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RUDIM ENTARY TREA TISE ON L O G A R I T H M S BY HE N R Y PA W . 59 . . CIV I L E NG IN EE R . L O NDO N : JO HN W E AL E . H IG H HO LBO RN 1 8 53 .

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h h ey h av e been so w r i tten wi t r efer en c e t h er . O LD Wrxnson . he acqui r ement of thatt one g ly r eat faci li at t he es t acqui remen tof he ot t her . While . PR E F A C E . o bec ome t who r eally wish es t h or ough ly ed i acqua n t wi th t he .er m i h tbe s e a g p ely st ra t u d ie d or r efer r e d t o t . sub ect j . . h ough a alt kn owled ge of h ematical pmpe r ties i s their mat n ot i al essent t o a knowledge of h e ir t u se . . d e m on st r at h eir e t n atur e an d r p po i er t e s h e matical for th e M at stu d en t . as e ach ot to for m buto ne ed c n n ec t i se . should ir e l e nt y p eruse . th e r efo r e . r eat t o o w hi ch he t u d en t st . ye t h ey t ar e so in t i ely mat connected . F or . 1 8 50 9t . h J u ly. H L . TH E followin g little work is int e nded to h av e a tw ofold — h to bot l i an d illu st l j o b ec t ex p a n r ate th e use an d a pp i ca i t on of logar ithms for h e practical calculat t o r an d to setfor t h and . th e Th eore tical an d ical Prac t par t s h av e been kep t di st in c t so t h at ei t h.

E xplana t ion and Definitions of Loga rithms : II . 62 Table by the aid of w hich the Number answ er ing t o 0 hm can be found t any Logar it o six places 64 . hms O f various Sy ste ms o f Logar it 8 IV . hmic A rithmetic V I Logar it 48 A PPE N D I X . Table of the Logar ithms of ev ery Pr ime Number from 2 t0 1 0 00 . of Pow ers 4 III . Descripti n of Logarithmic Tables o 31 . PM } C HA PTE R I . C O N TE N TS . M od e of ca lc ula t ing Logar ithms. O n the E xponen t . a nd Demonstra tion of heir Proper ties t 12 V .

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c o mme nc in g w i h t th e cyp h er th en w ill t h e n u m be rs i n t . an d a ‘ a. 1 . 243 m “. lo ws th a ti t . 3 a r i thmeti ca l ser ies fo r m e d by th e c o ntin u e d a d d ition of uni t y . wr i t i ng th em v er tically instead of i n h or izon t a l lin es N os . ar e the log a r ithms forming an Of Wh i ch 0’ 1’ 2’ 3’ 4 h met a rit i ca l ser ies " . and may h a v e a n y v al ue t hatw e pl ease a ssi g ned t o it Th u s . h e l o w er lin e e xpr ess th e n u m be r of r a tios o f u ni t y t o th e fir s t te r m. h me tical ser ies gi ven above a n d t h atth ey are th e refo r e th e logar i th m s of th e n um . . i n p lac e of w r itin g th e n umbe rs th e msel ve s w e m igh t i n tro . 3 . . of t h e p o wer s a r e th e sam e as th e ar i t . 27 . w ha t he e ver t value of a may be . ar e he t n u mber s forming a g eometr ica l 1 . of w h ich t h e r a tio o f u ni t y to a ll th e o th e r te r ms i s m a d e up . 1 0 00 . . 8 1 . { ar e the log a r it hms forming an a r i t h of wh i ch 0 l 2 3 4 6 meti ca l ser ies . 0N L O G A RITHM S . of w h ich . th r e e g eo me tr ical ser ies abov e . which m ay be d e Itmustbe borne in mind tha ta ° 1. th e pow er s ar e su cc essively taken i s te r med th e r o otor r a d ix . 13 2 . l l0 : 10 100 = l0 2 In th ese w e p ercei v e i mm ed iately th at th e n umber s d en ot e r m ed t in g th e p o w ers. L ogs . 9. a n d the r efo re th ey w ill be th e lo ga r i th ms of the n um bers i n t h e line abo ve th e m Fo r e xamp le . or . Nos . we der i ve an ot h er d efin i tion of a loga ri th m . ar e he t n u mber s for ming a g eo metr i ca l so . L og s. Nos . L ogs . 1 0 0 0 0 g eo metr ica l ser ies . fo r m in g th e m ultiplicat i on an d w e sh ould th us obtain for th e . N o w fro m th e v ery natu r e of a g eo m e tr ica l se r i es i t fol . s t erm s ar e all pow er s of t h e co n s tan tn u mber by th e m u lt i plicati on o f which th ey a r e p r od uced an d th e r efor e . he“ W e” “ { a re t w ins 8 And agai n. . 7 4 9’ 3 43 240 1 ’ ’ ser i es he log a r i thms for ming an a r i t { a re t h of W h l ch 0 ’ 1 ’ o 2’ 3’ 4 met i ca l ser ies . bers i n th e fir st colu mn s Th e con stan t n u mbe r . d u c e th e exp r essio n d enoting th e p ower wi th o ut ac tually p er . 1 00 . as th ey a r e t h e in d i ces or cap onent s .

t h e p ow er to which a g iven c on stan tbas e or r oo t m ust be i n . abo ve t . onent s o f Powers . . . O n the E c . wh i ch may be giv e n o f loga r it h m s d ep e n d ing u pon th e p a r . . h e follo win g page s c on sid er them . 8 Th e loga r it . th e same en r al r o er t i m ay be d e d uc e d as belon i n t l a r it h m g e p p es g g o o g s . la te th e se d efin i tion s be fo re p roce e din g far the r in or d e r to . . n n t s f t h w f m on st an tr oot A n d . i n or d er p o e o e p o e r s o s o e c . i n sur e th eir b ein g th or ou gh ly u n d ers too d . C HA P TE R I I . in o r d er to be equa l to the n um ber of wh ic h it is the log a r i t hm I ti s ther efor e .4 RU DI M E N TA R Y TR E A TIS E sc r ibed as he i nd ex t o r eap on en t . be r may h ave any n umbe r of logar ith ms c o r r espon ding w i th i t. o a mor e er fec tc on ce t i n of t h u b ect h ll fi t p p o e s j . ar e d en o t e d by th a t n u m be r b ein g w ri tt en See page 1 30 . or th a tth e sa me logar ithm may se r e for sev e r al d iffe r e n t v n u mbers . co n t in ed in t h e y a rat io of th e gi v en n u mber t o u ni ty . w e s a rs t c on si d er t h e pr op erties of th e expon en t s of o we rs n erall p g e y . g o a n o e r s er e r c a l p r og r essio n . h e n um ber of ra tios o f som e assu m e d c o n st a n t n u m be r t o un it . 2 L ogar i th m s ar e a ser ie s of n umbers in a r i thmet . w e sh all h ow ev er i n t . e v id en t th a t a gi v en n u m : . u n d er t h e n otion in volved in th e th i r d d efin i t ion as th e ex . t o which ba se a cer ta i n r oo to r mustbe in vo lved . ical p r o m r essi on an s we r i n t th i s o f n u mber s in g eo met i g . . q . . Wh ich e ver of th ese d efini tion s may be ad op te d . so t ake n t ha t0 in th e fir st corr e spond swith 1 i n th e la tter . h m o f a n umber is th e in d ex or expon en tof t . v olv e d t o be e ua l t o th a tn u mbe r . I n algebra. ticula r way in whi ch t h ey ar e r egar ded a nd we sh al l r eca p i tu . . th e powers of a quan t it y . 1 Th e lo gar ith m of a giv en n u mbe r is t . h e sam e th ing th e co mmon r a ti o . h e bases o r common ra tio s ar e 3 7 and 1 0 . . a ccor d in t o t h e val ue assu m e d for t h b ase o r r o o t g e to be i n volv e d o r wh a ti s t . o r th e n umber of t im es th at th a tquan tit h b n em lo e d as a fac t or t o p roduce y a s e e p y a gi ven quant ity. W e h ave th er efor e th ree d istinctly difie r ent d efini tio RS ‘ . of t h e geometri cal p r ogr essio n Thus i n th e exa m p l es .

a n d i s d en o te d by a 3 w r itten above th e x . p o w e rs . t i s on l ne ce ssa r y y t o s u b tr ac t t h e e x p o n e n t o f th e di visor fr om th e e x pon e n t o f th e d i vid en d to o btain the e x ‘ o n en of t h eir i T h l i i d d i vid e ’ p t q uot e n t u s . th e cub e of a w . t o w h i c h t h e q u a nt i t i e s 6 . c on v e rse of t h is r u le h old s goo d . a . instance th e square o f a . t h er efo r e . an d i s w r itte n The n u mber th us p lace d o v e r a n um ber . th u s . fo r e goi ng examples 2 3 . e t t b e re q u re t o " " " we h a ve " :s by :c . . 5 mewha t to ' so t h e r igh t and abov e th e n umber or le tter ex r essi n t h o r i in al qu an t i t r r oo t of t h w T h p g e g y o e p o e r us . c . or t he n th pow e r The quan tities . an d th e fifth " the s qua r e of 6 is w r i t 2 . th e sum of th e ex pone n ts o f th e two fac to r s A n d th e . o r t h e po wers of wh ich are t o be t ake n ar e t. . as 1 2 ' ' en ter s fiv e ti mes as a fact o r . as a: or b i n th e for ego ing . . u o e o r s o e u p e o an d re . . wh ich h ave t o be i n vol ve d . . . a “ and b . L e tus n extexa min e fo r he t val ue of h e p ower t of a p ow e r . i ti s t er me d t h e fifth po we r . a n d 1 2 a r e t o b e r e spec t iv ely rai sed or in vol ved . and by . to d e n o te th e p ower to wh ich i t is r equire d t o be r ai s e d is . . e x am p le s. as a: e n ters th r e e tim es as a fact o r . a n d or ’ " 3 x. F requ ently le tters a r e em ployed in st ead of n u mbe rs as ex p on en t e o f p o w e r s . as 6 e n t ers t w ice a s p . it is calle d t h e secon d p o we r . o wer o f 1 2 I n t h e fi rstexa m p le . te n 6 . ou L O G A RITHM S . t hat th e quan tit b h f ' y i s t o b e r a i s e d t o t e p o w e r o n . . " I n th is case we see ato nc e th atth e . and in th e las t ex a mple . fo r if itis r equir ed t o d i vid e o we r of a i nt i t b a ny o t h w f t h s a me a p g v en q u a y y e r p o er o e q ua nt i ty i . ter m e d th e i nd ea or exp o nen to f t h atp o we r . o s e e x p o is th e sum of th e two ex ponen ts of the powe r s to be mul ti p’ lie d F o r l e t s su p p s t h e p w e t b m l ti li d t b e . Wh en i tis d esired to multiply any two powe rs of a quan tit y . a nd 5 ar e th e ex po n en t s of t he ‘ . e r me d t h e r oo t s or ba s es . 3 2 . x m a ne a t he exp o nen tof w h ic h 5 is e qual to " . ‘ hing more than is mu ltip lied by it squa re of : c is no t " s elf. a vo i d a mbi u it w e sh all use on ly t h e las t Th s i n t h e g y u . as th e for m er of th es e t e r ms i s so me t i mes e mploye d in a d iffer e n t sen se to . a fac t or . a v e r y l i t tle c o n s i d e r a t io n w i ll s h o w th a t t h e i r p ro d u c t w i ll be equal t o we r of t h a tsa m e t it w h ne n t o a p q u an y . t i 3 he n c c a n . as he t expone n tof whi c h is e qu al o 6 t 2 . a n d is d en o ted by 2 w r i tte n o ve r th e 6 i n th e s ec o n d exam pl e. x “ d e n o te s t h a t t h e q u a n ti ty r e p r e se n t ed b x i t b e ra i e d t t h e w r r e n t d b y s o s o p o e p r e s e e y . i tis calle d t h e th ir d p o we r .

h ow e v er . m) . ‘ Th us th e squa r e r oo tof w is beca use 4 2 2 . or . i n th e a bo ve exam p les 3 x 2 3 th e r efo r e . the r oc e sses m a b r e ve r se d . y t o di vi d e th e e x on en t o f t p h e p ow e r by th e e xpo n en to f th e r oo t. tha t is. ” " a an d 3 x 3 . tial th a tth ey sh ould be p e rfec t ly co mp r eh e n d e d befo r e en te r in g o n t h e u se of logari th ms. 1 The m ult .6 a n DIM E NTA BY T E A R TISE our for mer rule for the m ultip lica tion of p o wer s w e ha ve a x . h atis . th a t th e n e w exp o n en t will be equal t o th e p rod uc t o f th e o th e r two . g p an o t h er 3 to th e ex po n en t. a x z “ " . th e n th pow er ' of : c is x " ” 4 The ex t . ax Th e con v e rse o f . a nd th e " s qua re r oo t of w i s beca use 6 -E 2 3 . we shall r ecap i tula te th e m in th e fo r m of rules . iplica tion of th e pow er s of any q uan t ity i s p e r fo r m ed by th e add ition of th e ir expon ents . wh ic h i s d en ot e d by t h e d en omina tor o f th e sa me . ‘ “ i i tw o uld ha ve b e e n x x . itis th er efor e obviou s th a tas th e . i t s u se . Th e fo ur p r oc esses which we h ave h e r e d escr ibe d ar e tho s e w h ic h a r e of th e m ostfr e que n toc cur r en ce an d as i t i s essen . wh ich is t g h a t th e . t h at is . o ex tr ac tan y r o o t of a p o w e r w e h a v e on l . e x p on en t. . 2 .fo r m e d by s ub t ra c tin g th e e x o n en tof t p h e d i v iso r f r o m t h e e x p o n e n t o f t h e d i v i d en d . . "' th a tis . ion of th e r oo tof any p ower is pe rfor m ed by r ac t d i vid in g i ts expon e nt by t h e expon en t of th e roo t r equir ed .Th e d ivision of t h e p ow er of a qua n ti t y yb a n y ot h er p o we r of t h e sa m e q u a n ti t y i s p e r. f r a c ti o n a . . " . 9 th e r efor e . th e n t t m h r oo tof m is w " I n th e la st exam ple w e h av e an ex pon en t d iffer in g fr om any w hich we ha ve r e v iousl m i h n a m ely a l p y e t w t . y p o e is p e rfor me d by m ultiplyi n g its exp on ent by th e expo n e n t of th e po we r to wh ich i tis t o be rai se d . l 3 Th e i nv olu t io n of any p ower of a quan tit to som e w r . th is r ule also h old s g oo d for if i ti s r e qu i r ed t . th u s. to obtai n th e e xpo n e n t d e sire d . an d t h en th e po wer r ai se d . q ua n ti t y to w h ic h i t i s at t ac h e d i s t o be r aise d t o t h e po we r d en o te d by th e n u me ra t o r of t h e frac tio n an d is th e n to h av e th e r o ot e xtr ac ted . in t h at ex ample sufficien t ly ex p la i ns i t s m ea n i n . e xpo n e n to f t h e o r iginal po we r h as to be ta ke n as ma n y ti m e s as t h e expo n e n to f th e po wer to which ith as to be ra is ed . a n d t h t fi t t t d p y e e r o o rs ex rac e . si nce th e o r d e r i n which these o erat i f r me d m akes no diffe ren ce i n t h fi l p ons ar e p e r o e n a re . f w e h ad r equir e d th e cu be of o we r of : 6 w e m us t a d d " " " " ar an d fo r e ver y h i h e r " t a w 23 .

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o r H p er bo li c lo a ri t h m Th l tt t m d i d f m y g s . which d epen d s u pon and v ar ies wi t h the v alu e of th e base . o a quan ti ty whic h . fo u nd to exist bet ween t h e logar ithms of this syst e m an d t h e asymp to tic spac e s o f th e hyper bola. a s w h i ch d e n o tes th e i n ver se n th powe r of v . u n d er t h e v ie w su ggeste d in th e th ir d d efinitio n giv en of th e m i n th e fir stch ap te r . a n d n o t som e si mp le i n t e ge r . y d i vid e d by th e h r oo tof : 17 d i r ectn t . It was t h er e sta te d th at any v al ue m igh t be assum ed for this base .h e r easo n w h y s u c h an i n tri ca te n u m b e r w a s a d o p te d .8 nunm nnu nr m m r rsn d ir ectn t h p ower of m. th e i n v en tor o f logar ith m s. is v e ry p ro p er ly falling i n to d isuse . 0f va r i o us ems Syst f o hms L og a r i t . m ore fre que n tly te r med N ap ie rean . C HA PTE R III . been e m ploye d . a s w 7 . in wh ich th e base e quals the . e a e r e r . t h at th e syst e m h avi n g t his n umbe r for i t s base . eg r a l ex o n en t p . it ma b p y ex p a y e s u ffici e n t a t p r esen t t o st at e. shall be e ua l t q o re . Th e fir st was th at a d op te d by Baron N api er . i ona l expon en t. bein g cap able of bein g e x r essed mor e p si m l . i on a l exp on en t. wh ich d e n o t 4 Th e neg a tive fr a ct es th e i nver se n th r ootof v. w h ic h w as c al cula t e d b y B ri gg s T . an d i s equal t o t h e prod uctarisin g fr o m the m ultiplication of n fac to r s each equal to w . 2 The n eg a tive i n t . w as th e r e aso n o f i t s ad op t ion . t hey have also been occasionally d esi n a t ed t h e na t u r a l sy st em o f lo a g g r it hm s th ey ar e h o wever . . t h an a n y o t h e r . a n d was em ployed by h im in th e first syst e m of lo a ri t g h m s . t ak en as a base for t h e system . h ow e v e r . WE n ext p roc ee d h wit th e co n si d erat i on of logar ith ms. two o nly h a ve . er ve ro c er ta1n r elations. as th e e xpon en t s of t h e p owers of so m e con st a n t n u mber . an d is e qua l t . a n d c alc ula t p y e d m o r e e asily . and is equal to unit . since i t is n ow kn o wn th at th e sa m e proper t ’ ' y belon gs to e ve ry syste m . fr om whi ch cir c umst a nc e . fo r i n th e system of which we a r e n ow sp eakin g . — . an d which w er e belie ved t o be p eculia r t o it . n amely. whi ch d e n o t The p osi tive fr a ct es h e d i r ectn th ro otof w and i s e qual t t . o u ni t d iv i d e d n y ti me s by l 3 . be i n g multi pli ed n ti mes by it s elf. th e only d iffe r en ce bein g i n the angle i nclud ed be tween th e as ympto te s. 2 7 1 828 1 8 - and 1 0 . will be r esen t l l i n e d . as min. .

li d b a con st n t n um be r . s e o n s r a on w i l l be gi ven in a subsequ en tchap t er * . 9 B ympmtes ar e at right an gles t o each o t her . Th e lo ga r i thms of an y par ticula r syste m a r e im med ia tely r e d uc e d t o t h ose of a ny oth er syste m by mer ely bein g m ulti . h e con sequ e n tly p ro po se d it to Ba r o n N ap ie r by . th e sam e thi n g. . p y th e p ur p ose s of calc ulat i on . an d t o oi nt ou t t h y p e ad v an tages th e r eby a ttai ne d . wh o se v alue d e e n d s o n t h t p e y a p e e la tiv e v al ues of th e bases of th e two syste ms Th us th e . ar e con v e r te d . I n th e table s belo w we h ave given th e n u mbers wh ose lo ga r ith m s a r e th e fir st six i n teger s i n both systems the . Ba se . i n to th e N apier ea n . by being m ulti p l i e d by 2 3 0 2 5 8 5 1 . and the righ th an d to th e base of 1 0 - . lo gari th m s belon gin g to th e commo n . bei n g m ult iplied by i t s r ecip rocal W e sh all n o tstO p h e r e to p rov e th is or to ex plain i ts r eason . Logs . a s i t would in v olv e co n si d er a t ions wi th w h ich th e stud en t i s s u p ose d at r esen t n ot t o be acquain t d i t d m t ti p p e . B 8 . 0N L O G A RITHM S . wh ose base is 2 7 1 8 28 . . le han d t f t - a ble be in g t ake n t o t h e N apier ean base . wh i le th e N a p ier ea n ar e r e d u c e d to t h e Briggean by being di vi ded by the sa me n u mber or . W e sh all n ow p r ocee d to explai n why th e bas e of th e s st e m w as al t e r e d fr om 2 7 1 8 28 t o 1 0 . syste m wh ose bas e is 1 0 . in th e o t ' h er S y stem . or as i t is som e time s te r m e d Br ig g ea n . fi rst table o f N a pi e rean logar i th ms . I t was so on p erceive d by Br iggs who ha d c alc ulate d the . that se v e ral i m p o r t ' a nt a d va n t a g e s w ere po ssesse d by a syst e m of lo ar it g h m s w h o s e b ase was 1 0 . Logs. N os . . h a vi n g 1 0 for i t s base t h ey m ake an an gle of " . . See pag e 22 . alth o u gh the N a p ie r ean loga r it h m s a r e always e m p loye d in th e Differ en tial a n d In teg ral C a lc ulus a n d th e o th e r h igh er br a nc hes o f a nalysi s . i vho m i t was a d o p t e d an d it i s n o w u n i v er sally em lo e d fo r . 0 2 2 7 1 8 28 3 2 7 1 8 28 4 5 1 48 4 1 2 7 1 8 28 By r efer ence to th e r igh t-han d table it will be seen tha t h e c om mon logari thm of l is 0 th atof 1 0 is l an d of 1 0 0 is t . wha ti s .

te r i stic . ly e m lo ed i n a diffe r e nt se n se . i t follo ws th a t th e loga r ith ms of a n y i a e n u mbe rs . in li ke ma n n er .10 a u nm nx ra a r RE A TISE ' T 2 . a n d of 1 0 i s I . Nos . h e i nd ex or cha r ac . bu t t h e loga ri t h m s of all th e in ter me d ia te n u mber s a r e c omp oun d e d of a n i n tege r an d a d eci mal fr ac tion . i s 0 . I n th e for egoing Table . . w e s hall h er e o n l us e th p y y e la tte r . bu t less tha n 1 0 . Logs . sinc e th e lo gar i th m o f l . fraction al v al ues of th e logar ith ms of th e i n te r me dia te n u m be r s . th e r eason of th is w ill be v e r y e vid e n t. wh o se v al u e lie s b etw ee n 0 a nd 1 . &c . logar ith m of 20 is equal to th e loga ri thm of 2 with thatof 1 0 . i f w e co mpar e th e loga ri thm of 2 w i th tha tof 20 we sh all fin d t . sinc e th e l oga r i thm o f 1 0 0 is 2. th e m a n tissa or d eci mal p or tio n be in g i dent e r is t ical in bo th . we obt a in t h e folld wi ug series : N as Log s . m u s t ha v e a valu e be twee n 1 an d 2 I n ter pola tin g . ha t th ey only d iffer in the c h a r ac t ic . a s fr o m 1 to 1 0 . the r efo r e . a n d i n l ike m a n n er tha t. 1 10 0m m 11 10 1 0 413 9 10 12 10 W 7 91" 2 0 4771 2 11-1 1394 3 10 13 10 4 10 I4 10 5 10 OM 15 10 “ 7 60 9 0 7 8 1 150 41 2 6 10 7 15 16 10 0 3 451 9 M 3045 7 10 17 10 8 10 “ 9 0309 18 = 10 1 9 5527 0 9 5424 1 9 7875 9 10 19 10 PM 10 10 20 10 &c . w hich p r ec e d es i t i s ca lle d t . th e se . . h ow e ver . . fo r th e p u rpo s es of ge ne ral c o m p u ta tio n th a t w e sh oul d kn o w the logar i th ms o f a ll th e i n te r m ed ia te n u mbe rs i nclud ed be twe en th e se . h e lo ga r ith m of 1 0 i s an in tegr al number . he n umber wh ich i tr e r ese n t s by 1 0 . g r eat te r m e d ia t er t h a n 1 . its a d d itio n on ly a ffec ts the value of th e ch a r ac te ri st ic In . A ll n u mber s w h ich a re w e rs o f h ave in 10 n ec e ssar ily p o . th e i n t ege r . i tis r equisit e . Th e d ec im al p o r tio n is ter m e d th e ma n tissa and . add e d t o i t a n d as t . fro m 1 0 to 1 0 0 . tege rs for th eir logar i th ms. fac t th e ad d i tio n of l to the ch ar ac te r istic is m u ltiplying t . as h o we v e r th e for m e r of th ese te r ms is fr equent . th e l o ga r ith m o f a ny i n te r me d iate n u mber be twee n 1 0 an d 1 0 0 . if we con si d e r t ha t 20 is 2 mul tiplied by 1 0 a n d th e refor e thatthe . m u s t be so me fraction . an d so on N o w . a ddin 2 p g .

. (be &c . 1 00. 4. o f th e logar ith ms of a ll n u m bers E qu al t o . &c . . 100 2 . t 10. Th e m an tissa or th e d ec im al p o r tio n of th e logar ith m . 6 7 8 54 an d successi vely d iv id e i tby 1 0 exa mini n g th e cha n ge . th u s in th e a bo v e exa m pl e w he n th e n u m be r is 20 th e . w h en the n um . a n d w hen 20 0 0 iti s 3 e r is t . i t i s o n ly the ch a r a cter istic wh ich ch an ges i ts v al u e w ith a c h an ge in th e p o sitio n of th e d eci m al p oi n t . m a ls o r i n tege rs . 0f2 x 100 20 0 is 0 3 0 1 0 3 2 2 30 1 0 3 . an d ' to th e er ist ch a rac t ' so o n . 6 7 8 54 48 3 1 57 6 38 3 1 57 6 6 7 8 54 28 3 1 57 6 6 7 8 54 18 3 1 57 6 6 7 8 54 08 3 1 57 6 ' 6 7 8 54 1 0 6 7 8 54 2 ' 0 0 6 7 8 54 ' . Logs . . . Of2 x 10 2 0 is 0 3 0 1 0 3 l 1 3 0 10 3 . e v a u e o e c a e r s c s d i minish e d by un i ty W e see fur ther tha t. is . Of2 x 1 000 20 0 0 is 0 3 0 1 0 3 3 3 30 1 0 3 . is m ultiplyi ng the n u mber by 1 0 0 . 3 We h ere pe rc e iv e . a l w ays t h e sa me wi th th e sam e figure s w h e th er th ey ar e d ec i . th er e fo r e . 1 0 00 0. ri t hm is n eg a ti ve . whe n th e fir st figur e afte r th e d ecimal . fig u r e s r e mai n ing una ltered n o ch an ge tak es p lace in th e . wh e n 20 0 it is 2 . m a n tissa o f the logari th m . be r is wh olly a d eci m al frac tion the cha ra c ter is tic of i ts le gs . 1 000 3. T he v al u e o f th e ch a r ac te ris tic of th e loga r ith m of a n u mbe r i s a lways o n e less th a n th e n u m be r of i n tegers i n th at n u m b e r . as w e h av e alr ea d y s tate d th at. 11 ic . 1 000 . T he ch ar ac ter istic . By way fur ther illustra tion w e w ill t of a ke th e n um ber . butthat as th e n umber i s succes s i v ely d i v id ed b 1 0 th l f th h r a ct i ti i y . butless th an 1 0 . on L O G A R I T HM S . th e . . c h a r ac t ic is 1 . or gr ea t er han 1 . Th us th e loga r i th m O f 2 is 0 3 0 1 0 3 . is l . . . th u s p r o d uce d in th e v alue of th e co rr espo n d ing logar ith ms : Nos .

th e c h a rac t e r i sti c o f th e lo ga r it h m o f a d e c i mal fr action is a n egati v e n umber . th us 3 8 3 1 57 6 The n ega ti ve sign is plac e d a bove. so th a t h t e fi r st s i g n i fi c a n tfi u r e is t g h e seco n d d e c i m a l fig ur e . &c . a nd Demons t i on rat of he ir t P r op er ties . y e e g g r u le s i t is usual t . as we h av e abo ve. ' 01 2. In t h e followi ng C hapt er t h e expressions. . 3 . 3. w h en a n oug htis in t e r pose d aft er t h e d e c i m al p o in t. t e h t a t i t is o n ly t h e cha r a cter i stic a n d n ot th e m antissa that is n e ga ti ve . Thus. the cha rac t er istic of th e log a rith m of 1 ' is 1 . greate r by u nity tha n th e n u mber of n ou h t f ll w i th d i m l i t I t d of g s o o n g e ec a p o n . d educe d . th e p All the numerals are significan t figures. o o mi t t h e m altogeth e r in th e tables of logar it h ms an d only to giv e th e man tissa o r d ecima l p or tio n. those notfamiliar wit h math emat i cal investigati on . n s e a w r iting . 0001 4 . . i n st e a d of be o r e t f h e c h arac te i r ist o d e no t c. M od e of ca lcu la t ing L og a r ithms. w i th t h e n e g a ti v e s i g n a bo ve i t. his was th e r eason of tha t nu mber bein g p roposed by Briggs for th e bas e of the common syst em of lo arit g h m s . wi t h the exception of the cypher. I t i s only logari th ms h avin g 1 0 for th eir bas e w h i ch p os sess t his i mp or tan tp rop er ty. o f h a vi n g th e sa m e m a n i ts sa for th e sa me figur es an d t . and g e ne r all y . th e ch arac te ri s ti c is p lac e d t o th e le ft o f t h e m an ti s sa . 00 1 3. Since th e charact er isti c of th e logari t h m of any n u mber d oes notd ep en d upon th e value o f th e figu r es composin g th at n u mbe r an d i s so easily foun d b at t nt ion to th for e oin . or formula) em p loyed for t h e c al c ula ti on of lo ar it g h ms ar e m at h e m atically . o . th e c har ac teri stic i s 2 w ith two noughts i t i s .12 r nm r rsn ‘ RUDIM E RIA BY * poi nt is a sign i fi c a nt figur e th e c h arac t e r is tic o f i ts 1 0 9 r ith m is 1 . C H A PTE R IV . an d d emonst r at i on s ar e giv en of all th e p ropert ies of logarith ms r efe rr ed to in any other por tio n of the work By . &c .

p to be e qu al t o th e nu mbe r . rithms of a seri es of n umbe r s. 1 The Power o f a n umber o r quanti ty is th e productar is . a n d t h at . . SC HOL I UM S uch a seri e s i s called a G eometri ca l Ser ies . is t . t o a ny o t h er ba se as s. 7 A se r ies of n umbe rs is i n A r i t . he n um ber of times . by th e co n t i n ua l multiplicat i on of which by i t self t hatpotte r . as t o t he r er u n d erst a n d in g of t h e remai n d er of t h e w or k . BO BO LI UM S uch a serie s is called an A ri th met . 6 A Syst . 2 The Ro o t o r ba se of a po we r is the nu mbe r o r quantity. 8 The Common Difier ence is t . o n L O G A R THM S ‘ I . i c” S ofi e} . W en t. p e t o ut i t. em o f L og a r i h t ms . 1 0 The Commo n Ra t . t h ey w ould become muc mo re fir mly fixe d i n th e me mo ry an d t h eir p rac t . is t he constan t numbe r . base b. is writ ten logs rt . h e ind exor ex on en to f t h e p o wer t o wh ich t h at base m ustbe in volve d . a m e re en un cia t ion of th e s everal propositions with outany p r oo f . . 4 The E ap mw nt o r in d ex of a r oo t i s t . he cont inual a d d itio n of w hich . i ng fr om th e multiplica t i on of that number . . t ha t i t m us t be e m l p y o e d a s a facto r . . is p r o d uce d . io . by the . an d it was consi d e re d t h at a r igi d d e mon stra tio n would be far mo re sa tisfac to ry to th ose by whom it could be followed tha n . DE FINITION S . 1 5 A L og a r i thm of a n u mber t . a: in like manner . as a to any . t a ken t o t h e sa me base . t imes th atth e r oo tof th a tpo wer enter s i nt o it . 9 A ser i e s of n u mbe rs is in G eometr i ca l Pr og r essi o n whe n . and any on e o f t h e n um be rs co m posin g i t. . a t “ en » . when eac h n u mbe r is de rived fro m th at Wh ich p reced es it. ch nu mber is d e riv e d fr o m t eit hatwhich pr eced es it. to p r od uc e a g i ve n P0 w er . he c onstan tn umber. . multip lica tio n by a con st a n tn u m ber . e . . by the ad d it io n of a co nstan tnumber . by t . i s t h e c ollec tio n o f th e logs . . is e xpresse d by lo g. hmetica l Pr ogressi on. t h e sub p p o ld h ve r ha ve be en h ar dly c om let w i h je c t w o u o w . . a s a fac t or . by the . Sc a o u mir Th e loga rith m of any n u mber . th e r e aso n o f th e se v er al p r o o si tion s an d r ules b ei ng u n d e rs t oo d . o r i nd ex of a p ower is t he nu mber of . 13 s hi p m msy be o mit i t is in nowise n ecessar y t ed . o any base. any n umber of tim e s by i t self . i cal applicat ion and ad a p t io n to pec ulia r cases r en d er ed far mo re easy ' at ‘ . 3 The. an a rithme tical ser ies i s fo rmed . the logar i thm of t he sam e nu mber .

the v alues of the coofi c ie n t ly i n d epen d e n to f th e v alue o f e a r e p e r fec t th e re fo re . is the integral n umbe r t o th e le fto f th e d eci mal po in t . tha tis. L et ther efore . . h m . . th e . A = a . t ion equat h en becomes t A a . wh ich the n bec om es B a: fi s ” + d z + + &c . THE ORE M . 1 3 A S ign ifica n tFt . . st royin g t h e equa tion . we may assu me a: t o hav e a n y v alue we please wit h o ut d e . 1 2 The M a n t . &c . h t e cyp her signifyin g n o act ual qua n t ity but bein g e mployed . l ) an d : A ( L A A 2 loo ked upon as th e co e ffici ents of a an d 2 i v ely "2 " ar e r espec t . B = b. a: he O. 1 1 The C ha r a cter istic o f a L ogar i t . C = c . . he i ci en tof a ny p o wer of x on o ne sid e of t h i o n . Beca use . o nly t o d eter min e t h e place of th e o th e r figur es . Pa or osu ' rou A . is t fl co e e equa t equa l t o t he coeflicie n tof the like p o wer of a: on the o ther sid e . a g eo me t r ical serie s is fo r med . m o re i m media t e ly u n d e r c o n si de ra tio n i s m ultiplied . ter m coefii cie nt is e mploye d i n a so me wha tex t e n d ed si n i fi g ca t io n. a n o m a uu a v r mu n sn ' 14 ' i nual con t multiplication by which . i n the abo ve ex pr ession . to mea n any qua n ti ty or expression ( ho we v e r com p li os ted ) yb w h ic h the q ua nti t y . Now . . sin c e A an d a ar e equal . g u r e i s e v e r fi y g u r e buta cyph e r .5 — 1 ? - 3 — 2 3 2 z + z . w e may r e mov e h em from t he t ori ginal equation . to th e r i ghto f th e d ec imal p oi nt . Thus in th e expr e ssio n. Sc a o u v u In i nv estiga tions si milar to th e followin g. In io n o a n equa t f he fo r m t + &c . g ) av a w ) . issa of a L oga r ithm i s the d ecimal n u mber . he quant t i tie s L A ( L A . .

.

o obt a in t he cob H he values of t i ciefité B. ‘ &c in terms o f A le tus pu t2: fo r A A . [ ] I Now in bi d er t . . 3 1 1 11 l § f § y + §z y I y ‘ l . we ha ve 1 + 3. ‘ A BA 3 11 A 2 ( 24 24 24 24 3’ A r ranging this last exp ression acco r d in t h t e p owers If g o A. we have. itbecomes g T ' b J . ' nofirnnm xar ‘ 10 raslm ' ‘ ’ sn A 2 A ' A ” i ll } . E xtr acting he t rooton bor nsid es. . . 6 4 t b — i die . G . BA2 CA 3 &c . b =( l an d rai sin h em to th e po we r t of r . r y + m 8w . then th e a bo ve exp r essid n becomes " 6 l z .e lg l + ' o o o &O .

.

A A A I 2 2 2 A A A I A 3. then expan ding n e r ms in t ’ p . . re + A1 2 . we ( P rop . % l 2 " 3 -- A . 3 A‘ + &c ’ 9 n 1 w + 0 2 in which ( P rop . er ms in t of b and n . B. A . 3 3 A A . l 1 A I : A lso a nd n i b“ i m of A t h av e ex p g n t e r s e. F r om the equa tion 2 3 A A + &c 2 3 . in th e ion n eqwat both si des a re rai sed t o the ” o we r of a i tbeco mes b . we ’ Ir b and obtai ned above . If.~ 18 s no w m a n! TBM ‘' IIBB 1 y h t e refor e : 1 o b l. h a ve A : 1 1 (b 1) a l) 4 &c . i tbeco mes “ &c . PRO BLE M . B. 2 3 If no w in th e e qua tio n 6 itute he values of ’4 subst t ’ n . 3 PRO POS ITI ON C . From which we h ave. we Obt ain ( P r op B. t er mi ne t o d et he va lue o f A . of x . by P rop . 2 3 A bl r =1 + A Aw + A 2 a 2 + ” A w + " &c . and if we it su bst ut e his t va u l e y w e .

. 1 abo ve ] h m becomes for th e logari t A= ( n g o . S ubstitu tin g th e assumed value of he above expres A . a l o ga r i th m page 4 th a tA is th e lo ga r ith m o f th e numbe r n . a s s ig n ed t o it . ed t h is base are o t ter me d N ap i er ea n Log a r it hms . y . i9 F r o m each of which we obt a in A 1 1 . This is th e v alue for A actually taken by Baron N a pier . i tfollow s from t he d efin i tion o f . 2 3 A I 1 (b — 1) . &c . er mi ne t o det he va lu e o b when A is ma d e equa l to u ni t t f . i tbeco mes See page 8 . an d e m ployed by h im in his firs tTa bl e of L oga r it h ms. an d as e ve r y d i ffe r ent valu e of 6 . PRO PO S I TI O N D . in t si on . . n . " S C HO L I O M Sin ce I) . W e m ay therefor e assu me such a val ue for b as shall give 1) — l 3 in A b ) l. to th e ba se 6 N ow as 6 may h av e any va lue tha tw e please . I n the equa tio n 2 3 ‘ A A A A 4 A + &c . g iv es a d i f fe r e n tvalue of A. e a ve . &c . 1 l ( n 1) — ( n - ( n l ) — &c ” . because a ny n u m b er o f v al u e s m ay be g i ve n to th e base b * . ON L o e aa tr n x s. PRO BLE M . h e expr ession w h ic h case t [ . 1) 2 — l 3 ) . A S u b stitut in g for A l i ts val ue g iven above . i tfollo ws th a t th ere may be a ny n um be r of l o g a r ith ms co rr espon d i ng w ith th e n um ber n . and for A i ts v alu e d ete r mi n e d i n P r o B w h p . fr o m w hi ch c i rc umst a nc e loga r ith ms c alculat .

. . 3 33 6 é: 889 ‘ v ” 7 ' O O O 1 98 . h ir tee n te rms t . we h av e A = log. IPBO PO SI TIO N E . 4 4 ' 0 4l 66 7 52 . Th e followi ng is th e calc ulatio n fo r t h e fir s t. ber i s us ually d eno te d by s a nd is as stated i n the sc holiu m' ‘ . n = — l ) — l )’ % ( n l) 3 he t . . es . wha tever he t he val u e o f i tisr t rue when A l i n which case i tbec om ’ ’ ‘ " ‘ A. to t h e p r eced ing prop osi tion . as t his expr ession is true . ( - . we m a y ’ ‘ ‘ ‘ ' - ‘ eas ily d e ter mi ne the v dlue o f b. the base of t . a s i ti sT apidly co nvergen t . 8 87 9 32 10 57 3 ll 0 52 12 088 I n w hich th e first n ine d eci mals a re c or rec t Th is mi m ' . 1 1 1 + &c " Fro m w hi ch expressfon . N ow . PRO BL E M . . m ‘ mr Isn ' go a Ben i m RY rré ax { A 1 l I b _ _ — 1 + __ As 3 c A + + 4 + A 2 573 2 3 4 . Se n o u mr . To o bta in a r ap idly converg ent ser ies. for ca l i ng the log a r i thms of n umber s cu la t . ' 1 4 00 0 a } a: 0 00 h « ! 3 2 6 67 s. h e Nap ierean sys te rn of logari t h ms . to da ny r eqd ire d d e gr ee o f ' ’ exac t n ess . i In the expression Pr op G « . 4 1 2 6 98 . if its p ut A f or i t t s e quita lentvalue i n the d enomin ator . .

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u = 1) — ( n l) 2 — ( n — l ) ‘ + 650 . d e pe n d e n t of n ( Pro p B an d w e h av e fur t . 1 1 1 log. bec a use i tcan be ex pressed in ter ms o f n al o n e an d h as un ity for i t . py e v e o u us o syst e m ha v i ng t he re qui r ed bas e See page 9 . h e r s h o wn P ro p D ) t h h h e base o f t h m f l i h m s is ( . by w h ich th ey be com e r e d uc e d to n atu r al . 6 A . h as bee n ca lled th e n a tu r a l system of lo g ar ith ms. To r ed uce th e lo g a r i thms of a system h a v i n g o ne ' ba se. N ow . t h valu e o f A i s r e d u ced q o e to uni t y . s c o e fli c ien t . an d t o r ed uc e logarithms ha vi ng on e base t o t hose h av in g a d iffe re n t on e . o r N apierean lo h ar i t m s an d t h en to m ult i l th m b th m d l f h t e g . We h a ve . Th e a n tcoe ffici en t co n st 1 is c alle d he modulus t A of he t t system o wh ic h i t belo n gs .22 BUD IM BNTA BT TRE A TIS E PRO PO S I TI O N F . . th er efor e . it is only n ece ssa ry to d i v ide them by th e m o d ul us of th ei r o wn system. w e h a ve log. — 1 — ) é ( b Ir fr om wh ich . in P rop E [ l ] . . by compa rison wi th [ 1 ] above we a n e xp r e ssion . we have ( Prob B. i mm ed iat ely p erce i v e t h atlo g. i n t hi s cas e th er e for e . . PRO BL E M . e qual t o th e N a pierean loga h m of r it b . J This system. l th e d e no mi nator of th e a nt co e ffic i en t con s t for em any syst of h ms t loga ri t o th e base 6 is . w en t e syst e o o ga r t ta ke n e u al t 2 7 1 8 28 1 82 8 e. a t . . or th a t th e v alue of A .. u = X ( { n 1) — l ' ) § ( n — 1 ) I n wh ich ex p r ession th e coe fficien t is an t fo r const e very K lo gari th m h av in g b fo r i ts base i t s v alue be in g e nt i r ely i a . . to those ha vi ng a d ifler entbase . h a v in g s for i ts bas e .

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multip lied by the lo g a ri th m of p to th e ba se q. Now w e ha v e . That is if in b n . an a r ithmetica l p r og ression . log. th en will n 1 . b log. ‘ ” Butq p . he F ] of t common syst em o f lo ari th m g s . t q. lo g q 10 1 y p ‘ n . : ll n n b b" n th e 6 " l‘ . uni t y . t 2 3 0 258 50 92 This is th er efor e t he mod ulus h e val ue of t [ P r op . geom e tr ica l p g r o r essi on . a . . wi th th e fo rm ula [ 2] in th e same . . lb fo r m . THE O RE X . I being the bas e of th e N api er ean syste m of lo gar i th ms . t h en 0 . l2. C omp ar in g th e ex p ressio n fo r the v alu e o f A . ther efor e lo g.The log a r i thm of a ny number q to t he base p . PRO BL E M . 25 PRO PO S ITI O N H . an d logq p h en h. . t t h a ti s. O N Lo s xa rr rn rs . Ther efor e log. p mustequal . the corr esp o nding n umber s will for m a ser i es i n g eo metr ica l p r og r essi on . I . n a. THE O RE M If a ser ies of log a r i thms to t . we h av e l . . values of t h e exp on en ts of b ar e such th at l1 . To deter mi ne the va lue of lo g . an d q " = 11 . PRO PosITI O N I . 11 b raised t he pow er o t of k. we see th at A is t h e N ap ier ean logar i th m of b or 5A =15 . gi ven in th e Scholiu m to P r o positio n 0 . 1. ‘ If th e firs tof th ese. is a lwa ys equal o u nit . Le tlogp q l. q log. he sa me base a r e in a r ithmeti ca l p r og r essi on . an d l k log. fro m Theo r em H . n2 . for m a . g e. a . . . For . n . h t p q . . PRO PO SI TI O N K . let 3 be t h e common d ifer ence of th e arith metical series. 6 A . h er efor e p t p .

I tshould be observed h at. sion . h t erefor e ( Def in g eom e t r ic a l p g ro re s . t 3 h ey are . g e r . n. . PRO BLE M To ded uc e a n exp r essio n for the li mit of the . &a .. t he co r r esp ond ing na tur a l number . na n n l .26 BUDII BM A BY m an he t . i tbecomes 1 1 l l 1 1 — 5 2 933 4 e . ) . h en . is . a qual t o t h e p r eced ing ter m mu ltip lied by th e con stant qua n ti t y n . n. duo . . I I f. &c . ain we obt . PRO Po srrI O N L . SO HO LI UM . t sin ce b 3. is th e lo d if er ence of t gari h t m o f the common r a tio of th e seri es of number s . . 3 ’ l' L et b n . s . &c . . &c . Or . a . ns. . l” l’ b b &c h e ir t equa ls n n 1 . = na n . n bein h t i commo n r a t io . . he t commo n h e series of logar ith ms. in th e exp r essi on P rop E w e put 1 for . n . t it subst ing ut he in t above for b . i ncr ement of a log a r it hm. p r od uced by a ny g i ven i ncr ease in . we that each ter m of t see he ser ies n . b .

i s l e ss th an th e m od ulus of h t e syste m divi d e d by th e lesser of t hose n u mber s . mod ulus i . ' han t t t 6 he re fo re t 1 l 1 lo g (3 + 1 ) log ( b 1) 10 g b - Th a ti s . Boa o mm r 1 In t he common system of logarith ms. as t e of increase of their logarith ms de crease . and ' . th e d ifier ence be t h ms of two num ’ ween t h e logarit ber s diffe r in g by u nity. th e r a t . 2 1 i s g r ea ter t he su m of all he succeed in g er ma. log b < — g log ( b 2) z —— Ei log ( b 3) from which we see t h at he n u mbers increas e. 0 N ro o m s . 27 equal th e i ncre mentoccasioned in t hm h e logar it of b. h e m od ulus of th e syste m. o 2 . hms of several consecut i v e nu mbers. t h us. by ia s val u e by un i t cr e as i n g i t y . the addi tion of unit y t o 6 in cr eases i t hm by s logar it whi le t he ad d it ion of m unit o b t 1 i nc reases i t hm s logar it only y b 1 howeve r b is a large number b and b . equal an d t . has been sh own [ P p i r o osi t on I] o be t equal t o 48 42944 8 2 ' in thi s case . each gr eater by u nit y tha n th e p r e cedi ng p ut t . h erefore the r a t e of i nc r ease of t he logari t hms may be consid ered “ propor tional to that of the correspond . h er efore . 1 1 I n th e e xp re ssi on —— he he firstterm. we have in g m for t ’ log ( b + 1 ) . t 56 5 . th e . we have t 43 429448 2 ' log ( 6 1) log 6 b SC R O LI UM 2 In th e case of the logarit . l are v er y n ear ly .

o t or . THE O RE M The sum of th e log a r ith ms . as compar ed wi t h th e n umbe r i tself . and ‘ b =n Now . L etA an d 1 d enote th e same as i n th e for egoin g proposi i on t Then . is . PRO PO SITIO N O . h m of the quo t ient of two number s is equa l to the log a r ithm of the d ivid end . h en b t l =m . A n d because th er efor e A l i s th e logarithm of m n. THE O RE M The log a r it . so long as th e incrementof the lat ter is sma l l . equa l t o t he loga r ithm of tha t number ." m b A __r n A nd because A— 3 m b n h erefor e A — l is t t h e logari th m of t h e base b . 28 - BUDII E NTA BY TRE ATIs: ing number s. t o th e base b . ’ THE O RE M The lo g a r ithm of a ny p ower of a number . PRO PosITIO N M . is equa l t o the hm log a ri t of m. L etA = log. t is he log a r ithm of their p r od uc t t . n. wi th the loga r i thm of n subtr a cted fr om i t . f o wo number s. an d l = log. wi t h t he log a r i thm of the d ivisor subt ed fr om i t r ac t . he t loga r ithm of the quot ien t of m d ivid ed by n . mult ‘ ip lied by the se p orrentof the p o wer . . o r . PRO PosITIO N N . m. the su m of the lo g a r i thms of m a nd n is the loga r ithm of thei r p r o d uct .

o n LO G A BITBHB . I n like m anne r . t n q A. . m. ” o r . t h e squar e r oo tof m = a2. d ivid ed by t of th e r o ot . an d l g . an d h e loga t . L e tx = Jog. an d h e loga t hm of z r it 9 th e n . n . 99 A n d b ecause h e r e fo r e t n ) is hm o th e loga ri t f m t o th e bas e 6 . TH E OR E M The log a r i thm of a ny r oo t of a nu mber i s . a nd q or . 8p t A. t A. if t he cube r oo to f m y. lett and th e logari thm of m= 1. . h root of nt m z. th e n m =y o y o y = b” = x h er efor e. h m of y r it p . " h en m b . h t e lo g a r i th m of he t nth r oo t of m . equ a l t o t he eap o n en t h e log a r i thm of tha tnu mber . PROPOSITION P . a nd p 3 And gen erally if th e . i s equ a l t o n ti mes t h e l og o r ith m of m . A h e r efor e . th e n A h e refor e . ‘ hm cy r it m divide d by . he loga r ithm t of he t h r o ot o nt f m is. equa l he o t t tg o a .

. be th e sa me as befor e i ts valu e w as alt e r ed A n d th e refo re . I n the syst hm whose ba se is 1 0. THE ORE M . t h en i t s value m ust be less than 1 0 . t h e mantissa of th e two logar ith ms wer e or iginally th e same . o f th e logar ith m o f the . p o w er of 1 0 . let th e n umber h e succ essively mul ti plied by 1 0. ce ssi vely add ed t o th e ch arac ter is tic of i t hm s logar i t . o the nu mbe r . h av e th e logarith m of th at po wer of 1 0 a d d ed to o r sub tr ac t ed fro m i t i n or d er t . now th e logar ith m of 1 0 is here t for e the logarith m of th e n u m ber m ust be less than th atof 1 0 an d th er e for e i t . I tt h e same t i m e tha ti twill add 1 to th e charact er istic of i ts logari th m . ‘ fig u r es i n h t a tn u m b er . t he two nu mbe rs may be mad e equal by alter in g th e po si tio n of ' t he point in one of th em whic h will in eflec t be m ul tip ly . o r some i ntegral p o wer of 1 0 m ust . s ch arac t ic must be 0 ( followed by e ri st so me d eci mal) an d in t . obvi ousl t h t f th t w d as th n n ti y e exp on en o a p o e r a n e. o be st ill th e corr ec tlo gar i th m of t h atn u mbe r .e x po e s i n tegral . h e place of th e d eci mal p o i n t. Th e a d . an d th ere for e as originally th e ch aract er» . THE ORE M . an d i twill th e n beco me equal to th e log ar i th m of t h e ot he r n um be r .80 s no w ma n ! n u n s: PRor osm on Q . i s a lwa ys o n e le ss tha n the n umber of i v eg r al . Now . the lo gar ith m of e v ery in t e ra l p ower o f 1 0 m ust g i tse lf be an in teger with n o d ec imals or man tissa . The logari th m of th e n u mbe r wh ich h as th us been m ulti t ip lied or d ivi d ed by 1 0 . Th en . If t h e figur es composi n g th e two numbe ' or d e r an d on ly d ifler in t . . th e logarithm of any powe r of 1 0 to th e base 1 0 is . PROPO SITION R . b y wh ich th e n u m be r h as b ee n m u ltip l ied or d ivid ed will n ot affect i ts m a ntissa w h ich will con se qu en tly . . . . i ng or di vi din g by 1 0 . an d at t h e sa me ti me let th e lo ga ri th m of 1 0 or 1 be such . wh et her eg er s or d ecima ls tho se figu r es a r e i n t . h is case is on e less th an th e n umber of i n te gers i n th e n u mb e r . The cha r a c ter istic of the log a r i thm of a number to t he ba se 1 0 . for eve ry place t hat th e d ec i m al is mo ve d to th e r ig htor to t h e left . em of lo g a ri t the ma ntissa is the sa me for the sa me or d er of fig ur es . di tio n or subtr ac ti on th er efor e. No w each multiplicatio n by 1 0 will ad d an in teger t . L etth e nu mber consistof only on e i n t eger .

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th e n u mbers i n t he left-h an d mar gin an d a t the h ea d o f th e table a re th e a r g u men t by wh ich we ar e d i r ect .e ca lc ula te d to a g r e a te r o r le ss n u m be r of d ec imal plac es accor d ing to th e p ur poses t . a e of t p g h e sam e .82 BUDIM E NTA B! mu m i n eith er of th e oth er series may be immediately fo u n d . said t o ent er t h a t c ol u mn wi th th e a rgu me n t Fo r ex a m p l e . if we ar e looking in th e table a tpage 3 3 for th e logar i th m o f 2 56 5 we ent . S i n es an d tan gen ts an d a gr eatvari ety of oth er tables wh ich . w i ll be foun d of fr equen t use W e hav e gi ven o n th e o ppo . h c in e s .h e sa me n u mber of plac es al so th e n a tur al . Th e best table s of th e L ogar ith ms of N um be rs to se ve n . h o se by B a b b a g e a l th ou gh f or g e n e r. at t he to p) wi th th e a r g umen t 256 5 a n d on th e same li n e in . si t a e as a s eci me n of t h t b l ti f e p g p e se a e s a p o r on o o n e . r esul t a nt s Wh en w e th us seek in any column O f a ta ble fo r . to t . the co n ti gu ous colu mn w e fin d th e r esu lt a n t40 90 8 7 4 wh ic h . on i nspect i on of th e t a bles The n um ber gi v e n i s ter m ed th e . . .a l u se w e sh ould r eco mm e n d H u t on s which con t ai n lo a r i t m i S ’ t g. e n er ally e m loye d t bl s t ven si x an d five l g p a e o se . Th us i n th e table a specim en of which is gi ven a tp ag e 3 3 . m e tr ical calc ula tio ns w her e consid erable acc ur acy is r equir ed . ta n gen t s &c . the ar gu m en t by wh ich t o fin d som e ot h er n umbe r w e a r e . . e d wh e r e t o fin d th e logar ith ms of those n umbe rs whic h loga ri th ms ar e th e . o whic h t h ey a re i nt en d e d t o be a li d F r A st ro n o m ical an d Tr i o n o g pp e o . . er t he co lum n of nu mbe rs ( d istin guish e d by N . tables ar e used in w hich th e lo gar ith ms ar e car r ie d to se v e n l ac es o f d e ci ma ls . . p a ce. a r g u men to f t he tables and th e n u m ber so ugh tthe r esu lta n t . p l ac e s ar e t . . s . Ta ble s of the L ogar ith ms of N u mbe rs ex ist u nd e r a g r e a t ety o f f or m s an d ar. fo r o r d in a r t b l f i l p yp p u r o s e s a es o S x p a c e s w ill be foun d a mple a n d e ven i n m any cases five l ac e s w il l p be sufficien t We Sh all d esc ri be so me of th e best an d m ost . is th e loga ri th m r equir e d .

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th e fir s tth r ee figu r es o r th e l st . . . or th e loga r i t h m it s elf w ill be foun d a tt . se r ve h o w e ve r . as e asily s u li e d fr o m t h e firs t column . 8 p lac es I . an d in the . u po n loo king a tt . s. h e case th a tth e in itial figu res foun d in th e . V ario us m eth o d s h av e bee n a d o p ted fo r d ir ec t in g atten t ion to ial fig ur e s . he i n itial figur es or th e 8 td d ec imal beco mes al ter e d in value . i n H u tt a bles i t i s ’ th is ch ange i n th e init on s t sh ow n by a lin e bein g d r a wn o v e r t h e firs tfigu r e of e a ch of th e logar ith ms to which th e alter e d in itial figu r es ar e t o be ' B i ’ ap lied . d eci mals ar e the sa me as tho se of th e loga . . an d bein g t h e sa m e i t i s co nsid er e d un n ec essar y to r ep ea t t h em a s th ey may be . n at u ral n umbers a r e gi ve n t h e last figu r e m ustbe so ugh t fo r . h e e x p l a n a t i o n of th is i s as follo ws th e fou r figu r es g iv e n in the se co lu mn s ar e . fir stcolu mn a r e th e cor r ec t i n i tial figur es fo r all th e o th er loga r ith ms in the sa m e line . a n d so con t inue u n til th e n in th li n e. We oh . as a .s i n o r d er to al lo w of this sav in g o f spac e . h o weve r . because as th e loga r i th ms s ue : c essi vely inc r ease . aft e r a c er t a in in t e rv al th e last o f t . ch a n ge . c olumn o f r es ult s h a vin fi an t ( g 0 a t t h e top) c o n ta i n s 7 g u r es . h e i nter secti o n of t h e two lin es i n wh ich th e two po r tio ns of th e arg u m e n t we r e fo un d tha t i s o n th e same line wi th t . h as been ad o pt ed I t i s not p . an d 3 rd . rit h ms i n th e fir st co lum n of r esul tant o s . d ecimals. alon g t h e top of th e ta ble . a nd t h is alteration is equ al ly li kely to occ u r in any o n e o f th e c o lu mns A n ex a mp le of this occ urs a tthe th ir d lin e o f th e . column h ea d e d wi t h 7 th ey h e r e . . n t h e ex tr e m e le ft-han d colu mn h ead ed N . t a ble . a nd t ” u nd er t h e r es ult ant . i n whic h th ey ch an ge to 40 8 in th e c olum n h ead e d wi th 6 . by wh ic h a p or t ion o f th e a r gum en t i s foun d in th e Sid e column an d a or ti on a tt h e h ea d o f t h e t able . o r log r i th ms ans we r in g t o t h em fr o m 40 0 0 0 0 0 0 to 5 0 33 4 1 97 3 . i n t h e lin e o f figu res i mmed ia te ly “ h e wo rd s L oga rith m of N u mbers . 5t h 6 th a n d 7 th . an d con si d e r able pp sa v i n g o f S p ac e i s e ffec t e d by t h e ir om ission I t i. o n ly t h e fo u r final figur es o f th e lo gar ith m tha t is th e 4th . by th e o missi on of th e si m ilar figu re s . th a tth e p eculia r ar ra n ge m en tof th e tables. . wh ic h is the c olumn of ar g um en ts only th e firstfour figur es of the . i n w h ich t h e i ni tial figur es 40 6 a pply as far as th e . h e fi r st f o u r figur e s a n d in th e sa me co lumn as th e las tfig ure . th e o th e r n in e co lumn s con t ai n o n l y 4 T . h o w e v e r always t . h e ta ble tha t wh ile th e first . an d i n som e ot h r t b l i n bb a g e s t p e a e s. t . 34 Run n u mn nr m m : The nat u ral n umbers wh ich fo r m t h e a r g uments of the t a ble ex t en d fr om 1 0 0 0 0 t o 1 0 7 999 t h e r es ulta n t . h e fo r me r being gi ven to 5 a nd 6 places w hile th e la tter e xte nd to 7 and . . n ex tcol um n becom e 40 7 . 2md .

85 s how n by th e first cyph er be ing pu t i n smal ler t yp e . t h e logari th ms cor respond in g with the m i a c rease in a r it h metical p r og r ession so that th e d iffer ence be . Prop L ) th atwi t . le t it be r equ ir e d to find th e lo ga rith m of 2558 46 . fo r each of t he nine unit s. Log of 2558 4 4 4 0 7 96 8 4 1 1 70 a n d as fo r any incremen tless t h an thi s. ' in d ica tes thatth e d ifler ence be tw e e n two su ccessi v e lo ga rith ms ha s ch a ng ed fr o m 1 7 1 to 1 70 i n the lin e in w hich i t s t a nds . I f no w w e d ivi d e th e wh ole di ffere nce 1 7 0 by 1 0 . as 0 33 8 I 0 1 68 ] I th as be en sh o wn . th e d ou ble o f th is o r 8 4 fo r t wo u n it . if th e incr e men to f the n at u ral n u mbe r h ad bee n 0 6 .A gai n . t ' h e c o rr espond in g in c r e men tof t h e logar ith m wo uld h ave been 1 0 2 . h er e th e i ncrem en to f th e n um ber be ing 6 we fo rm th e p ropor tion . w e obt a in 1 7. i t i s i n se r t g e d i n a colum n h ead e d D . n at u r al n umber d o es t o l for ex am ple . T he i ncremen t t h us obt ain ed . to asc e r t ai n th e lo gari th m of any n umbe r be t wee n those giv en a bov e .s th e tr eble o r 5 1 fo r th r ee un its a n d so . th e i n cre m e n t of th e logar ith m to be a d d e d to 4 4 0 7 96 8 4 will b ear th e same p ropo r tio n to 1 70 tha t th e inc r eme n to f th e . i v e n in t h is colu mn i s t h atd ue to an i nc r e me n tof one unit g in t he 5th figur e of th e natur al n umbe r th us . whi ch bein g add e d t o 4 4 0 7 96 8 4 g i v es 4 4 0 7 97 8 6 fo r th e logari th m o f 2558 46 . i s . n um be r 1 7 0 is i nsert e d in t h e column D on th e fifth line an d . h small inc re ments in the n at ural n u mbers. on t h e r ig h t of the t a ble on t h e lin e in w h ich th e chan ge occ u rs Thu s the . w e may consid er the lo gar i th ms to va ry in ar ith metical p rog r essio n . The d iffe re nces change much mo r e ra pid ly atth e comme nce m e n tof th e ta ble th an n ear its c onclusio n Th e d iffer e nce . o n ro o m . ' 6 z : 1 7 0 1 0 2. c h an e s . . tween th e successive logar it h ms r e ma in s const a n t fo r s e v eral ‘ l oga ri th ms in succe ssion Wh en eve r th e v alue of th e d ifie r ence . o n . t he d iffe r en ce cor respon d in g with an incr ease o f on e u nit i n th e Six th fig ur e of th e n atural n u mber . . i n th e conclusion of the p revi o us ch ap ter ( Sc holiu m 2 . a nd each of t h e n u mbers so ob tai n ed wi ll be th e i nc re men tof th e logarith m corr espon d i ng with a n i ncr ease of tha t number of un i t s in t h e six th figur e of th e n atu ral n umbe r . by ' wh ich we find that 1 0 2 is the cor r espon d in g i nc re men tof t he logar ith m.

wer e r e quired w e d er ive a ton ce fro m the t . d ivid e d by 1 0 0 Thus su p pose th e lo gar ith m o f 256 0 8 58 7 . e ve r t. th e M athemat i cal Table s. wh ich is th e six t h figu r e of th e n u mbe r requi r e d . to wh ich w e a d d th e p rop er i ncr e m en t fo r eac h ad d itional figu r e d erive d fr o m the little ta ble i n th e . n ea restn u m ber in t h e ta ble bein g 1 0 2. ing to th e log ar ith m 44 0 7 4249 is th er e fo r e 255524 6 . e x la in ed t h i m t f th l i th m f i i p . The n umbe rs con t ain e d i n th e se li ttle table s are as al r m dy . . fo r mi ng o ne of th e sa me ser ies ” as t h e p r esentwor k . . w h ich co r re sp o n d s wi th th e n a tur al n u mber 25552 th en s ubtrac ting the lo ga r ith m ta ke n fr o m th e t a ble . t h e n um be r 4. loo king in the ta ble w e see tha t th e n e x t less loga r i th m is 4 4 0 74249. Thus if th e lo gar i th m give n we r e 4 4 0 7 43 2 7 . to w h ic h a ddin g a n o u gh t we Ob t a in 1 0 0 a n d t he . a gain stwh ic h we fin d 6 thatis th e seve n th figu r e r equ ir ed . . p w e fin d again s t th e n e x t le ss d iffe re nc e 6 8 . a ble t h e logar i th m of t he firstfive figur es . ter ist ic i s o mitted t he tables co ntain in g o nly th e man tissa of . Th e ch ar a c te r istic must be ad d ed in acco r d a nc e wi t h t he m le gi ve n atp ag e 1 1 . s L og o f 256 0 8 0 0 0 is 7 4 0 8 3 7 57 Incr emen tfor 50 0 85 80 7 Th erefore th e log of 256 0 8 58 7 7 4 0 8 3 8 57 Th ese little tables of proportional par ts a r e of equ al ser v i ce in fin di ng th e n atural n umbers c orr esp ond i n g wi th an y gi ve n lo gar ith m . fro m th e give n loga ri th m we o btain th e d iffere nce 7 8 loo k .36 BUDI M KNTA BY TRE A TI SE i nse r ted i n a n adjoining colu mn (head ed an a bbr e via t i o n of P ropo r t ion al Pa r ts ) . h e i nc re men t s fo r t h e u ni ts i n th e seven th place o f th e nat ur al n u mbe r wh en d iv id ed b 1 0 o r fo r t h e ei g h th w h e n y . w e h a v e g ive n a p age fr o m . . r i ht -ha n d c olumn T h g u . I n th e se an d all th e bestt h m s th e ch ame a bles of lo ga r i t . h o w . e nc r e e n s o e og a r o r a n n c reas e n the six t h figur e o f th e n a tural n u mbe rs th ey ex p re ss . i n g in the se co nd co lum n o f th e t a ble of Pr o o r t i o n al Par ts. w e hav e st ill 1 0 left. W e n ext pass on to d esc r ibe tables o f logar ithm s to si v decimal pla ces A s a Sp ec im en . Th e n u m ber ans wer . on . the lo gari th m .

10 20 7 1 1 0 9 2 562 69 83 ' 10 1 1 1 1 4 . 1 7 6 8 4 8 82 6 51 1 6 6 74 . ar 2 6 40 8 1 48 1 47 6 - 16 4 8 9 4 1 . 3 2 2 6 56 8 - 1 6 1 46 7 3 32 6 97 2 6 72 1 6 2 3 4 2 1 4 58 2 6 89 6 9 6 1 l4 . I 3 S1 1 52 1 8 2 57 7 16 8 100 8 1 1 76 3 25 2 9 1 67 3 34 IOO 2 ° 1 50 - 3 9" 2 60 8 16 6 - 2 6 24 16 5 1 48 ' ' ' e 99 n 55 5 . 1 3 80 2 1 1 0 392 I 20 1 7 97 21 2 3 8 1 5 3995 5 6 0 6 57 8 5 4 7 39 ° 6 75 8 x6 6 5 9 934 3 6 390 93 5 1 1 1 2 7 2 6 97 8 44 5 2 4 6 2 7 9 6 1 99 6 3 74 so 7 940 81 14 I 6 9 74 984 7 1 57 3 3 3 12 1 3 92 2 4 8 4 34 0 5 5 0 s 6 540 6 71 ° 6 84 1 0 7 993 3 0 10 2 8 41 1 62 0 1 7 8 8 9 33 0 0 346 7 60 497 3 51 40 I 6 80 7 2 8 30 1 3 9956 0 12 1 4 42 1 6 0 4 . I SS 7 ' Q 2 50 3 - 2 51 7 17 2 1 37 6 1 54 8 10 26 1 36 8 ' 2 532 17 I I I 97 I S3P ° 2 54. 24 33 10 68 1 2 44 6 1L77 ' 354 10 62 12 39 6 ' I 59 2 5 ° - 24 60 - 17 6 35 ' s 10 5 6 1 2 32 1 58 4 ' 1 4 74 1 40 0 I S7 S 2 4 88 1 56 6 - a 1 397 1 3 8 21 . 8 1 3 5 82 97 97 5 7 99 41 N .

h e re we Obtain the loga r i thm of t h e firstfou r figur es a ton c e from th e body of the t a ble . an d o n th e sa me lin e with the firstfo u r fig ures of the gi ve n n umber an d i n th e sa me . . h e line i n wh ich th e d ifference ch anges i ts va lu e is Sh own but each line co ntains ten logar ith ms an d th e re is . .68 m m m m a ble This t he logar ith ms of ev ery ai ns cont t th an to six p laces of d eci mals. ot tables t . In th ese tables also the p ropor t ional ar e som e wh at d iffe re n tly arranged h er lo gar i th m ic ’ In H utto n s an . i n t he tables n o w be ing d esc ribe d . e sa e li ne will be fou nd t h e propo rt i o nal par ts for each uni tconstitu t i ng th e fift h figure of th e na tu ral n umber Thus let th e loga . r ith m of 24 6 0 57 be r equi r ed . co lumn as t h e fifth fi re o f the sa me we find the p ro r . not hi ng to ind icate be twe e n wh ich o f these logar i th m s the ch an ge occu rs . the n u m ber co rrespo n d in g wit h th e logari th m atwh ich the c han ge takes lace is g iv en in t h l f -ha nd col umn an d o n t h m p e e t . h er than atthe co mme nce men tof a line c ur s ot Th us i n th o . and his i s i nd icate d by th e line thus 98 7 5 t 00 5 1 . The log of 246 00 0 is 58 90 93 5 Incremen tfor 50 88 I ) 7 n h e log of 246 0 57 Therefore t 5 3 91 0 3 5 . midd le of the sixth li ne the i ni tial figures cha nge from 8 8 to 39. ' tional pa r t to be ad d fo r tha tfigure and on the same no . a nd in t h e sa me column as th e sixth figure w e find the p ro . o rtio na l t w hi h h vi n g fir stbeen d ivid ed b t m t b p par .c . and in fo r m an d arra n gementa re v e ry si milar to th ose j The natu ral numbers which for m t h e arg umen th e firs t th ree figures ar e to fac ilit ate t h e use of th e s ult an t s th e whole six figu - he loga ri th ms t ha ve d ifie rent i nitial figures the line be ing m ad e t ’ w hich . o break o r step up whe n th e ch an ge i n the ini tial figu r es oc . for th e inc re men t t o be add ed for t he other two figur es w e look i n th e table of pr opor tio nal pa r ts . a y en . the fo r me r of th ese being 8 8 98 7 5 and th e latter 8 90 0 51 . us e a d d ed for t hatfigur e Thus . . .

.

. fro m t he ublish ed in Fr ance .40 B O DIX E NTA BY m ums The able of n ex t t logar ith ms wh ich we those repri nte d . un d er th e supe rin t e nd ence th e Diffusion of Usefu l Knowle d ge . lo w.

A gai n . . th e r efor e . th is lo gar it h m an d th e one g ive n . a n d p oi n toff as ma ny d ec imals i n t h e p r od uct as th ere we r e fi g u r e s in th e m ultip lier . an d we w an t th e pr opor t ion al pa r tof th e wh ole d iffer e nce . fr o m w h i c h p o i nt i n g o ff t w o d e c i mals. T h us su ppose th e loga r it . In th ese tables n o p ropor t ional p ar ts of th e d iffer e n c e s ar e g i v en fo r t h e se ve r al un i t s i n t h e fif th p lac e o f t h e n a tu ral n u m ber . he t able . th e posi ti on of th e deci mal point will be d e ter mined by the value of the c h a . 1 1 . by th e d iffe r ence gi ve n i n th e th ir d c olu mn of th e ta ble . a m p le abo v e w e h av e 1 1 x 4 4 4. t h e logar ith m of 3 96 90 to he bu twe kn o w this to be too sma ll. the quo t ie n t will be th e fig ur e s to be ad d e d to th e firs t four a lr ead y d e ri ved fro m th e table s . l ea v es th e inte ge r 7 to be add ed . an d t h e d iffer e nce s betw een th e l ogari th ms bein g give n in a thi r d co lumn on th e r igh t ‘ han d . ‘ N ow. 1 1 . Th e r ul e. to this differ ence a d d as man y cy p hers as add i tional fig ur es ar e requir ed an d d i vid e . t he r e for e . butt hey ha v e to be foun d by p ropor tion in th e m a n n er e xp la in e d a t p age 3 5 . os L O G A RITHM B. h m of 3 96 94 w ere requir ed : we i m m e d ia t e ly fin d fr o m t . wha tis t he logar ith m of 40 3 56 7 Th e loga ri th m o f 4 0 3 50 0 is 56 0 58 4. 1 1 . an d t h e n umber an swe r i ng to i twi ll be th e first four figur es of t h e n u mber r equ ir ed The n take the d iffere nc e be twee n . t able s . i s 7 3 7 . fo w s — M u lti ply th e differ en ce giv en i n th e thir d colu mn by al l t h e fig u r es of the natur al n u mber exc ep t th e first four . th e logar i thm of 40 3 56 7 i s 5 6 0 591 . t h e d ifie r ence . bein g mu lt i p l ie d b y 6 7 . th e i nt eger of wh ich ' ' b e in g ad d e d t o gi ves 4 5 98 7 2 for t h e lo a r it g h m o f 3 96 94 . an d t h e differ en ce . p r oce ed as follo ws —Look fo r t h e n ex tless logari th m. th er efo r e as 1 0 . for fin d i ng th e p r opor tio nal p a rts is as fo l . co r r espon d s wi th an inc r ease of ten un i t s in t h e fifth figur e of the n umbe r . th e in tegr al po r tion will be th e p r o po r ti on al par t to be ad d ed to th e logar ith m In th e ex . 11 z: 4 which is th e p r op or t ional part r e qui r e d . 4! to five d ecimal car r i ed an d t h eir ‘ qu i te d i fiere n tfr o m h t a tof ey con t ai n th e logari th ms of every con secu ti ve n um b er fr om 1 to th e ar gu men ts a n d r esultan ts b e i n g p lac e d in pa ra llel c olu mn s . to be a d d ed fo r th e fo ur u n i ts in th e fifth plac e of th e n a tural n umbe r . To find a n u mber answe rin g to a logari th m fr om th e se .

any difie r cnce less t han th is migh t occur with ou t an y change i n the valu e of t he logarith m as given in th e table . a nd migh t the refor e d iffer fr om th e logar ith m given by v ery near ly ' 0 0 0 0 0 5 . h at can be kn own . an d ex plained the method of using th em. migh t for augh t t . the . th at th e d if ' fer ence be tween the logarit h ms of two n u mbers.42 nun mm m - r a nam : F or example. i t will be d esira ble to sho w h ow many figur es may be reli ed u pon as acc u rat e . . be anyth ing between 3 1 7 28 3 5 and 3 1 7 28 45. than 43 429448 2 d ivid ed by t ' h e lesser n umber Now. is less than th e mod ulus of th e system d ivided by the lesse r numbe r . . what is the number answe rin g to th e ga r ithm 3 6 0 423 ? Logar ith m gi ven 3 0 0 428 N extless logar ithm 8 6 0 423 40 20 50 -2 10 5 40 20 1 5 Therefore 4020 5 is he t number whose logari th m A gain . in ‘ the result ai n ed b t s obt a ble s of five. what is the number answeri ng h e logari o t t 4 1 50 7 1 9? Logar ithmgiv en 41 30 7 1 9 hm N extless logarit 13 40 470 6 0 0 -I 1 1 40 47 51 : Th er efor e 40 47 54 is the nu mber whose hm logar i t is 46 0 7 1 9 . or in t . si x. if ex p ressed to a g reat e r n u mbe r of p lace s. he case of co mmon logari th ms. . which d ifier only by unit y. I th as been shown i n Pro p L . which t h en is the e xtr eme li mit o f the ‘ ‘ difierence wh ich tables to five plac es will sho w . and sev en d ec imal y Pl ace s . ' L e tus have th e logari th m o five plac es of ven t d eci mals : now th e r eal value of this oga rithm . Ha ving descri bed some the p rinci pal t of ables. page 27 .

an d subtract 1 0 . in Which th e r ule i s d emonstrat . . sw an plac es will be accurat e . t ha t u nless the numbe r .G en e rally i n an. is l ess than 8 6 8 58 90 . i ns tead of sub trac ti ng a lo gari th m . only th e firstfive fig ur es must be kept A nd in . the fir st five figures d eri ved fr om th e table will be I n a similar way itmay be shown t h at. on l y s i c. wh en wor king wi th tabl e s of logar ith ms to six d eci mal places. for i f.o ' 0 0 0 0 0 5. s e r n e wi t h cer t ai n ty beyond four figu res . C HA PTE R v 1 . which is t her efor e less than 4342945 d ivide d by ' he t n u mber . th e firstsi x figures of t h e resultmay be d e pe nd ed upon if less t han 8 6 8 58 9. y t a ble s o f lo gar i h t ms . hmic A r ithmet Log a r it ic WE next pr oceedto t h e applica t ion of logarithms to t he or d inar y p r oc esses of ar it h metic . and t o illust r at e an d exp lain the ir gen er al u se for th e p u r poses of calcula tion Th e . p rov id e th e ma n tissa of th e logari th m is less th an 93 8 8 . By th e a r i thmetic a l comp lemen t of a logarith m is meant the r e maind e r left by th e subtraction of th e lo ga ri th m from 10 Th us th e ar ith metical co mplement of 3 2 4 1 7 35 is ‘ . o n L oo m 43 between th e t re u se hm and tha t gi ven to five r ue logari t p la c e s m ay. whose logar i t hm is i v e n . th e r esu l tobt ai ne d may be co nsid er ed acc ura te t o as man y fi r es as t h ere ar e d eci mal places in th e logar i t h ms. as we h a ve sho wn abo ve be n ea r ly equal t . ' th e n th e r esultwil l on ly be accu rate to one less nu mber of fig ur es than th e d ecimals in th e logari t hm . i s less t h n 8 6 8 59 i t value ca nno t be d et mi d g a . but if g r eater . we ad d i ts co mpleme n t. mentemploy t h e following r ule . or A . ed . 1 0 00 0 000 3 24 1 7 3 5 I ts grea t use is in divi sio n . but if g rea ter . bu t i f g r ea ter . as will be p r esen tly sh own . 0 00 0 05 ' Th a t i s t o say. th e case of lo gar i th ms to seve n d eci mal places if the r esult . refe r en c es followi n t g h e r ules sho w t h e p rop osi ti on in C ha p t er IV . or he t n umber is less than 8 6 8 58 9 . : Am To m m m m n C O M PLE M E N I ‘ L oo m r nu ’ r roar. bu t that if less than 8 6 8 5 9. we obt ain t h e same result To find the ari thme tical comple .

2 8 6 56 96 6 0 8 7 7 51 1 8 53 59 5 0 53 598 3 5 10 3 of 2 2 90 0 0 r 50 a 8 A nsw er 2 2 90 0 8 M ultiply 6 1 . when he ch a r a ct t e ri stic is nega t i ve . . Logar ithm of 6 1 1 7 8 5330 : 1 22 65 1 8 12 91 3 4 9 40 6 6 6 log of 8 72 30 . it m u s t be a dded o 9 t . 7 34. a nd a ll he her s i c lud i ng t e n p osi t ot ( n h e cha ra ct er i s tic wh ive ) t f rom 9 . 2 2 . o g et the i r log a r ithms the sum wi ll be t . Logar ithm of 52 7 34 . a nd 6 5 tog ether . . Em r m M ultiply 56 3 1 by Logarithm of 6 5 3 1 37 5 856 0 : 9 +1 “ 2 32 49 53 7 38 3 : a log of 2 36 50 0 4° 3 A nsw er I 2 6 3 50 2 M ult iply 52 .44 nunnrm anx m m 1 0. and her 6 toge t . En xr m The hmetical complementof ar it 6 5 3 1 6 4 2 is 3 6 4 3 358 8 2 1 70 6 30 7 8 2 9370 T 2 r 7o 3a 1 0 7 8 2 96 6 ' or to 3 7 3 80 1 6 1 2 8 2 6 32 0 3 6 0 7 8 2 1 6 3 92 7 8 2 RULE — TO multip ly two o r mo r e n umber s t her . he lo g a r i thm of p r o d uc t ( Pro p M ) .

i n th e example abo ve th e a ri th meti . . wh ich ad d ed t o 5 0 6 7 8 45 gives 1 3 42423 t h e same . i cal co mpleme nt of th e logar ithms of th e se veral divisors. . — B u m s To d i vid e one number by a no ther subtr a ctthe loga . .1°s of n o I n st ead of subt r acting the logarith m of th e di visor we may a dd its ari th me t ical co mplement the r esult. . r ith m of t he d ivisor fr om the lo g a r i thm of t he d i vidend a nd . Divid e 1 1 690 8 by 1 1 6 90 0 Prop partfor . 6 2 7 457 8 . log of 2 91 . . trac te d fro m t h e c ha racte ristic will as befo re he t he logar i th m . 0 DIV ISI ON . we h av e in such a case only t . 2 3. o ad d t . rit h m of the quotien t . wh er e it is d esired t o di vi de on e n u mbe r by several ot h ers . ans w er as before This meth od will be foun d very con ve ni ent . r e m a i nd er will be t he log a r ithm of t ie nt ( Prep N ) he quo t . on LO G A RITHI S . o t h e logar it h m of th e di vi d e n d th e ar ith met .9 23 n +7 8 : 4m7 90 2 ° 23 1 2 7 1 0 4. wi th 1 0 sub . and l o garithm of 5 7 6 2 97 8 Preport ional partfor 10 75 u u 6 o 579 4 1 6 = 3 57 6 2 990 4 » 3 93 9794" 9. . Divid e 1 r 64. ca l c omple men tof 3 7 2 5422 t hm of th e d ivi sor is h e logari t . of t h e quo tien t Th u . log of 1 34. Exu r ms . and subt ract fro m t he charac teristic as many tens as there were d ivisors the resultwill be the loga . 8 1 1 690 8 2 53 % 37 1 54-1 7 . . Divid e 57941 6 by 4. by Logarithm of 1 1 64 6 59 3 3 3 0 1: 4 .

‘ If an engine ot6 7 hor sesf pow er can rai se fr om a reser voir 57. . comp of log of 6 5 3 1 a 23 96 1 8 9as before . m h i r sum subt h e log a r it h m of t h e fi rstt a nd f rot e r ac t t en the r ema ind er will be t he log a r i thm of t he fou r t h ter m . da ys. 6o o cnhic “atof wat er i n a given t . qua n i t t y qr e u i r ed 0 r i n s . If 1 4 men. . tea d qf subtr.s a r ith me tical complemen t a nd subtr ac t 1 0 fr om the cha r ac t er ist ic .a c ti ng the lo g a r i h t m c the fi rst t er m ad d i t . ime . Rn x r nns . solved wit h gr ea tfacili y t wi h t h t e aid of lo a ri t g h ms . in . ex ca va t e 56 3 1 cubic yar ds . a unm r e names: ' 46 x rn r Pnor on rros . Q io ns uest inpropor tion or the r ule o f three may b e . what horn s power will be rsquimd ts tabs ’ cubic feet in the same time 1 O r. as 56 3 1 : 47 472 80 : 1 47 2 80 47 56 3 1 37 50 58 6 2 ' 6 59 89 1 log of By the sec ond met h od Logarithm of 6 6 4 74 7 7 P 6 72 0 8 or 9 A rith . ' on a n Buns or Ta n s. what leng t h of w ill i tt hem to excava te 472 80 cubic ake t yard e l ‘ O r. as : 67 o 67 39 98 8 8 7 10 8 0 ‘aaz4 e ' 6 9 332 34 67 P 8 2 60 7 5 52 39578 as before .

.

Whatis the cube rootof ' 00 0 1 951 1 2 1 L ogarithm of ' 000 1 951 1 2 $ 2 90 2 84. Wha tis the square r ootof ' 20 9 Logar ithm of 9 3 ' 20 1 20 1 46 A d ding A 1 to t he character istic w e have 2 2 - s 1 . an d th e qu o tien t w i ll be th e ch ar ac teri stic and m an tissa r esp ec tively of th e l o ga r ith m r equir ed Th e equal n u mbers ad d e d to th e ch ara c te r . En xr ms . ' he t cube o f 0 ' 8 5.Sho uld th e ch arac te ristic n o t be d iv isible by th e n u m ber by wh ich i t is r e qu i r e d t o d ivi d e t h e lo gar i th m . 4 . ther efore P32 0 1 46 2 P6 60 0 7 3 log of ne squar e rootof 2 0 9 ' . comp r ise a ct ually t he . In d ivi d in g a logar ith m wi th a e r i st i ve ch arac t neg a t ic by an y n um ber . h t e n ew ch aract er ic. r e me mbe ri n g . h ow e ve r . mu stof cou r se be consid e r ed neg a t i ve in t h e firstcase and posi t ive in th e o th e r . Ther efor e. 48 31 mm m u m m u m : Em ma . na mel . an d E vol ution . I: The four operation s just de sc ribed . sepa rat e t h e m an tissa fr om th e c h aracter is tic . 2 a 6 3 2 for the charact erist ic. and he sth power t of Losu ithm of ' 25 “ 3 97 940 2 5 7 95880 log of ' 0 62 5 . a n d ad d t o e ac h su ch a n um ber as will m ake t h e c ha rac ter istic d ivisible . i f th e ch arac te r istic is a m ulti ple of tha tn u m b e r . Wha tis the square of 2 5. M ulti y pl ication. 5 7 9 52 580 log of 8 96 56' . t hat the n ew c harac te r isti c will b e n eg a t ive . th e n d i vi d e eac h of th e su m s by the n um ber . or is d i visible by i t . i stic and man tissa . log of 1 10 0 1 951 12 . Logarithm of ' 0 58 3 33 2 90 2 84 . and 2 90 2 ' 84 + 2 . Division . Logar ithm of ' 8 97 4 . p r oc e e d to d ivid e i n t h e u su al ma n n e r . and ad ding 1 to the mantissa w e ha ve P 32 0 1 46 ist 2 6 60 0 7 3 £01 the ne w mantissa . 1 2 90 2 84 3. Involution. - 3 3 7 6 342 8 for the man tissa . 22 90 2 84.

M ultiply the loga r ithm of the ra tio by the t ime. S i mp le I nt er est A d d to g e th e r th e lo g ar i h t m s of t h e p rin c i pnl. fo r m u l a loga rith mically e xp resse d . a n d c l assify t he m u nd er th ei r p r o p er h ea d s In the follo wi ng . wi ll mean t . . a nd c be the three sid es. A s a n exercise i n t h e p reced i ng rule s. ‘ w h i c h a st a n d s . he w ill b e a ble to apply lo gar i th ms i n a v a rie ty o f case s i n wh i c h th eir use w ill be a t ten d ed wi t h th e sa ving of i mmense la b o u r . d a 5 (a b ). or 3 . t he r e ma i nd er w ill be the logarithm of th e in t er est . Comp o u n d I n t er est Find th e a mou n t of £1 at the gi ven ra te o f in t e rest fo r t he fir stterm thi s i s ca lled the r a t i o. a ll th e su p er ficial d i m e nsio n s in s uar e fe e t a ll t h e so lid d im e n q . e x e r c i s e s bu t ma p r ov e u sefu l fo r r efer e nce we sh a ll a rr an e . an d wh en th e stud e n tis con versantwi th these . h e lo gar ith m of a or th e qu an ti ty fo r . an d illustra te th ei r use . th e p rod uct the loga r it hm o f the p r in c ipa l . 49 wh ol e of the the perfor ma nce of which logarit p rocesses in hms a r e e m p l o yed . fo r m u l a the le tter A will be u se d to d e no te logari th m of “ th u s it a .w h e r e i t i s o th e rw i se e x p r ess ly s ta te d . b. y g . b y e x a m ples . o n L O G A BITHU S . e and A eq ua l the ar ea . M E N SUBA TI O N . s io n s i n c ube fee t a nd all t . atthe same time tha tth ey m ay n o t be m er ely . an d 2 h ( w m ea ns twic e t h e lo ga ri t h m o f th e q ua n ti y t i n c lose d w i th i n th e p a r e n h t e si s. an d m o re especially i n th e i r p rac tica l a pplica ti on we shall give a var ie ty of useful . I NTE RE ST . Tr i a ng le — Leta . and fr o m the sum subt r ac t 2 . a d d e d t o th e s qu ar e of c A ll th e lin eal d i me n sion s a r e g iv e n i n feet. th e sum is t he logar ithm of t he amount . and the ti me. a nd ad d to . and the lo gar ithm of the ra tio for such ra t es o f i nt er est as a re lik ely t o b e use d ar e g iv en in the a nn ex ed ta ble . then { a d + a (d — a ) + a (d — b) A (d D . h e we igh ts o r p ressu r es i n avoi r d u p o i s p o u n d s u n les s . t he r a te.

then 8 2 390 9 ah ' as 1 a 1 . C ir c a ul r a r ea — Le tr e qual the radi us. t here for e 1 0 mustbe subt e d from t rac t he cha ra cter istic given in the ta ble s . ° 2 ‘3 di l xa ‘ l ‘ 3 3-3 1 A 30' “ E llip se — Lett equal the trans ver se. l a x. ) 1. and g. a n d a eq ual the area . also le ta e qual the area . . 9 . a n d the corr e spo nd ing a bscissa . at equal he t measure of the arc in d egrees. . ~ 1 1 .50 nnnm s um nr Tax m an Th e logar ithm of the area equals twice t hm he logar it o f o ne Sq id ua r e of 0 s es. and be tw o abscissa . a nd a e qual the area . P a r a llelop ip ed on. an d 8 s solid it it y . g . Rect a ng le The logar ithm of the area equals the logar i t hm o f th e lengt h add ed to the loga ri thm of the height . and 0 eq u a l he ar ea . he t her letter s as in ot then A a = '6 98 97 + A r + z l 1 84 6 33 5 + ' 2 A d + 1 m 3 . and a e qual he t area. Ci r cle — Le td equal he diamet t er. Ci r cula r sect or s .60 2 0 6 + A ' ° a a l a ' 8 950 9 4 » z ad 1 2 1 c A d — 1 . and l i t h . y e ual an q y or di na t e . It it s per pend icular beig t . and e the conjugat e d iamet e rs. P o lyg on — Letl equal the len gth of one of th e sid es. and 12 equal the p er iphery then Ag = Ao + £ 1 3 . P mad orCone — Let a equal the area of the base. then A a 82 390 9 1 ( y. Formula [ 1 6] a pplies a lso i n the ca se of he t Hyper bola . or cyli nd er . h e nu m b e r of si d es. P a ra bo la — Le t . c equal he t circu mference. and s the surface then solidi t ' The logar ithmic tan musther e be tak en to a rad ius equal u n ity. equal he t y. p r i sm. c equal the cir cumference. he t corr esponding or d inat es. Sp he r e — Letd e qual the d iame t er . Let d equal the diameter . n equal t . then 9 —1 ( 5 ) a a = 397 94 + ° 2 1 l+ 1 n A t a ri 223 — - . The logarithm of the c ubic con t ent s eq ua ls t he logar ithm of the ar ea he ba se ad d ed to the loga r ithm of t of it s p er pe nd icular he i h t g . then t z d = 50 2 8 5 + A c ° A e = 497 1 5 + d . then s len g t A l A m 2 .

TR I G ON OM E TR Y . the sum w ith 1 0 su btracted from i tw ill he the loga ri thmic sine of the an le r equire d g . RULE — To the loga r i thm o f the d iffer enc e o f the g i ven sid e s a d d the arit hmetica l complemento f the logari t hm of th eir sum. To the logar ithmic sine of the give n angle a d d the ar ithme tical c om plement of t he logar ithm of the opposite sid e. an d th um g g p p s e g e s wit h 1 0 subtracte d w i ll be the loga r i thmic tangen t o f half the differ ence o f hose an g les t Then to the arith me tical compleme nto f the logar ithmic cosin e . the ar i thm e tica l complemen tof the logar ithmic sine of its oppo site angle . and d ed uct ed from i tw ill equal the shor ter one . a nd the logar ithmic an ento f ha lf the sum of t t h e a n les o o i t the ive n sid e s. — To the log ar i thm of the gi ven sid e ad d . e su 1 0 su bt ed wi ll be t ra c t he logar ithm o f the thir d sid e r e qu ir ed . o n LO G A RITHM S . t o fin d t he t hi r d Sid e. and the loga r ithm o f the o ther given sid e . P la ne Tri a ng les G i ven t wo si des of a t r ia ng le a n d a n a ng le op osi t p e t o o ne of them. When two sid es a n d t he i nclud ed a ng le a r e g i ven . ence a dd ed t o half the base wi ll e qual the lo nge r segment. an d t he lo ar i thm of th e su m o f th i v en sid e s th m w i th g g e g . t o fin d h t e a ng les Ru m — To t h e . and the loga r ithm of the d iffe rence of ho se t sid e s . e equal the of an y solid it y . ' s e qu a l he t sur face. t o fi n d th e a ng le o pp osi t e t o h t e o ther o ne R ULE . A s g a l +t . and a an d 6. of half th e sa id d ifference. the sum w ith 1 0 su btracte d w ill be the lo a ri th m f th id e requ ired g o e s . 51 Reg ula r Bo d ies — Let Z equ al th e lengt h linear ed g e. nu mbers o bt ained fr om the annexed a ble . and the logari thmi c sine o f the other angle . rithm of the su m of the o ther tw o sid es. G i ven t wo a ng les a n d a si d e op p osi t hem t o o ne of t e t o fln d the si d e . tWhen he t hr ee si d es a r e g i ven . ‘ D 2 . the sum w itfr om it i s th e loga r i thm o f the h 10 su bt racted difference of th e segments of the ba se or long est sid e Then h alf th is d iffer . a d d the log s . a d d th e logari thmic c osine o f ha lf the su m o f the same an le s. arithme tical complement of th e logarithm of th e longest sid e . o pp osit e t o th e oth er o n e RU a . t t he n 1 e = 2 1 l+ a.

a nd w e qua l the w eig htof half the arch . the n — [ 35. then A t a ' 2 5r 0 1 6 51 L ral Cen t f or ces . g a nd p e qual t he perpend icula r . a nd letf equal the r ifu a l C en t g force . o the velo city as abo ve . then A P . then A h i — p ’ ) (It — p) . to e qual ver tical w eig h t on e very squa r e fo ot o f the key-stone. P end u lu ms — Lettequal the time in . a nd b a n um ber obt a i ned fr om t he a nn ex ed table . ion second s of on e vi bra t in a v er y small c ircula r arc. a nd v i t vi v h then a v 1 10 the an d ‘ -Le ts A cti on of g m mt y .A1 f M a . “ [ 2 9. . and l the le ng th . A rd en — Let3 equ al rad iu s of curva tur e atcrow n. — Le t w equa l he w ei h t of t a bod y moving i n a circ le w hose rad ius is r. h a velocit w it y of v ee t per sec ond . then - Ar 6 9797 a at M y I ° 2 20 57oz + a nt I 1 90 2 38 2 2 w 2 . r equal the r ise of the ar c h. 6 equal bread th o f a rc h. z a h+b . a c ting horizo n ta lly a to n e -thir d o f the he ig hto f the w all abo ve i ts ba se. then A lso letd equa l hor izo nt a n ce o f ce n t a l d ist y of ha lf the arc h e r of grav it fr om its spr ingi ng. . 5 equal he t b a se. 0 it s velocit y in fee t p e r s vis ar o und . Re t a i n i ng wa lla — Le t ) 5 e qu al h e igh t of w a ll. q he thr u st or h or izo n t a l pr essure on the k e y -sto n e . V i: vied — Let to equal he w eig htof a bod t y . 1 1 1 ) b) 44 1 01 M E C HA N I CS . v A t ' 30 10 3 a: A t ' 904 88 1 ga s . . BUDI M E NTA BY m u l es e hy po t ' Rig id -a nled n a ng em t l — L etL equal t h enuse. P equ al pr essu re aga inst w a ll. then A r A w+ A d — J r . includ in g its o w n w ei ht and P e ual t g . e qual s ac e p pa sse d ov er in 1 secon d s.

0N L O G A RITHM S . and Q equal the cubic fe e tdi scharged in a second . 19 D ischa r g e t equal e d per ime t th e w e tt er . then M i a - a STRE N G TH or M A TE RI A L S . A . Tensi le ste g h — Le t (1 r n t . 1 e qua l the length. th e n A n 44 9 + 2 A u + za ° 2 30 Res s a — Th ion being the same h i t nce of w e r e e n ot at . h both end s r oun d ed . t en HYD RA UL IC S . then . qual ar ea in squar e inch es. Q equa l quan . er. wi t a w -s g G a - n — Ig a l -i. equa lth e d ep t h of water flowing over the w e ir. 1 equa l len th h equa l c or r es on d in f ll a nd 7: e qua l g . D u equal ex t ernal. hen w h en the heightof piece is betw een once and t imes it t s diame t er . Professor Hodgk inson s Formula ’ . D ischa rg e thr o ug h p ip es — Le t( 2 equal d iameter in inches. 53 Resi sta n ce of win — Leta eq ua l th e ar ea of a thin sur face moving through w ate r w ith a v eloci ty equal 9 fee t pe r secon d. I e qua l h t e len g th of h t e i p p . w the w eigh tprod ucing fract ur e. e d ucing fra cture. [ 41 [ 43 H = 1 ° 8 34 5 + A h + S U I — 2 1 9 s a d — 2. 6 equal it s br ead th. then — A p — l l } . A w = A a + B . a nd B number s in column 3 of annexed table . a w 1 a . an d d int er na l diame t h in in ch es. . C .e and h eq ua l h t e h ead . w equal w eightpro . a nd 3 equal the resist anc e . p g a . and A equal number in column 2 of annex ed t a ble . and its length notless than I 5 times its is solid . Di scha r g e over wei r s —Let d . then [ 4L ] — Ah — z } . St h f C l m — L r en t g q o u ns e t w e q al the br eaking w eigh t in tons. Str eng t o r esi st Cru shi ng — Let an equal t h t he ar ea in square inches. hr oug h ca nnu la — Let «5 equa l sectional ar ea of cana l. the veloci ty in fee tper seco nd . ti ty o f wa te r di schar ged in cubic fee t er min ut p e . bo t a nd 0 equa l number in column 4 of annex ed t able t hen w hen the column .

54 a unm nxr m v TRE ATE I S

When he
t mn is ho llow ;
c o lu then

aw d "
) 1 7 a t '
592 4 3
0 C .

When he
t co lumn is soli d bo th end s are fla t,
and th e le n
g th i s no t l e ss
than 30 time s the d ia meter ; th en
a w 3 6 a n I
°

7 a l + 47 I S43'
a .

When the colu mn i s ho llow ; then

)
"
a w a (n d l '
. 2. 47 32 1 7 C .

b
r ect
a an
n — Let 6 e ual the bread th
ula r and d
g q
the d e pth, both in inches, l equal the leng th , w the br ea ki ng w eight, an d D
the number in t he fifth c olumn of the annex ed t a ble ; then
a w
+ a ab o z a d — a z+ .
o

h of P r ofessor Hodg kin son s g ir d er — Le ta eq ua l a r e a

Tr a nsver se st
r eng t .

o f bo ttom flan e in inch es, and d , w, and l ha ve t
g h e s a m e m ea nin g as a b o v e

the n
a w : A o + A d A l .

D efla tio n — Let 3 equal the d eflexi on i n inches w i th the w e ightw . an d 3
he numbe rs in the si x th column of the ann ex ed ta ble ; the n
e qua l t

- 3 a l -f- a w — a b a d - I
g .

The followin g collection of examples apply to th e fore go i n g
fo r mula) , r efer en c e being m ad e by the nu mbers in paren t
h ese s .

O nly a por tio n of th e exam p les ar e wor ked outatlengt h , bu t
an swers ar e gi ven i n ev er y case .

Em r nns.

[ ]
I.Wha tw ould he interest at 4%
t per cent
. u on
p £36 53 for 7 y r a rs
a mountt
o l
Logarithm of 36 53
n «5 01 5532 1 3
o 84 so 8
'
7 9

2.

Log of II

A nswer is £I I 50 1 4s.

56 a u nm s xu a r TRE A TI S E

[ 7, 8 ,
an d What is he
t circu mference a nd ar ea of a circ le -
w h o Is
d iame ter is feetl

Loga rithm of d

497 1 50

Logar ithm of circ umference
2

4 5
3 6 8 8 1 0
2
.

Logarithm of area 2
°

56 8 8 1 0 370
°

52 .

[ 7, ,
8 a nd W ha t is h
t e d ia meter and circu mference of a circle wh os e
ar ea is 56 2 squa re fee t?

A ns Circ umfer ence is
. fee t, and diamet
er is feet
.

[ 10 . What is h
he lengt
t of a n arc o f 73
°
o f a cir cle, w h ose ra dius is
l A ns . feet
.

[ 1 1 .
1 Wha t i s th e or of a circle
a rea of a sect w hose rad ius is 2 6 fee t
,
an d
w hose Si d es i nclud e a n a ngle of A ns . 2 47
°

58 fee t
.

Wha tis the area of a para bola w h ose abscissa is and the cor
r e sponding ord inat
e 33
°

4 1

Log of (2 ya) 2 x
Log of z , 0 7 2 591 2
°
8 2 390 9

.
I

I 46 6 BO I 5 area of para bola.
°

In b l or d ina t imeasured
ts co nd ing an d
a para o a a n e ,
rr es
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sur face is fee t ‘ . w hatis t f th e e c orre sp ond ing wit ord ina t h a n abscissa of 2 0 festi Since =2 0 . 9 3 77 Lo ar it g hm o 10 ° Log sine of angle 0 p . solidity is ' Dod eca edron. since he three t angles o f a r iangle t are equal t o 1 80 ° we ha ve n 3 . tet one of whose lineal fee t. fee t. Log 20 LO 8 xs = 3 ll n +91 6 3 7 P 396 1 98 L0 8 6 3 8 1 3 797 84 Los t s: 1 7 0 7 570 Log g 12 684 1 2 1 8 553 ° he t e require d . and ° it s hei ht feetl A ns ’ g . [ 2 3 and I n a plane t riangle m o of it s sid es are and and the an le o pposi te t he longer side is 74 remaining angles an d ° g wha tare t he Th e length of th e other sid e i Then by [ 3] 2 Lo a rit g hmic sin of 74 39 ° ’ A r i thm com of log of 55 p ° I . [ 1 9 an d Wha t is he solid it the Sphe r ical sur fac e and t y of a sp h ere whose dia met ’ er i s feetl A ns Surface i . ' O ctaé d r on. of an oc t ed g e s is a ed r o n whose linea l ed g e is ' and of a d od eca é dr on w hose li n ea l ed e is g A ns Tet ra iéd r on. or dina t What i s h e cubic t con t s of a c ent ylind er w hose d iame t er is fee t. sur face i s . 51 20 3 1 . an d periphery is [ I In an ellipse w hose tw o d iamet he leng th ers are 51 and 38. sur face is 1 fee t. and its height 6 feet? A ns 3 . feet. feet solidi ty is feet . Wha t is he t cu bic co nt e nto f a con e w hose d iamet er i s 3 5 fe e t . . solid it y is 98 7 12 fee t . ON L O G A RITHM S . [ 2 1 an d What is he t su r fac e an d soli d it '' y of a '' ra e d r on. 67 [ x 4 an d I5 . Whatis the a r ea an d pe r iph er y of an elli se p w hose conjuga t e et er is 2 7 a nd its tran sv er se d iameter is 4 9 ! A ns A rea i s. solidit y is 1 5 fee t . other sid e 66 ° 4 I Then.

area by [ ] 3 35 3 is 8 5 square feet . and t 0 a ngle i nclud ed bet w ee n t of t he o ther sid e. A ns A nd the. In pla nea tr ian gle w h ose sid es are 54 5. comp . [ ihm Then by 24 L o ar g t of A r ith mp of log .68 acu m en m amas ° ’ ” he ’ (74 16 for t angle ° 1 80 ° 66 ° 4 39 4 op p o si t e h t sid e yet to be fou nd . log cos of 1 6 ° 33 14 Logar i thmic c i os ne o f 66 4 0 Logari thm of 22 7 1 493 I 10 ° Logarithm of third sid e A ns . of log 5 77 2 8 5 7 0 Logar ithmic t an of e.sid es of a tr ia ng ular p iec e of gr o u n d measure a n d hem is 47 w ha tis the leng th ° 1 0 57 5. and and t he th ree an gles op p o ively are 39 1 6 ° ’ ° ’ ° site to eac h respect 66 4 and 7 4 [ 5 2 an d 3 T w o. whatare t angles O posit e th os sid es r es ect i vely ? p e p A rithmetical comp. and the area of the p iece of gro un d ? Log I ° 81 7 391 A r ith . of ’ " h A rit . 66 ° 4 ’ 52 77 8 1 73 1 0 94 2 10 ’ ' Log t an of half the d ifierence of the angles ° ' ” 1 6 33 I4 osit h i . ° and he 6 2 . co . bf 66 ° 4 56 ’ ” Logari thmic sin of 39 1 6 4 5 ° 10 Logarithm of sid e required 5 47 9 1 . of log of 62 Log of 545 ) 2 o 27 3 L° 8 ° f 54 s ' 1 2 34 559 ' n 554-930 ° 10 ' Lo f th d fiere of he t : f th 88 8 e 1 3 54930 : Theref re he larger t segment is and he lesser t o 3 1 segmenti s 3 1 . . A ns Th e three sid es are 47 91 . e t e v en sides o pp g comp .

Wha tlength of time will a pend ulum inches in length be in making one vibra tion ? A ns . . 1 9. w ith a v e lo ci ty o f 1 2 7 fe et p e r sec on d . a c ircle w h ose radius is 1 5 feet . weighton each square fe e ta tthe key -st one is 974 lbs l A n s l bs . Wha tis he length t of he d iag onal t a ngle of a rec t wh ose tw o sid es are 34 and A ns . ose rea s . A bo d y ha ving been falling fr eely by the action of gravity for 7 5 ° sec onds i tis d esir ed t he space w hich ithas fallen throug h o kno w t . larg er segmen t 32 94 63 ° 54 ’ ” The n 90 ° ’ ” ° ’ ° 6 3 54 4 2 6 5 56 angle opposit e sid e w hic h measu re s A gain. [ 3 and 2 A b od y fa lls u n d e r h t e in flue nce o f gr a vi y t from a he i htof g 42 7 feet. neg ' lee ting the resist a nce o f the a i r l A ns Itw ill occ upy . o f lo g of = 1 L g h o a r it m of 1 ° I 1 8 5 43 10 ° Log sin o f an le op . of log of 54 5 ° Logar ithm of ' I 9 9S37- 94» 10 ° L o g aim of a n gle o pp. . c omp . Wha t is he t i v s vi v a of a railw a y train w e ighing 1 1 7 ton s. second s. A hou se is 47 fee t in h eig ht . 69 Th e n b y [ ] 3 2 1 ogari thmic sin of 90 ° A ri th .seco nd s . . Logarithmic si ns of 90 ° A rith . [3 A b o d y w ei ghi n g 53 lb s is w hir le d rou nd in . . a tw ha td istance mu stt he base of a lad d er 53 fee t lo ng be plac ed fr om the house i n o rd er tha tthe top of the lad d er may justmeettha tof the house ! A ns fe et . and ' ra vellin t g ata ra te of 33 miles per ho ur l A ns . A ns feet . . w h a t i s th e strai n u p on the rope by w hich i tis const rained t o move in t he circle ? A ns lbs . a nd acquire a v elocit y of 1 6 5 fee tpe r second . What isthe horizon t al pr essure at t he crown of an arch w ose h ure is 1 4 7 52 fe et radius of cur va t w h b d h t i f t nd th ver i cal t 35 ee a e ° . O N Lo e m tr mus . . comp . w ha t time w ill i tocc upy and w hat w ill be i ts final veloci t y. g p lesser seg men t 2 8° Th en 90 ° 2 8 34 2 3 ° 6 1 2 5 37 ’ ” ° ’ ” the angle o ppo site the sid e which measur es a nd 2 8 ° 34 ’ 63 ° 54 4 ’ ” = 92 ° 28 ’ the a n le o it th l t i d g ppo s e e on g e s s e . .

re h u ired t e r e w hich e v ery foo tin length of ithas t o sus ta in 3 q p essur A ns . oak. . [ 39 and What resistance w o uld a board w hose area is sq uare fee tex per ience in being moved thro ug h the air w ith a veloci ty of r 7 fee tper seco nd . .60 a um ma nn e r m a nn er: In an iro n br id ge ha v ing a span o f 2 1 2 feet. wrough tir on. I5 fe e tw id e a t h t e surface . . e . What is the pr essure aga ins t a sluice 20 feetwi d e . a nd und er a hea d of 7 5 fee t? Log cf d i 8 11 55 7 2 2 5 5 Log of h 75 18 7 50 6 1 3 P 34 8 50 0 0 h as (I an d) 5446 6 37 36 : s 2 3 6 7 3 798 3 Log of quantity per minut e 6 7 91 . wi th the sid es sloped at 1 to r. lbs 3 in water. a nd fir ? A ns . lbs . A retaining w all 37 feet in h eight suppor ts a loose sa n d y so il. Whatqua ntity of w ater w ill be d ischarged b y a i pp e 1 8 i n c hes in d iame t e r. the w e ig htof ha lf the arc h is 998 ton s. A ns 7 6 r 9 c ubic feetper minut . and the d ist anc e o f i ts ce n ter of g ra i it fro m t y he sp r in in g g is feet. Whatw eightw ill be n ec essar y to crush a block of cas tiron 3 inches square A ns . w ha tis the h a l th r u st o f t or izon t h e arch ! A ns 1 90 7 3 tons . ha t i s h t e gu n na ry o f w a t e r flo wing over a weir m 7 feetlong. a nd ha v ing s e r aga instit? d e pth o f 7 fee tw at A ns . What w eigh ts w ould be r equ isit e t o t ear asund er rods 2 inches squar e. lbs . . 47. fir. Wha tweightwill be required o break t a hollow col m h fist wit . lbs . and he inclination of the surface of the wate r in w hich is 6 inches per mile ? t A n s 1 3 8 3 feetper second . Cast iron. lbs . w i th a r is e o f fee t. Whatis the v elocit y wi t h w hich w ater w ill flo w through a cond uit . Whathead w ill be required t o forc e 350 fleet of wat cubic er per minu t hrough e t a pipe 1 inc hes i n diameter . lbs . . W ' [46 . 52 0 1bs . lbs . oak. of castiron. and 36 40 fee tlong ? A ns 2 2 7 39 foet . and w ha tw ould be t anc e in w at he resist er ? A ns In air . 4 fe e t d e e p. w hen t e surface of the r iver is 6 inch es abo ve the top of the weir ? A ns 3 cubic feetper second. . 537 1 fee tlong . wroug ht ir on. . .

the length of w hic h is 37 fee t. applie d i n t he ce nter ? A ns 2 56 inch . and m aths 6 fee t hea ring by a w e ig ht o f 2 7 30 lbs . 61 en d s . . . t w ha t w e ig h ta ppli e d in h t e ce n t e r w ould b r ea k it! A nne 6 1 3 5lbs . A bar of castir on inches wid e and 3 inches d eep is la id u pon sup 2 port s 6 fe eta ar p . in w hic h the ar ea o f the bo tto m ’ fla me is 2 6 sqm r e inches. . 3 i nc hes in d e pth. and its i n te r n a l d ia meter 1 0 inches l Log of n : 3 8 8 5 5 0 1 6 log of D " Log of d 10 log “ 3 60 0000 0 of d a d " d ) 6 3 93 s ' 33 6 74 3 8 0 1 1 7 3 1 86 0 4 7 32 1 7 5 z r 3 84 r ° Log I 37 x r 7 23 6 6 594 34 . Wha t d eflex ion will be pr od uced in a bar o f ca st ir o n 2 inc hes wid e . ' . a nd the d istance be tw een the suppor ts 2 3 fe et? A ns lbs . Whatweighta pplied in the center w ill be requi re d t o break a gir d er of Pro fessor Hod gkinso n s fo r m of sec tion. the d ep th 1 5 inches. its er nal ext d iamet er 12 inches. 0N O L GA RITHM S . 2 3 4 7 8 98 log of There fone the an swer is t ons . .

Ta ble f o he t Log a r ith ms of every P r ime r om 2 t o 1 0 00 f . . 62 A PPE N D IX .

.

th e n sub tr ac tthe r esult an t fr o m t h e r e m a ind er a nd loo k a gai n fo r the . th e t a ble for t h e n ex tless n u m ber to th e r em ai n d e r n o ting . an d th u s p roc e e d unt il th e gi ven lo gar i th m h a s bee n ex haust ed . th e fig u res foun d at t h e plac e of in tersectio n ar e 1 7 3 3 7 . an d w r i te d o wn t he n a tu r al n umbe r co rr espon din g w i th th e logar i th m t a ke n fr o m t he t a ble subt . each ti me n o ti ng th e n atur al n umbers cor . th e sa m e lin e wi t h th e fina l fig u re . or u ni ts fig ur e of t he sa m e in th e ex tr e me le ft-han d co lu m n . ract t h is lo ga r i th m from th e ma nt i ssa g iven a n d a ga in loo k a mon g th e resu ltan ts i n . . 64 APPE N D IX . to w h ich ad din g two cyph er s on t h e le ft h an d to m ake up th e s e v en fi ur es we h av e 0 0 1 7 3 3 7 w h i h i th m ti f ' g . In th e fiv e la s t colu mn s o nly t h e fin al sig n ific a n t fig u r es of the m a n ti ssa o f th e loga r i t h ms will be fo un d i n th e ta ble . as ma ny c y ph e rs m ustbe a dd ed to th e lefto f th e figur es gi ven as ar e n ec e ssa ry to m ak e u p se ven figu r es . an d in th e sa m e c o lu mn as the o th er fig ur es of t h e n a tu ral n u mber . c s e an ss a o the logar i th m of 1 0 0 4 . h e r esult a n ts i n th e ta b le f o r th e n ex tless n u mber to t h e m an tissa o f th e gi v en loga ri t h m. n e xt less r e sul t an t t o t h is r e main d er . I n th e above t able t h e ar gum en t s ar e n a t ur al mu m be r s . . Th us a t th e top o f th e fifth colu m n w e h ave 1 0 0 . The firs tfigu r e s of th e a r g u me n t s a re fou n d i n t h e to p h o rizontal li ne an d th e fin al . Table by the a id of which the n umber answa fng t o - a n y log a r ithm ca n be fo un d to si n: p laces . a nd t h e r esultan ts th eir lo ga ri th ms . t h a tis u n til no re . Th e m an ner of usin g th e table is as follo ws z— Hav ing g i v en a lo ar it g h m o f w h ic h i t i s d esi re d t o k no w t h e c o r re s on d in p g n u m be r lo ok a m ong t . ’ th e lo gar ith m is fo und at th e plac e of i n te r sec t io n th a ti s o n . an d on t h e fou r t h lin e we h a ve 4 th e n . th e n u mbe r a mo ng t h e ar g umen ts an swe r in g to i t. m aind er is left.

thus 220 0 0 0 0 220 0 x 1 000 90 0 4 220 0 x 4 227 50 0 4 This again has to be mul tiplied by 1 0 0 0 9. which h as to be n ex tmu lt i plied by 1 0 0 4. 227 50 0 40 0 0 0 2047 557 0 227 7 1 1 1 557 0 . g e r ul t i on may be v er y readily p erfor m ed iplicat . n u mb er s being th en multi pli ed togeth er . or by 1 0 0 0 and by 4. E X A M PLE . 1 4997 5 hm N ex tless logar it 1 28 3 7 2 log of 103 3rd r emai n d er 21 6 0 3 hm N extless logar it 1 73 3 7 log of 1 0 0 4 h 4t r e mai nd er 420 0 N extless logari thm 3 90 7 log of 1 000 9 5th r emaind er . 4 1 3 927 log of 11 2nd r emain d er . APPE ND IX . 3 59 N extless logar ith m 3 47 log of 1 0 000 8 6th r emain der 12 N earestlogarithm 13 log of 1 0 0 0 0 0 3 Then 2 x 1 1 x 1 0 3 220 0 . 65 r es on din g p to th e logar ithms taken from t he t able These . th e pr od uctw ill be n u mber corr es on di n gp t i o t g h e logari th m n ally or i i Th e se n umbers have been so arran ed t h at t h i g v en. —O f wha tn umber is 3 3 57420 2 t h e logarithm G i ve n logar ith m ' 3 57 420 2 N extless log i n t able 30 1 0 30 0 log of 2 ' l etr e main d er 50 3 90 2 N extless loga ri th m .

poin t 0 as m any figures fr om t h e r igh t as th e re ar e figur es be fo r e the n umber by wh ich you are a bo u tt o mult i ply an d perfo r m th e . i plica t . Th us. an d t h e r e mai n d e r to th e r ig h t m ay be cu t an d any fig ures i n t h e mult ipli ca tio n by 8 whic h would fall und er any of t h e figu res so cu t off be o mit ted . i n the exa m p le. an d on ly mult iply 227 by 8 . fig l 'l i l a . ain m or e t re t h an 8 figu r es. th e nu mbe r by wh ich we ar e goi ng t o mul t iply. an d the ans wer is true t h e r eal number being 227 7 8 o se ven p lace s. 1 8 22 22 7 7 293 7 68 227 7 8 0 0 5 Th e last multiplicat ion i s by 1 0 0 0 0 0 3 . w e po in t0 8 the five righ t h an d figur e s. 1 822 . as t h e figur es to be multiplied by 1 00 0 00 3 are no t ected by t he ad dition of 1 8 22. exam 1s above . we ad d i n . t hi s n eed nothave been perfo rmed until after wards . t o kn ow h ow ma ny figur es t hus to omit. m ult i on only u pon the r e maini ng figur es takin g c a r e . ho wev er . however . t o carry t o t h e mul tiplica ti on o f the first n umbe r wh a teve r would hav e been ca rr ied fr om t ha tof th e las t fig ure ‘ ou t ofi . t - In the .60 A PPE N DIX Th is h as again to be multiplied by 1 0 00 0 8 but we n e e d not . th er e bein g fins figur es be fo re 8. as below . 68 227 7 3 0 0 5 . 0 carr ied fro m t h e multiplication c ut0 5.

W hatis he t n umber whosé logar it h m i s 20 1 0 3 8 3 3 ‘ 5 6 1 0 38 3 3 101 6 0 20 6 0 0 log 0 f 4 4 8 3 23 3 40 40 0 0 43 2 1 4 log of 101 3 636 40 0 1 9 38 91 2 log of 1 00 9 8 1 53 20 3 8 1 107 20 4 86 9 log of 1 00 02 40 7 7 3 995 23 8 21 7 log of 1 0 0 00 5 21 log of 1 0 0 0 0 0 5 Th e answer is . A PPE NDI X . 07 W h a t is he t n umber cor r es on di n g p he o t t logari th m 4 8 55 1 0 7 l ° 8 551 0 7 1 7 x 1 02 7 1 40 0 0 8 4 50 98 0 log of 7 2 1 42 1 0 0 0 91 86002 log of 1 0 2 1 43 23 28 6 5 1 40 8 9 543 13 0 0 9 log of 1 0 0 3 7 1 6 3 20 3 1 1 0 80 869 10 8 Of 1 0 002 21 1 1 74 log of 1 0 0 0 04 37 log of 1 0 0 0 0 0 9 The num ber is 7 1 6 3 2 .

. 1 300 9 log of 1 00 3 4 2 1 25 3 51 30 65 3 0 39 log of 1 0007 6 0 22248 7 26 log of 1 0 0 0 0 0 6 Th e number requi red is 6 0222 48 .68 A PPE NDIX . Whatis he t nu mbe r hm i s whose logar it 97 58 7 ? 7 7 97 58 7 100 3 7 7 8 1 51 3 log of 6 6 1 6 0 74 8 0000 .