! I m. 9 N 168 m' q ~ I 0 8 1 3 I I ms0 8 ~ 937 O 0 5 o a4 m 8 o 479 g c n 5 u e b 5 c 107. 5 I ; s l ~s 252 r u8 o u3 c l I - a a a H5 b t 3 c Ems ~w e ca 026 - n I ! o t 8 b i 3 c an s a ~ O 11 24 801 c b 5 b H8 c
: o 355 o l t N a 133 l 4 1 6 I ! 0 c -O 9 c p 13 a c 11 23 694 u3 O nun q c - s 3 O HMS
n s g s Q o qa 258 c u u5 Q “E A o e H M$O ENE qG 042 827
11 22 613 ; » t » 8 b w a w 8 b 3 b a 9 M 189 d u e o H8 978 a fl b H8 "- fl i b H5 t 768 mH mH8 h 11 21 ‘- 559 o “ p l 8 h
p 145 e p 939 n fl9 mup a I fl 3 a g n m~ a e e r aQ 734 fi 49 c Q m - p fi 3 o o 530 l ‘ l u- a . 9 h nr b h " n -0 c -e 327 q i 0 9 a t o s o fl l l n ng fl w 125 w i 3 a i w ” M3 ” l S fl x8 a 924 1 31 x 5 8* 5 723 x8 : e 11 19 524 s “ 5 O 3 b 8 0 Q 3 O 9 a 3 0 h 3 N 3 w 3 0 i I x8 s 128 G 3 D 3 o 3 6 0 ” a ‘ - I 8 9 5 a 932 O o 5 O O 5 w “5 9 0 i - - l- 0 Q o 5 i H3 a i 5 6 8 9 5 i 736 n i O 3 b H5 a i 3 0 0 n 8 0 8 459 11 18 541 - O I 3 “ H5 m”q5 0
0 8 0 .“1 846 154 0 b ? n a Q 8 0 E 5 962 O h 8 b S 0 8 0 S 8 770 » b 3 p ,N Q 0 8 0 E 11 1 a b 8 o O 5 7 580 l s l I - F y a fl U a “ b F “ - fl q t b n8 n i N Q 0 mH k n5 mfl N 8182 701 9 5 8 201 u G 0 w ” N s QQ .a H 9 6 5 013 0 8 9 8 175 825 0 8 0 5 361 11 16 639 n o s » w o e 4 . m w 5 0 0 ) s 8 9 S q0 N 268 o m s 5 0 a . b “e 5 8 0 S 916 084 ! 0 q M u k r 5 ‘ G 8 9 S 100 900 -I - mi p t t 3 0 i i 8 9 S 282 11 15 718 o i q h s 3 ° 4 Logarithms of M onometric Functions [311
8 R 8 “ 8 8mm L 1 5 mV 8m m6 L 15 9 8g m
. 8 u 8 L 1 5 u4 2 m2 m8 u8 : 3 u3 a 8 m2 8 “ 6 L 44 m 4 n 4 n 8 n 8 u . 8 8 m5 8 m L l 4 m 6 w5 8 m8 6 m5 “ 8 m8 . 1 4 m m m . L a 7 3 8 m8 8 a 0 L 14 mo 8 u8 u8 m2 9 m8 m2 8m 5 8 n 7 L 1 4 m m m. : . 8 u0 L 44 m7 8 1 gm. 8 w w9 I 13 m. 5 2 uA u8 u8 3 J m8 m8 . . 4 m3 m8 w2 8w m 1 L 1 8m8 7 1 l 3 m 5 m5 m8 m8 8w a 4 8 8 1 6 m3 m8 m8 8 7 m9 m8 m8 8w m5 8fi 3 1 1 3 8 8 m£ m8 m8 1 8w a 6 8 w 35 ; 13 “ 9 m3 m8 m2
723 553 2 u m8 w8 3 m m8 m2 384 4 w w8 m8 8 87 661 11 12 2 15 5 u u8 a 8 6 1m 1m8 1m8 7 1m 1w8 1w8 8287 995 120 880 8 1w 1m8 1u2 9 1w 1w8 1w2 161 713 326 453 547 490 8 18 11 82 88 6 11 3 A mD m8 654 J mj m8 0 4 3 mn w2 817 948 052 6 £ wj m8 980 111 889 6 1 3 wn w8 142 726 7 1 3 1wj u2 8 1 £ 1n fi m8 304 563 1 w 9 1 3 . 2 402 760 0 080 0 u2 ” 8 m8 3 m2 g 8 u2 080 920 4 u2 u8 w8 102 240 5 w8 n 8 n 8 6 u8 m8 w8 260 399 601 7 1m2 1a 8 n 8 4 17 557 443 8 1m8 1n 8 w2 9 1m 1“ 0 u2 715 285 730 872 128 971 2 mA u8 m8 0 815 3 u1 m8 a 8 4 m8 m8 m3 660 5 n 5 n 8 m8 505 6 u3 w8 m3 1 350 7 m3 1 w8 m4 8 m5 1u8 m8
11 08 197 9 m3 1m8 m.7 %7 m . 8m% mn 0 1ma 0 8mu m%2 1mm 2 m2 m8 m8 3 u8 u8 u2 8m% 1m 4 m8 m8 m8 n 8mn u mm5 1 m?5 5 u8 758 74 8 6 w8 m8 m8 8 w n 6 m8 w mu6 . 7 1m2 m8 8 mn s mm6 1M m 8 1m8 m08 n 92 e 9 m m8 m1 8m% s mm6 18 6 m 8 w m a w m5 18 6 m5 z 8fi3 1 o o 1 w 3 s 1 3 2 ” 8 m8 a88 g 8 3 m2 1M mS 3 M 2 a 8 n 2 z 4 a 8 m8 s78 8 93 M a8 3 cm 18 6 s9l 5 n 8 728 n 8 88 n . 1 m 6 a 2 s78 w8 3 5 8 3 70 s 8 6 z n 7 1m8 m18 m8 88 M 4 3 903 1mo9l e 9 8 1a 8 u68 u52 8 93 s “ M9 1 oml 9 1” 8 m8 m8
° 52 6 Logarithms of Trigonometric Functions [III
° 83 Logarithms of Trigonometric Functions ° III] 7 Logarithms of Trigonometric Functions 53
914 086 092 019 981 795 123 877 8 897 227 773 g . 9 08 999 330 070 4 4233 f, 293
5 .0 0 566 202 537 403 9 zé 2 304 300 8 . 9 “ 9” 405 742 258 5 500 9 09 845 10 90 155
049 951 150 850 i $13 f; 9 09 907 4 393 g 9 10 000 9 10 353 2 5 540 g 333 g 205 555 9 9 304 050 344 750 244 501 9 10 850 10 89 144 2 097 050 944 795 155 845 5 49 0 893 254 740 58 8 9 10 990 353 10 88 047 452 184 551 281 351 377 747 253 9 11 474 9 11 845 10 88 155
000 040 900 701 138 802 857 235 705 952 9 12 332 008 572 142 525 475 236 621 379 2 331 717 283 g gg-g gg-g 33-3 9 12 425 9 12 813 10 87 187 5 4523 012 004 990 3 23 £23 £81-0 700 099 90 1 9 828 799 194 800 9 12 892 9 13 289 10 80 711
078 478 522 F" 7° 137” 171 573 427 203 007 333 355 9 13 701 239
539 948 052 030 04 1 959
3340 3 393 120 F" 82° 080 994 4 12 588 085 504 496 or read 175 597 403 9 14 200 9 14 088 312 220 ° 8 Logarithms of Trigonometric Functions [III
o i-5 : g 5 0 . 14 780 220 u7 o -5 c 0 14 2 u0 5 . 87 128 o - g 4 0 . 4 9 . u535 1 63 037 o -k 5 h g 0 0 u 0 . 15 054 946 n o -n e 1 fi “Hh 0 15 145 855 764 a . r 4 891 0 . 15 327 673 a e. 4 980 0 . 15 4 17 583 n- mr 9 0 . 5 08 2 . 5 06 1 5 49 a o 0 . 1 98 2 g t 5 157 5 5 40
0 . 15 688 312
0 . H5 333 0 . 15 777 223 9 ‘- 5 . U 5 42 1 0 . 15 867 133 0 5 . H5 508 0 . 15 956 044 9 5 . H5 596 0 . 16 046 954 o i- g O 3 0. 16 135 865 o ‘ g .L 5 770 0 . 16 224 776 p. H5 857 0. 16 312 688 o ‘-a s P c 0 . 16 401 599 ‘ eo F6 030 511
t- 0. fi6 116 0 .l 6 577 423 i a 0 . F 6 203 0 i. 6 665 335 “- i 0 . t 6 289 0 . b6 753 247 i 0 .a 6 374 0 . a6 841 159
0 . H6 460 0 . L6 928 072 ‘ n- 0. h 6 545 0 .i 7 016 984 ‘- — 0 I. 6 631 0 .i 7 103 897 ‘- t-, 0. F 716 0 6 —u I 8 810 0 . H6 801 0 .i 7 277 723 d 0 H3 0 i. 7 363 637
‘ 0. F 6 970 0 : 0 550
0 H7 055 0. 7 536 464 A-0 0 Hq p 0 0 0 t7 622 378 ‘-q ‘- 0 fi . 5 0. r 7 708 292 ‘ d ’ , 0 h 0 b Z 2 206
‘ 0. P 7 880 120 ‘- 0 H 0 . b 7 965 035 0 ‘- 0 ‘- l 949 ‘- 0 r 864 ‘- 0 i 779 : 694 2 609 525 440 356 220 272 9 188
9 . 104 465 021 0 1 9 0 3 ~ 937 854 771 688 605 10 80 522
643 357 725 275 807 193 111 971 029
uP - Lq pfimmmofiTfimmmmd fic q fimm a8 967 n 368 o3 039 o 294 c 3 110 o 221 o3 181 o 147 c 3 253 cc 074 ° 3 0 a0 000 o3 395 o s 3 466 c o g 3 536 o 781 o 1 o g 3 ao~ c t
. os) 563 : ! i 818 oam0 h 0 490 p wc 958 028 e 098 c 201 a 168 e 237 p 057 307 985 376 914 445 514 77l 699 628
486 415 345 3 274 203 133 937 063
, 9.27 008 992 o 078 922 148 852 o8 218 782 I c 8 C8 288 712 o8 357 643 c 8 2 573 o g 0 566 434 o g 0 0 w 635 365 c0 704 296 o0 773 227
o g 911 089 980 020 p 049 951 o c 117 883 p 186 8 14 a g 323 677 391 609 459 541 473 595 405 662 338 730 270 798 10 71 202 ° III ] 11 Logarithms of Trigonometric Functions 57
° 78 Logarithms of Trigonometric Functions ° 12 Logarithms of Trigonometric Functions [III
° 77 Logarithms of Trigonometric Functions
° 60 14 Logarithms of Trigonometric Functions [III
0 ° : 9 6 8 368 5 0 677 323 0 4 9 0 8 4 18 9 .0 731 269 0 9 “8 469 9 .0 785 215 54 53 52 : 4 2 9 e 8 519 9 0 838 16 N 9 w8 70 9 4 892 108 5 .0 Q O 4 ° o 9.0 8 620 5 o 0 945 F H 9 0 0 9 w0 001 5 0 3 O . 6 999 O ° 5 3 052 948 Q
9 9 .3 106 894 N
9 9 .3 159 Q ° O 9 5 3 212
9 .00 266
9 .00 319 681 51 50 49 9.00 372 628 o 0O425 10 59 575 t w ! 0 3 0C 0 wu 469 n 3 416 c b c .39 270 364 3 311 q o o 258 o 205 153 100 048 995 3 4 943 4 4 . 1 109 5 4 839 6 4 786 7
.41 266 8 38 4
O 9 909 4 682 4 630 4 578 4 526
422 371 319 267 216 1 836 164 113 061 10 58 010
l 907 3 1 856
23 805 $ 754
03 652
33 601 $ 550 6 0 3 448
£3 397 347 296 10 57 245 ° III] 5 Logarithms of h igonometric Functions 61
? 195 9 856 144 2 9 w5 906 094 “ 043 0 007 993 0 057 943 0 108 892 0 158 842 0 208 792 0 258 0 308 692 0 358 642 0 408 592 0 458 542 0 508 0 § 558 442 0 § 607 393 0 a 657 343 0 s 707 293 0 s 756 10 56 244 3 0 3 3 0 33 145 2 0 0 095 0 3 046 : f 0 5 i ? 996 t 0 5 i 5 947 t 0 9 i 898
' t 849 t wo. 99 0 i 7 t a n 0 i l ( O 10 55 750
0 0 g a g 0 652 o 0 a c 04 603 » 0 s O ? d 554 n 0 m3 a 10 05 3 N 55 5
! 0 8 0 4 6 8 408 0 0O0 4 $ 359 0 C 5 8 310 8 8 262 3 0 dF $2 213 3 I fl 0 3 h 164 0 3 § 116 3 0 2 wQ 067 0 8 w 10 55 019
0 3 412 922 n e 3 0oq 874 w3 0ow 826 3 « 5 g a 778 e 5 g m5 81 6 a s 3 G8 633 e 5 585 s 5 10 54 537 o s a o s 441 o s 394 o s 346 298 250 ° 62 6 Logarithms of Trigonometric Functions [III
: h 0 034 0. 1 5 750 250 0 h l 7 203 0 0q0 0 . 5 97 0 h 5 84 155 m12 . n 5 0 2 46 47 46 h 2 108 0 a 166 0 . l 5 89 5 1
0t 2 10 0 .fl5 940 060 0 0 !3 0t NO 0 0 . 5 987 0 13 0 t - 0 297 0 . 6 035 965 “ e 0 0a34 1 0 . 1 6 082 918 c i 5 0 h i 0 . 1 6 130 870 a 5 n 0 t 428 0 . 1 6 177 823 e 5 4 m 0 1 6 22 776 : 5 0 5 16 0 1 729 0 559 681 0m02 634 45 44 43 6 0 0E 587 l 0k 689 540 0 h 0 c 733 493 n 0 a 776 446 0 0t 399 0 1 ~ 0“s 352 0 0 306 0 0 259 0 2 12 0 035 165 0 077 119 0 120 072 3 1226 0 163 025 4 0 206 979 5 0 h i 7 932 t O 0 068 6 0 b 4 886 t 5 292 7 840 8 h 0 . 5 377 4 793 9 n 0 .fi 5 4 19 4 747 0 h 2 47 701 . n 5 46 . 299 h 0 . r 5 504 4 654 0 h 4 0 . l 5 547 6 8 h 0 . r 5 589 562 h 0 . 7 484 5 16 l 5 632 0 h 5 4 470 . n 67 0 h 424 t 5 716 378 0 5 332 . t 80 1 0 . 5 843 286
0 . t5 885 240 0 $5 5 927 194 148 h 0 . t 6 011 103 For or l 0 . fi 6 053 7 943 057
0 . 06 095 0 11
0 .06 136 965
0 .06 178 . 8 080 920
0 . 06 220 874
0 . 06 262 829
0 .06 303 783
0 0 4 ° . 6 3 5 738 F + “ or73 M 253 ,
0 .06 386 693 o 00 0 c 647 a : a o e 0 0 602 e d n a z n‘-- 0o c r 557 u m a o g 0oc c t 511 594 466 III] — Io mufihms oi Tfign mmmwk lhmd hms
: ! “ 00 C 9 0 0 tC 0 2 0 fi 466 0. 8 60 0 . 1 6 635 00 0 63 c 421 0 .98 056 5 2 0 . . 6 676 00 0 0fi 376 0 . 98 05 G £ c 5 0 . 6 717 0 331 0 98 04 0 6 . 8 w ? ” c S mm 5 ] i 4 0 $ 0 0mh 0 2 6 0 44 6 758 N 8 . 98 0 w p mHwb w4
' 24 4 — 1 0 . 98 0 0 0 p o v qb fl 3 0 h 00 1 0 . l 6 84 1 96 .98 036 mwNmwwb wr4 h 0 . r 6 882 00 15 1 0 . 98 032 a wfl o wa0 wv4 h 0 4 0. b 6 923 08 89 106 0 029 q wE mwoh wo0 0 h t 6 964 0 0 061 0 . 98 025 mwP o wmh ww0 o 0P mwob ww4 . ' 016 0 . 98 021
0 00 971 0 . 98 017 : t 5- 0 03 2 . 0 7 086 0q0 9 7 0 . 98 0 13 0 5 0 2 8 8 . 0 7 127 0 88 0 . 98 009 5 0 o ” o 0 . 0 7 168 0 837 0 . 98 005 t 0 t o ) a o 0 o o “ 00 0 1 nM n 0 D q 793 00 0 o u — 0 o P 0 v 0 . 7 249 00 748 0 . 7 997 n 0 n 0 00 704 0 1 c wP c w w a 0 o o 1 t g o w 0 . 7 330 00 659 0 l 5 0 7 371 0 0 615 0 s 1 0 w0 q w o 0 : o / o c c w0 o w 0 a ° 570 o o c « o w0 o w cc 0 h 2 0 0 2 . l 7 45 5 6 0 h 0 . h 7 492 00 481 0 h 0 0 . l 7 533 0 10 .50 437 0 h 0 t 7 573 10 .50 393
. 0 10. 8 9 97 2 50 34 . 96 11 7 5 3
00 fl 6 0 0 10 .50 304 9 .97 958 00 fl 02 0 260 954 1 00 q 30 0 0 2 16 0 07 774 0 172
. 0 128 : 00 s ? 0 084 00 g a 0 040
0 .07 934 0 . 0 004 996 0 07 974 0 952 092 908
0 . 08 054 0 864
0 .08 094 0 820
0 .08 133 0 777 0 08 173 0 733
.
0 .08 252 0 645
0 .08 292 0 602
0 .08 332 0 558 0 0 6 8 371 0 515 471 5 : 0. 0 8 450 03 428 For or 1
0.08 490 06 384 1 0 .08 529 03 34 1 1
0 .08 568 0 0 0 703 297
; : 00 15 0 s 0 746 254 7 861 n 0 . 08 647 0i 2 11 » 0 . 08 686 0 g c 167 r
0 .08 725 0 a 124 7 849 0 08 764 0 m 081 For or . " 038 5 00 8 0 . 1 005 995 w 0 + 00 $ 0 2 162 or read .c 1 048 95 - 0 . 08 920 0b 908 a 0 08 959 0 c 865 822 ° 18 Logarithms of Trigonometric Functions [III
° 71 l ogarithms of Trigonometric lfimctions ° III] 19 Logarithms of h igonometric Functions 65
Prop. Pts. 264 697 303 567 301 738 262 338 779 221 553 374 320 130 554 4 11 9 53 861 139 550 098 545 41 40 39 434 943 057 541 2 520 984 016 3 557 025 975 532 4 164 15 6 5 3 9 2 9 51. 9 54 065 935 5 8 5 394 523 6 666 147 853 519 7 702 187 813 8 738 228 772 9 9 51 774 269 731 506 309 691 501 347 350 650 37 33 35 883 390 610 919 431 569 2 7-2 7-0 9 51 955 9 54 471 529 3 11-1 10-8 10-5 4 10 45 433 9 97 479 5 027 552 448 6 21 0 063 593 407 7 099 033 367 466 8 135 327 9 171 714 286 207 754 246 242 794 206 273 835 165 34 5 4 314 875 125 2 350 915 035 3 10-2 1-5 1-2 385 955 045 4 2 -0 1-6 421 905 005 426 5 2-5 2-0 456 035 965 6 20 -4 2-4 492 9 55 075 925 9 97 417 g g-gg-g
527 1 o 1 5 9 9 52 563 155 345 593 195 305 634 235 765 399 669 275 725 394 705 315 635 390
740 9 .55 355 645 385
775 9 .55 395 605 9
811 9 .55 434 566 9 846 526 9 881 514 486 916 554 446 9 951 593 407 986 633 367 8 673 327 9297 349 056 712 288
092 752 248 9 .97 340
126 791 209 9.97 335
161 831 169 9 .97 331
196 870 130 9 .97 326
231 910 090 9.97 322
266 949 051 9 .97 317 301 989 011 312 336 028 972 308 9 53 370 067 933 303 893 299 66 ZWN—Jugmmhmwof Tfigwmmmukl hmd hms [III
1 ! o 0 0 0 6 107 893 g 3 1 O0 o t 0 : g e 0 3 o 0 06 146 854 r l a 1 3 1 p a0 0 ~ c 0 6 6 185 815 n4 a © i 0 0cc 0 06 224 776 o 3 1 m0 0 0 6 264 10 43 736 g
a : 1 0 ! o i s 0 0 q0 0 0 6 303 8 8 n 1 o : O g oo 3 O 0 0 6 342 658 o r ? m I 1 S q o a l g mc Q 0 N 0 0 6 381 619 — - w “C u 1 h ~ 0 o a : o e 1 N u g mc oo r 0 0 6 420 580 J - o 0 w0 t l 1 ° 0C b 3 t o 7 n c -a 0 6 6 459 10 43 541 n a «l - c w0 0 0 0& o 9 o z 0 O o . c g 0 1 Q 0 w3 o ! » 0 ! 0 : z a ua 0 0 6 537 463 0 » i 0 q w0 0 1 Q a 9 c 0 l 1 ‘ o n - 0 1 6 5 424 o 0 3 - i 0 ms c 6 76 c w1 O 0 h © 0 r 4 1 2 H 1 0 cb 0 g 0 6 6 615 385 o c w9 cc 6 O a nw o a2 1 c C 0 fi 0 0 6 654 346
. ! : ! a o0 s 0 0 6 7 2 268 t 3 8 8 0 o a- 0 6 1 2 8 a 0 77 29 1 4 0 ‘ s e m 0 Q o o1 m 0 0 6 810 190 1 1 0 0ob HO w cc c0 oc 0 0 6 849 151 § 0hb Hwa H 113 ‘- Q 0s mt fl o o a uk q 0 m6 926 074 0 ° o 0 n 0 1 5 t 1 ro g .0 o - — 1 c u 0 u 0 ’ t c 6 965 35 0 0 ~ 1 m1 r P m n o neg 0h 4 996 0 6 ‘ 0 1 mb 1 f w e c l g 0b 4 10 42 958 0 0 0 0 rm 0 P o
o ! c 2 3 0 0 7 120 880 o 1 c 2 03 0 0 7 158 842 o 0 1 c 2 0 3 0 0 7 197 o ! c 2 03 0 0 7 235 10 42 765 0 0b o
: 3 r 0 0h p e 3 3 0 0 7 274 1 0 1 t N 0 1 b u0 p3 0 3 0 6 7 312 688 n ‘ 1 - 0 o2 0 3 0 07 351 649 0 t 0 O 1 h w0 1 0 0 O o3 03 0 0 7 389 611 uQ 9 o w$ 1 ‘ 1 1 o - ~ t t 0 0 mw c $3 0 0 7 428 572 0 0 3 0 06 w0 534 1 I 0 3 Q 05 wG e 3 0 0q O8 496 1 1 e G3 016 7 5 a 457 1 c Q3 0 0 7 531 419 m 1 e Q3 0 0 7 619 381 o fl8o 342 o ! g 8 w 0 3 7 696 304 o 8 a 0 3 q3 266 1 1 p 8 o 0 ( 7 772 228 o c 8 w 0 7 810 10 42 190 ©3 8 0 3 1 1 3 8 0 6 3 113 8 5 0 07 925 075 1 1 8 8 0 6 3 037 1 8 3 0 0 8 001 999
1 3 102 0 0 8 039 961 ! 1 3 1 0 0 0 Q 0 0 8 077 923 1
0 5 169 00 g 115 885 n c 5 202 0 um153 847 i 0b 5 235 0 mm191 809 u
0 c 5 268 ’ 771 1 1 n 0 6 5 301 0 m é 3 733 1 - 0 0 5 334 0 ' om3 696 0 05 367 0 mm “3 658 1 0 0 5 400 0 mm380 620 582
° 68 22 Logarithms of Trigonometric Functions [111
358 641 359 389 577 323 420 714 285 451 750 250 0 57 482 785 10 30 214 37 36 35 9 57 545 9 50 859 10 39 14 1 575 895 105 3{ff 607 931 069 4 538 957 10 30 033 5 550 004 5 700 040 050 7 731 075 024 8 752 112 888 9 793 148 10 38 852
35 230 780 32 31 3° 885 255 015 292 708 2 6-0 328 10 38 572 3 9 -6 9-3 9 -0 4 008 400 500 5 156 181) 030 435 7 22 4 ' 211, 70 472 528 0 8 255 24 s M b 101 508 10 38 492 9 131 544 152 570 421 192 515 385 223 551 349 29 3 5 253 587 10 38 313 111 284 722 3 1-8 1 5 314 758 242 4 345 794 205 5 14 -5 2 -5 375 830 170 6 17-4 3-6 405 865 10 38 135 7 20 -3 4 -2 3-5 ‘ 435 901 332?fi jig 457 035 054 407 072 028 527 008 092 0 58 557 043 10 37 957
518 114 885 548 150 850 578 185 815 700 0 52 221 770
750 292 708 700 327 573 820 352 538 859 308 10 37 502
3333 33 33 “ 000 574 425 157 030 500 301 059 545 355 098 580 320 128 715 285 0 50 158 0 52 750 10 37 250 ° III] 23 Logarithms of Trigonometric Functions 69
5 5 03 1 215 5 2 6 5 180 3 5 3 1 145 9 3 2 0 o 2 5 0 c o 110 1 5 5 1 10 37 074
' 5 b 1 004 1 o5 969 8 934 ) 9 n- 3 5 1 oH 899 8 865 9 : Q 5 r so 830 3 G5 795 0 O5 §o 760 0 o1 G5 t ~ m 10 36 725
8 310 s a 0 8 3 655 -
0 3 379 621 w : 1-m o 5 p5 586 n 23 8 3 449 10 36 551 O 3015 3 0 3 519 481 03 553 447 63 588 4 12 5 : 3 0 5 Q 53 377
5 0 3 657 343 5 0 3 692 308 » 3 6 3 726 274 n 03 761 239 a 5 0 3 796 204 n 20 3 170 ) 6 3 135 5 20 3 899 101 5 0 3 934 066 5 6 3 968 10 36 032
: 2 8 5 3 8 « 963 I3 072 928 2 106 894 3 140 10 35 860
. 3 8 791 2 3 2 5 757 I2 278 722 3 312 688 654 “ I 381 619 For23 or 415 585 read as printed ; for 3 551 “ 113 or read 3 5 17 3 517 483 3 552 448 3 586 4 14 2 620 380 4 ” 2 Q 3 10 35 3 6 For 66 or
: ! a o 0 e B t 278 n m! 1 : 244 .o s5 a 9 oz a«c 210 824 176 o 3 0 c 3 8 0 142
° 111] 25 Logarithms of Trigonometric Functions 71
133 100 067 034 001 968 4 1312 935 5 15 5 902 869 7 10 32 837
771 738 705 673
640 . 607 » n 9 mH » 1 i 9 574 5 r9 uH e D- n 542 0 u9 c l 0 N -a - - 5 10 32 509 4 t 9 I 0 o 0 0 t “cN 0 444 4 11 378 10 32 346 3 2 1 281 248 215 10 32 183
118 085 053 10 32 020
956 923 891 10 31 858
794 761 729 697 Fromthe to : 664 p 632 600 568 115” or e 10 31 535 r ad
471 439 407 10 31 374 0 + For64 or
310 278 246 10 31 214 ° 72 26 Logarithms of Trigonometric Functions [III
o m 1 2 366 3 Q g o 8 0 o m 3 10 Q a 0 o c o 150 360 0 o 3 Q b 0 g l 118 354 n - 0 o -e 62 Q b 0 e r 086 348 0 n 3 Q a0 g o 054 9 . 95 34 1 l—o » 022 9. 95 335 » l o l u : 3 e — - 2 c n w 0 0 mI 39 Q 2 t 01 99 9.95 3 9 5 I 3 a e u 0 C Q 3 t 3 958 9 .95 323 a H o o : 4 l n 3 e N N fl 5 91 Q 2 074 926 9 .95 317 s u 5 ‘ : 4 a P 5 6 9 . 10 . s Q «c 10 95 3 o 0 t N 0 wS 0 862 9.95 304 s a 68 Q o O 830 9 .95 298 5 “ N 2 Q c 798 9. 95 29 0 Q t 1
0 19 Q o234 766 9.95 286 a 3 C Q c 266 734 9.95 279
’ 702 9 . 95 273 l : k °b -9 O 3 3 3 0 Q o329 671 9 . 95 267 H 2 3 - H 1 n 0 b i 3 C a : a n 1 3 2 Q o 639 9 .95 261 3 3 c 5 p — 5 0 M9 Q a fi 1 a 3 3 Q 3 c 393 607 9 .95 254 : 4 3 a3 Q 3 3 575 9 95 248 ° . o 0 o w t 0 6 t 9 a l Q 9 4 7 543 9.95 242 3 a3 . 5 Q £ 3 3 O 1
3 ~3 Q 9 488 512 9 . 95 236 1 3 ~3 p. 9 520 480 9 .95 229 1 3 ~ 3 o. 9 552 448 9 .95 223 o 3 3 c Q 9.95 217
“ 385 9 . 95 211 o 3 3 ” c . 9 647 353 9. 95 204 0 o 3 0 Q Q g . 9 679 321 9. 95 198 9 3 1 p. 9 710 290 9 .95 192 o 3 3 g . 9 742 258 9.95 185 7 16 8 o 3 3 c .09 774 226 9 .95 179
2 3 o.09 805 195 9 .95 173 3 3 3 p.0 9 837 163 9.95 167
3 3 p. 09 868 132 9. 95 160
3 3 o.09 900 100 154 5 5 . 9 932 068 148 5 5 . 9 963 037 141 5 5 . 9 995 005 9 .95 135 5 5 . 0 026 974 9 .95 129 9 « 0 058 942 9 . 95 122
p. 0 089 911 116
o. 0 121 879 110
o. 0 152 848 103
c. 0 184 816 097 o c . 0 215 785 090 o 3 1 c o$ ~ 753 084 o c . 0 278 722 078 o g . 0 309 691 071 o g . 0 34 1 659 065 2 059 o.q0 372 6 8 596 052 o g . 0 435 565 046 o g . 0 466 534 039 o g . 0 498 502 033 o c . 0 529 471 027 o r c .K 0 560 440 020 o e 2 408 014 . 0 59 “ o 153 or read g . 0 623 377 007 o e . 0 654 346 001 o e q0 685 315 995 283 988 ° III] 27 lo garithms of Trigonometric Functlons 73
9. 65 705 717 283
9 . 65 729 748 252
9 . 65 754 779 221
9. 65 779 810 190 9 65 804 841 10 29 159 4 853 904 096 5 878 935 065 6 902 966 034 7 927 003 8 25 6 952 028 972 976 059 941 001 090 910 025 121 879 050 153 847 075 184 816 4 099 215 785 g{28 {33 124 246 754 6 - 15-0 148 277 723 7 21 0 17 5 173 308 692 197 339 661 221 370 630 401 599 431 569 295 462 538 319 493 507 343 524 368 555 445 5 392 586 6 4 16 617 383 7 441 648 352 8 19 2 679 321 489 709 291 513 740 260 537 771 10 28 229
90 66 562 90 71 802 0 586 833 167 3 2 1 610 9 71 863 137 634 925 075 682 706 986 014 731 017 983 755 048 952
9.66 779 9 72 078 922
827 140 860
9 .66 851 170 830
9 .66 875 201 799 899 231 769 922 262 738 946 293 707 970 323 677 354 102 7 646 018 384 616 2 85383. 3323 {833 222 ” 090 476 524 153 “M 113 506 494 137 9 72 537 463 161 9 72 567 433 ° 74 28 Logarithms of Trigonometric Functions [III
161 567 433 185 598 402 208 628 372 232 659 34 1 9 67 256 689 10 27 311 4 303 750 250 5 327 780 220 6 350 811 189 7 21 7 9 67 374 841 128 42 1 902 098 932 068 468 963 037 492 993 10 27 007 515 023 539 054 946 ' 5 14 s 2 0 562 084 916 6 In' 14 4 586 114 886 zo s 1 8 609 144 856 633 175 656 205 795 680 235 765 703 265 735 726 295 750 326 773 356 644 4 796 386 614 5 820 416 584 6 9 67 843 9 73 446 10 26 554 7 8 18 4 890 507 493 913 537 463 936 567 433 959 597 403 982 627 373 006 657 343 3 2 l 029 687 313 052 717 075 747 253 098 777 121 807 193 144 837 163 167 867 133 9 68 190 897 103 927 073 23 orm8“ 7 957 043 , 260 987 013 283 017 983 305 953 328 077 351 107 893 374 137 863 397 166 834 9 68 420 804 226 466 256 744 489 286 714 512 316 684 9 68 534 9 74 345 10 25 655
' ° 76 30 Logarithms of Trigonometric Functions [III
o » g . 9 897 . 76 144 856 753 o n g . o9 919 . 76 173 827 746 o g s 3 .76 202 798 o a g . e 9 963 . 76 231 769 o g s Q3 .76 261 739 710 F H g . 0 028 681 O
g . 0 050 652 G
g . 0 072 623 “ 2 1 0
og qq0 093 594 Q 2710 ° 565 1- g . 0 137 536 x- g .' 0 159 507 x 4 g o - ° x 1 8 478 x g . ~ 202 449 oo° 0 o : g .1 0 224 420 o g .a0 245 391 ‘ O fl- Q o u l g 0 267 . 361 ’ ‘ o O - Q g .q0 288 332 fl HQ Q - P c a0 310 303 Q i Q i- i G 275 ‘- Q t Q
g .q0 353 246
g .q0 375 217
g . 0 396 188
oog . 0 418 159 130 i g l. 0 461 101
g .a0 482 072 3 554
g g 3 043 i g K o . 0 525 014 o 4 g .3 0 547 985 o : g . s0 568 956 o l g .l 0 590 927 o i g . s0 611 899 o g q0 633 870 841
g . 0 675 812
g . 0 697 783
g . 0 718 754
og q0 739 726 697
g . 0 782 668
g . 0 803 639
g . 0 824 610 : oog o3 o 582 3 427 1 og c 3 ~ 553 o “ g c 3 524 For30 or o g . 0 909 495 o g . 0 931 467 o or read g 0 952 438 409
g .a0 994 381
g .u1 0 15 352
g .q1 036 323 a oog 1 058 294 “ For59 or 266 o g .q1 100 237 4 4-s-N4- Q 1 1 g 1 209
9 . 71 142 180 9 .93 322
9. 71 163 151 9 . 93 314 184 123 307 ° III] 31 Logarithms of Trigonometric Functions 77
° 58 Logarithms of Trigonometric Functions ° 78 32 Logarithms of Trigonometric Functions [III
57 Logarithms of Trigonometric Functions ° 111] 33 Logarithms of Trigonometric Functions 79
9 .73 611 . 81 252 748
9.73 630 .81 279 721
9.73 650 . 81 307 693
9.73 669 .81 335 665
9 .73 689 .81 362 638
9 . 73 708 .81 390 610
9 . 73 727 .81 4 18
9.73 747 .81 445 555
9 .73 766 .81 473 527
9.73 785 .81 500 500
9 .73 805 .81 528 472
9 .73 824 .81 556 444
9 .73 843 .81 583 417
9 .73 863 .81 611 389
9.73 882 .81 638 10 18 362
9.73 901 .81 666
9 . 73 921 .81 693 307
9 . 73 940 .81 721 279
9 .73 959 .81 748 252
9.73 978 .81 776 224
.81 803 197
9 .74 017 .81 831 169
9 .74 036 .81 858 142
.81 886 114
.81 913 087
.81 941 059
Q . q4 113 .81 968 032
Q . 4 132 . 1 996 004
Q . 4 151 0 32 023 977 2 1 Q . q4 170 .8 05 10 17 949
2 Q . q4 208 .8 106 894
Q .44 227 8 867 4 Q .3 4 246 839 812 785 303 757 322 730 702 360 10 17 675 379 398 620 4 17 593 436 565 10 17 538 474 483 5 12 456 531 429 10 17 401
9 .74 587
9 .74 606 319
10. 17 292 10 17 265
210 183 156 10 17 129 ° 34 Logarithms of Trigonometri c Functions [III
9 .82 899 101 857 926 074 849
953 047 9.91 840
9 . 2 980 020 9. 91 832
008 992 9 . 91 823 g
035 965 9.91 815 4
062 938 9. 91 806 5 089 9 798 6
117 883 9 .91 789 7
9 83 144 856 9 .91 781 8 22 4
829 9.91 772
198 802 9 .91 763 225 775 9 755
252 748 9. 91 746
720 9.91 738
307 693 9.91 729
334 666 9 .91 720
361 639 9.91 712
388 612 9.91 703 7 18 2 13-3 415 585 9.91 695
558 9.91 686 9 23" 1“ 470 530 9 .91 677
503 9 .91 669
524 476 9 .91 660
551 449 9 .91 651
422 9.91 643 9 34 605 395 9 . 1 6 4 7 2
632 368 9.91 625
659 34 1 9 .91 617 314 608 713 287 599 260 591 768 582 795 205 573 178 565 849 151 556 876 124 547 903 097 538 930 070 530 957 043 521 984 016 512 011 989 504 962 065 935 9 91 486 092 908 9 91 477 119 881 469 146 854 460 173 827 451 200 773 9 91 433
280 720 693 334 666 398 361 639 388 612 381 585 558
469 10 . 15 531 354
496 10 . 15 504 9 .91 345
10 . 15 477 ° 111] 35 Logarithms of Trigonometric Functions 81
4 0.3 5 859 o8 477 1 o 0L. 5 877 e 8 450 « o 0 . -5 895 8 424 r m 0 .i 5 913 e 8 397 t o 0. s5 931 c 8 370 0 “ 49 o8 43 . 5 9 m 3 0 . Q5 967 e 8 316
0 . 5 985 o8 289 o 0 . 6 003 c 8 262 o 2 0 . 6 02 1 c 3 10 15 236 s 0 . 6 039 t 2 p 0. 6 057 e 8 182 8 0 . 6 075 c5 155 0 1- c 28 . 6 093 8 1 a 0. 6 111 « 5 10 15 101
0 . 6 146 02 048 2 0 . 6 164 03 021
0 . 6 182 08 994 10 14 0 . q6 200 08 967 : o 0 . 6 218 e 8 “ o 0 . s6 236 c 8 914 n o 0 .i 6 253 c 8 887
0 .a6 271 o8 860 t o 0 s6 289 c 8 10 14 834 0 e8 0 c8 780 0 a8 753
0. 6 360 e8 727
0. 6 378 p 8 8 o 700
0. 6 395 9 8 8 a 673
0 . 6 413 646 : 0.t6 431 s 620 4 0. . 6 448 p 8 n 593 1 a 0 46 466 g 8 fi 566
: o' fla 8 o 10 . 14 540
oh. 6 501 8 g 513 o r ph. 6 519 : 8 a 486 9 ph. 6 537 < 8 c 460 0 08 0 0q 433 : y . q6 572 9.85 594 406 0 p. -6 590 9.85 620 380 p t 07 47 353 . n6 6 6 oi g 8 326 p 4-a3 700 300
t oi a8 727 273 o l c . s6 677 754 246
o. 6 695 780 10. 14 220
o. 6 712 193 c 6 730 10 14 166
. 9 ! c 5 a q8 8 8 q 113 w 08 9.“6 782 c8 8 7 0 c 5 .' q6 800 8 8 o 060 0 17 c 033 5 ‘.q6 8 cc 8 8 q
. ° : : o 0 5 1 a 8 t 5 8 8 c 980 9 9 .76 870 5 8 2 a 954 9 9.76 887 5 . 86 073 927
9 .86 100 10 13 900 922 126 ° 82 36 Logarithms of Trigonometric Functions [III
o s q 8 126 874 o q m s - 8 153 847 o q- m8 179 821 s o s o8 206 794 p q c 8 232 10 13 768 o s : 9 8 8 o 0 s fl 5 8 ” 715 o 9 s fl 9 8 688 o 9 s fl 5 8 662 o l 0 s N 5 8 10 13 635 o o s s 8 o 582 o 555 o 529 ogg 10 13 502 s t c 449 e 423 p s 397 «a 10 13 370
o c 317 o s 291 p 264 o t 10 13 238
. ) s 185 p 158 p 132 a 106 o o 053 o 026 o 000 o qq 10 12 973
o c 10212 921 o o c 868 o c q 842 o 1 g ~q 815 1 oL 789 m1 g ~ qq 762 p 736 ; o z 710 317 683 657 369 631 396 604 422 578 552 525
10 12 446
394 367 341 10 12 315
° 84 38 Logarithms of Trigonometri c Functions [III
° 51 Logarithms of Trigonometri c Functions ° [III 39 Logarithms of Trigonometric Functions
° 50 h gafiM s of Trigonometric Functions ° 86 40 Logarithms of Trigonometric Functions [III
619 593 567 542 516 490 465 439 413 388
337 311 285 260 234 208 183 157 132 106
055 029 004 978 952 927 901 876 850 825 799 773 748 722 697 671 646 620 594 569 543 518 492 467 44 1 416 390 10 06 364
313 288 262 10 06 237
186 160 135 680 10 06 109 694
° 88 2 Logarithms of Trigonometric Functions [III
9 .95 444 556 9 . 87 107
9 .95 469 531 9 . 87 096
9 .95 495 505 9 . 87 085
9. 95 520 480 9 . 87 073
9 .95 545 455 9 . 87 062
9 .95 571 429 9 . 87 050
9 .95 596 404 9 .87 039
9 .95 622 378 9 . 87 028
9 .95 647 353 9 .87 016
9 .95 672 328 9 .87 005
9.95 698 302 993
9 .95 723 277 982 252 970
9 .95 774 226 959
9 .95 799 201 947
9.95 825 175 936
9 .95 850 150 924
9 .95 875 125 913
9 .95 901 099 902
9.95 926
9.95 952 048 879
9 . 95 977 023 867
9.96 002 998 855
9 .96 028 972 844
9. 96 053 947 832
9.96 078 922 821
9 .96 104 896 809
9 .96 129 871 798
9 .96 155 845 786 180 820 775
9 .96 205 795 763
9 .96 23 1 769 752
9 .96 256
9 .96 281 719 728 9 307 693 717
9 .96 332 668 705
9 .96 357 643 694 383 617 682 408 592 670 433 567 659 459 541 647 484 516 635 510 490 624 535 465 612 560 440 600 4 14 589 611 389 577 636 364 565 662 338 554 9 96 687 313 542 288 530 738 262 518 763 237 507 788 212 495 814 186 483 839 161 472 864 136 460 110 448 915 085 436 940 060 425 966 034 413 ° III] 43 Logarithms of Trigonometric Functions 89
98 9 8 9 8 9 8 009 9 8 9 9 984 98 9 8 958 9 8 9 8 10 02 933 98 9 9 n-t-0 2 98 9 9q g p 0 88
98 9 .97 143 857
98 9.97 168 832
98 9 .97 193 807 9 8 99 8 9 781 98 9 9 8 756 9 8 8 o 9 9 8 9 731 9 8 8 h 9 9 8 9 705 9 8 8 Q 9 9qq 88 680 o g 655 ° l 98 8g 0 _0q 6 ~ h 629 3 98 608 0 0q 0 00 604 98 62 1 579 98 8 553 9 8 8 528 98 661 503 98 674 477 98 Q g 452 l- 9 8 Q 9 l 427
1
98 4H0 . 402 0 98 Q N0 0 376 0 351 : ! 98 Qw0 0 326 0 98 Qc 0 0 300 I 0 9 8 N0 H 0 275 9 8 795 0 250 n 9 8 8 o 0 224 9 8 821 0 199 0 174
9O 8 8 0 0
90 8 8 0 0 123 0 3 8 098 98 8 0 0 073 98 08 0 0 0 9 8 08 0 S 0q0 022 0 9 8 08 0 00 003 997 0 98 8 0 00 029 971
98 08 0 .08 054 946
9 8 08 0.08 079 921 0 9 8 08 0 00 8 896
9 8 8 0 .08 130 870
9 8 8 0 .08 155 845
9 8 08 0.08 180 820 9 8 08 0 0m8 0 794
.
9 8 8 a 0 .08 231 769
9 8 8 0 0 .08 256
98 0 9 0 .08 281 719 1
9 8 0 0 0 .08 307 693 0 0 9 3 o 98 0 0 0 90 0 0 l 668
0 9 1 98 112 9 00 0 0 fi 643 i-° ! 0 9 98 t 1 0 9 00 0 8 617 i 0 1 98 h ( 0 592 98 151 567 9 8 “8 458 10 01 542 i 1 9 8 i 4 484 ° 90 44 Logarithms of Trigonometric Functions [ HI
8 8 516 693 8 8 491 681 8 8 98 466 669 3 8 8 95 440 657 8 8 98 4 15 390 8 8 8 365 8 8 8 339 8 8 8 314 8 8 8 289 263 0 t 8 8 [ p 238 8 8 8 213 8 8 8 188 8 8 8 162 373 137 112 087 8 061 8 8 036
38 011
38 985 8 8 e 960 8 8 w 935
38 O8 910 ‘ t-: 3 8 h t a 884 i-b t- i 85 8 3 8 l 9 i a n - 4 8 8 l c a 83 i- i 8 3 8 b 0 v 809 : t o8 38 Nu-a 783 ! 0 8 2 758 1 1 0 «0 38 8 fl 733
3 8 8 w 707 : : 3 Q 3 3 3 m 682 c o s o w 6 7 c Hc 38 8 5 8 632 98 606 98 581 8 556 a8 10 00 531
Q8 480 Q 8 455 9 8 3 Q8 430 9 8 Q8 404 9 83 ? 379 9 8 ! qz 354 9 8 “fi 8 328 9 8 5 w 3 8 303 9 8 5 o 8 278 9 8 087
3 253 9 8 08 8 773 227 9 8 08 2 8 202 9 8 8 3 177 9f8 037 q 3 W. 0 152 9 8 024 126 012 0 o 0 0o 101 999 - 9 h w 076 986 9g 051 3 9uO 025 961 000 949
92 Table V RadianMeasure Trigonometric Functions [V V] RadianMeasure Trigonometri c Functions 93
° ° ’ ‘l ' radians 180 l radian 57 17
" ' ° ‘l ' 3600 60 1 radian ° ‘ 1 right angle 90 t /2 radians 1 radians
Table Va Radians to Degrees
0° 0° 1° 0° 0° 0° 0° 0° 0° ° ' " ° ° ° 401 4 13 . o 40 4 0 0° 5° 0° 94 Table VI Powers Roots Reciprocals [VI
96 Powers Roots Reciprocals
4 50555
4252709 4 53872
4 61519
4203081 9 93838
1023002 10 5035
4 8841
11 2394
5 1076 1125432
521984
4 81664
5 6169 4 86826
6 1504 Powers Roots Reciprocals 97
6 3001
6 4516 5 03984
5205904 6 6049
1 63095
7 7284
7 8961 5 30094 1 41281 531977
5233854 1241780 1 69115
1209700
1 42931
8 0430 1 43257
87010 5 44059 1243581 98 Powers Roots Reciprocals
27 8181
2823720 28 0520
2922181
30 6643
10 4329
1025025 10 mn0
11 1556
5 79655
11 4244 5281378
Powers Roots Reciprocals 101
105 154
107 172 2 18174
2218032 1092215
2 19545 102 Powers Roots Reciprocals
27 . 144 1
145 532
2 30868
2231301 2 31517
2231948
104 Powers Roots Reciprocals
37 5769
3728225
40 0689
4023225 40 44 90 VI] Powers Roots Reciprocals 105
303 464 2 59422
2259808 106 Powers Roots Red proeals
381 078
8 57321
108 Powers Roots Reciprocals
9 08845 Powers Roots Reciprocals 109
74 . 1321
665 339
0092922
2 97153
2 97658
2298101
2 98831
2299100
112 Table VII Napierian or Natural Logarithms [VII Napierian or NaturalLogarithms 113
6991 7130 7269 8376 8513 8650 42 9877 ‘ 0013 1089 1223 1357 2551 2683 3732 3862 3992 5156 5284 6433 6560 7568 7694 7819
1263 1384 1505 2465 2585 2704 3653 3771 3889 4827 4943 5060 5987 6102 6217 7134 7248 7361 8267 8380 8493 9389 9500 9611
1594 1703 1812 2678 2786 2894 3751 3858 3965 4813 4918 5024 5863 5968 6072 6903 7006 7109 7932 8034 8136 8950 9051 9 152 9958 ‘ 0058 ‘ 0158 114 Napierian or Natural Logarithms 10 to 99 [VII
Above 409 f m 10 1 0 2 2 . use the or ula 3. 0a log. n log. 1 log. n 3 0 58509.
log. n log. 104 0 31; a 23 0258509 10 31.11.
116 Table 1X Logarithms of Hyperbolic Functions [IX
88692
.85214
.82696
1 2337 .81058 .32541
.80252 .34592 12 580 .09989
.77880
. 11292 .77105
.11726
12160 .75578 0 2837
.73345
.72615 .51254
.71177
.69768
.57807
.590 19
.67706 .60202 1X ] Values and Logarithms of Hyperbolic Functions 117
.26492
.26926
.27361
.29532
.29966
.88715
.31269
.31703
.32138 .47711 90817
2 1170 .32572 .47237
.46767 .92185
.92859 0 8615
.34309
0 9015 .95498
029280
.43171
.98051
.98677
.41478 .99916
.4 1066 .00528 118 Values and Logarithms of Hyperbolic Functions [IX
. 070 11
.43864 .36422 . 07580
.44298 .36060 .08146
1 .44732 . 35701 .08708
.45167 .35345
.4560 1 .34994 .09825
.46035 .34646 . 10379
.46470 . 34301
.46904 . 33960 . 11479
.47338 . 33622 . 12025
.47772 .33287
.48207 .32956 1 .3524 . 13111
.48641 .32628 . 13649
.49075 .32303 14186
.49510 .31982
4 44 . 1 21 2 3 . 99 3 664 5 5
.50378 .31349 . 15783
. 50812 . 31037 . 16311
. 5 1247 .30728 1 .4735 . 16836
. 51681 . 17360
.52115
. 52550 .29820 1 . 5276 . 18402
.52984 . 29523 . 18920
.534 18 . 29229 . 19437
. 53853 . 28938 . 19951
.54287 . 28650
. 54721 .28365 20975
.55155
. 55590 . 27804 221993
.56024 . 27527
. 56458
.56893 .26982 1 .7182 .23507
.57327 . 26714 .24009
. 57761 .26448 1 7583 .24509
.58195
.58630 .25924 1299 7 1 .25505
.59064 .25666
.59498 . 254 11
. 59933 .25158
. 60367
.60801
.61236 . 244 14 . 28464
.61670 . 24 171 1 .9477 .28952
.62 104 . 23931 . 29440
. 62538 . 23693 .29926
.62973 .23457 .304 12
. 63407 . 23224 .30896
. 6384 1 . 22993 . 31379
. 64276 .22764 .31862
.64710 . 22537 2 . 1059 . 32343
120 Values and Logarithms of Hyperbolic Functions [1x
.95979 . 10970
.96413 . 10861 .65795
.97282 . 10646
.97716 . 10540 .67128
.98151 10435 .67572
°
10228 .08459 IX ] Values and Logarithms of Hyperbolic Functions 121
07353
.07136
.06788
.17694
. 18128 .06587 .87836
15 333 . 18562 7 6338
2 4 1 15 043 . 19 3 .00393
15 800 . 19805 .00329 .89588
2 0 34 10 119 .2 7
.21108 .00142 .90901
06020
.05784
.05727
05613
.05393
.05340 05287 122 Values and Logarithms of Hyperbolic Functions [1x
124 Values and Logarithms of Haversines
.3979 .2513 .4001 .2525
2576 .4 109 .2589 .4 131 .2601
.4237 .2665 .4258 .2678
.2730 .4362 .2743 .4382 .2756
.4484 .2821
.4722 .2980
.4838 .3060 3073 .4876
. 3127 .4951 .3140
.5063 .3222
.3372 .5279 .3386
55 .5384 .3469
.3538 .5488 .3552
.3622 .5589 .3636
06 .5689 .3720
3790 .5787 .3805 .3819 .5819
.3904 .5915 23 900 .5977 .3975
4132 .6161 .4146
.4218 .6251 .4232 .6266 .4247 .6280
6339 .4319 .6353 .4333 .6368 2 4477 .6510 .4492 0 45“
.0021
1 .6676 .4666
.47 7 47 20783 38 .6 56 .4753 .0770 . 07
.4826 .6835 .4840 .0848
.4913 . 6913 .4937
.6990 .5015
.5087 .7065 .5102 2
5 174 .7139 .5189 .7151 .7163
.5262 .7211 .5276 .7223 .5291 .7235
9 .7283 .5363 .5378 .7306
5 . 5696 .7556 .5710 .7507 725
.5782 .7621 .5797
.7685 .5883
.5954 .5983 .7769
.7810 .6054 .6068 .7830
.6125 .7871 . 6139 .7881 .6153 .7891
. 6210 .7931 .6224 .7940 6238 .7950
.7989 .6308
. 6378 .8047 .6392
8159 .6559 8108 .0573 28177 . 2
. 6628 .8214 . 6642 .8223 . 0055 .8232
. 6710 .8267 .6724 .6737 . 8285
.6792 .8320 .6805 . 6819 . 8337
. 6873 .8371 .6887
.6954 .8422 . 6967 .8430 .6980 .8439
.7034 .8472 .7047 .8480 .7060
.7113 .8521 .7126 .7139 .8537
.8568 .7205 .8576 .7218
.8615 .7283 9 3 85 .8661 .7360 .866 .7 73 70 X ] Values and Logarithms of Haversines 125
"
.7500 .8751 .7513 .8758 .7525 .8765
.7575 .8794 .7588 .8801 .7600
.7650 .8836 .7662 .8843
.7723 .8878 .7735 .8885 .7748
.7820 .8932
.7939 .8998 .7962 .9010
.8009 .9036 .9042 .8032
.9073 .8090 .9079 .8101
.9110 .8158 .9116 .8169
.8214 46 .8225 .9151
.9180 .8291 .9186 .8302 .9192
9215 .8356 .9220 .8367 .9226
.8410 .9253 .8431 .9259
.9291
.9312 .9318
. 8597
.9374 .9379
.9432 .9430 .8793
.8940 .9513
.8993 9539 .9543 .9011 .9548 9568 9062 9572
9145 9612 .9153 .9616 .9161 .9620
.9638 .9209
.9657 .9660 9256
.9678 .9293
.9330 .9699 .9337
.9373 .9719 .9722 . 9387 9725
.9415 .9738 .9422 9757 9462
.9777 9507 .9780
.9808 .9811 .9579 .9813
.9826 .9614
.9636 .9839 .9841 9647
.9668 .9853 .9673
.9871
.9728 .9880 .9732 .9882 .9737
5 .9892 .9760 .9894 9764 9722 98 2 . 00 .9915 .9810 .9917 9814 .9919
9830 .9925 .9833 .9927 9
.9935 .9855 .9937 9858
.9908 .9900 .9911 .9901 . 9914
.9951 .9079 . 9953 .9980 .9955
.9981 .9992 .9982 .9992
.9988 .9995 .9989 .9995
. 9993 .9997 .9994 .9997 9995 126 Table XI Factor Table Logarithms of Primes [X I - [li l pdmeJts logalitllmls dm li t t lts factol aamdm]
130 Table XII c Amount of an Annuity
Anom or mAmrl'r or On Doma m Yn a am n Yn as X II d] Table XII d Present Value of an Annuity 131
Pusan Vanna or On Dow n mYua nos 1: Van s 132 Table XII e Logarithms forInterest Computations [XII e
F mun A of a rinci al P afte r 11 ears : A a P 1 or A o t , , ny p p , , y ( m A at the end of n ears : P as A l For present worth. P , of any a ount . , y ( o ms an tilo drithms of A and P to man si nificant fi ures use To find l garith d an g y g g .
able X I . 126 and able I a . 20. T , p , T , p
Table XIII Ameri can Experience MortalityTable
Based on living at age 10
134 Table XIVa Four Place Logarithms [X IV a
- hand side . The proportional parts are stated in full for every tenth at the right The logarithmof any number of four significant figures can be read dlrectly by add X IV a] Table XIVa Four PlaceLogarithms 135 136 Table XIVb Antilogarithms to Four Places [XIV b
X IV 0 ] Four Place Trigonometric Functions 139
[Characteristics ofLogarithms omitted— determine by the usual rule fromthe value]
1 4 108
1 4050
1 3992
1 3934
1 3584
1 3235
1 2712 140 Four Place Trigonometric Functions [X IV c
[Charactu‘lstics oi Logarithms omitted— determine by the usual rule fromthe value]