FORMULAS FOR AREA Rectangl e-base X altitude. * * Parallelogram-base X altitude. Triangk- Y2 base X altitude. Trapezoid- Y2sum of parallel si des X a ltitude . Par'8bola-% base X altitude . Ellipse-product of major and minor diameters X 0.7854. Regular polygon- Y2 sum of sides X perpendicular distance from center to sides. Lateral area of right cylinder = perimeter of base X altitude. o o Total a rea = lateral area + areas of ends. Lateral arca of right pyramid or cone = Y2 perimeter of base X slant height. Total area = lateral area + area of base. Lateral area of fru stum of a regular right py~amidor cone = Y2sum of perimeter s of bases X slant height. Surface area of sphere = square of diameter X 3. 141 6. o ! o o I o FORMULAS FOR VOLUME 1 Right or oblique prism-area of base X altitude. Cylinder-area of base X altitude. P yramid or cone-~area of base X altitude. Sphere-cube of diam eter X 0.5236. Frustum of pyramid or cone-add the areas of the two bases and add t o this the squar e root of the product (;f the areas of the bases; multiply by ~ of the height: V ~ h (B + b + ~B X b). IMPORT ANT CONSTANTS 7. = 3.1416. 'lt 2 = 9.8696.
~7(= 17724 1+7t = 0.3183. Base of natural logarithms = e = 2.71828. !vi = log!. e = 0.43429. 1 + M = loge 10 = 2.3026. Loge N = 2.3026 X loglo N. Number of degrees in I radian = ISO + -:t 57.2958. Number of radian s in I d"grec = 7. + ISO 0.01745. WEIGHTS AND MEASURES Avoirdupois Weight Long Measure Dry Measure 27 iVa? grs. = I dram 12 inches = I foot 2 pints = I quart 16 drams = 1 ounce 3 feet = I yard 8 quarts = I peck 16 ounces = I pound 5:1 yards = I rod 4 pecks = I bushel quar:~r = 25 pounds = 1 40 rods = I furlong 36 bushels I chaldron 4 quarter s = I cwt. 8 furlongs = I stat. mile Liquid Measure 2,000 Ibs . = 1 short ton 3 miles = 1 league 2,240 lbs. = 1 long ton 4 gills = I pint 2 pints = I quart Mariners' Measure Square Measure 4 quarts =1 gallon " feet = I fathom 31 J1 gallons = I barrel BY !LO fathoms = I cable 144 sq. in.,:. = I sq. ft. 2 barrels = I hogshead length 9 sq. ft. - 1 sq. yd. ROSS R. MIDDLEMISS 7:1 cable lengths = I mile 30~ sq. yds. = I sq. rod Surveyors' Measure 5,280 ft. = I stat. mile 40 sq. rods = I rood 7.92 inches = 1 link Associate Professor of Applied Mathematics 6,08.\ ft. = I naut . mile 4 rvods = 1 2~re 25 links = I rod WASHINGTON UNIVERSITY Troy Weight 640 acres = I sq. mile. 4 rods = I chain 24 grains = I pwt. 10 sq. chains or 160 sq. Technical Consultant To rods = I acre 20 pwt. = I ounce Cubic Measure 12 ounces = I pound 640 acres = I sq. mile Used for weighing gold, 1, 72S cu. in = I cu. ie. 3(' sq. miles (6 miles sq.) silver and jewels. 128 cu. ft. = I cord wood = I township 1 27 cu. ft. = I cu. yd. Miscellaneous Apothecaries' 40 cu. ft. = I ton (shpg.) Weight" 2,150.42 cu. inches 1 3 inches I palm 20 grains = I scruple standard bushel 4 inches = I hand 3 scruples = J dram 23 I cubic inches = 1 6 inches = 1 span 8 drams = I ounce standard gallon 18 inches = I cubit 12 ounces = I pound I cubic foot = about four 21.8 = I Bible cubit *The ounces and pound in this 2Y2 = are the same as in Troy weight. fifths of a bushel ft. I military pace 14 * * o o
o o o o
GENERAL DESCRIPTION This slide rule combines the convenience of the 5-inch pocket size rule with the ac curacy of the standard IO-inch rule. On the FRONT FACE ALONE it has all of the standard 5-inch scales-A. B. C , D, S, T, K; and DJ. The arrangement is more convenient than on other 5-inch rules be cause all of these scales are on the same side: it is not necessary to turn the rule over to read sines and tangents: triangles can be solved with a single setting of the slide.
o K o
A
B 5 T C o D o D I P"'I" IIIIIIII'"II"IIIII I '1"1'1"111" I I I ' I I I III 'I Q'l II' I'I 'I'I'IIIII'I' II " I q '1'P i ' I' l ' !! I I' I ' ) ' I' , I 11II 11I"IllIlIpll l 11 11 111 11 1111 q II' rI l l f ' I I I . I ' 1 9 e 7 6 5 4 I 3 ' } o o Fig.l On the back face there is a standard to-inch CoD scale combination folded at ",TO:
The two parts of C are called C, and C t , and the corresponding patts of D are called D, and D2 This combination gives the same accuracy as the standard to-inch rule for all problems in multiplication and division, percentage, proportion, etc. There is also an L scale which gives logarithms with to-inch accuracy and a 5-inch D scale. This scale, used with D, and D" gives squares and square roots with iO-inch accuracy.
0 o 0 o L
C 2
CI (}I 3456789 00 02 .J&"IIIIIIIIIII"'1""I""I"'~ l'I'I'I'I'I'I'I'I'~II'I'I'IIIIIII'I'II~'j1I'1'I'I'I'I'l'lt'll1'i1111'1'i'1'i~111'I11'1'I'I1111~'i'i'l'l'1'l1111111 o 0 1""1"111""""1""1"111'11111111"111""11 '1'1'1'1'1'111111111'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'.'1'11'1'11'111"1""1""1""1""1""1"'" 0 1 2 3 4 567891
'Stock SLid~ Fig 2
The names of the various parts of the rule are indicated in Fig 2. They are the BODY or STOCK; the SLIDE which moves in the grooves of the stock: and thB RUNNER, or INDICA TOR, or CURSOR. This cursor is molded of transparent plastic and carries a HAIRLINE or MARKER which is used in making accurate readings and settings.
Locating Numbers on the Scales 1. Reading a Slide Rule Scale. Anyone who knows how to read the scale on an ordinary ruler or yardstick can easily learn to read a slide rule scale. The only essential difference lies in the fact that the calibration marks on a slide rule scale are not uniformly spaced (except in the case of L). Fig. 3, which shows only the primary divisions of the D scale, illustrates this important point. It is much farther, for exampk, from I to 2 than it is from 8 to 9. The spacing is called "logarithmic" and it is based on the theory of logarithms. One does not need to understand this in order to use the slide rule-any more than he needs to know the theory of gasoline engines in order to drive an auto mobile.
2 o o Fig.3
The part of the 0 scale from 1 to 2 is divided into 10 secondary divisions each repre senting IfIO, and each of these is further divided into 5 parts as shown in Fig. 4: each smallest division then represents If. of 1/10 which is 1/ .. or 0.02.
~_O~~ ______D ___ I_II_"_I_"_"_I_I~_I_II_"_11_11_111_11_11_111_11_111_11_111_111_1I1_1_1_1_11_1_lt_I_I_II_I_'I_~_1_':_II_~_jl_I~_jl_I~_jl_jll_II_~._III_1I_1I1_1j1_II_jll_Ij~_II_II_I_II_"_~_11_"_11_1I_17_1I_"1_11_11_~_"_11i_"_1~_'! __ i~ ______~~() ___O-J
Between 2 and 5 each primary division is subdivided into 10 parts and each of these is divided into 2 parts; each smallest division then represents Yz of 1/ 10 which is Iho or 0.05. Between 5 and 10 (the right-hand) stands for 10), each primary division is divided into 10 parts so each smallest division represents '/10 or 0.1. When a person becomes familiar with this situation for one scale he can easily read any of them. o (0 l . 1'1 2. Locatin~a Number on the D Scale. In locating a number on the scale one dis o o o o re gards the deci mal poi nt entirely. lnu s the same spot on the scale serves [or 1.25 . 0.0 125. 12.5.1150. etc. To locate this number one may rega rd the D scale as running ~ from 1 to \0. the ri gh t-hand 1 standing for 10. He may then think of his number as ~ \ .25 regardless of the actua l position of the decimal point. 5l[~t&·'1~,~-::; ~ §lI96'_ZT'lf ...§ V) Since the first digit is I, the number is between main d iv isions I and 2 Since the m£~6"'1h§ next digit is 2. the number is bet ween \ .2 and 1.3 and is therefore located bctw~cnth~ ~cn sl~f:::~:~il~ ~ m06'-" ,il ~ second and third secondary calibrations. ~ow.each smallest division in this interval 519061"%. ....g ~~co SLI'-!I. ..§ represents 0.02. so the required point is half-wa y between the second and third of these. 51t6~i'''%." Figure 4 shows the location of this po int and also shows several others. ;lt~"-n,h~ ::-1'- SZm:::;!i,:1 ~ ~£8961''1~. O!..§ ='
5£il/1' 'l~I'"~ D,. The number s on C, and D, are printed in black while those on C, and D., arc in SI90H'''!, N~ 'O- :5 4 Slide Rule Operations 11. Multiplication usin& C and D. In wha t follo ws, the left·ha nd o f a sca lc is called it s LE FT INDEX. the right-hand I is wiled the RleHT INDE X. \Vc m ultiply t wo numbers as shown in the fo llowing t wo cX il mplcs : Example l. Multiply 2 X 17. S TEP I . Opposite 2 on D. set the L EFT index vf C STEP 2. Opposite J7 on C read the answer (34 ) on D . o o I< l'I'I ' I , \'I' I , I' IrI~! " , iIl 1l1 " "I",~ ""1",~",J" ,~,,, JII1I1 ' I'lrlrI,I,IrI'I'1 ~;"I""f""I", ""1""t""T,,, 1II~""I,I'I,I'I,I'I'IrI'I~" " I""1""I!II3""T,,,,t,,,,r,,, ,y, ,, ~, " ,I A 1 2 3D. 4 5 6 7 B 9 1 2 3 4 5 6 7 8 9 1 o o DI o o Step 1 Opposite 2017 D Fig.5 Step 2 Opposite 17011 C Set leTt index or C read 34 on 0 -' Example 2. Multiply 5.4 X 0.25. S TEP J. Opposite 54 on D. set the RIGHT index of C S TEP 2. Opposite 25 on C, read /35 on D. The decimal points have been disregarded in this operation. R ough m ental calcu lation shows that the answer must be 1.35. Note in this case that the reading would h ave been "off sca le" if the le ft index had been used. These examples illustrate the general rule for multip lying t wo factors, na mely: STE P 1. Locate one of the factors on D and set either the right or left index of C over it. STEP 2. Opposite the other factor on C, read the product on D. o o K !1 11 i l l l l l ll ll' h ! ~ ' 1 1 \1 1l~1H 1 111I;j II I t 1I t1~II IITII I ~mJn llt , l tI'hltl'h""I~ 1I1111111fll lllllld " ' lll1l1flllll"'~m~1II l'llhl'hl'hhhl~ 1111111111111111113 11 1 1~ 1I 1111l IlTIl t1'I I1 JI!! J 2 . D. 4 5 6 7 8 9 1 2 3 4 5 6 7 9 1 o ""1'''1''''1''''1 "'1""1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'I' o 7 6 5 4 3 o Step 2. Opp. 25 on C step 1. Opp.54 on D set read 135 anD Fig.6 right index of C 12. Division using C and D. This operation is the inverse of multiplication, and the division of 34 by 17 is shown by Fig. 5. The steps are : ~ STEP I. Opposite 34 on D. set 17 on C. STEP 2. Opposite the index of C. read l. on D. ~ 13 . The Number of Di~itsin a Number. If a number is greater than I the number ~ ~ of digi ts in it is defi ned to be the number of figures to the left of the decimal point. If a 01 (po:citive ) number is less than I the number of digits in it is defined to be a negative Co number equal numericall y to the number of zeros between the decimal point and the 0 ~ \} ~Il first significant figu re. 1~ ~ ~ Exa mples: 746.22 has 3 digits. 0.43 has 0 digits. 3.06 has I digit. 0.004 has - 2 digits. ~ ,~ Ru les can be gi ven for keeping track of the decima l in multiplication and di vision in - Ilt~IS"%' term s of the number s of digits in the number s but these will not be stressed here. An ! ~ u· I./'JU >. o:-ib: SI8SS'-'I~, ~ iS () 8 ~ ~01 ~ug. C1'I~ " examp le is : When two numbers are multiplied using C and 0 as described in section II, §Zlt~6'P~I' co-= CTI ~. - ~ U ~ til /l) [- ~ .S Co u- I the number of digits in t.he product is equal to the sum of the numb ers of digi ts in t he Il£i'-'~I u -~ ..c:'" " 0.. r,19IZS'''%s s;::: factors if the slide projects to the left--an d one less than this jf it projects to the right. -o~15~>c!~ . oocv 11901'-'1,;, 1 '~ ~ ~ C II E o.f5"'On ~:o<--5 14. Sq ua res and Sq u?r e Ro o ts . Opposite any number on D , rcad its square on A. mosl·''Jl. ~.--2 ~.s.s~-5~ = m :~'It - =- C\J /'t Examples: o~..::: u 1= "''' c is.s 0 Htu'_l~h r...::: Opposite 3 on D , read 9 on A, Opposite 1.47 on D, read 2.16 on A. . -;:....r::.s I:.) C1j. _ ..... ~ '1J '" -w -- ,3 c: ~ ~ll8Zfryh ~l18'_9~il ~ C onversel y, opposite any number on A, read its square root on D. Usc the LEFT ~ 0 o~ci ~ .f!O ~] 11896/"% ~ ~01) ~~ I-. ~ half of A if the number hasa n ODD numb er of digits and the RIGHT half if the number 3 - u b ~tl8l'_t~h - ~ .. -1'9 ' - has an EVEN numb er of digits. :. EI 8.J -:-.~~ El) : ? ',', :'1=: .;:; - c l0c::) ...r::;;. .- ~ C..c 5/,- C\i ~ Examp les: g ,..:;.- ....eol ~ u ro J-.I m~tl'''lh SL8tl'_I~'tr Opposite 5 on A (left), read 2.24 on D. Opposite 64 on A (r ight), read 8 on D. :) or>'" a .;: ""..c:.s " .,- c v u .s E '" ro ro SlItOt' 'j\, ~ _9~~1 ~ 15. Cubes a nd Cube Roots. Opposite an y number on D . read its cube on K. ~ 2 E~-5G'J) ~)~~ S£I9' ~ .,., - ~"'5<.g~-o 'I: . ]6 N Examples: x SZ9S9't~ 'lr -'" t3 C3 C/) ~ I./'J~~U,~C _u Opposite 2 on D, read 8 on K Opposite 4.2 on D. read 74 on K. N ~l9C~r"~h < ~c: '" ro..c:;: "' 0 Ci m·-V! Con versely, opposite an y number on K, read its cube root on D . Usc the right third . .: 2] ~g 0 ~ ~ ... '1J ILt6W"lI, -= ~~ ~ of K if the number of digits in the number is a multiple of 3 (- 3, 0, 3, 6, etc. ) : use the c;u ..c=-;:,d:-g :r,..cU C SLt6r 1~~1" ro IZlltl"%' middle third if the number of digits is one less than a multipl e of 3 ( -1,2,) ,8. etc ) : usc .. C .~ J u - ~ ';;'~ c: cJ-;, N ~~ OJ ~Z9r-' -O..c:r.'-~w \..c o the left third if the number of digits is two less than a multiple o f 3 ( - 2. 1. 4,7. etc .) . Q . ,. .,..; c c..c: 3 '- 5£ml'''li, x ==4: .- Opposite on (left), read 1.26 on D. ~ -"''''00 t:t u ~ a 1ZIt I' 'Ii, - -~~- Examples: 2 K '0 ro u"'U 1::,- 'C ~ 0 t N szms'''l!, LL c u ...... , Ci) Ow W ro -: =- == Opposite 64 on K (middle), read 4 on D . Opposite 125 on K (right ), read 5 on D . .- '-YI - eIS U CO J-.I {/: ..c ;>~ ' » / Ittla,' , It ,.. ~ o-o-sot: ~ ~... -"'O 16. Reciprocals . Opposite an y number on D , read its reciprocal on D! . a.. Il89r'H, U -5 .:;, 5;: 11ro onJ!§ .;:; 1 ImS~·"l', ..: c: u <-.J -:.; >< C"-' u ExOamples: D dl L 5 Df OPposite38 .40nD,read - =0026 0n01. '"3 " ...... '1111 .!2 ~-5 ~-is ~~o ~ pposlte 2 on . rca 7 2 = . on . 38 . 4 u 6 ~ - 'C.[I - ..Q t) J-.I ..... s: ,9- u ~ Th e decima l point is fixed by the rule that if a number which is not apower of 10 has , .S '"2 'S'5"::: ~w ~ ~-5 x digits, its reciprocal has I - x digits. T11us 38.4 has two digits and its reciproca l has en C'tI ''' ...c ::::I '"" Slt'- 'A "-\); () I - 2 = - I digits. :s -g ] ~~].eo E ~.~ ~ W6St' "lh V) ~ttlt'I~11 C CI:! C - ~ "'0 ~ I.. ~ Vl8,'" m8w' j'" ~N An~le.Opposite angle bt:tween 90° on S, read its sine ~ 'i: ~-g ~.:: -0 ~ ~ 17. Sine of an any 34' and 9 ·.2 1ZIt'-'li ...... C'tI 'c";-C ~ 6t\j r:...Jc... ". - on B (or A if rule is closed). I f the sine is read on the RIGHT half of B the decima l point , II ~ ~r- c· ~ c:: :l ;'; c. ~'" ~ goes immediatel y before the first figure. If read on the LEFT half of B there is one zero e o IZm·' 11 between the decimal point and the first significant figu,re, :.: 3 ~-0 .2 ~:. ] ~ 6 - OJ Im9Z");, - - .=';3t;~~~u g[1 ..c0-a IZ'-'A ~ ~ ..0..1 I./'J u C C" \.. IJ) ~ Examp les: '- r- e smtz' ''i, .~ Opposite 28 ° on S, read 0.47 on B . Opposite 3° 10' on S, read 0.055 on B. -; § .Ds - U 0 c:.o ..c SUlZ'lH u>=.D ,, ", . "'0: 5ZI£OZ'''\, 6- - 0 ~ '- '5"8E -g c t,,·~'0 ~ I, ~ 18. Tan~entof anAn~le. Opposite any angle between 5 ° 42' and 45 on T, read its ml' -' 1 =c 8,'IJ::r-;t; ~ HI It I' " '.. --1 .;:t: tangent on D . The decima l point goes before the first figure. Iml" l\ U ~ ~.£'§.0'cj ~ u 5 § ~ ~ mO~I·"l' C::)'...;;. Example: IZI'-~I Oppositc 20° 20' on T, .read 0.37 on O. smol' "c smo·'11 ~ Notc that cot 20° 20' = 1 = 2.7 can be read on Df with the same se tting. SZlilO' "l\ tan ·20° 20' SZ90'_9~, This is true because the number on 01 is the reciprocal of that on D and t.he cotangent I!mo'''l\ ~ mEO·'I·, _ OJ is the reciprocal of the tangent . $l9110"?', 00 ~ T o find the tan ge nt of an angle between 45° and 84° 18' use the fact th at tan 0 (90 _ /\ ) = Cot A = _ _1_ . T hus we find tan 58° by reading cot 32° on Dr. tan A OpposUe Jr on T, rea d cot 32° = tan 58° = J. 6 on Dr. Since the tan gent of an angle between 45° and 84° 18' is between I and 10, the decimal o o point comes after the first figure. o If an angle js between 0 and 5° 42' its sine ma y be used in place of its tangent . Th ey are nearly equa l for such small angles. 9 8 L Note that in this case the index was set opposite a number of its own color (black index opposite black number): when this is the case the product is read on the scale hav ing the SAME color as the second factor-black to black or red to red. When ei).her the left (black) index or the right (red) index of the C , C. combina tion is set over a number on D, or D, that has the OPPOSITE color. then the product is read on the scale whose color is OPPOS ITE to that of the second factor. Example: Multip ly 24 X 2. o o 8 9 y'iO o 02 III~II'III' 'l'I'I'I'I'~'III'Ill'l l l'l'l'lt'l' '1'1 '1' 1' I' ' IIPP I'I'I'I'I'I'I'I'I'I'I'I' I '1'1"" 1""111'11" " 1""1""1""1""1""1"": o o D 3 1 567 891 Step 1. Opp24onDl setriqhtindexuf'C2 Step 2. Opp.2 on Cl read 48 on D2. Fig.8 20. Division using C, C, and D, D,. To carry out the division !'--. locate x on the v q, D, combination and pull yon C, orC, over it: read the quotient on the proper one 01 the D, D, scales opposite the index of C , C,. If x and yare the SAME color. read answer on scale that has the SAME color as the index: if x and yare OPPOS ITE in color. read answer on the scale that is OPPOS I TE in color to the index. Example 1. Divide 75 by 6. . . - . .".__ on.-.". r..onr- ~ ~ ~ ~ ; ..., ...... , ::::;~;:~~ ----- _ ...... - 5on~ -;:;E ~:!!5_~ lIi:S=:::;~;:..,j :;::::;;~00_"_ ~- -~------=..... :g~:~;;;:;; 0 re~~;i;i :::;=:;:~;:~:= 0 ~~c;~~~~::: ~~~~~~ -~--- ~~~~~&~~~~~~~~~~~~~~ .~~~~~ ~~J~ L;;L~~ I ~~I~~~I ~~ ~I ~~~I~~#I~ ~~ I~~ (j ;;;:=~~"h~ ' ~#~~~ # ~~#~~'~~ .. :::.,.~ ... #b#~~=~~;~~~;~ ::::.::: :5:. - " '$." - " ~~" - " $: " - " .~" - " '$." =' ::r'· ~ L 0 5 .),8 11 111 11 11111111111111111 u t l,t llll l l l:llllh~~li:ltl'l1tll'ltlllllllll1lll,tllrtllltl , 1 I1l1 l t;J~ ltlllll l'l l 11111,' 1111111 11111 111 1 111 11111111111111111 111 1'lttlllttltll ll l1lt l l lll111111;I~i~lllllt lltl lll l l' llll1 IIIII i l ll ll' II I! llltl,;:, l C 2 tll~lljllll11tttllt dll!lh1llhlldll~ I I I I : I, I t II II I, II II~ t 111II 1,1,1,1,111, ,1,1,1,: ,I 11,11 1,1,1.1 11.1.1 ,I lid .I d~ddddlldddd~dddd , J,ld'I ' h~ Cl .11 .12 .e .I. ,I~ .I , ,I' .t " "I? ,I, 7,,1 .... 1~ .. 1; .. ~ ,ir, .t. . 1 • .t .1 .... 1. " .... 1" "1",,1,, d,\/iii l' ' f II. ' I) ' I ~ '1 '1 '1' 'IS'I' "ls"I"T '1 ' 1 1'1' ' I' ' I '1 '1 '1'3 v'iO - 0 Dl 6 7 " O~'V 02 ~'ml'l'l'll""l 111111"1"1111 [ 4'1111111 I'I'I'I'I'~'I'I'I'I'I'I'I'I'I'~'I'I'I'I'III' '111!1'I'I'I11' '1'I'I'I~'1II'I'1 l'I'III'I~II'I'I'I't'I'I'l'll 0 l""l"" !' II l ti l 11111" 11111111111111111111 l 'I' II I ' III'I'I'IIITI'I'III'I'I'I 'I 'I'~'1 '1'1'1 1I IIIIIIplili 111111111111111111111 1111 11 ' : 0 0 I 'I'~ 6 8 9 1 V'* STep.!. Opp. 75 0/7 02 Set 6'011 Cz STep 2. Opp. il7dex or C1 read .125017 f)1 Fig.9 ... ,--- ___ -.-Jf 0 ( o o 21. Squares and Square Roots usin~Dl D, and D. The D scole used here is the one under 0 1 and 0, on the back face. Opposite any number on 0 1 or D" read its square on O. '0 ~ ~ '--'-' Examples: \L(~I&·..y., ~~~- 'l.-.~ 11896'-";" Opposite 4 on 0" read 16 on O. IZIC,6"1\'HOO:=,lCS' _';" =- =cm , ~~ Opposite 16 on 0 1 read 256 on O. 111IZ6''1" - ::'= ~ , SZ90r _z}6t == =- O')posite J.47 on D 1, read 2.16 on D. SZ906&'~is ~l~CD SLB:~'/L -- ~~ Conversely, opposite any number on 0 read its square root on Dl or 0,. Lise 0 1 if ~ltt8'-Z~il =- ::-f"- .~~ the number of digits in the number is odd and D, if evcn. SZI8ZS"%s SZIS'_9Hl .~ ,189GC'j\, ~ =<:0- Examples: SZt8l'-z~iz ~~ ~ ~ N ~u Determining the number N when its logarithm is known is of course the revcrse 11/ operation. C. e Example: Find N if log N = 1.802. ~ ~ STEP J. Opposite the mantissa .802 on L, read 634 on C, (red). STEP 2. Since the characteristic is J there arc two figures to the left of the decimal point and N = 63.4. PROPERTIES OF CIRCLES Circumference = diameter X 3.1416. = J" Area = square of radius X 3.1416. o 1~3~O = square of diameter X 0.7854. 1 o 10 ::. Side of inscribed square == diameter X 0.7071. ." ~ (--= ci >= - Side of inscribed hexagon == radius of circle. -1 u U Length of arc == number of degrees in angle X diameter X 0.008727. Length of chord = diameter of circle X sine of % included angle. Area of sector == rength of arc X % of radius. h..-...-. Area of segment = area of sector minus area of triangle. 12 13