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IB IS A ^ S ^ I^ OH THEORETICAL AND PRACTICAL I c^ li.ir|jf^fji0j^ SUnVLVi ; I TO WHICH IS ADDEt GIVEN IN THE WORIT, With all the xNecessary tables. Br ALEXANDER MONRO, LAND SURVEYOR. PICTOU, N0VA-SC0TL\: ftlnm hj Gcldert & Pattsrsoii, Eastern Chronicle Off!.?, . Por the Aiitbor, WDCCCXLIT, the I DEDICATION. TO THE HON. AMOS EDWIN BOTSFORD, Membeu ok Her Majesty's Legislative Council, Memleu op the Board op Education, Lieutenant-Coeonel, &c. &c. Si.1, — Of tl,e .ncrits of tlie following Ti-eatiso it is not for „,e to judge. That poiut „,u.t he left to the cleci i™ "f .n.partK,l reader. I „,„j, however, be permitted to ex- 1 -.,n,y r,^re.thatitis„ot n.ore worthy of your patrt . tc U to one ,0 whose .nhul the urgent necessity which ex- L^l vl.;™ „ •'" "f '""" ""^-^ «o,ue„.ly s„.ges"a .K-ut:,W ■'■":'"=" "'° '■'^"'■^"f ■ho public, and ■I , . r •,• '"■"'-•""^'"S^ in <""■ Courts of Law,-to one o,,e lanuhar a„|ua,n.ance with the subjec: discussed, af! 1^,1 M "'",?"'"■'"""' "■"' "■" "■"'* "«<="• '^ "<" »lt"SC l,or u u , y „, notae, and favourab.e accep.ance,-to one from 'Mioni „s numerous in.i.erfcctious are sure to meet n-ith mos parental in.lu,ge,,ce.-and to one whose pa.",:.;' coLua of the „b,cun.y, and want of personal or relative ia- r ir DEOrCATIOW. fluenre on the part of the author, it might otherwise bo ex- posed, Oihor considerations have likewise induced mo to solicit the honour ofdedicating to yon this, my first attempt at au- thorsh.]). I wa,^ e.vcecdingiy desirous to avail myself of this opportunity puhliely to express my grateful sense of tho courtesy, kindness, and attention, which you have so gene- rously oxtMidcd towards the humble author of the following work, ever since he had first the happiness of being intro- duced to your notice. ]Jc assured, Honoured Sir, I am not ungrateful. Your name will ever be associated in my recollection with the most lively emotions of esteem and respect. lieside.s, the countenance and cncoura-emcnt which you have uniformly extended to the industrious and enterprising youth, and the interest which you have ever manifested in tlie cause of Education, and in every movement in which tho prosperity of British North America is involved, justly en- title you to this public expression of grateful approval. Hopuig- that tho work itself may not altogether disappoint your exj)ectation,-hoping even that it may meet with some degree of approbation,~.and praying that you may be Ion- spared to enjoy the confidence and respectful esteem of thoso wJio may be honoured with your acquaintancc,~an(l to wit- »ess w.th delight tho rapid progress of intoHectual improve- ment, unci tiie growing prosperity of your native country. I remain, Sir, With nmch esteem and respect, Your Most Obedient, And Very Humble Servant, ^ ,, ,. ALEXANDER MONRO. liRv De \ erf«, N. B., October, 1844, '"% iso bo ex* to solicit ipt at au- Llfofthis sc of tho so gone- rullowing ng intro- I am not d in my eein and lich you jrprising fosted in 'hich tlio ustly en- val. sappoint ith somo be long ofthoso [1 to wil- nprovo- Lintry. TOTHKL.vxD«:uvi.:vousormt.T,.sn.vouru AMERICA. GE.VTLnMEX, .ho following TrJle'rih' "'"''• ''''"''"^'" ""' ' '*'"'"• ence, !,„v.„vcr, I cSf" r, , '"""'• '''" ™'"-<-M.cri- ^<'" very we,, l r,^' 7,:7 r'"'', "^ ""'■' "''■'■'' '— • work, upo„ „„,, .«1,:\ '",;'';«'^ "''»l"«' <'-«n,i„, Lm,l Surveying _„;,",?, "!" l""-i>o^*'=« <"' Colonial --.raein.; of,),,-; „';,'"' '" ""■■ '""""'"• '"^'■"S. "■"'i -nts of i,,o p,,.,;, ,I:-'«™; of I.and.,_,„o £pr.n. -tood by,,,,, Colo,, i, V r«"""^'^' ""'■"'■'"'>■'"'"' '"•'■'"- investigation of ZLuorT-''" ' '""'" B'""" ■'>» •he reaso,,,, of.,, ,°':';'''»'t'l^v tl"M.or,>.al„f „-,,,„,, '- -:^iiy «n,,c,. o" ' f , "'" 'n ,'","•' "'■ "■" "-■'^ '"»^ worl< a„ ,1,0 TalivT'n " *"' ='"'' '"' "'<' <■"<' o'' tho ' have ,ho H'o„;:";.';t; '^" "'■'"'■"' '"'^I'"-'- ^*cntIonicn, Vour Obodirnc, And Very Humble Servant. JBayDeVcrt. Nn ,, /^^^^'-'^•^™^i^ -MONRO, '-'t.,, ,>, 1^,, October, 16 ij. I til o I TlIR xnn the nieasin n('f'0|)£arIoi tlir extcnsi of hndic.?. tiulc, distill Its of baflio- Respcotii liavc Ijooii ( Kjryptians t inotry. Ti their landm tlioir houiK origin of L Achiilo.s ': that the P:.r. and tlie eart ledge of tlie ^'"rodotu: to he the SI war up 0)1 ] reported to ] tlivided the lotinent, for Aristotle h ii PHE/'ACK. — "" » T '"-• v„.svrn.,r.l ofl,,c K,„-th, or cf Land. .„ i,, ,,,,i„,,^:^ Kl ;• " "'',''''"' '" "'" ""='•'■■''"■<""'=•" of tl.n ,na...i. .1.S of l,n,l,c.s on or „c,il' llic earth's si.rfa.'o Rc.,pccli„g ,1,0 oris^, of this „,,.:„„„, ,-„;,„„ i,,,. have heen entertained. The ancients n^ree in .min ■ t t e J.,,-p.,an, the cre<„t of heln. .he earhe;; .^^::::ZZ lo.., , he annual overflowings „f the Nih, ,Iis,n,Ci„.. on-h,n.h„arl„ren.,ere.nt necessary fre^nentlv to '^t X^^t::^i:^-r~- "^- -^ "•--- tlnt'the'rlT";.'"'' " M..hen,ati,.ian ef Greece, inf„r,„s „,, m tiieeaith. Moses .sevensaid to hnvcac,,„ire,l „ l<„,„v- l«lg of the se,e„ee, ,vhen he reside,! a. the l^vptia t 't enor;;' '"'"'"'"' "'" ■^''" "f S' .m.,. This King is 1 fide, , '■"■'; "'"•■'■■*""'' ''■^'J'l" •'-'' C-'.-.!^, and ,„ have ''>"lod the hind among his suhjeets, tjivin- to e.ch tn •,! .Ar,stotIe has attributed its origin to the Egyptian priests. viii W Hi PBRFACC. who nw„, ,co>u,.oa f,.„,„ .he .„.,„, h.U .,.„„ , ,.„„„ TJjc antiquity of this soionco nCToviU sn-.in ..: » r« uie o..s.^ of Land and Marino Sui vovin- ThMt. ,i- nil tie (nlcuhtinn. ;,. V • • '^"'^•'^'"o- -^ ncy direct \nude fhn . I >^'^Viivat.on and Astronomy. Thcv ocean, and the Minnr t^ . ^'^^■^''■^" ^''^ P'-^^'^lcs. rho .ul.jeot of ,ho p,-o,,o„t tre..i.e i, /.™,; ,%.„e,„-„. i'ns constitutes on«> of thn m«.-. • ^- ^t^ym^. branches of tlm M !l '^nportant and useful dill of ti. f ^^^.^'^'"f ^'«^- 'i^ho Surveyor, in the dis- 1 To ^^'•"^^•^^''^"^^' ^'"ti^^> ^i'-ects hi. attention ' Ji- 10 the tracing and measurement of lines: m1 ''V'", '''I'm'"'"" "'■'" ''»^""l'"i« "Pon« plan or .hr';„, ":;'"", ° """ "'r' "■"■" '■^^"'""''" "«"■-"" ™,e rf hi ' '■'='"•■"■'"'••'<' "''J-^"^ «i'lm.<'.' near ,ho h^!h. , "P"'-" »"».-to ...csun, their distances a,.I ;usht..-to a=.crta,„ .ho van,,;ion of ,ho compa.,,,-.ho la- mu a,,,, lo„.i.,.d„ „f p„.,i„,„,^^ __,^^, • - - o loh,...,.o „„, o,.lj' the boumlarie, c • an onlin,.,-y ^ :;' but also of coast, and harhour.,,_„r .o ?ivc a corroc. rlDro* =o„.a„„„ ,.,-.hc i„„,„ali.ic., of the earth'. JC "'""" Anions the ancicn..,. Arohin,edc,and Tan.agli, inade coi- pnirACK. nidcral)!( ix progress in evolving the pritirlnlea bv whi^^K are dct*innincd Tho r *" '"'P'''* oy «nich areas .00 much ;;: iio::,,'"' ,. r;z:^> - - f 7-. - -een 1. St, I more just ,„ ii, „pp|ica,i„„ ,„ ,, ,j,:,. /'""'' American Colonies W„<,d.L„„d Survey ,!» nftl', i "' For th,s re„so„ „.,„ o/'^: „: Lef;::!: d"; ""'"• ^• whh which I have „,e,, arcs,, s„i.ed X ,lhf r^'lh? vmces. I he necessity f„r so.ne „orI< on thesninec. ada r; to our condition, appear, to mo to bo trrea. 2: ' Fron, the frec,uency with which disp^e , t, t L'^'f'"'- are introduced into our Courts of Lav fcr I r- ""''"'' «ic.ent that son.o .-.e.uaintance wi^"l!:d t: ;;• rj:!^: tlio i.gal profession. Scarcely less necessary i, an ,c ri::rr:' ;,;:;:— 'rrir^r '■----- o.,,ht to have son,e Unowi^e 7 :t.':Zr7 :tX tie aequanuance with Land Surveying would hav. , .nany an individual from minors liticaZ I ^"^ only his property has been sauntfd ,:,;',:""" ""i ■mnd disturbcl, and strife and contonti „ t'pr d thrCh !: commun tv. Im')rcs=!Pr1 xviti. . i ' l"^eaa tnrough the TS, Befinial Fractions, . . _ Kxtraction of the Square Root, GKOMETRY^Definitions, Contractions, - _ _ _ Explanation of Signs, - . _ Geometrical Problems, Concerning Scales of Equal Parts, Logarithms, - . _ _' Trigonometry, _ . _ Mensuration of Heights and Distances, l^ANo SuRvEYmG-Instrument employed, L-sc of the Chain, . . . _ ' Circiimferentor, - . . Theodolite, - . . S3 r*rotractor, Field Book, --.."" Variation of the Compass, Running of Lines, - _ _ Mensuration of Land, - _ Division of Land, - , Location of Land, - _ _ ApPENDix-Demonstration of Problen.s, ' 1 romiscuous Probleins, Levelling, - - . 33 33 PAGE. 1 9 11 16 17 20 29 31 452 50 54 61 66 69 70 71 74 85 98 117 128 IS!) 153 17;> 4 Xll M CONTEXTg. PXrrZ. msc>:r.r.AJ.Eotrs^Re-e8tablish^cntoflost boundaries, 176 I- acts concerning Magnetiiim, . . . ,-- Meridian Lines, Concluding Hemarks, 179 180 I: itiaries, 176 - 177 179 - ISO DECIMAL FRACTIONS. IHE TERM FractioiV, literally denotes something broken To form a distinct conception of the nature of fraction's employed in calculations, let the Student suppose any object or quantity broken, or divided, into several equal parts. Any number of these parts, considered in their relation to tho whole objector quantity, constitutes a fraction. A fraction is expressed by two numbers, placed the one above the other, with a line between them, thus: — . 5 The figure below the line (5), called the denominator, ex- presses the number of equal parts into which any obje;'t or quantity is supposed to be livided; and the figure above the line (3), called the numerator, specifies the number of these parts which the fraction represents. Decimal Fractions are such as he . e for their dunomin.i- tors, 10, or some multiple of 10, that is, 10 multiplied into itself a certain number of times, as 100, 1,000, 10,000.. ix.c. Expressed in the common form, they appear thus: — ~ 1 000 ^^' wecimal form, the denominator, btini? etisi' — r- CI DECIMAL FRACTIONS, ly ascertained, is omitted; and its pKace is supplied hy a dot or decimal point, (.) prefixed to the numerator, thus: .3, .15, .261, &c. To ascertain the denominator of a decimal fraction, it is only necessary to write doAvn as many cyphers as there are iigures in the fraction; and then to place the figure 1 before them. Cyphers on the riglit hand of decimal fractions, do not ailect their value; but every cypher on the left hand dimi- nishes their value tenfold. * The value of figures in decimals as in whole numbers is determined by their position. The following table, in which fhe figures on the left hand of the decimal point are Avholc numbers, and those on the right are decimals, will illustrate the influence of position in determining their value: — Integers Deci mals 8 1 3 4 1^ 4 6 . 4 1 ^*0 3 5 7 X H H ffi H g H ffi H »^ X Is: o 3 5 ST housand: ens of T 5 o 57 2 2 Cm c c 2 O c -T" \J* J. r^ cc c ^ •-» o \f 7J p H^ urts. 4.O07 '• i'our and seven thousandth parts. 1 by a dot thus: .3, tion, it is tliere are 3 1 before s, do not md diiTii- iiiribcrs is in which ire Avholc illustrate Z^ECIMAL TRACTIONS ( ADDITION. RULE. Place the figure? dh-ectly underneath'those of the same va- lue, whether they be mixed numbers, or pure decimals, pay- ini]f particular attention to the separating points. These should always appear in a direct line, one under another. Then add as in whole numbers. EXAMPLES. 1. Add 2.81, 5.50, 1.6, 4.334, 6.3431, together. 2.81 5.50 1.6 4.334 6.2431 *&ns. 20.4871. 7 2. Add4. 28,3. 2187, .0024,342.501, .223, and 1 .2324101 5 together. X )^ 4.28 *— • 3.2187 r. n02-4 H> 3. 842.501 tr 7j % .223 c i 1.2324101 'Mns. 351.4375101. obvious th })arts, )usandth SUBTRACTION. RtTLE. Place the figures as directed in Addition, then deduct as in wJiole numbers. EXAMPLES* ! From 28.4 take 24.35. 28.4 24.35 *^ns, 4.05. DECIMAL FRACTIONS, 2. From 70,38 take .829. 70.38 .829 *fins. 69.551. MULTIPLICATION. KULE. m^frtn h. T ."''; P""" "^ "^ "^^"y PJ^<^^« for <^eci. r toTh r Tf th "' " ''"^ are decimals in both Fao- ductSt^ 1 T "'" ""' ^^ "^^"y %»»-^« ^" the pro. prcfo- ' r^' "' '^' '"'^ '^^"^^^^^ «WJy the defect by prehAing cyphers on the left hand. * EXAMPLES. 1. Multiply 3.141592 by 52.7438. 3.141592 52.7438 25189736 9424776 12566368 21991144 6283184 15707960 ^ns. 165.6995001296, 2. Multiply .15 by .3. .15 t^ns. .045 DIVISION. RULE. Divide as in whole numbers, annexing cyphers to the di- vidend when necessary, observing that the divisor and nl tienc musi together contahi as many decimal figures "as are M numbers. 3 for deci- both Fao- i the pro- defect by DKCIMAL FRACTIONS. 5 tontained in the dividend. If at the conclusion of the work the divisor and quotient do not contain as many decimal figures as are contained in the dividend, the deficiency must be supplied by prefixing cyphers to the quotient. EXAMPLES. 1 . Divide 66 , 993548 by 27 . 4. 87. 4)66. 99354»(2. 44602 w3ns. 548 1219 1096 1233 1696 1375 1370 548 548 «. Divide .45695 by 12.5. 13.5).45695Qa{.0S6ii56»an». 375 819 750 695 635 700 625 ^50 760 ft m f ' W W ? I <. f ® J^ECIMAL FRACTIONS. REDUCTION. To reduce a Vulgar Fraction to a Decimal of the same value. RULE. Annex cyphers to the numerator, and .livide hv the Ao EXAMPLES. 8 1. Reduce — to a Decimal Fraction. 4 4)3.00 . 75 ^ns. 5 2. Reduce _ to a Decimal Fraction. 64 64)5. 00(. 078125. ^W5. 448 520 512 80, &c. Every quantity may be considered as a fraction of a lar^ ger quantity of the sam. kind; as an inch is the 1 of afoot, a pole or perch is 1 of a rood, or -i- of an acre! ^c, and may be reduced to a decimal fraction by the preceding rule, observmg that the given quantity is the numerator ff the fractK>n and the number of that denomination contained in the higher denomination is its denominator. EXAMPLES. 1. Reduce 9 inches to the decimal of a foot In this example 9 is the numerator, and 12,'the number of inches in a foot, is the denominator; thus : -. Tho opera- tion is as follows ;— ^^ 1^)9.00 . 75 dnsi M DECIMAL fractions/ 7 2. Reduce 20 perches to the decimal of an acre. In this example 20 is the numemtor, and 160, the number of perches in an acre, is the denominator. Then 160)20. 0(. 125 Ans. 160 400 S20 800 800 When the given quantity is of different denominations, reduce them to the lowest denomination for a numerator. The number of the same denomination contained in the in- teger will be the denominator. Then proceed as above. KXAMPX.ES. 1. Reduce 1 rood 14 perches to the decimal of an acre. r. p. 1 14 40 rr,, , ^, H perches, Numerator. Then 160)54.0(.3375 w2n». 48 600 480 1200 1120 800 800 2. Reduce 21 min. 54 sec. to the decimal of ft dejiree. 21' 54" 60 60 3600) 1 31 4. 0(. 365 .^n*: 1080 13M Numerator. 60 60 %wQ JDtuoiiiinator* 23400 21600 18000 18000 3 ! DECIMAL PRACTIOKS, To determine the value of a Decimal. KVLZ. Multiply the decimal by the numhpr «<• inferior denomination contaTned r^h! /'"' '^ ^^' "^^* in the product as many pllce' for d '"T' ^^'"^'"^ ««* hand, as tho given decfm^rc J«ts o ".t' '^ *'' ^'^'^ gures on the left hand of thp .W , • ^ ^^"'^ «>• fi" ger number, xvhile the fi^ur.?'T ?"'"* ^'" ^« «» '"de- cimals. Then muldnfv Z^^ 'f ' ^'^"^ ^^'" ^^ de- parts c.mtained in tKLinl?"'' '^ ^^^ ""-'^er of off as bef,re. Pro e d" h tuUtt^ ^"^' "^^^ tlenomination. ^ " '' ^^^"^^^ to the lowest EXAMPLES. i What is the value of .6 of an acre? • 6 4 r. p. ^ins. 2 16. 2.4 40 16.0 «. What is the value of. 175 of a Pound? 9, d. *^ns, s 6 -175 20 3.500 12 6.000 «• What is the value of .« of a degree? .42 60 25.20 60 vJA*. 25' 13' 13.00 ^f the next pointing off J the right gure or fi- 56 an inte- viH be de- lumber of and point the lowest THE EXTRACTION OF THE SQUARE ROOT. The Square Root of any number is the quantity or num- ber, which, when squared or mulfiplied by itself, will yield the given number as its product. Thu.s, 4 is the square root of 16, as 4 squared or multiplied by itself will yield 16 aa it3 product. 16 is also the square of 4. To Extract the Square Root. RULE, Point the given number into periods of two figures each, beginning at the units place; then find the {greatest number the square of which shall be equal to or les than the first period, or the quantity before the first point towards the ^eft band. Place that number in the quotient. Write the square of that number under the first period, and subtract. To the remainder bring down the second period, and call the whole quantity the resolvend. On the left hand of the resolvend write the double of the figure placed in the quotient, after the manner of a divisor. Enquire how often this divisor is contained in the resolvend, omitting the figure in the units place of the resolvend. Write that number in the quotient and also on the right hand of the divisor. Multiply this divisor by the figure last placed in the quotient, and sub- tract the product from the resolvend. To the remainder bring down the third period for a new resolvend. To the last divisor add the figure last placed in the quotient and write the sum on the left hand of the resolvend. Then pro- ceed as before until all the periods are brought down. The quotient will be the square root required. iVofe.— When there is a remainder at the termination of the process after the last period has been brought down the operation may i,e continued at pleasure by annexing periods of cyphers for the formation of new resolvends: remember- ing always that all Inu figures placed in the quotient after the annexation of the first period of cyphers, are decimals. 10 ■ if THE EXTRACTION OF THE EXAMPLES. i' VVhat is the square root of the 5 S«ll/ABE HOOT. square numl>cr i(H5 ^ 20.35 (45 ^ns. 16 35)425 425 2. What is the square root of 22071 204.? 22.07.12.04 (4608 »^,w. 10 86)607 6 516 929)!)II> 9 8361 9388) 75104 75104 ^Vhat is the square root of 180000000? 1.80.00.00.00 (13416 ^rw. 23) 80 8 69 264)1100 4 1056 2681) 4400 I 2681 26826)1 71 900~ 160956 10944 '"■^ GEOMETRY. DEFINITIONS, 1. Geometry is that science which treats of the proper- ties and rehitions of inagnitu(le.s. 2. A Point is that Avhich has posi^ion, but not inafrnifiule. 3. A Line id that which 1ms lengtJi, without brcadtu or thickness. ^' B- — The extremities of a line are points. 4. A Straight or Right Line is the shortest line which can be drawn betweeen two points, 5. Every line which is neither strai-.. „ „i,vi,., „,. „ p„,, .,,. „ ,,„,, cr,l,e,l <„,„, ,l,e ver.c;. of „„y „„^,„ ,„ ;„ „,^ ^ be.- of degree.,. ,„,„„,«, &e., contained in ,l,c arc of .1,^, ••.rcle, ,„.ercep.e,l between the line, forming the angle, i, he measure of that angh.. Thus in the figure at definition R ho number of degree., minutes, &c., containe,! in the arc ^ U, IS the measure of the angle C F D. I 20 I^^'^ALL^i^ or Collateral Lines are lines equi-dis- J tant from each other in all their parts, or lines which, being . .n the same plane and produced ever so far both way. wiU never meet. • ' \ bytatr,;:::."" """"" '-' "■■'- -'-^ - "<-<-«'' 22. MixT,L.»EA.. F,o„BEs are those which are bounded partly by straight and partly hy curved lines. 23 Thilaterai. Figures or Triancrs ,re those which are contained by three straight lines 24. (JcADRiLATERAL FiouRE, arc thosc wlfich arc bound- ed by four straight lines. 25. Mur,TiLATERAr. FiouREs or P„,.vgom are those which are bounded by more that four straight lines 4T .ure of five sides is .sometimes called a ittg:; olsf," sides .Hexago,,, &c. If their si.lcs are all e.,mil they a ' .lulil'-^X^. I:;^-™" -'-»'"'■'■•■••-'.- -e. A B is called the t." ™i cT'Z"'"'' '" V- "' ■'''"• -''" angle nbt SO. An Obtuse- usf anfflo. (P ANGLED Trj^n lil\ Ui.) J>erpen(licular. i-r.K is that whioh h as an 'I ; *i^ : ^1^ f4 GEOMETRT. 31. An Acute-angled Triangle is that uhich has thre« acute angles. (Fi^. 17.) ^^^** 32 Of four^sided figures, a square is that which has all sfl equal, and all it. angles right angles. (Fig , V" 34. A Rhombus is that which has all its sides equal, hut •t« an^Ues are not right angles. (Fig. 20 ) ^ ' ■" 35.. A Rhomboid is that which has its opposite sides equal («^. ily"'" '"""'""''*'' """"^^ °™ ^■•'"'^'^ T„.„..„„». _37. A srraishtn„c, joining the opposite point., or angle, of a quudnlatoral %„rc, is called a D,Aoo^ „ Sr.a.e^thanr.vo'Ag ft al "te "i,t ^ /" f ^ """ «'''? '' '' o'"^ aiij^fcs, ir IS said to be rc-entraiit. ';'• ;^"^ ^^^•^ f ^ yocXa\no^.,l figure n:ay be called the base >1.K 1 he angular point, opposite to the base of a triangle 40. -rHE Altitude of any triangle or parallelogram, i« a I J 'it POSTULATES. 1- Let it be granted that a right line .na.v be drawn from any one point to any other point; 9. That a terminated straight line may be produced or '-•ont.nucd in a straight line at pleasure; and »i.h any radiu,,. '' "" "'"" ""'' """■<•■ ->* ^ withou I in the t 7 t5. A ' or inon '' 7.. A ^ ch has rhre« ^hich has all i^ig. 18.) has all itt« ' all equal. equal, bur sides equal r its angles an inclined 5 are paral- lAPEZIUMfr. or angle* s less thair when it 't» drant. 1 the BASE. a triangle. called the grarn, it) a side upon iwn trom duced or It re. and OEOMETRT, jj AXIOMS. 2. If equals be added to equals the wholes are equal equal ''" ''"'" '"" ^^"^^^' ^^« ---nders are 4. If equals be added to unequals, the wholes are unequal ^^^5.^ If^equals be ta.en fro. unequals, the re.aindrar. r^"^:""^ ""'' ''''''' ^' ''^ -- *^-^' are equal ro Le™X:.''^' ^" '^"'^^^ ^^ ^^^ -- ^^-^' - equal role'Sf '''''''' ^^^"'^ '" ^^^ -- ^P- - equal 9. The whole is greater than its part. 0. All nght angles are equal to one another, i. 1 wo straight lines cannot be drawn throimh th« , A rrr Explanation of Terms. - bythe letter B, a, dthTL{ ,■ '""""'''^ '*''"<'''''' « -r- is the sign of division r* - the two quantities betw:;" wh h'T''''. '''''' '''' '«"»-• «*' <^-' by the latter, and he "^ot '' ^''^''^ '^' ^« ^^ ^'^•- ' ^''^ ''^^'•^ expression denotes the quo- ^ IB OEOMETRF. ..;^i ii i II ^t m ■ -I'l t.ent ^vli,ch will result from the division of the former quan- tity by the latter, thus A ~ B indicates that the quantity re- presented r>y A, is to be divided by the quantity represented by C. Divjsion is also frequently expressed in the form of a fraction, by writing the quantity to be divided above the quantity by which it Is. to be divided, with a line between them; thus — expresses the quotient of A divided by B. : :: : is the sign of Proportion. Thus, A: B:: C- D denotes that A bears the same proportion to B, which C beai-s to D, and is read thus, as A is to B so is C to D. A= denotes the square described on a line A. If the line IS expressed by tAvo letters, A B, then the square described upon It IS denoted by the sign A B^. The principal signs employed in Algebra are the same with those exi,lained above. It may be observed, however, that Algebraists generaUy employ the small letters of the al- phabct in their calculations. Instead of X the sign of multi- plication, a dot ( . ) is frequently employed in Algebra, or the letters are written together without any sign between them. 1 hus aXb,a.b, and 06, all express the product of a mul- a-\-b — c tiplied by b, ax expresses the quotient which re- sults from the division of the excess of the sum of a added to 6 above c, by a multiplied into x. It is read thu.^, a plui 6 minus c, divided by a multiplied by x. x = a~{-b — c shews that the quantity represented by x is equa. to the excess of tlie sum of a added to b above c. It IS read x equal to a plus b minus c. y is called the radioal sign, and. denotes that some root of the quantity before which it is placed is to be extracted. 1 hus, Va denotes the square root of a. Wa denotes the cube root of a. Instead of the radical sign, a fraction is aom.etime.s employed; thus, ffi denotes the square root of a.. e extracted. denotes the a fraction is re root of a GEOMETfilCAL PROBLEMS, fVhi PROBLEM I, To binect a right line A B, (^Fig. 23.) From A and B, as centres, with any distance A6, Ba .roa,-- er than hal the line A B. describe two arcs, c a Ln/cZ cutting each other zn c and d. Through the points of inter- section . and rf, draw the line, . . d, cutling A B in . I hen \e =; eh. PROBLEM II. To raise a perpendicular from a given point C, in « ^ ,» ^«,.oj i- . :^n line A B, to make an angle equal to a given angle E. From the point E, with any radius describe an arc meet- ing the lines containing the angle E in a and h; with the same radius on the ])oint A as a centre describe the arc c d: apply !he distance a b on the former arc f. the arc c d, from d to c; then through the points Ac draw the line A D, which will form an angle with the line A B, equal to the angle at E. ^ •EOMKTar. PROBLEM V. To Hra. a .trm.kt line through a ,i,en point parallel to a ^'ven s'-aighl lijic. l,M A (Fig. .29,) bo the given point nn,l n r ,1. ■ Fron, the point A draw a stmij;ht line meeting the li„„ t ar.rra'T.^'i'o'c' "t" t •■""" - -' ■- -^ -' ^" I PROBLEM Vf f.ct At (Rff. 30,) be the given .straight line A C a„U C B td' h '«;';: r Bc'^wi,;?™,,'""'' ""' """ quired. oure A B c will be the triangle re^ M PROBLEM VII Vo construct a triangle, the sides of y^hich may he eaual to three g^.en straight lines A, B, and C. /i^. T ' I-'ay ofl'astrai'^ht lino DP « i '^ -aigh. ,i„e, aI'Ld a^fere'lwtr: °',""' ^'™" 'o another of the, iven line. B, descHh "'a:e™rET°' .iv„';:'trhT^r:;;'eTa*^L'r''a,iL":"^''"^ - "f--' PROBLEM VIII "" ^'^ ^^'"^ -^ ^-- '^ D perpendicular and equal to |i:K; OEOMKTnr, e« parallel to a 3 C the given 'ing the lino I angle D A K straight line o the straight straight line. t ^ B dejcril>e 'aw the lines triangle re«- ' be equal to Fig. SI.) f the given idius equal To/n E as a ven line C, ; then join luired. two of the d equal to A B: tli<..n from the points A and I) with the di.stanco A B or A D, describe arcs intersecting each other in E Draw the lines D E an.l B E, and the figure ABED will Ih- thf, «qui,re required. PHOBLKM IX.. I ToJinJa third line proportional to ttoo given straight lims I A and B. {Fig. 33.) I Froui any point C, draw two straight lines, tHe one C D -equal u> A, iho other C E equal to B running any ang:e;. .|.nnD E: produce C D and C E; lay oif D F equal to B. or C E, then draw F K paraMel to I) E, meeting C E pro- duced .n .., the line E K will be a third proportLnal. ' PROBLEM X. To find a fourth proportional to three given right line, A .. B, and C. {Fig. 34 ), Draw two straight linos D R, and D F fonnin. an> an^r K Bl, upon the line D E lay off D G equal to A, and GE jojn G H: from the point E, draw E F parallel to G II meeting the line D F in F; then H F will be a fourth pro: portional to the lines A, B, and C.f ^ PROBLEM XL To find a mean proportional between ttoo given strairht line. A and B. (Fig. 35). Draw any right line C D, on itlay off C P := A and P D ^. I {.e tru.„.le C F K. according to Euc^;" , 'r d • ^.' " r v' J^. K; '^at according to th. construction CD ~ k u' H r ' ^.^^ a|=K; therefore A : B : : B • E K " " ^ ^ut --risect the transverse axis A B in E, and also be perpendicu- lar to it.. On O P lay off the conjugate diameter C D bi- sected in E by the transverse axis A B; then find two points HI the right line O P such that if one foot of the compasse* jbe placed in them successively, the other will describe arcs |passu.g through the points C and D, the extremities of the jconjugate axis C D, and also touch the circles G B F and |(i A F in the points r, s, t, u. Draw the arcs t* C r, and HJ D s, and a figure will be tbrmed sufficiently near an ellipse Jfor the mo«t of practical purposes. Where greater nicety pr accuracy is required the following method may be adopt- I v'^ ■I 1*1 ■i^ GEOMETRY. I ^ m- Another method to describe an Ellipse. Draw the transverse and conjugate axis A B, C D, {Fig. 47,) bisecting each other perpendicularly, then with half th»^ longest diameter as a radius and centre C, describe arcs cutting A B in F, G; the points F, G, will be the foci of the ellipse. Then take two pins and fasten a thread upon them in such a way that when the thread is stretched the dis- tances between the pins shall be equal to the length of the transverse axis A B. Fasten the pins in the foci F, G, then by moving a pin or pencil round within the thread and keep- ing the thread always stretched by it, a curve will be traced out forming the ellipse required. PROBI \U XXIV. To project lines of Chords, Signs, Tangents, Secants, ^'C, to any Radius. On the line A B, (Fig. 48,) describe the semicircle A D B, Upon the centre C, erect the perpendiculr.r C D, continued at pleasure to F; through B draw B E parallel to C F, and consequently perpendicular to A B; and draw the right line D B. Divide the quadrant D B into 9 equal parts, and with one foot of the compasses in B, and the distances B 10, B 20, B 30, Stc, on the curve line B D transfer them to the right line D B, and it will be a Lvne op Chords. From the points 10, 20, 30, &c., on the arc B D draw a line parallel to D C, and it will divide the radius C B into a line OF Sines, reckoning from C to B, or of "Versed Sines, if reckoned from B to C. From the centre C through the several divisions of quadrants D B, viz: 10, 20, 30, Stc, draw right lines, until they meet the line B E, and it will be a line op Tangents. Transfer the distances between the centre C and the di- visions on the line of Tangents to the line D F, and it will give a LINE OF Secants which nmst be numbered from D to F. In the figure the divisions are only given to every tcntli degree; but by subdividing each of the 9 divisions, we niav 'i'lvo a ii -rt;e.s; a: 'i'lve sea Scales fiiiniatur( known m "'•ale an,si A ijl bf la ■1^ iiivator llicll he f; '-Pf't'ified ti'iucd oni fiif'la.st (Ii i first or pr ;«*is the first i'arts. ']'] ^0. -10, &.CV .4, &,c.; or than the s they may r Will stand 1 .? Of scales Ipery case C* ''Jiles of e( ^ f equal pa |d into Urn pressing the #ided on th? * usually cl **t the letter laces of fig determined j ide. The most inches, or ^'J having o C D, (Fig. nth half thr ;scribe arcs B foci of the 1 upon them led the dis- ngth of tho i F, G, then id and keep- ill be traced kcanls, fyc, ircle A D B, ), continued to C F, and tie right line rts, and with 5 B 10, B 20, to the right ) draw a line I into a LINE SD Sines, if iions of i, t lines, until Tangents, and the di- ', and it will ered from D every tentli ons, we mav GEOMETRY. C<,xcERN-,s„ Soles op E,ial P„.ts. fCiil-s of pq,j„| p„rts am nothino- more fh„. ™n,a..„, e,„p,„,„., r,. laying doC;™':"''"-^ '" Known measure ,„■ chains, var,l fee, &e , L "' -«le .uiswering to one el ,in „„ / '^ '"'" ™ ""' ' -■'.•..,.■ number of par s in „° teh " "'""'^ " ™»"'"- - ""•'«! <.nw,.r,l i„ ,he"a^ ' " ■• T'- "^ """* P""'' ''« ■•"- '"-on.Hn,arvaiv^:itt,c "/;'""' r'r^^ '•'■< fh<. first line has liPm ,. *• ™ ^' 2. 3, 4, &c., as far i«-. T,,;, :,!::,;" :" ":t:' "'" "^ " ^""'^ '"■«-' ».J., .., an., .„e„ ..e'^n^-.tt:; 'S I~ /"; ^«- "• «-c.i or fhey may represent 100 oflO Snn Tor, . ' ' ' •■•ale; of equaloar ,""■"""'""' »'•« '<> l« '"und o.;..a«a, pX '•::":■;;: ,:;'s™' -'« -'%' '^ « -'^ e'l into lines comn encinn^ w!. ..' ^ °"^ ''^^ ^'^ ^''^^^J- ^i'Jecl on that scale or 11^^^. T '"^'"^ "" '"^^ ^''^ ^'" fc usually changed in to a I'ine of hf f ^'^^ "P^'^^ '"- It the letter C When thl ^''"^'' '''^''^ oommences i-s of figures t^Ce if triro'^rf "' ^^'^^^ etermined accurately hv th. V " '^' P'"^*' '""^ ^^ ^^^ ately by the diagonal scale on the opposite 1 The most useftil scales for « «J,... inches, one of who e si •7"'''''' ''' '^"'^^ «^ ^2 or ''^''-'ingonthe t;^^^^^^^^^^ '/ -'^ ^'- other convex. ^ ^'tk th divisions and subdivisions ^0 GEOMETRY. marked on the edges, nnd continued to the end of the t^calo. On the centre Avill be found the numbers 20 and 40, 25 and 50, &,c., distinctly marked. (See Plate.) By means of the scale of equal parts it is easy to measure any line laid down upon a plan if we only know the scale by which the plan has been drawn, and also to lay down any distance upon any giver, scale. f the BC.alo. 40, Q 5 and to measure iW the scale y down any -H- I i i &P v,'-~- 2^ -^ ^ Pi I I ^ S2 :- I J J- LOGARITPMS. .1. About the end of the sixteenth century and the bopfinningf of the seventeenth, several Mathematicians began to con- eider by what means they might simplify the arifhmetic^il operations of multiplication, division, and the exLraction ot roots, which formed no inconsider.ible obstacles to the im- provement of those branches of knowledge; in the prosecu- tion of which tedious calculations were indispensable. For abridf'ing these calculations several ingenious expedient* were suggested. Of these by far the most complete, was the system of numbers called Logarilhms, invented by John Napier, Baron of Merchis^on, in Scotland, and afterwards improved and extended by Mr. Briggs, and others; forming, doubtless, one of the happiest and most useful contrivance!*, of modern times. Let two series of numbers be formed, the one in geome- trical progression, whose first term is unity or one, and the common ratio, 2; and the other in arithmetical progression, whose first term is 0, and the common difference 1, thus:— ^ Geo. Prog. Jlrith. Prop;. 1 2 1 4 2 8 S 16 , A 82 5 64 .6 J 28 7 256, &.C., . S. &c. .m •rilaM 4fi I 32 LOGARITHMS, The t.nm ,n the arithmetical series will l,e the lo.ar.th.ns of the corresporuhng terms in the geometrical series; that is .s he loganthm of I, and 1 is the logarithm of,, J ^Z the logarithm of 4, and 3 is the logarithm of 8, &<• ' s«mrf?V'"'T' ^'•'^'" ^ "^«^»««t''^ inspection,' that the sum of he logarithms of any two nu.nbers i.i the fo.e.oin. ser.es, .s ec^ual to the product of the numbei-.s themsSv "^ f^ or exan,ple the product of 4 multiplied by 32 is 1 28. Now he number m the arithnietical series coiTesponding to the ei-m 4 ,n the geometi-ical series, or in othe'r words oganthm of 4 is 2. In like manner the logarithin of V] is ^' Again 2 added to 5 is equal to 7, and the term in the I'lTr'tol '"'"'u "°''''^'^P'^»' «" quired. * "^'iAta ^\iii by the Log. re- Kv. Required the Log. of 36548. quired. "= i-5028Gi, the logarithm re- ». - aiiy j;;,:!!: f-^-^. -j'-^ ««. of ,,,„ , ,„. „^ „^ "aw ol'lhc rcnminilor. '""' '° "''" I to tl,e unit ■V. To Ma ae U,. of a Vulgar Fra.,Un, or „/„„.,„, number, o.ot':i;':;:;'S::™''''''' "" " '^-"""'' ">" "■- p™- V. ToMUme natural nu.,^,r corresponding ,o a ,f,en t( "ithm *'Ook for tJie jriven ^ mr u r 'he o.-s„c( i,„M.i,i„., ,i,. ' ""^ '" "'" '"'I'-"- It vou fin.) ■ lett hand of the page mafked \, I^.OOARITIIMS. .11? 6r No. TUni i{ thr index of the iriven Los?, be los-i than .% cutotr froi,. thori-lit haiul of the iiuiribcr ': ..,d nn many fif,'uro.^ as the index \h h>.ss than 3, the rigun . . c.it off uill he deeimals and the reniaiiuh'r a Nat. No. or Nut. Nos. Thus, if the Nat. No. eorrespoitdinir to the logurithin -2 S^Hi- 950 he required: the K.^.3i(i<>.50 being found in the taWes, opposite to it in the left column is 21i2, and at the top and bottom of the cohnnn in whieh it is found is 3, whieh, plaeed nfter '212, gives 2)ilS: and since the iiuh-x 2 is hss bv 1 than S, one figure is to be cutoff from the right hand as a'ch-eimaJ wh.eh wdl give 312.3 a^ the Nat. No. corresponding to the given h.g, 2.3-:J6950.. If however the in,h3x exceed 3, annex Its many cyphers to the number found as the index exceeds* •1, and you have the Nat. i\o. retjuired. But if a Logarithm exactly corresponding to the siven Log. cannot be found in the table, take; the Log.- next to the given one an(J less than it: then take the .litferonce between that arid the given Log., to which annex cvjihers and divide it by the tabular difference found opposite to tliteLo-. which you have taken iVom the table. Annex the quotient t.vthe Nat. Nos. corresponding to the F.og. taken from the tabic and place the decimal point wherever the index points c»ut and you have the Nat.- No. required, For example :-To find the Nat. No. corresponding to the Log. 4. 478309 • the Log^nearest to it, and less than it, is 478^78, answering to the Nat, No. .3008. The difference between 478278 and^he g.ven I og. 478309 is 31 . By annexing cyphers to this num- ber ai.d d.v.dmg by 145, the tab. diff. f„und opposite to th« Log. 478278 you have for a quotient ,:3 +, JLu an led o figures 300,3 mak,, 3008.132 +. 13nt the index 4 shows hat th.^e can only be five places of whole numbers. Tul decnnal ) mt bemg therefore placed after the fifth figure .v.s3.^2.3-Ha Jf the nmnber acquired is to consist altogether of decin.al figures, the sn? ne method must be used to obt Jiin it as direct- od above; only observe, that 9 cyphers, less the index »>4? prefixed to the No. found. Tb are to li us, to find the decimui I:i. 11 . .Sti LOGARITHMS. So. corrospomliiifr to the Log. 7.819083; look in th« Table <"or th«' Log. 819083, nnd you will find the corrcHporulinf? Nat. No. to he 6593 Now 9 cyphers les.s the inde.x 7, leave two cyphers to he prefi.xed to the No. found, giving .00()59.S HH the decimal number answering to the Log. 7.819083. VI J. To find the Arithmetical complement of a L „nnthm. The arithmetical complement of a Logarithm is the loga- rithm of the recii)rocal of the corresijonding natural number, or it is the number it wants of 10.000000 or 20.000000. To find it, begin at the left hand and subtract every figure from 9 except the la.st significant figure, which is to be subtracted tVom 10. If Uic index exceed 9 it is to be subtracted from 19, or if it be negative, it is to be added to 9, and the rest subtracted as bei;;re. In taking the sum of the Loga- rithms, observe that for every arithmetical complement em- ployed, 10 must be subtracted from the sum of the indices, in order to obtain the proper index of the result. The arith- metical comi)lement is frequently used in proportions, and in trigonometrical calculations, to change subtractions into additions. iMDLTIPLICATION BY LOGAllITHMS. RULE. Add the Logarithms of the numbers to be multiplied and Iheir sum will be the product in Logarithms. If there be negative and affirmative indices, their difference, with the proper sign prefixed will be the index of the Log. of the product. If, in any consequence of either of the factors or of both of them beinfe decimals, the index of the sum ex- ceed 10, reject the 10, and the remainder will be the index ^>f the Logarithm of the product. EXAMPLES. I. Required the product of 23.14 multiplied by 5 062 J he Log. of 23.14 is 1.364363 J' o*. 1 he Log. of 5.062 is 0.704322 Hroduct, U7.134. 2.068685 Log. ofthe ^roduct. A? Log. ( Log. o] Quo. Here ▼isor i^ I in thn Tabh; ^orrt'Mporulinjj index 7, leav«; iviiig .00()59.-{ 7.819083. !» L- „anthm. II is the loga- urul number, 000000. To y figure t'roin )e Hubtraeted e subtracted to 9, and the f the Loga- pleinent oni- ic indices, in The arith- ortion.s, and 'actions int(» IMS. iltiplied and If there be ;ej with the Liog. of the e factors or lie sum ex- 5 the index LOOtRITHMS. 37 y 3.06-2. Product. 2. What is the continued product of 9. 9n»J, 597 16 and 081173." The Log. of 3,002 is 0.. 591 ^287 or 0.50 1 -28" 1 he Log. of .':.:)7. IG i.s 2. 770091 or 2.776091 Tho Log. of .O.JM7.i i.s 2. 497938 or 8.4979.^8 Troduct, 73.3357 1.865316 1 .865316. rejecting 10 from the in«lex. , DIVISION DY LOGARITIP'S. RULE. From tho Logarithm of the Dividend sub;. act th<^ Loga- rithm of the Divi.sor, and the natural number answering to the remainder will bo the Quotient required. If tlic Log. of the divi.sor exceed the Log. of the divi- dend, proceed as before until you come to the index. If the decimal jiart of the Log. of the divisor exceed the decimal part of the Log. of tiie dividend add 1 to the index of tlie r.og. of the divi.sor. Change the .sign of the index of the r.og. of tho ,.', add the indox of the Lo-. of ;he divi- dend to it, and with its i)ropcr .«ign prelixcd it will be tho index of the Log. of tho quotient; or, when the index of the Log. of the divisor exceeds; the index of the Log. of the di- vidend, borrow 10, and the remainder will be the index of the Log, of the quotient. EXAMPLES.. 1. Divide 4768.2 by 36 954. Tho Log. of the dividend 4768.2 is 3.678.«J54 I he Log. of the divisor 36 . 954 is 1 . 567661 Quotient, 129.0307 2. Divide 4.6257 by .17608. Log. of 4.6257 is jO^^ 665 177 0.665177 of .17603 is 1.245710 or 9.24.5710 2. 110693, Log. of Quo. Log Quo. 26.2704 1.419467 1.419467, Lo^. ofQuotient. Here, in thr ^ st case, the J^ndex of the Log. of^ the di- Ti8orischangeuf»om-~l,or l,to-j-l,or 1, and the index of I.0GARITH5fS« the Log. of tho dividend, 0, being added to it gives Las the index of tho result. In the second case, 10 is borrowed for the nidex of the dividend, and the index of the divisor being Hubtractcd from it leaves the same result. 3. Divide .19876 by ,0012345. Log- of . 19876 isT.i>98329 or 9.298329 Log. of .0012345 is~3.091491 or 7.091491 Quo. l(U.O044 2.206833 2.206838. Log. of Quo ■ Hero again, the index of the Log. of the divisor is chang- ed fi-om - 3, to + 3, and this added to - 1, the index of the dividend, gives -t 2, or 2, as the index of the Log. of the Quotient. RULE OF TF!IEE, OR PROPORTION BY LOGA- RITHMS. nuLE. From the sitm of the logarithms of the second and third terms, subtract the lo-jfarithm of tho first term; the remain- der vviir be the logarithm of the fourth term: or, add the amhmeticar complement of the first term to the lorarithins of the second and third tc is, and the sum, after subtracting 10 from the index will b( he logarithm of the fourth term In any Compound Proportion the term sought may be fomid by subtracting tho sum of the logarithms of all those tcrnis which, when multiplied into each other, are to form the divisor, from the sum of the logarithms of all the term« which, when multiplied into each other, form the dividend »he remainder is the logarithm of the term required Instead of subtracting one logarithm from another, yoo may add the arithmetical complement of the subtrahend to the logarithm of the n.inucnd, and reject 10 from the index of the sum. EXAMPLES, SoiB 857 486 og* 2 55 259 Z l'""^' SSi^^a jiui,, ^.ojozjj or Los:, 2.553259 To 1 a, 1483 1 095103 1.095108 I \, BY LOGA: rOGARITHMS. 8. ^ind a third proportional to 12.796 and 8.24718. As 12.796 Arith. Coinn. 8.892926 3.24718 Log. 0-51I5C6 39 Is to So is 3.24713 Log. 0,511506 '^° r,. fT\ 1.915938, third propor. 3. I* md a fourth proportional to the three numbers, 36 48 and 6Q. Multiply 48 Log. 1.681241 by 66 Log. 1.819544 Divide the Pro. 3168 3-500785 l)y S6 Log. 1.556303 Quotient, 88 ,.944482, fourth proportional. INVOLUTION BY LOGARITHMS. nuLE. ' Multiply the logarithm of the given number by the index of the power, and the product will be the logarithm of the f ower sought. ga^TC l,v7„'"aSnn.f,'^'"'*' " \'>e^'''!'>^ "hose index is ne- «ne, as the ind^KTlh^^Tj',''''^ as many cyphers less EXAMPLES. J. Required the^square or second power of 25.791. 25.791 Log. 1.411468 25.791 Index. g 665.175 2.822936 ^XlHr *'r "^"^'^ "' *^'^^ Po^^er of 30.7146. 30.7146 Log. 1.487345 30.7146 Index q Cube, 28975.7 4.462035, Log. of Cube or Srd ptmer. ■ ) ! ; i w'. i is v^i i-^ 40 LOGARITHMS. 5. Required the cube or third power of .008. .008 Log. y. 903090 .008 Index .000000512 77709270 Here, the index of the Log. multiplied by the index of the power gives — 9, but as the number 2 is to be carried from the decimal part of the Log. this reduces it to ~ 7, as above. 4. Required the fifth power of .2. .2 Log. 9.S01030 Index 6 .2 .00032 46.505150. In this example the affirmative Log. for the decimal frac- tion is used. The excess of the product of 10 multiplied by 5, the index of the power, above 46 the index of the Log. is 4. This number, less one, that is 3 is the number of cy- phers which must be fixed to the natural quantity corres- ponding to the Log. .301030. EVOLUTION, OR THE EXTRACTION OF ROOTS BY LOGARITHMS. RULE. Divide the Logarithm of the given number by the index of the power, and the quotient is the root required. Note l.—When the index of the logarithm is negative, and the divisor is not exactly contained in it, increase the index by the smallest number that will make it exactly di- visible. Carry this l)orrowcd number as so many tens to the left hand figure of the decimal part of the Logarithm. Then proceed with the division as usual. Note 2.— When alHnnative indices are used for the loga- rithms of .decimal fnictions, prefix to the index of the Log. a figure less by 1 than the index of the power; then divide the whole by the index of the power. EXAMPLES. 1. Required the square root of 365. Index of the power 2)52,562293 Log. of 365. The root required 1.281146 Log. of 19.105 ^fi». the index of to be carried it to — 7, as [ecimal frae- nultiplied by f the Log. is imber of cy- ntity corres- IF ROOTS jy the index red, is negative, increase the t exactly di- y tens to the ;hmi. Then 'or the loga- of the Log. then divide 05 Jlnt, LOOlRITHKfl, 41 5. Required the Cube Root of 12345. Index of the power 3)4.091491 Log. of I2S45 1.363830 Log. of 23. 1116 the rwot required, 9. Required the Cube_Rootof .000000512. Index of the power 3)7.709270 Log. of .000000512 3. 903090 Lo^ of .008 the root re- quired. Here the index of the Log. is not exactly divisible by the index of the Power. Two, the smallest figure rvhich will render it exactly divisible, is added to it, making it 9. Thi« two IS then carried forward as so many tens to the decimal part of the Log. Say 3 is into 27, &c. 4. Required the fifth root of . 00032. Index of the power 5)46.505150 Log. of .00032 9.301030 Log. of .2 the root re- quired. Here the affirmative index to the Log. of .00032 is t) to which a figure less by 1 than the index of the power, that « 5 — 1 = 4 is to be prefixed, making AQ as above. Explanation op the^ Tables op Logarithmic Sines, Tangents, &c. From the manner in which Lines of Chords, Sines, Tan- gents. &c., .-ire projected, {See Prob. 23 of Geometry,) it is. evident that if the Radius consist of any number of equal parts the Sine, Tangent, Secant, &c., of every arc described on that Radius, bearing a determinate proportion to it, must also consist of a determinate proportional number of these equal parts. The computation of the number of these parts in the Sines, Tangents, &c., contained in every arc of the Quadrant, form Tables of Sines, Tangents, &c. In this lorm thnv nm n..n^.i Tvr„f i o- ^n, ' ---^. — ,„,!^vt iiatuiu! oincs, langents. So- cants, &c., and the Logarithms of these numbers give u^ 1 ables ot Logarithmic Sines, Tangents, Secants, &c... & s 41^ LOQAUITHMS. |! I To find the Logarithmic Sine, Tangent, ^e., of any num- ber of Degrees and Minutes, If the number of degrees given be less than 45, look for them at the top of the page, then look for the number of given minutes, in the left hand column; opposite to which, in the column marked Sine, Tangent, &c., you will find the Logarithmic Sine, Tangent, &c., of the arc proposed. If the number of degrees exceed 45, and less than 90, look for the given number of degrees at the bottom of the page, and for the minutes in the right hand column ; opposite to which, in the column marked at the foot, Sine, Tangent, &c., you have the Logarithmic Sine, Tangent, &c., of the arc of the specified number of degrees. If the number of degrees exceed 90, take out the Loga- rithmic Sine, Tangent, &c., of its supplement, that is of an arc consisting of the number of degrees contained in the re- mainder, wliicli will result from the subtraction of the given number of degrees and minutes from 180''. EXAMPLES. Arcs. Sine. Co-Sine. Tangents. Co-Tang. Secant. 18° 15' 9.495772 9. 97758G 9.518185 10.481815 10.022414 64° 5^' 9.957040 9.627030 10.330009 9.66999110.372970 The Natural Sine for any number of degrees and minutes will be found most readily from a Table of Natural Sines, the arrangement and uses- of which must be sufliciently ob- vious from the explanation e*" the Table of Logarithmic Sines already given. When the Natural Sine and Co-sine are known, the Natural Tangent, Secc-y^t, &c., are easily calculated. caiit. EXAMPLES. 1. Required the Nat. Sine of an arc of 23"* 20'. . 396080. 4. Required the Nat. Co-sine of an arc of 87^ 15'. ^n^ 047878. ^n*. V o f any nutn- 15, look for number of e to which, rvill find the posed, lan 90, look r the page, opposite to !, Tangent, &c., of the TRIGONOMETRY the Loga- hat is of an )d in the re- [)f the given PbAWE Trigonometrv treats of the relations sutwiBting- between the sides and angles of plane triangles. The prin- ftipal parts of a triangle are, the three sides and the three angles. The main object of Plane Trigonometry is to give rules by which, when some of these parta arc known, tb* others may be determined. Triangles are either right-angled or obliqut angled. Plane Trigonometry is therefore very naturally divided into twa parts. The first treats of right-angled triangles; and th« »«-, cond, of oblique-angled triangles. Paet I.. OF RIGHT-ANGLED PLANE TRL\NGLES, DEFINITIONS. 1. In a right-angled triangle the Hypotenuse is the sitl* opposite to tlic right angle. 2. The Base is the side opposite to the vertical angle. 3. The Perjjcndicular is the side which forms a right an- gle with the base. Notej,— The base and perpendicular are sometimes called Corq^ary. — in a right angled triangle, if one of the acute angles is given the other angle is given also. Dem^slration.— The three angles of any plane trianf?l» 44 TBIOONOMKTRT. If) H l»l H 1i ■ I isl3^n ^h. .f •• N«^^> when one of the acute angles IS known, the other is at once ascertained by subtractinir the known angle from 90° and the remainder is^ the rnea "f e of the other angle being the complement of the given angi; a e of'fhpth^'"^'r"^'^'/"/ ^^« «f th^ Principal %nn in/* r *^*'*' ^"^''^ ^^'^ «^ the three angles) beiL elven and one of these given parts being a side! the other S Sfth^oi^msfi::'^ '^ """^"^ easily iduced TromThe foC CASE I. t^hen a leg and the angle opposite to it, or when two sidet are given, to find the other part: rie^f^tTririh """^ f '^^ 'I ^. *^^ ^^^"^ «f ''' opposite an- gle, so is any other side to the sine of its opposite aM^le- «nH lots z'leV''' ^'"r' ^">; ^"^»« ^^^ to^^cs^«dt •0 IS tlw sine of any other angle to its opposite side. CASE 11. IVhen the legs, that] is the sides about the right ahgle are given, to find the angles and the hypotenuse : J^heorem.-^As one of the given sides is to the other ffiven Tn^'/?,'" "f'^'"' *^ '^^ tangent of the acute an^le ft the end of the side at which the proportion commenced! PROBLEM I. Oiven the angles and hypotenuse of a right-angled plant triangle to find the base and perpendicular : > EXAMPLE. In the triangle ABC, (Fig. 49,) right-angled at B, given the angle at C, 55'^ 30', and the hypotenuse A C 121 yard.-^ required the sides A B and B C. ' ' According to the preceding corollary / C 56° 80' — / B 90^ === Z A =. S4« 30'. , , To find the side A B. ' • i' As radius, (/. e. the sine of ^ B,) = to 00000 Is to the side AC 121 ^ '^ == "oS So IS the sine / C. fifio Qo' Z I^^i^ To the perpendicular A B 99. 72 yds. =. 7^99871^ .1! 1 TRIGONOMETRT. 46 180". A right ! must together ! acute angles mbtracting the he rneasuie of given angle. >rincipal parts ) being given, le other partsi )m the foilow- vhcn ttoo sidet i opposite an- ite angle : and >pposite side, 3 side. J^hf ahgle are •nuse : le other given e angle at the 3nced. angled plant ulitr : % >. d at B, given C 121 yards ^ '80' — ^ B 0.00000 2 . 08278 9.91599 1.9987X i Here we add the logarithms of the second and third term of the proportion, and subtract the logarithm of the first term from that sum. The remainder is the logarithm of tho fourth term, or the answer. To find the side C B. As radius 10.00000 Is to hyp. A C 121 2.08278 So is sine / A 34° 30' 9.75312 To the base C B 68.54 yds. 1.83590. PROBLEM 11. Given the angles and one side, to find the hypotenuse and the other side. EXAMPLE. In the right angled triangle ABC (Fig. 50,) right angled at B, given the angle at A 85° 30', and the side A B 294 chains, required the base C B and the hypotenuse A C. 90° — / A 35° 30' = / C 54° 30'. To find the hypotenuse A C. To find the base C B. As sine Z C 54° 30' 9.91068 As sine / C 54° 30' 9 91068 Js to per A B 294 ch. 2.46834 Is to per. A B 294 ch. 2.46884 bo isradms 10.00000 So is sine /A 35° 30' 9.7639& To hyp. AC 361. Ich. 2.55766 To C B 209.7 ch. 2.32161 PROBLEM III. Given the hypotenuse and one side, to find the angles and the other side. EXAMPLE. In the right angled triangle ABC, (Fig. 51,) right angled at B, given the hypotenuse A C 3 chains and 50 links, and the perpendicular A B 2 chains and 45 links; required tho angles A and C, and the base B C. To find the angle C. To find the side B C. As hyp. A C 8.50 0.54407 As radius lo 00000 Is to radius ^ 10.00000 Is to hyp. A C 3.50 o:54407 oc IS sme a jj a. 45 0.3891 / So is sine /A 45° 35' 9.85386 f I To sine ^ C 44° 25' 9.84510 To base B C 2.499 0.8979S ir 46 TRIOONOMETRY. The hypotenuse may befoiind independently of the angles; for, according to Euc. = V A B (A B -f BC . 47. we have A C = VA B'-f-BC* This latter form of expression A B ^ A C is by far the most convenient for logarhhmic calcula- tion. From the same property of aright angled triangle, viz: that the square of the hypotenuse is equal to the sum of the squares of the other two sido^ anyone of the two sidea about the right angle may he found independently of the angles, if the hypotenuse and the other side are given or as- certained. For since A C^ = A 13^ + B C=, it follows that B C^ = A C= - A B'=(A C + AB ). (A C - A B); and therefore B C = V (A C + A B). (A C ^ A B), from which expression B C is easily determined; or, Let H de- note the hypotenuse; B, the base; and P, the perpendicular- then, PP = B' -f- P2; and H= — B= = P'; and H^ — ps = B" Part II. OF OBLIQUE-ANGLED PLANE TRIANGLES. In an oblique-angled plane triangle, a side and any other two of the principal parts being given, the other principal parts may be ascertained. In every plane triangle the sum of the three interior angles rs equal to two right angles, or to 180°. Corollary 1. -.Two angles of any plane triangle being given, the thn-d is also given; for it is the supplement of the rtther two, and may be found by subtracting their sum from Corollary ^.—Onc angle of any plane triangle being given, the sum ot the other two is also given, and may be found by subtracting the given angle from 180°. The principles or rules by which unknown parts of ol>*. Uque-angled triangles may be determined from those which Are known, are evolved in the following theorems. TEIGONOMETRT. 47 CASE I. re ffiven or as- nterior angles IFhcn a side and two angles, or when two sides and the angle opposite to one of them, are given: rAeorem.--Tlie sides of a plane triangle are to one an- other as the sines oi their opposite angles, and vice versa, CASE ir. fVhcn two sides and the included angle are given: TA^omn --The sum of any two sides of a plane triangle IS to the dilirronco between then,, as the tangent of half th« Ibmue ^' "l'I»>«'te angles is to tlie tangent of half their dif- .SWio/mm --Having ascertained half the sum an.l half the difference of tlu^ unknown angle., the angles are easilv de- termined; for half the difll-rencS being added to half the sum ?rn''.'^H-.r"'''" ""^^'^^ Hnd half tlfe difference subtracte™ from half the sum gives the less.. CASE HI. When the three sides are given: Theorem.~M the base of any plane triangle is to th« «umof the other two skies, so is\he differ^.ce between these tvvo sides to the difference between the segments into which the base IS divided by a perpendicular let fall upon it from the opposite angle. *^ »SVAo/mm— Slaving obtained by the preceding theorem half the d.flerence of the segments, the segments fheiST. are easily found; for half the difference added to half the '^^Zf^T I ^H^T' ^^'«'"?"t, and half the difference sub- tracted from halt the sum gives the less.f PROBLEM I. Given the angles and one side of an oblique-angled. triofigU to find the other sides. EXAMPLE. In the oblique-angled triangle A B C, (Fig. 52,) given th*. angle at B 46^ 22', the angle at C 54= 15', and consequently the angle at A 79- 23'; and the side B C 1 ch.. 35/: required^ the sides A B and AC. ^_* Tor the demonelratiou of these theorems, ae« the App«... 41 TRIOONOMRTRT. To find the side A B. To find the side A C. As Sine / A 79° 23' 0.99250 As Sino / A 79° 28' 9.99250 .s to side li C 1.35 /. 2.13033 h to side B C 1.35 I. 2.13033 So is Sine Z C 54° 15' 9.90932 So is Sine / B 4G" 22' 9.«5960 To side A B lllJ /. 2.04715 To side A C 99.4 /. 1.99743 PROBLEM II. Given ttco sides and an angle opposite to one of them, to find the other angles and the remaining side. - EXAMPLE. In the oblifiuc-angled triangle ABC, (Fig. 53,) obtuse at B, given the side A C S ch. 18 /. the side B C 1 cA. 95 ^ and the angle at A 32° 40'; required the angles at B and C, and the side A B. To find the angle at B. To find the side A B. As the side B C 195 /. 2.29003 As Sine / A 32° 40' 9.73219 h to Sine Z A 32° 40' 9.73219 Is to side B C 195 /. 2.2900.'5 So is side AC 318 /. 2.50242 So is Sine Z C 29° 9.68557 To Sine of Z B 61° 40' 9.94458 To side A B 175.1 1. 2.24341 But by the data, B is an obtuse angle. It is therefore the eupplement of an angle of 61° 40', or 180° — 61° 40' = 118^ ao'=ZB. PROBLEM III. Given two sides and the included angle to find the other angles and the remaining side. EXAMPLE. In the triangle A B C, (F?;g-. 54,) given the side AC 919.95 I. the side xY B 500 /. and the included angle at A | a6° 52'; required the angles at B and C, and the side B C. To find the angles at B and C. 3.15227 2.62319 10.47716 A8AC + AB = 1419.95 IstoAC — AB = 419.95 So is Tan. i Z s B-t-C = 71°34' To Tan. A Z 8 B — C 41° 35' 9.94808 Then 71° 34' + 41° 35' = 113° 9' = Z B; and 71° 34' ~ 41« S5' = '29°59'=Z C. TRItiONOMETRY. 49 iide A C. = 23' 9.99250 SrW, 2.1303S IG" 22' 9.85960 ,4/. 1.99743 To find the side B C. As Sine Z C 29^ 59' 9.69875 Is to side A B 500 /. So is Sine ^ A 36"^ 52' To side B C 000.26 2.69897 9.77811 2.77833 rie of them, to <^ side. 53,) obtuse at 1 ch. 95 ^ and B and C, and side A B. , 32° 40' 9.73219 195 /. 2.2900.S C 29*^ 9.68557 75.11.2.24.341 5 therefore the 6^40' = 118^ find the other the side A C 3d angle at A ^e side B C. 3.15227 2.62319 0.47716 9.94808 I 71° 34' — 41'' PROBLEM IV. Given the three sides of an oblique-angled plane triangle to find the angles. EXAMPLE. In the triangle ABC, {Fig. 55,) given A B 5 c/i. 62/., A C 8 ch., and B C 3 cA. 20 /.; required the angles. To find the segments into xohich the base A C is divided by a perpendicular D B let fall upon it from the opposite angle B. As the base A C 800 /. 2 . 90309 Is to A B + BC = 562-1-320 = 882 2.94546 So is A B — B C = 562 — 320 = 242 2 . 4261^ 77 -* 71° 39'= 120° 5' = VD DISTANCES. I surveying may be ch fire inaccesaible. Liadrant is the least ■cle divided into de- iet suspended from ii: one of its radii. ,) from a wall, and of elevation is ob- of the wall. 9.98355 2. 19136 9.43142 1 . 93923 57.) i feel in height, 1 e, a straight line lo MtNSURATION OP HEIGHTS AND DISTANCES. l\ which from the top of the tower makes with the wall an an- gle of 67° 20'; required the distance of the tree from the bot- ^ torn of the tower. As Sine / A 22° 40' 9.58587 Is to the height of the wall 136.5/if, 2. 13513 So is Sine Z C 67° 20' 9.96509 To the distance A B 326.8^3?. 2..^;4^S5 PROBLEM III. Wishing to know the breadth of a river, (Fig. 58,) I mea- sured for a base a straight line, 250 links in length, close to the bank. From each end of this base line I found the an- gles subtended hy it and a tree at the brink of the river on the opposite side to be respectively 53° and 79° 12'; required the breadth of the river. Lot a perpendicular fall on the base from the opposite angle at A, the length of that perpendicular is the breadth of the river. Firet find the length of the side A B. Now we have / B 53°, and ^ C 79° 12', consequent! v ^ A is 47° 48' and the side B C 250 links. Then, To find the length of the side A B: As Sine Z A 47° 48' 9.86970 Is to side B C 250 I. 2.39794 So is Skie / C 79° 12' 9.99223 To side A B 33i .5 2.52047 Now we have the right-angled triangle B A D, in which ;»re knovv-li the side A B 331.5 links and the angle at B 53° Then As Rad. 10.00000 Is to side A B 331 . 5 ;. 2 52048 So is Sine / B 53° 9.90234 To sfdc A D 264 . 7 2 . 42282 The perpendicular breadth of the river accu.dingly in iu.li. PRORT.FIU 1X7 /I7»J~ rn V Wishing tu Know the height of, and my distance from an ol^ect apparently on a tevel with the place on which I 1, ■1 ^; hi MENSURATION OF HEIGHTS AND DISTANCES. Stood, oti the opposite side of a river; and being unable to mctisure backward on the same plane on account of the immediate rise of the bank, I placed a mark at the place on which I stood. I then measured a distance of 264 Imks up the ascendinnf ground in a straight direction from the object. At this station it was evident that I was above the level of the object. Looking through the sights of the quadrant first to the mark at my first station, I found the angle of depression (A E D) 42\ Looking in the same way to the bottoni of the object, I found the angle of de- pression (D h A) to be 27^ Directing the instrument in like manner to the top of the object, the angle of depression (D C F) was found to be 19°; required the height of the object and the distance between it and the mark placed at the first station. Let fall the perpendicular D A on the straight line A B, the angle at A will be a right angle. Find fu-st the length of the sides A D and A E. In the triangle A E D, right-angled at A, we have the hy- potenuse D E 264 links, and the / E 42°; rnd consequently the Z E D A 48°. Therefore As Rad. 10.00000 Is to hyp. 264 /. 2.42160 So is Sine / E 42° 9.8.2551 To side AD 176.7 And As Rad. is to hvp. 264 /. So is Sine / A D E 48° 2.24711. 10.00000 2.42160 9.87107 To side A E 1 96 . 2 2 . 29267. Find next the length of the line A B. Now we have in the right-angled triangle ADB the side A D 176.7 L, and the / B 27°, and consequently the / A D B 63°. Therefore As Sine / B 27° 9 . 65704 Is to side A D 176.7/. 2.24711 So is Sine / A D B 63° 9 . 94C38 To side A B 346 . 7 /. 2 . 539i«5 We found before A E 196. 2 I., tind now find A B 346. 7 /., NCES. MENSirnATlON OF HF.IGIITS AND DISTANCF.S. .'>;i being unable ccount of the at the place tancc of 264 rection from I was above sights of the I, I found the in the same angle of de- istrunient in )f depression might of the rk placed at fht line A B, it the length have the hy- ;onsequently 7 7. we have in D 176.7 /., 3 63°. B 346.7/., therefore BE must be 1.50.5 /., vvhicii is the distance be- tween the first station and the bottom of the object. We have still to find the height of the object, or the lengtli of the side C B. Draw C F parallel to A B, and because D A ana C B are perpendicular to A B, they must be paral- lel to each other; and because F C is parallel to A B, F A and C B must be equal. Having already found the lengtli of A D to be 176.7/. if we can ascertain the length of ][) F, the length of F A, and consequently of C B will be easily determined. Now in the triangle F D C riglit-angled at F, we have the base F C = A B, which we h e found to be 346. 7 /., and the angle at C 19°, and consequently the an- gle F D C 71°, to find the side D F. As Sine Z C D F 7P 9.97567 Is to side F Cor A B 346. 1 /. 2.53995 So is Sine / C 19<^ 9.51264 To .side DF 119.4 Z. 2.07692 It was before ascertained that A D is 176.7 /., and now that D F is 119.4 / in length, A F must therefore be 57.3 /. But A F and C B are equal: C B is therefore 57.3 /., which is the height of the object. The methods of ascertaining the heights and distances of inaccessible objects are numerous, if, however, the stu- dent fully comprehends the preceding examples, :^e will have r.o difficulty in applying the principles of Trigonome- try for the solution of any problem that may occur. Wherever sufficient level space can be obtained, the fol- lowing rule is applicable and expeditious: Let the observer retire from the object until the angle o elevation is 45°, the distance between the place of observa- tion and the object is equal to the height of the object. But if the object be inaccessible, let the observer find the point at which the angle of elevation is 26° 34'. Having marked the spot let him advance towards the object until he finds the angle of elevation is 45°; or let him retire from it until the angle is 18° 26', in either case the distance between the first and second places of observation will be equal to the height of the object. .*■ m I t i, LAND SURVEYlNrx. INSTRUMENTS EMPLOYED, I. The Chain. The Chain is a measure consisting of a certain nunib«^r pf links of strong iron wire, very generally employed in surveying for the purpose of measuring lines or distances. It is in length four poles, or sixty-six feet, or one hundred links. A link is therefore 7.92 inche:5 long. Hence it is easy to reduce any number of links to feet, and vice ver&a. To assist in counting the number of links, when any dis- tance does not amount to an exact number of chains, a'small piece of brass, generally marked, is attached to the end of every tenth link, dividing the chain into ten equal parts, Instead of the chain, a half chain, or, as it is often called, a two-pole chain is very frccjuently employed. For mea- suring lines rji cleared level land, such as marshes, inter- vales, &c., the whole chain or four-pole chair is the more convenient. In British North America, however, for which this treatise is principally designed, us a great part of the business of a Surveyor consists in running lines, and making surveys mi the forest, and on every variety of surface, the half chain is generally to be preferred. The reason is obvi- ous. It is much easier to keep a two-pole chain in a straight and horizontal position, than one of t^vice its length. By frequent use the rings which connect the links of tho chain are apt to open, and thus the length of the chain is in- creased. Before proceeding to measure any line, tho Sur- veyor should therefore carefully examine and measure his Cham. To this point too much attention cannct be paid. Two chain-men or chain-bearers are generally employed to Carrv tho. rha'in TTno" ♦*>"!»• ^"V"^'-l-^ r-. i -^-"-- ^. — _„ii-.i, ^^}i,,, ti£vir v.-.rtriuiiiusa una Kirici com- pliance with the directions of the Surveyor, the correctness LAND SURVEYING. 55 of ihe survey in a great measure dependg. Over their con- duct therefore it is of great importance that he exercise n careful supervision. Before commencing the measurement of a stationary dis- tance, an object easily seen is to be placed at one extremity of the line to be measured. The hindmost chain-man then takes up his position at the other extremity, holding the end of the chain exactly at the end of the line. The other cham-bearer being previously furnished with, and now car>- rying ten pins, sharpened at one end, about ten inches in length, if the surface to be measured is smooth, and about eighteen inches in length if the surface be uneven, or if it be 1,1 the woods, and holding the other end of the chain, proceeds towards the object placed at the farthest extremity of the line. It is now the duty of the hindmost chain-bear- er to direct the course of the foremost. Having advanced until the chain is stretcliQH, the latter is directed, if necessa- ry, by the former to move to the right hand or to the iei\ until he is in a direct line with the object toward which they are advancing. When he covers that object from the sight of the hindmost chain-bearer, the latter knows that he is in the proper course, and with a motion of the hand, or other-, wise, directs his companion to stick one of his pins exactly at the end of the chain. Both chain-bearers then advance simultaneously toward the object at the end of the line, un-, til the hindmost arrives at the place where the pin was de- posited, lie then directs the foremost chain-bearer as foi-s. merly, and pulling up the pin carries it carefully with him. Thus they proceed until either the whole line is measured or until all the pins carried by the foremost chain-man are exhau.«\d. In the former case, if the line do not contain an exact number of chains, the distance between the last pin and the object at the end of the line is measured in links. The number of pins held by the hindmast chain-bearer ex- presses the number of chains or half chains measured. This, with the odd links (if any) added to it, is the length of the line. lii iho hitter case, at the end of the first ten chAins, the hindmost chain-man returns all the pins tatihe foremo^t,.^ .)(] '\ 't 1% H 1 IS , :* iii lifl n w ''H 1 v 1 LAND SURVEVING, a note of the transfer is taken, which is sometimes called keepins; tally,— and the chain-bearers proceed as before, un- til the whole line is measured. Then the number of trans- fers, or tallies, each counting ten chains,-— the number of pins held by the hindmost chain-bearer, counting each a chain, and the number of odd links (if any) shew the length of the line. It must be very evident that, in a survey, much depends upon the hindmost chain-bearer. Inaccuracies very fre- quently occur in consequence of bad chaining. If, there- fore, a Surveyor cannot procure, for a hindmost chain-bear- er, an individual upon whom he can rely, he ought to act in that capacity himself. This is another point on which he cannot be too careful. The surveyor will requi-e likewise to caution the chain- bearers against losing any of their pins, and also to teach them that inclined planes, such a^he sides of hills are to be measured horizontally, and not along the incliiied plane. This subject however, we will discuss more particularly when we come to treat of the running of lines. n. The Circumferentor. This instrument is employed by surveyors for taking an- gles. It consists of several parts : 1. A brass box, about five or six inches in diameter. Within this box are; 1st, a brass graduated circle, the upper surface of which is divided into 360 degrees, and numbered 10, 20, 30, &c., to .360. The bottom of the box is divi- ded into four parts or quadrants, each of which is subdi- vided into 90 degrees numbered from the meridian, each way to the east and west points. And ^ndly. a steel pin in the centre, called a centre pin or pivot, finely pointed, upon which IS nicely balanced a needle, touched bv a loadstone, which, when at rest, and when the box contahiing it is in a horizontal position, always points in a North and South di- rection nearly. To the bottom of this box is also attached a Plidc, by means of which the nee.llo may be raised from the centre pin, or pivot, to prevent the fine point of the jat- '*■,! LAND St'RVETING. 57 mos called before, un- r of trans- ber of pins h a chain, igth of the h dependu very fre- If, there- hain-bear- it to act in which he the chain- to teach 1 are to be ed plane, irticularly aking an- diameter. the upper lumbered X is divi- is subdi- ian. each el pin in ted, upon aadstone, it is in a South di- attached ied from 'the lat- ter from being blunted when the instrument is carried from one place to another. The box is also covered with a glass lid to preserve the needle from being disturbed by wind or injured by rain at the time of using. When the instrument is to be carried from one place to another a brass lid or co- ver is placed over the glass to protect it from accident. •2. A brass index or ruler, about 11 or 12 inches in length, to the ends of which, and perpendicular to it, sights are at- tached, and to which the box above described is fastened by screws. In each of these sights are two apertures, or slits, the one above the other, and the one much wider than the other. In one of the sights the wide slit is uppermost; in the other, it is below. Through the widest of these aper- tures is placed longitudinally, a horse hair or fine silk threa E D. so is F U to B A. Note. — Tho above pro])ortion holds good in any paral- lelogram. Jlnother Method. Let t][ie distance between A and B {Fig. G2,) represent the width of a river, it is required to ascertain that distance by the chain only. Make A D perpendicular to A B. Bisect A D in C Draw D E perpendicular to A D. Measure along the line D E until you arrive at a point P2, in a direct line with C B.^ The distance between Dand E vyill bg e(]ual to the disUnce between A and B, the width of the river. Because A C and C D are equal, and the angles at A and D right angles, it is evident that the triangles ABC and DEC are not only similar but equal, and therefore D E is equal to A B. Note. — Tt J not neceiisary that the station A should be ex- actly on the brink of the river. It nuiy be taken at any con- venient distance from it. Hiiving deter. (Uiunl the distance between the station and tho inaccessible o])ject at the oppon site vA2,ii of the river, measure carefully the distance be- tween the station and the river's i)riid<. 'i'li .t distance being iiubtracted from the whole distance, tho remainder will be the breadth of the river. r I .. i, i if ^ > Use op the ClRCUMPERV.yTOE. To take field notes by the Circumfercnior. 1. By fore-sights. Tiace the instrument at any angle A, (Fig. 03,) as your first station; cause a otaff or ])ole to be erected perpendicuf larly at the next angle B: having levelled the Circumferen- tv)r, turn the flow^er-de-luce^ or nortii part of the box, to your eye. Looking through the small aperture, or slit in the sight, turn the index until you can see the staff at B, through the large slit in the opposite sight, and until the thread or hair which is in it divides or cuts that obicct; tli« i LIWD St'RVEYINO. 67- degrees pointed out by the south end uf tlic needle * n iir shew the number of degrees by wliich the stationary line i« distant from a north course, counting quite round witli the sun. Having entered the number of degrees, or tlie l)earing of ihc line A 13 in your field book, measure the line, and insert irs length in chains and links in the fiehl book likewise, un- der the title "distance.'* Being arrived now at your second station B, cause a pole to be erected at the next angle C. Place the instrument in ft horizontal positio". over the spot on which the object at B stood. Direct the sights to the object at C, with the north of the box to jour eye. When the instrument is in such a position, that, looking through the signts the thread in the large slit of the opposite sight exactly cuts the pole at C, count your degrees to the south end of the needle, and Re- gister them in your field book as before. Measure the line B C, and make the necessary entry in your field book. Pro- ceed thus from angle to angle, until you arrive at the place of beginning, noiing as you proceed the names of the own- ers of the contiguous lands, and the names of the roads, riv- ers, &,c., which bound the lot, or intersect the i)oundarie8 >vhich you are runninir. 'i. By back a7id fore-sight ft. Proceeding as directed above, before you leave tiie st:ition at A, set up a stake or pole in the spot over m hich the cir- cumferentor stood. Having arrived ?.t the second station B, and levelled your instrument, with the south part of the box to your eye, direct the sights to the obj(n hr, iajd ofi' into triangles by lines drawn from ary assigiie(! ptat F within the figure, as by the lines FA, F T], F C, &c. Now since the three angles of every triangle nrr togeth r equal to two right angles, (Euc. i. S2,') It is evi-. . - that the angles in all the triangles coatuined in this figure must be eq-inl to twice as many right andes as the figure 1 i^ side. But, according to Euc. i. IS, all the ^gles atout ,J- point F are together qual '-four right 70 Land SURVETINO. f 5.1 I angles. Therefore the remaining angles are equal to twice as many right angles as the figure has sides, less four. If, then, the angles of a field be correctly taken in any survey! their sum will be equal to twice as many right angles, less four, as there have been stations taken in making the survey. Use op the Protiiactor. To protract or draw a plan from a field book. It is required to protract, or draw a plan from the follow- ing notes in a field book: Commencing at the point F, (Pig. 66,) running thence K. 8 chains, thence N. 15° E. 8i chains, thence S. 80° E, 7i chains, thence S, 15° E. 7 chains and 90 links, thence S. 20' W. 10 chains, thence N. 75° W. 8 chains and 45 links, to the firs^ station at F, or place of beginning. Consider what part of your paper will suit best for the first station, aa at F. Frem that point draw a meridional, or a North and South line. Now from the field book it ap- I>ears that the first course is due North, it is only necessary that, on that line from F towards A, the given distance 8 chains be laid off, from a scale -qual parts. Lay now your protractor on the meridior * lii. , so that the centre may be cxa-tly on the point A. As jxt course is N. 15° E., from the meridional Jine prick oflf towards the East or right liand 15°, as ])()inted out by the chamfered edge of the pro- tractor, Through this point from the point A, draw a blank line A 13, formiii'r, of cmirse, with the meridional line, an annfle of 15°. On that blank lino 4 ay oft' the given distance 8^ chains to B. Next, through the point B draw another meiidian parallel to the neridionai line F A. Lay the centre of tiic ])rotractor on the ])oint B, .-tud prick off the next course S. 80° E. Through that mark or dot fr)m B, draw tlie blank line B C, on which lay ofl^'the given distance 7.i ciiains. Proceed in the same way until you have com- pleted the figure. Though it is the best way, in all protractions when the .ionrses arc given, to draw meridians parallel to eacJj otJier LA^fD SHRVKTING. 7r lal to twice four. If,' my survey, ingles, less the survey. through every angle of the field, as above; yet where the angle at each station is given, the centre of the protractor may be placed on the extremity of the distances when laid off, and the number of degrees contained in the given ano-le pricked off for the direction of the next line.. The method of using the Protractor is obvious, from the nature and use of the Circumferentor and Theodolite. ook, the follow- ing thence ) S. 80° E, ks, thence ns and 45 est for the leridional, look it ap- necessary distance 8 Lay now entre may N. 15° E., St or right f the pro- L, draw a iioridional the given It B draw ^ A. Lay ick off the )t fr')m B, n distance lavG coni- Of the Field Book. The accuracy of every survey, and the ease with which plans may be drawn depend, to a great extent, on the man- ner in which the field book is kept. The adoption of a con- venient and perspicuous method of keeping field notes is therefore a matter of great importance The subjoined form is simple, concise, and plain. JVo^^.—ln keeping the field book it is customarv and use- ful, where the distance line crosses a brook, lake, &c., to enter the same at the projjcr place by a line draAvn between the respective entries, representing the course of such stream, fccc, as between 35 and 40 in the 4th or N. W. course, and between 40 and 45 in the last course, in the adjoining field book, in which they are represented bv straight lines, in consequence of the Printer iwt having lines of the proper clirecticn. if when the ;ach otlier i i T* land 8umveving. Form of a Field Book. 3 38 ch . to the place of be$;innin$r. 35 70 70 ch. and 7 /. SO • C5 25 60 20 71 55 15 W 50 10 5 45 ■ 40 ch. • < o 40 35 30 NarrowLake Thence N 50^ E i 40 tJ 35 "Cross ro 25 s 30 Lake CO o 20 ^ 25 tf 15 t^ 20 10 5 bo o 15 10 Thence S3^54'E 5 30 ch. 4 30 25 34 cA. Thence N45^W i 30 i' 25 20 1 20 15 i 15 10 .5' 10 5 25 ch. 5 35 ch. f Thence S 81° E Thence S85^ W 35 30 5 20 15 10 5 25 20 15 10 5 Thence South Thence V 60^ E ■s^- 4 c 20 24 ch. 3 i J8 eh. 4> . 15 35 ^ u 10 80 Wg 5 25 20 15 10 Thence West efD. Parish of C. , and County 5 tate of A B., Esq., in the Thence I ^25°W easterly angle of the Es- 1 Cominei iced at the most 11 (COK TINUBD.) LAND SURVKYINO. ■;;{ fining. ro ck. and 7 /. NarrowLake 4 cL t ck. eh. Remarks o.n the preceding Form of a Field Book, Tim procctliiig uotf>s urn sii])];osed to be entered in the field book, wliile surveying the Estate of A. B., Es((,, (.S'.v- Fig. ()7,) the counscs being taken from the meridian l>y ;i '•In-umferentor, and the .stationary distances mea-^ured by a half or two-pole chain. Determine rirst at what part of the Estate it will be most eonvenient to co/nmence the survey. Having conclnded to begin at the angle No. 1., which is the m,)st Easterly corner in the Estate, you insert in the field book the place of be- gimiing, thus: " Commenced at the most Easterly corner of," ike. In keeping a field book it is found most conveni- ent to begin at the bottom of the page, and to write .ip- ward.s. Set the instrument on the angidar point, and take the course to the second station at tiie angle No. 2, which is seen to be due West. Write in your field book " Thence West," or " Thence W\" Proceed to measure the distance between / 1 and Z % and every tin.e that the ten pins cur- ried by the foremost chain-Jjoarer are all transferred to the hindmost chain-man, insert 5 in your field notes, writing upwards; for 10 half or tw^o--pole chains, only tnake 5 chains^ When you have arrived at the station write down the odd chains and links, if any, and place the sum of the whole in the right hand column, thus: «' 24 cA." Observe whoso land lies to the left of the land which you are surveying, and enter a note of it in your field book, in the left haml colunm, thus: " Estate of Mr. A W." Observe also if any particular objects a])j)ear in the immediate viciinty ai.'l ad- joining the stationary line, such as stream, fence, road, &e., and insert it also in the left hand colmnn of your field jmok' thus: '-Kryad." Proceed in this nmnncr from station to station, uniil you leturn to the place of beginning, always ••arefully noting to whom tl^e adjoining land belongs, tluv r..ads, waters, &c., which bound, the Field or Estate'^whicli you are surveying, and also the streams, lakes, Lr., which you cross in running the lines, and at what part of the line you cro.ss them; as all these particulars must be expressed T! 74 LAND SUKVEYING. and iiccurately laid down upon the plan which is afterward^ to be drawn. I'Jik' ii H r ■! { I 1 ■ VARIATION OF THE COMPASS. The natural magnet or loadstone was for a long time sni.- posed to be the only body which possessed inagnetic proper- ties. It ,3 an ore of iron, whose specific gravity is about five times that of water. Its colour is iron-black, a.id its lustre metallic. It is found in almost every part of the world, and occin-s in beds, often of vast thickness, and of great extent. Its attractive power over small piece, of iron has been known from the remotest antiquity. It is sai.l to be distinctly referred to by Hon.er, Pythagoras, Aristotle, i'lmy, !kc. Iv has been asserted that its directive i)ow.M- or polarity was known to the Chinese in the earliest ages, a.Kl that the needle was employed to guide travellers by lan.l a thousand years before the commencement of the Christian Ji.Ba. To whatever amount of importance these statements may be entitled, it may nevertheless be confidently asserted, that nearly all our knowledge of the magnetic virtue, and nearly all Its ai,plications to practical purposes, are discoveries and mvontions of comparatively modern date. Notwithstand- mg, however, of the numerous and valuable additions, which have recently been made to our acquaintance with the laws by which magnetic influences are regulated, many interestin-r and important points still remain to be determined. To the dotermmation of some of these points the attention of scien- tific men has recently been directed. Magnetic observatories have been erected, and powerful and delicate instruments hav-e been constructed for the advancement of this brand. ot Science. Members of the British Government were re- questeu some years ago, to establish magnetic observatories not only in Lntain but also in these Colonies. About the same ime Baron Humbolt addressed an interesting letter to the late Duke ot Sussex, President of the Royal Society, VARIATION OP THE COMPASS. 75 1 is afterwards long time sup- ijnctic proj)er- tivity is iiho'.n hlack, and its y part oC th(; kncss, and of pif3cc,> of iron It is said to •as, Aristotle, tive j)(nv T-\NI) .sun\ EYrwo, ami varying innuiMicrs and (lisfniliiinccs, uhich tl or nnnt Im« iil»Ic to diMcct inul cstiniiitp. 'rhn^duivoof tl><«h'i> u' siirvov influotn'cs is still invi)|\M!il in olwcnritv, hut tlir cd'c^ts ars not point Nonli mid South cx- srrvi'd th.il llir iimj'nrt d acliv : or, in other words, that the niiinnrt terrestrial meridian seldoi ma-fnetie needle from a true North and Sontli I a true meridian, is called its vnrinli eontinnally varyinjr, Aecoidiiiirlv, the va; .at in ( lill les. on. ie meridian and the n eoineide. Th,' deviation of the ine, or iront on. This deviation is ion is (lili'erent brent jdaees, and in the same plare at ro is little or no pereeptihh^ variati In otluM- places the variation is great. At some times it ap- pears to h(> stafi^mary at a particnlar place, at otiier times in the same j)hice it increases or diminishes with ein-U> 17'K. The Easterly rom 1()57 to 1G(>2, no dity. At Lomhm, in the year 1 ;')?(), th 15' Kasterly. In 1580, it had jjreat«>st Easterly deviation, 1 \ariation then be. I variation was percejuible. In KUHs it >vas S-l' V\ Csterh The Westerly variat ion continnetl to increase until 1815, when it arrived at its maximum or ^n-eatest Westerlv d ation, heinj; then '24'' i27' 18" W. It 1 V tievi- 1112 In 18;>1 it was ^21 \V las since been decrcas- At Paris, in 1541, the north end or pole of th die pointed 7^ to the East of iNortlr/ In 1580 it had its maximum of Easterly variat c nni'nietfc nee- attained ion, Avhich was 11^. SO'. 'J'l le easterly variation declined ffrad about Hj[)6, when it became variation was 15' W, The Westerl Ji-radualiy until 1814, when it had since been decreasiui;. In 18'iO it frradually from that time until imperceptible. In nm thtt Accordino- to the report of Dr. (Jesner I- y variation increased iirrived at 2\>^' 54'. It has wasi2\>-^ Ui' W. fist, made to His Ex rovincial (ieolo- Colebrooke,K.H., Lientenunt-C celleucy Sir ^Villiam MacLean ( leorife he Governor of Nevv-Brunswicl< ariation iu that Province ranges from 17-to ^IV ^V V Mir \TTON or Trn. roMrx- 1 1 An.l on tlK- lirli nCN.pf., |h|.{, I (oiiixl lli<' variniion nt lluv (If VcltO to l)«! IS' 10' W., lU'Jlllv. IN siflfs tlipsi' pn^'iTssivp cFmnj^'cs in tl 10 v.'iiiiiiinn (I -•"mpri-^, tlir ii.TiMr is ,il nr<'rial»||. osrilhitioiis, nt ijiti; -ill tf .nn «M-r-iit timrs of the flay jiikI iii-Wif. 'I'Im; oI \\\o('t \n inorp iriiriiiti> yif. ii|»- rent soasfUH «»r ilirycMr. ami i-^crvailMii- whi.-ii havo Urvn ina nnnntr- T 111 s.ininuT, wlioii it is aliuii. anic discropanpy o\ist,s in the ruiiiicron-! -•'ivations niaiK! tu dptorniino tlio mnni of tlip daily variati(»n. hi f.ondon tl niax'itnntn and mini- daily variation is «,(>' -M' In ( iV'lco t>. :» ajpf it IS only 10' .W.. I", in Ka-a.i it 10 ,',M'nf'ral moan ioiiova it is not rpiito !(/. In amounts to I ' 10' 'I'iic lollowini; diurnal variations liavo hoon obsorvcd Paris. Durinsf tlio ni^rht it was noarly stationary. At m rise its North cvtromit SIUI- y movod to tlio VVostwanI, as if uvoid- in-,Mlio <;(dar indup.n.'o. Towards noon, or j trom nor)!! to A o"c| iioi'o ironorallv^ dcvialion. Tl ock, it attainod its maxinii o'clock, when it had r H'li it roturnod Kastward, till i», lo. im VVo.st»;i-|y ur n 'fMP liiH-d its orijjMnal jxtsition, whoro '•'•"iaino.1 until moi-nin- In April, May, .Tuno, Jnlv, A :-nist, and Soptomhcr, the ^ IMAGE EVALUATION TEST TARGET (MT-S) 1.0 1 5 '""^™ «" lllitt '- 1. I.I i.25 1.4 2.5 2,2 12.0 1.8 1.6 Photographic Sciences Corporation ^ <^>^> c?-* «^^ ^<^A "% V «^ >^ '<9^rN\ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 "-b' «i* y the circumferentor, or quartered compass; or it is the num- l)er of degrees, &c., which the sun or star rises or sets to the North or South of the magnetic East or West points. To find the variation of the Compass by Amplitudes. The Latitude of the place, the Sun's Declination, and the magnetic amplitude bemg given, find first the true ampli- tude by the following rule : As the Co-Sine of the Latitude Is to the Sine of the Sun's Declination,* So IS radius To the Sine of the true Amplitude. Then, if both the magnetic and the true amplitudes be either North or South, their difference is the variation; but If the one is North and the other South, their sum is the variation. To know whether the variation is E. or IV. Let the observer's face be turned towards the suii, and if the true amplitude is on the right-hand side of the mao-uetic , in Lat. ^}(r N.. r)„; brarinj. of tho sun at its rising was ohsprvod to be N. 85^ K. by the rornpass; roqiiirod tho true uniplitwlo and tho variation ot rlif? r)(.'0{|lo. To find the true ^mplUude. A.s tho Co-sino of tho Lat. 46^ Is fo tho Sino of tho .Sun's Doc. 15^ .S.V So is Radius 0.81177 i>.4-l!)17 lo.oonrij tho To tho Sine of the truo Amplitude 22^ 45' 9.58740 Tiio bearing of tho .sun by the conipass being N. 85^ E.. fMagnetio amplitude is flO^ — 85^ = 5'^ N. Thoroforo, Truo Amplitude E. 2-r 45' N., for the Do.r. is N. Mag. Amplitude E. 5'^ 0' N. Variation 17*^45'; whioh is V/e.st, because the observation beijig talvoa in tho morning, and the observer's face being turned towards tin. sun, tho truo Amplitude is on tho left of tho ma-neti<' ..r oljsorved Amplitude. ^^l. Suppo.so the bearing of the sun at setting to bo S. 80' V/., and con.scquently tho magnetic Amplitude W. 10^ S nhde the truo Amplitude is found to be VV. 25^ S. ^Vhat is tin; variation? True Amp. ^Y . 2-P S. Mag. Amp. W. IQ^ S. Variation 15- W., becausothe true Amplitude is to the left of tho Magnetic. '"5. At the time of rising, suppo.se the .sun's bearing to bo > . 7;)^ E. by the compa.ss, and consequently tho, Mag. Amp. h. 1.)^ S., while the true Amp. is found to be E. S' 15' N • required the variation. '' True Amp. E. 8^ 15' N. Mag. Amp. E. 15^ 0' S. Variation -25^1.5'W,, because the truo AinplitudG is to the left of the Marrnetic. VARIATFON OF THE COMPASS. g] 4. ^^"FPOSO the sun':^ honring at setting to 1)0 S 83 W Ml.en the true Amp. is found to be W. 7^ .'50' N.; re„nir.Mj the variation-. i ■'<« True Amp. W. 7° 3(7 N Mag. Amp. W. 7= 0' S. Variation 14°30'K 1.ecau.se the true Amplitude i. to the right" of the Mag.oti.. Tofmd the variation by Concentric Circles mu^loTZ^ '""', ''' '"""^'"^^ ^^^^' (about a foot M naie,) diaw several concentric circle.. In the centre p aTe "^r'tV^V ''"^'^^-^ ^""«^' PerpendicuL t , J on any of the crdes, upon ^vhich the shadow of tiie heac of the pm rests, an. ,„ark it carefully. Observe aNo in he rmke a ma.k. A right line joining these snot.s or mark, nil! be m East and West linp k;.o^: .u- .- "laiKs, mil ne an,l u win be a iVo.W and South line. 7'ZZtl n fron, .„o centre, and in i,« p,„ec .,„„,:„ ,e „„ X ■ r t^ uit, ^tne needle of a compass suits verv uoli ^ n» i i nhus.„,ea„.,ea™„H.„i,e,.,,eNr.i::::<^ By tins method, with care and a little experience the v. -^^'-' '"ay be detennined with consiclerable "euSc' Tofnd the variation by the Nokth Sxah and Aooxn 1 be constellation Ursa Major, or the Gr.^f « known by the name of Charles^ vLn , '"' ''''" i'4 82 LAND SURVEYING, Ming a rectangle, or square, which is considered as forming the body of the wain. The two hindmost of these are call- ed pointers, because a right line drawn from them towards the North will pass near the polfe star in the tail of the Lit- tle Bear. Following this square arc three other stars of about the same size, forming the tail of the Great Bear, or the handle of the jdough, visible every clear night. The one next the body of the bear is Mioth. Now it so hap- l)ens that the pole star and Alioth come to the meridian at the same time. This star is accordingly often employed by Navigators to determine the latitude. By its assistance also we may ascertain the variation of the compass with consi- derable accuracy, by the following method:— About midnight, in the beginning of October, or about 8 o'clock, P. M., towards the end of November, the pole star comes to the nuoidian, and it the same time Alioth is at the meridian below the pole, and consequently both stars are due North. Alioth is also exactly perpendicular to the pole star, or directly below it. While Alioth is still to the West- ward, let the compass be duly levelled and prepared to take its bearing. Then suspend a plummet by a white thread, in some convenient place, so that without moving the body much the plummet and compass may both be attended to. Bring the thread to cut the North star. Watch carefully until Alioth comes to the thread, and is cut by it. Then take the bearing of A ,th by the compass, and its diverg- ence from the North wdl be the variation. Note.— By causing the plununet to fall into a vessel of water, the wind (if there should be any at the time) will be prevented from moving it. In this way the object may be gained with greater case and accuracy. Besides these progressive, annual, and diurnal variations, the magnetic needle is liable to disturbances from local and from temporary causes. Of these the most important are, the existenceofniasses of ferruginous matter in the vicinity, the Aurora Borealis, and thunder storms. It is well known that the magnet is powerfully affected VVniATlON OP THE COMPASS. gg •ho whole sui.: „";r^ V ,: ;:,r"7^ '■••™'^' ?~"^"-" -no., are wi.hou, „„ .ZZT^r ^r^'r, .'r!''?'; >n clays, and sands, and even in n-,v, ■ ,.(■ , ''""''•""' -1 suManee., and in the a.J ,^C"' ^S" ^ ""'V','"- ».se, ,„ eertain conditions aflc« tl e' ne^dt W ' T therelore rcasoimhlv evnee. ,h., .. ^"-' '""''" ""•"once. DomC " '"'^"'""°»"" "f *■".,,„„„ oc- vonenee orsu^e "r. Z ^u" ""',"""""■"' """ "'" '"'"1- whena survey IXa.ie l.^^ "'"' '"'""'" «'■""'"'' ■""' "thor .notallic's ,i" 7; d'' TT " '"' "'' "'"" "'•'•■ '"■ ■■aWe, amounting oft«:::;e'^:Te:it!:'" T, V '"'"""- of (loteotinff denarfiirn . r. '" "^fe'^^''- ^ iie bast inetJ„„| ..";; i.;>eed!,. r:r:i«r rjitrr ""-^ -- affect the n,„ ""e Su, v """='• '™'»'"'^ -'•■^'ances, which -"«ioninthe;:;hLro? :i;Znrf'7''T;''^"' '.i-"oney i„r™rtlr;:r;:^^:e*-" ''""'"'''"'•'' try it hy the following. ,cs, « '""*"""= "wtrumcnt, to in»t">,nent, and ,11^1,! T"™ ""^ ""<''"'= *■'•"■" "'» in a hoard. Wl e t ,he ne'T ° f '"^ "'""""' P'»' '-""-'I vibrate, then W ,t ins r ' ""'"™'"'' ''"■' ^<^"-'' "' «he presence of "he ittn.n T", ''"'^'"'^'' "'"' "> "• "• ".» «.nner i^ in tti:": ^/^r^^f .-'>o "r""-' appear that the presence of th„ i„ «<« if >t evidently 'i.e instrnnien. sLuld 'i: :I,:d:,re:;""""' '"'''°" ""^ "^•""'•' '-le ■■'«^"": "r::;- ':;:: i:::;rzr"'r ""'-" "" "ary »-ove,„ents, to which 5. ^ H ™S T ""'""■"''- '■apncious movements th„„„ i """"« «"''■■<'' apparently "'otion, and fre!,"";, "".'""'= "--crseswith „ shiveri frequently oscillates several degrees on each 84 LAND SURVEYING. side ot' its moan position. Whon the Aurora only rise?? a lew degrees' above the, horizon, the disturbance is Hniaii und ot'ten inappreciable. But when the Aurora rises to the ze- nith, the di.sturban(!e is generally very considerable M. Ara- go, wlio has stuilied the influence of the Northern liights upon the needle with particular care, states that the part ot' the heavens at which all the beams or radiations of an Au- rora meet and unite, is precisely the })oint to which a nuig- wetic: needle directs itself when suspended by its centre of gravity. The needle may be affected by an Aurora which is iiivisi- ble at the place of disturbance. M. Arago assures us that Auroras visible oidy in America, at St. Petersbnrg, and in Si])eria, produced very perceptible derangements of the nee- dle at Paris. It seems however, that some kinds of Aurora, though exceedingly brilliant and rapid in their movements, .scarcely affect the ntagnetic needle; producing oidy at most, a slight trenmlous motion. The needle of the compass may lie afiected also by the electri(r fluid, previous to or during a thunder-storm. The effect of an electric shock is i)eculiar. Soinetinies it com- nuinicat'^s the nuignetic virtue to unnnignetised iron or steel. Sometiines it entirely destroys the m^ignetic virtue o|' u magnetised body. And sometimes it reverses the poles of a magnet. It may therefore be worth while to examine mag- netic instruments after violent thunder-storms. There is still another affection of the magnet, called tht dip of Ihe needle, to which we nmst shortly advert. This expression denotes the angle which a well l)alanced needk; forms with the horixon^ after it has been mugnetiseo, and w hen it is allowed to move freely in the plane of the mag- netic meridian. This angle, like the angle of variation, has different values in difierent j)U\ces; being, generally speak- ing, very small at the Equator, and increasing towards the j>oles. At the magnetic pole, which Connuander Ross found to be situated in North Latitude 70° 5' il", and We.-;t Lon- gitude 9G^ 45' 48'', the dip was 89' 59', or within one mi- nute of being perpendicular. There is much reason to sup- THE RUNNING OP LINES gjj, poso that every placo Iims itij own magnetic axis, with its own polo, and 0(iuut()r. Like the variation, the tlij) of the needle also undergoes a continual chamro, increasing in .some places and dindnish- ing in others. At Lon.lon, in 17^?0, the okscrved dip wa« 74" 4-1', and the computed dip was 7b' 27, while in 1830 the observed dip was GO- 38', and ir. 1833 the computed dir. was 69'^ 21'. Whenever the needle of a compass is perceived to vary from the horizontal position Avhile resting freely on the pivot, it becomes unf.t for service until it is corrected. For this purpose, make the instrument perfectly h>vel by mcan.s of the spirit-level, which will bo indicated by the air bub- bles remaining in the centre. Then supjily the end of the needle with an additional quantity of magnetism, until it re- sumes Its horizontal position. Then both ends ot the nee- dle Will bo equally distant from the bottomof the instrument. . Then too, if the compass be a good one, both ends will point to the same degree on the graduated brass circle within which It revolves. This operation should be repeated every three or four years. THE RUNNING OF LINES. , In the runningof lines fourthing.s are to bo observed, viz- —Course, Distance, Difference of Latitude, and Departure. 1. The Course is the angle which the lino run forms with the meridian of the place from which you started. * 2 The Distance is the length of the line run, reckoned m chams and links, rods, &c. 3. The Difference of Latitude is the distance of the one end of a line from the other end. North or South, and IS reckoned on a meridian, 4. The Departure is the distance between one end of a hne, and the meridian passing through its other end. It i.s East or West, and is measured on a parallel of Latitude H Land scrvkviwo. PROBLEM I. The Course and Distance of any line being given, to Jind the Difference of Latitude and the Departure. The Distance the Difference of Latitude, and the Dcpar- ture, from a right-angled triangle. Therefore, To find the Difference of Latitude. As Radius Is to the Distance, I" ^;V''^;^?°-S'"^ of t'je Course i o the Difference of Latitude. And, 2^0 find the Departure. As Radius Is to the Distance, So is the Sine of the Course 1 o the Departure. N. 2. \\ * the Distance, or length of the line 71 cA 20/ • required the Difference of Latitude or Northing V^t Departure or Westing. ^ To find S. N. the Difference of Latitude. As Ra To find A N, the Departure. As Radius S0° ,^ .^^„^ l8tptheDist.SA7l20/. '^^8 So IS the Sine of the Course 25" 9.G2595 To the Departure A N 3009 / 3 473^3" Thus the Difference of Latitude, or the Northing i« a. curtained to be G152 /., or (M ./. «,a 5. I., a„u t e D p . t«re, or W estmg, to be 3009 /. , or 30 eh. and 9 I ^ By the same i.ethod, the Difference of Latitude and the ^^•Tbo Course of the line S. A. «« laid don-,, on diagram 68,. Tilt nuifNINO OP Lll-iEf. 17 Departure may bo Tound af^er running any line, when the Course and Distance arc known. Note.--\Vhm the course is tlue No-th or South the rjir vn^^'lii^^'lifT T'-^'^"^'^^ Distanco'^rand he ot artuicisO; and when the Course is due Ka.st or WesT PROBLEM H. The Difference of Latitude and the Departure being given to fmd the Course and Distance. To find the Course. To fmd the Distance. fs^r^'h?n ^^I* -^"^ ^' the Sine of the Counje Sottcm^^"'^""' IH to the Departure Tr.tv, 'r ^. ^ So IS Radius 1 o the Tangent of the Course. To the Distance. EXAMPLE. Suppose I run in the North East quarter until my Diff. of Lat. IS C ch. 83 /., (Fig, c,9,) and my Dcp. 4 ch. 76 /., wh"' was the Course and Dist. To find the Course. As the Diff. Lat. C85 I. Is to tlie Dep. 476 1. 'So is Racijua C0° 2.8S569 2.67661 10.00000 To the Tan. of the Course 34° 44' To fmd the Distance. As the Sine of the Course 34° 44' IS to the Dep. 476/. So is Radius 90° 9.84092 9.75569 2.67661 10,00000 2.92092 dls^ To the Dist. 833. 5 I. Hence the course is N. 34° 44' E., and the strtionar tance, or length of the line is 8 t/i. 83 . 5 /. PROBLEM IIL To find the direct Course and Distance made good in Traverse Running. Make a Table divided into six colums. In the first colunm m TAN') 8URVCYINQ. ! >«■ *et down the acvcml rour.sos, and opposito tofliem set down in the hccoiuI roluiim tlit;ir (•orrosjxjtidin;? distfiiice?. The rhiid and tourth cohDiuiH arc to -rontain the Difibrenco of l.atiiiidc, th« third being marked N., and thn fourth, S. Tho fifth and nixth cnlunni s are to contain tho Departure, the fifili Ueiug niark(;.l E., and tho sixth, W. Having' entered tho Course and Distance in their rcspec- live colunniK. find l)y thoprecodinj]^ problems, the Diflerence uf Latitude and the Dejjarlure, for each Course and Dis- tance, and insert them in the Table, in their prope. columns opposite to the Course and Distance; observing that the Diff. of Lat. must be placed in the colmun marked N. when the course is Northerly, and in he column marked S. when it is Southerly; and (Uat the Departure must be inserted in the column marked E. if the Course i.- Easterly, and in the column marked W. if it is Westerly. Add up'tho columns of Northing and Southing, and of Easting and Westing. The Diflerence between the sums of the N. and of the S. columns will be the whole Diff. of Lat. made good, and will be of the same name with tho greater: and the diflerence be- tween the sums in the E. and W. colutnns will be the whole Departure made good, of the same name with the larger sum. With this Diff. of Lat. anc; Departure find a corres- ponding Course and Distance. The defective columns of the Table will show the direction of the home Course. EXAMPLE. A surveyor on an exploration, run the following Courses and Distances, viz:-~S. 2'4° W. 54 ch., (Fig. 70,) thGnco ti. 783° W. 39 cL, thence N. SSr W. 40 ch., thence N. 56^ E. 69 cA.,* thence N. 22^° W. CO cA.; required the direct Course and Distance to the place of beginning. * The distance on iho 4iti slatiora.y line of the diagram belojw- lug to Ftg. 70, is wrong. " ^ *lia UOKHlH.i or LINKA. TRAVERoF TABLE. Courses. S. 7Hr W, N. 33.^ W. N. 5G.1° K. N.2-2i«W. Oopnrturr. I Diff. of Lat. 09.G'lN 104.09 57.30 To find the Course, Astho DiT. of Lat. G952 /. Is to the Dep. 4673 1. So is Radius 90° To Tan. of Course 33° 55' 2'o find the Distance. As Sine of Course 33^ 55' lis to the Dep. 4G73 t. So ia Radius 00^ 3.84204 3.6(i959 10.00000 9.82755 9.746G2 3.66959 10.00000 To the Dist. 83G1 /. S.r.l2&7 As the dcf eiency in the Table is in the S. and E. column.s, rhc home course is S. 33° 55' E., and the di^^tance to the place of beginning 83 c/i. 61 L iingrarn belong- PROBLEM ly. To lay out a straight road from A !o D, (Fig 71,) instead of the crooked line iV B C D. Set your compass at A , and take the course anct dis- tance to B, wiiich, insert in a Traverse Table as below, ac- cording to the preceding problem. In the same manner tAke tlie courses and distances from B to C, and froui C to D. Then find the course and distance from D to A, as di> rected in the foregoing example, and run it accordingly. 'I'k'' iQ LAND SUUVEYINO. Couives. Dist. Diff. of Lat. N. ) S. ~ Departure. E. ! W. y. 80° E. jP. 15^ E. ih. 20° W. 7.ft0 7.00 10.00 1 . 80 7.63 9.40 7.3!^ 2.05 1 1 5J.42 18.33 9.41 3.42 1 1 -. l'^'^2 Diff. of Lat. S.' 18.33 1 5.02Dep.E.I ■fhe course from D to A is N. IS^ G' W., and the distance 19 cA. S9/, By traverse runnin^ir, a Surveyor being furnished with a correct outline of the boundaries of any County or Province, may lay off the courses and distances of roads through the forest, or explore a body of Wilderness Land. Traverse running may also be applied to the determination of inac- cessible distances. PROBLEM V. Tofmd the Diference of variation* on an old line. _ ff the mark.s made at the time of the first survey remain Visible, c/ace the original line with your compass. Then the Difference between the Course now given bv the compass, and the Com o laid i\o^^■n ui)on the plan or ip.ap is the Dif- fcircnee of variation, since the time at which the fr /mer sur- vey was made, Tut if only tiio cvtrcmitio: of the old line, or if only twa pomts in tlie line not visible from each othe-, can be satis- fdctordy determined, .;et your compass to tiie course laid oown on the jilan, or mentioned in the grant or deed, and run out the distance. Tills course and distance will gene- rally bring you out in sight of the end of tlie original ^line. From the end of the old line> and at a rigjit angle thereto! measure the e\uct distance to tijc line which yo»i have just run. Then, * DiJJh-cncc of Variation denotes the diflereiico between the variiitioij of tl)e compass at tlie lime when the line w.-i, fir-. r„,. «ndil,e l.meuhenit beco.r.s necessary to run it ngain, or ra- ■# THB RCKWIKO OP LIWEfl. d the distance • ngain, or ra- As the length of the old line Is to the Distance, SI So is 57.3 1 o the degrees, minutes, &c., in the Diff. of Variation. it the course bring you out to the left of the old line, the difference of variation is Westerly; if to the right, it is East- erly^, and must be allowed accordingly., iVo/c—In changing the course by the compass to suit th« d fferenee of variation, remend)er to add when the cou se of the line to he run s n the N P or «; "ur V ^°"V^® ?* tract when it is n tlio N n^' S I' ' ^"^"e'-, and sub- EXAMPLES. 1. The original course of the old line A B, (Fj> 72 > was N. 550 W ,„d length, 100 ch. Running that^course and distance I eame out 6 ch. to the left of B. Required the difference of variation, and the home course. a/' it '^'k ''MZ '■ ''''^ = ''' ~^' + ^^'^ I^iff- of Var. S 410 s/e r r-'° ^^'^^• = '^^^ S4'. Therefore, o. '11 34 L., 13 the home course. The difference of variation may also be found by the 2nd case of right angled Trigonometry. Thus, As the distance A B 100 ch. o noooc Is to the Distance BCCcA. 5 77815 bo IS Radius 90 o j^-^^^^ ToTan. of/BACS=26' V^TsIs giving 3 o 26' for the difference of variation as before 2. Uuiming the old line A B, (Fig. 73,) by its orij^iral required the correct present course from B to A ch. ch. As 50 : 4 : : 57. S'^ : 4^^ 34' = Diff. of Var W Tf,n« v 10*^ R ~U Ao QA' wr XT ^'"« "I var. ». lhenls<. A iU^^lc • a \. '• -^ ne course from B to •a, tnereioro is .9 i.ic c>.ay be employ! In running line, over hilly ground, a two-pole or half chain s preferrable to a ^hole chain, i: i, much easier to stretch he former, until it becomes nearly s-raight, than thettter meZr"" """•»••>'' P^OP" '0 employ even a shorter Jh ^'""r """""^ "f "="'•"« 'he boundaries of land, upon , ac 1«' 160 155 150 145 140 135 ISO 125 120 Ji 115 4-> 110 a> 105 100 95 CO 90 '3 ,85 X! 80 CO 75 f^ 70 .2 G5 60 J3 55 s 50 W 45 40 35 80 25 20 15 10 5 Thence N. land surveying.. Example of a Field Book. and 1840. 160 chs. to a juniper trco marked H, W. on four .sidej'. 144 chs. to a maple squared and hacked 128 chs. to a pine s(piared. 112 cAs. to a birch squared. 96 chs. to a spruce squared. 80 chs. to a pine do. G4 chs. to an ash marked do, 43 chs. and hounded by a spruco stake 32 chs. to a hemlock marked S. B. 16 chs. to a fir marked T. D. . W. 20' Commenced at a spruce tree marked H. W, and 1840 . THE RUNNING OF LINES, 97 !0 murkod il. W. W, and 1840 Additional Remarks on the Running of Lines. The great secret, the grand security of success in survey- ing, is to run the correct course, to run the lines exactly straight from bound to bound, and to measure with accuracy the distance between them. To obtain these results, every thing ^depends upon the quality of the surveyor's instru- ments, and on his skill and dexterity in using theiiL As lands are frequently bounded by curved, as well as by straight lines, it is the duty of every surveyor to make him- self thoroughly acquainted with the properties of curves. It is customary to bound lands on rivers, roads, &c. When lands are to be bounded by rivers, great care should be taken to place the boujids at a proper distance from the edge of the stream or shore. Banks are liable to be under- mined, and if the bound bo too near the brink it may fall in, and leave its exact position in uncertainty. From this source disputes frequently arise, leading to lawsuits, and resulting in the loss of the property, peace, and character of the parties, and the reputation of the surveyor. Difficulties likewise frequently arise from bounding lands upon roads. The road is liable to be changed for the pur- pose either of straightening or levelling it, and their boun- daries are in danger of being removed. Surveyors therefore should, as far as possible, measure from known and well esta- blished boundaries, and run in straight lines. It is particu- larly desirable that the place of beginning be tlistinctly marked, and not liable to be removed. Lines then would be easily retraced either by the surveyor by whom they were run, or by his successors. In the tracing and retracing of lines, great care sJiould al- so be taken to follow closely the original line, especially if it be straight. No trees should be blazed except those which were blazed before. When individuals unacquaint- »'d with the properties of lines, mark trees which stand per- haps two rods to the right or left of the original line, inte- I r I 08 LAND SLRVEYING. rested parties may be deceived, or an opportunity afforded to the litigious to enibroif his peaceful neighbour in the anx.eties and lo.ses of a lawsuit. The surveyor too comc3 in for a full share of the blame. No unqualified or unau- thorized person has any right to take such liberties. By such improper conduct peaceful settlements are frequently thrown into confusion, and evUs of incalculable magnitude have been produced. MENSURATION OF LANDS. The area or contents of any figure is the measure of the surface contained within its lines or bouiKlaries. In Land Measuke. m feet, or 25 links make , Rod, Po,e, or Perch. 4 Hods, or 66/^, or 100 1., or 22yds. 1 Chain. 10 square poles or perches i Rood 4Roods.o. 10sq,chs.,orI60sq.poles 1 Acre.* 10,000 Square Links i Sn..n,.« ri • ,.1- t; T- , i^quare Cham. ():Jo square Links it- r. . or. AftftQ T , 1 Square Perch. iij,000 Square Lmks i s^„^,.„ p , innnnna t • , Square Rood. 100,000 Square Lmks i c^......^ * nn u A ^ Square Acre. «i4n Square Acres i «„., iv,., un r'l • • , . 1 Square Mile. HO Chains in length j j^jjj^, ent^ft;ro"irr^'''"^ Umd itwmbe found most convrni- tnt to take all the measures m four-pole chains and link. PROBLEM r. To Ml the area of a Parallelogram, whether it be a Hanare a hectangle, a lihomhus, or a Rhomboid. Mnliinly the length I the product v.-ill be th" - RULE J. >y the per[)endicular breadth, arul e area. ortunity afTortlcd icighbour in the veyor too comes lalified or unau- h libertioss. By s are frequently la])Iy magnitude MENSURATION OP LANDS. 99 DS. measure of the rie- '•■ ,Pfi lie, or Perch. in. xl. 1 are Chain. are Perch. are Rood. ire Acre. ire Mile. 1 n)o.st conveni- ins and l\u\i!i . it be a Sqnare. mboid. r breadth, an»l RtJLX U. As radius Is to the sine of any angle of the parallelogram, So is the i)roduct of the sides containing the angle To the area of the parallelo^nam. ' nVLE. III. Multiply the product of any two adjacent sides of the parallelogram by thejiatural sine of the included angJe. EXAMPLES. 1. A square tract of land, B A C D, {Fig. 79,) fronts on the road B A, which runs N, 80^ W., and the length of its side 44 cA. 72 /.; required the area, the courses o/the other sides, and a plan of the same. AB= = 44.72 X 44.72 = 1999.8784 ch. ~ 10 = 199 .98784 ac. = 199 .?c. 3 r. 38;). = area. The course of A C is N. 10° E.— of C D, S. 80^ E - nd ofDB, S. 10°W. Draw the road B A. Take any point on that road as at B, and lay off 44 ch. 72 I. towards A, according to the given scale. At A raise the perpendicular A C, and according to the same scale lay off 44 ch. 72 /. Draw C D equal and parallel to C A. Then join D B, and the plan is drawn. 2. Requn-ed the area of the Rectangle A G F E, (F»g- 80 ) whose length A G is 80 ch., and breadth A E 30 cA ' AGXAE = 80XS0= 2400 ch. = 240 ac. = area. 3. Required the area of a Rhombus A B C D, (Fig 81 ) whose front A B runs E. 15 cA., and the side B C, N. 10- E. and the perpendicular D K is 14 ck. 77 I. ' 1477Z. = DK. 1500Z. = AB = Ba 22.15500 4 . 62000 40 24.80000 ^n«. 22 ac. r. 24/). 4, Requ^ ' the area of a Rhomboid A B E F, {Fig. 82,) m 100 LAND SURVEYING. •n \vh')>v front A B is 15 ch., the side B E, 50 ch. 86 /., and the per|)eijdicular, 50 ch. 8 /. 5008 /. X 1500 /. = 7512000 /. = 75 ac. r. VJ p. =nrea. 5. Required the area of the above Rhomboid the angle B A F being 80°. By Rule 2: As Radius 90° 10.00000 Is to Sine / B A F 80° 9 . 99S.S5 Sois B A X A F 762.9cA. 2.88246 751.3 cA. 2.87581 751 .3 ch. = 75 ac. Or. 21 p. = area. Or, by Rule 3: B A X A F X Nat. Sine / 80° == 762.9 cA. X 98481 = 751. SI 1549 ch. = 75 ac. Or. 20.98;>.j or 75 ac. r. 21 p., nearly. PROBLEM 11. To find the area of a Triangle. RULE 1. Multiply one of its sides by a perpendicular let fall upon it from the opposite angle, and half the product will be the area. RULE n. Multiply the product of any two of its Bides by the natu- ral sine of their included angle and half the product will be the area, EXAMPLES. I. In the triangle ABC, (Fig. 83,) A B is 15 cA., and the perpendicular C D let fall upon it from the opposite an- gle at C is 14 ch. 77 l.-, required the area. A B = 1500 Z. CD = 1477/. 2)2215500 A rea = 1 1 . 07700 I. 4 .30800 40 12.35000 X 1 ac. Or. 12 p. MCN8URATI0N OP LANDS. 101 ^.»u t. 8 cA., and A C runs N. 20*^ E. 26 ch. (J5 /.• lo- quired the area. S. 80^ E. N. 20^ E. 180' 100' 100 Sum, 80° = ^ B A C Then by Rule 2, A C X A B X Nut. Sine / 80' ^ ,HnZn ^ ' '^- ^ ■'^'^' = 209.9614920 cA. ^ 2 = lO-i .9807460 ch. = 10.4980746 ac. = 10 ac 2 r. nearly, ^ urea. I he area may al«o be determined by Logarithn.s, accord- nig to the following rule: As Radius I^ lo the sine of any antjle of a i ian-rle So is the product of the sides containing the anglo 1 o twice the area of the triangle. Then, As Radius 90° Is to the Sine of / B A C 80' .So is ABXAC= 26.65X8= 213.2 10.00000 9 99.335 2 . 32878 To twice the area 210. ch. 2.32213 Then 210 ^ 2 =: 105 cA. = 10 ac. 2 r. = area. PROBLEM III. The three sides of a Mangle being given to find the arm. RULE. From half the sum of the three sides subtract each sido separately. Then multiply the half of the sun. of the sides by .he three remainders successively, and the ..nuare root oi the product will be the area. . EXAMPLE. ..^T'ri'}^ ^'"^ °^ "" '"^"'^^^ ^^^°«« three «ide.. arc re> spcctively 20, 30, and 40 ch. (Fig. 85.) 10:^ LAND SURVEYING, eh. 30 80 40 2)90 sum, 45 half-surn, 25 1st rem. 1125 15 2nd rem. 16875 5 3rd rem. 45 — 20 = 25 l8f rem. 45 — 30= 15 2nd rem, 45 — 40 = 5 3ni rem. V 84375 = 290.47 ch. = 29.047 ac. == 29 ac. r. 7 p. area. By L0UARITUM.S. Half-.suni 45 1st rem. 25 2nd rem. 15 3rd rem. 5 Log. 1.G5.S21 1.39794 1.17G09 0.69897 2)4.92021 sum. 290.4 cA. 2.463105 square root. Now 290.4 ch. = 29.04 ac. — 29 ac. r. Gj3. 5= area. PROBLE ' IV. Tcjind ih:- irea of v Trapczoii^. nuLE. Add together the two parallel sides, and multiir- their sum by the perpendicular breadth, or by the distan be- tween them. Then half the product will be the area. EXAMPLE. Required the area of a Tr.npnzoid A BCD, {Fig. 86,) whose parallel sides are 12 ch. 41 I. and 8 ch. 22 /., and whose perpcndicuicir distance is b ch. ID I. MENSURATION OF LAMDS. A B= 12.41 CD= 8 22 lOA Sum of parallel sides 20. G3 And rectangular breadth =: 5.15 2)100.2445 = product. Half prod. =: 53. 12225 cA. = 5.S12225 »c. 10 p. nearly. = '» uc. I »• PROBLEM V. To find the area of a Megular Polygon. RULE. Multiply the perimeter of the polygon or the sum of its Bides, by the perpendicular let fall from the centre ui)ori one; of the sides, and half the product will be tho area. EXAMPLE. The sides of the regular pentagon A B C D P: A, {Fig. 87,) measure each 25 ch., and the perpendicular O P mea- Burcs 17 ch. 2 /.; required the area. 25 cA. X 5= 125 ch. := perimeter. Then 125 ch. X 17- .02 ch. = 2127.50 cA. -^ 2 = 1063.75 cA.= 10G.S75ar, = 100 Of. 1 r. 20 j9. = area. PROBLEM VL To find the area of a Trapezium. RULE. Draw a diagonal dividing the trapezium into two trian- jrles; then find tho areas of these triangles, and their sum will he the area required. iYo^e. -If two perpendiculars be let fall on the diagonal Jrom the opposite angles, and their sum multiplied bv the diagonal, halt the product will be the area. EXAMPLE. -n the trapezium A B C D, {Fig. 88.) the diagonal A C is 13cA. 50/., the perpendicular Da 6 cli. 50 /., and B6 5 -/i. 70 /.; required the area. J 04 LAND SURVEYING, 5.70 + 6.50= 12.20 X 13.50 = 164.7000 - 3500 ch. = 8.235 ac. = 8 «c. r. 37 p. = area. •2 = S2. ffi! I, ' PROBLEM VII. To find the area of any Rectilineal Figure. RULE. Divide the figure into triangles, find their areas separately, then the sum of their areas will be the area required. EXAMPLE. Required the number of acres contained in the farm, thr, field notes of which are contained in the following Feld Book. (Fig. 89.) No. OF Triangle. Bases in links. Perpend. in links. 1 o 3 4 5 Double Areas in links. 5210 6400 6100 6100 4500 1700 2500 2900 2500 1450 8857000 16000000 17690000 15250000 6525000 Sum of douiile areas Area in links 64333000 321 61000 4 2.44000 40 .ins. 321 ac. 2 r. 17;?. 17.60000 In running the J)ase line, the perpendicular may be taken by the cross stafl^l Set up the cross staflTat that point in the base at which, while one of the lines c ' the cross ranges with the base, the other points exactly to the opposite angle. Measure the distance from that point to the opposite angle, and enter in the Fold Book in the column marked perpen- diculars. Fini.«h the running of the base line, and insert its entire length in the column marked bases. Multiply the base by the perpendicular, and insert the product in the column marked double areas. Proceed in this manner until all the triangles of which the figure i.s frjinnoRfid are mea^ .sured and computed. Divide the sum of the double areas by 2, and you have the area of the whole figure or farm, 00^0 = Si, •ea. Fissure, MENSURATION OF LANDS. Vf)h In order to plot tlio survey it will be necessary to note the distance of the point at whicJ^ the perpendicular was taken from the end of the base line. i^"i^r~'^ *^'°"^^ staff with sights is to be preferred to one whicli has only small points at the extremities of the cr<)-sed are mea- D double areas ire or farm.. PROBLEM VIII. To find the area of a Mixtilineal Figure, hy means of eqiii- distant ordinates. RULE. Measure the perpendicular breadths of the figure in seve- ral places, equidistant from each other, then divide the sum of these perpendicular breadths by their number, the ((uo- tient multiplied by the whole length of the line will give a near approximation to the area of the figure. example. The length of the base of a field, curvilinear on one side, is 7 ch. 20 /., {Fig. 90.); and the lengths of seven equidi?:- tant ordinates erected upon it are respectively 200/., 225/., 230/., 248/., 260/., 280/., and 300/.; required the area of the field. . . ^inks. Then, 249 mean breadth. A I = 200 7C70 D A = 225 L fjf = 230 1.79280 area in link*. F/ = 248 4 G e = 260 Hdr== 280 3.17120 (3 C = 300 40 7)1743 6.84800 249 mean breadth. Ans. lac. 3 r. 6/?. nearly. If greater accuracy l)e re(|uired, take half the sum of the two extreme breadths for one of the ordinates, and add it to Uie others as before; then divide the sum by the number of parts in the base, instead of bv the mimbRv nf ni.Hin,.t«. '•Hid this mean breadt lultiplied by the length of the ba so ii 106 LAND SURVEriNG. will give the area. It is still, however, only an approxima- tion, but sufficiently near the truth in ordinary circumstances. It may be observrl, farther, that the greater the number of onlinates employed, the nearer the result will be to the ex- act area. When the curved boundary meets the base, as in often the case in surveying, the area is found by dividing the sum of all the ordinates by the number of parts in the base, and then multiplying the quotient by the length. If it is particularly inconvenient or imiu-acticablc to erect ordinates at equal distances, perpendiculars may be raised at unequal distances, and the parts into which the figure is then divided may be computed as so many trapezoids, and the^sum of their areas taken as the area of the whole. PROBLEM IX. To find the diameter of a circle whose circumference is given, or the circumference when the diameter is given. RULE I. As 7 is to 22, so is the diameter to the circumference. As 22 is to 7, so is the circumference to the diameter. RULI^II. As 1 is to 3. 1416, so is the diameter to the circumference. As 3. 1416 is to 1, so is the circumference to the diameter. EXAMPLES. 1. If the diameter of a circle be 1 ch. 12.68 /., what is the length of the circumference. Asl : 3.1416 : : 112.68/. : 354 /., or 3 ch. 54 L, Jlns. I. If the circumference of a circle is 3 ch. 54 /., what is the diameter. As 3.14ir> : 1 : : 354 I. : 112.68 l, or 1 ch. 12.68 /. Jns. By this problem the number of degrees contained in the radius of a circle may be determined. For since the radius is half the diameter, and the cirenmference cnntaing 360° it follows that half the quotient of 360° divided by 3. Hie^ill cive the number of degrees contained in the radius. MENSURATION OF LANDS, PROBLEM X. To find the area of a circle whose diamcicr is given. !07 RULE, Multiply the s~- X .07»53 = 78«.6.9C8O0 L == 7 „.. 3 ,, ,3^. ^ /., what is the PROBLEM XIL To find the circyference of an Ellipse, the transverse and conjugate diameters being given. RULE. Multiply the square root of half the sum of thn the two diameter. 1^ 3.14,(i, and the ^h ■ villT"?' '•ircumfereuee, nearly. l"-0(ni( t udl be the EXAMPLE. Rrquired the circumference of -.n Vii:. trans- v^^L±_Ji'l ^ X 3.1410'= V 5iljf_36^ xa. Mifi==. 108 LAND SURVEYING. V58.5X 3.141G = 7.648 X 3.1416= 23.9269= the vircumfVionce. PROBLEM XIII. To determine the area of an Ellipse, the transverse and conjugate diameters being given. RULE. Multiply the product of the transverse and conjugate di- junolers by . 7854, und the result will give the area. EXAMPLE. Required the area of an Ellipse whose transverse diame- ter is 9 eh., and conjugate 6 ch. 9 X 6 X .7854= 42. 4116 cA. = 4ac. Or. 3B p. Ans. Note. — In practice, a surveyor is seldom required to mea- sure a circle or ellipse. PROBLEM XIV. To find the area of a farm by drawing apian of il from a scale of equal parts. RULE. On good paper, draw a correct plan by a scale of equal parts, (say a scale of 2 ch. to an inch, — the larger the scale employed the more accurate the work will be). Draw large lines dividing the figure into triangles, on these bases let perpendiculars fall from the opposite angles. Measure the length of these bases and perpendiculars by the same scale by which the plan was drawn, and enter them into their re- spective columns in the Calculation Table. Multiply the several bases by their corresponding perpendiculars, and insert their products in the column of double areas. Then half the sum of this column will be the area of the field, nearly. EXAMPLE. Required the area of the farm, the field notes of which are contained in the following Field Book. (Fig. 91.) MENSUR.VTION OP LANDS. 109 23.9269 = the transverse and ?n. [1 conjugate di- e area. lasverse diame- 3Sp. Jlns. jquired to meu- un of it from a scale of equal larger the scale ). Draw largo these bases let . Measure the the same scale m into their re- Multiply the mdiculars, and 3 areas. Then ea of the field, lotes of which (Fis.dl.) Fie i-i) Book. Nu. OF S TATION. Course. |D ISTANCE. 1 at A B N.18'^34'E. 4892 2 BC N. 20^ E. 5000 3 CD East 4000 4 dp: S. 20^^ E. 5100 5 EF N. 80^ VV. 4500 6 FG S. 15° E. 5500 7 GA West. 6000 Calculation Table. No OF Trian. Base. |Perp. jDouBLE Area. 1 2 3 4 5 CED . C K F C B F B G F A GB j 7503 7503 5000 6404 6000 1 2506 2300 2600 2603 4609 18802518 1 7256900 13000000 16669612 27654000 2)93383030 466.91515 4 3.66060 40 t ... ^ 26.42400 ^'Ins. 466 ac. 3 r. 26 p.. Kemarks.-.ln the preceding Field Book the first column contain.s the No. ot the stations, and consequentiv the num- ber ot the sides of the farm; the second, the bearings of the Imes from the meridian; the third, the stationary (fistances m links. 1 he second and third columns contain all the data necessary to protract the plan of the farm, A BCD E F G A i he first cokmmof the Calculation Table contains the iNos. of the triangles into which the lignre is divided- the second the base of caclMrianglo; the third, the porpemJicu- ur; and the l.)urth, the product of the base and perpendicu- lar, mid IS called the column of double areas, half the sum of which gives the area, nearly. This method, however, will only give an approximation lil. IV''"'- 1^^ '^ ^^^\poss\h\c by it to determine the length, ot base and perpendicular lines within several links, tvspocially It the-e lines arc of considerable len-rh 110 LAND SURVEYING. PROBLKM XV. 1 find Ihc'avca of am, Rectilinear Fis;urc (the courses and distances roimd the same beinu; friven,) without the assist, ance of a plan, by Rcctanii;ulur Surveying-; i. e. by calcu- lation from Tables of Northing, Souihing; Easting, and il esting. KULE. Prepare a Tablcj with ten columns. In the first, lioaded '' No. of Stilt.," write the nimiber of the staiioii, 0, 1, 2, 3, &c. Ill the second, headed " IJeurinir," write the eour/ie,' In the third, n.arked " Di^st. in ch. /.," insert the distanec hi chaniH and links. In the fourth and fifth eohunns, headed " Difi". Lat.," insert the Difierence of LutitiKh.-, a.-eordin.r to the directiojjw contained in the HI. and IV. Prob. ol" Transverse n^tniinjr of lines. Fill up likewise, ae(;ording to tlu' same directions, the sixth and seventh colunms, heu(hMl " Half Dep.," observing I only that inste-ad of the whole, the half of the Departure is to be entered. Place the si-.i -|- before the Hastings, and the .si-n- before the Westi..-s. W the field notes have been ('orrectly taken, and the entries of Difi". of I.at. and of Half Dep. been accurately nuule, the sum of the North and S()uth C(dunuis will be equal. The sum of the East and ^V est colunuis will also be etiual. In the eighth column headed " Mer. Dist.," and oppositu U) in the first colunm, insert the whole Departure, or doti- l>le the sum uf the Half-Kastin-s, or Half-VVestin-^ con- tained in the sixth and seventh columns. Obscrre what sum has been entered in thefi'rst lineof eitiierof the columns headed "Half Dep.," opposite to the fi-ure 1 in the fast column. Observe also whether the v.niry has been nutde in the E. or W. cohunn. If the entry Jias been made in tlw r:. colunm, add the smn to the whole Departure. If the entry has been made in the W. column, snbtract'the sun. irom the whole Dej.arture. Insert the smn or difierence in the ei-dith culuiim in, iik.ul " Mer. Dist." 01 kiriy, that the sun) now iu.sertef the. middle of the first line. To this sum add or subtract accordinjr as it is P^ast or W(^st the same sum or Hjslf De- parture, and you have the meridional distance at the end of the fi-st line. Observe a,i,rain the sum that i'* entered in the column of Half Dep., oj)j)osite to the figiii-.' 0. in the iirst column ^f the Table; and accordinpr as it Ici- !he sign 4- or — pr-^fixcd, add it to or subtract it from the in. ildional dis- tance at the end of the first line. This will ive the meri- dional distance of the wuV/r/Zt' of the secojid i!;)f. Proceed in this manner until the column is completeh :i!l, d. if the operation has been correctly performed, the last sum will be equal to the sum at the head of the column. Next, multiply the several meridional distances at the middle of each line into the Northing or Southing, which will be found opposite to that meridionnl distance hi one of the columns marked Dim Lat. in the Table. Y\heu the sum has been taken from the column marked " N." the pro- duct is to be inserted in the ninth colunm, marked '' North Areas;" but if it have been taken from the column mnrKed "S." the product must be entered in the tenth coiinim, headed "South Areas.^' Then the difference between the sum of the products contained in the column of North Areas and the sum o" the products in the column of South Areas will be the area of the Figure. t. Idionul dislauGi EXAMPLES. 1- Required the aroa of a farm whose field notes are as follows, viz:— N. 20° E. 50 ch., P^ast 10 ch., S. 20° E. 51 eh., N. 80° W. 45 ch., S. 15^ E. 55 ch., West 60 ch., and N. 18° 34' E. 48 ch. 92 /. Having prepared your Table, and entered your stations, courses and distances in their resjjective columns, by the di- rections and principles laid down for the running of linos, or from the annexed Tables, find the DitT. Lat., and the Half Dep., and insert them in their proper i.dace in the Tn- ble. Thus the first course is N. 20^^ E., and distance 50 ch., the Northing is 40 ch. 98 /., and the Half Easting 8 ch. 55 /. Insert the former in the column marked N.; and the latter Ml l.KTil) SURVEYING. n the column marked E., and place the sign + before U. Proceed in this manner until the columns of DifT. Lat. and Half Dep. are filled u]). Add up the colunm.s of DifT. Lat. and of Half Dep. The sum of the Northings is 101.05. The sum of the Southings is also 101 .05. The Eastings and Westiygs arc likewise equal. These agreements shew that the survey 'las been correctly taken. Proceed next to fill np the column headed " Mer. Dist," In this column, in the same line with in the first column, write 104 ch. 32 /., the whole departure or double the sum of the Half Eastings or Half Westings. Under the head- ing «' Half Dep.," and in the column marked E., you will find the sum 4-8.55 has been entered. Then 104.32-4- 8.55 = 112.87. This sum is the meridional distance at the middle of the first line. Insert it in the column of Mer. Dist., opposite to 1 in the first column. Then to this sum add 8.55, and you have 121.42, the meridional distance at the end of the first line, or at the beginning of the second line. Place this sum in the column of Mer. Dist., perpen- dicularly below the Mer. Dist. at the middle of the first line. Again to the sum last entered add 20.00, which you will find in the E. Column, of Half Dep., and it will give you the meridional distance at the middle of the second line. Thus 121.42 + 20.00 = ]41.'t2. Insert thi.j sum in the column of Mer. Dist., immediately below the last entry, and directly opposite to 2 in the first column. Proceed in this manner until the column is filled up. The last sum must be equal to the first, or the sum at the head of this column. Next, multiply the meridional uistanct at the middle of the first line, which in this example is 112.87, by the differ- ence of latitude which will be found under « Diff. Lat." in the column marked N., and which in this Example is 46.98; and insert the product 5302.6326 in the column of North Areas, because the Diff. Lat. ij N. The second meridional distance in the middle of the line in this column of Mer. Dist. is 141.42, but as the course is due East there is no dif- lerence of Latitude, und conscn[UGntly no product to be in- MBKSURATIOM OF LANDS, U.i Uff. Lat. and •crted in either column of areas, The third meridional dis- tance in the middle of the line is 170.14, and the different*^ of Latitude is 74.92 in the column marked S. The product of these numbers is 8153.1088, and is to be placed in the Column of South Areas, because the Difl*. Lat. is S. Pro- ceed in this manner until these columns are completed. ^' B.— The meridional distance for the middle of the line will always be found in the column of Mer. Dist., opposite to, and m the same line with, the No. of the Station, the Bcarmg, Distance, DifT. Lat. and Half. Dep. ♦ The sum of the products contained in the column of North Areas is 10989.0006, and the sum of the products in the co- lumn ot South Areas 15679.5043. The difference between them 4690.5040 ch., or 469 ac, r. 8 ». is the area of the r arm. "^ 114 tMfD SURVETING. o tf ax o as u ?5 W u Q fa ao H ■< O 1 — ® CO CO iljjA in •^ Vi« OS S <2 CO IM lO iO CO i CO r>. P o CO 8 o o CO »-« 'I" c?> CO o !>• »?» c» (n rf (r> o >* CO 00 CO o) vn e» 00 -^ -n- '^ »-< CO t-» »n «0 »^ t^ r^ »f5 GO G« -H _■ r. o CO «o Tf •-< CO »-^ (D CO »* 1—1 5» TT to 1^ t- iOCO'<3'-^COaD<5iO I-* »^ I-« 1— ( T-H »-< 1—* ""^ w o CO < urn o »-4 s o n» 1 Gt CO o I 1 •n o r« c< r-. 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" ~ 75 ,« 75 ' i a -S a o a. a b O ••« «a Q> .£3 fa .. ° s '^ ^ rt 3 "" s ■s 2 ID U «-i • CO -n" CO c» rr rr T cr CO Xin CO 'o o J^ • • • • • m G» G« o in f^ .-^ CO t>» Oi -r ll £ &. " a. p . a. 5 u = CD •• W jl; • r » u ^ (-.•'• o '^ " o, c • Hi)'- a. ^ > fl - - C4 *^ "1 r; (S (— ! tK (-» ■•iw ■S;o §2 s = -3 bo-3 C 3 ~ O 7J o ^rf M^ o «■« o «-J so m _c na u. IS O a. u Q} ti O O 'I" CI T »-< O "V C? G» • • • , • !>• C t^ •^ f G« G« G« G» C« xxxxx in o in o G« CO G» t^ -< ,-4 00 o X -^i r- an e "S-S cur 5 2 OS « O a u 2 fi M-i n - - a - Q « 3 ^0,0 CO CQ 2 " . 5 3 < 'o S $ cd o ^ «> DIVISION OF LANT>. U7 (5* 00 -r --rj II II II II ■^ 00 CO Q «5 O r- O '^i O irJ o »>• crs Tf CO •5 t o w E b •' c b> s ■ a. E u 3 dS'S m fl - - *" o "5 4) a> c 1" T (T) CO CO CO o J^ tI e< o »n CO I— OJ 'T + -f-|-}- O -^ CI T o T CO <;;» C t>-* Tj! TP T3< rl Tf — < 5« G« &» C« xxxx D CI CO i5« • • • • s X c« r- ■^ *-• OS r: « . -O 60- S S 3 OS § C — la O a u =^ I ° 2 "s S «•- « - OS « « a - Q^ 3 O *» . o u "Jl .s ^ S •° .2 & a .s = .fc'u the Norf lungs a.ul Southing, into a w w,^^^^^^^^^^ "/'"'I. 'I'" « f the .olm , ;\ ■ Sou h te wiVr^'lr''? '" DIVISION OF LAND. PROBLEM J. To divide a parallelogram in any propor-l, on hy a line run- nmg parallel to a given side. RUtE. Since parallelograms of the same altitude are to one ano- ther as their bases, (Euc. vi 1 > fir,o .. . I ^^"*- *'• ^>>' "'^<' nift the {>rea or quan- tity of land contained in the whole figure. Then, as the area of the whole parallelogram is to its base, ho is the area or quantity of land to be laid off, to its base, EXAMPLE. i 20 1a' P'f '^'^«^!r .^ ^ ^ ^' <^^- ^4') the base A B 1.. 20 ch., and the side A D is 16 cA. It is required to de- raised from these points, to the East or West acconlin. .0 .1 m,.p IS to be drawn on the East or West side 0^^^ ^ . ^ -^^''^ /'I, I IB LAND SUnVEYIKG. terminc the point in the base A B, from which a right line must coinnicnco, which, runnin^r i.arallel to b C until it strikes the opposite side D C, shall form a rectangle E F B C, containip-r lo ac. A D X A B = 16 X 20 = 320 cA. = 32 ac. = whole area. Then, as 32 ac. : 20 ch. : : 10 ac. : G ch. 25 ;. = B F, the base of a rectangle E F B C, con- taining 10 ac. Then from the point B lay off* 6 ch. 25 /. to- wards A, and you will have the point required. The preceding rule applies to all parallelograms, whether Squares, Rectangles, Rhombuses, or Rhomboids, PROBLEM ir. From a ginen point in the boundaries of a Square or Rcc- tan^le, to run a line which shall cut off a given quantity of land in a giv."n direction. EXAMPLE. In the Rectangle A B C D, (Fig. 95,) containing 28 ac. the line A B runs E. 14 ch., and the line AD runs N. 20 cA. It is requirf'fl to lay off 15 ac, to the east by a line commcnc- cing at E, 8 ch. from the / at B. The course and distance of this division luie are also required. RUIiE. From the given point E run the line E F parallel to A D or B C. Then determine the area of the rectangle A E F D; thus, A D X A E = 20 X C == 120 c/t. =12 ac. = area of A E F D. Subtract this area from the area of the whole filture; thus, 28 — 12 = IG crc. = area of the remaining rectangle E F C B. Find the difference between this area and the area to be laid off; thus, 16 — 15 = 1 ac. Now it will bo seen at once that this area is in the form of a riirht angled triangle. Of this triangle there are known or deter- mined the area which is 1 ac.,and the base line E F, which is 20 ch. From these data, determine the length of :he per- pendicular F S, by the following rule: x)ivide the area by half the base, and the quotient will he tho length of the perpendicular; thus, 1 ac. or 10 ch. -r EF DIVISION OF LAND. UO ch a right line o h C until it angle EFBC, !0 = 320 ch. =r ch. : : 10 ac. : E F B C, oon- ' C ch. 25 /. to- ll. rams, whether jidsj quare or Rce- fen quantily of taining 28 ac. run-s N. 20 ch. line comnicnc- e and distance arallel to A D nglcAEFD; ac. = area of of the whole he remaining eeii this area 1 ac. Now it rni of a right own or dcter- e E F, which th of he per- lotient will he EF or 10 ch = 1 ch., which is the length ol' the pcrj.endicular i" b. As the area contained in the roctan^-le E 13 C F cut of}- by the line E F, exceeds the orth from tlie base or front A B. From the angle at D let a perpendicular D S fall upon ho base A B. By Rule I., Right-angled Trironon.etrv, we ;;;;•' that as is l ch. 39 /., and that D S is Hch. in length. Iheu by Prob. II., Mensuration, we find thn nma of the tnangie A D S to be 55000 /. Now the area of the whole i rape.oul is 12 ac. 3 r. ^ p. or 1280000 /. Half the area of no LAND SURVEYING. tho wliole Trapezoid is therefore 640000 /. But we have already the ,area of the triangle A D S = 55600, which, sub- tracted from 640000 leaves 584400 /. as the area remaining to be laid off, which from an inspection of the figure, it in evident must be laid off in the shape or figure of a rectangle. In this rectangle we have the area 584400 /., and one of the sides, D S, 8 ch. Hence, 584400 8.00 7 ch. 30 /. Then from the point S run off 7 ch. SO I. due East to E,; and from E run E F due North and the the trapezoid is divided by it as wan required. PROBLEM IV. To divide a prn-altelogram into two equal parts by a line. * riminng from a given point. EXAMPLE. Let A B C D, (J%. 97,) be a parallelogram. It is re- quired to run a lino from the point F which shall divide the parallelogram into two equal parts. Draw the diagonal' D B, and bisect it. Draw a line from the point F through the point of bisection of the diagonal, and continue it till it strikes the opposite side D C iu the point E; the line F E will divide the parallelogram A B C D into two equal parts, as was required. (Euc. i. 34.) PROBLEM V. To divide a Trapezium.. EXAMPLE.. In the trapezium A B C D, (Fig. 98,) the two sides B C and A D are parallel. It is required to cut off one third ol" the whole area by a line running from the point A. Produce the line B C to E, so that C E may be equal to A D. From J hiy off B G equal one third of B E, and join A G; the triaTigle A G B is the third part of the trapezium A BCD. (Euc vi. 1.) DIVISION OP LAND. /. But we have j600, which, sub- 3 area remainiinj the figure, it is re of a rectangle. ., and one of the /. Then from E,; and tVom E is divided by it ur PROBLEM VI. To divide a Trapezium into two equal parts, h, « line drawn from one of its aiigles. EXAMPLE. Let A B C D, (Fig-. 99,) be the given trapezium, and A the angle from which the dividing line is to be drawn. Draw the diagonals A C and B D. Bisect D B in E. Throufrh E draw G E F parallel to A C. Join A F and it will di- vide the trapezium A B C D into two equal parts. parts by a line It is re- shall divide the gram 'raw a line from )f the diagonal, side D C iu the logram A B C D le. i. 34.) ; two sides B C oft' one third of oint A. nay be equal to if IJ E, and join the trapeziuiiv PROBLEM VIL To divide a Triangle into any proposed number of equal parts, by lines running- from a given angle. EXAMPLE. It is required to divide the triangle ABC, (Fig. iOO,) into three equal parts, by lines running from the ano-le at C According to Euc. i. 38, triangles upon equal bases,' and between the same parallels, are equal to each other, there- fore divide the line opposite the angle at C into three equal parts, and from the points of division D, E, draw the lines D C and E C. The areas of the triangles A D C, D E C, and E B C, will be equal to each other. PROBLEM Vm. To divide a Triangle by lines running parallel to a given side. EXAMPLES. 1. In the triangle ABC, (Fig. 101,) the side A B mea- sures 40 cA., the side A C 53 ch., and the side C B 56 c^i It Ks requu-ed to divide the triangle ABC into two equal parts, by a hue running parallel to A B According to Euc. vi. 19, similar triangles are to each other m the duphcate ratio of their homologous sides: there- tore as the area of the whole triangle is to the square of its mle, so ,s the urea to be cut oft' to the square of its side" 122 LAND SURVEYING. The square root of this sum will he the side of rho trJanHe to be cut ofl^ thus: the area of the triangle A B C is 106 of niul consequently th(^-ea to be cut off is 53 ac. Then, us AJB C or lOG ac. : A C^- or 3364 ch. : : C D E or 53 ai, : C L)» or 1683 ch.', and V 1682 = 41 = C D. Then, draw I) E from the point D, parallel to A B, and the trian-le is divided into two equal parts, as was required. 2. Divide the above triangle into three equal parts, by lines runninjr parallel to the base A B, (Fig.. 102.) The whole area being 100 ac, one third of it is 35.3333 -f nc; therefore as ABC oi^06 ac. : ATC^ ov 3364 ch. : : C E U or 35.3333 -\- ac. : C E= or 1121.33 A-, and V J121 .33 + = 33 ch. 43 /. = C E. Again as 106 : 3364 : : C G II or 70.6666 -f- • C~G- or 2242.65; then V 2242.65 = 47 ch. 35 /. = C G. The:i through the points E and G draw ED and GH parallel to A B, and the triangle is divided into three equal jiarts, as was required. ^ By the same principle any specified amount of area may oo laid off in a trapezium by a line running i)arallel to one oi Its sides, for by producing some two of its sides until they meet, a triungle Mill be formed. EXAMPLE. In the trapezium A B C D, (Fig. 103,) the side A D runs S. 17- E. Ibch., A B runs E. ^20 ch., and B C runs 8. 20^ ^'V . 1 7 ch. It is required to lay off 6 ac. towards the North by a line running parallel to A B. Produce the sides A D and B C until they meet in F i'hcn ISO^ - (Z E A B -]- Z E B A) := 37^ = / A EB. Bv Obli^ur Angi>ed Trigonometry. To find the side \ E. As Sine Z A E D 37° Is to A B 20 Ho is Sine /ABE 70^ To A E 31.23 9.77046 1.30103 9.97299 1.4945G DIVISION OF I. VND. m3 To find the side B E. As Sine Z A E B 37^ Is to A B 20 80 is Sine Z B A E 73° 9.77046 1.J0103 9.980G0 To BE 31.78 1.50217 Next, by Mensuration, Prob. II. Rule 2, the area of the triangle A B E is found to be 29.86331 ac, from which take the area to be cut off 6 ac, and the remainder is 23.8633 lac. Then as 29.86334 : B E^ or 1009.9684 : : 23.88334 : ET-^ or 807.0507, and V 807.0507 = 23 ch. 40 I. Lay off B F =-- 28 ch. 40 I, and from F run F G due West, and A B F (i will contain 6 ac. By the same rule we find that the triangle E C D contains 7 ac. 2 r. and 35 jo., and that the trapezium G F C D con- tains 16 ac. and 23 p. PROBLEM IX. To divide land hy Calculation. EXAMPLE. It is required to divide the farm A B C D E F A, {Fig. 104,) into two equal parts by a line running from the point A, Calculate first the area of the whole farm. Then draw a line, or suppose a line to be drawn from the given point to some other known station as at D, which will divide the farm in the required proportion, as nearly as you can judge. Then fill up the columns of a Calculation Table with the courses and distances from A round to D. The difference be- tween the sum of the North and of the South columns will show the Difterence of Latitude, and the diflerencc between the sum of the East and of the West columns will show the Departure of D A. From these data find the area of the part cut off A B C D A. Find the difference between this area and the area of the half of the whole faru). That dif- ftn-cncc will be a triangle. If the area of the piece cut off exceed the half area of the whole figure, that triangle lies within the figure cut off, but if it be less than half the area ^-24 LAND SURVEYING. <)(■ the whole figure, the triangle lies on the opposite side of the lin(^ A D. Suppose that tho quantity cut off exceed the half of the whole area. Find the area of the triangle ADC. Then as the area of the triangle A D C is to the square of the side D C so is the area of the triangle A D H to the square of the side D H. Then having the course and dis- iuv ca of A D and D H, by a Traverse Table find the course anl iistance of A H, the true dividing line. Set your com- pass at A, and run the course and distance A H, and the farm is divided as required. Suppose the farm to contain 16 ac. 1 r. 21 p., and the field notes of the survey to be as follows: Commenced at station A, and run thence S. 80° E. 7 ch. 50 Z., thence S. 15° E. 7 ch. 90 /., thence S. 20° W. 10 ch., thence N. 75° W. 8 ch. 45 L, thence N. 8 ch., thence N. 15" E. 8.50 /., to the place of beginning. Let the plan be laid by a scale of 10 ch. to an inch. DIVISION OP LAND, 125 pposite side of off exceed the riangle ADC. I the square of A D H to the jurse and dis- find the course Set your com- A H, and the .J and the field a s« H < S U o a: CC 5 h o 6 1?; en o 4- -f o o O CD !^» o GO O 50 G^ CO •^ GO • 1— « h- Ci CD CO eo GO GO • CO CD _,„. T-t i-H o o Ci o o o» • 1^ o 05 »-i • o ^ 1-^ I— ( ira & G« CD m c« »-< GO •=? CO 05 co'^ 00 o 12; o Q o ^ rr»o C^l G« '-' (?» OS OS CO X> >o •^ G-« SI Oj OS 1-H 1— ( T-^ X) GO CO 03 UO c< u Q •«1 triangle E. E. ls^= 1-4 {j^ o o O O CO i]; -i u >■ ii i3 CO Q.S a — < e S3 s )i S 2 ^ 2 il § ^ ' O ti *>» M . _, •-< >» •• « 2 "^ tr. ^ CO aJ a> — ;-* " ' Ph » ^ a> « o (=> c ca ".23 -w '^ "^ TJ c « o o o CO ci in m MM 3 o C -J .23 i'-2G T.AND StIRVKVING. Cnfr.ula/ionor of Conslniction. "><'"ion of liy Calcutation wo can ascortain flio true course- -in,! rJi« PROBLEM X. To divide a Triangular Lot of Land in certain proportions. EXAMPLE. Being employed to divide a ten acre lot of Marsh between h CO clannants,-A, B and C; A clain.in, 4 acres, and B and C 3 acres each. The lot is triangular. The base A B ( IS- 105,) measuring 20 ,/,., ,„j ^ perpendicular let fall .hereon fr-om the opposite angle measuring 8 ch. 50 /. The lot, therefore, it is evident, >vill not hold out its .neasure- ment. Now snj)posing .;.. division lines to run from the angle opposite to the base, how much land, and what nro^ portion of the base should each claimant receive C D X A B = 20.00 X 8.50 = 170.0000 ^ 2 =, g^.oOOG = 8 ac. 2 r. = area of the lot. AslO:8^:.4:3ac. lr.24p. = A'sshare. :3:2«c. 2n8;,. = B'sorC'sshare. 20 ch. : : 3 ac. I r. 24;,. ; 8 ch. = A's share As 10 : 8^ : As 8.5 ac. : of base. As 8.5 ac. : share of base. 20 ch. 2 «c. 2 r. 8 ^. : 6 ch. = B's and C's PROBLEM XL To divide by Calculation a lot of land of a certain amount of value <^nong different claimants, in proportion to the amount of their claims and the estimated value of the land. EXAMPLE. A Testator leaves by ys\\\ a lot of land containing 500 ac., the value of which he estimates at £1470, to be divided f»»,u DIVISION OP LAND. V27 fiinong h.s servants A B C D E F in the following propor- uom, ucnonling to the value of the hind, viz: to A he bo- T'lnn''' V'''"" "^""'^^ •^^^' t«B £20, to C £10, to D ^100, to E £400, and to F £1,000. Now the vah.e of the land most convenient for A, B, and C, is estimated at 7s per acre, vs le the land most convenient for D is worth 10s per aero, for E 15.. per acre, and for F 12.. per acre [t is roqu.re.l to determine the quantity of land which falls ta the share of each. nULE. Divide the sum bequeathed to each by the value of the and per acre which is to Le allotted to him, add the quo- ■ont, together, d.vule the whole given quantity of lani by .ha sum; th.s quotient will be a common .nultiplier, by « ch multiply each particular quotient and the jn-oduct h/d viz, ' ''T"'"''r' "'""^^ '^''^ '^ '"^^^ .shareof each ndmdual, or: Say as the sum of all the quotients is to the Thus: A 7) 40( 5.714281 B 7) 20( 2.85714 7) 10( 1.42857 [ . 20( 2.85 f' 7) 10( 1.421 D JO) 100(10. E 15) 400(26. GC666 * 12)1000(83.33333 Quotients. Now, Then, Sum of Quotients, 129.99998 or ISO 500 -. 130 = 3.846153, the common multiplier. 5.71428X3.846153= 219778 a/. - \». i 2.85714 X 3.846153 = lo 9918 ac ~ R L'"'''' 1.42857 X 3.846153 = .5 4941 ac - T' 'h""'^' X 3.846153= 38:' 1 ^ : - g;; h'a::,^- 10. 26.66666 X 3.846153 = 10^ 5641 ar - F.'"r'"" 83 S'n*?'^ V Q QAi-ir.o MZ ^^' J^^^s share* ^''^.dJJj., X 3.846153 = 320.5127 ac. = F's share. Sum of the whole shares, = 500~o"o20 «c. ^in the same princinln if pny '^incrl- -hnv-. \ j . land of different vah.^^. " ^^ ^ '^'""'^^ '''''^'' Again if there be different quantities of land as well as xi,ltl. .nultipliod hy tho .qroator nn.l divi.lod l>y ;'.o loss nu„.!,or of the pro- j)ortiou will be tl.o leiigih. R^ 'I.E. If it H ro(pnred to lay ou m ar. 17 p. in a rortnnjrnlnr form, (Fig. Ill,) so that the l.roa.lth may l.o to tho lonuth as 5 IS to8, uhat must tho ionjrth and tho hroadth l.rn- speot.vcly? Draw u map of the lot hy a scale of .J5 ch. to an inch. I09«c.l7;>. = 109l0625/. X5=.54553l25^8=:f,H10Ma the square root of uhioh is iitill = breadth of tho farm' Ihon 2G11 X 8= 20888 -^ 5=.4179. Tho lon^th A |{ t .-^relore is 41 ch. 79 /., and the breadth A D 20 ch. 1 1 /. PROBLEM VII. To lay out land in the form of a Rectangle, so that the Un'-th may exceed tkc breadth, by a certain given quantity "' RULE. Add the squai-e of one fourth of the given difloronoo to the area, . .d f,-om tho square root of the sum .subtract half theg.ven differonee for the less side. To the remainder add tho nhole difference for tho greater side. EXAMPLE. It is required to layout 200 a., "n a rectangular form {tig 112,) so that the length may exceed the breadth by 10 ch. Y- = 250000 + 2000000 area, = 20250000, the 1000 .'•quart: ^■oot of >yhich i. 4500 - -^ or half the giyen difference =, rS'i LAND SURVEyiNG. 4000 /, = tho leas side, and 4000 -f- 1000 the whole difter- ence = 5000 /., the greater side. The side A D therefore must be 40 ch., and the side A B 50 ch. PROBLEM VIII. To lay out land in the form of a Rhomboid^ so that the length may be a given multiple of the breadth. RULE. Divide the area by the product of the given multiple and the natural sine of the given angle, and the quotient will he tho breadth; and the breadth multiplied by the given multi- jde '^'ill be ihe length. EXAMPLE. Required the sides of a Rhomboid containing 10 ac, {Fig. 113,) Avhose acute angle is 80°, and whose length is three times greater than the breadth. Nat. Sine 80^ to Rad. 1 is .98481 X 3 = 2.95443, and 10 ac. = 10.00000 /. -f- 2.95443 = 338474, the .square root of which is 32 /. or 5 ch. 82 I. = width, and 5 ch. 82 /. X 3= 17 cA. ij /. = Icuffth. PROBLEM IX. To lay out land in the form of a Rhomboid, so that the length may be to the breadth in any given proportion. RULE. Multiply the area by the less number in the given pro- portion, nnd divide the product by the product of the Nat. Sine of the given angle multiplied by the greater number of the proportion: the sipiare root of the quotient will be the •breadth; and the breadth multiplied by the greater and di- vided by the less number in the given proportion gives the length. EXAMPLE. l3(Mn2f employed to lay out 10 ac. in tho form oS a Rhoni- btiid whose length shall be to its breadth as 2 to 5, and whose DIVISION OF LAND, ISS \g 10 ac, (Fig. \ of a Rhnm- included angle shall be 80°; required the length and breadth. (See the preceding Figure.) Nat. Sine 80« is .98481 X 5 = 4.92405, and War =. 1000000 /. X ^ - 2000000 ~ 4.92405 = 4061(59, the square root of which is C3S /. or G ch. 33 I., = the hrea.lth. and 038 X 5 —^-=1595 /. or 15 ch. 95 /., = the length. PROBLEM X. To lay out land in the form of a Trapezoid* Mho.e cenlral length shall be any given multiple of the xmdlh. RULE. Divide the area by the given multiple, and the squnre root ot the quotient will be the width; which multiplied by the given multiple will give the central len-th EXAMPLE. It is desired to lay off 200 ac. on the East side of a roa..Jt0:: tm LAND SURVEYINO the differenco or half A N, the remainder will be the length of the shortest side DC. 6324 = 5411 or 54 ch. IW. = D C, the shortest side. Wherefore commencing at A, A B runs East 72 cA. 37 l. i^i"? "^^' ^^ '^^•' ^2 ^•' ^ ^ '•""^ W«st 54 ch. Ill and D A runs N. SO'^ W., and the distance is ascertained by' 1 ngonometry to be 3(> ch. 51/. PROBLEM xr. To lay out land in the form of a Trapezium, having one of its sides given. , RULE. Divide the given area into two parts, either equal or un- equal, and then find the perpendicular that will lay out one ot these parts in a right-angled triangle upon the given side as a base. 1 his perpendicular will be a diagonal of the tra^ pczium, and a ba.e upon which theren,aining triangle must be constructed. Then find the perpendicular, which, fall- mg upon the-opposito side of this base, will lay out the other part. » ,f^;.5:^---'^''^f^, perpendiculars are found by dividing dou- Ue the area ot the triangle by the given or known shfes EXAMPLE., sure 8 c?' '^^' """" °^ ''^''''*' '''^'' ^ ^ '^^" '"«^- Let the area be supposed to be divided into triangle., one of which contams 5 ac. and the other 3. 5 ac. or 500000 I. X ^ S'ch'.'i^i^QCO ir'~ ^ ^-^^ '• = perpendicular. Then from the poi.u U in tho given side B C, and perpendicular to tt draw hA = laso L, and join A C; the triangle A B C C0nt2,Hiri 5 uc. Next --"''- °'' ^^^^^^' XJi * I5r,0 = '^^O'- == perpendicular of LOCATION OF LANDS. ISS remaining triangle. From A, and perpendicular to A'B, draw A D 4S0 /. ; join D B, and tiic triangle A D B will contain 3 ac. The trapezium A C B D will also contain 8 ac, and the side B C is 10 ch., as was required. Pi.. >BLEM XII. To lay out land in the form of a Triangle, of which one side and the an^le at one of its erzlremities are given. RULE, Divide double the area by the product of the given side multiplied into the Nat. Sine of the given angle, and tho quotient will be the other side, including the given angle. EXAMPLE. From the Northern extrcTiiity of the line N C, (Fig-. 116,) which runs due North 25 ch., it is required to run another line C O, S. 34° 41' E., so that the triangle N O C may contain 80 ac. Nat. Sine of 34° 41' is .56C04 X 2500 or N C = 1422, a«d 80 oc. X 2 = leOOOQOO I. -M422 = 112457. or 112 ch. 45 /., = side C Q. PROBLEM XIII. To lay off any quantity of land in a triangular form, be- tween two lines forming an angle, one of the sides of the triangle being given. RULE. Divide double the area to be laid off by the length of the given side, and the quotient will be the length of a perpen- dicular let f\ill from the opposite angle upon some part of the base or line given. From the extremity of the given line raise a perpendicular of the ascertained length. From the end of that perpendicular run another line^ parallel to the given line until it intersects the other line. Then a line drawn from the point of intersection to tlie point frpm 136 I-\ND Sl'UVEYING. which the porpendicular Avas raised will complete the tri- angle containing the required number of acres. EXAMPLE. hi the corner or angle formed by the road A B (Fig. 117,) and A C, I am required to lay off 4 ac. fronting on the road A n 10 cL; rciiiiircd the termination of the lino A C. Double of the area = 400000 X 2 = 800000 -f- 1000 /., (the length of A 13) = 800 /. or 8 ch. = length of perpen- dicular. Then from B and perpendicular to the line A B run a lino B S, 8 ch. in length. From the end of this per- pendicular or from the point S run a line S C parallel to B A, until it strikes or intersects the side line A C aforesaid in C. Join C B. The triangle ABC contains 4 ac, as was required. N. B.— In' this case the quantity of the included ande does not aitect the accuracy of the rule. It may be an an- gle ol 80^ as C A B, or of 75° as C A B, or of 50^= as C" A B, PROBLEM XIV. To lay off land in the form of an Isosceles Triangle, the an- gle contained between the equal sides and the area being given. RULE. Divide doulle the area by the Nat. Sine of the given angle, and the square root of the quotient will be the length of one of the equal sides. EXAMPLE. It is rrq Mired to lay out 38 ac. 2 r. 18 p. in the form of an Isosceles Triangle A B C, {Fig. 118,) the course of A B is S. 25° W., and the coiuve of A C is S. 26° E.; What must be the length of thf! sides end the course of B C. 38 ac. 2 r. IS p. = 33(11250/. X 2 = 772250, the double nrea, ~ .77715, the Nat. Sine of / A, (25° + 20° = 51°) = 9936940, the square root of which is 3151 = A C or A B. Then 180^ - / A or 51- = (/ B -|- / O^or 129°. Now since the / B and the / C are equal each of them is 64'^ SC. Whercforo, by Trigonometry: LOCATION OF LA5DS, 137 plete the tri- H (Fig. 117,) J on the road A C. -f- 1000 /., h of perpcn- he line A B 1 of this per- ^ parallel to C aforesaid ins 4 ac, as ;luded angle ly be an an- »°asC"AB, ngle, the an- '■ area being f the given e the length ! form of an le of A B is What must /. the double 2G° = 51°) \CorAB. 29°. Now them is 64'' As Sine of / C 64° 30' : B A 3151 : : Sine / A 51^ • B C 2713. The courhe of A B is S. 25° W., and the / B is IJI'- 30'; the course of B C is N. 89*-^ 30' E. Hence the sides A B and AC are each 31 ch. 51 /., and the side B C runs N. 89'^ SO' E. 27 ch. 13 /. PROBLEM XV. To locate land in the form of a Circle. E,ULE. Divide the area by .7854, and the square root of the quo- tient will be the diameter. EXAMPLE. Required the diameter of a circle containing one acre. {Fig. 119.) V 1 oc. = 100000 /. -h 7854 = V 127323 . 65 =r 356 . 8 /. or 3 c/^. iJG.S /, PROBLEM XVI. To lay out land m the form of an Ellipse. CASE I. When the Transverse Diameter exceeds the Conjugate by a given quantity. RULE. Divide the area by .7854, to the quotient add the square of half the difference between the diameters, from the square? root of the sum subtract one half of the difference between the diameters, and the remainder will give the Conjugate. The difference added to the Conjugate will give the Tnuis- verse. EXAMPLE. ^ Required the Transverse and Conjugate diameters of an ^"llliprfo containing one acre, whose Transverse diameter shall exceed thu Conjuo-ate by one chain. (Fig. 120.) 1' 138 LAffD SURVEYIlfO, 1 ac, = lOOOOO /. 4- . 7854 = 127323 -j- V — or 2500 « 2 129823, the square root of which is SCO — ~ or &0 ^ 310 = Conjugate, and 310+ 100 = 410= Transverse. The Transverse diameter therefore is 4 cA. 10 /., and the Conjugate 3 ch. 10 /. CASE II. When the Transverse mid Conjugate Diameters are to each other in a certain ratio. RULE. Multiply the area by the greater number in the propor^ tion, and divide that product by the product of the less num- ber multiplied into .7854, and the square root of the quo- tient will bo the Transverse diameter; then multiply the Transverse by the less number in the proportion and di- vide by the greater, and it will give the Conjugate. {See the preceding Figure.) EXAMPLE. It is required to lay out one acre in the form of an Ellipse whose Transverse diameter shall be to its Conjugate in the ratio of 5 to 3, 100000 X 77854 X 3 » 276 = the Conjugate* 4C0 = the Transverse, and l^l^L? APPENDIX. It is presumed that in the preceding treatise nothing oe^ curs requiring a formal demonstration until the etudeoi ar. rives at RECTANGULAR TRIGONOMETRY. THEOREM. The sine versed-sine, tangent, and secant, of an arc which IS the measure of any given angle, is to the «ine, versed-sme, tangent, and secant, of any other arc which is the measure of the same angle, as the radius of the fo-.st arc 18 to the radius of the second. Let B D (Fiff. 121,) a„,, p H j^^. ^^^^ ^^^^ ^^.^^.^j^ ^^^_ sure the same an.^le BAG; and let A B be the radius of the arc B D, and A If tlic radiu.. of the arc F H. Let D C be the Sine, B C the tangent, and A C the secant, of the arc l^aL'n/Tu ""V^ "'"^' " ^^ '''' ^''^"^^"^' -'* ^ F. the parallel, accordmg to Euc. vi. 4, tang. B C : tang, il E a'd V n''- ^ '^' ^"'^ ^^"" ^ ^ = «'"^ J^^ I ^^ rad. ; r A « „ ' '"'^ '^' '^'-''^- ^^ ^ •' '''' ^^ E : : rad. A B , Tdii. A H. Hence the trutli of the 'J^heorcm is obvious Froni this Theorem if is evident, that, as the Trigonome^ trical lables exhibit in numbers, the sines, tangents, bo- eants, &c., of certain angles to a given radius, they exhibit THEOREMS. i'.O also the ratio of the sines, tangents, eecantfi, Stc, of the same angles to any radius whatever. Upon this principle the solutions of the difierent cases of right-angled plane triangles depend; and from this Theoroin the Rules for Ilcctangular Trigonometry are deduced. HI OBLIQUE-ANGLED TRIGONOMETRY. THEOREM I. The sides of a plane triangle are to each other as the sines of the angles opposite to them. Let ABC (Fig. 123,) be a triangle, and C D a perpen- dicular let fall from the vertical angle at C upon the oppo- site side A B; because the A C A D is right-angled at D, C A : C D = R : Sine / A. For the same reason C B : C D = R : Sine /.' B; and inversely, C D : C B = Sine B : R; therefore, by indirect equality, C A : C B = Sine / B : Sine / A. In the same v/ay it may be demonstrated that C A : A B = Sine / B : Sine ^ C. THEOREM II. . If to half the sum of two quantities be added half their dif- ference, that sum will be the greater quantity; and if from half the sum of two quantities be subtracted half the differ- C71CC, the remainder will be the less quantity. Let the two quantities be represented by A E and E B, ( Fig. \2?,,) A E being the greater, and E B the less. Then it is ovidint that A D is the sum, and C E the diftercnce of the two quantities, and A D or D B their half sum, and C D or D E their half difference. Now if to A D we add D E Ave have A E, the greater quantity; and if from D B we take D ]•: we have E B, the less quantity. THEOREM in. The sum of the two sides of a triangle is to their difference as the langmt of half the sum of the angles at the base is to the tangent of half the difference. Let A B C {Fig. 121,) be any triangle. From A as a 140 Sec, of the int cases of •s Thporoiii lui'ed. 'RV. s the sines > a perpen- I the oppo- jlcd at D, ason C B : i = Sine B = Sine / rnonstrated ' iheir dif- nd if from ^ the differ- \ and E B, ss. Then ftercncc of sum, am! D we add from D B difference the base in m A as a 141 APPENniX. centre wilh the radius A 15 describe the semicircle D B E Produce C A to D. Join D B and 13 E, and draAv E F pa- rallel to B C. Then because the anj?le D A B is the exte- rior angle of the triangle A B C, it is equal to the 6um of the two interior and opposite angles ABC and A C B. But the angle D E B is equal to half the angle D A B; therefore the angle D E B is equal to half the sum of t'ho an-lcs A B C and A C B. Now since A B is equal to A E, the angle A B E is equal to the angle A E B. But the angle A E B IS equal to the two angles E B C and B C E- where- fore, also, the angle A B E is equal to the sum of the an-les E B C and B C E. To each of these add the angle E B C- then the whole angle A B C is equal to twice the angle E B C together with the angle B C E. Whence it is plain that the angle E B C or the alternate angle B E F is equal to half the difFe.ence of the angles ABC and B C A Now the angle D BE is a right angle. (Euc. iii. SI.) Therefore to the same radius E B, D B will be the tangent of the an- gle D E B and F B will be the tangent of B E F; so that B D : B 1. : : tan. DEB:: tan. B E F : : tan. ^ (A B C + A C B) : tan. ^A B C - A C E). Also, A D and A E are each equal to A B, it is evident that D C is the sum of the sides A B and A C, and that C E is their differonc. But because E F is parallel to B C, D C : C E : : D B • B F; that is, the sum of the two sides of the trianHo ABC IS to their difference, as the tangent of half the sum of tho angles opposite to these sides is to the tangent of half their uiflerence. THEOREM IV. Ina7iy right lined plane trianp;le the base h to the sum of the other sides as the difference of these sides is to the the difference of the segments of the base, made by a per^ pendieular let fall upon it from the angle opposite to it In the ol)lique-anfflcd triangle A B D, {Fig. 125,) pro- duce B D until B G is equal to A B, the si B as a centre, with the distance B G or B H describe lortest side. On a cir- If' 149 MCN6CRATI0N OP StJPEnFICIEB. cle A H G, cutting B D and A D in the points H and F.. Then D G is evidently equal to the sum of the sidca D B and B A, and li D their difference. And since A E is equal to E F, D F is the difference between D E and E A, the segments into which the base is divided by the perpendicu- lar let fall upon it from the opposite angle. Now (by Euc. •iii. SC,) the rectangle contained by D G and D H is equal to the rectangle contained by D A and D F; therefore, A D .: G D : : H D : F D, i. c. the base is to the sum of the other sides as the difference of these sides is to the differ- ence of the segments of the base. nOLE IT. This is me'-'^ly another application of the same principle. ROLE. III. Let a = D A, 6 = D B, c r= A B, and a = D E, the great- er segment; then a — a; = E A, the less segment. Then, a : b ~\- c : : b — c:2x — a and 2 ax — a^ = 6= — c' Hence, 2 ax — o'-f6' — c' X = a^-\-b'^~ c* and Hence the Rule. 1 -" a 1 Let i perpen 1 C B E, the lust B : :C 2a MENSURATION OF SUPERFICIES. Ill PROBLEM I. nULE I. The measuring unit of a superficies may be one inch, one foot, one yard, one chain, or any determinate figure and mag- nitude. Let A B C D {^Fig. 126,) be a rectangle, and M the unit of measure. When M is contained a certain num- l>er of times in A B and B C, it is only necessary to multi- ply together the figures which exjjress the number of times the linear unit M is contained in A B and B C. The pa- rallelogram A B E F is equal to the rectangle ABC D- li?-*?-= area of the trhuiglc. Thc-eforo tho inula expressed in words, is tho rule. his for- PROIJLEM IV. The area of the triangle A D D, = -^-— ^(Fj-. 128,) DC DC and the area of tho A B D C = -^ - X B F --p^X DK. Then A B X D C X -;r^ = the sum, from which fornmla tho rule is sufficiently obvious. PROBLEM V. In the Pentagon A B C D E, (Fig. 129,) let a perpcn- dicular full from the centre II. Then A B X -^= area of A A B R. Now tho area of the polygon is plainly equal to the areas of as many As, each equal to the triangle A B R, as the polygon contain.i sides. Hence the reason of the Rule is manifest. PROBLEMS VI & VH. These two problems are simple applications of the rule for determining the area of triangles. The area A i — m tXn s PROBLEM VIII. A DXmt \-Ani, (Fig. 130;) the area A D-f A D - Hence Thes land St vestiga This ascertai ms = 2 \- m n> &c. The area of the whole figure If a SI Northin; sum of t Let a I first Stat line. T fi c, and TJipn ■exactly f sum of tl Also, tl n g -f k j Ite added to the lut mm of tl: In any sides [yrv^j^ sum of tl] they stand APPENDIX. ^Jl± i!*J_fni -r ° >• -h £ g -;- B C A D -;- B C ]45 6 X A B. Now , .i is an arithmetical mean between the two ends. Hence the Rule. PROBLEMS IX-XllI. These Problems are of so little importunco in practical land Hurvey.ng that I think it unnecessary to eivi the in- vestJL'ations. •' ^ '^ '" PROBLEM XIV. This Problem is merely the application of the rule for ascertainmg the area of u triangle. PROBLEM XV. THKOREM I. If a survey has been accurately made the sum of tho ^orthmgs u.ll e.,ual the sum of the Southings, and the sum 01 the Eastings will equal the sun, of the wLstin^s Let a b c de, (Fig. m,) represent a f .Id. Let a be tho ^t stat.or, 6 the second, &c., and let N S be a meridian hue. 1 hen a n b h, and c p, will be n.^ridians, and n i, h ., and p d, Will be departure, or East and We.t lines Ihon It IS evident that the Northings (//J- ,. ,vill* be •exactly equal to the SotUhings, « n -^ /. A '- c ;r T th. sum of the Northings is equa- .o the sum of 'the'souU;;,^;:; Also the Departures, c h -' a o- = 6 w J- » ,/ • / ,. /„, bo added to the first part of the preceding equation, and b n to the latter, then c h -\- ag = n h -L pd^-fe i e tho .^um of the Eastings is equal to the sum of ihe Westlnfeo. THKOREM II. 1" any trapezium, as A B C D, {Fig. 1S,2,) having two ^i- perpendicular to a given side, the product of half the side- sum of the parallel sides multii they stand will be th e area of the I'ljure. 'I"'^ 'V^ the ba?c on which n 146 MENSC7RATI0N OP SUPERFICIES, 1 ii^ Let Ds and m n be drawn parallel to A 13 , and R F pa- rallel to B C or A D; Cs is the diiTerence between the sides A D and B C, and C n = 7is = w D z::^ their half difference, and the perpendicular E F will l)iscct Ds and in n. Now a» the angles D F m and n F C are equul, Euc. 1. 2'2, and the side F n :^ the side F in, and 7iC = mD, the ii iungles are equal and similar. Now if A B be multiplied by -= E F, the product will be the area of the I Trapezoid, THEOREM HI. In rectangular surveying (m which the work is always on one side of the first Meridian,) if the departure of any sta- tionary distance is East, and the work lios on the East side of the North imd South line, the farther that that course is run the greater will bo the departure or the distance from the f'rst meridian. But the farther that a stationary dis- tance having West departure extendi, the nearer will the first meridian be a})proaclied. Draw N S (^'Vg-. 133,) for a first meridian. Let a, b, c,. and d, be stations on the East side of N S, Let also the perpendiculars I a, h i, s; c, and e d, be raised upon the N. and S. line. Draw a r, b t, and c p, parallel to N S. Now the departin-e of the first stationary line rt 6, is 6 r, and lies to the West si" ""^ Northings; and C I, Dp, and Ex,' will be the Southmgs. - ' The meridian distances to •'.e middle of the stationary « '''T^'r'^' "" Northward, are oa,dn,andgk Now by Theor. II, Z H X » o = area Z A F H l-L H/X,in = areaHFB/I,;„nd/«Xff*=area/BC6/ and the sum of the areas will bo the areaof Z A FBCJZ or the whole North area. Again the meridian distance, ..l warT ;"??" '''"T' """ '''•''' ■=— -» «»«". wai(i, are (x R, e and X = width; then as ^ •' Y~ C B, and S 0. = D E, the perpendicular breadth; hence ^ + Sx=anr^ and x'= -- therefore a; = v ~ - A n n , ^^ — Vgg— AD, vv hich affords t>ie 8 X 5 PROBLEM X. Lot * — i»;^..i, iK -. ..... then 2 ^ = central long,!,. Then 2 :c X *"-"=«''»''»=vf which is the R„,„. !50 LOCATIOK. PROBLEM XI. The reason of this rule appears from the rule given at Prob. II, because a triangle is just half a rectangle of the same base and altitude. PROBLEM XIL Let b = given side, S = the sine of the given / , and « =» bxS the side required; then, — tt"— ^> nence we have fis x = aa 2« and X =7—' from which the reason of the rule is obvious. * PROBLEM XIIL This Problem is merely a particular application of the rule given at Prob. XI. PROBLEM XIV. Let S = the nat. sine of the given / , and x = one of tho equal sides; then s x^ = 2 a, and x^ = — and x = V y» which is the Rule. PROBLEM XV. According to Euc. xii. 2, circles are to each other as the souares of their diameters. Now the area of a circle whose diameter is 1, according to the calculation of the celebrated Van Ceulen, is .78c>3i)8 -\-, but for practical purposes . 7854 is sufliciently near the truth. When therefore x = diameter, x' X -7854 = a. Thenx" = a .7854 > and X = V -78(^4 ' which gives the Rule. Ui PROBLEM XVI. Let 6 = .7854, and x = the Conjugate diameter, then APPENDIX. 151 X -f- 100 /. = the Transverse diameter; then {x -\- 100) X X X 6 = a, antl 6 a;^ -f- 100 6 i = a: and x* + 100 x = ?- b Complete the Square, and x' -|- 100 z + 2500 = ^^ -j- 2500. Thenx-f-50 =V ^-1-2500 andx = v|-f 2500- 50,which affords the Rule. PROBLEM XVII. Let X = the Transverse diameter, and 6 = , 785'!, Then no 3x 5 b x^ as & : S : : I : --= the Conjugate; hence -^-p — = o, and 5a 5a 3 6 x' = 5 a and x'= "f^- Therefore x = V ||, which for- mula expressed in words is the Rule, FOR T In the B D = S AP=i. length of and B to that the t Make t tre Coft and A B A B F gles is cq must conl n^ent A F •'cnti-e of (Euc. iii. angle A ABC nil A COLLECTION OF PROMISCUOUS PROBLEMS, FOR THE FARTHER ILLUSTRATION OF THE PRECEDING RULES. PROBLEM I. In the triangle A B D, (Fig. 135,) arc given A B = 5, B D =: S, and A D = 4, and the line P D in position, viz: A P _ 1. Required the construction of the figure, and the length of the lines, A 0, and B, drawn from the angles A and B to a vvnidmill at O, on such a point in the line P D that the angle A O B shall contain 120°. CONSTRUCTION. Make the base, A B, the chord of 120^ and find the cen- tre C ot the corresponding circle by making the angles BAG and A B C each S0=. Because in every quadrilateral, a, A O B F , inscribed in a circle, the; sum of the opposite an- gles IS equal to 180-, (Euc. iii. 22;) and as the angle A O B must contain 120° by the terms of the Problem, its supple- n^ent A F B must contain 60^. And since the angle at the ocmve of a circle is double the angle at the circumference, (Euc. ni. 20,) the / A C B must be 120^ And as the tri- angle A B C is an isosceles, each of the angles CAB and ABC must contain 30^ The point in which the ciroumfe- 154 APPENniX, rcnce of the circle cuts th«; line V D, given in position, will be the situation of the windmill, and the angle A O IJ will contain 120°, as was required. Proportion hy which to find the segments A H and H B. As A B (5) : A B + D B (7) : : A D — D B (!) : A H — H B(1.4). Now A B = 5, and5+ 1 .4 = 6.4, the half of which is S.2 = A H; and 5 — 1 .4 = 3.6, the half of which is 1.8 = H B. To find the perpendicular D H. As the triangle A D B is right-angled at D, the perpen- dicular D H is a mean proportional between the segmenta A H and H B; therefore V 3.2 X 1.8 =/v/5.76 = 2.4 = D H, and AH — PAz:=:2.2 = PH. Proportion to find the / D P H. As the side P H = 2 . 2 . 34242 Is to HD = 2. 4 0.38021 So is r ad. 90° 10.00000 To the tan. Z D P H = 47=^ 29' 10.03779 Hence 180° — 4^ 29' = 132° 31' = / D P A. To find C B. As the sine / B C A 120° 9.93753 Is to A B = 5 0.69897 So is sine / C A B 30° 9.69897 To the side C B = 2 . 88 . 46041 To find (Ae Z 5 B C P and C P B. As B P + B C== 6.88 0.83759 IstoBP — BC = 1.12 So is tan. i sum = 75° 0.04922 10.57195 To tan. h diif. ^ 30° 17' 9 . 78358 Then 75° -\- 31° 17' = 106^ 17' = Z B C P. And 75^^ — 31° 17' = 43° 43' == Z C P B. PROMISCUOUS PKODLEMS. To find C p. As the sine / B C P = 10(J° 17' fl.OS'iiJiJ Is to side B P = 4 O.OOiOtJ So is sine ^ C B P = 30^ 9 69897 152^ To the side C P = 208 0.81881 Tojind ZsP O C andP C O, As side C O =2.88 Is to sine / C P O 91" 12' So is C P = 2 . 08 O.lfiOIl 9.99991 0.81881 To sine / P O C 4G'' 1 1' 9.85831 Then 130° — - 46° 11' — 91° 12' = 42"" .S'7' = / P C O.. To find the dist. P O. As sine / C P O 91° 12' 9.99991 Is to the side C O = 2.88 0.4G041 So is sine / P C O 42" 37' 9.880H5 To the side P O = 1.95 0.29115 To find the Z s P O H and P B O. As PB-fP = 5.95 077452 Isto P B— P O =2.05 0.31175 So is tan. ^ sum ^ s 6G° 15' 10.35654 To tun. -i difr. 38^4' 9.89;}77 Then 66° 15' -j- 38= 4' = 104° 9' = / P Q «. And (j6'-' 15' -- 33^ 4' = 27'Mr = / P B O. To find the side O B, As sine / P O B 104° IG' 9.986SO Is to side P B = 4 0.60206 J?o is sine / B P O 47" 29' 9.86752 To side O B = 3,04 0.48828 ii !l 156 APPENDIX, To find the side A 0. As sine Z A O IJ li20' h to side A B = 5 So is sine Z P B O 27^1' 9,93753 0.60897 9.65976 To side A O = 2.63 0.4'il'iO yote.—'Vh'is Problem might have been solved more ex- peditiously by Algebra. ii i PROBLEM II. Near the middle of a certain farm or tract of land A B (', (Fig-. 136.) whose form is that of an equilateral triangle, and whoso side A C runs due North, is a spring of water at 0, so situated that the perpendiculars O T, O D, and S. let fall from it upon the three sides of the triangle, are re- spectively 18. '20, and 24 chains, viz: T r=: 18 c/i., O D :=r 20 ch., and OS — 24 eh. The owner of the farm has be- queathed it to his three daughters, O, P, aiul Q; and to is bequeathed the triangle A O C, to P the triangle A O B, and to (i. the triangle B O C. Required the area of the whole farm, the area of the portion bequeathed to each of the daughters respectively, and the courses and distances of the division lines. It can be proved that the perpendicular C R is exactly equal to the sum of the perpendiculars O D, O T, and O S, in whatever part of the triangle the point O is situated. The Problem is then solved by Trigonometry and Mensura- tion of Superficies, as follows : To find the side A C, (md consequently the other sides. As the triangle A B C is equilateral, each of its Z s — (iO'-; therefore • As the .sine of the angle at A 60° 9.93753 U to C R = T + O D + O S = G2 1 .79239 Soisrad. 90= 30.00000 To the Bide A C r- 71.59 1.85486 PROMISCUOUS PROBLEMS. <57 Then 71.59 c/t. (A B) X , 5, 7. From the i)oints A and B, draw the parallel lines A C and B D, on opposite sides of the Koc A B. Prom a scale of equal parts lay ofl' ?. iVom A towards E, and 7 from B t'l- wards F. From the same scale lay off also 5 from E to- wards C, and fr(,in F towan^s D. Tiien draw the iines E D and C F, intersecting A B in H and R. Then A H : H R : : 3 : 5, and 11 R : R B : D 7. In this %vay A B may be divided similarly to -any g' van di- vided line, JS'olc. — This method of dividing lines might be advantag'^- ously employed lo find the point 1) in the preceding Prob., nni\ a-ho t^niiif P. in thif vvllu'Jl fnllllWS- I for do- id exten- ;nts of a jnnation ed in de~ )y skilful es on tho by a lino 5, so t!!at I B being d to each ded have ) to each L = A D. ?. = D B. PKtMISCUODS PROBLEMS. gg PROBLEM VIII. The side A B of a triangular field containing 6 acre, is 4C0, and A C 420. It is required to lay ol 2 ac. Ty a frS. '""'""^ ^"'" '^" ^'''''' ^' "'"^'^ '^ 22° *^-Lt f E als between A and D, the point F will he in A C;but ^r not, tho^ponu F will be in B C. Now if wo join D C and A 1< E would be parallel. Therefore, As A D : A E : : A C : A F. To find A E. By Prob. VI., As G : 2 : ; A B (4GC) : A E (156). To find A F, As A D (230) : A E (155) : : A C (420) : A F (2S3) Lay off 233 from A to F, in the direction of C, and ruu the line l D, and the work is completed If E has equi-distant from each other. When you arrive oi)posite to the end of the original line, th«; marks upon which you wi^h to restore, remove your instru- ment thereto, and let a perpendicular fall uj)on the random line. Then as the length of that line is to the perpendicu- lar let fidl upon it iV.j'u the extremity of tho olil line, so is the distance to the first .stake to the distance which it must be moved to the old line. 'J'hcjn if this distance be multi- plied by 2, 3, 4, &,c.j as circumstances may require, tf* pr - ducts will bo the several distances which the stakes must jo moved, in order to st^nd on the old line. Let C B, {Fig, 145/) be an old line, the mark.s upon which it is rcquireil to re-establish, while at the same time it is in- convenient to run the same. Run the random line C A 20 ch. The length oi" the perpendicular A i^, or the distance of A X: liflcrencr, itie of tho [uantity of h »hull be PROMISCCOUS PROBLEMS. 165 from tho end of tho old line is found to be 5 ch. On this lino, A C, set up stakes at a, c, and e, each 5 ch. distant from each other. Then, As C A (20) : A B (5) As C A (20) : A B (5) AsC A (20) : A B (5) Or thus: Having found hy^tho first proposition that c fL 1.25, then 1 .25 X 2 = 2.50 = c d, and 1 .25 X 3 = 3.7D=: a b. Ce( 5) :(?/(!. 2.5.) C c (10) : c rf(2.50.) C a (15) : a h (3.75.) I given Z, S X- ^ 2 a ICC results ; of the in- will be the )'unning it. [ line, upon 5r. When 1 line, tho our instru- lie random erpendicu- [ line, so i.s ich it must ; be multi- re, th pr. • tes must bo ipon which ime it is in- 5 C A 20 ch. •tancc of A PROBLEM XII. A surveyor having run .30 ch. in a direction between South and East, finds that .the sum of hi.s difference of latitude and departure is 40 ch. Required the course, and the area mcluded within the line run, the dilTerence of latitude, and the departure. INVESTIGATION. Let a = the distance B C, (Fig-. UG,) and h = diff of lat. + dep., or A B + A C; also let x = the dep. or A C. then B — a- = diff. of lat. Now, B C"- = B A= + A C"' (Euc. i. 47,) i. e. a^^b^^ 2 6 a; -I- 2 x= or 2 .T^ - 2 6 a; = «= _ h\ Therefore, x^ - — 2/,07 = A C the dep. Then -10 •— 27 07 = ]f> 93 = A B the diff. lat. Honcc we have the following rule to find tho remaining f^idcs of a nght-angled triangle, when the hypotenusp and the sum of the other two sides are given, viz: Fioni twice the square of the hypotenuse, or distance run --ubtract the square of the sum of the other sides, viz. the' 'i'«. lat. and dep.; divide the remainder by 4, and to tho square root of the quotient add half the sum of the two ^-Jucs, and that sum will bo the departure. m Ik If- ;. : u If 166 ArpsNDix. To find the Course . AsthedifT. of hit. 12.02 latothsdep. 27.07 So is rad. CO'' 1.11125 1 . 4S248 10.00000 To the tan. of course G4' SO' 10.32122 7^0 find the Jlrea. zIJU-^-J^-H^ = 17.\ ac. nearly. The course, therefore, is S. C^° 30' E., and the area 17.i ac. nearly. PROBLEM XIII. To locate land in tbeform of a Bight-angled Triangle, the Area and the Hypotenuse being given. Let a = area, b — the hypotenuse A C, {Vig. 147,) and "a a; = base A B, tlien -- = perpendicular C B. Then, by . 4 a' Euc. i. 47,x--{---2- = i'^; then x» -\~ 4 a- ~ b"^ x-, and x< . i« x5 ^ _ 4 a'. Next, x^ — 6= x' -j- --- = -^ — 4 o» S64 Then, z»~6= = V J^—^a^^ and x= = 6*-;-V 4 a' ---4a»J ' 4 > which cx- Whercforc x==vj6^iV[-^ prcssion afTtsrdo us the following rule: From the hiquadrate of the hypotenu.se divided by 4,"sub- tracl four times the -^quarc of the area, and extract the pquarc root of the remainder. To the root add, or from it subtract (according as vou wish to ascertain the lon<,^c.st or shortest side of the triangle) the square of the hypolnnu.so divided by 2, the square root of the sum and of the differ- ««co Avillgive the length of the sides respectively. •I therefore, '.angle t the 147,) and Then, by x^, and x< 4 o« which cx- 64 4 " :4 i by 4, "sub- extract the , or from it 5 lon<^c.st or hypotenuse ' the diflcr- y- paoiiiscuous PROHLEMa. J€7 PROBLEM XIV. Required the length and breadth of a rectangular meadow, whose perimeter is GO chains, and area 20 acres. Let a =. area, b = perimeter, and x = length, then --~ breadth, and 2x -;- ~ rrr b. Then 2 a-' -!- 2 « rrr 6 r. This equation reduced tjives x '4 "'-^ rro-^- i ^''c" CO a c™20cA. r= length, and -- — 10 ch. — breadth. This fur- mula, expressed in words, affords the following rule; Divide the square of the perimeter by IG, and from the quotient subtract the area. To the square root of tha re- mainder add one fourth of the perimeter, and the sum will be the length, and the area divided by the length will b«tho width. PROBLEM XV. An easy method of locating land in the form of a llhcmb'oid. Let the figure A B C D, {Fig. 143,) represent a Rhom- boid whose area and front A D arc ^iven. By Trigonometry find the base and perpendicular A e and c D, of the right-angled triangle A E D = C 6 B. Next, find the areas of these triangles; then by Prob. II., Loc., by out the remainder in a rectangle as A 6 C e, and D e -f- « C «= D C, a side of the Rhomboid. PROBLEM XVL The owner of a square field A B C D, {Fig. 149,) con- taining 10 ac, wishes to lay off a walk half way around which shall take up one acre. Required the width of the walk. Let a = one side of the square, and x = width of tho H-alk; thon 2 a z — x' = 6, the area of tho walk. New this t I u 168 APPENDIX. equation rcrluccd gives a; = a + V (o'^ — ■ 6) = 52 /., the width of tho vvulk. Henco tho From the area of the square subtract tho area of the walk, sulitra<;t tho square root of the remainder from the side of the square, and this last remainder will be the width of thtj walk. JS^nte. — Perhaps it may not be altound hy Trigonometry, thus- .. ^"i. z\ R fe A, you have all " the / s and the side R S to imd^ *70 APPZIODIS. tho other side.?. Again in the A R S B, you have oil tho / s and i side II S to find the other sides. Having as- certained ti.c sidc^i R A and R B, and knowing the / A R 13, the side A B is easily determined. In the same way all tho remaining sides of the field may ho ascertained. PROBLEM XXr Jlnoihcr method cf fmdins the area of ri^hUlined figures by Calculalioii. Find the DilTcrcnee of Latitude and tho whole Departure, and fill up the Table as in f rob. XV. of Mensuration. Hav- ing filled Ujithc column of Meridian Distances, add thcfirsi M. D. to the soeond, for the first or upper number in the Column of Sim.s; add the second M. D. to the third for the second number in the colu:nn of Stms, &.C.; and add tho lowest M. D. to tho uppci-mosl for the last number in the column. Then multiply each number in the column of sums by its respective Northing or Southing. Insert tho product in the column of North or South areas, and half the difercncc be- tween these columns will be the area of the field. Tho following example will .servo to illustrate tho Rule* « .M a. >n c ai 00 1 o CI . ^<. • M ^5 C *^ C5 C Q CI c ^ u at K> rt O M O C o O '^.> _ ^ _^ < o <: >^ ID O u prtoMir;cuoc3 phselcmc. IT CO o o o crj O O O «r o ui cj »^ 1^ m 00 1^ 00 O (XJ 00 rr> TT i-« C» »^ CO ^.' « «> C< V O I- I- ^ iO -JJ ■n CO »>• — o -. c< 'u"» •:« O V ;W J l-i l^r IO U:< _» t' OD CO \,' QTt -<•; •-• f« t~ •J* r7;§ TP O CTJ CJJ 30 O TT re 1CC( H u H C/! H O CO o o •-? oco r* po •-4 '; OT ■V lO vn C3 o> m TT ^ w4 OO~C0nr);2G0-; »n -J c« c» CO m ffj CO cr. G» — CO c» ■ ®3JI!***^oeoco o o « — r^ — i ^ o —1 — < « i-» no CO « « o c< en CO c; t^ 1" CO CO r- ""a" »n oj « ^^ 05 CTJ |>. ^4 c« c» c;? rf CO lO O tj ^1^ O in o CI CO ;j CI O C5 •-< Tf O O G< ci irj «—• lO »— 1 00 oo >n o CI •-< rr CI 'CO t^ G« CO '-' OO •«,» !OOOOOOOt>« ocooocoo Cr3O«if5OO GO 0-5 in CI cyj GO GO o - ., ^ ^ t> r* r. o o O O • • o 4-^ o o o O C^ 00 •-» CD lO o ^."^ C« c/f (/: ^i CO •-• "• 7 O V c 171 *0 O M S; ^ ;.^ ?:^^ •*• ** ~ «^ W « ^ S m >-■ r rt 4* t* > c fl 's «^-S— < « "•5 -'« ,, re T3 - 5 -3 an ca •- _n w _ dj •-• '^ tn « nr "^ ♦J t; -^3 '/! H 5 Oj J, O U -;:; « r p »- « 0) 2 2 c.^"S '- o '' ^ i -9 -^ ^'^S '-I *3 ^ r- o o es =: 3 — ^" O rt O rt o jj w G, o r; «5 c I;? 3 o J ^ J o o o rJ C o P o J3 C S to o s 3^ - -r d a V. ^ ■c ^•' :'' P =: 2 '■'■< o o •-< c^ 2 - -^ & X *-* o r— J^ I o s: « c> « c^ I" ^ g S ^^ ">. « e o o *► H ■« > ^MAGE EVALUATION TEST TARGET (MT-3) /. / ^ t-?/ & f/x 1.0 I.I 1.25 IM 112.5 1144 _ j50 la IM 1112,2 II: 1^ 2.0 1.4 1.8 1.6 V^ <^ /a 'a I c^. C), ^m • j^^ .>' A >^ w/ °'W Photographic Sciences Corporation M 'h -b w \\ ^9) V .^^ ^^o^ 6^ ^ % '^ 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 172 s^" APpnrfDix. :m hnve not thou /> lit it advi.-vablr. to swell the sizo or increase the price ot thi;* Treatise by their insertion. A Surveyor will hnd his advantage in solectii-g- some particular system, in rnakin- liirnsclf thorou,o-hly and faniiliarly acquainted with It, and adhering to it with very few deviations. Before concluding this branch of the subject, it may bo serviceable to lay down a few Hxjlf.s for finding CGniained anp^les. 1. If the first letters of the course nre unlike, and the last likewise, subtract the lesi course from the greater; Thus: S. 6'rW. N. 15^ E. 47'= -. included /. 2. If the first letters arc unlike and the last alike, add the bearings; * Thus: S. SO'^W. N. 40'-^ W. 70=" :.— included /. 3. If the first and l;ist letters are both alike, add the less bearing to the supp]cn,ont of the greater. 4. If the first letters are alike and the last unlike, subtract the sum of the bearings from 180°, General Ra!e.~-\n oi,lcr to finu the quantity of an ans:le suppose yourself stauiling at the angular point. Then re- verso one or both courses, (as necessity may require,) and the (pumtity of the angle will be easily ascertained. LEVELLING. Though it is not expected that every Land Surveyor will possess all the qualifications, ;uid provide himself with all the instruments necessary f >:• (;ivil engineering, yet he will often find an acquaintance with the ger,crarprinciples of Levelling of very great service. lie may bo employed to 8urvey a courr for condn.^tlng supplies of water, to deter- mine the most suitable si;> f >r the erection of a mill, or the most eligible route for the lunation of a road. If he make hirwsolf completely master of the subject he is in this respect LEVELLING. I'r, "•-heap and simnlv ■•„„ , , ^- ■'''"''° Circumfbre;,tor. a '"eon.,-,,,,,,, o,...,,.:,', 'theta't':"' ■ '" ""'''•"" "" b' ">- -curo'thn a. ■ t„ eeof uT"""" """"''""^ "» -^"""^^ staves to nlac- thorn ,7,. '^■""'«'"'" '""" to carry the an erect „,' e-nc ' ' !"'"\'="""'°"^' '" '«'op the,, i„ or ,n„,.e to ,0 Cc oi;:: ll!l"'"'.'7 '"^'-'-. -" ""^ iKuvc obstruction.^j from the ^vay. ii; ij 174 APPENDIX. Being thus furnished, the Surveyor directs the chain- bearers to proceed, and measure or run tlie stationary dis- tance. In consequence of the curvature of the earth\s sur- flice the stationary distances shouM not exceed a few chains. (The curvature of the earth in 10 ch. nmo«nts to an 'ghth or . 1-25 of an inch.) He then plants his instrument in the centre of t'lis distance, or midway between the two stations. Then having carefnlly adjusted and levelled his instrument, observing particularly that the air bubble is exactly in the centre, he directs the men at the stations to raise or de- press the vane until it is cut by the hair of the instrument, and to note the height upon the staves. Suppose the height of the vane ujjon one staflf is G feet from the surface, and upon the other 3; then it is evident that the rise between the places is exactly .'3 feet. In a continuous ])roccss of levelling, or what is termed compound levellir.g, it is not necessary to find by Subtrac- tion the differences between every stationary distance. It is sufficient to oi>ter each observation in its respective co- lumn, under its title of f )re-sight or back-sight. Having made the necessary entries, the back-stafl:" should be car- ried f )rward and placed at a convenient distance before the fore-stafl", which now becomes the back-stafi". Then place the instrument in the centre, and proceed as before. In this manner complete the survey. Add up tho column of back-sights and the column of fore- sights. If these sums are ef|ual, and the survey has been correct, the first station and the last have the same elevation, or they arc on the same level. If the sum of the back- sights exceed the sum of the foresights, the terminating point or last station is higher than the place of beginning or first station, and vice versa, as will appear by the follow^ ing examples: 1 2 3 4 5 G chain- nary dis- rtlrs sur- IV chains, m ghth nt in tho stations, trument, ly in the 10 or de- truinent, le height ace, and ween tho 3 termed Suhtrac- mcc. It :tive co- Having he car- !forc the m placo ore. In of fore- las been evation, le hack- ninating 3ginning ! follow^ LCVELLIWa. EXAMPLn I. 17a No. OF Stat. COUR SE. Back-sights. || Fore-sights. 1 o 3 5 G 7 N. 10^ E.I J> N. 15° E. Disi. 300 SCO soo soo £00 SOO 300 £00 c 7 2 7 3 G 8 5 4 4 3 2 3 o o 5 S 4 'y.00 1 1 42 I 4 f 37 3 8 1 ? 3 cy G 4 7 G 8 5 4 9 7 4 4 4 II 37 I 3 I 2 o 5 I o The terminating station is therefore 5/.'. l in. 2 fcn'Aa higher than the first. EXAMPLE ir. No.o^ Stat. 1 2 3 4 5 G Course. S."45nv: )> s. ooHv. DlST. IN LIN. Back -sights. || f ore-sights. 300 300 300 SOO 300 .FT.;lN.iTKNTHS||FT.jN.!'rENTH& 4 18C0 1 i^ 1 G 2 5 1 o 4 5 1 5 1 3 4 7 o o •4# ii i S2| , 25 8 8 7 Hence it appears ^hat the terminating point is ? feet low. er than the first station. The height of hills maybe ascertained likewise by Tri^o. nometrical Calculations, by taking the angles of elevatTou and depression, and measuring the slant sides. M { S C E L L A N E IT S • .V Re-,-:stabi.i3:ime.nt of lost BouNnxRir-s,— Rem:.',?;'. in attemp^s to .settle disputes about nkl lines, it often hap- pens that tJio parties who were pre,ent at the ori-inal sur- v-ey have forgotten tlie circuiT.stances or iuivo removed fron, the country, or h:;ve thcniocivr ; l.eon removed by deatli and consequently eufficlent cvidcneo cannot be procured to restore the original boundaries. When a new line has been ngrced to, in order to prevent liti-ation afterwards, it is ad- visable that the parties formally release to each other alJ the land quite to the newly established line. The following i.s a copy of a Legal llclcasc for this pur- pose : — ^ KNOW ALT MEN, by the.e presents, that I, Jl. «,, of m the County of ,,ud Province of tor ai". jn consideration of the sum of good and lawi- money of tlic said Province of r n r.r ■ ^'' ""^^ ^'' ^'^-''^h ^-'^'^^ ^ni] tru]v xmid by in tlie County of and P^i-iucc of at and l;eloro the ensealing and deliveiv of these Zntn '^:^{y''lP\^l^-^^>^ { jo hereby acknowledge and a^n,;^ -^li^ lf'^ 'r^^^^^^' i^^^VK remised, rdeLed, am quit clain>, and by these presents DO remise, release and quit clan,,, to the said C^l)..^ his heirs and ^.l^'Zl vdntinf"-:^n-^'V'^^"' ''''T'l^ '^'^^"^" «»^1 ^'^'^''^"''' of md tot .o'^.n' ^^-'."^^^-'^.^^•'■^'•' V, both -r in law orequiiv,of, in, . n to tae .oilounig p.eco or parcel of Land, situate', Wing '•'''' ^'''"- '''-' -'^n' -• ^--- other circumstances ..cquamted with the lines, is genoralk- the most safe and satisfactory course to pursue. In d en however, it is frequently difficult, and often utterlv i m ^ : dete. """"''' '" '"^''^^'"» ""^^"-"^'^ ^"-" - -i- Notwithstandiug the heavy responsibility which devolve* on ho Surveyor, frequently he must depend, to n great ex ent, upon his own judgment and discretion. I„ Tse cir camstances an intimate and extensive knowledge of Mathe' matical principles, and a general acquaintance tvith colla c labyrinth, by suggesting to him numerous expedients whi<-h -;^^ never occtir to the mind of . mere Lr;::^;- 1 4 ' '' ,<• ERRATA. Page 7f», third line from tlie bottom— for '•' Sir William MacLcan George Colebrooke, R. H. ^ 8lc., read Sii William MacBeaii George Colcbiooke, K. H. &c. Pag9 116, last line of tlic Calculation Ta!)!c, in the column Half Departures — for "= 13.95 ' read — 13. OS, Page 133, at the head— for <' Division of Land" read Lo- cation of Land. TRAVERSE TABLE, CONTAINING TIIE DIFFERENCE OF LATITUDE AND HALF DE- PAIITUIJE, TO EVEUY ^UAllTEU DECHliE OF TIIL COM VPS; THE DIS- TANCE SUPPOSED 10 BE OWE CHAIN, OK FOUR RODS. To find the Dijf. of La', and Dcp. by the following Table. In tlic fi'-sit column tmder D. M. find the Degrees rncl Minutes contained in the bearing; W) w line Avitli wliichj un- der N. S. and Ft. W., you liavc the Diff. of Lut. and halt' the D'^p. ior that hearing. ISIidtiply tliis Difl". of Lut. and Dep. by the length of the Stationary Distance; and the pro- duct will 1)0 the DiiT. of Lat. and half Dcp. for that line. Thus, let the bearing be N. 10'^ 15' W., Distance 20 ch. 30 /. llcqnircd tlie Diif. of Lat. and Departure. .9840 N.S. .0839 E.W. 20.30 20.30 295200 19G800 2(;G70 17780 1.9975200 Northin,?. 1.801(370 half Westing. 3.G09340 Westing. N. 3. — In the following Table, if the Dep. be multiplied by two, the Dei), and Diif. of Lat. will be the natural sine.-^ and co-.-^ines of the corresponding Courses, respectively; the Radius being supposed to be one. r ?MFFERENCn or LATITUDE AND DEPARTURE. 15 30 45 .8988 .8968 .8949 .892!) i>j 90 ,2210 ,2230 ,2250 5. .(» .99(32 .043(; 1(>. .9()13 .1378 i27. .8910 .2270 15 .99.58 .0457 15 .9{;00 .1399 15 .8890 .2239 ,50 .9954 .0179 30 .9588 .1420 30 .8870 .2308 45 .09i;i .0501 45 .957«) .1441, 45 .8850 .2323 C\ 0; 15: 3(»i M5| 7.1 01 i!n! '30, ^15" : ! 5; '30 .!5! 1 9 JO! ro,= 0, .99451.0522 .99 10. or U .993(>j.05{i6 .9931 L 0587 ^91)25.01109 .9920!. On31 .99J4i.Oi352 .9'^0:!|.0ti74 '.1545' 29. .1500 15 .15S'ji 30 .l()07i 45 j;^0 .9903]. oi;9(; !19. .989!.;. 071 7 .9890|.0739 .98?'3 .0700 ■38:7Tom2|:20, .9870 .0801'! .:'8!i3'.{;825 .9SJ5I.0847 .981S;.08i)8 .♦»-• IOJ.088:' .'»^32j.OiJll .9824'.0^;33 15 30 45 .8829 .8309 .8788 .8767 .8725 .8703 .8084 .2347 ,23b"f) ,2335) ,24051 ,2424 ,-.;443 ,2462 ,2481 30.1 15 It'U 45 .8(;(io .81)38 .St-IG .8594 ,2500 ,2519 ,2537 ,2556 21. 1 15 .-JO 4 J 31. .8571 .2575 15 .8549 .2594 30 .8527 .2612 451 .8503 ,2031 32. .8480 .2(i49 !5 .8457 .26(:8 30 .8431 . 'IbSO ^45 .8410'. 2705! 33. ■ 15 30 ' 45 34. 1 01 '15 30 i 45 35. 15 30 45 3(3. 15 30 45i 37. 15 30 1 4'5 38. 15 ;}0 . 45 . 39. . 15 . 30 . i -In 41. 15 30 45 ^' DIFFERENCF • OP LATITUDE AND DEPARTt'RE, ID. |.M.! iN.H. |lvVV.|iI).|M.| N.S. |E.W.||1). |M • 1 N. «. |E.VV.| 33. (] .8387 .2723 ;44. .7102 .3473 55. f ..5731 .4095 ■ 10 .83()3 .2741 15 .7163 .3489 15 .5706 1.4108 3i .833J) .2759 30 .7132 .3.504 30 .5664 .4120 ■ib .8315 .8290 .'2718 '.2796 45 .7102 .3520 45 .5628 .4133 34. 45. .7071 y r* '.> Tv 56. .5592 .4155 If) .8266 .2814 15 .7041 .3551 15 .5555 .4157 .id .8241 .2832 30 .7009 .3.566 30 .5519 .4169 i 4b .82K) .2850 45 .6978 .3581 45 .5483 .4181 35. .81SI1 .2667 46. .694(i .3596 '57. .5446 .4193 lb .816(i .2885 15 .6915 .3612 15 .5408 .4205 30 .8141 .2903 1 30 .6883 .3627 1 30 .5373 .4216 45 .8116 .2921 1 1 45 .6852 .3642 45 .5336 .4228 3(3. 6 .80!.t0 .2939 147. .6820 .3656 58. .5299 .4240 15 .8064 .2956 15 .6788 .3671 15 .5269 .4251 30 .8058 .2974 30 .6756 .3686 30 .5225 .4263 45, .8012 .2991 45 .6724 .3701 45 .5188 .4274 37. .71)8() .3009 43.1 .6691 .3715 :59. .5150 .4286 15 .7960 .3026 15 .6659 .3730 15 .5113 .4297 30 .7933 .3044 30 .6626 .3745 30 .5075 .4308 45 .7907 .3061] 45 .6593 .3759) 45 .5033 .4319 .4380 38. .7880 .3078 49. .6560 .3773 60. .5000 15 .7853 .3095 15 .6527 .3788 15 .4962 .4341 30 .782(i .3112 30 .6494 .3802 30 .4924 .4352 45 .7799 .3129 35 .6461 .3816 45 .4886 .4362 74373 '69. .7771 .3146 50. .6428 .3830 01. .4848 lo .7744 .3163 15 .6394 .3844 15 .4810 .4,383 30 .7716 .3180 30 .6361 ..3858 30 .4771 .4394 1 1 i 45 .7(i88 .3197 45 .6327 .3872 45 .4733 .4405 ■10. .7660 .3214 51. .6293 .3885 62. .4695 .4415 lo .7632 .3230 15 .6259 .3899 15 .4656 .4425 30 .7601 .3217 30 .6225 .3913 30 .4617 .4435 45 .7575 .3263} 45 .6191 .3926 45 .4579 .4445 ■u. .7547 .3280 52. .6157 .3940 m. .4540 .4455 lb .7518 .S!i9,6 15 .6122 .3!>53 15 .4501 .4465 30 .7489 .3313 30 .6087 .396i] 30 .4462 .4475 4'2? 45 .7460 .3329 45 .6052 .3980 45 .4423 .4485 .7431 .3345 53. .6018 .39931 64. .4383 4494 15 .7402 .3361 15 .5983 .40o;>| 15 .4344 4503 'id .7373 .3378 30 .5948 40191 ^0 .4305.4513 15 .7313 S394I 15 .5912 4032, i 65. 15 .4265 .4522 •13.1 .73 131.. 34 10 54. .5878{ .4045 .4226 4532' 15 . 7232 j. 34 20 15 .58421 .40.58 15 .4186 1541 30 .72.54 ,3441| 30 .5807 4070 JO .4140 4550 45 .7224 .3457 15 .5771 4083; 1 15 .4177|. 4559 4" 'I I- ■ ^B ! H^^^B DJFFEUEKCR OF LATI TUDE A ND DEPARTURE. 1E.\V.||1)-|IV».| N. S. jE-VV-l fD:j¥p. sT ii:.w.! |D.l->i.|N.s. f)G. 15 30 45 .40G7 .,4027 .3087 .3947 G7. or .3907 15 30 45 .38G7 .38i>7 .373G .4507 .157G .4550 .4594 .4011 .401!) .4027 74. 75. 15 30 45 ~0 15 30 45 ,275(5 .2714 .27G2 .2G30 ^2588 .2540 .2504 .2401 .4800 .4812 .4818 .4823 4829 4835 4840 4840 82. 83. 15 30 45 ~d 15 30 45 .1392 .1348 .1305 .120 2 T2I8 .1175 .1132 .1088 .4951 .4954 .4957 .4900 .4902 .4905 .4908 .4970 08. 15 30 45 .3740 .3705 .3005 .3024 4030 4044 ,4052 ,4u0t» 70. 15 30 15 .2419 .2377 .2344 .2292 ,4851 ,4850 ,4802 .4807 84. 15 30 45 .1045 .1002 .0958 .0915 .4972 .4974 .4977 .4979 09. 15 30 45 .3581 .3543 70. 71. 15 30 45 .4008 .407(i .ooy^ .4( 8.3 _^GJU001_ .4!)98 .» .3420 .3379 .3338 o 2207 15 30 .3255 211 f.i .47(H) .4713 .4720 .4727 .4734 77.1 0' 15 45 "0 15 3fi 73. 45 .3173J.4741 .3131 .474S 79. 15 45 .2249 .2207 .2104 .2122 T2079' .203(; .1993 .1951 TTlK>S'.4908 .18051.4912 .18221.4910 15 45 .3090 3007 ,4755 ,4' ,47; H'^Z .29C^5l.4775 50. 15 73. 15 30 45 29241.4781 ,2882'. 4 /.'^S| .2840|.47it4i .2798'.4y^'0 15 30 .4872 .4870 .4881 .4880 ^4890 .4895 .4899 .4904 85. SO. 15 30 45 15 30 45 .0871 .0820 .078'! .0741 .0097 .0054 .0010 .0500 .4981 .4983 .4884 .4980 4988 ,4989 ,4990 .4992 4920 .177! .l73iJl.4!i24| 87. .0.523 .4993 15 .0480 .4994 30 .0430 .4995 45 .0392 .4590 .1 .':oa ,4927 30 .1050.4931 45 .1007 .49.35 88. 15 30 45 .15(^4 .1521 .5 478 .14 o'-i 4938 ,4941 ,4945 .4948 89. 1 15 30 45 .0349 .0305 .0202 .0218 T0174 .0131 .0087 .0043 4997 49!I7 4998 4998, 4999 ,4999 , 5000 .5000 90.1 0! .0000;. 5000 S. lE.VV. 32. 4951 48 . 4954 [15 . 4957 G2. 49G0 18 49G2 75 49G5 3:1 4908 8S .4970 Tb .4972 02 .4974 58 .4977 15 .4979 71 .4981 20 .4983 •m .4984 r4i .4980 J97 .4988 J51 .4989 ilO .4990 3G() .4992 323 .4993 ISO .4994 43G .4995 3i'2 .4596 34f .4997 305 . 49!I7 ■2{)-i ! .4998 21b ) .4998; 17^ I .4999 13] .4999 08' 7 , 5000 104 } .5000 )000,.5000| Fr;i L-j T'l. S / \ ■'''(' iJt. Tfi -• --ir — f'M; A- E -i L\, / V M;iMroe n ■T. N / .-^a?iw f! I 111 I! : )!) ,\ >A' C^ 11; 5^ /.,>V '■ I q J. xU a*. \':i ; 11 '.i I .1" * * # m nk-- f ':'W^ i I' f '• ' k ill HI V A m •ill 'V . > \ 'V! 1 i .,i / iu/ 77 ^^ * 3L. li:^^ fji ill It I- rv' ^f '''/. • -/. ::■/' ,/^ \ Jl. i^.«.' ^i rx: )■(. j; /' - — 4^^. T~ f-- '■<'l ,'Mi -^ J'l. 1. /'<) ,^»r, ^1' ~j—~ 1 K' ' - a \ )(. ,/•' > '"',';■'' u' ■ V y^ 'J JJ /•M/'t'.>. (r' - -J 1- 1. 1: K . — . '•'mj. irv, ■' It" — ^ „___, W) ^A^.A I i W '■ 1 y 1 E—- *^ iT'i \ \ I i A ^ 1 1 ^' I •- D L.. Si (ill' .'li'' (Ji.((i tin iiii'l t'lj IIV , ■ S' ruli- i'.'i rh tr n n m rh — -'H T Li,. ii>. lf)■i- n-'^. \. .1* ir \A- n* T'ni n i,. F A -I 'c /- t'nj.U! yri, .1 (It: A' T^'.f/ l]^ /)-=^- / 4 ■ V. , C P=! /•Vf/.xpfi, 1 :-f I . ■ ' i- I -^L .._ . ' I ! t i I I / — i~ :_..„_ J M K II c w I' "I- I \',S \ 7{ T.,h ' y ftlCW: .Vt'.iu ly / I ' 5V" ? ^~ irt^- ni. //I— = V ^^*-li^^ vV If, .».-'. y'i /• ^ ft I 11 ir I I PI i^.;r r :, If i <3' A<~- VI. r.) trN^ .^- TO li I ;• IT rvh.Try Mev '' :Atl N. 1 o" 2 0. 3 0. 4 0. _5 0. 6 0. 7 0. 8 0. 9 0. 10 1. 11 12 13 14 15 16 17 18 19 - 20 21 22 23 24 25 A TABLE OP LOGARITHMS OF NUMBERS FROM 1 O 10,000. N. 1 Lo?. N. 26 Log. N. 51 Loi?. N. 76 liOg. 0.000000 1.414973 1.707570 1.880814 2 0.301030 27 1.431364 52 1.716003 77 1.886491 3 0.477121 28 1.447158 53 1.724276 78 1.892095 4 0.G02060 29 1.462398 54 1.732394 79 1.897627 5 0.698970 30 1.477121 55 1.740363 80 1.903090 6 0.778151 31 1.491362 56 1.748188 81 1.908485 7 0.845098 32 1.505150 57 1.755875 82 1.913814 8 0.903090 33 1.518514 58 1.763428 83 1.919078 9 0.954243 34 1.531479 59 1.770852 84 1 . 924279 U) li 1.000000 1.041393 35 36 1.544068 1.556303 60 61 1.778151 1.785330 85 86 1.929419 1.934498 12 1.079181 37 1.568202 62 1.792392 87 1.939519 13 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 39 1.591065 64 1.806180 89 1.949390 15 1.176091 40 1.602060 65 1.812913 90 1.954243 16 1.204120 41 1.612784 66 1.819544 91 1.959041 17 1.230449 42 1.623249 67 1.826075 92 1.963788 18 1.255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.643453 69 1.8.38849 94 1.973128 "■M 1.301030 45 1.653213 70 1.845098 95 1.977724 21 1.322219 46 1.662758 71 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 23 1.361728 48 1.681241 73 1.863323 98 1.991226 24 1.380211 49 1.690196 74 1.869232 99 1.995635 2b 1.397940 50 1.698970 75 1.875061 100 2.000000 N. B. In the following table, in the last nine columns of each page, where th« first or leading figures change from Q's to O's, points or dots are introduced instead of the O's through the rest of the line, to catch the eye, and to indicate thai from Ihence the annexed first two figures of the Logarithm in the second column stand in the ne.\t lower line. A TABLE OF LOGARITHMS FROM 1 TO 10,000. '"'I S N. 1 |l|2|3i4i5i6|7|8|9|D. 1 100 000000 0434 08()8 1301 1734 2166 2.'i;'S. 3029 3461 3S91 432 101 4321 4751 5181 5609 6038 6406 6894 "321 7748 8174 428 103 8600 9026 9451 9876 .300 .724 1147 1570 1993 2415 424 103 012837 3259 3680 4100 4521 4940 5360 5779 0197 6616 419 104 7033 7451 7868 8284 8700 9116 9532 9947 .361 .775 416 105 021189 1603 2016 2428 2841 3252 3664 4075 44S6 4896 412 106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 107 9384 9789 .195 .600 1004 1408 1812 2216 2619 3021 404 108 033424 3826 4227 4628 5029 5430 5830 6230 6629 7028 400 109 110 7426 7825 8223 2182 8620 2576 9017 291)9 9414 3302 9811 3755 .207 .602 4540 . 998 4932 390 3i)3 041393 1787 4148 HI 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 389 112 9218 9606 9993 .380 .766 1153 1538 1924 2309 2694 386 113 053078 3163 3816 4230 4613 4996 5378 5760 6142 6524 382 114 6905 7286 7666 8016 8426 8805 9185 9563 9942 . 320 379 11 -J 000698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 372 117 8186 8557 8928 9298 9668 ..38 .407 .770 1 145 1514 369 118 071882 225(t 2617 2985 3352 3718 4085 4451 4816 5182 366 119 120 5547 5912 6276 9904 6640 .266 7004 .626 7368 ,987 7731 1347 8994 1707 8457 8819 363 360 079181 9543 2067 2426 121 082785 3144 3503 3861 4219 1576 4934 .5291 5647 6004 357 122 6360 6716 7071 7126 778 1 8136 8490 8845 9198 9552 355 123 9905 .258 .611 .963 1315 1667 2018 2370 2721 3071 351 124 0931221 3772 4122 4471 4820 5169 5518 586() 6215 6562 349 125 6910 7257 7604 7951 8298 8644, 8990 9335 9681 ..26 3''e 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 343 127 3 804 4146 4487 4828 5169 5510, 5851 6191 6531 6871 340 128 7210 7549 7888 8227 8565 8903' 9241 9579 9916 .253 338 129 130 110590 0926 1263 4611 1599 4i)14 1934 5278 2270 2605 5943 2940 6276 3275 (5608 3(509 6940 335 333 113943 4277 5611 131 72 n 7603 7934 821)5 ,8595 8926 9256 958() 9915 .245 330 132 120574 09()3 1231 1560 .1888 2216 2544 2871 3198 3525 328 133 3852 4178 4504 4830 *5156 548 1 5806 6131 645(5 6781 325 134 7105 7429 7753 8076 8399 8722 9045 9368 9690 ..12 323 135 130331 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 136 3539 3858 4177 4496 4814 5133 545 1 5769 6086 6403 318 137 6721 7037 7354 7671 7987 8303' 8618 8934 9249 95(54 315 138 9879 .194 . 508 .8-.2 1136 1450; 1763 2076 2389 2702 314 139 140 143015 3327 6138 3639 3951 7058 4263 73(57 457114885 519(5 8291 5507 8603 5818 8911 311 309 146128 6748 7676 7985 141 9219 9527 9S35 . 142 .449 . 756 1063 1370 1676 1982 307 142 152288 2594 2900 3205 3510 3815 4120 4421 4728 5032 305 143 5336 5640 5943 6246 6549 6852 7154 7457 7759 806 1 303 144 8362 8664 8965 9266 9567 9868 .1(58 .469 .769 10(>8 301 145 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 147 73 J 7 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 148 170262 0555 0848 1141 1431 1726 2019 2311 2603 2895 293 149 150 3186 3478 6381 3769 4060 6959 4351 7248 4641 4932 5222 5512 5802 8689 291 289 176091 6670 753(5 7825 8113 8401 151 8977 9264 9552 9839 .126 .413 .699 .985 1272 1.558 287 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 154 7521 7803 8084 8366 8647 8938 9'2(IM 9490 9771 ..51 '281 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 156 3125 3403 368 1 3959 4237 4514 4792 5069 5346 5623 278 157 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 158 8657 8932 9206 9481 9755 ..29 .303 .577 .850 1124 274 159 201397 1670 19'13 2216 2488 2761 3033 3305 3577 38481 272 1 .N. 1 1 1 1 2 1 3 1 4 i 5 1 6 i 7 1 8 1 9 1 D. 1 A TABLE OP LOGARITHMS FROM 1 TO 10,000. 9 D. S[H 4:32 174 428 415 424 616 419 775 416 896 412 978 408 O'Jl 404 01^8 400 998 396 iVM ;393 8:30 389 (594 386 5-^4 382 •S-20 379 08:3 376 815 372 514 369 182 366 819 363 4:^6 360 004 357 552 355 071 351 562 349 .26 im 462 343 871 340 25:3 338 609 335 940 333 245 330 525 328 781 325 .12 323 219 321 40:3 318 564 315 702 314 818 311 911 30;) 982 :}()7 0:52 305 06 1 3U3 068 301 055 29;> 022 297 968 295 895 293 802 291 689 289 558 287 407 285 2:39 283 .51 '281 846 279 )623 278 !:382 276 1124 274 5848 272 9 D. N. I 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 D. 1 160 204120 4391 466.1 1 4934 520'] t 54751 574f ) 60 le . 628( > 6.55e > 271 161 6826 709(] . 736!; 7634 7904 I 81 7.'^ t 8441 871f > 897t 1 9247 ' 269 162 951.^ 978L 1 ..51 .319 .58f » .85S 1 112! 138? 1 16.54 t 1921 267 163 212188 2454 c 272(J 2986 325S ! 351fe 378:^ I 404S 4314 4579 266 164 4844 510ii 5:37;^ 5638 5902 , 6l6r 643f 1 66f>4 - 6957 ' 7221 264 165 7484 7747 ' 801(J 8273 853P 8798 906C 9323 958.1 984G 262 166 220108 037(J 0631 0892 11. 5L 1414 167.'5 1936 2I9f 2456 261 167 2716 29 7() 323fi 3496 3755 4015 4274 453.1 4792 .5051 259 168 5309 5568 582(j 6084 6342 660U 6858 7115 7372 7630 2.58 169 170 7887 8144 0704 84O0 0960 8657 1215 891:] 1470 9170 1724 9420 1979 9682 2234 993S 2488 .193 2742 2,56 2.54 230449 171 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 172 5528 6781 6033 6285 6537 6789 7041 7292 7544 7795 252 173 8046 8297 8548 8799 9049 9299 9550 9800 ...50 .:300 2,50 174 240549 0799 1048 1297 1546 1795 2044 2293 2.541 2790 249 175 3038 3286 35:34 3782 4030 4277 4525 4772 50 1 9 5266 248 176 6513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 177 7973 8219 8464 8709 8S4i 0712' 0840i 0968' 1096 1223 13.'; I ' 128 1 1 1 2 1 3 i 4 1 5 1 r, 1 7 1 8 ! M 1 I). 13 il ' t I ' c A TABLE OP LCdARITHAlS FliOM 1 10 10,000. N. I I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I U. 340 311 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 ~n"." 531479 2754 4026 5294 6558 7819 9076 540329 1579 2825 544068 5307 6543 7775 9003 550228 1450 2668 3883 5094 556303 7507 8709 9907 561101 2293 3481 4666 5848 7026 568202 9374 570543 1709 2872 4031 5188 6341 7492 8639 579784 580925 2063 3199 4331 5461 6587 7711 8832 9950 591065 2177 3286 4393 5496 6597 7695 8791 9883 6()()!»73 1607 2882 4153 5421 6685 7945 9202 0455 1704 2950 4192 6431 6666 7898 9126 0351 1572 2790 4004 5215 6433 7627 8829 ..26 1221 2412 3600 4784 5966 7144 8319 9491 0660 1825 2988 4147 5303 6457 7607 8754 9898 1039 2177 3312 4444 5574 6700 7823 8944 ^._61 1176 2288 3397 4503 5606 670'; 7805 8000 9992 1082 1734 3009 4280 5547 6011 8071 9327 0580 1829 3074 4316 5555 6789 8021 9249 0473 1694 2911 4126 6336 6544 7748 8948 .146 1340 2531 3718 4903 6084 7262 8436 9608 0776 1942 3104 4263 5419 6572 7722 8868 ..12 1153 2-291 3426 4557 5686 6812 7935 9056 .173 1287 2399 3508 4614 5717 6ft 1 " 7914 9009 .101 1191 1862 3136 4407 5674 6937 8197 9452 0705 19,53 3199 4440 5678 6913 8144 9371 0595 1816 3033 4247 5457 6664 7868 9068 .265 1459 2650 3837 5021 6202 7379 85.54 9725 0893 2058 3220 4379 5534 6687 7836 8983 .126 1287 2404 3539 4670 5799 6925 8047 9167 .284 1399 2510 3018 4724 .5827 6927 8024 9119 .210 1299 1990 3264 4534 5800 7063 8322 9578 0830 2078 3323 4564 5802 7036 8267 9494 0717 1938 31.55 4368 5578 6785 7988 9188 .385 1578 2769 3955 5139 6320 7497 8671 9842 1010 2174 3336 4494 56,50 6802 7951 9097 .241 1381 2518 3652 4783 5912 7037 8160 9279 .396 1510 2621 3729 4834 .5937 7037 9228 .319 1408 2117 3391 4661 5927 7189 8448 9703 0955 2203 3447 4688 .5925 71.59 8389 9616 0840 2060 3276 4489 6699 6905 8108 9308 .604 1698 '3887 4074 .5257 6437 7614 8788 9959 1126 2291 3452 4610 .5765 6917 8006 9212 .355 1495 2631 3765 4896 6024 7149 8272 9391 ..507 1621 2732 3840 4945 6047 7146 8243 9337 .428 1517 2245 3518 4787 6053 7315 8574 9829 1080 2327 3571 4812 6049 7282 8512 9739 0962 2181 3398 4610 5820 7026 8228 9428 .624 1817 3006 4192 5376 0555 7732 8905 ..76 1243 2407 3568 4726 5880 7032 8181 9326 .469 1608 2745 3879 5009 6137 7262 8384 9503 1732 2843 3950 .5055 6157 7256 8353 9446 .537 2372 3645 4914 6180 7441 8699 9954 1205 2452 3696 4936 6172 7405 8635 9861 1084 2303 3519 4731 5940 7146 83^19 9.548 .743 1936 3125 4311 5494 6673 7849 9023 .193 1359 2.523 3684 4841 5996 7147 8295 9441 .583 1722 2858 3992 5122 6250 7374 8496 9615 .7,30 1843 2954 4061 5165 626/ 7366 8462 9556 .646 2.500 3772 .5041 6306 7567 8825 ..79 1330 2676 3820 .5060 6296 7529 8758 9984 1206 2425 3640 4852 6061 1 I 2 16251 1734 6 I 7 7267 8469 9667 .863 2055 3244 4429 5612 6791 7967 91 ■ .309 1476 2639 3800 4957 6111 7262 8410 9555 .697 1836 2972 4105 5235 6362 7486 8608 9726 .842 1955 3064 4171 5276 6377 7476 2627 3899 5167 6432 7693 8951 .204 14.54 2701 3944 5183 6419 7652 8881 .106 1328 2.547 3762 4973 6]82 7387 8589 9787 .982 2174 3362 4548 5730 6909 8084 9257 .426 1592 2755 3915 5072 6226 7377 8525 9669 .811 1950 3985 4218 5348 6475 7.599 8720 9838 .9.53 2066 3175 4282 5.386 6487 7586 8572' 8681 96651 9774 .7.55 .864 18431 19.'-.1 128 127 127 126 126 126 125 125 125 124 124 124 123 123 123 122 122 121 121 121 8 9 120 120 120 119 119 119 119 118 118 118 117 117 117 116 116 116 115 115 115 m 114 114 114 113 113 113 112 112 112 112 111 111 HI 110 110 110 110 109 109 109 D. N. 1 400 60 401 402 403 404 405 406 407 408 61 409 410 61 411 412 413 414 415 416 417 62 418 419 420 62 421 422 423 424 425 1 426 ( 427 63( 428 429 430 63; 431 / 432 , 433 < 434 435 i 436 ( 437 64( 438 439 A 440 64r 441 A 442 f 443 f 444 •3 445 f 446 s 447 65(J 448 1 449 2 450 0.53 151 4 452 6 453 6 454 7 A r f i.y.-j 8 456 8 457 9 458 660 45!) 1 N. I ( A TABLE OF LOGARITHMS FROM I TO 10,000. N. 1 |l|2|3|4 5|6|7!8|9|D 400 (j0206(] 2169 2277 238f 2494 2603 2711 2819 2928 3036 108 401 3144 3253 3361 3461J 3577 3686 3794 .•^902 4010 4118 108 4oy 4226 4334 4442 455C 4658 4766 4874 4982 5089 6197 108 4U:j 5305 6413 .5521 5628 5736 5844 .5951 6059 6166 6274 108 i 404 6381 6489 6,596 6704 6811 6919 7026 7133 7241 7348 107 « 405 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 406 8526 8633 8740 884? 89.5-3 906 1 9167 9274 9381 9488 107 407 9594 1 9701 9808 991^ ..21 .128 .234 .341 .447 ..5.54 107 40M 610660 0767 0873 0979 1086 1192 1298 1405 1611 1617 106 40'J 410 1723 1829 2890 1936 2996 2042 3102 2148 3207 2254 3313 2360 3419 2466 3525 2572 3630 2678 3736 106 106 612784 411 3842 3947 40.53 4159 4264 4370 4475 4.581 4686 4792 106 412 4897 5003 5108 6213 .5319 5424 5529 6634 6740 6845 105 413 5950 6055 6160 6265 6370 6476 6.581 6686 6790 6895 105 414 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 106 41,0 8048 8153 8257 8362 8466 8.571 8676 8780 8884 8989 106 416 9093 9198 9302 9406 9511 9615 9719 9824 9928 ..32 104 417 620136 0240 0344 0448 0552 0656 0760 0864 0968 1072 104 418 1176 1280 1384 1488 1592 1695 1799 1903 2007 2110 104 419 420 2214 2318 3353 2421 .3456 2525 3559 2628 3663 2732 3766 2835 3869 2939 3973 3042 4076 3146 4179 104 103 623249 421 4282 4385 4488 4591 4095 4798 4901 5004 5107 .5210 103 422 6312 5415 5518 6621 6724 5827 5929 6032 6135 6238 103 423 6340 6443 6546 6648 6751 6853 6956 7058 7161 7263 103 424 7366 7468 7.57117673 7775 7878 7980 8082 8185 8287 102 425 8389 8491 8593, 8695 8797 8900 9002 9104 9206 9308 102 426 9410 9512 9613 9715 9817 9919 ..21 .123 .224 .326 102 427 630428 0530 0631 0733 0835 0936 1038 1139 1241 i.342 102 ■ 428 1444 1545 1647 1748 1849 1951 2052 21.53 2255 2356 101 4ii9 430 2457 2559 3569 2660 3670 2761 3771 2862 2963 3973 3064 4074 3165 4176 3266 4276 3367 4376 101 100 633468 3872 431 4477 4578 4679 4779 4880 4931 .5081 5182 5283 5383 100 43^ 5484 5584 5685 5785 .5886 5986 6087 6187 6287 6388 TOO 433 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 100 434 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 99 435 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 99 436 9486 9586 9686 9785 9885 9984 ..84 .183 .283 .382 99 437 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 99 438 1474 1573 1672 1771 1871 1970 2069 2168 2267 2866 99 439 2465 2503 3551 2662 3650 2761 3749 2860 2959 3946 3058 4044 3156 4143 3255 4242 33.54 4340 99 98 440 643453 3847 441 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 442 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 98 443 6404 6502 6600 6698 679G 6894 6992 7089 7187 7285 98 444 7383 7481 7579 7676 7774 7872 7969 8067 8166 8262 98 445 8360 8458 8555 86.53 8750 8848 8945 9043 9140 9237 97 446 9335 94:J2 9.530 9627 9724 9821 9919 ..16 .113 .210 97 447 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 97 44 S 1278 1375 1472 1.569 1666 1762 1859 19,56 2053 21.50 97 449 2246 2343 3309 2440 340.5 2.536 3502 2633 3598 2730 3695 2826 ,379! 2923 3888 3019 3984 3116 4080 97 96 450 653213 45 1 4177 4273 4369 4465 4562 4658 4754 4850 4946 6042 96 452 5138 5235 5331 5427 5523 .5619 57 15 .5810 5906 6002 96 4o3 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 4o4 7056 7152 7247 7343 74381 7534 7629 7725 7820 7916 96 liJO son 8107 8-^03 8298 8393 8488 8584 8679 8774 8870 95 456 8965 9060 9155 9250 9.346 9441 9536 9631 9726 9821 95 457 9916 ..11 .106 .201 .296 ..391 .486 ..581 .676 .771 95 458 660865 0960 1055 11.50 1245 1339 1434 1529 1623 1718 96 45') 1813 1907' 20021 2096 2191 228(i 2380 24751 2569 2663 96 N. 1 <» 1 1 1 2 1 3 1 4 ! 5 1 6 1 7 1 8 1 9 1 D. 1 u ' ■mffiSi I I i f ( ■ : IK r m '!!•'' i 8 N. 4fi0 461 462 463 464 465 466 467 468 469 476 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 612 513 614 5!5 516 517 618 519 N. A TABLE OP LOGABITHMS FROM 1 TO 10,000. T 1 I a I 3 I 4 I 5 I 6 i 7 I 8 I 9 | D. 662758 2852 2947 3041 3135 3230, 3701 3795 3889 3983 4078 4172 4642 4736 4830 4924 5018 5112 5581 5675 5769 5862 5956 6050 6518 6612 6705 6799' 6892 6986 7t63 7546 7640 7733 7826 7920 8386 8479 8572 8665 8759 8852 9317 9410 9503 9596 9689 9782 670246 0339 0431 0524 0617 0710 1173 1265 1358 1451 1543 1636 C72098 2190 2283 2375 2467 2560 3021 3113 3205 3297 3390 3482 3942 4034 4126 4218 4310 4402 4861 4953 6045 6137 5228 6320 5778 5870 5962 6053 6145 6236 6694 6785 6876 6968 7059 7151 7607 7698 7789 7881 7972 8063 8518 8609 8700 8791 8882 8973 9428 9519 9610 9700 9791 9882 680336 0426 0517 0607 0698 0789 681241 1332 1422 1513 1603 1693 2145 2235 2326 24 T 2506 2596 3047 3137 3227 3317 3407 3497 8947 4037 4127 4217 4307 4396 4845 4935 5025 5114 5204 5294 5742 5831 5921 6010 6100 6189 6036 6726 6815 6904 6994 7083 7529 7618 7707 7796 7886 7975 8420 8509 8598 8687 8776 8865 9309 9398 0285 9486 0373 9575 0462 9664 0550 9753 0639 690196 1081 1170 1258 1347 1435 1524 1965 2063 2142 2230 2318 2406 2847 2935 3023 3111 3199 3287 3727 3815 3903 3991 4078 4166 4605 4693 4781 4868 4956 5044 5482 5569 5657 5744 5832 5919 6356 6444 6531 6618 6706 6793 7229 7317 7404 7491 7678 7665 8101 8188 9057 8275 9144 8362 8449 9317 8535 9404 698970 9231 9838 9924 ..11 ..98 .184 .271 700704 0790 0877 0963 1050 1136 1568 1654 1741 1827 1913 1999 2431 2617 2603 2689 2775 2861 3291 3377 3463 3549 3635 3721 4151 4236 4322 4408 4494 4579 5008 5094 5179 5265 5350 5436 6864 5949 6035 6120 6206 6291 6718 6803 7655 6888 7740 6974 7059 7911 7144 7996 707570 7826 8421 8506 8591 8676 8761 8846 9270 9355 9440 9524 9609 9694 710117 0202 0287 0371 0456 0540 0963 1048 1132 1217 1301 1385 1807 1S92 1976 2060 2H4 2229 2650 2734 2818 2902 1 2986 3070 3491 3575 3G50 3742 ! 3826 3910 4830 4414 4497 4581 1 4665 4749 5167 5251 5335 5418 ! 5502 5586 3324 1 4266 5206 6143 7079 8013 8945 9875 0802 1728 2652 3574 4494 5412 6328 7242 8154 9064 9973 0879 1784 2686 3587 4486 5383 6279 7172 8064 8953 9841 0728 1612 2494 3375 4254 5131 6007 6880 7752 8622 9491 .358 1222 2086 2947 3807 4665 5522 6376 7229 8081 8931 9779 0625 1470 23 1 3 3154 3994 4833 5669 3418 436': 5299 6237 7173 8106 9038 9967 0895 1821 2744 3666 4586 5503 6419 7333 8245 9155 ..63 0970 1874 2777 3677 4576 5473 6368 7261 8153 9042 9930 0816 1700 2583 3463 4342 5219 G994 6968 7839 8709 9578 .444 1309 2172 3033 3895 4751 5607 6462 7315 8160 9015 9863 0710 1554 2397 3238 4078 4916 5753 3512 4154 6393 6331 7266 8199 9131 ..60 0988 1913 2836 3758 4677 5595 6511 7424 8336 9246 .154 1060 1964 2867 3767 4666 5563 6458 7351 8242 9131 ^19 0905 1789 2671 3551 4430 5307 0182 7055 7926 8796 9664 .531 1J«5 2258 3119 3979 4837 5693 6547 740 8251 9100 9948 0794 1639 2481 3323 4162 5000 5836 3607, 4548 5487 6424 7360 8293 9224 .1.53 1080 2005 2929 3850 4769 5687 6602 7516 8427 9337 .245 n5l 2055 2957 3857 4756 5652 6547 7440 8331 9220 .107 0993 1877 2759 3639 4517 5394 6269 7142 8014 8883 9751 .617 1482 2344 3205 40G5 4922 5778 6632 7485 8336 9185 ,.33 0879 1723 2566 3407 4246 50S4 5920 1 I 2 I 3 5 ! 6 I 7'T 8 ! 9 94 94 94 94 94 93 93 93 93 92 92 92 92 92 91 91 91 91 91 90 90 90 90 90 89 89 89 89 89 89 88 88 88 88 88 87 87 87 87 87 87 86 86 86 86 86 86 85 85 85 85 85 85 84 8j 84 84 84 84 "57 N. 1 620 71 621 522 623 624 625 72 626 627 528 529 630 72 631 i 532 i 533 ( 534 •■ .535 « 536 ( 637 J 538 73( 639 ] 540 73i 541 542 r 543 < 544 £ 645 e 546 7 547 7 648 8 549 9 550 74(J 561 1 562 1 653 2 554 a 555 4 566 5 .557 5 55» 6 559 7 560 748 561 8 662 9 563 750 564 1 565 2 566 2 567 3 568 4 ,569 6 570 755 671 6 572 7 673 8 574 8 CVR n £7 576 760 577 1 578 1 579 2 N. 1 9 1 D. fior 94 548 94 187 94 124 94 300 94 293 93 224 93 153 93 080 93 005 93 929 92 8n0 92 769 92 ()87 92 002 92 516 91 427 91 337 91 245 91 151 91 055 90 957 90 857 90 756 90 652 90 547 89 440 89 331 89 220 89 107 89 993 89 877 88 759 88 639 88 517 88 394 88 269 87 142 87 014 87 883 87 751 87 617 87 482 86 344 86 205 86 065 86 922 86 778 86 632 85 485 85 336 85 185 85 .33 85 879 85 723 84 566 8j 407 84 -246 84 0S4 84 )920 84 9 1 D. A TABLE OF LOGARITHMS FROM 1 TO 10,000. N. 1 |l|2|3|4|5|fi 7|8|»|D. 1 620 71600C \ 608 < r 6170 6251 [ 6337| 6421 650.i [ 6.58(1 6671 6754 83 621 683^ 1 692 J 7004 708!: 1 7171 725^ 733f- 1 7421 7504 758? 83 522 7671 7754 i 7837 792C 1 8005 1 8086 816L 1 825."] 8336 «4H 83 623 850S ! 858£ » 866^ 1 8751 883^1 891? ' 900(J 9083 9165 9248 83 624 9331 9414 t 949? ' 958C 9663 974.': 982« 9911 9994 ..77 83 625 720168 • 024'^ 0325 040? 049U 0573 065.'] 0738 0H2I 0903 83 626 0986 ima 1151 1233 131C 1398 1481 1563 1646 1728 82 627 1811 i89;j 197.'5 2058 2140 2222 2305 2.387 2469 2552 82 628 2634 2716 2798 2881 2963 3045 3127 3209 .329 1 3374 82 629 630 3456 3538 4358 3620 4440 3702 4522 3784 4604 3866 4685 3948 4767 4030 4849 4112 4931 4194 50 1 3 82 82 724276 631 5095 6176 5258 5340 5422 5503 6585 6667 6748 5830 82 532 6912 5993 6075 6156 6238 6320 6401 6483 6564 6646 82 633 6727 6809 6890 69Y2 7053 7134 7216 7297 7379 7460 81 634 7541 7623 7704 7785 7866 7948 8029 8110 8i91 8273 81 535 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 536 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 637 9974 ..55 .136 .217 .298 .378 .459 •540 .621 . 702 81 538 730782 0863 0944 1024 1105 1186 1266 1.347 1428 1508 81 639 540 1589 1B69 2474 1750 2555 1830 2635 1911 2715 1991 2796 2072 2876 2152 2956 2233 3037 2313 3117 81 80 732394 541 3197 3278 3358 3438 3518 3598 3679 3759 3839 39 1 9 80 542 3999 4079 4160 4240 4320 4400 4480 4560 4640 47'>0 80 543 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 80 544 5599 5679 5759 6838 5918 5998 6078 6157 6237 6317 80 545 6397 6476 6556 6636 6715 6795 6874 69.54 7034 7113 80 546 7193 7272 7352 7431 7511 7590 7670 7749 7829 7908 79 547 7987 8067 8146 8225 8305 8384 8463 8543 8622 870 1 79 548 87>il 8860 8939 9018 9097 9177 9256 9335 9414 9493 79 549 9572 9651 9731 9810 9889 9968 ..47 .126 .205 .284 79 550 740363 0442 0521 0600 0678 0757 0836 0915 0994 1073 79 551 1162 1230 1309 1388 1467 1546 1624 1703 1782 1860 79 552 193! 2018 2096 2175 2254 2332 2411 2489 2568 2646 79 553 2725 2804 2882 2961 3039 3118 3196 3275 3.353 3431 78 554 3510 3588 3667 3745 3823 3902 3980 4058 41.36 4215 78 555 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 78 656 5075 5153 5231 5309 6387 5465 5543 .5621 5699 5777 78 557 5855 5933 6011 6089 6167 6245 6323 6401 6479 6.556 78 55H 6G34 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 559 560 7412 7489 8266 7567 8343 7645 8421 7722 8498 7800 8576 7878 8653 7955 8731 8033 8808 8110 8885 78 77 748188 561 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 562 9736 9814 9891 9968 ..45 .123 .200 .277 .3.54 .431 77 563 750508 0586 0663 0740 0817 0894 0971 1048 1125 1202 77 564 1279 1356 1433 1510 1587 1664 1741 1818 1895 1972 77 565 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 566 2816 2893 2970 3047 3123 3200 3277 3353 3430; 3506 77 567 3583 3660 3736 3813 3889 3966 4042 4119 4! 95] 4272 77 568 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 569 570 5112 5189 6951 5265 6027 5341 6103 5417 6180 5494 6256 5570 6332 5646 6408 5722 6484 5799 6560 76 76 755875 671 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 76 572 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 76 573 8155 8230 8306 8382 8458 8533 8609 8685 8761 8836 76 574 8912 8988 9063 9139 9214 9290 9.366 9441 9517 9592 76 575 9688 V 1 'to 9819 9894 9970 . .45 .121 .196 . 272 .347 75 576 760422 0498 0573 0649 0724 0799 0875 0950 1025| 1101 75 577 1176 1251 1326 1402 1477 1552 1627 1702 1778 18.53 75 bV8 1928 2003 2078 2153 2228 2303 2378 2453 2529: 26041 75 579 2679 2754 2829 2904129781 3053 3128' 3203 3278: 3353' 75 N. 1 1 1 2 1 3 1 4 1 5 1 H 1 7 1 8 1 9 1 D. 1 I in A TABLE OF LOOARITIIMS PflOM 1 TO 10,000. i t , '*:- .!, 1.^ N. 1 |lf2|8|4|filfi 7|H|9|D. j 580 763428 3503 3578 3653 3727 3802 3877 3952 4027 4101 75 581 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 75 58'2 4923 4998 6072 5147 5221 5296 5370 5445 5520 5594 75 583 6669 5743 5818 5892 5966 0041 6115 6190 6264 6338 74 584 6413 6487 6562 6636 6710 6785' 6859 "933 7007 7082 74 585 7156 7230 730-1 7379 7453 7527 7601 7675 7749 7823 74 58« 7898 7972 8046 8120 8194 8'<»6.^ ' 8342 8416 8490 8564 74 587 8638 8712 8786 8860 8934 90(1 n 9082 9156 9230 9303 74 588 9377 9451 9525 9599 9673 9746 9820 PS 94 9968 ..42 74 58 a 770115 0189 0263 0336 0410 0484 0557 0631 0705 0778 74 590 770852 0926 0999 1073 1146 1220 1293 1367 1440 1514 74 591 1587 1661 1734 1808 1881 1955 2029 2102 2175 2248 73 592 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 593 3055 3128 3201 3274 3348 3421 3494 3567 3640 3713 73 59'1 3786 3860 3933 4006 4079 4152 4225 4298 4371 4444 73 595 4517 4590 4663 4736 4809 4882 4955 5028 5100 5173 73 596 6246 5319 5392 5465 5538 5610 5683 5756 5829 5902 73 597 5974 6047 6120 6193 6265 6338 6411 6483 6556 6629 73 598 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 7427 7499 7572 7644 7717 7789 7862 7934 80^6 8079 72 600 778151 8224 8296 8368 8441 8513 8585 8658 8730 8802 72 601 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 602 9596 9669 9741 9813 98S5 9957 ..29 .101 .173 .24=i 72 603 780317 0389 0461 0533 0605 0677 0749 0821 0893 0905 72 604 r037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 605 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 606 2473 2544 2616 2G88 2759 2831 2902 2974 3046 3117 72 607 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 71 608 3904 39VO 4046 4118 4189 4261 4332 4403 4475 4546 71 609 610 4617 4689 5401 '.'760 5472 4S31 5543 4902 5615 4974 5686 5045 5757 5116 5828 5187 5899 5259 5970 71 71 785330 611 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 612 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 613 7460 7531 7602 7673 7744 7815 7885 7956 8027 8098 71 614 8168 8239 8310 8381 8451 8522 8593 8663 8734 8804 71 615 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 616 ^581 9651 9722 9792 9863 9933 ...4 ..74 .144 .215 70 617 790285 0356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 0988 1059 1129 1199 1269 1340 1410 1480 1550 1620 70 619 1691 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 620 792392 2462 2532 2602 2672 2742 2812 2882 2952 3022 70 621 3092 3162 323 ij 3301 3371 3441 3511 3581 3651 3721 70 622 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 623 4488 4558 4627 4697 4767 4836 4906 4976 5045 5115 70 624 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 625 5880 5949 6019 608f 6158 6227 6297 6366 6436 6505 69 626 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 627 7268 7337 7406 7475 7545: 76141 7683i 7752 7821 7890 69 628 7960 8029 8098 8167 8238' 8305 8374 8443 8513 8582 69 629 8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 799341! 9409 9478 9547 9616 9685 975^ 9823 9892 9961 69 631 800029 0098 0167 0236 0305! 0373 0442 0511 0580 0648 69 632 07n 0786 0854 0923 0992' 1061 1129 1198 1266 1335 69 633 1404 1472 1541 1609| 1678 1747 1S15 1884 1952 2021 69 634 2089 2158 2226 2295! 2363! 2432 2500 2568 2637 2705 69 635 2774 2842 2910 2979 1 3047: 31161 3184 3252 3321 3389 68 636 3457! 3525' 35941 36621 3730 3798! 3867! 3935 4003 4071 68 637 4139 420814276 43441 4412; 4480 4548 4616 4685 4753 68 638 4821 4889' 4957 5025 5093 5161 5229 5297 5365 5433 68 639 5501 5569' 5637 5705 5773' 5LU 590Si 5976 6044 6112 68 N. I I 1 1 2 I G 1 4 i 5 1 6 1 7 8 I 9 1 D. i F 1 640 806 641 6. 642 7. 643 8' 641 8t 645 9. 646 810; 647 01 648 ].' 619 2: 6.'j0 812< 651 3f 652 45 653 4< 654 5f 655 65 656 6( 657 7J 658 85 659 8f 660 819f 661 8205 662 0* 663 1.' 664 2 665 2.'' 666 3 667 4 668 4- 669 5-1 670 826( 671 6' 672 1{ 073 8( 674 8( 675 91 676 9J 677 830.' 678 15 679 U 680 83?? 681 3 682 3- 683 4-1 684 5{ 685 66 686 6? 687 6r 688 7f 689 82 690 838S 691 94 692 8401 693 0? 694 i:^ r)H;i H 696 2r 697 32 69,S 38 699 44 N. 1 1 D.j 01 Tf) 18 75 J'i 75 m 74 H'2 74 i'.i 74 R'l 74 [):j 74 12 74 78 74 14 74 18 73 il 73 13 73 14 73 ra 73 )2 73 i9 73 ■A 73 m 72 )2 72 24 72 I'S 72 >5 72 M 72 )1 72 7 72 J2 71 6 71 )9 71 ro 71 )0 71 )0 71 »8 71 )4 71 71 5 70 8 70 ,0 70 2 70 2 70 1 70 8 70 5 70 1 70 5 6g 8 69 69 2 69 2 69 i 69 8 69 5 69 1 69 5 69 9 68 1 68 3 68 3 68 2 68 lD.i A TAHLE OF LO(;AniTUMS FiJOM f TO 10,000. f! 64() |0 1|2|3 4 5|6 71819 U. 1 806180 6248 6316 63-14 6451 65 19 6587 6655 6,23 6790 68 641 6858 6926 6994 7061 712J 7197 7264 7332 7400 7467 68 612 7535 7603 7670 773ft 7806 7873 7941 8008 8076 8143 68 643 8211 8279 8346 8414 8481 8549 8616 8684 8751 881S 67 644 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 645 9560 9627 9094 9762 9829 9896 9964 ..31 ..98 .165 67 046 810233 0300 0367 0434 0501 0569 0636 0703 0770 0837 67 647 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 648 1575 1642 1709 1776 1843 l.tlO 1977 2044 2111 2178 67 619 650 2245 2980 237C 3047 2445 3114 2512 3181 2579 3247 2646 3.il4 2713 3381 2780 3448 2847 3514 67 67 812913 651 3581 3648 3714 3781 3S48 3914 3981 4048 4114 4181 67 652 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 67 653 4913 4980 5046 5113 5179 5246 5312 5378 .5445 5511 66 654 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 86 655 6241 6308 6374 6440 6506 6573 6639 6705 6771 «)838 66 656 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 657 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 66 658 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 659 660 8885 8951 9610 9017 9676 9083 9741 9149 9807 9215 9873 9281 9939 9346 ...4 9412 ..70 9478 . 136 66 66 819544 661 82020! 0267 0333 0399 0464 0530 0595 0661 0727 0792 66 662 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 66 663 1514 1579 1645 1710 K75 1841 1906 1972 2037 2103 65 664 2168 2233 2299 236-1 2430 2495 2560 2626 2691 2756 65 665 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 666 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 65 667 4126 4191 4256 4321 4386 4451 4516 4581 4646 4711 65 668 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 65 669 670 5426 5491 6140 5556 6204 5621 6269 5686 6334 .^75' 639J 5815 6464 5880 6528 5945 6593 6010 6658 65 65 826075 671 6723 6787 6852 6917 6981 7046 7111 7175 7240 7-->()5 05 672 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 65 673 8015 8080 8144 8209 8273 8338 8402 8467 85:31 8595 64 674 8660 8724 8789 8853 8918 8982 9046 9111 9i75 9239 64 675 9304 9368 0432 9497 9561 9625 9690 9754 9818 9882 64 676 9947 ..11 ..75 .139 .204 .268 .3.32 •396 .400 .525 64 677 830589 0653 0717 0781 0845 0909 0973 1037 1102 1166 64 678 1230 1294 1358 1422 1486 1550 1014 1678 1742 1806 64 679 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 83?509 2573 2637 2700 2764 2828 2892 2956 3020 3083 64 681 3147 3211 3275 3338 3402 3466 3530 3593 3657 3721 64 682 3784 3848 3912 3975 4039 4103 •'^.66 4230 4294 4357 64 683 4421 4484 4548 4611 4675 4739 4805> 4866 4929 4993 64 684 5056 5120 5183 5247 5310 5373 5137 5500 5564 5627 63 685 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 686 6324 6387 6451 6514 6577 6641 6704 6767 6830 6894 63 687 6957 7020 7083 7146 7210 7273 7336 7399 746^ 7.525 63 688 7588 7652 7715 7778 7841 7904 7967 8030 8093 SI 56 63 689 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 838849 8912 8975 9038 9101 9164 9227 9289 9352 9415 63 691 9478 9541 9604 9667 9729 9792 9855 9918 9981 ..43 63 692 840106 0169 0232 0294 0357 0420 0482 0545 0608 0671 63 693 0733 0796 0859 0921 0984 1046 1109 1172 1234 1297 63 694 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 63 r)H;i 1 nsp. 2047 2110 2172 2235 2297 1.1 fiO 2422 2484 2547 62 696 2609 2672 2734 2796 2859 2921 2983 3046 3108 3170 62 697 3233 3295 3357 3420 3482 3544 3606 3669 3731 3793 62 rm 3855 3918 3980 4042 4104 4J66 4229 4291 4353 4415 62 699 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 62 N. t 1 1 1 2 3 i 4 1 5 1 6 ' 8 1 " D. I 1 *< ! H il rii •I Hi ,f !i 12 A TABLE OF LOOAKITIIMS FROM 1 TO 10,000. N. 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 'H 9 1 D. 1 700 845098 5160i 5222 5284 5346 5408 5470 5532 5594 5658 62 701 5718 5780 5842 5904 5966 6028 6090 6151 6213 6275 82 702 6337 6399 6461 6523 6585 6646 6708 6770 6832 8894 82 703 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 82 704 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 705 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 706 8805 8866 8928 8989 9051 9112 9174 9235 9297 9358 61 707 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 708 850033 0095 0156 0217 0279 0340 0401 0462 05':4 0.585 •:i 709 710 0646 0707 0769 1381 0830 1442 0891 0952 1014 1075 1138 1747 1197 1809 6~1 851258 1320 1503 1564 1625 1686 711 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 712 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 3090 3150 3211 3272 3333 3394 3455 .3516 3577 3637 61 714 3898 3759 3820 3881 3941 4002 4063 4124 4185 4245 61 715 4308 4367 4428 4488 4549 4610 4670 4731 4792 4852 81 716 4913 4974 5034 5095 5156 5216 5277 5337 5398 5459 81 717 5519 5580 5640 5701 5761 5822 5882 5943 6003 6084 61 718 6124 6185 6245 6306 6366 6427 6487 6548 6808 6888 80 719 720 6729 6789 6850 6910 7513 6970 7031 7634 V091 7894 7152 7755 7212 7272 7875 80 00 857332 7393 7453 75'-''4 7815 721 7935 7995 8056 8116 81'3 8236 8297 8357 8417 8477 60 722 8537 8597 8657 8718 8778 8833 8898 8958 9018 9078 80 723 9138 9198 9258 9318 9379 9439 9499 9559 9619 9879 80 724 9739 9799 9859 9918 99 /•'8 ..38 ..98 .158 .218 .278 80 725 860338 0398 0458 0518 0578 0637 0697 0757 0817 0877 60 726 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 80 727 1534 15')4 1654 1714 • 773 1833 1893 1952 2012 2072 60 728 2131 2191 2251 2310 2370 2430 2489 2549 2608 2688 80 729 730 2728 2787 2847 2906 2966 3025 3085 3680 3144 3739 320 i 371*9 3263 3858 60 59 863323 3382 3442 3501 3561 3620 731 3917 3977 4036 40O6 4155 4214 4274 4333 4392 4452 59 732 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 59 733 5104 5163 5222 5282 5341 5400 54.': 1 5519 f)578 5837 59 734 5696 5755 5814 5874 5933 5992 6051 6110 6169 (1228 59 735 6287 6346 6405 6465 6524 6583 6842 6701 6760 6819 59 73(» 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 737 7467 7526 7585 76'44 7703 7762 7821 7880 7939 7998 59 738 8056 8115 8174 8233 8292 8350 8409 8488 8527 8588 59 759 740 8644 8703 8762 .8521 8879 9466 8938 9525 8997 9584 9056 9642 9114 9173 9780 59 59 809232 9290 9349 9408 9701 741 9818 9877 9935 9994 ..53 .111 .170 .228 .287 .345 59 742 870404 0462 0521 0579 0638 0896 0755 0813 0872 0930 58 743 ^^989 1047 1106 1164 1223 1281 1339 1398 1456 1515 58 744 1573 1631 1690 1748 1808 1865 1923 1981 2040 2098 58 745 2156 2215 2273 2331 2389 2448 2506 2504 2622 2681 58 746 2739 2797 2855 2913 2972 3030 3088 3146 3204 i'^-'82 58 747 3321 3379 3437 3495 3553 3611 3669 3727 3785 3814 58 748 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 749 750 4482 4540 5119 4598 4658 4714 5293 4772 5351 4830 5409 4888 5466 4945 5003 5582 58 58 87506 1 5177 5235 5524 751 5640 5698 5756 5813 5871 592'^ 5987 6045 6102 6160 58 752 6218 6276 6333 6391 6449 6507 6564 6822 8680 6737 58 753 6795 6853 6910 6968 7026 708;' 7141 7199 7256 7314 58 754 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 58 755 7947 8004 8082 8119 8177 8234 8292 8349 8407 8464 57 750 8522 8579 8637 8634 8752 SHOO 8866 S924 8981 9039 0/ 757 9096 9153 9211 9268 9325 9383 9440 9497 9555 9812 57 758 9869 9726 9784 98J' 9898 9956 ..13 ..70 .127 .185 57 759 880243 0299 0356 J413 0471 052S 0585 0642 0899 07h8 57 N. 1 1 1 2 1 3 1 4 1 5 !() 1 7 1 8 1 9 1 D. 1 # N. i < 760 88u 761 1 762 1 763 2 764 3 765 3 766 4 767 4 768 5 769 5 770 886 771 7 772 7 773 8 774 8 775 9, 776 9i 777 890' 778 Oi 779 1.' 780 892( 781 2f 782 35 783 3< 784 4r 785 4S 786 5'j 787 5( 788 or 789 7C 790 897P 791 81 792 87 793 92 794 98 795 9003 790 09 797 14 798 20 799 25 800 9030 801 36 802 41 803 47 804 52 805 57 808 63 807 68 808 74 809 79 810 9084 811 90 812 95 813 9100 814 06 815 111 8!H 18' 817 22" 818 27, 819 32{ N. 1 U i 1 D. of) 62 75 62 91 62 11 62 28 62 4:3 02 58 61 72 61 85 •:i 97 f'l 09 61 19 61 29 61 37 61 45 61 52 61 59 61 04 61 (58 60 72 60 75 60 77 60 78 60 79 60 78 60 77 60 75 60 72 60 68 60 Si'A 60 i58 59 t52 59 )45 59 )37 59 !28 59 il9 59 t09 59 )98 59 )86 59 73 59 row 59 J45 59 )30 58 )15 58 )98 58 581 58 M52 58 HI 58 124 58 )03 58 i82 58 1(50 58 m 58 514 58 ■IS9 58 164 57 )?. 344 f ) 49 87 , 349£ > 3544 359r ( 364.^ 3695 ! 374S , 3791 3841 3891 ) 393i ) 49 879 ' 398t > 403t ) 4088 !Ul3? ' 4186 ) 423f ) 428f ) 433f ) 438^ \ 4433' 49 1 N. i 1 i 2 1 3 1 4 1 5 1 1 7 1 8 1 9 1 13. ! I l< N. 880 94 881 882 883 884 885 886 887 888 889 890 94 891 892 95 893 894 895 896 897 898 899 900 95^ 901 902 1903 1 904 905 906 907 908 8 909 8 910 959 911 9 912 9 913 960 914 915 ] 916 1 917 2 918 2 919 3 920 963 921 4 922 4 923 6 924 6 925 6 926 6 927 7 928 7 929 8 930 968 931 8 932 9 933 9i 934 970. 935 0) 936 1' ao-y r 6313 3835 4307 4778 5249 5719 6189 6658 7127 7595 8062 8530 8996 9463 9928 0393 0858 1322 17S6 2249 2712 6745 7224 7703 8181 8659 9137 9614 ..90 0566 1041 1516 1990 2464 2937 3410 9536 ..24 0511 0997 1483 1969 2453 2938 3421 3905 4680 5173 5665 6157 6649 7140 7630 8119 8609 9097 4387 4869 5351 5832 6793 7272 7751 8229 8707 3882 4354 4825 5296 5766 6236 6705 7173 7642 8109 8576 9043 9509 9975 0440 0904 1369 1832 2295 2758 9185 9661 .138 0613 1089 1563 2038 2511 2985 3457 9585 ..73 0560 1046 1532 2017 2502 2986 3470 3953 4729 5222 5715 6207 6698 7189 7679 8168 8657 9146 4435 4918 5399 5880 6361 6840 7320 7799 8277 8755 9634 .121 0608 1095 1580 2066 2550 3034 3518 4001 4779 5272 5764 6256 6747 7238 7728 8217 8706 9195 3929 4401 4872 5343 5813 6283 6752 7220 7688 8156 9232 9709 .185 0661 1136 1611 2085 2559 3032 3504 3977 4448 4484 4966 5447 5928 6409 6888 7368 7847 8325 8803 9683 .170 0657 1143 1629 2114 2599 3083 3566 4049 4828 5321 5813 6306 6796 7287 7777 8266 8755 9244 4877 6370 5862 6354 6845 7336 7826 8315 8804 9292 9280 9757 .233 0709 1184 1658 2132 2606 3079 3552 4532 5014 5495 5976 6457 6936 7416 7894 8373 8850 9731 .219 0706 1192 1677 2163 2647 3131 3615 4098 8623 9090 9556 ..21 0486 0951 3415 1879 2342 2804 4024 4495 4919 4966 5390 6437 5860 6329 6799 7267 7735 8203 8670 9136 9602 ..68 0533 0997 1461 1925 2388 2851 5907 6376 6845 7314 7782 8249 9328 9804 .280 0756 1231 1706 2180 2653 3126 3599 0716 9183 9649 .114 0579 1044 1508 1971 2434 2897 4071 4542 5013 5484 5954 6423 6892 7361 7829 8296 4580 5062 5543 6024 6505 6984 7464 7942 8421 8898 9375 9852 .328 0804 1279 1753 2227 2701 3174 3646 9780 .267 0754 1240 1726 2211 2696 3180 3663 4146 4628 5110 5592 6072 6553 7032 7512 7990 8468 8946 4927 5419 5912 6403 6894 7385 7875 8364 8853 9341 9829 .316 0803 1289 1775 2260 2744 3229 3711 4194 9423 9900 .376 0851 K}26 180.1 1 1848 4677 5158 5640 6120 6601 7080 7559 8038 8516 8994 9471 9947 .423 0899 1374 8763 9229 9695 .161 0626 1090 1654 2018 2481 2943 4118 4590 6061 5531 6001 6470 6939 7408 7875 8343 8810 9276 9742 .207 0672 1137 1601 2064 2527 2989 2275 2748 3221 3693 4165 4637 5108 6578 6048 6517 6'>86 7454 7922 8390 8856 9323 9789 .264 0719 1183 1647 2110 2573 3035 2322 2795 3268 3741 2212 4684 5155 5625 6095 6564 7033 7501 7969 8436 8903 9369 9835 .300 0765 1229 1693 2167 2619 3082 49 49 49 49 49 49 49 49 49 J9 49 49 49 49 49 48 48 48 48 jlS 48 48 48 48 4 .-1 to 48 48 48 48 48 48 48 48 48 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 40 46 /L(i 46 46 46 tai 16 A TABIiE OP tOGABITHMS FROM 1 TO 10,000. 1* i ' t : In- i ^ f •^ ! i 1 ' i t 1 N. 1 |l 2|3|4|6|6|7|8l9|D. 1 940 973128 3174 3220 3266 33131 3359 .3405 3451 3497 3543 46 941 3590 3636 3682 3728 3774 3820 3866 .3913 3959 4005 46 942 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46 944 4972 5018 5064 5110 5156 5202 5248 6294 5340 5386 46 943 5432 5478 5524 5570 6616 5662 5707 5753 5799 5845 46 946 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 947 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 46 948 6808 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 949 950 7266 7312 7769 7358 7815 7403 7861 7449 7906 7495 7952 7541 7998 7586 8043 7632 8089 7678 8135 46 46 977724 951 8181 8226 8272 8317 8363 8409 8454 8500 8546 8591 46 952 8637 8683 8728 8774 8819 8865 8911 8956 9002 904V 46 953 9093 9138 9184 9230 9275 9321 9366 9412 9457 9o03 46 954 9548 9594 9639 9685 9730 9776 9821 9867 9912 99.18 46 955 9S0003 0049 0094 0140 0185 0231 0276 0322 0367 0412 46 95fi 0458 0503 0549 0594 0640 0685 0730 0776 0821 086Y 45 957 0912 0957 1003 1048 10931 1139 1184 1229 1275 1320 45 958 1366 1411 1456 1501 1547 1592 1637 1683 1728 17V3 45 959 960 1819 1864 2316 1909 2362 1954 2407 2000 2452 2045 2497 2090 2543 2135 2588 2181 2633 2226 2678 45 45 982271 961 2723 2769 2814 2859 2904 2949 2994 3040 3085 3130 45 962 3175 3220 :265 3310 3356 3401 3446 3491 3536 3581 45 963 3626 3671 3716 3762 3807 3852 3S97 3942 3987 4032 45 964 4077 4122 4167 4212 4257 4302 HP 47&7 4392 4437 4482 45 965 4527 4572 4617 4662 4707 4752 4842 4887 4932 45 966 4977 5022 5067 5112 5157 5202 5^47 5292 5337 5382 45 967 54'* 6 5471 5516 5561 5606 .5651 5696 5741 5786 5830 45 968 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 969 970 6324 6369 6817 6413 6861 6458 6906 6503 6951 6548 6996 6593 7040 6637 7085 6682 7130 bV2V 7175 45 45 986772 971 7219 7264 7309 7353 7398 7443 7488 7532 7577 V622 45 972 7666 7711 7756 7800 7845 7890 7934 7979 8024 8068 45 973 8113 8157 8202 8247 8291 8336 8381 8425 8470 8Di4 45 974 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 975 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 45 976 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 9895 9939 9983 ..28 ..72 .117 .161 .206 .250 .294 44 978 990339 0383 0428 0472 0516 0561 0605 0650 0694 0V38 44 979 980 0783 0827 1270 0871 1315 0916 1359 0960 1403 1004 1448 1049 1492 1093 1536 1137 1580 1182 1625 44 44 991226 981 1669 1713 1758 1802 1846 1890 1935 1979 2023 2067 44 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2!509 44 983 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 984 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 44 986 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 4317 4361 4405 4449 4493 4537 4.581 4625 4669 47 13 44 988 4757 4801 4845 4889 4933 4977 5021 5065 5108 51.W 44 989 990 6196 5240 5679 5284 5723 5328 5767 5372 .5811 6416 5854 5460 5898 5504 5942 5547 6986 5591 603C 44 44 995635 991 6074 6117 6161 6205 6249 629S 6337 6380 6424 6461^ 44 992 6512 6555 6599 6643 G687 6731 6774 6818 6862 690t 44 993 6949 6993 703? 708C 712^ 7l6g 7212 7255 7299 734t 1 44 994 7386 7430 7474 - 7517 7561 760.^ . 764S 7695i 7736 77VL 1 44 995 7823 7867 791C 1 7954 - 7998 8041 8085 8128 8172 82 U ) 44 996 fi9?10 sfiny 8347 ' 8?iur 8434 L R4.7'; r Hii21 8564 . 8608 86.5i J 44 997 8695 8731J 8785; , 8826 886C 1 891? \ 8956 » 900( 1 9043 gos-i f 44 998 9131 9174 921.^ 92-61 930f ) 934^ i 9395 ! 9435 . 9479 952i J 44 999 9565 960r 1 965i t 969( . 973' ) 978[ ) 98261 9870 ) 991.*3 995'/ f 43 N. |0|l|2|3l4l5 6 7|8 «! D.J ^ 46 46 46 46 46 46 46 46 46 46 € 932 4b 45 382 45 830 45 279 45 727 45 '175 45 '622 45 !068 45 S514 45 ^960 45 )405 45 )850 44 294 44 )738 44 1182 44 1625 44 2067 44 2509 44 2951 44 «92 44 3833 44 1273 44 4713 44 5152 44 5591 44 6030 44 6468 44 6906 44 7343 44 7779 44 8216 44 8652 44 9087 44 9522 44 9957 43 9 i P. 1 j A TABLE OP LOGARITHMIC SINES *AND TANGENTS FOR ETEHT DEGREE AND MINUTE or THE QUADRANT. N. B Tae minutes in the left-hand column of each page, increasing downwards, belong to the degrees at the top ; and those increasing upwards, in the right-hand column, belong to the degrees below. i t Mi •\\ I h I! il' 18 (0 Degree.) a table of logarithmic M. Sine D. Cosine Tang. D. CoianR. 1 2 3 4 5 6 7 8 9 10 11 12 \Z 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 GO O.OOUUOOI 6.463726 764756 940847 7.065786 162696 241877 308824 366816 417968 463725 7.505118 542906 577668 609853 639816 667845 694173 718997 742477 764754 .'ioirn 293485 208231 161517 131968 111575 96653 85254 76263 68988 7.785943 806146 825451 843934 861662 878695 895085 910879 926119 940842 7.955082 968870 982233 995198 8.007787 020021 031919 043501 054781 065776 8.076500 086965 097183 107167 116926 126471 135810 144953 153907 16 2681 8.171280 179713 187985 196102 204070 211895 219581 227134 234557 241855 Co.'iine I 10.090000 000000 000000 000000 000000 000000 9.999999 999999 999999 999999 999998 62981 57936 53641 49938 46714 43881 41372 39135 37127 35315 33672 32175 30805 29547 28388 27317 26323 25399 24538 23733 9.999998 999997 999997 999995 999996 999995 999995 999994 999993 999993 22980 22273 21608 20981 203901 19831 19302 18801 18325 17872 9.999992 999991 999990 999989 999988 999988 999987 999986 999985 999983 9.999982 999981 999980 999979 999977 999976 999975 999973 999972 999971 00 00 00 00 00 01 01 01 01 Oj_ 01 01 01 01 01 01 01 01 01 21 01 01 01 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 0.000000 6.403726 764756 940847 7.065786 162696 241878 308825 366817 417970 463727 501717 293483 208231 161617 131969 111578 99653 852.')4 76263 68988 7.. 505 120 542909 577672 609857 639820 667849 694179 719003 742484 764761 62981 57933 53642 49939 46715 4388 41373 39136 37128 35136 7.785951 806155 825460 843944 861674 878708 895099 910894 926134 940858 iiiiiiiiio. 13.536274 235244 059153 12.934214 837.304 758122 691175 633183 582030 536273 12 .494880 4.57091 422328 390143 360180 332151 305821 280997 257616 235239 17441 17031 16639 16265 15908 15566 15238 14924 14622 1433 3 14054 1.3786 13529 132.S0; 130411 12310 12587! 12372 12164 11963 9.999969 999968 999966 999964 999963 999961 999959 999958 999956 999954 9,'jy9952 999950 999948 999946 999944 999942 ; '9940 9;i;>'138 i; 99936 *ii!'j'934 7.955100 968889 982253 995219 8.007809 020045 031945 043527 054809 065806 33673 32176 30806 29549 28390 27318 26325 25401 24540 23735 02 02 02 03 03 03 03 03 03 1 03 03 03 03 03 (^3 ('4 04 04 04 04 8,076.531 086997 097217 107202 116963 126510 135851 144996 1.53952 162727 12.214049 193845 174540 156056 138326 121292 104901 089106 073866 059142 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 12 39 38 37 36 35 34 33 32 31 30 7 22981 22275 21610 20983 20392 19833 19305 18803 18327 1787 4 17444 17034 16642 16268 15910 15568 15241 14927 14627 14336 12.044900 031111 017747 004781 11.992191 9799.55 968055 956473 945191 934194 8.171328 179763 188036 196156 204126 2119,53 219641 227195 2.34621 241921 11.923469 913003 902783 892797 883037 873490 864149 855004 846048 837273 29 28 27 26 25 24 23 22 21 20 19' 18 17 16 15 14 13 12 11 10 14057 13790 13532 1.3284 13044 12814 12590 12376 12168 11967 11 828672 8202371 811964 8038441 795R74 788047 780359 772805 765379 758079 Sine (J'jlang. Taiic. M. 8U Uogreoii. M. e 8.2 1 2 2 2 3 2 4 2 5 2 6 7 2 9 51 8.50 52 51 53 51 54 52( n n (J 472454 7066 476693 6998 480892 6931 485050 6865 489170 6801 493250 6738 497293 6676 501298 6615 505267 6555 8.509200 6496 513098 6439 516961 6382 520790 6326 524580 6272 628349 6218 532080 6165 535779 6113 539447 6062 543081 6012 Cosine Sine Cotaiig. 11 . 7.08079 750898 74383.' 736885 730044 723309 716677 710144 703708 697366 691116 11.684954 678878 672886 666975 661144 6.55390 619711 644105 638570 633105 11.627708 622378 617111 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 6119081,36 606766 601685 696662 591696 586787 581932 11.577131 6723821 567685 563038 558440 553890 649.387 644930 640519 536151 11 531828 527.546 623307 519108 614950 510830 506750 502707 498702 491733 11.490800 486902 483039 479210 47.5414 471651 467920 464221 460553 456916 35 34 33 32 31 30 29- 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 B 5 4 3 2 I 4 Tang. Degrees. M. i : i 1 ' 1 "'* !;; 20 M. (2 Degrees.) a table op logarithmic diue !). ('osiiie D. Tftnp. D. ("lotaiiR. S 1 2 3 4 5 6 7 8 9 10 U 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 5^ 56 67 58 59 60 .542819 646122 549'J95 553539 557054 560540 663999 567431 570836 574214 577566 6004 6955 5906 68)8 5811 5765 6719 6674 f630 />587 5544 .5808!)2 584193 687469 6907il 693948 59715'i 600332 603489 606623 609734 .612823 615891 618937 621962 624965 627948 630911 633854 636776 639680 .642563 645428 648274 651102 653911 656702 659475 662230 664968 667689 .670393" .73080 675751 678405 681043 683665 686272 688863 691438 693998 8.696543 699073 701589 704090 706577 709049 711507 713952 716383 718800 6502 5460 5419 5379 6339 6300 5261 6223 5186 5149 6112 6076 5041 6006 4972 4933 4904 4871 4839 4806 4775 4743 4712 4682 4652 4622 4592 4563 4535 450 6 4479' 4451 4424 4397 4370 4344 4318 4292 4267 4242 4217 4192 4168 4144 4121 4097 4074 4051 4029 4006 9.999735 999731 999726 999722 999717 999713 999708 999704 999699 999694 999689 9.999685 999680 999675 999670 ' 999665 999660 999655 999650 999645 999640 .999035 999629 999624 999619 999614 999608 999603 999597 999592 999586 9.999581 999575 999570 999564 9995.1S 9995^3 999547 999541 999535 999529 9.99~9524 999518 999512 999506 999500 999493 999487 999481 999475 999469 .999463 999456 999450 999443 999437 999431 999424 999418 999411 999404 07 07 07 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 u 11 11 8.543084 646691 550268 553817 557336 560828 564291 667727 571137 574520 577877 8.581208 684514 587795 .591051 694283 597492 600677 603839 606978 610094 8, 613189 r 16262 619313 622343 625352 628340 631308 634256 637184 640093 .642982 645853 648704 651537 654352 657149 659928 662689 665433 668160 8.670870 673563 676239 678900 681544 684172 686784 689331 691963 694529 8.697081 699617 702139 704646 707140 709618 712083 714534 716972 719396 6012 6962 5914 5866 5819 6773 6727 6682 6638 5.595 6552 111, 5510 5468 6427 6387 5347 5308 5270 5232 5194 5158 5121 5085 5050 6015 4981 4947 4913 4880 4848 4816 4784 4753 4722 4691 4661 4631 4602 4573 4544 4526 4488 4461 4434 4417 4380 4354 4328 4303 4877 4262 4228 4203 4179 4155 413--4 4108 4085 4062 4040 4017 456916 453309 449732 446183 442664 439172 435709 432273 428863 425480 422123 11 .418792 415486 412205 408949 405717 402508 399323 396161 393022 389906 11.386811 383738 380687 377657 374648 371660 368692 365744 362816 359907 11.357018 354147 351296 348463 .345648 342851 340072 337311 334567 331840 11.329130 326437 323761 321100 318456 315828 313216 310619 308037 305471 11.302919 300383 297861 295354 *rr2S60 290382 287917 285465 283028 280304 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Cosine Sine Cotanc. Tang. 87 Degreefl. M 1 Sii d.7U 1 72 2 721 3 72{ 4 72t 6 73( 6 73r 7 73£ 8 737 9 73£ 10 745 11 8.744 12 746 13 74fl 14 761 15 752 16 75S 17 757 18 760 19 762 20 21 764 8.766 22 768 23 770 24 773 25 775 26 777 27 779 28 781 29 783 30 31 786 8.787 32 789 33 791 34 793 35 795 36 797 37 799 38 801 39 803 40 41 805 8.807 42 809 43 811 44 813 45 815 46 817 47 819 48 821. 49 823' 50 825 51 8.827( 52 828J 53 830' 54 832( 55 834^ 56 8365 57 838] 58 839f 59 8417 60 843£ R- 1 1 )916 60 1309 f)9 1732 58 )183 57 5664 56 )172 55 )709 54 5273 53 i863 52 vl80 51 5123 50 ^792 49 i486 48 2205 47 ^949 46 5717 45 i.008 44 J323 43 1161 42 3022 41 3906 40 7018 29 4147 28 1296 27 8463 26 5648 25 2851 24 0072 23 7311 22 4567 21 1840 20 9130 19 6437 18 3761 17 1100 16 8456 15 5828 14 3216 13 0619 12 8037 11 5471 10 2919 9 0383 8 7861 7 5354 6 2860 10382 4 7917 3 5465 2 3028 1 !0304 ig- jmJ SINES AND TANGENTS. (3 Degrees.; fit M. Sine D. Cosine I). 'i: .■•lie. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 8.718800 721204 723595 725972 728337 730688 733027 735354 737667 739969 742259 n. 8.744536 746802 749055 751297 753528 755747 757955 760151 762337 764511 8.766675 768828 770970 773101 775223 777333 779434 781524 783605 785675 8.787736 789787 791828 793859 795881 797894 799897 801892 803876 805852 8.807819 809777 811726 813667 815599 817522 819436 821343 823240 825130 8.827011 828884 830749 832607 834456 836297 838130 839956 841774 843585 4006 3984 3962 3941 3919 3898 3877 3857 3836 3816 3796 3776 3756 3737 3717 3698 3679 3661 3642 3624 3606 3588 3570 3553 3535 3518 3501 3484 3467 3451 3431 3418 3402 3386 3370 3354 3339 3323 3308 3293 3278 3263 3249 3234 3219 3205 3191 3177 3163 3149 3135 3122 3108 3095 3082 3069 3056 3043 3030 3017 3000 9.999404 999398 999391 999384 999378 999371 999364 999357 999350 999343 999336 9.999329 999322 999315 999308 999301 999294 999286 999270 999272 999265 9.999257 999250 999242 999235 999227 999220 999212 999205 999197 • 999189 1118.719396 9.999181 999174 999166 999158 999150 999142 999134 999126 999118 999110 9.999102 999094 999086 999077 999069 999061 999053 999044 999036 999027 9.999019 999010 999002 998993 998976 998967 998958 998950 998941 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 U 14 14 14 14 14 14 15 15 15 15 721806 724204 726588 728959 731317 733663 735996 738317 740626 742922 8.745207 747479 749740 751989 7.54227 756453 758668 760872 763065 765246 8.767417 769578 771727 773866 775995 778114 780222 782320 784408 786486 8.788554 790613 792662 794701 796731 798752 800763 802765 804758 806742 8.808717 810683 812641 814589 816529 818461 820384 822298 824205 826103 8.827992 829874 831748 833613 837321 839163 840998 842825 844644 ("otanB. 4017 3995 3974 3952 3930 3909 3889 3868 3848 3827 3807 3787 3768 3749 3729 3710 3692 3673 36t)5 3636 3618 11.2806041 60 278194' 5!» 275796 273412 271041 268683 266337 264004 261683 259374 257078 11 3600 3583 3565 3548 3531 3514 3497 3480 3464 3447 254793 2.52521 250260 243011 245773 243547 241332 239128 236935 234754 58 57 56 55 54 53 52 51 50 11.232583 ,230422 f'28273 r 26 134 224005 221886 219778 217680 215592 213514 3431 3414 3399 3383 3368 3352 3337 3322 3307 3292 3278 3262 3248 3233 3219 3205 3191 3177 3163 _31^50_ 3136 3123 3110 3096 ouoo 3070 3067 3045 3032 3019 11.211446 209387 207338 205299 203269 201248 199237 197235 195242 1932.58 11.191283 1893J7 187359 185411 183471 181539 179616 177702 175795 173897 11.172008 170126 168252 166387 1 64529 162679 160837 159002 167175 155366 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 II 86 Degrees. 14 23 (4 Degrees.') a table op LooAniTiiMicj M. Siiiti ' I. i / » I- 1 2 ;j 4 5 7 R 9 10 11 12 i;} M if) Ifi 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 [34 35 36 37 38 39 40 S.HiarvM;') Hir,3M7 847183 818971 8r)07r)l 852.W5 854291 8r)(;()19 85780 1 85951fi _8(512t[3 8.S(';iOri 8(i4738 8(56455 808165 869868 871565 873255 874938 876615 878285 8.879949 881607 883258 884903 886542 888174 889801 891421 893035 894643 I). '3005 2992 29H0 2967 2955 2943 293 1 2919 2907 2896 288 i_ ',1873 2861 2850 2839 2828 2817 2806 2795 2786 2773 ,896246 897842 899432 901017 902596 904169 905736 907297 908853 910404 41 8 42 43 44 45 46 47 48 49 50 61 8 52 53 54 ub 56 57 58 59 60 ,911949 913488 915022 916550 918073 919591 921103 922610 924112 925609 ,927100 928587 930068 931544 933015 934481 935942 937398 938850 94029 6 Cosine 27(53 2752 2742 2731 2721 2711 2700 2690 2ti80 2670 2660 2651 2641 2631 2622 2612 2603 2593 2584 2575 2566 2556 2547 2538 2529 2520 2512 2503 2494 2486 CoHliie i 9 . 99894 i 998932 9!)H923 998914 998905 99889(5 998887 998878 9988(59 9988(50 998851 I). 'I'aiiR. 9 ,998841 998832 998823 998813 998801 998795 998785 998776 9987(56 99 8757 9.998747 998738 998728 998718 998708 998699 998689 998679 998669 998659 2477 2469 2460 2452 2'i'io 2435 2427 2419 2411 2403 9.998649 998639 998629 998619 998609 998599 998589 998578 998568 9 9 8558 9.998548 998537 998527 998516 998506 998495 998485 998474 998464 998453 9.998442 998431 998421 !5 15 15 15 15 15 15 15 15 15 ]5 15 15 16 16 16 16 16 10 16 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 18 8.844644 81(5455 818260 850057 85 181 (5 853(528 855403 857 1 7 1 858932 860(58(5 862433 8.864173 86590(5 867(532 869351 8710(54 872770 874469 8761(52 877849 879529 8.881202 882869 884530 886185 887833 889476 891112 892742 894366 89 5984 8.897596 899203 900803 902398 903987 905570 907147 908719 910285 911846 "^3019 3007 2995 2982 2970 2958 2946 2935 2923 29 1 1 2900 8 998410 18 998388 998377 998366 998355 998344 18 18 18 18 18 .913401 914951 916495 918034 919568 921096 922819 924136 925649 927156 8.928658 930155 931647 933134 934616 936093 937565 939032 940494 941952 "117155356"' 153.545 151740 149943 1481.54 146372 144597 142829 1410t;8 2888 2877 2866 2854 2843 283?. 2821 2811 2800 2789 2779 2768 2758 2747 2737 2727 2717 2707 2097 2687 2677 2667 2658 2648 ^638 2629 2620 2610 2601 2592 11. 139314 _nr/5(57 135827 134094 132368 130(549 128936 127230 12.5.531 123838 122151 120471 11.118798 117131 115470 113815 112167 110524 108888 1072.58 105634 101016 2583 2574 2565 25.56 2547 2538 2530 2521 2512 2503 2495 2486 2478 2470 2461 2453 2445 2437 2430 2421 11.102404 100797 099197 097602 096013 094430 092853 091281 089715 088154 GO 59 58 57 56 55 54 53 52 51 50 4!> 48 47 46 45 44 43 42 41 40 39 38 37 3(5 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 11.086599 085049 083505 081966 080432 078904 07738 1 075864 074351 072844 11.0 1342 009845 068353 066866 065384 063907 062435 060968 059506 058048 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 M "IT 1 2 3 4 5 6 t 8 9 10 IJ 12 13 14 15 16 17 i 18 19 '^0 21 2i> 23 24 25 26 27 28 29 .30 31 32 33 34 35 36 37 38 39 40 8.9 9 9' 9 9'1 91 91 9: 9.^ 9^ 8.95 d5 95 9(i 9(i 9(5 I 96 9(5 96 JM5 8". 96 97 97 97; 97 97( 97' 97* 98( 98] 8.98i 98^ 98f 98f 98fc 981] 990 991 993 991 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sine Colang. Tang. M. 8.995 997 91; s 099 9.000 002 003 OOti OOfil 00 7( 9.008; 009.' 010' 0111 013] OW, 01. 5f oioe 018C 019S Cosine 85 Degree*. K' i •AM] f.O Mf) na 710 58 91 :j r)7 IM f>r) M-Z nr, r)'j7 r)4 Hi>U M OfiH r)a :n4 r)()7 f)! f)!) BINES AND TANOKN'M. (5 DogroOS.) 23 M Hino I). ('osliio I). 1 2 .1 4 6 fi t 8 i) H) 11 12 U) 11 IT) Hi 17 I 18 lit 'ii) 21 2!) 21 25 20 27 28 2a 30 iii .12 33 31 3;-) 30 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 TaiiK. I>. 8.910296 91 1 738 913171 911606 916034 917456 948874 950287 95169( 953100 _9544J>9 8.955894 957284 958670 960052 961429 962801 964170 965531 966893 96'I249 8.969600 970947 972289 973628 974962 976293 977619 978941 980259 981573 8.982883 984189 98.5491 986789 988083 989374 990660 991943 9932221 994497 2103 2394 2387 2379 2371 2363 23.55 2348 2340 2332 2325 2317 2310 2302 2295 2288 2280 2273 2266 2259 2252 8 ,99.5768 997'>36 91; ,i99 099560 000816 002069 003318 004563 005805 007044 2244 2238 223 1 2224 2217 2210 2203 2197 2190 2183 2177 2170 2163 2157 3150 2144 21.38 2131 2125 2119 .008278 009510 010737 011962 013182 014400 01,5613 016824 018031 019235 2112 2106 2100 2094 2087 2082 2076 2070 2064 20.58 2052 2046 2040 2034 2029 2023 2017 2012 2006 2000 9,998314 998333 998322 998311 998300 998289 998277 998266 998255 998243 99 8232 9.998220 998209 998197 998186 998174 998163 998151 998139 998128 998116 9.998104 998092 998080 998068 998056 998044 998032 998020 998008 997996 19 19 19 19 19 19 19 19 19 19 j9 19 19 19 19 19 19 19 20 20 20 9.997984 997972 997959 997947 997935 997922 997910 997897 997885 997872 .997860 997847 997835 997822 997809 997797 997784 997771 997758 997745 .997732 997719 997706 997693 997nrt0 997667 9976.54 997641 997628 997614122 20 20 20 20 20 20 20 20 20 20 20 20 20 20 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 22 22 22 22 22 22 8.9419.52 94310J 944852 . 946295 917734 949168 9.50597 952021 95.34 1 1 954856 956267 8.9.57674 959075 9()0473 961866 963255 964639 966019 967394 968766 970133 8.971496 972855 974209 9755C.0 970906 978248 979.586 980921 98225 1 983577 8.984899 986217 987532 9888421 990149 991451 992750 994045 995.337 996624 8.997908 999188 9.000465 001738 003007 004272 005534 006792 008047 009298 9.010.546 011790 01.3031 014268 ni?i5n9 016732 017959 019183 020403 021620 4'()tiiti|{. 2421 2113 2405 2.397 2390 2382 2374 2366 2360 2351 2344 2337 2329 2323 2314 2307 2300 2293 2286 2279 2271 2265 2257 2251 2244 2237 2230 2223 2217 2210 2204 2197 2191 2184 2178 2171 2165 21.58 2152 3146 2140 21.34 2127 2121 2115 2109 2103 2097 2091 2085 2080 2074 2068 2062 2056 3051 2045 2040 2033 2028 2023 Jopine Sine Cotang. 111.0.58048 I 056596 055148 053705 05226(i 0.50832 049403 047979 040559 045144 0^3733 11.042326 040<)25 039527 038134 036745 035361 03398 1 032606 031234 029807 11.028.501 027145 025791 024440 023094 0217.52 020414 019079 017749 016423 11.015101 013783 012468 0111.58 009851 008.549 007250 005955 004663 003376 10 11,002092 000812 , 999535 998262 996993 995728 994466 993208 9919.53 9 90702 10.9894.54 988210 986969 985732 98.3268 982041 98081/ 979597 978380 60 59 58 57 56 55 54 63 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 .30 29 28 27 26 25 24 23 23 21 20 19 18 17 16 15 14 13 12 11 10 ~9 8 7 6 u 4 3 2 1 X 64i)egrees, I Tang. I M 24 (6 Degrees.) A TABLE OF LOGARmiMIC iih M Hliie D. Comno 1 1). Tang. 1 D. Cutang. 1 1 9.019235 2000 9.997614 22 9.021620 2023 10.978380 '60 1 0204;ir) 1995 997601 22 022834 2017 977166 69 2 O'ZUy.VZ 1989 997588 22 024044 2011 975956 58 3 022825 1984 997574 22 025251 2006 974749 57 4 024016 1978 997561 22 026455 2000 973545 56 6 025203 :?73 997547 22 027655 1995 972345 55 6 0263H6 1967 997534 23 028852 1990 971148 54 7 0275f)7 1962 997520 23 030046 1985 969954 53 8 028744 1957 997507 23 031237 1979 968763 52 9 029918 1951 997493 23 032425 1974 967575 51 10 11 031089 9.0;J2257 1947 997480 9.997466 23 23 033609 1909 966391 10.965209 50 49 1941 9.034791 1964 12 033421 1936 997452 23 035969 1958 964031 48 13 034582 1930 997439 23 037 44 1953 96285^ 47 14 035741 1925 997425 23 038316 1948 96168. ifil 15 036896 1920 997411 23 039485 1943 960515 4.1 16 038048 1915 997397 23 040651 1938 959349 44 17 039197 1910 997383 23 041813 1933 958187 43 18 040342 1905 997369 23 042973 1928 957027 42 19 041485 1899 9973{;^ 23 044130 1923 955870 41 20 042625 1894 997341 23 24 045284 1918 954716 40 39 9.043762 1889 9.997327 9.046434 1913 lO.tJ.^3566 22 044895 1884 997313 24 047682 1908 952418 38 23 046026 1879 997299 24 048727 1903 951273 37 24 047154 1875 997285 24 049869 1898 950131 36 25 048279 1870 997271 24 051008 1893 948992 35 20 049400 1865 997257 24 052144 1889 947856 34 27 050519 1860 997242 24 053277 1884 946723 33 28 051635 1855 997228 24 054407 1879 945593 32 29 05274;; 1850 997214 24 055535 1874 944465 31 30 31 0538.')9 1845 997199 9.997185 24 24 056659 9.0,57781 1870 1865 943341 30 29 054966 1841 10.942219 32 056071 1836 997170 24 058900 1869 941100 28 33 057172 1831 997156 24 060016 1855 939984 27 34 058271 1827 997141 24 061130 1851 938870 26 35 059367 1822 997127 24 062240 1846 937760 25 36 060460 1817 997112 24 063348 1842 936652 24 37 061551 1813 997098 24 064453 1837 935547 23 38 002639 1803 997083 25 065556 1833 9344 '4 22 39 063724 1804 997068 25 066655 1828 933345 21 40 41 064806 9.065885 1799 997053 25 25 067752 1824 932248 20 19 1794 9.997039 9.068846 1819 10.9311.54 42 066962 1790 997024 25 069938 1815 930062 18 43 068036 1786 997009 25 071027 1810 928973 17 44 069107 1781 996994 25 072113 1806 927887 16 45 070176 1777 996979 25 073197 1802 926803 15 46 07 1 242 1772 996964 25 074278 1797 925722 14 47 072306 1768 996949 25 075356 1793 924644 13 48 073366 1763 996934 25 076432 1789 923568 12 49 074424 1759 996919 25 077505 1784 922495 11 50 51 075480 1755 996904 9.996889 25 25 078576 i 9.079644 1780 921424 10 9 9.076533 1750 1776 10.920356 62 077583 1746 996874 25 080710 1772 919290 8 53 078031 1742 996858 25 081773 1767 918227 7 64 079676 1738 996843 25 082833 1763 917167 6 55 080719 1733 996828 25 088891 1759 916109 5 56 081759 1729 996812 26 084947 17.55 Q15053 4 57 0a27fl7 1725 996797 26 OAROOO 1751 M/lfiOQ s 58 083832 1721 996782 26 087050 1747 91295C 2 69 084864 1717 996766 26 088098 1743 911902 1 60 085894 1713 996751 26 089144 1738 910856 1 Cosine Sine 1 (Jotanij. I Tang. 1 M. 1 lif ; 83 Degrees. M 1 8 U y.ol 1 Oi 2 Of 3 1 4 1 5 r .J. b I ')' 7 8 (," 9 \J. 10 rr 11 •K. 12 13 14 Ud 09 10 -5 10 16 10 17 10 18 10 19 10 20 10 21 9.10 22 10 23 10 24 10 25 11 26 11 27 11 28 11 29 11 30 11 31 9.11 32 11 33 11 34 11 35 12( 36 12 37 12 38 12; 39 12' 40 12i 41 9.12f B-^^ 12' 43 12' 44 12f 45 12f 46 13< 47 131 48 1.3S 49 l.^.*^ 50 VM 51 9.13E 52 136 53 137 54 138 55 13t] 56 13fJ r r/ kM 14u 58 141 59 142 60 143 1 1 so 60 w 60 )(\ 58 1!> 57 If) 56 IS 55 18 54 54 53 ^n 52 75 51 91 50 D'J 49 n 48 5r 47 ?. i6| 15 4a 19 44 87 43 27 42 70 41 16 40 0(5 39 IH 38 73 37 31 36 92 35 50 34 23 33 93 32 65 31 41 30 19 29 00 28 84 27 70 26 60 25 52 24 47 23 'A 22 45 21 48 20 54 19 62 18 73 17 87 16 03 15 22 14 44 13 68 12 95 11 24 10 56 9 ,90 8 !27 7 67 6 09 5 )53 4 >0Q 3 )5C 2 )02 1 i5G |M. ♦aWEfl AND TANGENTS. (7 DcgrCf)3.) 2b M. Sine I D. I r'^024 >!»;>, ^37 0.»l9:;)65 U98066 099065 100062 101056 102048 103037 104025 105010 __105992 9.106973 107951 108927 109901 110873 111842 112809 113774 114737 115698 9.116656 117613 118567 119519 120469 121417 122362 123306 124248 125187 .126125 127060 127993 128925 129854 130781 131706 132630 133551 134470 9.135CS7 136303 137210 138128 139037 139944 1408.j0 1417.54 142656 143555 1713 1709 1704 1700 1696 1692 1688 1684 1680 1676 1673 1668 1665 1661 1657 1653 1649 1645 1641 1638 1634 1630 1627 1623 1619 1616 1612 1608 1605 1601 1597 1594 1590 1587 1683 1.580 1576 1.573 1569 1.566 1.562 1,559 1.5.56 1552 1549 1545 1.542 1539 1.535- 1532 1.529 1.525 1.522 1519 1516 1512 1.509 150G 1.503 1500 1496 9.996751 996735 996 720 996704 996688 99667 1 996657 99664 1 996625 996610 986594 9.996578 9965C2 996546 996530 996514 996498 996482 996465 996449 996433 9.996417 996400 996o84 996368 99635 1 996335 996318 996302 996285 996269 9.9962.52 996235 996219 996202 996185 996168 996151 996134 996117 996100 .996083 996066 996049 9960.32 J'^6015 995998 995980 995963 995946 995928 9.99.5911 995P94 99J876 995859 995841 995823 995806 995788 995771 995753 26 26 26 26 26 20 26 26 26 26 26 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 28 28 28 28 28 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 I <^>l«tlR, ( 9.089144 090187 091228 092266 093302 094336 095367 096395 097422 098446 09946S 9.100487 101504 102519 103532 104.542 105550 106556 107559 108560 109559 9. 110.556 1U55I 112.543 1136.33 114.521 115.507 116491 117472 118452 1 19429 .120404 121377 122348 12.3317 124284 125249 126211 127172 128130 129087 9.130041 1.30994 131944 132893 133839 1.34784 135726 136667 137605 lo8542 9.1.39476 140409 141340 142269 143196 144121 145044 145966 146885 147803 1738 734 1730 1727 1722 1719 1715 1711 1707 1703 1699 1695 1691 1687 1684 1680 1676 1672 1669 1665 1661 1658 16.54 1650 1646 1643 1639 1633 16.32 1629 1625 1622 1613 1615 1311 1607 1604 1601 1.597 1594 1591 10.910856 909813 908772 907734 906698 905664 904633 903605 90 .;i78 9015.54 900532 10.899513 898496 897481 896468 895458 894450 893444 89244 1 891440 890441 1587 1.584 1581 1577 1574 1571 1.567 1564 1.561 1658 10.889444 888449 8874.57 886467 885479 884493 883509 882528 881.548 880571 10.879596 29 878623 28 8776.52 27 376683 26 87.5716 26 874751 24 873789 23 872828 22 871870 21 870913 20 60 59 58 57 56 55 64 53 62 51 50 ''9 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 19 18 17 16 16 14 13 12 1.555 1551 1548 1545 1.542 1539 1535 1532 1.529 1.526 10.8;/>959 86'i, 06 86ft .06 867107 866161 865216 864274 863333 862395 11 861458 10 9 8 7 6 5 4. 3 2 1 10.860624 8.59591 858660 857731 856804 8.55879 854956 854034 863115 852197 Oosini! Cotang. Tung. |M. Wf r- '-' t 26 (« Degrees. ) a table of LOGARixiimc ■ M. Sine D. 1 Cosine 1 D. | Taii^'. 1 D. 1 Cotai^;. 1 i prr 9.143555 1496 9 . 995753 30 • 9,147803 1526 10.8.52197 60 019 1 — 1 9 M4453 1493 995735 30 148718 1523 851282 59 1 145349 1490 995717 30 149632 1520 8.50368 58 r 3 140243 1437 995099 30 150541 1517 849456 57 .I 4 147136 1484 995681 30 151454 1514 818.540 56 1 4 M 5 148026 1481 995604 30 1.52303 1611 847637 55 5 C 148915 1478 995040 30 153269 1608 846731 54 6 t 7 149802 1476 995028 30 154174 1505 845820 53 7 8 150686 1472 995610 30 15.5077 1502 84^1923 52 8 9 151569 1469 995.591 30 155978 1499 844022 21 9 nM * 1 10 11 152451 1466 1403 995573 9.9955,55 30 30 156877 9.157775 1496 843123 49 10 ] 1 Ml 9 153330 1493 10.842225 9 f;" 12 154208 1460 995537 30 168671 1490 841329 48 12' r . lo 155083 1457 995519 30 159565 1487 840435 47 13 14 155957 1454 995501 31 160457 1484 839543 46 14 1 15 156830 1451 995482 31 161.347 1481 838653 45 15 16 157700 1448 995464 31 102236 1479 837764 44 16 1 17 158509 1^*45 995446 31 163123 1476 830877 43 17 1:^ 18 150' 35 1442 995427 31 164008 1473 835992 42 18 1 '■ 19 100301 1439 99.5400 31 104892 1470 835108 41 19 ■ 20 21 161164 1436 1433 995390 3' 31 165774 9.166054 1467 1404 834226 40 39 20 21 9.162025 9.995372 10.833340 9. !;■■ \- 22 162885 1430 995353 31 16i'532 1401 832408 38 22 '^1 23 163743 1427 995334 31 168409 14.58 83159' 37 23 l! 24 164600 1424 99.5316 31 109284 1455 830716 36 24 25 165454 1422 995297 31 170157 1453 829843 35 25 26 106307 1-^19 995278 31 171029 14.50 828971 34 26 27 167159 1416 995260 31 171899 1447 828101 33 27 28 168008 1413 995241 32 172707 1444 827233 32 28 2j 168850 1410 995222 32 173634 1442 826366 31 29 30 31 169702 1407 1405 995203 32 32 174499 1439 1430 82.5.501 CO 29 80 31 9.170547 9.995184 9.175362 10.824038 9? 32 171389 1402 995165 32 1 70224 1433 823770 28 32 33 172230 1399 995140 32 177084 1431 822916 27 33 34 173070 1396 995127 32 177942 1428 822058 26 34 35 173908 1394 995108 ijf-* 178799 1425 821201 25 35 36 174744 1391 995089 32 1796.55 1423 820345 24 36 37 175578 1388 995070 32 180508 1420 819492 23 3/ 38 176411 1386 99.5051 32 181360 1417 SI 8640 22 1 1 38 « 39 177242 1383 996032 32 182211 1415 817789 21 39 < 40 41 178072 9.1789«)0 1380 1377 99.5013 33 32 1830.'^,9 1412 816941 10.810093 20 19 40 41 ( 9.994993 9.183907 1409 Q i • ■ ; 42 179726 1.374 994974 32 184752 1407 81.5248 18 a: 1 42 i7 * A s . i 43 180551 1372 994955 32 185.597 1404 814403 17 43 A 44 181374 1369 994935 32 180439 1402 813501 16 44 45 182196 1366 994916 33 187280 139 J 812720 15 45 A K 46 183016 1304 99-1896 33 188120 1390 811880 14 40 S 2 2 47 183834 1361 C94877 33 188958 1393 811042 13 47 48 184651 1359 994857 33 189794 ' 1391 810200 12 48 49 185460 1350 994838 33 190629 1389 809371 11 ''9 50 51 186280 ] 353 1.351 994818 33 33 191462 9.192294 13S6 f384 808538 10 9 50 51 'I 9.187092 9.994798 10.H0770r 972 52 187903 1348 994779 33 193124 1381 800876 8 52 53 188712 1346 994759 33 193953 1379 806047 ' 7 53 54 189519 1343 994739 33 194780 1376 80522( I 6 54 2 2 2 55 190325 1341 994719 133 195600 1374 80439' \ 5 55 56 191130 1338 99470( 33 - 190430 1371 803571 » 4 56 57 191933 1330 994G80 33 197253 1369 802747 ^ 3 VI 1927-W 1333 994060 33 1 08074 1306 «0192f ) 2 * f A, 69 193534 1330 994640 33 198891 1304 soiioe ) 1 60 194332 1328 994620 33 199711 ! 1361 800287 1 ( io 2: i 1 Cosine 1 1 ^'"^' i 1 (Jiitaiig. 1 'J'liiii.'. j M- 1 (Jot ' !l tjl Uesjf ees. ^"P- 1 1 )2197 60 )1282 59 J0308 58 94r)(; 57 8540 56 .7637 55 16731 54 15826 53 t4923 52 14022 21 13123 5t> 12225 49 H329 48 10435 47 39543 46 38653 45 37764 44 36877 43 15992 42 35 108 41 34226 40 33346 39 32468 38 3159' 37 30716 36 29843 35 28971 34 28101 33 27233 32 26366 31 25501 CO 29 24638 23770 28 22916 27 22058 26 21201 25 20345 24 19492 23 18640 22 17789 21 16941 20 19 16093 515248 18 i 14403 17 ! 13561 16 112720 15 ! 11880 14 ! 11042 13 n0206 12 W9371 11 ^08538 10 <0770r 9 ^06876 8 ^06047 ' 7 B0522( I 6 30439' t 5 303571 ) 4 302741 f 3 ■i0192f ) 2 BOiioe ) 1 300281 r 'iii.i.'. j M- 1 f SINES AND TANGFNTS. (9 DcgrCCS 1 2 3 4 5 6 V 8 9 U) II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 80 31 32 33 34 35 36 3/ 38 39 40 9.19433 195129 195925 196719 197511 198302 199091 199879 200666 201451 20223'i 41 42 43 14 45 40 47 48 19 50 51 52 53 54 55 56 57 9.203017 203797 204577 205354 2<)6131 206906 207679 208452 209222 209992 9.210760 211526 212291 21.3055 21.3818 214579 21.5.338 216097 216854 _2r7609 K 2 18363 219116 219868 220618 221.367 222115 222861 223606 224349 225092 9.22,58.33 226573 227.111 228048 228784 2295 1 8 230252 230984 231714 _232444 9.ii3.3172 233899 234625 23.5.349 236073 236795 237515 1328 1326 1323 1 32 1 1318 1316 1313 1311 1308 1.306 J^.304 1301" 1299 1296 1294 . 1292 1289 1287 1285 1282 1280 1278 1275 12V 3 1271 1268 1206 1264 1261 12,59 J257_ 12,55 1253 1250 1248 1246 1244 J 242 1239 1237 1 235 12.33 1231 1228 1220 1224 1222 1220 1218 1216 _1^2M_ 1212 1209 1207 1205 1203 1201 1199 1197 1 I or: 9 9 94600 9945H0 994560 994,540 994519 994499 994479 994459 994438 _994418 . 994397 994377 994357 994336 994316 994295 994274 9942.54 994233 994212 ^994 191 994171 9941.50 994129 9941 OH 994087 994066 994045 994024 994003 .993981 993960 993939 993918 993896 993875 993854 993832 99.3811 _ ^9.3789 9.993768 993746 993725 9!)3703 9: 13681 993860 993638 n936l6 993594 __99;j572 9.993550 993528 993506 993484 99.3462 993440 993418 y93396 993374 993351 33 33 33 34 34 34 34 34 34 34 34 34 34 34 34 34 34 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 36 36 36 36 36 36 36 36 36 36 36 36 36 36 37 .37 ■37 37 37 37 37 37' 37 37 37 37 201.345 202159 202971 203782 204592 205400 206207 207013 _207817 9.208619 200420 210220 211018 211815 212611 213405 214198 214989 215J780 9.216568 217356 218142 218926 219710 220492 221272 222052 222830 _223606 E». 224382 225156 225929 226700 227471 228239 229007 229773 23053'! 231,3(»2 1.3.54 1352 1.349 1.347 1345 1342 1340 J338_ 1.3.35 1.333 1,331 1328 1326 1324 1321 1319 1317 1315 797841 797029 796218 70.5408 794600 793793 792987 792183 1312 1310 1308 1305 1303 1301 1299 1297 1294 1292 10.791381 790580 789780 788982 788185 7873S9 786595 78,5802 785011 784220 10 9.232065 232826 233586 234345 235103 235859 236614 237368 238120 238872 1290 1288 1286 1284 1281 1279 1277 1275 1273 1271 ,783432 782644 7818.58 781074 780290 779508 778728 777948 777170 776394 60 69 5S 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 .30 9 1269 1267 1265 1262 1260 12.58 12,56 1254 1252 12,50 10.775618 29 774844 28 774071 27 773300 26 772529 25 771761 24 770993 23 770227 22 769461 21 .23'96;;i' 1248 240371 1246 ',41118 1244 241865 1242 242610 1240 243354 1238 244097 1236 244839 12.34 245579 1232 2463 1 9 1230 768698 20 10.76793,5 19 767174 18 766414 17 765655 16 764897 15 764141 141 763386 13^ 762632! 12 761880 11 761128 10 9 8 7 6 5 4 3 o 1 10.760378 759630 7588«2 7.58135 757390 7566-16 5003 Cotaiii;. 7550 755161 754421 7.5.368 1 IM. fcO Degree's. 28 (10 Degrees.) a table op logarithmic M. smu D. I Cosine D. Taiic. D. Coiaiig. I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ao 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 14 45 46 47 48 49 50 9.239670 2403^" 241 iO 2418x4 242526 243237 243947 244656 245363 246069 246775 .247478 248181 248883 249583 250282 250980 251677 252373 253067 2.53761 1193 1191 1189 1187 1185 1183 1181 1179 1177 1175 1173 1171" 1169 1167 1165 1163 1161 1159 1158 1150 1154 9.2544.53 255144 255834 256523 2.57211 257898 258583 259268 259951 260633 9.261314 261994 262673 263351 264027 264703 265377 266051 266723 267395 9.268065 268734 269402 270069 270735 271400 272064 272726 273388 274049 51 52 53 54 55 56 58 59 I 60 I 9.274708 275367 276024 276681 277337 277991 278644 279297 279948 280599 11.52 1150 1148 1146 1144 1142 1141 1139 1137 J135 1133 1131 1130 1128 1126 1124 1122 1120 1119 1117 1115 1113 nil 1110 1108 1106 1105 1103 1101 1099 .993351 993329 993307 99C285 993262 993i:0 993217 993195 993172 993149 993127 9.993104 993081 993059 993036 993013 992990 9r,'>967 9.f2944 992921 992898 9.992875 992852 992829 992806 992783 992759 992730 992713 992690 992660 9.992643 992619 992596 992572 992549 992525 992501 992478 992454 992430 1098 1096 1094 1092 1091 1089 1087 1086 1084 1082 9.992406 992382 992359 992335 992311 992287 992263 992239 992214 40 992190 40 37 37 37 37 37 37 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 40 40 40 40 40 40 40 40 40 40 '0 9.246319 247057 247794 248530 249264 249998 2.50730 251461 252191 252920 253648 9.254374 2.55100 255824 256.547 257269 257990 2.58710 259429 260146 260863 .261.578 262292 263005 263717 264428 2651.38 26.5847 5^66555 267261 267967 1230 1228 1226 1224 1222 1220 1218 1217 1215 1213 1211 9.268671 269375 270077 270779 271479 272178 272876 273573 274269 274964 9 ,992166 992142 992117 992093 992069 992044 992020 991996 991971 99194? 40 40 41 41 41 41 41 41 41 41 9.2756.58 276351 277043 277734 278424 279113 279801 280488 281174 281858 1209 1207 1205 1203 1201 1200 1198 1190 1194 1192 9.282542 283225 283907 284588 285268 285947 286024 2H?301 287977 288652 10.753681 752943 752206 751470 7.50736 750002 749270 748539 747809 747080 746352 10 1190 1189 1187 1185 1183 1181 1179 1178 1176 1174 1172' 1170 1169 1167 1165 1164 1162 1160 11.58 11.57 ,745626 744900 744176 743453 742731 742010 741290 740571 7398.54 739137 10 ,738422 737708 736995 736283 735572 734862 7341.53 733445 732739 732033 1155 1153 1151 1150 1148 1147 1145 1143 1141 1140 1138 1130 1135 1133 1131 1130 1128 il2G 1125 1123 10.731329 ?30625 729923 729221 728.521 727822 727124 726427 72573 1 725036 10 ,724342 723649 722957 7222( 6 721576 720887 720199 719512 718826 718142 10 ,7174.58 716775 716093 71.5412 714732 7140,53 713376 7i2x';99 712023 711348 Cosine I Bine Cotani;. Tanc i 97 Degrees M. ~0 1 2 3 4 5 6 7 8 9 10 U 12 13 14 15 16 17 18 119 120 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 /:j 'M ■t ■;■ I ■*bi 4V 48 49 50 IS 54 59 jOO^ IZJI H "^- 1 1 J681 60 IMS 59 5206 68 470 57 )736 56 )002 55 )270 54 ^539 53 ?809 52 ro80 51 3352 50 )626 49 1900 48 1176 47 3453 46 2731 4"= 2010 'i i 1290 43 0571 42 9854 41 9137 40 8422 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 W5 15 14 13 12 1' JO 9 8 7 6 5 4 3 1 01 IM. I i SIMES AND TANGENTS, (ll Degrees.) 9.280599 281248 281897 282544 283190 283836 284480 285124 285766 286408 287048 9.287687 288326 288964 289600 290236 ?90870 291504 292137 292768 293399 1082 1081 1079 1077 1076 1074 1072 1071 1069 1067 1066 . 294029 294658 295280 295913 296539 297164 297788 298412 299034 299655 1064 1063 1061 1059 1058 1056 1054 1053- 1051 1050 'M .300276 300895 301514 302132 302748 303364. 303979 304593 305207 305819 1018 1046 1045 i043 1042 1040 1039 1037 1036 1034 9.991947 991922 991897 991873 991848 991823 991799 991774 991749 991724 991699 9.991674 991649 991624 991599 991574 991549 991524 991498 991473 991448 1.288652 289326 289999 290671 391342 292013 292682 293350 294017 294684 295349 o')7041 307f:50 308^59 ■■U(88G7 :''^9474 3 1 0080 3106S5 311289 31 1893 9.312495 31^097 313698 314297 314897 3! 5'! 05! 316()<>'jj 316689 3 17284! 317879' 1032 1031 1029 1028 1026 1025 1023 1022 1020 lOiO .991422 991397 991372 991346 991321 991295 991270 991244 991218 991193 1017 1016 1014 1013 1011 1010 1008 1007 1005 1004 1003 1001 1000 998 997 99n 994 993 991 990 .991167 991141 991115 991090 991064 991038 991012 990986 990960 990934 9.990908 44 990882! 44 990855! 44 990829; 44 990803 44 990777, 44 990750 44 990724 44 990697; 44 990671 9 44 .990644 44 990618 44 99«»59I 44 990565 44 9905-'8 44 90;..:; 1 45 .-^90485 4f 990458 45 990431 45 990404 45 1.296013 296677 297339 298001 298662 2i;9322 299980 300638 301295 301951 9.302607 303261 303914 304567 305218 305869 306519 307168 307815 30 8463 9.309109 309754 310398 811042 311685 312327 312967 313608 314247 __31.188oj }.3155'?3 316 1. ')9! 3167951 317430 3180641 3186971 319329 3199611 320592; _3'>!2J22! .3i>l85I 322179 323106 323733 324358 124983 .::5607 3'362.'l 326853 •J27475i 1123 1122 1120 1118 1117 1115 1114 1112 1111 1109 1107 10.711348 710674 710001 7093J29 708658 707987 707318 706650 705983 705316 704651 1106 1104 1103 1101 1100 1098 1096 1095 1093 1092 1090" 1089 1087 1086 1084 1083 1081 1080 1078 1077 10.703987 703323 702661 701999 701338 700678 700020 699362 698705 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 1<^75 1074 1073 1071 1070 1068 1067 1005 1061 1062_ 1061 1060 1058 J 057 055 1054 1053 1051 1050 1048 1047 1045 1044 1043 6980491 40 39 38 37 36 35 34 33 32 31 30 10.697393 696739 696086 695433 694782 694 1. J 1 693481 692832 692185 691537 10.690891 29 690246 689602 688958 688315 68 : ;373! 24 687033 23 G8639?' i2 •8 27 26 2''^ 685753 685115 10.G8447' 683S41 68320 ;^ .•<36 681303 680671 680039 ?^7.«ree». M.t ! 9.: 1 2 3 A ; ■ 9.3 52 3 53 3; 54 3! 55 3; 56 3J 57 3J [iS 3^ 59 3.' 60 3i! SINES AND TANGENTS. (^13 Degrees.) 31 252"rv 190f. 1285 0606 0047 9430 i88l3 18197 i7582 16967 ^354 1574 1 >5129 )4518 53907 53298 32689 i208l 51473 30867 B026 1 59656 59052 58448 57845 5/243 56642 560421 i55442| ;54843 15424 5 153647 i.53051 )i>2455 5518.59 551265 550671 550078 549-180 B48894 648303 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 261 251 24 23 22 21 20 (547713 647124 646535 645947 645360 644773 644187 643602 643018 642434 '641851] 641269 640687 640107 639.526 638947 •698368 63'^t790 637213 6 36636 Tuns I M. 19 18 17 16 15 14 13 12 9 8 7 6 5 4 3 2 1 M. 1 Sine 1 D. 1 Cosine 1 1). 1 'I'aiiK. I 1^- (!()iaii^. j 9.3.52(W8 911 9.988724 49 9.. 363364 960 I0.6.3663r 60 I 352635 910 988695 49 363940 9.59 636060 59 2 353181 909 988666 49 364515 958 6354815 58 3 3.53726 908 988630 49 365090 957 6.34910 57 4 ,3.54271 907 988607 49 365664 955 634336 56 5 3.54815 905 988578 49 366237 9.54 633763 ' 55 6 355358 904 988.548 49 366810 953 6.33190 54 7 355901 903 988519 49 367382 952 632618 53 8 356143 902 988489 49 367953 951 632047 52 9 356984 901 988460 49 368524 950 631476 5! 10 11 357524 9.358064 899 988430 49 49 369094 9.369663 949 630906 50 49 898 9.988401 948 10.630.337 12 358603 897 98B371 49 370232 946 629768 48 13 359141 896 988342 49 370799 945 629201 47 14 359678 895 988312 50 371367 944 628633 46 15 360215 893 988282 50 371933 943 628067 45 16 3607.52 892 98S252 50 372499 942 627.501 44 17 301287 891 988223 50 373064 941 626936 43 18 361822 890 988193 50 373629 940 626371 42 19 362356 889 988163 50 374193 939 62.5807 41 20 21 362S89 888 887 988133 9.988103 50 50' 374756 938 625244 40 r,9 9.363122 9.375319 937 10.624681 22 363954 885 988073 50 37.5881 935 624119 38 23 304485 884 988043 50 376442 934 623558 37 24 365016 883 988013 50 377003 933 622997 36 25 365.546 882 987983 50 377563 932 622437 35 26 366075 881 987953 50 378122 931 621878 34 27 366604 880 987922 50 378681 930 621319 33 28 367131 879 987892 50 379239 929 620761 32 29 367659 87r 987862 50 379797 928 620203 31 30 31 368185 9.368711 876 987832 9.987801 51 51 380354 927 619646 30 29 875 9.380910 926 10.619090 32 369236 874 987771 51 381466 925 618534 28 33 369761 873 987740 51 382020 924 617980 27 34 370285 872 987710 51 382575 923 617425 26 35 370808 871 987679 51 383129 i;22 616871 25 36 371330 870 987649 51 383682 921 616318 34 3/ 371852 869 987618 51 384234 920 615766 23 38 372373 867 987588 51 384786 919 615214 22 39 372894 866 987557 51 385337 918 614663 31 40 41 37.3414 9.373933 865 987526 9.987496 51 .51 38,5888 917 614112 20 19 864 9.38643S 915 10.613562 42 374452 863 987465 51 386987 914 613013 18 43 374970 862 987434 51 387.5,36 913 612464 17 44 375487 861 987403 52 388084 913 611916 16 45 376003 860 987372 52 388631 911 611369 15 46 376519 859 987341 52 389178 910 610822 14 47 377035 8.58 997310 52 389724 909 610276 13 !^ 377549 857 987279 52 390270 908 609730 12 49 378063 856 987248 52 .390815 907 609185 11 .h!.. . 378577 9.379089 854 9S7217 9.987186 52 52 391360 906 608640 10 9 853 9.391903 905 10.608097 i)2 379601 8.52 9871.55 52 392447 904 607553 8 i)3 380113 851 987124 52 392989 903 607011 7 54 380624 850 987092 52 393531 902 606469 6 55 381134 849 987061 52 394073 901 605927 5 56 57 381613 848 987030 52 394614 900 605386 4 382152 847 986998 52 395154 899 604 S4 6 3 58 3S2661 846 986967 52 395694 898 604306 2 59 383 1 68 845 986936 52 396233 897 603767| 1 1 bO 1 3S367;')| 844 986904 52 396771 896 603,:2;>' .?; ■!■ -It .1. CiwitiH hiiip Cotiiiu;. I am'. M. 70 Dt'greea. Ill 1 ■ I i 32 (14 Degrees.) a table of logarf' '"ic M.| SiriR I I). 1 () 9.383075 844 1 384182 843 2 384687 842 3 385192 841 4 385697 840 fi 386201 «»39 6 396704 838 7 387207 837 8 387709 836 » 388210 835 10 388711 834 Cosine D. Tang. I). Cotung. 11 12 13 14 15 16 17 18 19 2a 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.389211 389711 390210 390708 391206 391703 392199 392695 393191 393685 9 833 832 831 830 828 827 826 825 824 823 y.9H6904 986873 986841 986809 986778 986746 986714 986683 986651 986619 986587 .394179 894673 395166 395658 396150 396641 397132 397621 398111 _398600 '.399088 399 )75 400062 400549 401035 401520 402005 402489 402972 403455 822 821 820 819 818 817 817 816 815 814 813 812 811 810 809 808 807 806 805 804 9.403938 404420 404901 405382 405862 406341 406S20 407299 407777 408254 iT 40,^7311 409i.H)7i 409682' 4101571 410632 4! 1106 411579, 4120521 4125241 412996' 52 53 53 53 63 53 53 53 53 53 ^ 63 9.9865551 53 986.5231 53 986491,53 9864591 53 986427 986395 986363 986331 986299 986266 .396771 397309 397846 398383 398919 399455 399990 400.524 401058 401591 402124 803 802 801 800 799 798 797 796 795 794 794 793 792 791 790 7S9 V88 787 786 785 53 53 54 54 54 54 97986231 54 986202 .54 986169 986137 986104 986072 9S6039 986007 985974 985942 9.402656 403187 403718 404249 404778 405308 405836 406364 406892 407419 .985909 985876 985843 98.5811 985778 985745 985712 98567S 985646 98.5613 54 54 54 54 54 54 54 54 55 55 55 55 55 55 55 55 55 55 .407945 408471 408997 409.V21 410045 410.569 411092 411615 412137 412658 9.985.580 985547 985514 985480 985447 985414 9H5380 985347 985314 985280 9 . 985247 98.5213 985180 985146 985113 985079 985045 98.5011 984978 984944 9.413179 413699 414219 414738 41.5257 415775 416293 416810 41V326 417842 55 55 55 55 56 56 56 56 5_6 56 56 56 56 56 56 56 56 56 56 9 896 896 895 894 893 892 891 890 889 888 887 .418358 418873 419387 419901 420415 420927 421440 4219.52 422463 422974 .423484 423993 424503 49.5011 425519 426027 426531 427041 427547 42805 886 885 884 883 882 881 880 879 878 877 876 875 874 874 873 672 871 870 869 868 867 866 865 864 864 863 862 861 860 859 8.58 8.57 858 855 8.55 854 8.53 8.52 851 850 lU. 603229 602691 6021.54 601617 601081 600545 600010 599476 598942 598409 697876 10.597344 696813 696282 595751 695222 694692 594164 593636 593108 592.581 60 59 58 67 56 65 64 53 52 51 60 49 48 47 46 45 44 43 42 41 40 10.. 592055 591,529 691003 690479 689955 589431 .588908 688385 587863 587342 10.586821 586301 58578 1 685262 584743 584225 583707 583190 682674 .582158 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 10 849 848 848 847 846 845 844 843 843 842 ,.581642 .581127 580613 680099 679585 579073 578560 578048 577537 577026 10 ,576516 676007 575-^97 .574989 674481 573973 5734f)0 672959 {;72453 571948 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 75 Degrees. M,l t 9.4 1 4 2 4 3 4 4 4 ii*^9 G91 154 617 081 545 010 476 942 409 '876 344 1813 1282 )751 )222 1692 U64 3636 J108 2581 60 59 58 67 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 2055 1529 1003 0479 9955 9431 8908 8385 7863 7342 6821 6301 1578 1 15262 !4743 !4225 13707 53190 J2674 J2158 ^642 31127 30613 30099 79585 79073 78560 78048 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 77537 11 77026 10 9 76516 76007 8 75197 7 74989 6 74481 5 73973 4 734611 a .72959 2 ,72453 1 .71948 ang EJ SINES AND TA1SGENTS. (15 Dcgrees.) MUJ aiiie I D. I Coaine | D. | Tang. 33 D. C(.taiie. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 49 50 51 52 53 54 55 56 5V 58 59 60 9.412996 413467 413938 414408 414878 415347 415815 416283 416751 417217 417684 9.418150 418615 419079 419544 420007 420470 420933 421395 4218.57 422318 9 422778 423238 423697 424156 424615 425073 425530 425987 426443 426899 9.427,354 427809 428263 428717 429170 429623 430075 430527 430978 431429 9.431879 432329 432778 433226 433675 434122 434569 43.5016 435462 785 784 783 783 782 781 780 779 778 777 770 775 774 773 773 772 771 770 769 768 767 9.984944)67 984910 984876 984842 984808 984774 984740 984706 984672 984637 984603 .984569 984535 984.500 98446s< 984432 984397 984363 984328 984294 984269 767 766 765 764 763 762 761 760 760 759 758 757 756 755 754 753 752 752 751 750 9.984224 984190 9841.55 984120 984085 984050 984015 983981 983946 983911 435908 742 9.436.3.')3 741 436798 740 437242 740 437686 739 438129 738 438572 737 /iQQm.i •yon 439456 736 439897 735 440338 734 1 749 749 748 747 746 745 744 744 743 9.983875 983840 983805 983770 9837351 59 67 57 57 67 57 57 67 67 57 ^ 67 57 57 57 58 58 58 58 58 58 58 58 58 58 58 58 58 68 58 58 58 59 59 69 983700 983664 983629 983594 983558 9.983.523 9834S7 983452 98.3416 98.3381 983345 983309 983273 60 59 59 59 69 59 59 59 59 59 69 59 59 9.4280.52 428557 429062 429666 430070 430573 431075 431577 432079 432.580 4.33080 9.433580 434080 4,34579 435078 435676 436073 436570 437067 437663 43 8059 9.438.5.54 439048 439543 440036 440529 441022 441514 442006 442497 442988 9.443479 443968 444458 444947 445435 445923 446411 446898 447384 447870 983238 983202 60 60 9.983166 60 983130] 60 983094 983058 983022 982986 60 60 60 60 9.448366 448841 449326 449810 450294 450777 451260 451743 452225 452706 982914 982878 982842 60 60 60 9.453187 453668 4.54148 4i34628 4.55107 4.55586 456542 4.57019 457496 I C(wine | Sine 842 841 840 839 838 838 837 836 835 834 883 832 832 831 830 829 828 828 827 826 825 824 823 823 822 821 820 819 819 818 817 816 816 815 814 813 812 812 811 810 809 809 808 807 806 806 805 804 803 802 802 801 800 799 799 798 797 795 796 795 794 10.571948 671443 570938 570434 669930 669427 668925 568423 567921 567420 56G920 10.666420 665920 56.5421 564922 564424 563927 563430 562933 562437 561941 10.561446 660962 560457 559964 659471 558978 558486 557994 557503 567012 10.6.56521 556032 655542 555053 554565 654077 553589 553102 552616 552130 10.551644 551159 560674 650190 549706 549323 548740 548257 547775 647294 iO.. 5468 13 546332 545852 545372 544893 514414 643930 54.3458 542981 542504 i Tail?. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 M. I li Degrees. MV '1 4f 34 (16 Degrees.) a tatile op logarithmic M. Sine I). Cosine D- TnuR. I D. I Cot:ii!(j ( 1 2 3 4 6 6 7 8 9 10 II ly 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.440338 440778 441218 441658 442096 442535 442973 443410 443847 444284 444720 9.445155 445590 446025 446459 446893 447326 447759 448191 448623 44 9054 9.449485 449915 450345 450775 451204 451632 452060 452488 452915 453342 .453768 454194 454619 455044 455469 455893 456316 456739 457162 457584 .458006 458427 45S848 459268 459688 460108 460527 460940 461364 461782 .462199 462616 463032 463448 463864 464279 4fi4r)04 465108 465522 465935 734 733 732 731 731 730 729 728 727 727 726 725 724 723 723 722 721 720 720 719 718 717 716 716 715 714 713 713 712 711 710 710 709 708 707 707 706 705 704 704 703 702 701 701 700 699 698 698 697 696 695 .982842 982805 982769 982733 982696 982660 982624 982587 982551 982514 982477 9.982441 982404 982367 982331 982294 982257 982220 982183 982146 982109 9.982072 982035 981998 981961 981924 981886 981849 981812 981774 981737 9.981699 6 60 60 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 62 62 62 62 62 62 62 62 62 62 62 62 02 62 981662 981625 981587 981549 981512 981474 981436 981399 981361 .981323 981285 981247 98 J 209 981171 981133 981095 981057 981019 980981 695 9.980942 694 980904 693 980866 693 980827 692 980789 691 980750 690 980712 690 980673 689 980635 688 980596 63 63 63 03 63 63 6o 63 63 63 63 63 63 63 64 64 64 64 64 64 64 04 64 64 64 64 64 64 9.457496 457973 458449 458925 459400 459875 460349 460823 461297 461770 462242 .462714 463180 463658 464129 464599 465069 465539 466008 466476 466945 .467413 467880 468347 468814 469280 469746 470211 470676 471141 471005 794 793 793 792 791 790 790 789 788 788 131. 786 785 785 784 783 783 782 781 7S0 780 9.472068 472532 472995 473457 47,3919 474381 474842 475303 475763 476223 9.476683 477142 477601 478059 779 778 778 777 776 775 775 774 773 7T3 772 771 771 770 769 769 768 767 767 766 765 765 764 763 478517 763 478975 762 479432 761 479889 761 480345 760 480801 759 9.481257 759 481712 758 482167 757 482621 757 483075 756 483529 755 483982 755 484435 754 484887 753 485339 753 10.642504 542027 541.551 541075 540600 540125 639651 539177 538703 538230 6 37758 10.537286 530814 536342 .53.5871 63.5401 .534931 534461 633992 533524 5330.55 10.532587 532120 .531653 531186 530720 530254 629789 529324 528859 528395 10.527932 527468 527005 526543 526081 525619 625158 624697 524237 623 777 10.. 5233 17 522858 522399 521941 621483 521025 520568 520111 519655 51919 9 "518743 518288 617833 617379 516925 516471 616018 515565 515113 5M66I 10. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 I 351 b^ 33 32 31 30 29 '^8 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 I (Jo.~iue I Sine Cotang. Tang. M. 73 Degrees. 1 2 3 4 5 6 7 8 9 10 55 4«/ 488 56 488 5''' A DO •iOO .')8 489' 59 489. 60 489! 1 Cusin( K« (iO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 I 351 i>4 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 0111 i2 9655 11 9199 10 8743 9 8288 8 7833 7 7379 6 6925 5 L6471 4 6018 3 5565 2 15113 1 MfiBI ang. M. SINES 7iND TANGfeNTs. (l7 Dcgrees.) 35 M. >r 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 f.O 9.512642 513009 5 1 3375 513711 514107 611472 6M837 515202 615566 515930 516294 SINES AND TANOENT8. (10 DeglCea.) 1>- I THny. I 1). I Ci„^i^ a? Conine 9 516657 517020 517382 517745 518107 518468 518829 519190 619551 519911 520271 520631 520990 521349 521707 522066 522424 522781 623138 623495 .523852 524208 524564 624920 525275 625630 525984 526339 526693 527046 9.527400 527753 628105 528458 528810 629161 629513 529864 530215 530565 9.530915 531265 531614 631963 532312 632661 533009 533.?.57 533704 534052 Cosine I 612 611 611 610 609 609 608 608 607 607 606 605 605 604 604 603 a03 602 601 601 6oo 600 599 599 598 598 597 596 596 595 595 594 594 593 693 592 591 591 590 590 589 589 588 588 687 687 586 586 685 585 584 584 583 582 582 581 581 580 5m 579 578 .975670 975627 975583 975539 975496 975452 975408 975365 975321 975277 975233 9.975189 975145 975101 975057 375013 974969 974925 974880 974836 974792 9.974748 974703 974659 974614 974570 974525 974481 974436 974391 974347 9.974302 974257 974212 974167 974122 974077 974032 973987 973942 973897 9.973852 973807 973761 973716 973&n 973625 973580 973535 973489 _973444 9.973398 973352 973307 973261 973215 973169 973124 97307S 973032 972986 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 74 74 74 74 74 74 74 74 74 74 74 74 74 74 75 75 75 75 75 75 75 75 75 75 75 75 75 75 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 77 77 9.536972 637382 537792 538202 5.38611 539020 639429 639837 640245 540653 .54 1061 9.. 54 1468 541875 '4228 1 .)42688 543094 643499 643905 644310 644715 545119 9.54.5524 545928 546331 546735 547138 547540 647943 648345 548747 649149 9.649550 549951 550352 550752 5511.52 651.5.52 651952 652351 552750 563149 9.. 5,5,3548 553946 654344 554741 5,55139 555536 555933 556329 656725 557121 .5.57517 .5.57913 558308 658702 659097 5.59491 659885 5G0279 560673 561066 684 683 683 682 '82 680 680 679 679 10 I Siiio Until ngr. 678 678 677 677 676 676 675 675 674 674 673 673 672 672 671 671 670 670 669 669 .463028 462618 46220H 461798 461989 460980 460571 460163 459 A5b 459347 458939 668 668 667 667 666 666 665 665 665 664 664 663 663 662 662 661 661 660 660 659 659 6.59 658 658 657 657 6.56 656 655 655 10.4.58.532 4.58125 457719 457312 456906 4.56.501 456095 465690 4.55285 454881 10.4.54476 454072 453669 453265 452862 452460 452057 451655 461253 4.50861 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 16 14 13 12 11 10 9 8 7 6 5 4 3 2 I 10.4504,50 450049 449648 449248 448848 448448 448048 447649 447250 446851 10.4464.52 446054 445656 445259 441861 444464 444067 443671 443275 442879 10.442483 442087 441692 441298 440903 440509 440115 439721 439327 438934 Tana. M. 70 Dct'rees u ^> # ^ IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 • 56 I 3.2 |||ll /^ n I! iSi^ ~ ? ■- Ills 111= U IIIIII.6 Photographic Sciences Corporation m iV ^^^ \\ ^ <* % n> #1 9>'- 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 ¥ m >' i^.- px 1 I ■P—HjHI— ■ 38 (20 Degi ees.) A TABLE or LOGARITHMIC M. 1 Bine 1 D. ! Cosine IX 1 Tang. 1 n. CotiinR. 1 1 9.534052 578 19.972986 77 > 9.561066 6.55 10.438934 60 1 534339 577 973940 77 561459 654 438541 59 2 534745 577 972894 77 ')61351 6.54 438149 58 a £35092 577 972848 77 563244 653 437756 57 4 535438 676 972803 77 533636 653 437364 56 5 535783 576 973755 77 563028 653 436972 55 6 536129 575 972709 77 563419 652 436,581 54 7 536474 574 973663 77 563811 652 436189 53 8 536818 574 972617 77 564302 651 435798 .52 9 537103 573 973570 77 564592 651 435408 51 10 11 537507 573 973534 9.973478 77 77 564983 650 435017 50 49 9.537851 572 9..56537S 650 10.434027 12 538194 573 972431 78 565763 649 43423" 48! 13 538538 571 972385 78 5661,53 649 433847 47 14 538880 571 973338 78 566542 649 433458 46 15 539333 570 972351 78 56G932 648 433068 45 16 539565 570 973245 78 567320 648 432680 44 17 539907 569 972198 78 567709 647 432291 43 18 540249 669 973151 78 568098 647 431902 42 19 540590 568 973105 79 568486 646 431514 41 20 2T 540931 568 973058 78 78 568873 646 431127 10.430739 40 39 9.541272 567 9.973011 9.569261 645 22 541613 567 971964 78 569648 645 430353 38 23 541953 566 971917 78 570035 645 429965 37 24 542293 566 971.870 78 570433 644 429578 36 25 542632 505 971833 78 570809 644 429191 35 26 542971 565 971776 78 571195 643 428805 34 27 543310 564 971729 79 571.581 643 428419 33 28 543649 564 971682 79 .571967 642 4300J3 32 29 543987 563 971635 79 572353 642 437648 31 30 31 541325 9.544663 563 971588 79 79 573738 642 427262 30 29 562 9.971540 9.573123 641 10.426877 32 545000 562 971493 79 573.507 641 426493 28 33 545338 561 971446 79 573892 640 426108 27 34 545674 561 971398 79 574276 640 425734 26 35 646011 560 971351 79 674660 639 425340 25 36 546347 560 971303 79 575044 639 424950 24 37 546683 559 971256 79 575427 639 424573 23 38 547019 559 971308 79 575810 638 424190 22 39 547354 558 971161 79 576193 638 423807 21 40 41 547689 558 971113 9.971066 79 80 676576 637 423424 20 19 9.548024 557 9.576958 637 10.423041 42 548359 5.57 971018 80 577.341 636 422659 18 43 548693 556 970970 80 577723 636 422277 17 44 .549027 550 970933 80 578104 636 421896 16 45 549360 556 970874 80 578486 635 421514 15 46 549693 555 970837 80 578867 635 421133 14 47 550036 554 970779 80 579348 634 420752 13 48 550359 554 970731 80 579629 634 420371 12 49 550692 553 970683 80 580009 634 419991 11 50 51 551024 553 970635 80 80 580389 63,: 633 419611 10.419231 10 9 9.5513,56 552 9.970586 9.580769 52 551687 552 97'\538 80 581149 632 418851 8 53 552018 5.52 970*90 80 581528 632 418472 7 54 552349 551 970443 80 581907 632 418093 6 6b 552680 .551 970394 80 582286 631 417714 5 56 553010 550 970345 81 583665 631 417335 4 bV 553341 550 970397 81 583043 630 416957 3 58 553670 649 970349 81 583432 630 416578 2 59 554000 549 970200 81 583800 629 416200 1 ou 004339 r)4S 970152 81 584177 629 415N33 u ma (^'osiue 1 Sine I 1 Cotanp. 1 Tani'. M. 1 M.| t 1 9 1 2 3 4 6 6 7 8 9 10 11 9 13 13 14 15 16 17 18 19 20 21 22 9. |23 24 25 26 27 28 29 30 31 9. 32 33 34 35 36 37 ( 38 ( 39 t 40 t 41 9.f 42 r 43 f- 44 r- 45 f 46 R t^ 47 .'i 48 ,'■ 49 5 50 5 51 9.5 52 5 53 5 54 5 55 5 56 5 57 5 58 5 59 5 60 5 1 Co 09 Det^rtes. inp. 1 1 38934 60 38541 59 38149 58 37756 o7 37364 56 36972 55 30581 54 36189 53 i5798 S2 35408 51 35017 50 49 34027 3423" 48 !j 33847 47 33458 46 J3008 45 J26S0 44 }229i 43 U902 42 U514 41 J1127 40 i0739 39 10352 38 •9965 37 59578 36 19191 35 J8805 34 !8419 33 S80J3 32 !7648 31 J7202 30 ,6877 29 !6493 28 ,6108 27 ,5724 26 ,5340 25 ,4950 24 4573 23 4190 22 3807 21 3424 20 3041 19 2059 18 2277 17 1896 16 1514 15 1133 14 0752 13 0371 12 9991 11 9611 10 9231 9 8851 8 8472 7 8093 B 7714 5 7335 4 6957 3 6578 2 6200 1 5823 u ns- M. SINES AND M. TANGENTS. (21 Degrees.; 39 Sine D. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 50 57 58 59 60 9.554329 554658 554987 655315 555643 555971 556299 550626 556953 557280 557006 Conine | D. | Tnng. | D. 9 557932 558258 558583 558909 559234 659558 559883 560207 660531 _560855 .561178 561501 561824 562146 562468 562790 .563112 563433 563755 56407.' 9 . 564396 .')64710 505036 66,5356 565676 565995 .560314 500032 500951 507209 9.. 50 758 7 567904 508222 508539 508850 509172 509488 509804 570120 57043S 9 ,570751 571060 571380 571095 572003 572323 572036 5729.^0 573203 6735751 548 548 647 547 546 546 545 545 644 644 643 543 543 642 542 641 .541 540 540 539 539 9.9^0152 970103 970055 970006 969957 969909 969860 909811 969702 969714 969665 538 638 637 537 636 636 536 635 635 534 634 633 533 .532 532 531 531 531 530 _530_ 529 529 528 628 528 527 627 526 626 525 81 81 81 81 81 81 81 81 81 81 81 82 82 82 82 82 82 82 82 82 82 82 82 82 82 83 83 83 83 83 83 9.968028 83 968578 83 968528 83 Cotang. 9.969616 969507 909518 969469 969420 969370 969321 969272 969223 969173 9.969124 969075 969025 968976 968920 908877 968827 968777 908728 908078 908479 908429 908379 908^29 908278 908228 968178 525 524 524 523 523 623 522 522 521 621 83 83 83 83 83 84 84 84 84 84 84 84 84 84 84 84 84 9.908128 968078 968027 967977 967927 967876 967826 967775 967725 96 7074 9.907024 967573 Gr,7.522 907471 907421 967370 967319 967268185 907217! 85 967166185 84 84 85 85 86 85 85 .684177 584555 684932 685309 685686 586062 686439 586815 687190 687566 587941 9.588316 688091 689066 689440 589814 690188 690502 590935 .591.308 .591681 9.592054 692426 592798 693170 693542 593914 594286 694656 59502V 695398 9.59.5768 696138 696508 696878 597247 597610 597985 598354 598722 699091 9 ''199459 •99827 000194 600502 000929 601296 001002 002029 60239.-) 602761 9.603127 603493 003858 604223 604588 004953 605317 605682 000040 0004 lu 629 629 628 628 627 627 627 026 626 625 625 625 624 624 623 623 623 622 322 622 621 10.416823 415445 41.5068 414691 414314 413938 413661 4131S5 412810 412434 621 620 620 619 619 618 618 618 617 617 617 616 616 616 615 615 615 614 614 613 613 613 612 612 611 Gil 611 610 610 610 609 609 609 608 608 607 607 607 GOG 606 60 59 58 57 56 65 54 53 62 5 412059 I 60 49 48 47 46 45 44 43 42 41 40 39 38 37 36 36 34 33 32 31 30 29 28 27 26 26 24 23 22 21 20 19 18 17 16 lo 14 13 12 11 10 9 8 7 6 5 4 3 2 1 10.411684 411309 410934 410560 410186 409812 409438 409085 408692 4 0831 9 10.407946 407574 407202 406829 400458 406086 405715 405344 404973 404602 10.404232 403862 403492 403122 402753 402384 402015 401646 401278 400909 10.400541 400 17r 39980b 3994.38 399071 398704 398338 397971 397605 39723 9 10.396873 390507 390142 395777 395412 395047 394683 394318 393954 393590 m 40 (22 Degrees.; a VABliE OP LOGARITHMIC M. 1 Bine 1 D. 1 C'oaine j D. Tang. D. Coiling. •^ U 9. 573575 1 621 9.967166 86 9.606410 606 10.39.3590 60 1 673888 620 967115 85 606773 606 393227 59 2 674200 620 967064 85 607137 605 392863 58 3 574512 619 967013 85 607500 605 392500 57 4 674824 619 966961 85 607863 604 392137 56 5 575136 519 966910 85 608225 604 391775 55 6 675447 618 966859 85 608588 604 391412 54 7 575758 618 966808 85 608950 603 391050 53 8 576069 517 966756 86 609312 603 390688 52 9 576379 617 966705 86 C09674 603 390326 51 10 11 576689 616 966653 86 86 610036 9.610397 602 389964 10.389603 :,o 49 9.576999 616 9.966602 602 12 577309 516 966550 86 610759 602 389241 48 13 577618 615 966499 86 611120 601 388880 47 14 677927 615 966447 86 611480 601 388520 46 15 578236 514 966395 86 611841 00 r 388159 45 16 578545 514 966344 86 612201 600 387799 44 17 578853 513 966292 86 612561 600 387439 43 18 579162 613 966240 86 612921 600 387079 42 19 579470 513 966188 86 613281 699 386719 41 20 21 579777 9.. 580085 612 966136 86 87 613641 599 386359 10.386000 40 39 612 9 966085 9.614000 598 22 580392 511 966033 87 614359 698 385641 38 23 580699 511 965981 87 614718 598 38.5282 37 24 581005 611 965928 87 616077 697 384923 36 25 581312 510 965876 87 615435 697 384565 35 26 681618 510 965824 87 615793 597 384207 34 27 681924 509 965772 87 616151 596 383849 33 28 582229 509 965720 87 610509 596 383491 32 29 582535 509 165668 87 616867 596 383133 31 30 31 582840 508 965615 87 87 617224 595 382776 10.382418 30 29 9.583145 508 9.965563 9 617582 595 32 583449 507 965511 87 617939 595 382061 28 33 583754 507 965458 87 618295 594 381705 27 34 584058 606 96.5406 87 618652 594 381348 26 35 684361 606 965353 88 619008 694 380992 25 36 684665 "^06 965301 88 619364 593 380636 24 37 584968 605 965248 88 619721 593 380279 23 38 585272 505 965195 88 620076 593 379924 22 39 585574 604 965143 88 620432 592 379568 21 40 41 685877 604 965090 88 88 620787 592 379213 10.378858 20 19 9.. 586 179 603 9.965037 9.621142 592 42 586482 503 964984 88 621497 591 378503 18 43 686783 503 964931 88 621852 591 378148 17 44 587085 502 964879 88 622207 690 377793 16 45 687386 502 964826 88 622561 590 377439 15 46 687688 .501 964773 88 622915 590 377085 14 47 587989 501 964719 88 623269 589 376731 13 48 688289 501 964666 89 623623 589 376377 12 49 688590 500 964613 89 623976 689 376024 11 50 51 588890 9.589190 500 964560 9.964507 89 89 624330 9.624683 588 375670 10.375317 10 9 499 588 52 589489 499 964454 89 625036 588 374904 8 53 689789 499 964400 89 625388 687 374612 1-/ 1 54 590088 498 964347 89 625741 687 374259 6 55 590387 498 964294 89 626093 587 373907 5 56 590686 497 964240 89 626445 586 37355a 4 57 590984 497 964187 89 626797 586 373203 3 58 591282 497 964133 89 627149 .586 372851 2 59 591580 498 964080 89 627501 585 372499 1 60 591878 496 9G4026'89 627852 .535 372 M-! u Co .le 1 Sine C(it;iii),'. 1 Taiip. 1 M. [ M. '~0 1 2 3 4 6 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 3: 34 35 36 37 38 39 40 44i 44 45 46 47 48 49 50 61 153 I 53 54 55 56 57 58 6y 60 (iT Dcuit'ci irig. 359(r 60 3227 59 2863 58 2500 57 2137 56 1775 55 1412 54 1050 53 0688 52 0326 51 9964 :.o 9603 49 9241 48 8880 47 8520 46 8159 45 7799 44 7439 43 7079 42 3719 41 6359 40 GOOO 39 'i641 38 5282 37 4923 36 4565 35 42G7 34 3849 33 3491 32 3133 31 2776 30 3418 29 2061 28 1705 27 1348 26 0992 25 0636 24 0279 23 9924 22 9568 21 9213 20 8858 19 8503 18 8148 17 ?793 16 7439 15 7085 14 6731 13 6377 12 6024 11 5670 10 6317 9 4904 8 4612 1-/ 1 4259 6 3907 5 3555 4 3203 3 2851 2 2499 1 2M-i iifi. M. M. Sine 1 2 3 4 5 6 7 8 9 10 11 9.695T37 12^ 695432 595727 596021 596315 596609 596903 597196 597490 597783 9.591878 592176 692473 692770 693067 593363 693659 693955 594251 694547 594842 SINKS AM> ■^A^'(M;lv^s. (23 Degrees.; _l !>■ I (.'osiiic I ]i, I Tarii.' | D. 41 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 3C 34 35 36 37 38 39 40 41 ^2 43 44 45 46 47 48 49 50 496 495 495 495 494 494 493 493 493 492 492 9.964026189 C()taii)(. 9.598075 598368 598660 698952 r'9244 099536 699827 600118 600409 600700 9.600990 601280 601570 601860 602150 602439 602728 603017 603305 603594 491 491 491 490 490 489 489 489 488 488 963972 963019 903865 963811 963757 963704 963650 963596 963542 963488 9.963434 963379 963325 963271 963217 9631631 90 963108 91 89 89 90 90 90 90 90 90 90 90 90 90 90 90 90 9.627852 628203 628554 628905 629255 629606 629956 6303(]o 630656 631005 631,355 487 487 487 486 486 485 485 485 484 484 484 483 483 482 482 482 481 481 481 480 963054 962999 962945 9.962890 962836 962781 962727 962672 962617 962562 962508 962453 962398 9 61 53 53 54 55 56 57 58 6y 60 .603882 604170 604457 604745 605032 605319 605606 605892 606179 606465 9 606751 607036 607322 607607 607892 608177 608461 608745 609029 609313 480 479 479 479 478 478 478 477 477 476 9.962343 962288 962233 962178 962123 962067 962012 961957 961902 961846 9.961791 961735 961680 961624 961569 961513 961458 961402 961.346 961290 476 478 475 475 474 474 474 473 473 473 9.961235 961179 961123 961067 961011 9609.55 960899 960843 960786 960730 91 91 91 91 91 91 91 91 91 91 91 91 92 92 92 92 92 92 92 92 92 92 92 93 92 92 93 93 93 93 93 93 93 93 93 93 93 93 93 93 94 94 94 9.631704 6320.53 632401 632750 633098 633447 633795 634143 634490 634838 585 685 585 584 584 683 583 583 583 582 582 lO.. 372 148 371797 371446 371095 370745 370394 370044 369694 369344 368995 368645 9.636185 635532 635879 636226 636572 636919 637265 637611 637956 638302 9.6.38fi47 63K992 63i)337 639682 640027 640371 640716 641060 641404 641747 9.642091 642434 642777 643120 613463 643806 644148 644490 644833 645174 Cosine .645516 64.5857 646199 646540 646881 647233 647563 647903 648243 648583 Sine I 583 581 581 581 560 580 580 679 579 _57o 578' 578 678 577 577 577 577 676 576 576 10.368296 367947 367599 367250 366902 366553 366205 365857 365510 365162 10.364815 364468 364121 363774 363428 363081 362735 362389 362044 361698 575 676 676 674 674 574 673 673 673 672 10.. 36 135;, 361008 360863 360318 359973 359629 359284 60 59 58 57 66 56 64 53 62 61 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 36 34 33 33 31 30 29 38 27 26 26 34 23 358940 23 673 5^3 572 671 571 671 670 570 570 669 669 .569 569 568 568 568 667 5B''' 567 .566 368696 368353 10.357909 357566 357323 356880 356537 3.56194 3r.5852 35.5610 355168 354626 10.354484 354143 353801 353460 353119 352778 352438 21 201 19 18 17 16 16 14 13 13 11 10 9 8 7 G 5 4 3 Cdtang. 3517571 I 351417. 1 riim- fir 6& Degrees, HP 42 (24 Degrees.") a table of logarithmic M. Sino 1 D. 1 Cobine 1 D. Tang. D. Colarip. 1 1 U 9.609313 473 9.960730 94 9.648583 566 10.351417 60 1 609597 472 960674 94 648923 666 351077 59 2 609880 472 960618 94 649263 666 350737 58 3 610164 472 960561 94 649602 566 350398 57 4 610447 471 960505 94 649942 665 350058 56 6 610729 471 960448 94 6.50281 565 349719 55 6 611012 470 960392 94 650620 665 349380 54 7 611294 470 960335 94 650959 564 349041 53 8 611576 470 960279 94 661297 664 348703 52 611858 469 960222 94 661636 664 348364 51 10 11 612140 469 960165 94 95 651974 563 348026 10.347688 50 49 9.612421 469 9.960109 9.652312 663 12 612702 468 960052 95 652650 663 347350 48 13 612983 468 959995 95 652988 663 347012 47 14 613264 467 959938 95 653326 662 346674 46 15 613545 467 959882 95 653663 662 346337 45 1« 613825 467 959825 95 654000 662 346000 44 17 614105 466 959768 95 654337 661 345663 43 18 614385 466 959711 95 654674 561 345326 42 19 614665 466 959654 95 655011 661 344989 41 20 21 614944 465 959596 95 95 655348 661 344652 10.344316 40 39 9.615223 465 9.959539 9 . 655684 560 22 615502 465 959482 95 656020 560 343930 38 23 615781 464 959425 95 656356 560 343644 37 24 6 J 6060 464 959368 95 656692 659 343308 36 25 616338 464 959310 96 657028 559 342972 35 2(5 616616 463 959253 96 657364 559 342636 34 27 616894 463 959195 96 657699 559 342301 33 28 617172 462 959138 98 6580,?4 658 341966 32 29 617450 462 959081 96 658369 658 341631 31 30 31 617727 462 959023 96 96 658704 568 341296 10.340961 30 29 9.618004 461 9.958965 9.6.59039 658 32 618281 461 958908 96 659373 557 340627 28 33 618558 461 958850 96 659708 557 340292 27 34 618834 460 958792 96 660042 557 339958 26 35 619110 460 958734 96 660376 557 339624 25 36 619386 460 958677 96 660710 556 339290 24 37 619662 459 958619 96 661043 556 338957 23 38 619938 459 958561 96 661377 556 338623 22 39 620213 459 958503 97 661710 656 338290 21 40 41 62048P 458 958445 97 97 662043 655 337957 10.337624 20 19 9.620763 • 458 9.958387 9 662376 655 42 621038 457 958329 97 662709 554 337291 18 43 621313 457 958271 97 663042 554 336958 17 4-1 621587 457 958213 97 663375 654 336625 16 45 621861 456 958154 97 663707 554 336293 15 46 622135 456 958096 97 664039 663 335961 14 47 622409 456 958033 97 664371 553 335629 13 48 622682 455 957979 97 664703 553 335297 12 49 622956 455 95'!'921 97 665035 663 334965 11 60 51 623229 455 957863 97 9V 665366 9.665697 552 3346S4 10.334303 10 9 9.623502 454 9.957804 552 52 623774 454 957746 98 666029 552 333971 8 53 624047 454 957687 98 666360 551 333640 7 54 624319 453 957628 98 666691 551 .333309 6 55 624591 453 957570 98 667021 651 332979 5 56 624863 453 957511 98 667352 551 332648 4 67 625135 453 957452 98 6676S2 550 332318 3 68 625406 452 957393 98 668013 550 331987 2 59 625677 452 957335 98 668343 550 tin -I ffr »v 1 i 60 625918 451 957276 98 668672 550 33132S 1 Cosine J Sine 1 Coiling. 1 1 Tang. 1 M. 1 M. "o" 1 2 3 4 6 6 7 8 9 12 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2b 26 27 28 29 30 31 32 33 34 35 36 S7 38 39 40- 41 42 43 44 45 46 47 48 49 50 53 54 55 66 57 58 I 60 I 65 Degrees. '^. 1 1 417 60 077 59 737 58 398 57 058 56 719 55 380 54 041 53 703 52 364 51 026 50 S88 49 350 48 012 47 674 46 337 45 OOO 44 663 43 326 42 989 41 652 40 316 39 980 38 644 37 308 3G 972 35 636 34 301 33 966 32 631 31 296 30 961 29 627 28 292 27 958 26 624 25 290 24 957 23 623 22 290 21 957 20 624 19 291 18 958 17 625 16 293 15 961 14 629 13 297 12 965 11 684 10 303 9 971 8 640 7 309 6 979 5 648 4 318 3 987 2 f^rrt^f 1 i}o; i 328 iP- |M. SINES AND TANGENTS. (25 Degrees.) 43 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2b 26 27 28 29 30 31 32 33 34 35 36 S7 38 39 40 41 42 43 44 45 46 47 48 49 50 9.625948 626219 626490 626760 627030 627300 627570 627840 628109 628378 _(i28647 9.6289161 629185 629453 629721 629989 630257 630524 630792 631059 _ ^1^?^ 9.631593 631859 632125 032392 632658 632923 633189 633454 633719 633984 451 451 451 450 460 450 449 449 449 448 448 .957276) 9572171 957158 957099 957040 956981 956921 956862 956803 956744 956684 447 9 447 447 446 446 446 446 445 445 445 9.634249 6.34514 634778 635042 635306 635570 635834 636097 636360 636623 444 444 444 443 443 443 442 442 442 441 . 956625 956566 956506 956447 956387 956327 956268 956208 956148 956089 51 52 53 54 55 56 57 58 ou 60 9.636886 637148 637411 637673 637935 638197 638458 638720 638981 639242 441 440 440 440 439 439 439 438 438 438 .956029 955969 955909 955849 955789 955729 955669 955609 955548 95.5488 9819.668673 981 669002 98 669332 98| 669661 669991 670320 670649 G70977 671.306 671634 671963 9.672291 672619 672947 673274 673602 673929 674257 674584 674910 675237 .955428 955368 955307 955247 955186 955126 955065 955005 954944 95488? 9.639503 639764 640024 640284 640544 640804 641064 641.324 G4 1 584 6418421 437 437 437 437 436 436 436 435 435 435 434 434 434 433 433 433 432 432 432 431 I 9.54823 954762 954701 954640 954579 954518 954457 954396 954335 954274 .954213 954152 954090 954029 953968 953906 953845 953783 953722 9536601 ). 675564 675890 676216 676543 676869 677194 677520 677846 678171 67 8496 9.678821 679146 679471 679795 680120 680444 680768 681092 681416 681740 9.682063 682387 682710 683033 683356 683679 684001 684324 684646 68496 8 9.685290 685612 685934 686255 686577 686898 687219 687540 687861 1 688182 5.50 549 549 549 548 548 548 548 547 547 647 10 547 546 546 646 546 645 545 545 544 544 .331327 330998 330668 330339 330009 329680 329351 329023 328694 328366 10 544 644 543 543 643 643 642 542 542 642 60 59 58 67 66 55 64 63 52 61 . 328037 50 .327709 327381 327053 326726 326398 326071 325743 32.5416 325090 324763 10 641 641 641 641 540 540 640 640 639 639 ,324436 324110 323784 323457 323131 322806 322480 322154 321829 321504 10 639 639 638 638 638 538 537 537 637 537 .321179 320854 320529 320205 319880 319,556 319232 318908 318584 318260 536 536 636 636 535 635 535 534 534 10 .317937 317613 317290 316967 316644 316321 31.5999 315676 31.5354 315032 10.314710 314388 314066 313745 313423 313102 312781 oi*4DU 312139 311818 49 48 47 46 4o 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 G 5 4 3 1 64 Degrees, [IK •*' 44 (26 Degrees.) a TABLE OF LOOABITIIMIC M. Bine D. t CoHinu 1). ■IV.tIL'. 1 I) 1 C.tttirig. 9.641842 431 U . 953660 103 9.688182 634 10.311818 60 1 642101 431 953599 103 088;j)2 534 311498 59 2 042360 431 9.13537 103 688823 634 311177 68 3 642618 430 953475 103 089143 533 310857 67 4 642877 430 953413 103 689463 633 310537 66 5 643135 430 953352 103 689783 533 310217 55 6 643893 430 953290 103 690103 533 309897 54 7 643650 429 953228 103 690423 533 309577 53 8 643908 429 953166 103 690742 532 309258 52 9 644165 429 953104 103 691062 632 308938 51 10 11 044423 9.644680 428 953042 103 104 691381 9. 69 1700 632 308619 10.308300 50 49 42S 9.952980 531 1*2 644936 428 952918 104 692019 631 307981 48 13 645193 427 952855 104 692338 631 307662 47 14 645450 427 952793 104 692656 531 307344 46 15 645706 427 952731 104 692975 631 307025 45 16 645962 426 952669 104 693293 530 306707 44 17 646218 426 952606 104 693612 530 306388 43 18 646474 426 952.544 104 693930 .530 306070 42 19 646729 425 952481 104 694248 630 305752 41 20 21 646984 425 952419 9.9.52356 104 104 694566 9.694883 ,529 305434 10.305117 40 39 9 . 6-! t7240 425 529 22 6^ 7494 424 952294 104 69.5201 629 304799 38 23 6^ 17749 424 952231 104 69.5518 529 304482 37 24 6' 8004 424 9.52168 105 695836 629 304164 36 25 6^ 8258 424 952106 105 6961.53 528 303847 35 26 6^ 18512 423 9.52043 105 696470 528 303530 34 27 6' 8766 423 951980 105 696787 528 303213 33 28 G4 9020 423 951917 105 697103 528 302897 32 29 6^ 19274 422 9518.54 105 697420 627 302580 31 30 31 G-i 19527 422 951791 105 105 697736 9.6980.53 527 302264 10.301947 30 29 9.649781 422 9.951728 527 32 650034 422 951665 105 698369 627 301631 28 33 650287 421 951602 105 698685 .526 301315 27 34 650539 421 951.539 105 699001 626 300999 26 35 650792 421 951476 105 699316 526 300684 25 36 651044 420 951412 105 699632 626 30036b 24 37 651297 420 951349 106 699947 .526 300053 23 38 651549 420 951286 106 700263 625 299737 22 39 651800 419 951222 106 700578 525 299422 21 40 41 652052 9.652304 419 951159 106 106 700893 9.701208 625 299107 20 19 419 9.951096 524 10.298792 42 652555 418 951032 106 701.523 524 298477 18 43 652806 418 9.50968 106 7018.37 524 298163 17 44 653057 418 950905 106 702152 524 297848 16 45 653308 418 950841 106 702466 .524 297534 15 46 653558 417 9,50778 106 702780 52. 297220 14 47 653808 417 950714 106 70.3095 623 296905 13 48 G54059 417 950650 106 703409 523 296.591 12 49 654309 416 950586 106 703723 523 296277 11 50 51 654558 9.654808 416 950522 107 107 704036 9.7043,50 623 295964 10.29.5650 10 9 416 9.9.504,58 522 52 655058 416 950394 107 704663 522 295337 8 53 655307 415 9.50330 107 704977 522 295023 7 54 655556 415 950266 107 '^05290 522 294710 6 55 655805 415 950202 107 705603 521 294397 5 56 656054 414 9.501.38 107 705916 621 294084 4 57 656302 414 950074 107 706228 ,521 293772 3 58 656551 414 950010 10'^ 706541 521 293459 2 CO 658799 413 949945 10? 706854 S'- 1 60 657047 413 949881 107 707166 520 2928.34 _! Cosine Sine 1 Oota.ig. 1 I'ang. 1 M. J 63 Degrees. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 191 20 21 22 23 24 25 26 27 28 29 30 31" 32 33 34 35 36 37 38 39 »40 41 42 43 44 45 46 47 48 49 60 51 9. 52 53 54 65 56 57 n» t 59 ( 60 ( ) C( «■ 318 60 li»8 59 177 58 if>7 57 537 56 il7 55 397 54 577 53 ^58 52 ^38 51 619 50 100 49 981 48 662 47 344 46 1)25 45 707 44 188 43 [)70 42 752 41 434 40 117 39 799 38 482 37 164 36 847 35 530 34 213 33 897 32 580 31 264 30 947 29 631 28 315 27 999 26 684 25 368 24 053 23 737 22 422 21 107 20 792 19 477 18 163 17 848 16 534 15 220 14 905 13 591 12 277 11 964 10 650 9 337 8 023 7 710 6 397 5 084 4 772 3 459 2 146 1 834 ?• 1 M. SINES AND TANOENTa. (27 Degrees.) 4.1 01 p. 657047 1 657295 657542 657790 f>^8037 658284 658531 658778 659025 659271 659517 9.659763 660009 6602.55 660501 660746 660991 661236 661481 661726 661970 .662214 662459 662703 662946 663190 663433 663677 663920 664163 664406 .664648 664891 665133 665375 ^>65617 oG.5859 666100 666342 666583 666824 9.667065 667305 667546 667786 668027 668267 6(38506 668746 668986 669225 9.66'9464 669703 669942 670181 670419 670658 670896 413 413 412 4)2 412 412 411 411 411 410 410 410 409 409 409 409 408 408 408 407 407 9.949881 U49816 949752 949688 949623 949558 949494 949429 949364 949300 949235 407 407 406 406 406 405 405 405 405 404 404 404 403 403 403 402 402 402 <102 401 398 398 398 397 397 397 897 671372.1 396 60 1 671609 396 401 401 401 400 400 400 399 399 399 399 y. 949 170 949105 949040 948975 948910 948845 948780 948715 948650 94 8584 9.948519 948454 948388 948323 948257 948192 948126 948060 947995 947929 9.947863 947797 947731 947665 947600 947533 947467 947401 947335 947269 107)9.707166 1071 707478 707790 708102 708414 708726 709037 709349 709660 709971 710282 107 108 108 108 108 108 108 108 108 108 108 108 108 108 108 109 109 109 109 109 109 109 109 109 109 109 109 110 110 9.710593 710904 711216 711.'^25 711836 712146 712456 712766 713076 713386 9 947203 947136 947070 947004 946937 946871 946804 946738 946671 946604 110 110 110 110 110 110 110 110 110 110 9.713696 714005 714314 714624 714933 715242 715551 715860 716168 716477 9.946538 946471 946404 946337 946270 946203 946136 S46069 946002 945935 110 111 111 HI 111 111 111 111 Ul m 111 111 111 111 112 112 112 112 112 112 9.716785 717093 717401 717709 718017 718325 718633 718940 719248 719555 9.719862 72C169 720476 720783 721089 721396 721702 722009, 722315 722621 I Cosine 1 9.722927 723232 723538 723844 724149 724454 724759 725066 725369 725674 520 520 520 520 619 519 519 619 519 518 518 518 618 618 547 517 617 517 616 516 516 10.2U28;M|f3u 292522 69 292210 ftfl 291898 57 291586 66 291274 65 290963 54 290651 53 290340 52 290029 61 289718 .50 516 516 615 4:5 6lN 615 614 614 514 514 614 613 613 613 613 613 612 612 512 612 612 511 511 511 511 511 510 510 610 510 510 509 509 509 509 509 608 808 508 608 10.289407 289096 288785 288475 288164 287854 287544 287234 286924 286614 10.286.304 285995 285686 285376 285067 284758 284449 284140 283832 2S3523 10.28.3215 282907 282^599 282291 281983 281670 281367 281080 280752 280446 10.280138 279831 279524 279217 278911 278604 278298 277991 277685 277379 10.277073 276768 276462 276156 27.5851 276646 275241 274835 : 7463 11 2743261 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 »5 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 I 5 4 3 2 I \ 62 Degrees. \r^ s 46 (20 Degrees.; a TABLK of LOOAniTIIMTO 1 ■11 M. 1 Kino 1 '»■ I UoHhU! 1 U. 1 Taiiu. 1 l>. ("..taiii;. 1 1 9.671009 396 "97945935 112 9.725674 608 10.274323 I5(r 1 671847 395 945868 112 725979 508 274021 59 2 672084 395 945800 112 7262H4 507 273716 5« n 672321 395 945733 112 726588 507 273412 57 4 672558 395 945666 112 726892 507 273108 56 5 672795 394 945598 112 727197 507 2?2803 55 f) 673032 394 945531 112 727501 507 272499 54 7 673268 304 945464 113 727805 506 272195 53 8 673505 394 945396 113 728109 506 271891 52 9 673741 393 945328 113 728412 506 2V1588 51 10 ifi 673977 393 945261 9.945193 113 113 728716 506 271284 50 49 9.674213 393 9.729020 506 10.270980 12 674448 392 945125 113 729323 505 270677 48 i;{ 674ti«'i 392 945058 113 729626 505 270374 47 14 674919 392 944990 113 729929 505 270071 46 15 675155 392 944922 113 730233 505 269767 45 If) 675390 391 944854 113 730.')35 505 269465 44 17 675624 391 944786 113 730838 504 269162 43 18 675859 391 944718 113 731141 504 268859 42 19 676094 391 944650 113 731444 504 268556 41 20 21 676328 390 944582 114 114 731746 504 268254 40 39 9.6765(52 390 9.944514 9.732048 504 10.267952 22 676796 390 944446 114 732351 503 267649 38 23 6T7030 390 944377 114 732653 503 267347 37 24 677264 389 944309 114 732955 503 267045 36 25 677498 389 944241 114 733257 503 266743 35 26 677731 389 944172 114 733558 503 266442 34 27 677964 388 944104 114 733860 502 266140 33 28 678197 388 944036 114 734162 502 265838 32 29 678430 388 9:3967 114 734463 502 265537 31 30 31 678683 388 943899 114 114 734764 502 265236 10.264934 30 29 9 678895 387 9.943830 9.735066 502 32 679128 387 943761 114 735367 502 264833 28 33 679360 387 943693 115 735668 501 264332 27 34 679592 387 943624 115 735969 501 264031 26 35 679824 386 943555 115 736269 501 263731 25 36 680056 386 943486 115 736570 501 263430 24 37 680288 386 943417 115 736871 501 263129 23 38 680519 385 943348 115 737171 500 262829 22 39 6S0750 385 943279 115 737471 500 262529 21 40 41 680982 385 943210 115 115 737771 500 262229 20 19 9.681213 385 9.943141 9.738071 500 10.261929 42 681443 384 943072 115 738371 500 261629 18 43 681674 384 943003 115 738671 499 261329 17 44 681905 384 942S34 115 738971 499 261029 16 4ft 682135 384 942864 115 739271 499 260729 15 46 682365 383 942795 116 739570 499 260430 14 47 682595 383 942726 116 739870 499 260130 13 48 682825 383 942656 116 740169 499 159831 12 49 683055 383 942587 116 740468 498 259532 11 50 51 683284 382 942517 116 116 740767 498 259233 10 9 9.683514 382 9.942448 9.741066 498 10.258934 02 683743 382 942378 116 741365 498 258635 8 53 683972 382 942308 116 741664 498 258336 7 54 684201 381 942239 116 741962 497 258038 6 55 684430 381 942169 116 742261 497 257739 5 56 684658 381 942099 116 742559 497 257441 4 57 684887 380 942029 116 742858 497 2.57142 3 58 685115 380 941959 116 743156 497 256844 2 09 GS5343 38U 941889 117 743454 49/ 496 256540 1 60 685571 380 .. 941819 117 743752 256248 1 30i!J 3248 g- SINES AND TANOENTS. (20 Degrees.) 47 .685571 685799 68602V 08C254 686482 686709 686936 687163 6S7389 687616 687843 . 688069 688295 68*<5«1 688747 688972 689198 689423 689648 689873 690098 .690323 690548 690772 690996 691220 691444 691668 691892 692115 692339 .692.562 692785 693008 693231 693453 693676 693898 694120 694342 694564 380 379 379 379 37« 378 378 378 378 377 _?77_ 3/7 377 376 376 376 376 375 375 375 _375 374 374 374 374 373 373 373 373 372 372 9.W18I9 ■Ml 74 9 941679 941609 941539 941409 941398 941328 9412.581 941187 941 U7 9.941046 940975 940905 940834 940763 9406931 940622 940.551 940480 940409 9 J. 694786 6950071 6952291 695450 695671 69.5892 696113 696334 6965.54 696775 .696995 697215 697435 6976.54 697874 69S094 698313 698532 6987.51 698970! 372 371 371 371 371 370 370 370 370 369 .940.338 940267 940196 940125 940054 939982 939911 939840 939768 939697 9.713752 744050 741348 744645 744943 745240 745538 745835 746132 746429 746726 9.747023 747319 747616- 747913 748209 748505 748801 749097 749393 749689 496 496 496 496 496 496 495 495 495 495 495 10.256248160 255950 59 , 25.5652 58 255:i55 57 255057 254760 2.54402 £.54165 25.3868 253.571 253274 56 65 Ui .531 52 i 51 6«i 369 369 369 368 368 368 368 367 367 367 9.939625 939554 939482 939410 939339 939267 939185 939123 939052 938980 9.749985 750281 750576 750872 751167 751462 751757 752052 752347 752642 367 366 366 366 366 365 365 op- 365 364 9.938008 938836 9.38763 938691 938619 938547 938475 938402 9.38330 __938258 9.9.38185 9.38113 938040 937967 937895 937822 937749 ucs/fivOi 937604 9375311 9.752937 753231 753520 753820 7.54115 754409 754703 7.54997 7.55291 755585 9.7,55878 756172 766465 756759 757052 757345 757638 7579:^ 1 758224 758517 493 492 492 492 492 492 ■1 4l>. 490 490 490 490 490 490 489 10.2.50015 249719 249424 249128 218P33 248538 ^ .243 ^7948 "653 '58 39 38 37 36 36 34 33 32 31 30 •3 29 9.758810 759102 759395 759687 759979 760272 760564 760856 761148 761439 489 489 489 489 489 488 488 488 488 488 .1 1 2'l^ucJ5 24.5591 246297 245003 244709 244415 10 488 487 487 487 487 487 487 486 486 486 244122 243828 243535 243241 242948 242655 24236S 242069 241776 241483 10 .241190 240898 240605 240313 r'i 48 (30 Degrees.) a table op logahithmic M. I Hiim I I). 3 4 5 6 V 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 38 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 159 [60 (.'odillR TaiiR. f). 9.698970 099 1 89 699107 699626 699844 700062 700280 700498 700716 700933 701161 9.701368 701585 701802 702019 7022:16 7024.'i2 702669 702885 703101 _7033_17 9.703533 703749 TK)3y64 704 179 7C4395 704610 704825 705040 705254 _ 705469 9 705683 705898 706112 706326 706539 706753 706967 707180 707393 707606 9.707819 708032 708245 708458 708670 708882 709094 709306 709518 709730 9 7099-il 710153 710364 710575 710786 710997 711208 711419 *y I 1 con s 1 i \T ,g J 711839 364 364 364 364 363 363 363 363 363 362 36S 363 362 36] 361 ,%1 361 360 360 360 360 359 359 359 359 359 358 358 858 358 357 357 357 357 356 356 356 356 355 355 355 355 354 354 354 354 353 353 353 353 353 352 352 352 351 351 351 351 350 9.937531 937458 937385 937312 937238 937135 937092 937019 936946 936872 936799 9,936725 936652 936578 936505 936431 936357 936284 936210 936136 __r "6062 9.93598b 9359 1'i 935840 935766 935692 935618 935543 935469 935395 93 5320 9.935246 935171 935097 935022 934 48 934873 934798 934723 9.'«4649 934574 9.934499 934424 934349 934274 934199 934123 934048 933973 933898 933822 9.933 '7 933/1 933596 933520 933445 933369 933293 933217 nnn ^ A t 933066 21 22 22 22 22 22 22 22 22 22 22 i,2 23 23 23 23 23 23 23 23 23 23 23 23 24 24 24 24 24 24 24 24 24 24 24 24 24 25 25 25 25 25 25 25 25 25 25 25 25 26 26 26 26 26 26 26 26 26 26 26 (-((laiit' r 9.761439 486 761731 486 762023 480 762314 486 762606 485 762897 485 763188 485 763479 485 763770 485 764061 485 764352 484 9.764643 484 764933 484 765224 484 765514 484 76580r^ 484 766095 484 766335 483 766675 483 766965 483 767255 483 9.767545 483 767834 483 768124 482 763413 482 768703 482 768992 482 769281 482 769570 482 769860 481 770143 481 9.770437 481 770726 481 771015 481 771303 481 771592 481 771880 480 772168 480 772457 480 772745 480 773033 480 9.773321 480 773608 479 773896 479 774184 479 774471 479 774759 479 775046 479 775333 479 775621 478 775908 478 9.776V05 478 776482 478 776769 478 777055 478 777342 478 777628 477 777915 477 778201 477 1 1 cid 1 'I i 1 778774 477 10 238.561160 238269 59 237977 237686 237394 237103 236812 23652! 236230 235939 235648 10.235357 235067 234776 234486 234195 233905 233615 233325 233035 232745 10.232455 232166 231876 231587 231297 231008 230719 23043 J 230140 229852 10.229563 229274 228985 228697 228408 228120 227832 227543 227255 226967 10.226679 226392 226104 225816 225529 225241 224954 224667 224379 224092 10.223805 223518 223231 222945 222658 222372 222085 221799 221226 Cosine j Sine I I Cotang. I 58 57 56 55 54 63 52 51 50 49 48 47 4b 45 44 43 42 41 40 39 38 37 38 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 i I Thiin- I M. uQ £)'^(eui 1 2 3 4 5 8 7 8 9 ]0 11 12 13 14 15 16 17 18 |l9 I 20 ^21 122 iV3 24 TiO 26 27 28 129 30 'i^ 32 33 34 35 36 37 38 139 40 41 9.' 42 43 44 45 7 40 7 47 7 48 7 49 7 50 7 152 '53 5.; 55 .'J6 57 58 59 I 601 lit' n S5«nj60| S2()9 59 7977 58 768fi 57 ■/•39-1 56 7103 55 1812 54 j52! 53 ?230 52 VJ39 51 J241 14 1954 13 1667 12 t379 11 1092 10 3805 9 3518 8 3231 7 2945 6 2658 6 2372 4 2085 3 1799 2 ibl)i I 1226 M. Hire 9 7^18391 7120501 71226C* 712469 712679] 712889 713090 71330] 713517 713726 __713935 9.714144| 7143521 714561 714769 714978 715186 715394 715602 715809 716017 ). 71 6224 716432 716639 716646 717053 717259 717466 717673 71 7879 718085 1^1—L. !i!i:i!i« I «>. I Tmll'. I 348 347 347 347 347 347 346 346 346 346 '9.9330661 126 93J!')90| 127 9329141 127 932838! 127 33i:7f)'> 127 93268.) 93*i609 932533 932457 932380 932304 ''.718291 7i0497 718703 718909 719114 719.320 719525 719730 719935 720140 345 345 345 345 345 344 344 344 344 343 9.9.32228 9,32151 y32075 931998 931921 931845 931768 931691 931614 931.537 9.J3146 931.383 931306 931229 9311.52 931075 93099S 930921 930843 930766 127 127 128 128 128 128 128 138 128 128 9.720345 720.549 720754 7; 9958 721162 721366 721570 721774 721978 72!3181 9.930688 930611 930533 930466 9.S0378 930300 930223 930145 930067 929989 x28 !28 128 129 129 129 129 129 129 129 .778774 779060 779346 779632 779910 7R020? 7804891 7N0775 781060' 7813461 __78163l| 9.781916 78',520l 782486 782771 783056 783341 783626 783910 784.95 784479 9.72238.'>l 722Ji88| 722791 722994 723197 723400 723603 723805 7240071 724210 1 Cosine I 9.929911 929833 929755 929677 929599 929521 929442 929364 929286 929207 9.929129 929050 928972 928893 928815 928736 928657 928578 t\i>OAf\n 928420 9.784764 7850481 785332 78.5016 785900 786184 786463 786752 7870361 78 73191 9.78760' 787886 788170 788453 788736 789019 789302 7895851 789868 _790151 1.790433 790716 790999 7912811 791563 791846 792128 792410 792692 792974 10. 221 22BnKr 220940| 59 8206.'i4 58 !i203f)8 57 82008r> 66 2197971 f,o 219511 - 219225 218940 2186,54 2]83^ 10.218084 217799 217514 217229 216944 216659 216374 210090 21. -1805 215.521 40 1.215236 214952 214668 214384 214100 21.3816 213.')32 213248 212'J64 212681 9.793256 793538| 793819 794101 794383' 794664! 794945I 795227] 795,008! 7957891 .212397 212114 211830 211,547 211264 210981 210698 210415 210132 209849 10.209567 209284 209001 208719 298437 208154 207872 ., 207590 12 207308 II 207026 1 10 10.206744 206462 206181 205899 205617 205336 205055 2O4773I 2044921 204311 UJ^ no (32 I)c<;frces.) a tablk of LofiAHiTHMic M. Sii:e I). ('osiiiH I I). I Tiing. [ P. CoiniiR. 1 2 3 4 5 6 7 8 1) 10 11 12 1:3 14 15 16 17 \H 19 20 21 22 2:3 24 25 2fi 27 28 20 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 4f] 47 48 49 50 51 52 53 54 55 56 57 58 59 60 .724210 724412 724014 ""■24810 725017 725219 725420 725622 725823 726024 726225 ,726426 72»)626 726827 727027 727228 727428 727628 727828 728027 _728227 .728427 728626 7^28825 729024 729223 729422 72962 729820 730018 730216 .73041.5 730613 7308 1 1 731009 731206 731404 731602 731799 731996 732193 9.732390 732587 732784 732980 733177 733373 733569 733765 733961 7341.57 9.7.?4353 734549 734744 734939 735135 735330 73552 735719 735914 736109 :«37 337 336 336 336 336 335 335 335 335 334 334 334 334 334 333 333 833 333 333 332 332 332 332 331 331 331 331 330 330 330 330 330 329 329 329 329 329 328 328 328 328 328 327 327 327 327 327 326 326 9 . 928120 928342 928263 928183 928104 928025 927946 927867 927787 927708 92762t> 9.927549 927470 927390 927310 92723 1 92TI51 927071 926991 926911 92683 1 9.92'6751 926671 926591 926511 926431 926351 926270 926190 926110 926029 1321 132 132 132 132 132 132 132 132 132 132 132 133 133 133 133 133 133 133 133 1.33 326 326 325 325 325 325 r ■.. C 324 324 324 9.925949 925868 925788 925707 92.5626 925545 925465 925384 925303 92522 2 9. 925141 925060 924979 924897 924816 924735 924654 924572 924491 9244(t9 133 133 133 134 134 134 134 134 134 134 134 134 134 134 134 135 135 135 135 135 1.35 135 135 135 135 136 136 136 136 136 .795789 796070 796351 796632 796913 797194 797475 797755 798036 798316 798596 9.798877 799157 799437 799717 799997 800277 800557 800836 801116 801396 .801675 801955 802234 802513 802792 803072 803351 803630 803908 804187 468 468 468 468 468 468 468 468 4f\'T 467 j467 467" 467 467 467 460 466 466 466 466 466 9.804466 804745 80.5023 805302 805580 805859 806137 806415 8066J3 806971 9 9.924328 924246 924104 924083 924001 923919 92375.5 923673 923591 1.36 136 136 136 136 I3r. I'M. 137 137 137 ,807249 807.527 807805 808083 808361 808638 808916 809193 809471 809748 9.810025 810302 810.580 810857 811134 811410 811687 81196. 812241 812517 466 466 465 465 465 465 465 465 465 465 464 464 464 464 464 464 464 463 463 463 463 463 463 463 463 402 462 462 462 462 462 462 462 462 461 461 461 461 461 461 10.204211 203930 203649 203368 203087 202806 202525 202245 201964 201684 201404 10.201123 200843 200563 200283 200003 195)723 199443 199164 198884 198604 10.198325 198045 197766 197487 197208 196928 196649 196370 196092 195813 10.195534 19.5255 194977 194698 194420 194141 193863 193.585 193307 193029 10.192751 192473 192195 191917 191639 191362 191084 190807 190529 190252 10.189975 189698 189420 189143 188866 188590 188313 ; 88036 187759 187483 60 59 58 57 56 55 54 .53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 .3 4 fi 7 8 9 10 I u 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4f 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 nS 59 60 Cosine Sine t Cotansf. 'J'ang. M. S7 Decrees. HiR. 1 1211 60 M)30 59 3649 58 13fiH 57 5087 56 i80«) 55 252.') 54 2245 53 PJfil 52 l()8'l 51 1404 50 1123 49 )843 48 )5f.3 47 ^283 46 1)003 45 .(723 44 9443 43 9104 42 8884 41 8604 40 8325 39 8045 38 7760 37 7487 36 7208 35 6928 34 6649 33 6370 32 6092 31 5813 30 5534 29 5255 28 4977 27 4698 28 4420 25 4141 24 3863 23 3585 22 13307 21 >3029 20 12751 19 12473 18 )2195 17 )1917 16 )1639 15 )1362 14 H084 13 )0807 12 )0529 11 )0252 10 •!9975 9 ^9698 8 39420 7 39143 6 38860 5 88590 4 88313 3 S803fc ) 2 87755: 1 1 874 8r 1 ang. 1 M. m SIXrs AND TA.VOKNT8. (33 DegrPGS.) fil 9.736109 736303 736498 736692 736886 737080 737274 737467 737661 737855 _72H0J8 '738241 738434 738627 738820 739013 739206 7393;J8 739590 739783 _739975 .740167 740359 740550 740742 740934 741125 741316 741508 741699 741889 9 . 742080 742271 742462 742652 742842 743033 743223 743413 743602 743792 9.743982 744171 7443G1 744550 744739 744928 745117 745306 745494 745683 9 .745871 746059 746248 746430 746624 746812 746999 747187 747374I 7475621 324 324 324 323 323 323 323 323 322 322 322 322 322 321 321 321 321 321 .320 320 320 320 320 319 319 319 319 319 318 318 318 318 318 317 317 317 317 317 316 316 316 316 316 31.5 315 315 315 315 314 314 314 .']14 314 313 313 313 313 313 312 313 312 y. 923591 923509 923427 923345 92.'}26;i 923181 923098 923016 922933 922851 _9227J58 9.922686 922603 922.'>20 922438 922355 922272 922189 922106 922023 921940 9.921857 921774 921691 921607 921.524 921441 921357 921274 921190 9 21107 9.921023 920939 920856 920772 920688 920604 920520 920436 920352 920268 9.920184 920099 920015 919931 919846 919762 919677 919.593 919.508 919424 9.9L9339 919254 919169 919085 919000 918915 918830 918745] 918659 918574 .812517 812794 y 13070 813347 813623 813899 814175 8144.52 814728 81. -3 004 _ 815279 9.81.5555 81.5831 8i6107 816.382 8166.58 816933 817209 817484 817759 8J8035 9.818310 818.585 818860 81913; 819410 819684 819959 820234 820.508 820783 9 821057 821332 821606 821880 8221.54 822429 82270 822977 823250 _823524 .823798 824072 824345 824619 824893 825166 825439 825713 825986 826259 lAO 9.826532 826805 827078 827351 827624 827897 828170 828442 828715 828987 10.187482 187206 186930 186653 186.377 186101 185825 18.5.548 185272 184996 184721 10 ,184445 184169 183893 183618 183342 183067 182791 182516 182241 181965 i0.18Ki90 181415 181140 180865 180590 180316 180041 179766 179492 179217 457 10 457 178943 178668 178394 178120 177846 177571 177297 177023 176750 176476 . 176202 175928 17.5655 17.5381 175107 174834 174.561 174287 174014 17.3741 .173468 173195 172922 172649 172376 172103 171330 1715.58 2 171285 1 1710131 60 59 58 57 56 55 64 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 » 14 13 12 11 iO 9 8 7 6 Til I 111. M. 56 D'!gre«;s. 52 (34 Degrees.) a TABLE OP LOGARITHMIC "mT Sine D. 1 Cosine | D. 1 Tang. 1 D 1 Cotaiig. 1 1 9.747562 312 9.918574 142 9.828987 454 10.171013 60 1 747749 312 918489 142 829260 454 170740 59 2 747936 312 918404 142 829532 454 170468 58 3 748123 311 918318 142 829805 454 170195 57 4 748310 311 9182.33 142 830077 464 169^^23 66 5 748497 311 918147 142 830349 453 169(351 56 6 748683 311 918062 142 830621 453 169379 54 7 748870 311 917976 143 830893 453 169107 53 8 749056 310 917891 143 831165 453 168835 52 9 749243 310 917805 143 831437 453 168563 51 10 11 749429 310 917719 143 143 831709 9.631981 453 453 168291 50 49 9.749615 310 9.917634 10.168019 12 749801 310 917548 143 832253 453 167747 48 13 749987 309 917462 143 832525 453 167475 47 14 750172 309 917376 143 832796 453 167204 46 15 750358 309 917290 143 833068 452 166932 45 16 750543 309 CI 7204 143 833339 452 16G661 44 17 750729 309 917118 144 833611 452 166389 43 18 750914 308 917032 144 833882 452 166118 42 19 751099 308 916946 144 834164 452 165846 ^.1 20 21 751284 308 308 ■ 916859 144 144 834425 462 165575 10.165304 40 39 9.751469 9.916773 9.834696 452 22 751654 308 916687 144 834967 452 165033 38 23 751839 308 916600 144 835238 452 164762 37 24 7^2023 307 916514 144 835509 452 164491 36 25 752208 307 916427 144 835780 451 164220 35 26 752392 307 916341 144 836051 451 163949 34 27 752576 307 916254 144 836322 451 163678 33 28 752760 307 916167 145 836593 451 163407 32 29 752944 306 916081 145 836864 451 163136 31 30 31 7.53128 9 753312 306 915994 145 145 837134 451 162866 30 29 306 9.915907 9.837405 451 10.162595 32 753495 306 915820 145 837675 451 162325 28 33 753679 306 915733 145 837946 451 162054 27 34 753862 305 915646 145 838216 451 161784 26 35 754046 305 915559 145 838487 450 161513 25 36 754229 305 915472 145 838757 450 161243 24 37 754412 305 915385 145 839027 450 160973 23 38 754595 305 9152P7 145 839297 450 160703 22 39 754778 304 915210 145 839568 450 160432 21 40 41 754960 304 915123 146 146 839838 9.840108 450 160162 20 19 9.755143 304 9.915035 450 10.159892 42 755326 304 914948 146 840378 450 159622 18 43 755508 304 914860 146 840647 450 159353 17 44 755690 304 914773 146 840917 449 159083 16 45 755872 303 914685 146 841187 449 158813 15 46 756054 303 914698 146 841467 449 158543 14 47 756236 303 914510 146 841728 449 158274 13 48 756418 303 914422 146 841996 449 158004 12 49 766600 303 914334 146 842266 449 157734 11 50 51 756782 302 914246 147 147 842535 449 157465 10 9 9.756963 302 9.914158 9.842805 449 10.1,57195 62 757144 302 914070 147 843074 440 156926 8 53 757326 302 913982 147 843343 449 156657 7 54 757507 302 913894 147 843612 449 156388 6 55 757688 301 913806 147 843882 448 1.56118 6 56 757869 301 913718 147 844151 448 15.5849 4 57 758050 301 913630 147 844420 448 155580 3 KO •vcQoon om 147 aAAdan AAQ '559,1 I o rf' 7 ? ?,-\.' vjrf «-?VF 1 •w'-r-i'-f* ■•^^ TT-J? 59 758411 301 913453 147 844958 448 155042 1 60 758591 301 913365 147 845227 448 154773 CoKirie 1 1 Sine 1 1 Coiang. Tang. M j 11 12 13 14 15 16 1 17 18 19 20 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 142 43 44 45 46 47 48 49 50 51 52 53 54 55 57 58 59 60 55 Degrees. ■Jllg. 1 013 60 )740 59 »468 58 (195 57 K>23 56 (iol 55 1379 54 1107 53 1835 52 663 51 1291 50 iOl9 49 '747 48 '475 47 '204 46 .932 45 .661 44 .389 43 (lis 42 .846 '^.1 .575 40 .304 39 .033 38 1762 37 t491 36 1220 35 1949 34 1678 33 1407 32 tl36 31 !866 30 ,595 29 !325 28 !054 27 784 26 513 25 243 24 »973 23 »703 22 »432 21 1162 20 1892 19 1622 18 1353 17 1083 16 !813 15 >543 14 !274 13 !004 12 '734 11 r465 10 '195 9 1926 8 .657 7 .388 6 ill8 6 .849 4 .580 3 .Ol 1 o .042 1 t773 8- M SINES AND TANGEWT9. (35 Degrees.) 9.768591 758772 758962 759132 759312 759492 759672 759852 760031 760211 __760390 9.760569 760748 760927 761106 761286 761464 17 761642 i« 761821 761999 _ 762177 9.762356 762534 762712 •^62889 763067 763245 763422 763000 763777 763954 TaiiE. .913365 913276 913187 913099 913010 912922 912833 912744 912655 912566 912477 D, 9.912388 912299 912210 912121 912031 911942 911853 911763 911674 911584 9.845227 845496 845764 846033 846302 846570 846839 847107 847376 847644 847913 Cotaiig. 9.764131 764308 764485 764662 764838 765015 76519J 765367 766644 785720 >. 766896" 766072 766247 766423 766698 766774 766949 767124 767300 767475 9.911496 911405 911316 911226 911136 911046 910956 910866 910776 91068 6 9.910596 910506 910415 910325 910235 910144 910054 909963 909873 909782 9.848181 848449 848717 848986 849264 849622 849790 860058 850325 860593 448 448 448 448 448 447 447 447 447 447 447 9.860861 861129 851396 851664 861931 852199 862466 852733 853001 853268 447 447 447 447 447 447 446 446 446 446 10. J 54773 154604 154236 163967 1636S8 153430 153161 162893 152624 152366 152087 1.767649 767824 767999, 768173 768348 7686971 768871 769045 9.909691 909601 909510 909419 909328 909237 9091^^6 909056 908964 908873 9.863635 863802 864069 854336 854603 854870 855137 865404 866671 855938 446 446 446 446 446 446 446 445 445 445 9.908781 908690 908699 908507 908416 908324 908233 908141 908049 907958 151 9.866204 8.56471 856737 857004 857270 857637 867803 1S2| 858069 152 858336 152| 86 8602 162 162 152 152 153 163 445 445 445 445 445 445 445 445 444 444 9.858868 8.59134 859400 859666 85993'^- ---, 86019^ 163 86046 1 153| 860730 860995 861261 444 444 444 444 444 444 444 444 444 443 60 59 58 57 66 65 54 63 62 51 ^ 60 10.151819 49 151551 48 151283 47 151014 46 150746 150478 160210 149942 149675 149407 10.149139 148871 148604 148336 148069 147801 147534 147267 146999 146732 10.146465 146198 145931 145664 145397 146130 144863 144596 144329 144062 10 443 443 443 443 4/ia 443 44? 443 443 443 143796 143529 143263 142996 142730 142463 142197 141931 141664 141398 10.141132 140866 140600 140334 139802 139536 139270 139005 138739 1!» I 54 (3G Degrees.) a TIBLE OF L06AU1T1IMIC M.l Sine D. fosine 1 D. | 'Pnnff. 1 D. 1 Ci'ianp. 1 1 9.769219 290 9.9079.581 153 9.861261 443 10.1.38739 60 1 769393 289 907866 153 861527 443 138473 59 2 ' 769566 S189 907774 153 861702 442 138208 58 3 769740 289 907682 153 862058 442 137942 J>7 4 769913 289 907590 153 862323 442 137677 56 5 770087 289 907498 153 862589 442 137411 55 6 770260 288 907406 153 862854 442 137146 54 7 770433 288 9073 M 154 863119 442 136881 53 R 770606 288 907222 154 863385 442 1,36615 52 9 770779 288 907129 154 863650 442 136.3.50 51 10 1 1 770952 288 288 907037 154 154 863915 442 136085 50 49 9.771125 9 906945 9.864180 442 10.135820 !2 771298 287 906852 154 864445 442 1355.55 48 13 771470 287 906760 154 864710 442 13.5290 4'/ 14 771643 287 906667 154 864975 441 13.5025 46 1ft 771815 287 906575 154 865240 441 134760 4.'d 16 771987 287 906482 154 865505 441 134495 44 17 77'>.159 287 906389 155 865770 441 134230 43 18 772331 286 906296 155 866035 441 133965 42 19 772503 286 906204 155 866300 441 133700 41 20 21 772675 286 906111 1.55 155 866564 9.866829 441 133436 40 39 9.772847 286 9.906018 441 10.133171 22 773018 286 905925 155 867094 441 132900 38 23 773190 286 90.5832 155 867358 441 132642 3V 24 773361 285 305739 155 867623 441 132377 3b 25 773533 285 905645 155 867887 441 132113 3b 26 773704 285 905552 1.55 868152 440 131848 34 27 773875 285 905459 1,55 868416 440 131584 33 2S 774046 285 905366 156 868680 440 131320 32 29 774217 285 905272 156 868945 440 131055 31 30 31 774388 284 905179 156 1.56 869209 440 130791 30 29 9.774558 284 9.905085 9.869473 440 10.130.527 32 774729 284 904992 1.56 869737 440 130263 28 33 774899 284 904898 156 870001 440 129999 2/ 34 775070 284 904804 1.56 870265 440 129735 2b 35 775240 284 904711 1.56 870529 440 129471 2b 36 775410 283 90*617 156 870793 440 129207 24 37 3S 775580 775750 283 283 904523 904429 1.56 157 871057 871321 440 440 128943 128679 23 22 39 775920 283 904335 157 C71.585 440 128415 21 40 41 776090 283 904241 157 1.57 871849 439 439 128151 20 19 9.776259 283 9.904147 9.872112 10.127888 42 776429 282 904053 1.57 872376 439 127624 18 43 776598 282 903959 1.57 872640 439 127360 17 44 776768 282 903864 1.57 872903 439 127097 lb 45 776937 282 903770 1.57 873167 439 126833 lb 46 777106 282 903676 1.57 873430 439 126570 14 47 777275 281 903581 1.57 873694 439 126306 13 48 777444 281 903487 1.57 873957 439 126043 12 49 777613 281 903392 158 87422C 439 12578C 11 50 51 777781 281 1 281 90329*= 1.58 1 158 874484 9.874747 439 125516 10 1 9 9.77795C 9.90320? ' 439 10.1252.5S 52 778 11 f » 281 903106 t 1.5f! 8750 K t 439 12499( ) 8 53 778281 r 280 9030 H [ 15S 87527J i 438 12472T 7 54 77845{ ) 280 9029H i 1.5S ! 87553( ) 438 12446' 1 6 55 77862'i 1 280 90282^ 1 1.5(- ) 87580( ) 438 12420( ) h i)(i 77879'. i 280 902721 ) 1.5^ I 438 12393' 1 4 57 778961 ) 280 90263-^ 1 1.55 I 87632( 5 438 12367^ 1 3 58 779 1 2J i 280 902535 ) 15i ) 87658! ) 438 12341 1 2 5M 77929. i 279 90244' t 1.55 ) 87685 1 438 12314< :> 1 60 77946. 5 279 90234! ) 1.5( ) 87711- 1 438 1 12288 b 1 Cosine 1 t Sine 1 1 Cotiini;. 1 j Tauf!. 1 M. i •Jl)e« rc'CS. .\I J__ 9. 1 2 3 4 5 6 7 8 9 10 11 9. 12 13 14 15 16 17 18 19 20 21 9. 22 23 24 25 26 27 » 28 29 r 30 r 31 9.' 32 33 c 34 35 36 .• ( 37 "i 38 7 39 7 40 7 4i 9.7 42; 7 43 7 44 7 45 7 46 7 47 7 48 7 49 7 50 7 51 9.7 52 7 53 7 54 7 55 7 'jfi 7 57 7 is 7 i9 7 ( :;o 7! 1 (Jo. ^ 3 39 60 1 73 59 08 58 42 57 .77 56 m 55 46 54 !81 53 515 52 150 51 )85 50 i-ZO 49 355 48 J90 47 )25 46 reo 45 195 44 J30 43 )65 42 700 41 im 40 171 39 300 38 B42 37 377 36 113 35 348 34 584 33 320 32 055 31 791 30 527 29 263 28 999 27 735 26 471 25 207 24 943 23 679 22 415 21 151 20 888 19 624 18 360 17 097 16 .833 15 >570 14 1306 13 5043 12 )780 11 J516 10 1253 9 199(1 8 1727 ' 7 146'] [ 6 420C ) 5 ^93' r A 367^ 1 3 341 2 3141 ) 1 288( i 11^. 1 M. SINES AND TANGENTS. \^S7 Dcgraes.) 65 M. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2G 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 5S 59 60 Sine 9.779463 779631 779798 779966 780133 780300 780467 780634 780S01 780968 781134 .781301 781468 781634 781800 781966 782132 782298 782464 782630 _782796 .782961 783127 783292 783458 783623 783788 783953 784118 784282 784447 .784612 784776 784941 785105 785269 785433 785597 785761 785925 786089 . 786252 78G416 786579 786742 786906 787069 787232 787395 787557 7S7720 9.787883 788045 783208 78S370 788532 7.'^^694 788856 789018, 789180 7893421 T>. Cosine D. 'r-int;. n. 279 279 279 279 279 278 278 27S 278 278 278 277 277 277 i;r7 277 277 276 276 276 276 27'4 276 275 275 275 275 275 275 274 274 274 274 274 274 273 273 273 273 273 273 272 272 273 272 272 272 271 271 271 _271_ 271 271 271 270 270 270 270 270 270 269 9.902349 902253 902158 902063 901967 901872 901776 901681 901585 901490 901394 Cotiiiin. .901298 901202 901106 901010 900914 900818 900722 900fS26 900529 900433 .900337 900240 900144 900047 899951 899854 899757 899660 899564 899467 9.899370 899273 899176 899078 898981 898884 898787 898689 898592 898494 9.898397 898299 898202 898104 898006 897908 897810 897712 897614 897516 O.S97"418 897:i20 897222 897123 897025 896926 896828 896729 896631 896532 .877114 877377 877640 877903 878165 878428 878691 878953 879216 879478 879741 1.880003 880265 880528 880790 881052 881314 881576 881839 882101 882363 .882625 882887 883148 883410 883672 f 83934 ^'84196 8.H457 8S4719 88'i.4 434 434 434 434 112145 111884 111623 111361 II 1100 110840 110579 110318 1 10057 109796 10.109535 109275 109014 108753 108493 108232 107972 107711 107151 107190 Toil''. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 30 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 I 52 Degieus 5« (3fi Degrees.) a table or logarithmic ^^B M.l Bine 1 0. 1 (.'osine 1 D. Tang. | D. Cotanp. M 1 iB 9.789342 269 9.896532 164 9.892810 434 10. 1071901 80 1 ^^^■t 1 789504 269 896433 165 893070 434 106930 69 9 V, 789665 269 896335 165 893331 434 106669 68 1 I^K' ^ 789827 269 896236 105 893591 434 106409 67 2 3 4 6 6 7 8 9 10 1 |«'i 4 789988 269 896137 165 893851 434 106149 56 5 790149 269 896038 165 894111 434 105889 55 6 790310 268 895939 165 894371 434 105629 54 7 790471 268 895840 165 894632 433 10.5368 53 8 790632 268 895741 165 894892 433 105108 62 ■^Hi q 790793 268 895641 165 895152 433 104848 61 H • 10 790954 268 895542 9.895443 165 166 895412 433 104588 50 49 48 ^B IT 9.791115 268 9.895672 433 10 104328 11 9.) 12 ( 13 f 14 ^ ^H 12 791275 267 895343 166 895932 433 104068 '■ 791436 267 895244 166 896192 433 103808 47 ^^H 14 791596 267 895145 166 896452 433 103.548 4b ^H 15 791757 267 895045 166 896712 433 103288 4b J I IB 791917 267 894945 166 896971 433 103029 44 lu c 16 6 17 g 18 8 17 792077 267 894846 166 897231 433 102769 43 18 792237 266 894746 166 897491 433 102509 42 19 792397 266 894646 166 897751 4.33 102249 41 20 ?,1 792557 266 894546 9.894446 166 167 898010 433 101990 10,101730 40 39 I 2 n a 8 9.792716 266 9.898270 433 1 r\ t> 9,^ 792876 266 894346 167 898530 433 101470 38 'il J>.C5 22 8 23 8 ?,3 79Q035 266 894246 167 898789 433 101211 3/ 24 793195 265 894146 167 899049 432 100951 36 ?,5 793354 265 894046 167 899308 432 100692 36 O c 1 <-i 86 793514 265 893946 167 899568 4a2 100432 34 26 8 27 8 28 8 on o 27 793673 265 893846 167 899827 432 100173 33 28 793832 265 893745 167 000086 432 099914 32 ifl^^i 29 793991 265 893645 167 900346 432 099654 31 30 81 794150 264 893544 167 168 900605 432 099395 30 29 3 3 3; 3; 3^ 3f 3( 3^ 3f 3£ 4C 41 42 43 44 4*1 D 8 jSH 9.794308 264 9.893444 9.900864 432 10.099136 1 9.8( 32 794467 264 893343 168 901124 432 098876 28 33 794626 264 893243 168 901383 432 098617 2/ ^ ai 34 794784 264 893142 168 901642 432 098358 2b 3.'i 794942 264 893041 168 901901 432 098099 2o 1 o/ 36 795101 264 892940 168 902160 432 097840 24 37 795259 263 892839 168 902419 432 097581 23 1 or 38 795417 263 892739 168 902679 432 097.321 22 1 fit 39 79557C 263 892638 168 902938 432 097062 21 1 Of\ 40 41 795733 263 892536 168 169 903197 9.903455 431 431 096803 10.096545 20 19 ' oil 80 9.795891 263 9.892435 9.80 an 42 796049 263 892334 169 903714 431 096286 18 43 796206 263 892233 169 903973 431 096027 17 fi/t 44 790364 262 892132 169 904232 431 095768 16 fin ^^Hl 45 796521 262 892030 169 904491 431 095509 16 fin ^^Ht ^^ 796679 262 891929 169 904750 431 095250 14 46 fin ^^^^H i 47 796836 262 891827 169 905008 431 094992 13 47 fin 48 796993 262 891726 169 905267 431 094733 12 48 49 50 51 fin \ i. 49 797150 261 891624 169 905526 431 094474 11 fin ■ .1^ 50 51 797307 9.797464 261 891523 9.891421 170 170 905784 9.906043 431 094216 10 9 80 £ .^? 261 431 10.093957 9.80 fin -, * 52 797621 261 891319 170 906302 431 09369S 8 52 53 797777 261 891217 170 906560 431 093440 V 53 80 An aH 54 79793'!: 261 891115 170 906819 431 093181 6 54 JHj 55 798091 261 891013 170 907077 431 092923 5 55 80 60 80' Hn' SB 261 890911 170 90733P 431 092664 4 ;5fi j^i 57 79840:i 260 890803 170 907594 431 092406 3 57 ^H ' 58 798560 260 89070? 170 907852 431 092148 2 58 ^H 59 798716 260 89060f 170 908111 430 091889 1 69 act' |H 60 798872 260 89050:3 170 908369 430 091631 60 80^ |l. 1 Cosine 1 1 Sine 1 1 Cotang. 1 j 'I'ang. M. . Cosi 51 Dcgrsea. 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 74 11 16 10 57 9 9S 8 40 7 81 6 23 5 64 4. t06 3 48 8 89 1 31 |M. n SINES AND TANGENTS. (39 Degrees,) 57 Sine. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 9 .798872 7IJ9028 799184 799339 799495 799651 799806 799962 800117 8002721 800427 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.800582 800737 800892 801047 801201 801366 801511 801665 801819 801973 3.802128 802282 802436 802589 802743 802897 803050 803204 803357 803511 9.803664 803817 803970 804123 804276 804428 804581 804734 804886 80 5039 9.805191 805343 805495 805647 805799 805951 806103 806254 806406 806557 9.806709 806860 807011 807163 807314 607465 807615 807766 807917 808067 260 260 260 259 259 259 259 259 259 258 258 9.8905031 170 890400 171 258 258 258 258 258 257 257 257 257 267^ 267 266 256 256 256 256 256 266 256 255 690298 890195 890093 889990 889888 889785 889682 889579 889477 9.889374 889271 889168 889064 888961 888858 888755 888651 888648 888444 171 171 171 171 171 171 171 171 171 255 265 255 256 254 264 254 254 254 254 9.888341 888237 888134 888030 887926 887822 887718 887614 887510 887406 172 172 172 172 172 172 172 172 173 9.9083691 908628 908886 909144 909402 909660 909918 910177 910435 910693 910961 172 9.911209 9.887302 887198 887093 886989 886885 886780 886676 886571 886466 886362 173 173 173 173 173 173 173 173 174 911467 911724 911982 912240 912498 912766 913014 913271 913529 173 9.913787 254 253 253 253 253 253 253 253 262 25a 9.886257 886152 886047 886942 885837 885732 885627 885522 885416 885311 174 174 174 174 174 174 174 174 174 175 914044 914302 914560 914817 916076 91.5332 915590 915847 916104 Cosine 9.885205 885100 884994 884889 884783 884677 884572 884466 884360 884254 sine i 176 175 175 175 175 175 176 176 176 176 176 176 1761 176 17R 176 176 176 176 177 9.916362 916619 916877 917134 917.391 917648 917905 918163 918420 918677 9 ,918934 919191 919448 919706 919962 920219 920476 920733 920990 921247 1.921503 921760 922017 922274 922530 922787 923044 923300 923557 923813 Cotang. 430 430 430 430 430 430 430 430 430 430 430 10.01,16311 60, 0913721 59 091114 090856 090598 090340 090082 089823 089665 089307 089049 430 430 430 430 430 430 430 429 429 429 429 429 429 429 429 429 429 429 429 429 429 429 429 429 429 429 429 428 428 428 428 428 428 428 428 428 428 428 428 428 428 428 428 428 A no 428 428 428 427 427 10.088791 088533 088276 088018 087760 087502 087244 086986 086729 086471 10.086213 085956 085698 086440 085183 084925 084668 084410 084153 083896 10.083638 083381 083123 082866 082609 082352 082096 081837 081680 081.323 10.081066 080809 080552 080295 080038 079781 079524 079267 079010 078753 10.078497 078240 077983 077726 077470 077213 076956 076700 076443 076187 58 57 56 56 54 53 512 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Tang. IM. 59 D«gree«i 68 (40 Degrees.') a table op logarithmic M.| B'.ne I). Costne 1 I) 1 Tung. 1 I>. 1 Coliintt. 1 j 9.808067 251 9.884254] 177| 9.923813 427 10.076187 ()0 1 808218 251 884148 177 924070 427 073930 39 3 808368 251 884042 177 924327 427 073673 38 3 808319 2.30 883936 177 924383 427 073417 37 4 808669 230 883829 177 924840 427 073160 36 5 808819 230 883723 177 92.'J096 427 074901 33 6 808969 250 883617 177 923352 427 074648 34 7 809119 250 883510 177 923609 4'57 074391 53 8 809269 230 883404 177 923865 427 074133 32 9 809419 249 883297 178 926122 427 073878 31 10 11 809569 249 883191 9.883084 178 178 926378 9.926634 427 073622 10.073366 30 'To 9.809718 249 427 12 809868 249 882977 178 926890 427 073110 48 13 810017 249 882871 178 927147 427 072833 47 14 810167 249 882764 178 927403 427 072397 46 15 810316 248 882657 178 927659 427 07234 1 43 16 810463 248 882330 178 927915 427 072083 44 17 810614 fJ48 882443 178 928171 427 071829 43 18 81076'j 248 882336 179 928427 427 071373 42 19 810912 248 882229 179 928683 427 071317 41 20 21 811061 248 882121 9.882014 179 179 928940 9.929196 427 071060 10.070801 40 39 9.811210 248 427 22 811338 247 881907 179 929432 427 070348 38 23 811507 247 881799 179 929708 427 070292 37 24 811633 247 881692 179 929964 426 070036 .36 25 811804 247 881384 179 930220 426 069780 33 26 811932 247 881477 179 930473 426 069323 34 27 812100 247 881369 179 930731 426 069269 33 28 812248 247 881261 180 930987 426 069013 32 29 812396 246 881153 180 931243 426 068V37 31 30 31 812544 246 881046 180 180 931499 9.931735 426 426 068301 10.068215 30 29 9.812692 246 9.880938 32 812840 240 880830 180 93'^,010 426 06/990 28 33 812988 246 8807'^r2 180 932266 426 067734 27 34 813135 246 880613 180 932522 428 067478 26 35 813283 246 880305 180 932778 426 067222 23 36 813430 245 880397 180 933033 426 066967 24 37 813578 245 880289 181 933289 420 066711 23 38 813725 245 880180 181 933545 426 066433 22 39 813872 245 880072 181 933800 426 066200 21 40 41 814019 9.814166 245 879963 9.879853 181 181 934056 9.9.M311 426 426 063944 20 19 1 245 10.0636H'J 42 814313 245 879746 181 934367 426 063433 18 43 814460 i 244 879637 181 934823 426 063177 17 44 814607 1 244 879329 181 93.3078 426 064922 16 45 814753 244 879420 181 933333 426 064667 15 46 814900 ' 244 879311 181 933389 426 064411 14 47 815046 : 244 879202 182 935844 426 004136 13 48 815193 ; 244 879093 182 936100 426 063900 12 49 815339 , 244 878984 182 936353 426 063643 11 30 51 81 5485 9.815631 243 878873 182 182 936610 9.936866 426 063390 10.0631.34 10 9 i 243 9.878766 423 52 813778 243 878656 ; 182 937121 425 062879 8 53 813924 243 878547 182 937376 425 062624 7 54 816069 243 878438 182 937632 425 06236H (] 53 81621fi 243 878328 ; 182 937887 425 062113 5 50 HlGoOi 1 243 878219 ; 183 938142 423 061H5H 4 37 81630? i 242 878109 : 183 938398 423 061602 3 58 816632 , 242 877999 183 938633 423 061347 2 59 81679S 1 242 877890: 183 93H908 42,5 06I0!»2 1 60 81694? t! 242 877780 183 939163 1 425 0608:57 j Cohirie I 1 Sine 1 1 Coiaiig. 1 1 Tani,'. 1 M. 49 Uegreva M. 1 2 8 4 6 6 7 8 U U) II 12 13 14 15 16 17 !8 19 2() 21 22 23 24 2!) 26 27 28 29 30 31 32 33 34 33 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 f)\ 32 33 34 53 56 82 57 82 58 82 59 82 00 82 1 1 i7 HO <(l f)5) r.) M 17 ru )() M >.i f):) tH M H M Jf) r)2 78 51 f)0 ')(') '10 10 4H ')H 47 )7 '!<) 11 'If) htf) 41 !i«» 4:j 73 4 a 17 41 60 40 01 •A'J AH :w !»-2 :r<' 'ir. 34 CO 33 i:{ 32 r)7 31 01 30 If) 20 00 28 ;m 27 78 20 TZ 25 (57 24 11 23 r)5 22 00 21 44 20 8'J 10 :j:) 18 77 17 aa 1(5 i\7 15 11 14 5(5 13 00 12 45 11 00 10 34 U 70 8 24 7 f)H (5 13 5 5H 4 02 3 47 2 !»2 1 37 |MT SlWrS AND TANOKNTf. ^41 DogrOCS.) BO HIlKt 0.81(5043 81708N HI 7233 HI 7370 HI 7524 HI 7(5(58 817813 817058 818103 818247 ^818302 0.818530 818(581 818H25 HI 80(51) H10II3 8 1 0257 8 1 0101 HI 054 5 810(580 HI 0832 „i__'L„J <'"Hiim i l». I TaiiH. ( p | 0.81007(5 820120 H202(53 820400 820550 820(503 H2083(5 820070 821122 _82I2(55 31 9.827407 8215.50 821(503 821835 821077 822120 8222(52 822404 822.54(5 822088 .822830' 822072 823114 8232.55 823307 823530 823(580 82.3821 8230(53 824104 0.8 24 24 .5 1 8243801 8245271 824(5(58 824808' 8240401 825000 8252301 82.5371 j 8255111 a4a 242 242 242 241 241 241 241 241 241 241 240 240 2-10 210 240 240 240 230 230 _ ?''^'^ '2.30 230 230 230 238 238 238 238 238 __238 2.38^ 2.38 237 2.37 237 237 237 237 237 23(5 230 23(5 23(5 230 2.3(5 23r» 235 235 235 235 23.5' 235 235 234 0.87? 780 877(570 8775(50 H77450 8773'|(t H7 7230 877120 877010 87(58!M> 87«78<) 87(5(578 0.87(55(58 87(5457 870347 87023(5 87(5125 87(50 M 875004 87570.3 875082 875571 0.87.5'150 87534 H 875237 875120 87.50 M 874003 87470! H74080 874508 __874450 0.874344 874232 874121 874000 873800 873784 873(172 873500 87.3448 87.3335 183 183 183 183 183 IH'l 184 184 184 184 IS4 ]84 184 184 185 185 185 185 185 185 185 l85 185 185 180 180 180 180 180 180 180 18(5 187 187 187 187 187 187 187 187 187 'J 87.3223 187 873110 188 872008 188 872885 188 872772 188 872050 188 872547 188 872434 188 872321 188 872208 188 234 234 234 234 234 0.872005 87 1 08 1 87I8()H 871755 87!fM! 871528 871414 871.301 871187 871073 180 180 180 180 IHO 189 180 180 180 190 0.0301(53 030418 030(573 030028 040183 040438 040(504 W40040 04 1 201 0414.58 041714 0.041008 042223 042478 042733 042088 04324.3 043408 043752 044007 0442(52 li. 0145 1 7 044771 iM5020 045281 045535 04!, "0 04(5045 04(5200 040554 040808 ». 04 7003 047318 047572 047820 048081 04833(1 948500 048844 040090 040353 .04'0«07 0108(12 9.50 no 9.50370 950025 950870 9511.33 95 1 388 951042 _951890 9.0.52150 952405 952059 ii529l3J 953 If;?; 9,5342)1 953075! 9.530291 9.54183 9.544371 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 425 424 424 424 424 424 424 424 424 424 424 424 424 424 424 424 424 424 424 _424 424 424 424 424 424 424 424 424 424 424 424 424 424 424 423 423 423 423 423 423 • '(ll(IM((. '0.000837 0(50582 000327 0(1007 0508 1 7 05il5(l2 050300 05905 1 05879(1 05H542 05828(1 10.0.'i8(»32 057777 057522 0572(1/ «)570I2 05(5757 05(5502 050248 055993 055738 i 0.055483 055229 051074 051710 0.544(15 054210 053955 0.53701 05341(1 053 1 92 10.052937 052(582 052428 052174 051910 0510041 24 (10 59 58 57 50 55 54 53 52 51 50 40 48 47 40 45 44 43 42 41 40 39 .38 37 3(5 35 34 33 32 31 .30 29 28 27 20 25 051410 051150 0.5(»90 1 050047 T0.05?)i«9.3 05(»i.38 019884 049030 049375 049121 0488(57 048012 048358 _ 048104 l(). 04 7850 047595 047.341 047(;87 040833 040579 010325 04007) 0458 1 7 0455031 t'tiiiji. I 23 22 21 20 )9 )8 17 10 15 14 13 12 II 10 9 8 7 6 4 3 2 1 hi. 4« U<-greeN. eo (42 Degrees.) a table of LOGARirHMio M. 1 8iMe I). 1 (,'o»in(i 1 I). TniiL'. 1 D. 1 Colnnc. 1 1 9.825511 234 9.871073 190 9.9.54437 423 10.045563 60 1 825051 233 870960 190 9.54691 423 045309 59 2 82.5791 233 870846 190 9.54945 423 045055 58 3 82.5931 233 870732 J 90 955200 423 044800 57 4 826071 233 870618 190 95.54.54 423 044546 56 5 826211 233 870504 190 955707 423 044293 55 S 826351 233 870390 190 955961 423 044039 54 7 F26491 233 870276 190 9.56215 423 043785 53 8 826631 233 870161 190 956469 423 043.531 5?, 9 826770 232 870047 191 956723 423 043277 51 10 11 826910 232 869933 191 191 956977 423 043023 60 49 9.827049 232 9.809818 9.957231 423 10.042769 12 827189 232 869704 191 957485 423 042515 48 13 82732ft 23i 869.589 191 957739 423 042261 47 U 827467 232 869474 191 957993 423 042007 46 15 827606 232 869360 191 958246 423 0417.54 45 16 827745 232 869245 191 958.500 423 041.500 44 17 827884 231 8691.30 191 958754 423 041246 43 18 828023 231 869015 192 959008 423 040992 42 19 8'>8162 231 868900 192 959262 423 040738 41 20 21 828301 9.828439 231 868785 192 192 9.59516 423 040484 40 39 231 9.868670 9.9.59769 423 10.040231 22 928578 231 868555 192 960023 423 039977 38 23 828716 231 868440 192 960277 423 039723 37 24 8288.55 230 868324 192 960531 423 039469 36 2f) 828993 230 868209 192 960784 423 039216 35 26 829131 230 868093 192 961038 428 038962 34 27 829269 230 867978 193 961291 423 038709 33 28 829407 230 867862 193 961545 423 038455 32 29 829.545 230 867747 193 961799 423 038201 31 30 31 829683 230 867631 193 193 962052 423 037948 10.037694 30 29 9.829821 229 9.867515 9.962306 423 32 829959 229 867399 193 963560 423 037440 28 33 830097 229 867283 193 962813 423 0.37187 27 34 830234 229 867167 193 963067 423 036933 26 35 830372 229 867051 193 963320 423 036680 25 36 830.509 229 866935 194 963574 423 036426 24 37 830646 229 866819 194 963827 423 036173 23 38 830784 229 866703 194 964081 423 035919 22 39 830921 228 866586 194 9643.35 423 035665 21 40 41 831058 228 866470 194 194 964.588 422 035412 10.0351.58 20 19 9.831195 228 9.866353 9.964842 422 42 831332 228 866237 194 965095 422 0.34905 18 43 831469 228 866120 194 965349 422 034651 17 44 831606 228 866004 195 965602 422 U34398 16 45 831742 228 865887 195 965855 422 034145 15 46 831879 228 86o770 195 966109 422 0.33891 14 47 832015 227 865653 195 966362 422 033638 13 48 832152 227 865536 195 966616 422 033384 12 49 832288 227 865419 195 966869 422 033131 11 50 51 832425 227 865302 195 195 967123 422 032877 10 9 9.832561 227 9.865185 9.967376 422 JO. 032624 52 832697 227 86.5068 195 967629 422 032371 8 53 832833 227 864950 195 967883 422 032117 7 54 832969 226 864833 196 968136 422 031864 6 55 833105 226 ^64716 196 968.389 422 031611 5 833241 226 864598 196 968643 422 4 57 8.33377 226 864481' "6j 068896 422 031104 3 68 833512 226 86436; , 969149 422 030861 2 59 833648 226 864245 969403 422 030597 1 60 833783 226 864127' X 969656 422 030344 _J Cosine 1 Sine 1 1 Cotang. 1 Taiig. 1 M. 1 M. 47 Dsgrees. 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 40 47 48 84 84 49 84 50 84 61 9.84 62 84 53 84 54 84 55 84 56 84 57 84 58 84 59 84 60 84 1 Cosi ■ 1 1 'i63 60 109 59 055 68 '^00 57 51(5 56 J!)3 55 ):}9 54 785 53 331 52 277 51 )23 60 rog 49 515 48 261 47 J()7 46 ('54 45 JOO 44 M6 43 )93 42 r.is 41 184 40 J3l 39 )77 38 r23 37 K)9 36 >16 35 J62 34 mo 33 r55 32 401 31 M8 30 594 29 140 28 87 27 )33 26 580 25 126 24 73 23 )19 22 565 21 U2 20 58 19 )05 18 551 17 J98 16 45 15 <91 14 538 13 584 12 31 11 577 10 524 9 571 8 117 7 mi 6 511 5 «<-~ j;ji '1 04 3 551 2 597 1 544 1 M. "NESANDTANOENTS. (43 DcjrreCS.) 61 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 9.833783 833919 834054 834189 834325 834460 834595 834730 834865 834999 835134 9.835269 835403 835538 835672 835807 83594 1 836075 836209 836343 836477 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 40 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.836611 836745 836878 837012 837146 837279 837412 837546 837679 837812 9.837945 838078 838211 838344 838477 838610 838742 838875 839007 839140 9.839272 839404 839536 839668 839800 839932 840064 840196 840328 840459 9.840591 840722 84085* 840985 841116 841247 841378 841509 841640 841771 226 225 226 225 225 225 225 225 225 224 224 224 224 224 224 224 224 223 223 223 223 9.864127 864010 863892 863774 863650 863538 863419 863301 863183 863064 862946 19619.969656 223 223 223 222 223 222 223 223 223 222 9.862827 862709 862590 862471 862353 862234 862115 861996 861877 861758 9.861638 861619 ^61400 861280 861161 861041 860922 860802 860682 860562 196 197 197 197 197 197 197 197 197 198 198 198 198 198 198 198 198 198 198 199 969909 970162 970416 970669 970922 971175 971429 971682 971935 972188 222 221 221 221 231 221 221 221 221 220 220 220 220 220 220 220 219 219 219 219 9.860442 860322 860202 860082 859962 859842 859721 859601 859480 859360 219 219 219 219 318 218 318 218 218 218 9.859239 859119 858998 858877 858756 858635 858514 858393 858272 858151 .858029 857908 857786 857665 857543 857422 857300 857178 857056 856934 199 199 199 199 199 199 199 199 200 200 200 200 200 200 200 200 201 201 201 201 201 201 201 201 202 202 202 202 202 202 9.972441 972694 972948 973201 973454 973707 973960 974213 974466 974719 9.974973 975226 975479 975732 975983 976238 976491 976744 976997 977250 9.977503 977756 978009 978262 978515 978768 979021 979274 979527 979780 202 202 202 203 203 203 203 203 203 203 9.980033 980286 980538 980791 981044 981297 981550 981803 982056 982309 9.982662 982814 983067 983320 983573 984079 984331 984584 984837 423 422 423 433 433 433 422 422 423 423 422 423 422 422 422 422 423 423 433 423 422 422 422 422 422 422 422 422 423 432 422 423 422 422 422 422 422 422 422 423 423 433 433 423 431 431 431 431 431 431 431 421 421 421 421 421 421 421 421 421 10.030344 U30091 029838 029584 03933 1 029078 02HH25 028571 028318 028065 027812 10.027559 027.306 027052 026799 026546 026293 026040 025787 025534 025281 10.025027 024774 024621 024268 024015 023762 023609 023256 023003 022750 10.022497 022244 021991 021738 021485 021232 020979 020726 020473 020220 10.019967 019714 019462 019209 018966 018703 018450 018197 017944 017691 10.017438 017186 016933 016680 016427 016174 015S21 016669 0164161 I 0161631 60 69 58 67 56 56 54 63 63 61 60 49 48 47 46 45 44 43 42 41 40 39 38 37 36 36 34 33 33 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 IG 15 14 13 12 11 10 46 Degrees. i 6S hi, i« Si. (44 Degrees.) a table of LooARtTiirnc M.| Hiiifi 1 "• 1 Ccmine i>. j T;mi?. i n. 1 ('(llMIIS 1 9.841771 218 9.8.56934 203 9.98 '.837 421 10.015163 6U I 841902 218 856812 •..M)3 985090 421 014910 59 2 842033 218 850690 204 985343 421 0146.57 58 3 842163 217 856568 204 985696 421 014404 57 4 842294 217 856446 204 985848 421 0141.52 56 » 842424 217 856323 204 986101 421 013899 55 6 842555 217 8.56201 204 986354 421 013646 54 7 842685 217 856078 204 986607 421 013393 53 8 842S15 217 855956 204 986860 421 013140 52 9 842!)4<) 217 855H33 204 987112 421 012888 51 10 ll 843076 9.vS43206 217 216 855711 205 205 987305 421 012635 50 49 9.855.''j88 9.987618 421 10.012382 12 843336 216 85.5465 205 987871 421 012129 48 13 843466 216 855342 205 988123 421 011877 47 14 843595 216 855219 205 988376 421 011624 46 1ft 843725 216 855096 205 988629 421 011371 45 in 843855 216 854973; 205 1 988882 421 011118 44 17 843984 216 854850 205 989134 421 010866 43 18 844114 215 8.54727 206 989387 4.1 010613 42 19 844243 216 854603 206 989640 421 010360 41 20 21 844372 9.844502 215 854480 9.854356 206 206 989893 421 010107 40 39 215 9.990145 421 10.009855 22 844631 215 154233 206 990398 421 009602 38 23 S44760 215 854109 206 990651 421 009349 37 21 844889 215 8i;3980 206 990903 421 009097 36 2r) 845018 215 853862 206 991156 421 008844 35 26 845147 215 853738 206 991409 421 008591 34 27 845276 214 85.3614 207 991662 421 008338 33 28 845405 214 853490 207 991914 421 008086 32 20 845533 214 853366 207 992167 421 007833 31 30 31 845662 214 853242 207 207 992420 421 007580 30 29 9.845790 214 9.853118 9.992672 421 10 0G7328 32 845919 214 8.52994 207 992925 421 007075 28 33 846047 214 8.52869 207 993178 421 006822 27 34 846175 214 852745 207 993430 421 006570 26 35 8*6304 214 852620 207 993683 421 006317 25 3fi 846432 213 852496 208 993936 421 006064 24 37 846560 213 8.52371 208 994189 421 005811 23 38 846688 213 852247 208 994441 421 005559 22 39 84C316 213 852122 208 994694 421 005306 21 40 41 846944 213 851997 9.851872 208 208 9S»4947 421 005053 20 19 9.847071 213 9. 95199 421 10.004801 42 84T"09 1 213 6 ^747 208 995452 421 004548 18 43 8't,,: ' ?13 851622 208 995705 421 004295 17 44 847iti'. ■ 2.2 851' f»7 209 995957 421 004043 16 4f> P,47;;82 1 2i2 -^513/2 209 996210 421 003790 15 4R 84770;* 212 851246 209 996463 421 003537 14 47 847836 212 851121 209 996715 421 003285 13 48 847964 212 8.50996 209 996968 421 003032 12 49 818091 212 850870 209 997221 n 002779 11 50 51 848218 212 212 850745 209 209 997473 1 421 002527 10 9 9.848345 9.850619 9.997726 10.002274 52 848472 211 850493 210 997979 421 002021 8 53 848599 211 650368 210 998231 421 00176S 7 54 848726 211 860242 210 998484 421 0015U 1 6 55 848852 211 8.50116 210 998737 421 00126S 1 5 56 84S„79 211 849990 210 99^:89 421 001011 4 57 849106 211 849864 2i0 999242 421 000758 i 3 58 849232 211 849738 210 999495 421 00050? ) '^ 59 849359 211 849611 210 999748 421 0002.5C ) 1 leo 849485 211 849485 210 10.000000 421 00000( ) Iz Cosine 1 1 Sine 1 1 Cotaiig. 1 1 Tang. M. 4 i5 D^ ;ree8. "*" > M 8 00 1 00 2 00 3 00 f 00 6 7 S 9 10 11 12 13 14 16 17 18 19 20 21 22 23 24 , I 29 30 38 39 40 46 47 48 49 50 51 54 55 56 57 58 I 59 M " 1 i )ifi;j «U 910 59 M7 58 1404 57 H5'2 56 J899 55 Jt)4fi 54 33!):) 53 }14() 52 WHH 51 HV.i^ 50 i-.iS'Z 49 il29 48 1877 47 1624 46 1371 45 1118 44 0866 43 0613 42 0300 41 0107 40 9855 39 9602 38 9349 37 9097 36 8844 35 8591 34 8338 33 8086 32 7833 31 7580 30 29 7328 7075 28 6822 27 6570 26 6^17 25 6064 24 15811 23 15559 22 )5306 21 )5053 20 19 )4801 )4548 18 )4295 17 )4043 16 )3790 15 )3537 14 )3285 13 J3032 12 )2779 11 32527 10 9 32274 32021 8 31769 r 315H] 6 01263 5 01011 4 00756 3 0050f » 2 00255 \ 1 OOOOC ) ng. ■ ■ W A TABLE OF JVATUUAL SIIVES. M 1 2 3 4 5 6 7 S 9 10 11 12 13 14 Dt•^r "NatT j pT (;o- 00000 00029 ooor>8 00087 00116 00145 001751 0020 1 00233 0026-^ 0029 1 0032') 00349 00378 00407 _[5 00436 16 17 18 19 20 21 22 23 24 Unit. 00000 00000 00000 '00000 00000 00000 iooooo I'OOOOO Iooooo 00000 99999 99999 99999 99999 39999 1 Deg. 00465 00495 00524 00553 00582 006 1 1 00640 00669 00698 25 00727 26 00756 27 00785 28 00814 29 00844 00873 Nut. Sine 0l7'15 01774 101803 ()1832 !01H62 01891 019201 1019491 101978 02007 02036 02065 02094 02123 02152 02181 N. (;o. nine 99985 99984 99984 99983 99!)83 999'>t2 [99982 99981 99980 99980i 99979 99979 99978 99977 99977 999/6 ,> Dog. 30 31 32 33 34 00902 00931 00960 00989 35'0I018 36|01047 37 01076; 38 39 40 41 01105 01134 01164 01193 42101222 43' 01 251 44 01280 45 01309 46 47 48 49 50 51 53 54 55 56 57 58 59 01338 01367 01396 01425|99990 01454 99989 99996 99996 99995 99995 99995 99995 ,99994 99994 99994 99993 99993 99993 99992 99992 9999! 02211 02240 02269 0:.^298 02327 02356 02385 02414 02443 02472 02501 02530 02560 02589 02618 99991 99991 99990 01483 01513 01542 01571 01600 01629 01 658 99989 99989 99988 99988 99987 99987 99986 02647 02676 02705 02734 i»2763 02792 02821 02850 02879 <(2908 02938 02967 02996 03025 03'»54 99976 99975 99974| 999741 99973 99972 99972 99971 99970| 99969! 99969 99968| 99S67 999661 9 9966 , 99965 i 999641 99963! 99963 99962 9'J961 99960 99959 99959 99958 99957 99956 999551 999541 99953; iVut. Hiiio o:ri90 035 1 9 03548 03577 03606 03635 (3664 03693 03723 0375t; 0378 1 03810 03839 03868 03897 03926 03955 03984 04013 04042 04071 04100 04129 04159 04188 04217 04246 04275 04304 04333 04362 9993U 99938 39937 99936 99935 99934 99933 99932 99931 99930 99929 39927 99926 99925 99924 99923 04391 04420 04449 04478 04507 04536 04565 04594 04623 04653 04682 [04711 04740 04769 04798 99922 9992 1 99919 99918 99917 99916 99915 99913 99912 99911 99910 99909 99907 99906 99905 3jl)uir. "Sat. iNrTJo^ J^ine I Hiiie 05234 052(53 0.5292 05321 05350 05379 05408 05437 k)546r 0549 05524 05553 05582 05611 05640 05669 05698 05727 05756 05785 05814 05844 05873 05902 05931 05960 05989 06018 06047 06076 06105 99863 9986 1 99860 9985S) 99857] 99855 99854! 9985v;' 99851 99849 99847 99846 99844 99842 99341 99839 4 Ueir. 01687' 99986 01716 99985 89 Deff. 03083 03112 03141 03170 03199 03228 03257 03286 03316 03345 03374 03403 03432 03461 99952 99952; 99951;! 99950 i 99949 i 99948! 99947 99946, 99945 9 9 004 99902 99901 99900 99898 99897 99896 99894 99893 99892 99890 99889 99888 99886 99885 04827 04856 04885 04914 04943 04972 05001 05030 99883 99882 99881 99879 99878 99876 99875 99873 06134 06163 06192 06221 06250 06279 06308 06337 06366 06395 0(5424 06453 06482 06511 06540 99SiS 9!i-.."; 99 99833 99831 99829 99827 99826 99824 99822 9982J 99819 99817 99815 99813! Or.OPiQ Q0fl70 N. Co- Sine 99944;;05088'99870 05117 99869 99943 99942 99941 99940 Nat. Sine 88 Deg. 05146 05175 05205 N. Ci)- Sine 99867 99866 99864 Nat. Sine 87 Deff. 06569 06598 06627 06656 06685 06714 06743 06773 99812 99810 99808 99806 99804 99803 99801 99799 99797 99795 99793 99792 99790 99788 99786 99784 99782 99780 99778 99776 99774 99772 199770 Nat. Hine 06976 07005 07034 07063 07092 107121 07150 07179 07208 07237 072(.'' 0729.j| 073i54 07353 0/382 (37411 07440 07469 1)7498 07527 07556 07585 07614 07643 07672 07701 07730 07759 07788 07817 07846 99756 3754 99752 !"»750 99748 99746 99744 99742 99740 99738 99736 99734 99731 99729 99727 99725 M 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 06803 193788 0683' '99766 06860 06889 06918 07875 07904 07933 07962 07991 08020 08049 08078 08107 08136 08165 08194 08223 08252 08281 99723 99721 99719 99716 99714 99712 99710 99708 99705 99703 99701 99699 99690 99694 99_692 99689 99687 99685 99683 99680 99678 99676 99673 99671 99668 99666 99664 99661 99659 99657 06947 99758 N. Co- Sine 99764 99762 99760 Nat. Sine 86 Peg. 85 Deg, 08310 08339 08368 08397 08426 08455 08484 08513 A Q »^ Ac\ 08571 08600 08629 08658 08687 \. Co- Sine 99654 99652 99649 99647 [99644 99642 99639 99637 99632 99630 99627 99625 996?2 Nat. Sine 44 43 42 41 40 39 38 37 36 35 34 3? 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 1^ 14 1?. 12 11 10 9 8 7 G 5 4 3 2 _l 'n i 64 A TABLE or NATtTRAX SINES. - o Deg. ! 6 L>eg. 7 Dog. i « l^eg. 1 9 D t'g. M 60 >I N. S. 08716 N. CS. A.S. 10453 N. CS. iN. S. N. CS. 99255 j N.S. 13917 1 N.CS.' 99027 N.S. 15643 N. CS. 99619 99452 12187 98769 1 08745 99617 10482 99449 12216 9925:. 13946 99023! 15672 98764 59 'Z 08774 99614 10511 99446 12245 99248 113975 990191 15701 98760 58 8 08803 99612 10540 99443 12274 99244 114004 990151 15730 98755 57 4 08831 99609 10569 99440 12302 99240 ,14033 9901 Ij 15758 98751 56 b 08860 99607 10597 99437 12331 99237 114061 99006! 15787 98746 55 b 08889 99604 10626 99434 12360 99233 il4090 990021 15816 98741 54 V 08918 99602 10655 99431 12389 99230 J14119 98998 15845 98737 53 8 08947 99599 10684 99428 12418 99226 114148 98994 15873 98732 52 y 08976 99596 10713 99424 12447 99222 114177 98990 15902 98728 51 10 09005 99594 10742 99421 12476 99219 14205 98986i 15931 98723 50 ii 09034 99591 10771 99418 12504 99215 14234 98982 j 15959 98718 49 12 09063 99588 10800 99415 12533 99211 14263 989781 15988 98714 48 la 09092 99586 10829 99412 12562 99208 14292 989731 16017 98709 47 14 09121 99583 10858 99409 12591 99204 114320 98969J 16046 98704 46 lb 16 0Q150 99580 99578 10887 10916 99406 126:20 12649 99200 99197 14349 14378 98965 98961 10074 98700 98695 45 44 09179 99402 16103 IV 09208 99575 109*5 99399 12678 99193 114407 98957 16132 98690 43 18 09237 99572 10973 99396 12706 99189 14436 98953 16160 98686 42 19 09266 99570 11002 99393 12735 99186 14464 98948 16189 98681 41 2U 09295 99567 11031 99390 12764 99182 14493 98944 16218 98676 40 5il 09324 99564 11060 99386 12793 99178 14522 98940 16246 98671 39 "Z'Z 09353 99562 : 11089 99383 12822 99175 14551 98936 16275 98667 38 23 09382 99559 11118 99380 12851 99171 14580 98931 16304 98662 37 24 09411 99556 11147 99377 12880 99167 14608 98927 16333 98657 36 2b 09440 99553 11176 99374 12908 99163 14637 98923 16361 98652 35 26 09469 99551 11205 99370 12937 99160 14666 98919' 16390 98G48 o4 2V 09498 99548 11234 99367 12966 99156 14695 98914: 16419 98643 33 28 09527 99545 U263 99364 12995 99152 14723 98910 16447 98638 32 2'J 09656 99542' j 1129 1 99360 13024 99148 14752 98906 16476 98633 31 31 09535 09614 99540 '11320 11349 9935? 99354 13053 13081 99144 14781 98902, 98897 16505 16533 98629 30 29 99537 99141 14810 98624 32 09642 99534 ! 11378 99351 13110 99137 114838 98893 16562 98619 28 33 09671 99531 11407 99347 13139 99133 il4867 98889 16591 98614 27 34 09700 99528 11436 99344 13168 99129 14896 98884 16620 98609 26 3b 09729 99526 11465 99341 13197 99125 i 14925 98880 16648 98604 25 36 09758 99523 11494 99337 13226 99122 i 14954 98876 16677 98600 24 3V 09787 99520 11523 99334 13254 99118 114982 98871 16706 98595 23 38 09816 99517 11552 99331 13283 99114 15011 98867 16734 98590 22 39 09845 99514 11580 99327 13312 99110 15040 98863 16763 98585 21 40 09874 99511 11609 99324 13341 99106 15069 98858 16792 98580 20 41 09903 99508 ] 1638 99320 13370 99102 15097 98854 16820 P«575 19 42 09932 99506 11667 99317 13399 •9098 i 15126 98849 16849 98570 18 43 09961 99503 11896 99314 13427 99094 ; 151 55 98845 16878 98565 17 44 09990 99500 11725 99310 i 3450 99091 15184 98841 16906 98561 16 45 46 10019 99497 99494 11754 11783 99307 13485 13511 99087 15212 98836 98832 16935 16964 98556 15 14 10048 99303 99083 15211 98551 47 10077 99491 11812 99300 13543 99079 ,15270 98827 16992 98546 13 48 10106 99488 11840 99t:'37 13572 99075 115292 98823 17021 98541 12 49 10135 99485 11869 99293 13600 99071 15327 98818 17050 98536 11 50 10164 99482 11898 99290 13629 99007 15356 98814 17078 98531 10 51 10192 99479 11927 99286 13658 99063 ; 15385 98809 17107 98526 9 52 10221 99476 11956 99283 113687 99059 115414 98805 17136 98521 8 53 10250 99473 11985 99279 ^13716 99055 115442 98800 17164 98516 7 54 10279 99470 12014 99276 113744 99051 15471 98796 17193 98511 6 55 10308 99467 12043 99278 113773 99047 45500 98791 1722298506 5 56 10337 99464 12071 99269:^13802 99043 15529 98787 17250 ! 98501 4 57 10366 99461 12100 99205 j 13831 99039 15557 98782 17279 198496 3 58 10395 99458 12129 99262 13S60 99035 115586 98778 17308 198491 2 59 M 10424 99455 12 ■58 N. CS. 99258 13889 99031 N.S. 15615 N. CS. 98773 N.S. 17336 98486 1 M N. CS. N. S. N.S. 1 N.CS. N. CS. N.a H4 Desr. U-i Dog. 1 Ut 1 Jeg. 81 Deg. 80 Ueg. 41 185 42 185 43 185 44 186 45 186. 46 1'86( 47 187 48 187; 49 187( 50 1871 51 1885 62 188f 53 188^ 54 1891 55 189r 56 189f 0/ iSHi 58 190S 59 1905 M N.Ci 79 , cs. M 769 60 764 59 760 58 755 57 751 56 746 55 741 54 737 53 732 52 728 51 723 50 718 49 714 48 709 47 704 46 700 45 695 44 690 43 686 42 681 41 676 40 671 39 667 38 662 37 657 36 652 35 G48 34 643 33 638 32 633 31 629 30 624 29 619 28 614 27 609 26 604 25 600 24 !595 23 !590 22 !585 21 1580 20 i575 19 !570 18 !56o 17 !561 16 i556 15 S551 14 i546 13 !541 12 !536 11 5531 10 *526 9 i521 8 J516 7 ^511 6 ^506 5 ^501 4 ^496 3 ^491 2 =1486 1 ^. a M S- A TABLE OP NATUHAL SINKS. 65 U 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 10 Ueg. N. 8. 17365 17393 17422 17451 17479 17508 1 75371 17565 17594 17623 17651 17680 17708 17737 17766 17794 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 62 53 54 55 56 Hi 58 69 M 17823 17852 17880 17909 17937 17966 17995 18023 18052 18081 18109 18138 18166 18195 18224 N.CS. 98481 98476 98471 98466 98461 98455 98450 98445 98440 98435 98430 98425 98420 98414 98409 98404 11 N.8. De^. 18252 18281 18309 18338 18367 18395 18424 18452 18481 18509 18538 18567 18595 18624 18652 98399 983S4 98389 98383 98378 98373 98368 98362 98357 98352 98347 98341 983361 98331 98325 19081 19109 19138 19167 19195 19224 19252 19281 19309 1^338 19366 19395 19423 19452 19481 19509 1'8681 18710 18738 18767 18795 18824 18852 18881 18910 18938 18967 19024 19052 98320 98315 98310 98304 98299 98294 98288 98283 98277 98272 98267 98261 98256 98250 98245 N.CS. 98240 98234 98229 98223 98218 98212 98207 98201 98196 98190 98185 98179 98174 98168 N. 8. 19538 19566 119595 119623 19652 19680 19709 19737 19766 19794 19823 19851 19880 19908 19937 19965 19994 20022 20051 20079 20108 20136 20165 20193 20222 20250 20279 20307 20336 20364 N.CS. 98163 98157 98152 98146 98140 98135 98129 98124 98118 98112 98107 98101 9S096 98090 98084 98079 12 \)eg. N.S. N.CS. 98073 98067 98061 98056 98050 98044 98039 98033 38027 98021 98016 98010 98004 97998 97992 20791 20820 20848 20877 20905 20933 ^20962 20990 21019 21047 21076 21104 21132 21161 21189 21218 97987 97981 9797[ 97969 97963 97958 97952 97946 97940 97934 97928 97922 97916 97910 97905 20393 20421 20450 20478 20507 20535 20563 20592 20620 20649 20677 20706 20734 21246 21275 21303 21331 21360 21388 21417 21445 21474 21502 21530 21559 21587 21616 21644 97815 97809 97803 97797 97791 97784 97778 97772 97766 97760J 97754 97748 97742 97735 97729 97723 97899 97893 97887 97881 97875 97869 97863 97857 97851 97845 97839 97833 97827 21672 21701 21729 21758 21786 21814 21843 21871 21899 21928 21956 21985 22013 22041 22070 97717 97711 97705 97698 97692 97686 97680 97673 97667 97661 97655 97648 97642 97636 97630 N.CS. 13 P eg N.S. 22495 22523 22552 22580 |22608 '22637 22665 22693 22722 22760 22778 22807 22835 22863 22892 22920 22098 22126 22155 22183 22212 22240 22268 22297 ,22325 J22353 22382 97623 97617 97611 97604 97598 97592 97585 97579 97573 97566 97560 97553 97547 97541 97534 22948 22977 23005 23033 23062 23090 23118 23146 23175 23203 23231 23260 23288 23316 , 23345 123373 23401 97437 97430 97424 97417 97411 97404 97398 97391 97384 97378 97371 97365 97358 97351 97345 97338 97331 97325 97318 97311 97304 97298 97291 97284 97278 97271 97264 97257 97251 97244 97237 U Dbg. N.S. 24192 24220 24249 24277 24305 24333 24362 24390 24418 24446 24474 24503 24531 24559 24587 24615 97230 97223 23429197217 2076397821 N.CS. N.S". 22438 22467 N. CS. 97528 97521 97515 97508 97502 97496 97489 97483 97476 97470 97463 22410 97457 97450 97444 N.S. 79 Peg. 78 Peg. [| 77 Peg. || 76 Peg. {"irP^. 23458 23486 23514 23542 123571 23599 23627 23656 23684 23712 23740 23769 23797 97T27 23825 97120 23853 97113 23882 97106 23910 97100 97210 97203 97196 97189 97182 97176 97169 97162 97155 97148 97141 97134 24644 24672 24700 24728 24756 24784 24813 24841 24869 24897 24925 24953 24982 25010 25038 N. CS . 97030 97023 97015 97008 97001 96994 96987 96980 90973 96966 96959 96952 96945 96937 96930 96923 AI 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 23938 23966 23995 24023 24051 24079 24136 24164 97093 97086 97079 97072 97065 97n.'iS 25066 25094 25122 25151 25179 25207 25235 25263 25291 25320 25348 25376 25404 25432 25460 96916 96909 96902 96894 96887 96880 96873 96866 96858 96851 96844 96837 96829 96822 96815 96807 96800 96793 96786 96778 96771 96764 96756 96749 96742 96734 ^6727 96719 96712 96705 24108 97051 97044 97037 25488 25516 25546 25573 25601 25629 25657 25685 25713 25741 257fia 25798 25826 25854 N.CS. N.S. I N.CS. N.S 96697 96690 96682 96675 96667 360 o53 96645 96638 96630 96623 96615 96608 96600 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 6 4 3 2 J. M ! ( 66 A TABLE OF NATURAL SINES. 15 De^. M 1 2 3 4 5 6 7' 9' 10 11 12 13 14 15 K) 17 18 19 20 21 22 23 24 25 26 27 28 29 30 N.8. ■)882 259 10 25938 259r)G 25994 20022 2(5050 2(5079 2(5107 20135 20163 26191 26219 26247 26275 26303 26331 26359 26387 26415 26443 26471 26500 26528 26556 26584 26612 26640 26668 26696 26724 16 Ueg. 31 32 33 N. CS. 96593 96585 96578 96;" 70 96.. .2 96555 965471 96540 I 96532'; 96524 96517 9(5509 96502 96494 96486 96479 96471 96463 96456 96448 96440 96433 96425 9(5417 96410 96402 16394 96386 96379 96371 96363 27564 27592 27620 27(548 27676 7704 27731 ;J7759 znsi 27815 27843 27871 27899 27927 27955 27983 N. 8. N. CS . 96126 96118 96110 96102 96094 96086 9(5078 96070! 96062 96054 96046 96037 96029 96021 96013 96005 17 L>eg. N.S. IN.C'S. 28011 28039 28067 28095 28123 28150 28178 28206 28234 28262 28290 28318 28346 28374 128402 26752 26780 ,26808 34126836 35126864 36 26892 37 269;>0 38 26948 26976 27004 27032 27060 2708M 271 i( 39 40 41 42 43 44 95997 95989 95981 95972 95964 95956 95948 95940 9593 1 95923 95915 95907 95898 95890 95882 45127144 96355 96347 96340 96332 96324 96316 96308 96301 96293 96285 96277 96269 96261 9(5253 96246 29237 29265 29293 2932 1 29348 2937(5 29404 29432 29460 29487 29515 29543 29571 29599 29626 29654 29682 29710 29737 29765 29793 29821 29849 29876 29904 29932 29960 29987 30015 30043 95630 95622 95613 95605 95596 95588 95579 95571 95562 95554 95545 95536 95528 95519 95511 95502 _m Peg. NTJT^N. C8. 309(V2 30929 30957 30985 31012 31040 31068 31095 3 1 1 2:» 31151 31178 31206 31233 31261 31289 31316 2H429 28457 28485 28513 28541 28569 28597 28625 28652 28680 28708 28736 28764 28792 28820 46 1 27 172 47127200 48127228 49 1 -^7256 50127284 51127312 52;27340 53 127368 54:27396 55127424 56 27452 57j274S0 58127508 59 27536 96238 96230 96222 96214 96206 96198 96190 96182 96174 96166 96158 96150 96142 96134 M N.CS 95874 95865 95857 95849 95841 95832 95824 95816 95807 95799 95791 95782 95774 95766 95757 30071 95493 95485 95476 95467 95459 95450 95441 95433 95424 95415 95407 95398 95389 95380 95372 95106 95097 95088 95079 95070 95061 95052 95043 95033 95024 95015 9500(5 949i)7 94988 94979 94970 30098 30126 30154 30182 30209 30237 30265 30292 30320 30348 3037(5 30403 30431 30459 30486 28847 2^875 28903 28931 28959 28987 29015 29042 29070 9098 29126 29154 29182 29209 N. 8. 74 Dejr. 95363 i)5354 1)5345 95337 9532S 95319 95310 95301 95293 952H4 95275 95266 95257 95248 95240 31344 31372 31399 31427 31454 31482 31510 31537 31565 31593 31620 31(548 31675 31703 31730 9496 1 94952 94943 94933 94924 94915 9490(5 94897 94888 94878 9486i) 94860 194851 94842 94832 19 Peg. N. 8. "IsTCiB 94552 94542 91533 94523 94514 94504 94495 94485 94476 94466 94457 <)4447 94438 94428 94418 94409 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 95749 95740 95732 95724 95715 95707 95698 95690 95681 95673 95664 95647 95639 nTcsT N_8^ 73 PefT. 31758 31786 51813 31841 31868 31896 31923 31951 31979 3200(i 32034 32061 32089 32110 32144 ;52997 33024 3305 1 33079 33106 33134 33161 33189 33216 33244 3327 1 33298 3332(5 33353 3338 1 30514 30542 30570 30597 30625 30653 30680 30708 30736 30763 30791 50819 30846 30874 95231 95222 95213 95204 95195 i)5186 95177 95168 95159 95150 95142 95133 95124 i95115 94823 94814 94805 94795 94786 94777 94768 94758 94749 94740 94730 94721 94712 94702 94(593 94399 94390 94380 94370 94361 94351 94342 94332 94322 94313 94303 9429;5 94284 94274 94264 3340S 3343(5 334(53 33490 33518 33545 33573 33(500 33627 33655 33682 ;53710 33737 33764 33792 32171 32199 32227 32254 32282 32309 32337 32364 32392 32419 32447 32474 32502 32529 N.CS. I N.8. 72 Peg. 94684 94674 94665 i)465(5 91646 94637 94627 94618 94609 94599 94590 945«0 ,94571 94561 94254 94245 94235 94225 94215 94206 i)4196 941 ^'6 94176 94167 94157 94147 94137 94127 )4118 14 43 42 41 40 39 38 37 3 O'lO^H 9 94049 8 94039 7 94029 « 94019 5 94009 4 93999 'A 93989 'i 93979 I N.S. AI )epr. A TAHLE OF NATURAL SINES. JVl 1 2 3 4 f) <; 7 H 9 10 11 12 13 14 ll> l(i 17 18 19 r>7 iJ02 y39("iy _N 34 34229i93!)r)9 34 2r) 7 93949 34284 93939,1 3r)9 18 31311 !>;i()i;891 31393 34421 34448 l^38H9 93879 34 '17.01 93809 34.')03 938r)!> 31 r)30 93849 34 r)r)7 93839 34r>84|93829 34012 93819 22 24 20 27 28 29 30 31 32 33 34 3.'> 30 34039 34000 3'i094 34721 20134748 21 3477.0 34833 .34830 34857 34884 34912 34939 J4900 34993 35021 3004 8 i 93H()9 93799 93789 93779 !i3709 9.3759 93748 93738 93728 93718 93708 93098 93088 93077 03(i07 93(J57 I 30027 3i;0.04 30081 30108 30135 30102 30190 30217 ^30244 30271 30298 .30325 3(i352 .30379 :3(i400 ,30434 ;3040l 30488 30515 3(i542 30509 30590 3fi()23 30050 :KWi77 39 35200 40 35293 41 35320 4213.5347 43 '3.5375 4413.5402 N. VH. 93'35N 93348 93337 93327 13310 93300 !».3295 93285 93274 93204 93253 93243 93232 93222 »32ll 93201 !i3l90 93180 93109 J3159 93148 93137 J3127 93110 93100 93095 930H4 93074 9300.3 93052 93042 9303 1 93020 93010 92999 J2988 92978 92907 92950 J.5075 9.30471 30704 }5102|93037i 30731 35130193020; 307.58 35157;930ini3078.'y 35183.93000! 30812 37 35211|93590|30839 38 i 35239 93.585,30807 93575 30894 93505130921 9.3.555| 30948 93514:30.975 93534' 37002 92902 93524 37029 92892 I 45|;i5429i935^[_4;,3705<; !i^H81 40i35450l935():j''37O83 <)2870 47 35484i93'193 37110 48 3551 119.3483' 37137 49{3.5538 93472*37104 .50 35505 93402 37191 51i35592 9.3452 37218 52j35019i93441!:37245 .53j35047 93431, 37272 .54135074; 93420 37299 55135701 9341 0i!37320 50 1 35728 ' 93400 37353 57! 35755 ' 93389 ! 373.MO 58,35782'9.3379 59 1. 3 58K) 193308 Miiv.cH. iior !1294.0 1)2935 :i2924 92913 12859 92849 92838 92827 928 1 112805 !i2794i 92784 i 92773; 92702 j^r.ii N. H. 37401 37488 375 1 5 37512 37509 37595 37022 3701!) 3707( 37703 37730 37757 37784 17811 17838 .37805 37892 37919 37940 .37973 37999 mvM 38053 38080 38107 38134 3810! 3818H 18215 23 Dtjr. 3824 I 38208 :J8295 138322 1.38349 38370 38403 38430 38450 18483 38510 38.537 38504 138591 380 1 7 38044 38071 38098 38725 38752 38778 38805 N.Cis. 92718 92707 92097 !)20N0 92<)75 920041 920531 !)2042 92031 92020 92009 925!)H 92.5N7 92570 925()5 92554 92.543 92532 9252 1 925 1 92499 92488 92477 92400 924.55 92444 92432 92421 92410 I N.H. '39073 1^39100 139127 39153 '39180 39207 .39234 39200 J39287 39314 39341 39307 39394 39421 39418 39474 19.501 39528 39555 3958 1 39008 N.rrt.' 92(')50 92039 92028 'M Dog. N.N. 40074 10700 40727 91833 ill 822 39035 91810 1)2399 !)2388 92377 92300 92.355 92343 92332 92321 92310 92299 922H7 92270 .3900 1 39088 .39715 3974 1 39708 39795 39822 39848 92OI0ti40753 40780 4080(1 40833 40800 108N0 40913 40939 4090J^8^ N.(J8. 4M99 45425 45451 45477 45503 45529 45554 45580 45606 45632 45658 45684 45710 45736 45762 45787 43471 43497 43523 43549 43575 43602 43628 53J43654 64143680 55143706 56143733 67 43759 43785 43811 58 59 M 90246 90233 90221 90208 90196 90183 90171 90158 90146 90133 90120 90108 90095 90082 90070 90057 90045 90032 90019 90007 89994 89981 89968 89956 89943 89930 89918 89905 89892 N. C8. N. 8, 64 Deg. 44646 44672 44698 44724 44750 44776 44802 44828 44854 44880 44906 44932 44958 44984 45010 45813 45839 i5865 45891 45917 45942 45968 45994 46020 46046] 460721 46097 46123 46149 46175 89480 89467 89454 99441 89428 89415 89402 89389 89376 89363 893ii0 89337 89324 89311 89298 89101 89087 89074 8906 1 89048 89035 8902 1 89008 88995 88981 88968 88955 88942 88928 88915 88902 88888 88875 88862 88848 88835 88822 88808 88795 88782 188768 88755 88741 88728 88715 88701 46947 46973 46999 47024 47050 47076 47101 47127 47153 47178 47204 47229 47255 47281 47306 47332 45036 45062 45088 45114 45140 45166 45192 45218 45243 45269 45295 45321 45347 45373 N. CS. N. S, 46201 46226 46252 46278 46304 46330 46355 46381 46407 46433 46458 46484 46510 46536 46561 89285 89272 89259 89245 89232 89219 892061 891931 89180 89167 89153 89140 89127 89114 46587 46613 46639 46664 46690 46716 46742 46767 46793 46819 46844 46870 46896 46921 N. C8. 88688 88674 88661 88647 88634 88620 88607 88593 88580 88566 88553 88539 88526 88512 88499 88485 88295 88281 88267 88254 88240 88226' 88213 88199 88185 88172 88158 88144 88130 88117 88103 88089 ±'^ Ueg. N. 8. IN.CS. 47358 47383 47409 47434 47460 47486 47511 47537 47562 47588 47614 47639 41665 47690 47716 88075 88062 88048 88034 88020 88006 87993 87979 87965 87951 87937 87923 87909 87896 87882 48481 48506 48532 48557 48583 48608 48634 48659 48684 48710 48735 48761 48786 48811 48837 48862 47741 47767 47793 47818 47844 47869 47895 47920 47946 47971 47997 48022 48048 48073 48099 63 Deg. 88472 88458 88445 88431 88417 88404 88390 88377 88363 88349 88336 88322 88308 N. 8. 87868 87854 87840 87826 87812 87798 87784 87770 87756 87743 87729 87715 87701 87687 87673 M 87462 00 8 7448; 59 87434158 87420'57 87406 56 87391 87377 87363 87349 87335 87321 87306 87292 87278 87264 87250 55 54 53 53 51 50 49 48 47 46 45 48888 48913 48938 48964 48989 49014 49040 49065 49090 49116 49141 49166 49192 49217 49242 87235 87221 87207 87193 87178 87164 87150 87136 87121 87107 87093 87079 87064 87050 87036 44 43 42 41 40 39 38 37 36 35 i 34 33 32 31 30 49268 49293 49318 19344 49369 49394 49419 49445 49470 49495 49521 49546 49571 49596 49622 48124 48150 48175 48201 48226 48252 48277 148303' 48328 48354 48379 148405 48430 4845 6 N. csT 87659 87645 87631 87617 87603 87589 87575 87561 87546 87532 87518 87504 87490 87476 N.S. 62 Deg. 1) 61 Deg 87021 87007 86993 86978 86964 86949 86935 86921 86906 86892 86878 86863 86849 86834 86820 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 49647 49672 49697 49723 49748 49773 49798 49824 49849 49874 49899 49924 49950 49975 86805 86791 86777 86762 86748 86733 86719 86704 86690 86675 '86661 86646 86632 86617 N.CS. N.S. 14 13 12 11 10 9 8 7 6 6 4 3 2 _1 M 60 Deg. 48 512 49 512 50 512 51 512 52 513 63 513 54 513 55 613 66 514 57 614 58 514 69 614 M N.C 5t A TABLE OP NATURAL SINES. fi9 21 1 2 3 4 5 6 7 30 Deg. 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 I 47 48 49 50 51 52 53 54 55 66 57 58 59 N. S, I N. C8. 8^603 86588 86573 86559 86544 86530 86515 86501 86486 86471 86457 86442 86427 86413 86398 86384 50000 50025 50050 50076 50J01 50126 50151 50176 8 50201 50227 50252 50277 50302 50327 50352 50377 50403 50428 50'153 50478 50503 50528 50553 50578 50603 50628 50654 50679 50704 50729 50754 86369 86354 86340 86325 86310 86295 86281 86266 86251 86237 86222 86207 86192 86178 86163 51504 51529 51554 51579 51604 51628 51653 5 J 678 51703 51728 51753 51778 51803 ,51828 51852 151877 31 Peg. N. 8. [ n.cb; 85717 85702 85687 85672 85657 85642 85627 85612 85597 85582 85567 85551 85536 85521 85506 85491 50779 50804 50829 50854 50879 50904 50929 50954 50979 51004 51029 51054 51079 51104 51129 51154 51179 51204 51229 51254 51279 51301 51329 51354 51379 51404 51429 51454 51479 86148 86133 86119 86104 86089 86074 86059 86045 86030 86015 86000 85985 85970 85956 85941 51902 51927 '51952 51977 52002 52020 52051 52076 52101 52126 52151 52175 52200 52225 52250 85926 85911 85896 85881 85866 85851 85836 85821 85806 85792 85777 85762 85747 85732 52275 152299 52324 52349 52374 52399 52423 52448 52473 52498 52522 52547 52572 52597 52621 85476 85461 85446 85431 85416 85401 85385 85370 85355 85340 85325 85310 85294 85279 85264 85249 85234 85218 85203 85188 85173 85157 85142 85127 85112 85096 85081 85066 85051 85035 52992 53017 53041 53066 53091 53115 53140 53164 53189 53214 53238 53263 53288 53312 53337 53361 53386 53411 53435 53460 84805 84789 84774 84759 84743 847281 84712 84697 84681 84666 84650 84635 84619 84604 84588 84573 84557 84542 84526 84511 53484184495 5350984480 53534 84464 54464 ,54488 (54513 154537 54561 54586 54610 54635 154059 54683 54708 54732 54756 54781 54805 54829 83867 83851 83835 83819; S3804 83788; 83772' 837561 83740,1 83724 83708 83692 83676 83660 83645 83629 "sTn eg. 53558 53583 53607 53632 53656 53681 53705 53730 N. CS. I N. 8. 59 Deg. 52646 52671 52696 52720 52745 52770 52794 52819 52844 52869 52893 529 IS 52943 52967 N. CS. 85020 85005 84982 84974 84959 84943 84928 84913 84897 84882 84866 84851 84836 84820 N.S.'I 53754 53779 53804 53828 53853 53877 53902 53926 53951 53976 54000 54024 54049 54073 54097 84448 84433 84417 84402 84386 84370 84355 84339 84324 84308 84292 84277 84261 84245 84230 84214 84198 84182 84167 84151 84135 84120 84104 54854 54878 54902 54927 54951 4975 j4999 55024 55048 55072 55097 55121 55145 55169 55194 54122 54140 54171 54195 54220 54244 54269 54293 54317 54342 54366 54391 54415 54440 58 Peg. 84088 84072 84057 84041 84025 84009 83994 83978 83962 83946 83930 83915 183899 83883 N.