ALGEBRA. A LG E B R A. ſn Algebra we employ certain characters or letters to represen ese characters are separated by signs, which describe the operatio ht means, simplify the solution. 1. Whatever the value of any quantity may be, it can be represe haracter, as a. Another quantity of the same kind, but of different vº ng represented by b. The sum of these two quantities is of the same k\ of different value. For Addition we have the algebraical sign +, (plus) which, when between quantities, denotes they shall be added; as a -i-b, reads in t algebraical language, “a plus b,” or a is to be added to b. Another algebraical sign =, ſº that quantities which are placed on each side of this sign, are equal. Let the sum of a and b be denoted by the letter c : then we have, a-Hb=c, This composition is called an algebraical equation. The quantity on each side of the equal sign is called a member, as a 4-b, is one member, and c, the other. When one of the members contains only one quantity, that member is generally placed on the first side of the equal sign, and its value commonly unknown; but the value of the quantities in the other member being given, as a+4, and b=5, then the practical mode, to insert numerical values in algebraical equa- tions, Will appear; as, Equation, c=a+b, 4+5=9, the value of c. 2. The sum of three quantities a, b, and c, is equal to d, then Equation, d-a-i-b-i-c, - w . 4+5+9=18, the value of d. 3. For Subtraction we have the algebraical sign, -, (minus) which, when placed before a quantity, denotes it is to be subtracted as, a-b, reads in the algebraical language “a minus b,” or from a, subtract b. Let the difference be denoted by the letter c, and a-8. b-3 - Equation, c=a—b, =5, the value of c. 4. From the sum of a and b, subtract c, and the result will be d, then, Equation, d-a-Hb–c 8+3–5–6, the value of d. 5. When two equal quantities are to be added, as a+a, it is the same as to take one of them twice, and is marked thus 22. The number 2 is called the coefficient of the quantity a. If there are more than two equal quantities to be added, the coefficient denotes how many there are of them; as, - Equation, - - - - a--a=2a, 66 a-Ha-Ha-3a, £6 a-i-a-Ha-Ha =4a, déc., d c. When the quantities are separated by the signs, plus, or minus, they are called terms. 6. Multiplication.—When a quantity a, is to be multiplied by another quantity b, then a and b are called factors; and separated by no sign as ab ; which denotes that a is to be multiplied by b : but when the values of a and b are expressed by numbers, they are separated by the sign X (Multiplication); the result from Multiplication is called the product. Let a-8, and b-6, and the p duct of a and b, to be c, then, - . Equation, c=ab, - - 8X6=48, the value of c. 7. The product of a and b, is to be multiplied by c, and the latter product will be equal to d, then, - Rºuation, d-abc, - - 8X6X4S=2304, the value of d. d ALGEBRA. ºf a and b, is to be multiplied by c, and the product will wation, d = c (a+b Eq ’ ”TA: § # '-672 the value of d. the sum of two or more quantities is to be multiplied by another qu 'e sum is to be enclosed in parentheses, and denotes itself to be one facto er factor is to be placed on the outside of the parentheses, as seen in th ing example. *To the product of a and c, add b, and the result will be d: them, Equation, d = ac +b, 8X48-1-6 = 390 the value of d. Be particular to distinguish the two Examples 8, and 9. - - 10. The sum of a and b, to be multiplied by the sum of a and c; the product will be d, then, º - . Equation, a-gº §– 784. 11. The sum of c and b, to be multiplied by the difference of c and a; the re- sult will be d ; then, Equation, d = §§ &– 2160. | 12. Division.—When a quantity a, is to be separated into b equal parts, the | numbers of parts or b, is called the divisor, and the value of each part, is called | the quotient. The sum of the parts or the whole quantity a, is called the dividend; a and b, is separated by the sign : (Division); as a : b, reads in the algebraical | language, “a divided by b.” Let the quotient be denoted by the letter c, and a-gl&, b=6, then, Equation, c = a + b, - 18 : 6 = 3 the quotient c. In Algebra it is found more convenient to set up Division as a fraction, then it will appear as, \ 13. Divide a, by c, and the quotient will be b. Then, Equation, b =#. 18 - T= 6 the quotient b. nº The product of a and b, to be divided by c.; and the product will be d. Qn, Equation, d = *. 3. 18X6 15. The sum of d and b, to be multiplied by c, and the product divided by a; 36. then the result will be e. Equation, e = 5 c (d+b) —- 3 (36+6) 18 l 16. From the product of a and c, subtract 35; divide the remainder by the = 7. erence of a, and c; the result will be h. * - PROPORTION. +- - equin, A-º-; - 18×3—3×6 18–3 An old man said to a smart boy, “How old are you?” to which he replie" | “To seven times my father’s age add yours, divide the sum by dou'le Šh | difference of yours and his, and the result will be my age.” = 2.4. Letters will denote, a = the old man’s age, b = the father's age, c = the boy’s age. Then, & 7b--a y Fquation, c = 2 (a. b) the boy's age. - Now for any number of years of the old man and the father, will be a corres- ponding age of the boy; suppose, f - a = 73 years the age of the old man, b = 57 years the father's age. Require the boy’s age. _ 7×57+73 T E (73-57) =143 years. —h. A P R 0 P 0 R T I 0 N. THE relative value of two quantities, is obtained by dividing one into the other, and the quotient is called the ratio of their relationship. If the ratio of two quantities is equal to the ratio of two other quantities, they are said to be in the same proportion; as, - - a : b = c : d, roads in the algebraical language “a is to b as c is to d.”—a, b, c, and d, are call- ed terms, of which a is the first, b the second, c the third, and d the fourth term. The first and fourth are called “the outer terms,” and the second and third, “the inner terms.” The whole is called an “analogy.” A property in the nature of analogies is, that the product of the outer terms, #. ºual to the product of the inner be. Suppose a = 4, b = 9, c = 12, 4: 9–12 : 27, ad=bc, 4X27=9X12. If any one of the four quantities are unknown, its value can be calculated by the other three; as, a = — = ~ = 4 d 27 º b= ** = &ºi=9, C 12 _ ad 4×27 º •= —-----12. d= —” –** = —z---H---. s SIMPLE INTEREST. I M P L E IN T E R E S T. rest is a profit on money which is lent for a certain time. s Letters will denote. c == the standing capital, or lent money. . r = interest on the capital c, p = per cent. on 100 in the certain time. Analogy. - c: r = 100: p. If p is the per cent. on 100, in one year, then t = time in years for the stand- || , ing capital c, and the interest r. - ; : . Analogy, c: r = 100: pt. From this analogy we obtain the equations, . . . Cpt Interest, r = -īn-, • - - - 1, 100 * - Per cent., p = −, - - - - 2, & & 100 2- . . . Capital, C = –H– y º tº , sº • tº 3, 100 ºr - Time in years, t = ---, • - - - 4. Now for any question in Simple Interest, there is one equation which gives the answer. If the time is given in months, weeks, or days, multiply the 100 cor- respondingly by 12, 52, 365. . - Example 1. What is the interest on $378935, for 3 years and five months, at 6 per cent. per annum ? - t = 3×12+5 = 4.1 months, from the Equation 1, we have, 378935×6×41 Interest, r = Tiāxī00- Example 2. A capital c = $469.78, gave an interest r = 150.72 dollars, in a time t = 4 years and 7 months. Require the per centage per annum ? t = 4×12+7 = 55 months, from Equation 2, we have, _ 12×100×150.72 469.78 × 55 =776.81 Dollars. Per cent, = 7 per cent. Eacample 3. What capital is required to give an interest r = 345 Dollars in 6 years, at 5 per cent. per annum ? From the Equation 3, we have, Capital, C = tº: = $1150. Example 4. A capital c = $2365 shall stand until the interest will be r = 550 Dollars, at p = 6 per cent. per annum. How long must the capital stand? From the Equation 4, we have, 100X550 2365x6 | 12×0.876 = 10.512 months, 4×0.512 = 2.048 weeks, the time t = 3 years, 10 Time, - t = = 3.876 years. months, and 2 weeks. REBATE or DISCOUNT.-FELLOwsHIP. 1A. RE BATE OR DISCO UNT. ebate or Discount is an allowance on money which is paid before duo. : amount of money to be paid in the time t. By agreement the amount is paid h a capital c, at the beginning of the time t, but discounted a Rebate r, at p cent., so that the interest on the capital c, at p per cent., should be equal to Rebate r, in the time t. & = c + r. a p t tº 100(a —c) b * = -—. º & e Tºm t = – s © * 8. • afe, 100 + pt 5 €, c p - ºpilot, c=-“–. 6. Amount, a = *-(100+p 9. , 9. . 160 + p t 100 Percent, p-tº-o. e 7. Airtownt, a = i. (100 + p t). . . 10. - C p - * Now, for any question in Rebate or Discount, there is one equation that will give the answer. Example 5. A sum of money, a = 78460 dollars, is to be paid after 3 years and 6 months, but by agreement payment is to be made at the present time. What will be the Rebate, at 7 per cent.” 78460 × 7 × 3.5 Rebate, r= tº 2^ ZS ºf = $15439.91. 100 + 7 × 3.5 F E L L () W S HIP. , Fellowship or Partnership is a rule by which companies ascertain each fellow’s profit or loss by their stock. Each fellow’s part in the stock is called his share. The sum of shares is called the stock. Fellowships are of two kinds, Simple and Dowbie. Simple Fellowship, when there is no regard to the time, the shares or stock is employed. Letters will denote, : A = share of either one fellow. S = stock or the sum of the shares. : a = profit or loss on the share A. s = gain or loss on the stock S. Then, 4 : G = S: S. | Stock, s= **. . . 11. Share, A = *š. . . 18. r Gº, S | Gain or loss, s = *ś. . . 12. Profit or loss, a = #. . . 14. Example 1. A person had invested A =$11645 in a stock S-$64800, which gave a gain of s = $13864. What will be the profit of the person’s share : 11645 × 13864 & Pro a = tº: & tººf =$2491.45. yfit, 64800 Double Fellowship. When the different shares are employed at a differ- ‘ent length of time, each share is multiplied by its time employed, and the product is the effect of the share. Petters will denote, ** = time for the employed share A. a = profit of the effect e. T= mean time for the employed stock S. E = effect of the stock. e = effect of the share 4. s = gain of the effect E. * Then, e : a = E : S. 2 18 PERMUTATION. Formulas for Double Fellowship. Effect of A, -e-*.*. . . 15. | Time, = #. tº º E} Profit := es. e tº • 16. Share => 0, & ºn © oft of , a =# tare, t S Effect of S, E= *š. © º 17. Meantime, T= 6 s. gº G 0. a S Gain of E, s–º. . . 18. Stock, S= **. . . 6 a T' | Example 2. A canal is to be dug, and requires an effect E = 76850 (men and days) to be accomplished; after that it will give a gain s = 12390 dollars. An em: . ployer has A = 168 laborers. How many days must those laborers be employed at the canal, that the employer will obtain a profit a = 5000 dollars? Time, t = 9000X76880 -1846 days. - 168 × 12390 P E R M UTATION. . Permutation is to arrange a number of things in every possible position. It is commonly used in games. § - Bacample 1. How many different values can be written by the three figures 2, 3. - : 'I X 2 × 3 = 6 different values, namely, - 123, 132, 213, 231, 312, 321. - With any three different figures can be written six different values. Any thre things can be placed in 6 different positions. - Ezample 2. How many names can be written by the three syllables, mo, ta, la # The answer is, Motala, Molata, Tamola, Talamo, Lamota, Latamo. Example 3. How many words can be written by the five syllables, mul, tip, li, ca, tion f 1 X 2 × 3 × 4 × 5 = 120 words, the answer. COMBINATION. Combination is to arrange a less number of things out of a greater in every possible position. It is commonly used in games. Example 1. How many different numbers can be set up by the nine figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, and three figures in each number ? 9 × 8 × 7 1 X 2 × 3 Example 2. How many different variations can a player obtain his cards, when | the set contains 52 cards, of which he receives 8 at a time? 52 X 51 X 50 X 49 × 48 × 47 × 46 × 45 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 If there are four players, and p r . 4 = 24, they can play 24 × 752538150 = 18,060,915,600 different plays. If it takes half an hour for each play, and they play 8 hours per day, it will take 18060915600 2 X 8 = 84 different numbers. = 752538150 variations. ! = 1128807225 days = 3092622 years. ALLIGATION. 19 AL LIGATION. Alligation is to mix together a number of different things of different price or value, and ascertain the mean value of the mixture; or from a given mean value of a mixture ascertain the proportion and value of each ingredient. Let the different things be a, b, c and d, etc., their respective price or value per unit, 2, y, z and w, etc. A = a + b + c + d, etc., the sum of the things. JP= mean value or price per unit of A. Then, A P= a z + b y + c 2: + d w --, etc. © º º © 1. and P-a-row Hatawi, i. - - - 2. Example 1. If 3 gallons of wine, at $1.37 per gallon, 2, at $2.18, and 5, at $1.75, be mixed together, what is a gallon worth of the mixture? - A = 3 + 2 +5 = 10 gallons. P_3}× 1.87 + 2 × 2.18 -- 5 × 1.75 - - 10 Alligation of two ingredients, a and b, with their respective prices or value per unit, z and y. zX PX y. A = a + b. a : b = (P –y): (2—P). e © e e e 3. a =9CP-2, and a = 4GF=9 . . . 4 & 5. - ... (z = P) (2–3) Example 2. A silversmith will mix two sorts of silver, one at 54 and one at 64 cents per ounce. How much must be taken of each sort to make the mixture worth 60 cents per ounce? (Formula 3.) P= 60. a = 54. y = 64. a : b = (60–54): (64–60) = 6:4, or 4 ounces, at 54 cents, and 6 ounces, at 64 cents. =$1.72 per gallon. # Alligation of three ingredients, a, b and c, with their prices or value per unit, 2, y and 2. aſ : c’ == (P–3): | -P). . . . . . . 6. a/ ; b = (P–y): (2 — P) when ££% 35. o • 7. b: c’’ = (P-2): º when z jº: <º o 8. a = a + a”, c = c/+ cº. - Example 3. A farmer will mix wheat, at 94 cents per bushel, with barley, at 72 cents, rye, at 64 cents per bushel. How much of each sort must be taken to make the mixture worth 80 cents per bushel? (Formula 6.) z = 94, y=72, a = 64, and P=80. aſ : c’ = (80–64): (94–80) = 16:14. a/: b = (80–72): (94–80) = 8: 14. The wheat a = 16 +8 = 24 bushels, at 94 cents per bushel. &g barley b - 14 46 66 72 {{ 4t &4 rye C = 14 & 4 & 64 4& $º Alligation of four ingredients a, b, c and d, respective prices or value per unit, 2, 3), 2, and we a : d = (P–w): (2 — P) 9. b: c EğTº;;−}} when z > y) - P -a, -w- {1. a': d = (P–w): (z—P) 11. a’’: b = (P — y ićI; when z > PX y > a. Pw - 12. %: EğTº z—P) 13. a = a + aſ + a”. a : d = (P–w): (z—P) 14. b: d" = (P–w): (y—P)}-when zX y > a. P. PX w 15. c: d" = (P–w): (2 – P) 16. d = d/ -- d" + d”. - In the same manner, formulae can be set up for any number of ingredients. & e ARITEIMETICAL PROGRESSION. AR IT H M ETICAL PR 0 GRESSION. Arithmetical Progression is a series of numbers, as 2, 4, 6, 8, 10, 12, &c., or 18, 15, 12, 9, 6, 3, in which every successive term is increased or dimin ished by a constant number. Letters will denote, a = the first term of the series. b = any other term whose number from a is n. n = number of terms within a and b. 3 = the difference between the terms. S = the sum of all the terms. In the series, 2, 5, 8, 11, a = 2, D = 11, n = 4,3 = 3,and S= 26. Mºº-When the series is decreasing, take the first term = b and the last term = 0. The aecompanying Table contains all the formulas or questions in Arithmeti- cal Progressions, and the nature of the question will tell which formula is to be used. Formulas for Arithmetical Progressions. b—a a = b–3 (n-1), - - 1, 3 = n–i’ * - ºn 9, 2S s_Q tº)(?-?), a =#–8, - - - 2, * TºS-2-ET , - - 10, _ _S § & 2 (S.–an) Q = n —g(n-1), 3, 3 = 7G-T)" ſº gº II, 2 (bm— b = 2+3 (n—1), tº tº 4, 3 = # =} º * * • 12, *-*-a, - - - 5, s=****) - - - 13, 72 2 , b = #4; (n-1), - • 6, S = Głºgº-º, º gº 14, 3. n = ~ +1, - - - 7, S = * [at; (n-1)] 15, 2S § *-ār; . . • 8, s= aſh-a-D] - - 16, 3. * 3 Nº •-ºv/gº-as º º wº gº 17, _ 3 + 3. *-*VG-5) as . . . . . _1_2 1 aw” 2S n = g-F# ;-) ---, - - - 19, =1--" 1 b \* 2S º 2 tº i. ++) ~5-> sº tº tº º 20. 5 ARITIIMETICAL PROGRESSION. 21 Jºrample 1. A man was engaged to dig a well at one dollar ($1) for the first foot of the depth of the well, $1.84 for the second, and 84 cents more per every successive foot in depth, until he reached the water, which was found at a depth of 25 feet. How much money is due to the man? This will be answered by the formula 15, in which a =1, d = 0-84, and n = 25, then the sum, S = 25 [1+ **as *ºr 1)] = $277 the answer. Example 2. A Propeller ship which is to run between Philadelphia and Charleston, cost $116500, of which the company agreed to pay on account $14075 at her first trip to Charleston; and per every successive trip, they paid $650 less than the former. How many trips must the vessel make until she is fully paid : S. Th; will be answered by the formula 20, in which b = $14075, d = 650, and 1 14075 iſors TV 2×116500 g Arithmetical Progressions of a Higher Order. Arithmetical Progressions are of the first order, when the difference 3 is a constant number, but when the difference 3 progresses itself with a constant number, the Progression is of the second order. When the difference 3 progresses in a second order, the Progression is of the third order, &c., &c., and is thus explained: 1, 2, 3, 4, 5, 6, . . . . n, - - Arith. Prog., first order. ?: (n+1) 1, 3, 6, 10, 15, 21, . . • - 2d. order. n (n+1)(n+2) 2X3 p 1, 5, 15, 35, 70, 126, . tº: * - - - 4th order. Here you will discover that the sum of n terms in one order, is equal to the same nth term in the next higher order. Arithmetical Progressions of the first, second, and third orders, are applied to PILES OF BALLS AND SHELLS. Example 1. A complete triangular pile of balls has n = 12 balls in each side. Require how many balls in the base, and how many in the whole pile? 1, 4, 10, 20, 35, 56, • - 3d. order. In the base, e = lagº) = 78 balls, • * ~ 2d. order. TWhole pile, - - -*#2-sº balls, - - 3d. order. Square Piling. 1, 4, 9, 16, 25, 36, - - - - nº • - - - - 2d. order. 1, 5, 14, 30, 55, 91, - - natºr, - - - 3d, order. [See Examples 2 and 3 on page 23] 22 GEOMETRICAL PROGRESSION. —t— G E 0 M ETR I C A L P R O GR E S S I ON. Geometrical Progression is a series of numbers, as 2:4: 8:16:32: &c., or 729:243:81:27:9: &c., in which every successive term is multiplied or divided by a constant factor. Letters will denote, a = the first term of the series. b = any other term whose number from a is n. n = number of terms within a and b. r = ratio, or the factor by which the terms are multiplied or divided. S = Sum of the terms. In the series 1: 3: 9:27: a = 1, b = 27, n = 4, r = 3, S = 40. The accompanying Table contains all the formulas or questions in Geometrical Progressions. The nature of the question will tell which formula is to be used. - Formulas for Geometrical Progressions. b rt-l b *=== º tº- e l, r-R/; e º º 7, S — a = S-r (S – b), - - 2, r=#=# - - - 8, r—1 a = S -i, , e e 8, ar"+S- rS — a = 0, º 9, br— b = arº-1, tº - e 4, s=#EH, tº º º 10, b-s—tº, - - 5, s-ºr-9. . . . 11, * r – 1 r — I b (r" — 1) b = s(+)-, 6, s= #F#, - - 12, º, + I+ *: logº, º & - tº º º º 18, bog.b — log.a. *-i-Hºº-º-y - - - - - 14. ſº =m log.ſa-ES(r-1)] —log.a. log.r. 9 _1 , log.b-log.[br— S(r—1) * = 1-1 log.r. “H), - - - - 16, *R/5 – "Wa GEOMETRICAL PROGRESSION. 23 Example 1. Required the 10th term in the Geometrical Progression 4:12:3t....? Given a = 4, n = 10, and r = 3. We have, Formula 4. b = arm-1 = 4X39 = 78732, the tenth term. Eacample 2. Required the sum of the 10 terms in the preceding example? Formula. 11, S = -1 = —g—— 118096, the sum. Example 3. Insert 6 proportional terms between 3 and 384? Given a = 3, b = 384, and n = 6+2 = 8. rt—l 7 Formula. 7, * = V/# F *–2, a 8 then 3: 6:12:24:48: 96: 192: 384, the answer. Example 4. A man had 16 twenty dollar gold pieces, which he agreed to ex- change for copper in such a way, that he gets one cent on the first $20, two on the second, four on the third, and eight on the fourth, &c., &c.; until the sixteen $20 pieces were covered. How many cents will come on the sixteenth gold piece, and what will be the whole amount of copper on the gold? In the progression 1 : 2:4: 8: &c., we have, Given n = 16, r =2, and a = 1, then, 216 48 4. Formula 4. b = 1X2*-* = 3 = :-- 1. - *- 32768 cents, on the sixteenth piece. - The total sum of cents will be found by the Formvºula 10. s=** = C5585 cents = $65535. Piling of Balls and Shells-(From page 21] Example 2. IIow many balls are contained in a complete square pile, n = 10 rows 7 10(10+1)(2X10+1)_10×11X21 2X3 ======885 balls. Rectangular Piling. Let m be the number of balls on the top of the complete pile, and n = num- ber of rows in the same, then the number of balls in the whole pile will be, n(n+1)(2n+3m—2) 2×3 2 The number of balls in the longest bottom side will be = m—-n — 1. tº gº Sd. order. Bæample 3. The rectangular pile having 15 rows and 23 balls on the the top, how many in the whole pile? —2 15 gºtº }= lºg = 2680 balls, 24 COMPOUND INTEREST. 0 0 MP 0 UN D IN T E R EST. Compound Interest is when the interest is added to the capital for each year, and the sum is the capital for the following year. % */2 Amount, a = c(1 + p)". • 1. Percentage, 2 = ižº — 1. . 3. capital, c=–4–. . . 2. Number of years, n = **="#. (1+p)” log.(1 + p) AEG In these formulas p must be expressed in a fraction of 100. Example 1. A capital c = 8650 standing with compound interest, at p = 5 per cent. What will it amount to in m =9 years? Amount a = 8650 (1.05).9 = 13419 dollars. Example 2. A man commenced business with c = 300 dollars; after n = 5 years he had a = 6875 dollars. At what rate did his money increase, and how soon will he have a fortune of 50000 dollars? The first question, or the percentage, will be answered by the formula 3. p=i/ § –1 =1/22,9166–1=0.81, or 87 percent. The time from the commencement of business until the fortune is completed will be answered from the formula 4. * % = log:50000–log.300 4.69897—2.47712 log.1.87 0.27 20048 or 8 years and 2 months. = 8.169 years, Compound Interest Table, CALCULATED FROM ForMULA 1. This table shows the value of one unit of Q? COMPOUND INTEREST. money at the rates of 5, 6 and 7 per cent. per annum, compound interest, up to 60 years. Example 1. What is the amount of 864 pounds 1.1025 | 1.1236 | 1.1449 º 12 years, at 6 per cent. compound 1.1576 | 1.1910 | 1.2250 | T T. 4. - I ºf * 1.2155 | 1.2625 | 1.3.108 lº,201.19% sai-Iissiºns, or £1738 1 2 3 4 § 1.2770 | 1.3382 | 1.4025 JEacample 2. What is the amount of 3450 dollars 7 8 9 Years. 5 per ct. 6 per ct. 7 per ct. 1,0500 | 1.0600 1.0700 1.3400 | 1.4.185 | 1.5007 for 18 years. at 5 per cent. compound interest? jºij | jºg |iº || “...ºs. 1.4774 #; 1.7182 When the interest is compounded in more or 1.5513 | 1.6895 1838; less than one year, at the rate of interest per 10 1.6289 | 1.7908 lº. year, and m = the number of months in which 11 1.7.103 || 1.8983 || 2.1048 the interest is compounded; 12 1.7958 2.0122 || 2.2522 & e 777, 13 | 1.8856 2.1329 2.4098 || Then, instead of p in the formulas, put º y 14 1.9799 2,2609 || 2.5785 - º # |###|###|##. and instead of n, put + 16 || 2.1829 || 2.5403 || 2,9522 JEacample 3. A capital of 500 dollars bears # ; ; ; cº interest ºuis, at 5 per . º & .d5! er annum : What will it annount to in 10 years # : | tº: | 3:... per annum; what will it y 20 || 2.6533 3.2071 || 3.8697 m = 6 months, p = + = –H–– 21 2.7859 3.3995 || 4.1406 12 12 22 || 2:3252 || 3:3035 | 4430; and n=1}^ 19–20, 23 3.0715 ; ; 6 24 3.2251 .0487 5.072 then (Z = c(1 + p 31, = 500(1 + Q.025)20 - 25 3.3864 4.2919 5.4274 šig3.11 dollars, the ºr. 35 5.5166 7,6861 |10,6766 20 40 7.0400 10.2858 14.97.45 -** 45 8,9850 13.7646 |21.0025 9214.89 50 |11,6792 (18.4190 |29,4570 log. 500 = 2.69897.00 60 |18.6792 |32.9878 |57.9466 Amount, 8.193.11 = 2.9134480 ANNUITIES. 25 A N N UIT IES. Annuity is a certain sum of money to be paid at regular intervals. A yearly payment or annuity b is standing for n years; to find the whole amount a, at p per cent. interest. Amount, a-vaſiº gº ºsmyloist • - - - 1. b Amount, 0. =p' [d +p)"— 1] Comp. Int. . . . . 2. A yearly payment or annuity b is to be paid for n years;...to find the present Worth, or the amount a, which would pay it in full at the beginning of the time 7, deducting p per cent. interest. Amount, a = bn | -º-; #( * + 1 ) Simple Int. . . . 3. 2\1+p? 4mount, Q = b [i * Tººl Comp. Int. & tº gº • 4. p (1 + p.)” A debt D, standing for interest, is diminished yearly by a sum b : to find the debt d after 7 years, and the time z when it is fully paid 7 The debt d after ºn years will be— % d=92 —b)(1 + + b Comp. Int. . . . 5. £9 The time n until fully paid will be— m = {299–1990-PP). . . . . . 6. log.(1 + p) If b = D p, then n = o, or the debt D will never be paid. If b&D p, the debt D will be increased. To find the yearly annuity b which will pay a debt D in n years, at p per cent. compound interest? b = D p(1+p)”. e tº & tº { } * 7. (1 +p)”—1 Annuity Table, Showing the present worth of an annuity or rent of one unit of money, at 5, 6 and 7 per cent. compound interest for years up to 60, calculated from formula 4. Years. 5 per ct. 6 per ct. 7 per ct. Years. 5 per ct. 6 per ct. 7 per ot. , 0.9524 0.9434 || 0,9345 17 II.2741 10.4772 9.7.632 1.8594 1.8333 | 1.8080 18 II.6896 10.8276 10,0591 2.7232 2.6730 2.62.43 19 12.0853 11.1581 10.3356 3.5459 3.4651 || 3,3872 20 12.4622 11.4699 10.5940 .. 4.2123 4.1001 21 12.8211 II.7641 10.8355 5.0757 4.9173 || 4-7665 22 l3.1630 12.0416 II.0612 5.7864 5.5824 5,3892 23 13.4881 12.3034 11.2722 6.4632 6.2098 || 5,9712 24 13.7986 12.5503 11.4693 7.1078 6.8017 | 6,5152 25 I4.0939 12.78.83 11,6536 10 7.7217 7.3601 || 7.0235 30 l6.3724 13.7648 12,4090 4 3 2 9 5 11 8.3064 7.8868 || 7.4986 35 16.3742 14.4982 12.94.76 12 8.8632 8.3838 7.9426 40 17.1591 15.0463 13,3317 13 9.3936 8.8527 S.3576 45 17.7741 15.4558 13.6055 14 9.89S6 9.2950 | 8.7454 50 18.2559 15,7618 13.8007 15 10.3796 9.7.122 || 9,1079 55 18.6334 15,9905 13.9399 16 10.8378 || 10.1059 || 9,4466 60 18,9292 16.1614 14.0389 * 26 "Calculus of Differentials. By the Differential Calculus we ascertain the simultaneous progress of variable quantities depending on one another. The variable quanti- ties are designated by the last letters w, v, ac, y, z, and the constant quantities by the first, a, b, c, e,f, of the alphabet. The letter d is placed before variables to denote the instantaneous progress of that quantity, as day, and called the differential of a. d’ reads differential. Let the side of a square be denoted by a and the area by 2: ; when aſ increases uni- formly, z will increase more rapidly. When a = 1, 2 = 1, but when a = 2, 2 = 4. When we know the instantaneous increase of a, what will be that of 2%. If we add, say only a point to the side º there will be added two lines or 2 as to the square. We know that z = a,", the d" or increment of the square will be d2 = 2a: dæ, of which da' is the point added to a and 2a is the two lines added to the square, called the differential coefficient. Let v denote the volume of a cube, and ac its side, we have v = a," and dv=3a*da, which shows that if a point da; is added to a there will be 33% or three squares added to the cube. The dº of any power of a variable is equal to the power diminished by 1, multiplied by the primitive exponent and the product by the dº of the variable. The d" of a constant is = o. When the constant is a factor to the variable it appear unchanged in the d' coefficient, but when a term it disappears. I. The d' of length w of any line defined by a formula of rectangular co-ordinates w and y, is du = x/ dº-Fdy”. II. The d of area 2 of any plan figure bounded by a curved line and rectangular co-ordinates is d2 == y day, y = ordinate, a = abscissa. III. The d" of solidity v of any figure bounded by a plan rotating round its abscissa ar, is dv = try”da, gy = ordinate of the outer line of the plan. IV. The d' of surface 2 of any solid bounded by a plan rotating round its abscissa aſ, is dz = 2 try du, in which w = length of the outer edge of the plan. successive is is when the first i coeff. is considered a function of a new function. du, du,\ dºw. - 4 g sº- * e amº *-* =ms, * 2 w = a act. 1st. d * =4a a ", 2nd. d’ coeff. a(; )= # = 12 a wº, 3d. d’ coeff. aſ dº...) == #: =24 a ac, etc., etc. dºu means the second, dºw, the third d' coefficient of u. daº" means the square of the d' of a, etc. Eacample 1. The diameter of a sphere increases at a rate of da = 2:31 inches per second, when a = 9-5 inches, at what rate (dv=?) does the Traº volume v increase? v=": = 0.523a*. du = 0.523X3 aſſº da = 1-569X2:31 = 327-1 cubic inches, the answer. Eacample 2. It is found that the displacement of a ship increases as a's the draft of water. At the load draft a = 18 feet the displacement is T= 2000 tons. Required the displacement (t=?) when a = 12 ft. and how much (dt. 1) can the vessel be loaded per da = 1 inch or 1–12 foot, at that draft, I 5 1.3 * t = w” T_12"X2000_ 1088.6 tons, at a = 12 feet draft. a 1.5 -º-º: 18, 5 g % e /T· dt = 1.5 T2%da 1 5X2000XV12 11:34 tons per inch. zº T - T1STSX13 in. The following page contains the differentials of formulas and trigo- nometrical functions. l means the Naperian logarithm. The common logarithm log. multiplied by 2-302585 gives the Naperian logarithm l. Calculus Differential Formulas. 27 FORMULAS, DIFFERENTIALS, FORMULAS, DIFFERENTIA LS, ri; dy=da, 1 Glº = a l a da, 21 da: g=aaº d y= 2 a acda, 2 d'l'a: : -i- 22 g=atº dy= man-i da, 3 a: l'ac e (1-Hl'a)da, 28 3 abaº = 9a baºda, 4 lº – (1-º)º, 24 a:n a:n-l-l 4 a bºaº = 4n a bºan-1 da, 5 º st: (la-1)da: 25 l'a. (l'ac)º –a acº d a-t-arº = 3aºda, 6 º – – ºººº ºº 2s A/aº-Hyº vGF) (a+b)aº = 2a (a+b) da, 7 | a-2ba 2b a da (a-Hb a)º (a-Hba)º 6a bºaº–c = 18 a bºaºda, 8 vºa=aº - da: 28 2.Ma a +32º–v = da-H62 dz-dv, 9 | (aa-Haº)” = n(aa-Haº)n-1(a+2a)da 29 63 º 2;2 2,3 7,3 ba: da: 30 +4aº–3a-(18aº-H3 aa –3)da,10 v aº-Hba, Ma Ebaº a: viº = v da-H2a v dv, 11 dº(a vº) = 6 aa daº, 31 da . dv . dz 2 = - - - - - - - 3C 5 avz(")ia dº(aaº) = r 6 a daº, 32 a (aº–a bº) = (3 aº–bºa) da, 13 dº(aaº) = ob a aº-ida –o, 33 a 2 2 a: v da –acº d v serie 14 sin. v = +cos. v dv, 34 6, a da: a - acº 15 cas.t = – sin. v dv, 35 (l 72 a acn-1 da: dv an 2- – IT 16 tan. v -. cos3 v 36 3 a)ºda (ervo – º 17 | coto – – ri- si va SiIn, º ) I - - --Il cos. v dv an 1 - (a+ Va)m = m (a+ Va i dac. sec. v = + i se | 1 da: cos. v dv l M(a x) ti: (ata) TFT 19 COSeC, 29 t: - sinºv,v, 39 sa- sms al 2 2y2 a ctaº - 2 d da 20 | Tant. for any curvet=ys 1+i 40 2, - ºv 2 aa –aº se- 28 Calculus of Integrals. The Integral Calculus is the reverse of the Differential, or to find the original formula of a given differential. The symbol fis placed be- fore tae d’ to denote that the integral is to be taken out of it, or that the original formula is to be found. 2+1 3 da; The d" of a a "=3 a aº day, and ſ 3 a aº da = —--a aº. Rule to find the integral. Add 1 to the exponent of the variable as in the d', divide the d' by the new exponent, da will disappear, and the quotient is the integral. The integral ſ does not effect a constant. A constant term in a formula disappears in its d', ..", any inte- gral may have a constant term, whose value is determined by making the variable in the integral = o, when the first member in the formula will be a constant. It is therefore customary to add a constant C to the integral. When it is known that the first member is = o at the same time the variable in the fis = o, then C= 0. When a differential is to be integrated between two limits of the invariable, say a = a and a = b, it is indicated by ſ ".. or f "3 c & de = c(b°—a”). Successive differentials are accompanied with the same order of inte- grals, as ff.6 a da” = ſ.3 aſſº da = a ". The integrals of the differentials gives the formula for the problem. Eacample 3. It is required to find by the calculus a formula for the area 2 of a rightangled triangle. Proposition II, page 78, d2 = y d; the formula for the hypothenuse is y =da, dz- a a day, 2 = ſa a da = *-** or the area z is half the rectangle of the sides aſ and y. Ea'ample 4. Find a formula for the convex surface 2 of a cone, whose side is w, and r = radius of the base? Prop. IV, page 78. dºz = 2 irr du, and z = ſ? T r dw = m ru, the answer. Eaxample 5. Find a formula for the area 2 of a circle, when it is known that the circumference y = 2 tra º Prop. II, page 78, d2 = y da =2 it a day, and z = f 2 tra da = traº the answer, a = radius of the circle. Eacample 6. Find a formula for the area. 2 of a parabola of a = abscissa or height, and y = ordinate, or half the base ? Formula for a parabolº g = Vºa, in which p = the constant parametric diameter, or p = #. º 2 */3 2 -- w=4, aw-*.*.*. Prop. II, dz = yde-3 ºi. and z = *2 y” dy _*º- p p 19 p 3 p 2 a y , the answer, or the area of a parabola is % of the base by the height. Ea’ample 7. Find a fºrmula for the volume v of a paraboloidº Prop. III. dw-try” da = ** way, and v=ſ**** = " ', but p = *and 19 - - ſp 70 2p 3C #!" ac, the answer. V. The center of gravity s from the origin of ac, of any plan figure bounded by a curved line and rectangular co-ordinates is s—ſº lº when z area of the plan. y; º center of gravity 8 from the origin of a of any solid figure is Q: 2, S= 1) Ea’ample 8. Find a formula for the centre of gravity (s = ?) of a cone. 5 when z = ordinate cross section and v volume of the same. The ordinate cross-section z = try”, and v = +y a, when a = height and Q y radius of the base of the cone. Prop. vi. s = {*** = 3 fºr a y” da; Tr ºf a; As the center of gravity is not influenced by the proportion of a; and y, We can make y = a, when s = *Jº” ºf º-s: ac, from the top. Trac" 4 tra." *** * * * * Caleulus limtegral Formulas. 29 I) IMFFERENTIALS, INTEGRALS, I DIFFERENTIALS. INTEGRALS, gº da: Jaz-z-Hº Jºão---C, 1 ſºvia - reirº, a J4 a a "da: = 4a ſa" da = a a "+ C, 2 jamaa, = m b%–m a”, 22 +1 & 771, ..fair day, = #: + C, 3 fm a da; 2-3 #(º-a”), 23 CO zºº.” 2a.” J. da —t- 24 o - - da: ſº-ſº *a*= 2}/a;+ C, 5 ſº- #, ya; : Vas–3 da; -1 º 6 b a --- a da = l'ar-H C, 6 =–ſ J.-J.-Hſ 26 & da; –2 1 g da: cos.ac-i-C 27 -a-- a "da: = — ++C, 7 j sin. a. -ac * ** 2 #-- 3'da' = +a+ C, 8 Jºcos. a da = sin.a;+C, 28 Q} ~4 ſ aw-Hº-)a== *#4-by--C,9 J'tan. a da = — tº cos. 4.—HC, 29 2)/a. 4 a dº are to to ſcot was - – B sinº FC, so &; 2 bda: da: e &: a-Ha. T bl’(a+a;)+C, 11 J; = 1 tan, ; + c, 31 3 a aº da; *— — i. (; #) *# = totazºc, *| ſiz -*(i+3)+** 2 * 1 . faacda;+3aºdaº—bºda = º:+a+-bºa-HC, f sim.a cos.a. da = 2. sin”. a-HC, 33 CO sin,ba, Tr J(a^+b”) da = x(a^+b^)+C, 14 ſ==da = Tº 34 £os bac º 2 º far-ºydz-zº-az-i-º)+c. ſ==da = CºC), 35 dt - - J3(a a-aº)*(a—22)da =(aa-aº)"+C, f1-Hº = circle arc of which testant. ſº = y/aº-Fºr-HC 17 /*= = circle arc of which Ma-Fºr - n-HC, A/22–23 a = sin. Versus. 37 #. 1:4 C, 1s |ſſföadaº–ſſGazdaº–ſsadºda-ar"--C 98 ſyāº-R*a*=#Vºf- *H: *(*-i- |ſſ? (a+b) da' = (a+b)++C, 39 Yaº-H22), /yº-Fººds – #2(VFFF+c, 20 Jºſzvºda"-i-8 va, dz dw-H2 zºdvº-a wº,40 30 Maxima and Minima. Two variable quantities a and y depended on one another, to find the value of one, when the other is a maazima or minima. * lis a maxima or minima when its Jay T" first differential coefficient dy + = 0, dº da, When the second d' coef. a’y is positive, y is a minimum, and when da:3 negative y is a maximum. The variables may have both maximums and minimums, as formulas will indicate. Eacample 1. Find the value of a when y is a maximum or minimum, in the formula y = a –12a:-H22? dy= (3a*—12) day, #– 3a*— 12 = 0. - d? of which w = V*=2 the answer. # = 6a, which is positive, conse- quently y=2°–12X 2-H22=6, a minimum, when a = 2. Eacample 2. It is required to cut out the strongest possible beam of height h and breadth b, from a log of diameter D, *:::::: page 174? The strength of a beam is in proportion to bh” which is to be a maximum. D*=b2+h”, h°– D*—b°, b h9– b (D9—h”), d(bh”)=(D°–3b*) db. d (bh?) b # =D"—80%–0, of which the breadth b--D}/33=0.577 D, and height h = V/D*—5% = D.V0.6666 =0-8164 D, the answer. The second dº coef. d2(b h9 ( º = —6 b, which is negative, and therefore b hº is a maximum When b = 0°57'7 D. Eacample 3. It is required to know the proportion of heighth h and diameter D of a cylinder, having the greatest cubic containt v, with the 2 smallest surface ‘2 including top and bottom 7 2–º + ºr D h=} + trl) h, which is to be a minimum. Set v = 1 and D=1, then ==}+rh, and 4. dz 4 T az-(1-#) dh, ji=t-Ha- o, when A-Jº = 1-1284D, the answer. d2 2 The second dº coef. # - +% which is positive, and 2 a minimum when h = 1°1284 D. Maclaurim’s Theorem. . Maclaurin's Theorem, explains how to develop into a series a function With One variº, as ..., d2 tº /dº ຠ/07/ a; Qſ, t 20, a;n dnut 1, =(u)++(#. + 3\d. +2×3 da; “tºx3x. 70, #) etc. where the factors in the parenthesis is that which it assumes when al-o. The function u, *::: developed into a series will be 1 1 a , ºº acº acn —— = -—--|--—-...— ... etc a-Ha. T a a as Ta'an-Fi' " Taylor’s Theorem. Taylor's Theorem, explains how to develop into a series a function of the sum or difference º: two yº. w = *H, _2,+%, Lºº; /*-ī-9" 9 L *** 9" F@#y)=w:#y+...+. ăzătº 3x3 ...xn. where w represents the value of the function when y = o. -* Interpolation. 31 * * Interpolation is to insert numerical values between given data, for constructing tables or empirical formulas expressing the probable rela- tive variation of quantities. Let a, and y be two variable quantities de- | pending on one another and measured in simultaneous stages of their progress, as all a’, as a , and as 91 ya ya y, and y, We have y = Ayººby, FCys-HDy, HEy, F&c. - - - - 1 2 3 4 5 given \/ \/ \/ \/data. A_{*-*) (*-*) (*-*) (*-*). (a 1–2.) e-º (a 1–2,) (wi-a:) , _(2–2) (2–2) (2–2) (w-º). º T(cº-w,) (a, -w,) (w-w,) (22–w.), § 2 - - - - - | º 3 C (a – a 1) (a —a,) (2–2) (ac — 2.) • § (ws—w,) (ws—w,) (als—w,) (ws—w.) : ‘ā a- - - - - - - - -] . . . . . Cº- _(2–aci) (a –a.) (2–2) (2–2), : D=== a’i) (2-2) (2-2's) (wº- a'.) Ä 4- - - - - - - - - - - - -- s # g_{*-*) (*-*) (*-*) (*-*): 2. (als— ah) (a),— a.) (aºs— a's) (acs- w) D The values of the coefficients A, B, C, D, and E, with their given data, inserted in formula 1 gives an empirical formula for the variation of 2: and y. The number of observations or given data of a and y should be one more than the order of progression. In arithmetical progression two observations are sufficient for a correct formula. For all curves in the conic sections, or others which are of the second order, there should the at least three observations. Pressure of steam progresses with the temperature in the 6th order, for which requires seven observations to make a correct formula. When the order of progression is not known, the more observations gives the most correct result. Eacample. Let y represent the boiling-point of salt water and a the percentage of salt in solution. It is found in three experiments, that a 1–3, a 2–18, a's=36 per cent. salt. when y=2132 ya=219°, g3=226° boiling-point. Find a formula that will give any intermediate value of a; and y? _(2–18)(x-36) =sr (2-3)(x-36). C= (3–3)(3–18) (3–18)(3–36).” (18–3)(18–36)' (36–3)(36-18)” y=2132A-H219 B+226 C. y=0'42-H212 32 UNITED STATES STANDARD MEASURES AND WEIGHTS. *. UNITED STATES STANDARD MEASURES AND WEIGHTS. MEASURE OF LENGTH. - THE Standard Measure of Length is a brass rod = 1 yard at the temperature of 32° Fahrenheit. The length of a pendulum vibrating seconds in vacuo, at Philadelphia is 1-08614 yards, at + 32°Fahrenheit. The surveying Chain is = 22 yards = 66 feet. It consists of 100 links, and each link = 7-92 inches. - - ROPES AND CABLES, 1 Cable length = 120 fathoms = 720 feet. 1 fathom = 6 feet. GEOGRAPHICAL AND NAUTICAL MEASURES. 1 Degree of the great circle of the Earth round the Equator = 69.032 statute miles = 60 Nautical miles. - 1 Statute mile = 5280 feet = 0-86875 Nautical miles. 1 Nautical mile = 6037°424 = 1°150 Statute miles. - LOG LINE. - The Log Line should be about 150 fathoms long, and 10 fathoms from the Log to the first knot on the line. If half a minute glass is used, it will be 51 feet between each succeeding knot. For 28 seconds glass it will be 4.7-6 feet = 7-93 fathoms per knot. This is the length of knot by calculation, but prac- tically it is shortened to 7-5 fathoms per knot for 28 seconds glass. - MEASURE OF CAPACITY. Gallons The standard Gallon measures 231 cubic inches, and contains 8:3388822 pounds Avoirdupois = 58372-1757 grains Troy, of distilled water, at its maximum density 39.83° Fahrenheit, and 30 inches barometer height. Bushel. The standard Bushel measures 2150-42 cubic inches = 77.627413 pounds Avoirdupois of distilled water at 39.83°Fahrenheit, barometer 30 inches. its dimensions are 18; inches inside diameter, 19% inches outside, and 8 inches deep; and when heaped, the cone must not be less than 6 inches high, equal 2747-70 cubic inches for a true cone. . - Pousade. The standard Pound Avoirdupois is the weight of 27-7015 cubic inches of distilled water, at 39.83° Tahrenheit, barometer 30 inches, and weighed in the air. - * MEASURE OF LENGTH. Miles. | Furlongs. Chains. Rods. | Inches. * Yards. . Feet. 1. 8 80 320 1760 5280 63360 0-125 1 10 40 S20 660 7920 0.0125 0-1 l 4 22 66 792 0-0.03125 0.025 * I 5.5 16-5 198 0-00056818 00045454 (0.045454 |0-181818 I 3 * 36 0.00018039 ||0.00151515 (0.015.15151 (0.0606060 Q33333 I I2 0 0000157830.0001262620 0012626260-00505050 0:0277777 0.083333. 1. MEASURE OF SURFACE. Sq. Miles. Acres S.Chains. Sq. Rods. Sq. Yards. Sq. Feet. Sq, Inches. . I 640 6400 102400 3097600 |27878400 | 40.14489600 0.001562 1 10 160 4840 43560 6272640 0.0001562 0.] I I6 484 4,356 627264 0.000009764. |0.00625 0.0625 I 30-25 || 272-25 || 39.204 0.000000323 00002066 0.002066 0.0330 1 - 9 1996 00000000358 0:00002296 |0.0002296 |0.00367 (0.1111111 || 1 144 000000000025,00000001590:00000159 0.0000255220.0007716 0-006944 l MEASURE OF CAPACITY AND WEIGHTS. MEASURE OF CAPACITY. Cub. Yard. Bushel. Cub. Feet. Pecks. Gallons. Cub. inclu. T. 21-6962 27 100.987 201-974 46656 0°03961 1. 1-24445 4. 9°30918 2150°42 0-037037 0-803564 I. 3-21425 7°4805 *28 0-0092.59 0-25 0°31114 I 2-32729 537-605 0.10742.1 - 0-1336S1 0.429684 I 231 gºssº 0-000547 0:00.1860 0-004329 1. MEASURE OF LIQUIDS. Gallon. Quarts. Pints. Gills. Cub. inch. I 4. 8 32 231 0.25 I 2 8 57-75 0-125 0-5 1. 4. 28’S75 ().03125 0-125 O-25 I 7.218.75 0-004329 0-017315 0-03463 0-13858 l i AVOIRDUPOIS. Ton. Cwt. Pounds. Ounces. Drams. 1. 20 2240 35840 573440 0-05 I I12 1792 2S672 0-0004:4642 0-00SS285 1 16 256 0-00002790 0'000558 0.0625 l 16 0-00000174 ():0000348 0-0016 0.0625 1. TROY. Pounds. Gunces. Dºyt. Grains. Pound Avoir. I 12 . 240 5760 0°822861 0.083333 I 20 480 0-06S5'T1 0-004166 O-05000 I 24 0.0034285 0-0001736 0-002083333 0-0416666 I {}-00014285 1-2152.75 14:58.333 2.91"6666 TO00 l APOTHECARIES. Pounds. Ounces. Drams. Scruples. Grains. I 12 96 28S 5760 {}-08333 1 8 24 480 0-0104.1666 O-125 Fl 3 60 0.0034722 O'0416666 O-3333 1 20 0-00017861 0.0020833 ().016666 0-05 I 34 MONEY. MONEY AND COINS OF THE UNITED STATES. 10 mills = 1 cent. 10 dinnes = 1 dollar. 10 cents = 1 dimo. 10 dollars = 1 eagle. The standard gold and silver coins contain 900 parts of pure metal and 100 parts of base metal in 1000 parts of the alloy. The remedy of the Mint is the allowance for deviation from the exact standard fineness and weight of coins. The nickel cent contains 88 parts of copper and 12 of nickel. The new bronze cent contains 95 parts of copper and 5 of tin and zinc. Pure gold, 23.22 grains = $1, or $20.67.183 = 1 ounce. Pure silver, 357.03 grains = $1, or $1.36.166 = 1 ounce. Silver coins of less value than one dollar are issued at the rate of 384 grains to the dollar. Standard alloyed gold = $18.60.465, and silver = $1.22.5 per ounce. Gold coins. Grains. Silver coins. Grains. Copper coins. Grains. Double eagle, . . 516. One dollar, . . 412.5 Cent (old), . . 168. Eagle, . ſº . 258. | Fifty cents, . . 192. Cent (new), * 72. Dollar, . . . 25.8 Twenty-five cents, 96. Cent (bronze), . 48. For silver and gold tables see pages 000. WEIGHT AND FINENESS OF DIFFERENT COINS, AND TFII3][R, VALUE IN AMERICAN MONEY. * tº * UNITED weight |...|sº Grains. Grains. $ Cts. e Crown, . we o * . 171.36 900. 154.22| 6.64.19 Austria, . . {º, & º e ºs 190.56 900. 171.5 0.48.63 Country. Jºiece and Divisions. Baden, . . Ducat, . te tº * tº 47.5 987. 46.9 .00.70 Belgium, . 25 Francs, . & e we 121.92 | 899. 109.6 .72.()3 Brazil, . . 2000 Reis, e e Q . . .393.6 918.5 || 361.5 .02.53 Canada, . . . 20 Cents, 1851, . © e 96.0 925. 88.8 .18.87 China, . . Tael, . . . . . . . . . . . . . . . . . .43.00 Chili 10 Pesos, 1855, . . . 236.16 900. 212.5 .15.35 5 e > * 1 Peso, 1854–6, . . . . 384.48 || 900.5 || 346.2 98.17 Denmark, 2 Rix dollars, . º º 444.96 || 877. 390,2 :10.65 England, . . Pound sterling = 20 shillings, 123.21 || 916.5 | 112.9 86.34 East Indies, Company’s Rupee, . tº . 180, 892. 16.5 10.49 France, . . . Napoleon, 20 Francs, . e 99.5 898. 87.4 S5.00 Greece, . 20 Drachms, . * * gº SS.S 900. 80.9 44.29 IIamburg, . ; Dollar, e ſe g 450. 860. 397.5 7.66 s ucat, . tº tº iº e 53.75 982. 52.77 9.7 Holland, |{ij. . . . . . . . 50. 787. 39.32| 1.69.30 Italy, . . . 20 Lire, . . . 99.36 S98 89.22 !.26 l Doubloon = 8 Escudos, . 416.4 870.5 362.5 Peso = 8 Reals, . º . . 415.68 901. 374.5 Norway, . . . 2 Rigsdaler, * * & 444.96 877. 390.2 Peru, . . 1 Sol = 100 Centavos, . . 385.82 900. 347.24 Mexico, . 6 2 {} i i : 55 5.41 *I fl (r Corona (Crown), 1838 * 147.84 || 912. 134.8 80.66 Portugal, |{iº. y . . . . . 45.6 gig. 41.6 | 1.18.00 Prussia, . Thaler, . . . . 268.46 900. 243.6 72.89 Rome, . . . , 2.5 Scudi = 250 Bajochi, . 67.2 900. 60.5 60,47 Russia Imperial = 5 Roubles, . 100.8 916. 92.3 97.64 *Sacº • Rouble silver = 100 Copecks, 320.16 875. 286.8 79.44 Spain 100 Reals, * e g . | 128.64 896. 115.2 96.39 P* ' ' | {80 Reals = 4 Dollars, . 103.2 869.5 89.73| 3.86.44 Sweden Ducat. . gº g ſº g 53. 979. 51.9 23.50 5 * Rix Dollar = 100 Ore, tº 112.3 873. 97.15 0.26.10 Turkey, . . . Piastres, 1845, . . . . 110,88 || 900. 99.79 Foreign MoxEY. 3× | THE CURRENCY OF DIFFERENT COUNTRIES COMPARED : WITH ENGLISH AND AMERICAN MONEY. | | France. •º re | Gar- ssi: - |Ingl’nd. Belgi’m. | Prussia. Austria. Sweden, º: Russia. Ham- U. S Sw’land. (in notes.) | * : |many. (in paper) burg. | £ s. d. | Frs. Cts. Th. Sgr. Pf. Fl. Kr. Rix. Ore. F1. Kr. Rhl. Kop. Mrk. Sch.|$ Cts. | 0 0 1 || 0 10%| 0 0 10 || 0 5 0.07 0 3| 0 3 || 0 0 || 0.02 | 0 | 0 2 || 0 21 || 0 1 8 || 0 10 0.14 || 0 6|| 0 5 || 0 2 0.04 0 0 3| 0 32 || 0 2 6 O 16 0.21 - || 0 9| 0 8 0 23 0.06 10 0 4 || 0 , 42 || 0 3 4 || 0 21} 0.28 0 12| 0 12 || 0 23, 0.08 0 0 5|_0_53 || 0 + 2 || 0 27 0.36 0 15|| 0 16 || 0 4# 0.10 | 0 || 0 6|| 0 64 || 0 5 1 || 0 31, 0.44 || 0 18| 0 19 || 0 53 0.12 0 - 0 - 7 || 0 74 || 0 5 11 || 0 36+| 0.51 0 21| 0 22 || 0 63 0.14 0 0 8 || 0 85 || 0 6 10 0 42+ 0.59 0 24 || 0 26 0 7 || 0.16 0 0 9 || 0 96 || 0 7 7 || 0 47# 0.66 0 27 || 0 27 || 0 | 8 || 0.18 0 0 10 || 1 | 6 || 0 8 6 || 0 53 0.73 || 0 30| 0 33 || 0 83 0.20 0 0 11 || 1 16 || 0 9 5 || 0 57#| 0.80 0 34|| 0 36 || 0 93 0.22 0 1 0 || 1 27 || 0 10 3 || 0 62 0.89 0 36|| 0 39 || 0 11 || 0.24 0 2 0 || 2 55 || 0 20 6 || 1 25 1.78 1 13| 0 79 || 1 6 || 0.4S () 3 O 3 S2 1 0 9 || 1 S7 2.67 1 49 || 1 18 2 1 || 0.72 0 4 0 || 5 10 || 1 10 11 || 2 50 3.56 2 24 || 1 58 || 2 12 || 0.96 () 5 () 6 36 || 1 21 3 || 3 12 4.45 2 59] 1 97 || 3 7 | 1.21 0 6 0 || 7 64 || 2 || 6 || 3 74 5.34 3 38; 2 37 || 4 2 | 1.45 0 7 0 || 8 92 || 2 11 9 || 4 36 6.23 4 12| 2 77 || 4 12 | 1.69 () 8 6 || 10 20 2 22 () || 4 95 7.12 4 47 || 3 |18 || 5 || 7 | 1.93 0 9 0 || 11 46 3 2 0 || 5 58 8.09 5 22| 3 58 6 2% 2.18 0 10 0 || 12 72 || 3 12 4 || 6 25 8.90 5 58; 3.94 | 6 13+ | 2.42 0 11 0 || 13 99 || 3 22 6 || 6 87 9.79 6 34|| 4 38 || 7 | 8% 2.66 0 12 0 || 15 27 4 2 9 7 49 || 10.68 T 11 4 75 8 33. 2.90 0 13 0 || 16 55 || 4 13 0 || 8 12 || 11.57 7 46|| 5 || 5 || 8 14; 3.14 0 14 0 || 17 84 || 4 23 3 8 75 | 12.66 8 24 5 55 9 9 || 3.39 () 15 0 || 19 8 || 5 3 5 9 37 || 13.45 8 57 || 5 96 || 10 4} 3.63 0 16 0 || 20 40 || 5 13 8 || 10 0 || 14.24 9 33 6 35 | 10 15% 3.87 0 17 0 || 21 66 5 23 11 || 10 65 | 15.13 10 9| 6 74 11 10} | 3.12 O 18 O || 22 92 || 6 4 2 11 28 || 16.02 || 10 46 7 14 || 12 5& 4.36 0 19 0 || 24, 18 6 14, 4 || 11 8S 17.01 11 21 7 44 13 0; 4.60 1 0 () 25 45 || 6 24 6 || 12 50 17.80 || 11 57 || 7 88 || 13 9 || 4.84 2 0 0 || 50 90 13 19 0 || 25 0 || 35.60 23 54 15 77 27 2 | 9.6S 3 0 0 || 76 35 | 20 13 6 37 50 53.40 35 51 || 23 65 40 11 ||14.52 4 0 0 | 101 80 27 8 0 || 50 0 || 71.20 47 48 || 31 54 54 4 17.36 5 0 0 127 25 34 3 0 || 62 50 | 89.00 59 46. 39 42 | 67 11 24.20 6 .0 (, ; 152 70 40 27 6 75 0 || 106.80 71 42| 47 3i Sl 4 |29.04 7 0 () || 178 15 47 22 6 || 87 50 | 124.60 83 39| 55 20 94.13 33.88 8 0 0 |202, 60 54 16 6 100 0 || 142.40 95 36 63 9 108 6 |38.72 9 () 0 |229 5 61 11 6 112 50 | 160.20 107 34| 70 96 |121 15 |43.56 10 0 0 |254 50 | 68 6 0 125 0 | 178.00 (119 301.78 84 1135 8 |48.40 The mark of Finland is equal to the French franc. . - DIAMOND. - Carat. Grain. Parts. . Grains (Troy). 1. 4. . 64 3.2 0.25 1. 16 J.8 0.015625 0.0625 i 0.05 0.3125 12:5 20 1. ſ 36 RULE MEASURE. . Conversion of Inches and Eighths into Decimals of a Foot. - - FRACTIONs of AN INCH. - — . 3 1 5 Inches. 0 # # 3. 7 || 5 # # 0 .000() .01041 .02083 || .03125 .04.166 .05208 .0625 .07291, 1 . . .08333 .09375 .10416 | .11458 .125 .13541 .14588 || . .15639 2 .16666 .17707 .1875 .19792 .20832 .21873 .22914 | .23965 3 .25 .26041 | . .270 .281.25 | .29166 | .30208 .3125 | .32291 4 .33333 .34375 | .35416 .364 .375 .3S541 | .39588 .40639 5 | .41666 .427 ()7 .437 .44792 .45832 .46873 | .47914 .48965 6 .5 .51041 .520. .53125 .54166 .55208 || .5625 .57291 7 .5S333 | .59375 / .60416 .614 .625 .63541 .64588 .65639 8 ,66666 .67707 | .6S5 .69792 | .70832 .717.73 .729.14 || 73965 9 .75 .76041 .770. .78125 | .79169 | .80208 || 8425 .82291 10 .83333 | .84375 .8541 .864 .875 .88541 .89588 .90639 11 .91666 .927.07 | .937 .94792. .95832 .96873 .97914 .98965 I2 1 foot. foot. foot. foot. foot. foot. foot. foot. # in. = 0.005208 ft.; sº in. = 0.00265 ft.; ºr in. = 0.001375 ft. Angle Measurement by the opening of a Two-foot Rule. Opening 1 1 Fºotiºns OF an INCH. 5 3 7 Rule, | 0 TS Aſ 3. º '8 Zſ º Inch's. O / O p o t O / O / O / o / O / IS 97 II 98 5 || 99 0 §§ 55 100 5i | 101 47 | 103 44 || 10: 43 19 104 40 105 39 105 39 || 107 40 108 41 || 109 43 || 110 46 || 111 49 20 112 53 | 113 58 || 115 4 || 116 11 || 117 20 | 118 30 119 41 | 120 53 Conversion of Vulgar Fractions into Decimals. . Fract’ns. Decimals. ||Fract’ns. Decimals. IFractºns.T.Decimals. Fract’ns. Decimals. 1: 2 .5 1 : 16 | .0625 1 : 32 | .03125 1:64 .015625 1 : 3 .33333 3 : 16 | . .1875 3: 32 .09375 3:64 .046875 2:3 .66666 5 : 16 .3125 5 : 32 .15625 5:64 .0781.25 1:4 .25 7 : 16 .4375 7: 32 .21875 7:64 109375 3: 4 .75 9 : 16 .5625 9:32 | .28125 9: 64 .140625 1 : 5 .2 11 : 16 .6875 ... } 11:32 .34375 11 : 64 .171875 3: 5 .6 13:16 | .8125 13: 32 .40625 15:64 . .234375 1 : 6. .16666 15: 16 .9375 15:32 .46S75 19:64 .296875 5 : 6 .83333 || l: 24 ,04166 17:32 .53125 23 : 64 .3593.75 1 : 8 .125 5: 24 .20833 19: 32 .59375 27: 64 | .421875 3.: 8 .375 7 : 24 .29166 21:32 .65625 || 31:64 .484.375 5 : 8 .625 11:24 . .45833 23: 32 .71875 35 : 64 .546875 7 : 8 .875 13:24 .54166 25 : 32 .78125 39: 64 .609375 5 : 12 .41666 17:24 .70833 27 : 32 .84375 43:64 ,671875 Hä .58333 19: 24 .79166 || 39: 32 .90625 57:64 | 891625 ,925 23:24 .95833 || 31:32 .96875 61:64 .953125 METRICAL SYSTEM. 37 To Determine an Angle by the Aid of a Two-foot Rule. b = opening of the rule in inches; v = angle formed by the rule; Sin. #v = # and b = 24 sin. #v. E.cample 1. How much (b = ?) must a two-foot rule be opened to form an angle of 48° 40/? b = 24 + sin, 24° 20' = 24 × 0.412 = 9.888 inches. I Barample 2. A two-foot rule is opened to b = 8 inches. Required the angle formed by the rule. Sin. #v = + = 0.3333 = sin. 19° 30', and v = 39°. i THE FRENCH METRICAL SYSTEM. The French units of weight, measure and coin are arranged into a perfect deci- mal system, except those of time and the circle. The division and multiplication of the units are expressed by Latin and Greek names, as follow: Latin, Division. Greek, Multiplication. Milli = 1000th of the unit. Deca = 1.9 times the unit. Centi = 100th of the unit. Hecato = 100 times the unit. I)eci = 10th of the unit. Kilio = 1000 times the unit. Metre, Litre, Stere, Are, Franc, Gramme. Myrio = 10000 times the unit. French Measure of Length. 1 Millimetre = 0.03937079 inches. 1 Metre (unit) = 3.280899 feet. 1 Centimetre = 0.3937079 inches. 1 Decametre = 32.80899 feet. 1 Decimetre = 3.937079 inches. I JI ectometre = 328.0899 feet 1 Metre (unit) = 39.37079 inches. 1 Kilometre = 3280.S.99 ft. : < 0.62138 1 Sea mile or \ **) lº-fi mile knot }= 1.847.2 kilometre. 1 Statute mile = 1.609315 kilometres. 1 Kilometre = 0.541343 sea miles. I Kilometre = 49.7106 chains. French Measure of Surface. 1 Sq. metre = 10.7643 square feet. 1 Are = 1076.43 square feet. 1 Are = 100 square metres. 1 Decare = 107.643 square feet. 1 Decare = 10 ares. 1 Hectare = 2.47 114 12ng. acres. 1 Hectare = 100 ares. 1 Sq. mile = 258.989 hectares. French. Measure of Volume. 1 Stere (cubic l 1 Stere = 35.3166 Eng. cubic feet. metre) } = 10 decasteres. 1 Litre = 61,0271 ing. cubic inclues. 1 Stere = 1000 litres. 1 Gallon = 3.7852 litres. 1 Litre = 1 cubic decimetre. l Decistere = 2.84 bushels. 1 Decistere = 3.53166 cubic feet. French Measure of Weight. 1 Ton = 1 cubic metre dis- || 1 Gramme = 10 decigrammes. tilled wafer. 1 Decipramme = 10 centigrammes. 1 Ton = 1000 kilogrammes. 1 Centigramme = 10 milligrammes. 1 Kilogramme = 1000 granmes. 1 ICilogramme = 2.2047 pounds avoir- 1 Hectogramme = 100 grammes. dupois. 1 Decagramme = 10 grammes. 1 Eng. pound = 0.45358 kilogrammes. 1 Gramme = 1 cubic centimetre distilled water. 1 Gramme = 15.48315 grains troy. 1 French ton = 0.984274 Eng. tons. 1 Inglish ton = 1.01598 French tons. French Coinn. 1 Franc 100 centimes = 19.06 cents of an American dollar. FEET AND METRES. Conversion of English Inches into Centimetres. Inch's O I 5 G 7 Ct.mt. Cl.mt. Ct.Imt. 0.000 || 2.540 || 5,080 25.40 27.94 || 30.48 50.80 || 53.34 || 55.88 76.20 || 78.74 || S1.28 101.60 | 104.14 || 106.68 || 1 127.00 | 129.54 || 132.08 152.40 154.94 | 157.48 || 1 T77.80 | 180.34 182.88 || 1 228.60 231.14 || 233.68 || 2 251.00 |256.51 |356 oš 09.22 111.76 134.62 60.02 | 162.56 85.42 187.96 203.20 205.74 208.28 210.82 36.22 238.76 261.62 114.30 116.84 139.70 || 1:42.24 165.10 | 167.64 190.50 | 193.04 215.90 218.44 241.30 243.84 266.70 || 269.24 Ct.nt. | Ct. mt. 12.70 || 15.24 38.10 | 40.64 63.50 | 66.04 88.90 || 91.44 Ct. mt. 17.78 43.18 68.58 93.98 119.38 144.78 170.18 195.58 220.98 246.38 271.78 17536 226.06 251.46 276.85 Comaversion of Centimetres into 0. Il 2 3. 4. 5 6 7 English Inches. 8 9 Inches. | Inches. Inches. 0.000 || 0.394 0.787 3937 || 4.331 || 4.742 7 S74 || 8.268 8.662 11.811 | 12.205 || 12.599 15.74S | 16.142 | 16.536 | 16.929 19.6S5 | 20 079 20.473 23,622 24.016 24.410 27.56() 27.953 || 28.347 31,497 || 31.890 || 32.284 35.434 || 35.827 | 36.221 39.370 || 39.764 40.158 Inches." 12.992 20.867 24.804 28.741 || 29.134 32.678 36.615 40.552 1.181 5.118 9,055 Inches. 1.575 5.512 .9.449 13.386 17.323 21,260 25.197 33.071 37.009 40.945 Inches. Inches. 13.780 || 14,173 17,717 | 18.I.11 21.654 22.048 25.591 25.9S5 29.52S 29.922 33.465 33.859 37.402 || 37.796 41.339 41.733 1.969 2,362 5.906 || 6.299 9.S43 || 10.236 Inches. 2.756 6.093 10,630 14,567 18,504 22,441 20.37S 30.316 34.253 3S.190 42.126 Inches. 3.150 7,087 11.024 14.961 18,89S 22.835 26.772 30.709 34.646 38.583 42.520 Inches. 7.480 11.418 15.355 19,292 23.229 27.166 31.103 35.04() 38.977 42.914 Conversion of English Feet into Metres. O T 3 4. 5 6 7 8 Met. Met. Met. 0.000 || 0.3048 0.6096 || 0 3.0479 || 3.3527 | 3.6575 3 6.0359 || 6.4000 6.7055 || 7 9.1438 || 9.4486 || 9.7534 12.192 || 12.496 12.801 15.239 15.544 15,849 18,287 | 18.592 | 18.897 | 19.202 21.335 | 21.640 || 21,945 24.383 || 24.688 24,993 27.431 27.736 || 28.04.1 30.479 || 30.784 31,089 Conversion of Metres into English MetrôS. O Il Feet. Feet. 0.000 || 3.2809 32.809 || 36.090 65 618: 68 S99 98.427 | 101.71 131.24! 134.52 164 04|167.33 196.85| 200.13 229.66 || 2:32.94. 262.47 || 265.75 295.2S 298,56 328.09|| 3:31.37 2 .9144 .0102 10.058 13,106 16.154 22.250 25.298 28.346 31,394 Met. 3 Met. I.2192 .9623 || 4.2671 7.3150 10.363 13.411 16.459 19.507 22.555 25.602 28,651 31.698 Foet. 6.5618 39.371 72.179 104.99 137.80 170.61 203.42 236.22 269.()3 391.84 334.65 Feet. 9.S427 42.651 '75 461 10S 27 141.0S 173.89 206.70 239.51 272.31 305.12 337.93 4. Feet. 13,123 45.932 '7S.741 111.55 144.36 177.17 209.98 242.79 275.60 308.40 341.21 1.5239 1.82ST 4.5719 || 4.8767 7.6.198 || 7.9246 13.716 || 14,020 Met. Met, 10.66S | 10.972 16.763 17.06S 19.811 20.116 22.859 || 23.164 25.907 || 26.212 28.955 ; 29.260 32,003 || 32.308 5 6 Feet. Feet. 16.404 || 19.6S5 49.213| 52.494 S2.022| 85.303 II.4.83; 118.11 147.64. 150.92 1S0.45 183.73 213.26 216.54 246,07| 249.35 27S.S.S. 282.16 311.69 || 314.97 344,49| 347.78 Met. 2.1335 5.1S15 8.2294. 11.277 14,325 17.373 20.421 23.469 26.517 29,565 32.613 Feet. 2. Feet. 22.966 55.775 88.584 I21.30 154.20 187.01 219.82 252.63 285.44. 31S.25 351.06 Met. 2.4383 5.4863 8,5342 11.582 14.630 17,678 20.726 23.774. 26.822 29.870 32.91S 8 Feet . 26.247 59,056 91.865 124.67 157.48 I90.29 223 I0 255.91 288.72 321.53 354.34 30.174. 33.222 Teet. 29,528 62,337 95.146 127.96 160.76 193.57 226.38 259.19 292.00 324.81 357.62 * MILES AND KILOMETRES. 39 Conversion of English Statute-miles into Kilometres. Miles. O I 2 3 4. 5 6 7 8 9 ſ Kilom. Kilom. Kilom. Kilom. Kilom. Kilom. I Kilom. Kilom. Kilom. Kilom. || 0 0.0000| 1.6093 3.2186| 4.8279| 6.4372; 8.0465||9.6558||11.2652|12.8745|14.4848 10 16.093; 17.702; 19.312| 20.921 22.530 24,139; 25.749| 27.358; 28.967 30.577 20 32.186|| 33.795 || 35,405| 37.014| 38.623| 40.232| 41.842|| 43.451 45.060|| 46.670 30 48.279; 49.888 || 51.498 || 53.107 54,716 56.325|| 57.935 | 59.544, 61.153| 62.763 40 G4.372 65.9SI 67,591 || 69.200|70.809 || 72.418| 74.028; 75.637 || 77.246 78.850 50 80.465| S2.074| 83.684| S5.293| 86.902| S8.511| 90.121} 91.730| 93.339; 94.949 60 96.558 9S.] 67 90.777 || 101.39|| 102.99 || 104.60. 106.21 J 07.82| 109.43 111.04 270 112.65 114.26 115.87 117.48 119.08 || 120.69| 122.30 123.91 125.52| 127.13 80 | 128.74 130.35||131.96 || 133.57|135.17| 136.78 || 138.39 140.00; 141.61| 143.22 90 144.85; 146.44|| 148.05] 149.66; 151.26; 152.87 | 154.48} 156.09| 157.70: 159.31 100 160.93| 162.53| 164.14|165.75; 167.35||168.96 || 170.57| 172.18| 173.79| 175.40 Conversion of Kilometres into English Statute-milies. Kilom. O I 2 3 4. 5 6 7 S 9 Miles. | Miles. | Miles. Miles Miles. | Miles. Miles. | Miles. | Miles. | Miles. () 0.0000 || 0.6214| 1.2427 | 1.8641 2.4855|3.1069| 3.7282| 4,3497 || 4.9711; 5.5924 10 6.2138 6.8352| 7,4565 8.0780| 8.6994| 9.3208 9,942] | 10.562 11.185| 11.805 20 12.427 | 13,040 13.670 14.292] 14.913| 15,534|| 16.156|| 16.776|| 17.399 || 18.019 30 18.641 19.263. 10.884] 20.506] 21.127| 21.74S 22.370 22.990] 23.613| 24.233 40. 24.855 25.477| 26.098| 26.720 27.341| 27.962| 28.584] 20.204| 29.827|30.447 50 31,069 || 31,690|| 32.311| 32.933| 33.554 34.175|| 34,797 || 35,417 | 36,040|| 36,660 60 37.282 37.904| 38.525 39.147 39.768; 40.389; 41.011. 41.631|| 42.254; 42.874 'º () 43.407 || 44.118, 44.739; 45.361| 45.982| 46.603| 47.225 47.845 || 48.468|49.08S S0 49.71.1 : 50.332| 50.953| 51.575 52.196 || 52.817 53.439 54.059 || 54.682| 55.302 9() 55,924 56.545 57.166 ſ 57.7SS 58.409 59.030| 59.652, 60.272, 60.895 61.515 100 62.138 || 62.759| 63.380| 64,002| 64.623 65.244|| 65.866| 66.486 || 67.109| 67.729 Connversion of Sea-miles, Knots or Minutes into Kilometres. Knots. O I 2 3 4. 5 Kilom. Kilom. Kilom. | Rilom. Kilom. || Kilom. 0 ().0000: 1,8472. 3.6944 5,5416, 7.3SSS| 9.2361 }{} 18.472| 20.319| 22.166 24.013; 25.861 || 27.708 20 36.944|| 38.791; 40.638 42.485 44.333| 46.180 30 55.416; 57.263 59,110 || 60.957 || 62.805 || 64.652 40 73.888| 75,735 | 77.582| 79.429| 81.277| S3.124 50 92.361. 94 207 96.054, 97.901 || 99.749| 101.59 60 J10.83; 112.68||114.53 116.37 118.22| 120.06 7() 129.30} 131.15 133.00||134.84 136.70 || 138.54 80 347.77|| 149.62 151.47 | 153.31 155.18) 157.02 90 166.25 168.09 | 169.94| 171.78 173.65 175.49 100 I84.72 186.56 18S.41 || 190.25 192.12 193.96 6 Kilom. 11.083 29.555 48.027 66.499 84.971 I03.44 121.91 140.39 15S.87 177.34 195.81 7 Kilom. T2.930 31.402 49.874 68,346 S6.818 105.29 123.76 142.24 160.72 I’79.19 198.66 8 9 Kilom. 14.777 33.249 51.721 Kilom. 16.625 35,097 53.569 72.041 90.513 108.98 127.45 145.94 164.43 1S2.90 201.37 Conversion of Kilomet res irato Sea-mail es, Knots or Minutes. Kilom. () I Knots, l Knots. O 0.0000|| 0.5413 IQ 5,4134|| 5.9547 20 10.827 | II.368 30 16.24 | 16.781 40 21.653i 22.194 b0 27.066 27.607 60 32.480i 33,020 70 37.894; 38.433 80 43.307; 43,846 90 48.72] 49.259 100 54.134} 54.672 2 Knots. 1.0827 6.4961 11.909 17,322 22.735 28.148 33,561 38.974 44,387 49.800 55.213 3 4. 5 6 7 S 9 Knots. 1.6240 '7.0374 12.451 17.864 23.277 2S.690 34,103 39,516 44.929 50.342 55.755 Knots. 2.1653 '7,5787 12.992 18.406 23.819 29.232 34.645 40.05S 45.471 50.884 56.297 Knots. 2.7066 S.1200 13,533 18.946 24.359 20.772 35.1S5 40.598 46.01.1 51,424 Knots 3.24SO 14.075 19.4SS 24.901 30,314 35.727 41.140 46.553 5].966 56,837 57.379 Knots. 3.7894 S.6614. 9.2028 14.616 20.029 25.442 30.855 36.26S 41.681 47.004 52,507 57.920 Knots, 4.3307 9.7441 15.157 20,570 25,983 31,396 36.809 42.222 47,635 53.048 58,461 Knots. 4, S721 10,285 15.702 21. IL5 26.528 31.9-11 37.364 42.777 48.190 54.603 60,016 40 FOOT-MEASURES AND POUNDS. Comparison between Foot-measures of Different Nations. LINEAR FEEy. English. Metre. Prussia. | Saxony. Baden. Austria. Hanover Sweden. I 0.3048 0.97.11 F.0763 1.0160 0.9642 | 1.0435 | 1.0265 3.2809 I 3.1862 3.5312 3.3333 || 3.1634 3.4235 | 3.3678 I.0297 0.3138 1 1.1083 1.0462 0.9929 | 1,0745 | 1.0572 0.9291 0.2S32 0.9023 Il 0.94.40 || 0.8959 || 0.9695 || 0.9538 0.9843 0.3000 0.9559 1.0594 | 0.9490 | 1.0271 | 1.0164 1.0371 0.3161 I.0072 1,1163 1.0537 Ji 1.0822 | 1. § 0.95.83 0.2921 0.9307 1.0314 0.97.36 0.9240 1. 0.98 0,974.1 0.2969 0.9459 1.0484 0.9S38 || 0.9122 | 1.0].65 1 SQUARE FEET. 1. 0.0929 0.9431 1.I584 1.0322 | {}.9297 I.OSSS ( 1.0537 10.764 I 10.152 12.469 II.III | 10.007 || 11.72} | 11.342 1.0603 0.0985 l 1.2283 1.0945 || 0.9858 || 1.1545 1.1130 0.8603 0.0802 0.8141 i 0.8911 || 0 8026 0.9400 || 0.9097 0.9688 0.0900 0.913'ſ 1.I.222 I 0.9007 || I.0549 | 1.0330 1.0756 0.0999 1.0144 1.2460 I.]103 1 I.1712 | 1.2019 0.91S4 0.0853 0.8661 1.0639 0.94.80 || 0.8538 I. 0.96.79 0.9489 0.0881 0.8947 1.0941 ().9679 0.832]. 1.0331 1. CUBIC FEET. I. 0.0283 0.9159 1.2468 1.0487 || 0.8964 I.1362 | 1.1018 35.316 1. 32.346 44,032 37,037 || 31.658 | 40.126 || 38.198 1.0918 0.0309 Tº 1.3613 1.1450 0.978'ſ 1.2405 1.1816 0.8021 0.0227 0.7346 I 0.8411 || 0,7190 || 0.9113 || 0.8677 0.9535 0.0270 0.8733 1.1889 I 0.8548 || 1.0834 || 1.0501 I.0756 0.0999 1.0144 1.2460 1.1103 I 1.1712 | 1.3176 0.8801 ().0249 0.S061 1.0973 0.9230 0.7890 l 0.9522 0.9243 0.0262 0.8483 1.1444 0.9522 || 0.7590 | 1.0501 1. Conversion of Pounds of Different Nations. Eng. av. Kilogram. Prussia. Austria. | Spain. Hanover Russia. | Sweden. I 0.4536 0.9072 0.8110 0.9839 || 0.9320 | 1.1076 | 1.0664 2.2046 1. 2.0000 1.7857 2.1692 1.9842 || 2.4419 || 2.3511 1.1(323 0.5000 I 0.8929 1,0857 1.0271 1.2209 | }.1755 I.2346 ().5600 1.1200 I 1.2132 | 1.1490 | 1.3675 | 1.3166 1.0164 0.4610 0.92.11 0.8243 I 0.94.70 | 1.1257 | 1.0839 1.0730 0.4696 0.9752 0.8596 1.0557 I 1.1884 1.1442 0.902S 0.4095 0.8190 0.7313 0.8883 0.8414 l 0.9628 0.9377 0.4253 0.8508 0.7595 0.9226 || 0.8738 || 1,0386 1 Ancient Measures of Length. º Scripture. Feet. I Inches. Hebrew. Feet. l Inches. Digit, . gº ºn . . . . 0.912 || Cubit, * - e ſº 1 9.S68 Palm = 4 Digits, . . . . 3.648 || Sabbath day's journey, 3648 | . . . Span = 3 Palms, . . . . . 10.94 || Mile = 4000 Cubits, . . 7296 | . . . Cubit = 2 Spans, . 1 9.888 || Day's journey = 33.164 mi. . . . . . . . Fathom = 3.46 Cubits, . 7 || 3.552 || Sacred Cubit, . . . 0.24 Egyptian. Finger. . . . . .7374 Roman. Nahud Cubit, . º 1 5.71 || Digit, - • o . . . . . . .7257 Royal Cubit, e e 1 || 8.66 § (Inch), . . . . . . . .967 es (foot) = 12 Uncias, . . . . . 11.60 Grecian. Čubitº 24 Digits, ". 1 || 5.406 #_ 16 Digits. . . . . 'I º: Passus = 3.33 Cubits, . 4 || 10.02 § - $5 tº e i i. |Millarium (mile), . . 4842 | . . . stadium, . . . . . 604 || 4.5 || Arabian. Foot, . | 1 | 1.14 Mile = 8 Stadiums, . . 4835 | . . . Babylonian. Foot, 1 T.68 FOREIGN WEIGHTS AND MEASURES. Foreign Measures of Length Compared with American. Places. Amsterdam, Antwerp, . J3avaria, . IBerlin, . . Bremen, . Brussels, . . China, . . £ 6 {{ {{ * tº Copenhagen, Dresden, . England, . Florence, . France, . {{ © e Geneva, . Genoa, . . Hamburg, IIanover, . Leipsic, . Lisbon, e e { Measures. Foot. . tº {{ builder’s, surveyor’s Braccio. e Pied de Roi, . Metre, . e Foot, g º Palmo, e Foot, e º {{ {{ ſº © e ** º mathematic, tradesman's, Palmo, . tº 11-14 11:24 II-29 II-45 Il-11 I2-96 Inches.| 8-64 Places. Malta, . . Moscow, . Naples, . . Prussia, . Persia, . . Rhineland, Riga, . . Rome, Russia, . . Sardinia, Sicily, . . Spain, . $6 {{ Strasburg, Sweden, . Turin, . Venice, . . Vienna, . Zurich, . . Utrecht, . Warsaw, . Measures. Foo t, e e {{ & Palmo, . º Foot, . e Arish, . te Foot, . e {{ $6 & © Palmo, * £w º e Foot, . we Toesas, . e Palmo, iº Foot, º ſº {{ Inches. 11-17 13-17 10°38 12:36 38-27 12:35 10'79 11'60 13-75 9-78 Foreign Road Measures Compared with American. Places. Arabia, . . Bohemia, . China, . . Denmark, England, . {{ Flanders, . France, . $4 {{ Germany, . Hamburg, Hanover, . Holland, . Measures. Mile, . ge 66 ę Q Ili, dº e Mile, º e “ statute, “ geographical, €4 League, marine, common, post, . Mile, long, . {{ {{ &{ Yards. 2148 10137 629 8244 I760 2025 6869 6075 4861 4264 101.26 8244 11559 6395 Places. Hungary, Ireland, . Netherlands, Persia, . Poland, . Portugal, Prussia, . Rome, . Russia, . Scotland, Spain, . . Sweden, Switzerland, Turkey, . Measures. Mile, . & 66 £6 Parasang, © Mile, long, . League, . g Mile, . e {{ e ſº Verst, . e Mile, * * League, common, Mile, . te 4 & * Berri, . * Yards. . 9113 303S . 1093 60S6 . S101 6760 846S 2025 . | 1167 1984 7416 . 11700 9153 I826 Foreign Measures of Surface Compared with Americam. Places, Amsterdam, Berlin, . {{ Canary Isles, England, . Geneva, . IIamburg, . Hanover, . Ireland, . . Naples, . Measures. Morgen, . {{ great, {{ small, Fanegada, Acre, . ſº Arpent, ſº Morgen, . &ć e º Acre, & Moggia, . . Sq. Yds. 97.22 67S6 3054 2422 4840 6179 II545 3100 TS40 3998 Places. Portugal, Prussia, . Rome, . Russia, . Scotland, Spain, . . Sweden, Switzerland, Vienna, . Zurich, . & * Measures. Geira, * Morgen, tº Pezza, * Dessetina, . Acre, . ſº Fanegada, . Tunneland, Faux, . * Joch, Cº. Common acre, º Sq. Yds. 6970 42 FoREIGN WEIGHTS AND MEASUREs. Foreign Liquid Measures Compared with American. Places. Measures. Cub. In. Places. Measures. - Cub. Iu. Amsterdam, . . . Anker, . . . ] 2331 || Naples, . . . Wine Barille, 2544 . (4. . Stoop, . . . . 146 {{ . Oil Stajo, . I133 $f. Antwerp, . . . . . . . 194 Oporto, . . . Almude, . . . 1555 . Bordeaux, . . . . Barrique, . 14033 || Rome, . . . Wine Barille, 2560 Ǻ Bremen, . . . . Stubgems, . . 1945. . . . Oil Cº. 2240. Canaries, . . . Arrobas, . 949 &G º Boccali, . . 80 Constantinople, Almud, . . 319 || Russia, . . . Weddras, . 752 Copenhagen, . Anker, . . 2355 . . . Kunkas, . . 94 Florence, . . . . Oil Barille, . 1946 Scotland, . Pint, . . . 103.5 {{ . . . Wine “ . . 2427 || Sicily, . Oil Caffiri, . 662 | France, . . . Litre, . . . 61:07 || Spain, . . . Azumbras, . 22.5 Geneva, . . . . Setier, . . 2760 Q & . Quartillos, . 30.5 Genoa, . . . . . Wine Barille, 4530 Sweden, . l’imer, . . . 4794 “. . . . . Pinte, . . . 90.5 & 4 . . Kanna, . . . 159:57 Hamburg, . . . Stubgen, . 221 Trieste, . . | Orne, . . . . 4007 Hanover, {{ . . 231 Tripoli, . . Mattari, . . . 1376 ifungary,. . . . . . Eimer, . . . . . 4414 || Tunis.” . . . oil º'. . . . iişi Leghorn, . . Oil Barille, . 1942 Venice, . Secchio, . . . . 628 Lisbon, . . . . Almude, . 1040 Vienna, . Rimer, . . 3452 {{ Malta, . . . . Caffiti, . . . 1270 | . | Maas, . . . S6'33 Foreign Dry Measures Compared with American. Places. Mcasures. “Cub. In. - Places. Measures. Cub. In. Alexandria, . . . Rebele, . . 95S'ſ Malta, . . . Salme, . . . , 16930 gº . | Rislos, . . 1041S Marseilles, Charge, . . 9411 Algiers, . . . | Tarrie, . . . . 1219 Milan, . . . Moggi, . . . 84.44 Amsterdam, . . Mudde, . . 6596 Naples, . Temoli, . . 3122 {{ . . Sack, . . . . . 4947 Oporto, . . . Alquiere, . . 1051 Antwerp, . . . Viertel, . . 4705 || Persia, . Artaba, . . 4013 Azores, . . . . Alquiere, . . 731 Poland, . . Zorzec, . . 31.20 Berlin, . . . Scheffel, . . 3180 || Riga, . . . Loop, . . . 3978 Bremen, . . . {{ . . . 4339 Tºome, . . . Rubbio, . . . 16904 Candia, . . . . Charge, . . . 9288 e Quarti, . . 4.226 Constantinople, Kislos, . . . . 2023 Rotterdam, Sach, . . . 6361 Copenhagen, . Toende, . . 8489 || Russia, . . . Chetwert, . 12448 Corsica, . . . | Stajo, . . . . 6014 Sardinia, Starelli, . . 29.SS Florence, . . . . Stari, . . 1449 Scotland, . Firlot, . . 2197 Geneva, . . . . Coupes, . . . 4739 Sicily, . Salme gros, . 21014 Genoa, . . . . . Mina, . . '73S2 “. . . “ generale, 16886 | Greece, . . . . Medimni, . . . 2390 Smyrna, . Kislos, . . . 2141 | Hamburg, . . Scheffel, . 6426 || Spain, . . . Catrize, . . 41269 Ilanover, . . . Malter, . . . 6868 || Sweden, . Tunna, . . 8940 Leghorn, . . . . Stajo, . . 1501 Trieste, . . . Stari, . . . 4521 {{ . . . Sacco, . . . . 4503 || Tripoli, . Caffiri, . . . 19780 Lisbon, . . . . Alquiere, . 817 || Tunis, . . {{ tº e 21855 “. . . . | Fanega, . 3268 Venice, . Stajo, . . . . 4945 Madeira, . . . . Alquiere, , 684 || Vienna, . . . Metzen, . . 3753 Malaga, . . . . Fanaga, . . . 3783 English Measures of Capacity. The Imperial gallon measures 277-274 cubic inches, containing 10 lbs. Avoirdu- pois of distilled water, weighed in air, at the temperature of 62°, the barom- eter at 30 inches. - . For Grain. 8 bushels = 1 quarter. - 1 quarter = 10:2694 cubic feet. Coal, or heaped measure. 3 bushels = 1 sack. 12 sacks = 1 chaldron. Imperial bushel = 2218-192 cubic inches. * Heaped bushel, 19; ins. diam., cone 6 ims. high = 2812'4872 cubic ins. I chaldron = 58-658 cubic feet, and weighs 3136 pounds. 1 chaldron (Newcastle) = 5.936 pounds. FoREIGN WEIGHTS AND MEASURES. 43 Foreign Weights Compared with American. * Ibs. per - Lbs. per Places. Weights. 100 avoir. Places, Weights. 100 avoir. Aleppo, . . . . Rottoli, . . . 20.46 || Hanover, . . . Pound, . . . 93.20 “. . . . . Oke, . . . 35.80 || Japan, . . . . Catty, . . 76.92 Alexandria, . Rottoli, . . . 107. Leghorn, . . . . Pound, . . . 133.56 | Algiers. . . {{. * * 84. Leipsic, . . “ (common)| 97.14 | Amsterdam, . | Pound, . . . 91.8 || Lyons, . . . “ (silk), . 98.81 Antwerp, . “ . . . . 96.75 || Madeira, . {& tº tº 143.20 Barcelona, . . {{ . . . . 112.6 Mocha, . . . Maund, . . 33.33 Batavia, . . . Catty, . . 76.78 || Morea, . . . Pound, . . 90.79 Bengal, . . . . Seer, . . . . . 53.57 || Naples, . . . Rottoli, . . 50.91. Berlin, . . . Pound, . . 96.8 Rome, . . . . Pound, . . 133.69 & £ lłologna, . . . . “ . . . ; 125.3 || Rotterdam, . tº gº tº 91.80 Bremen, . . {{ • • 90.93 || Russia, . . “. . . . . 110.86 Brunswick, . . . “ . . . . 97.14 || Sicily, . . . . “. . . . 142.85 Cairo, . . . . . Rottoli, . . . 105. Smyrna, . | Olke, . . . 36.51 Candia, . . . {{ ge e 85.9 || Sumatra, . . . Catty, . . 35.56 China, . . . Catty, . . . 75.45 || Sweden, . . . Pound, . . . . 106.67 Constantinople | Oke, . . . 35.55 {& tº º “ (miner's), 120.68 Copenhagen, Pound, . . 90.80 || Tangiers, . “ . . . 94.27 Corsica, . . . { % 131.72 || Tripoli, . . . Rottoli, . . 89.28 {{ Cyprus, . . . Rottoli, . . I9.07 || Tunis, . . . & Cº Lº 90.09 Damascus, . . $6 . . . . 25.28 || Venice, . . . Pound (heavy) 94.74 Florence, . . . Pound, . . . . 133.56 || “ tº º “ (light) || 150. Geneva, . . “ (heavy), 82.35 || Vienna, . . & 4 ge º 81. Genoa, . . 46 {{ 92.86 || Warsaw, . * . . . . 112.25 Hamburgh, . gº {{ 93.63 * A Uniform System of Metrology much Needed. The preceding variety of tables of weights, measures and coins shows the great need of a uniform system of metrology throughout the world. The French are the first in adopting a uniform decimal system of metrology, and an International Decimal Association has been formed for the special purpose of advocating the introduction of the French system into other countries, which Association has now labored on that subject for some twenty years with but little success. Only Belgium, Switzerland, Germany and Italy have adopted and enforced the French metrical system. It has been made legal in some other countries, but not enforced. The principal difficulties in the way appear to be prejudices and jeal- ousy. It must be admitted that the introduction of a new system of metrology causes some temporary inconveniences, but tho objection is only temporary. Some few countries have decimated their old units in preference to adopting the French system. - One difficulty of the decimal system is, that the base 10 does not admit of more than one binary division without fraction. See A New System of Arithmetic, page 44. - 44 ARITH METICS. A NEW SYSTEM OF ARITH METIC, Weights, Measures and Coins. Our present Arithmetical system is very inconveniently arranged for the general requirements of mankind; it causes an international difficulty and discordance in the adoption of a uniform system of Weights, Mea- sures and Coins. An International Association for obtaining a wniform decimal system of Weights, Measures and Coins, has been in existence several years, but as yet, has accomplished very iłłie. They meet with the most naturai and reasonable objections, namely, that the Arithmetical base 10 does not admit of binary divisions, as required in the shop and the market. In practice, we want our units divided into the most natural fractions namely, quarters, eighths, sixteenths, &c., &c., for which the decimal system is not suitable. A most common fraction 1/8 expressed by decimals will be 0-125; if this number is shown to the majority of the people there will be comparatively few who understand the true meaning of it; it will then be necessary to explain that the unit is divided into 1000 parts of which 125 is 1/8 of the whole. The people will then surely remark that this is a roundabout way of doing things, and that they are not willing to cut up their things into 1000 parts in order to get it into 8. Even among the educated classes and among the best arithmeticians, there will be few, if any who have it clearly located on the mind that 125 is 1/8 of 1000, but it is very well known to be so, by practice in calculation. There- fore, our present arithmetical system is a great burden on the student, and very frequently exceeds the limits of the power of the human mind, beyond which solutions are performed mechanically, like a musician who plays the crank organ. The base ten has often been complained of, and more suitable numbers proposed. Charles the XII., of Sweden proposed the number twelve for the arithmetical base; to use his own just expression that, “it is quite rediculous to use ten as the base for arithmetics, it can be divided once by two and then stops.” It is not sufficient merely to propose or say that 8, 12, or 16 would be better as a base, but in order to make a correct im- pression of its utility, it is necessary to enter into details with examples that any one may be able to see its advantages without taxing his own mind. The Author laid before the above mentioned International Associa- tion which met at Bradford in Yorkshire, on the 10th, 11th, and 12th of October, 1859, a new system of Arithmetics, Weights, Measures, and Coins, founded uniformly throughout on the number 16, as the base. This would become the most simple system to the mind, and it would embrace all requirements of the different classes of mankind. In that system it is proposed to add six new figures, thus, 1 2 3 4 5 6 78 : 9 & U) 8 & ‘f 10 The new figures will appear strange at the first glance, but a little re- ſlection will soon convince one of their simplicity and utility. A complete description with numerous examples of this new system is now published by J. B. Lippincott & Co., Philadelphia. In one example it will be found that our present arithmetical system requires, four additions, seven multiplications, and one division, em- ploying in the calculation 215 figures; while the new system requires only one multiplication and employs only 39 figures for the same solu- tion. JOHN W. NYSTROM. Philadelphia, January, 1862. GEOMETRY. 45 G E 0 M ET R Y. DEFINITIONS. Demonstration is a course of reasoning by which a truth is established. It consists of, , Thesis, the truth to be established, and, Hypothesis, the foundation for the demonstration. 42-ton is that which is self-evident and requires no demonstration. Theorem is something to be proved by demonstration. º is something to be done, but is self evident and requires no demon. SUT<10I). Problem is something proposed to be done, and requires demonstration. JProposition is either a Theorem or a Problem. b ºary is an obvious conseqence deduced from something that has gone ©IOre., Scoltum is a remark on preceding propositions, commonly demonstrated by algebraical formulae. Lemma is something premised for a following demonstration. Geometrical Quantities, Point is a position, but no magnitude. A Line is length, without breadth or thickness. A Straight Line is the shortest distance between two points. Curved line is a length which in every point changes its direction. Superficies, Swrface, Area, is that which has length and breadth, but no thickness. Plane surface is a plane which coincides with a straight line in every dire tion. Curved surface is a plane which coincides with a curved line. Solid has length, breadth and thickness. Circles Circle, Cirwmference, Periphery, is a curved line drawn on a plane surface, anº bounded at a common distance from one point in the plane, (centre.) Radius is a line's drawn from the centre in a circle to the periphery. Diameter is a line drawn through the centre to the periphery, or the longest line in a circle. Chord is any line extending its both ends to the periphery of a circle, and doer not go through the centre. Arc is a part of a periphery. Circle plane, is a plane surface bounded within a circumference. Sector is a part of a circle-plane bounded within an arc and two radii. Segment is a part of a circle plane bounded within a chord and an arc. Zone is a part of a circle included between two parallel chords. Lune is the space between the intersecting arcs of two eccentric circles. Oval is a round figure having one long and one short diameter at right angles to one another. Semicircle is a half circle. Quadrant is a quarter of a circle. Anglese Angle is the opening or inclination of two lines which meet in one point. If two radii being drawn from the extremities of a circle arc, to the centre; the arc, is a measure of the angle at the centre. Right angle is when the opening is a quarter of a circle. Acute angle is less than a right angle. Obtuse angle is greater than a right angle. * Lºne by itself means a straight line. CONSTRUCTIONS, 1. To construct an ellipse. With o as a centre, draw two concentric circles with diameters equal to the long and short axes of the desired ellipse. Draw from o any number of radii, A, B, &c. Draw the line B b' parallel to m and b b' parallel to m, then b' is a point in the desired ellipse. 2. To draw, an ellipse with a string. Having given the two axes, set off from c half the great axis at a and b, which are the two focuses in the ellipse. Take an endless string as long as the three sides in the triangle a, b, c, fix two pins or nails in the focuses one in a, and one in b, lay the string round a, and b, stretch it with a pencil d, which then will describe the desired ellipse. 3. To draw an ellipse by circle arcs. Divide the long axis into three equal parts draw the two circles and where they intërsect one another are the centres for the tangent arcs of the ellipse as shown by the figure. 4. To draw an ellipse by circle arcs. Given the two axes, set off the short axis from A to b, divide b B into three equal parts, set off two of these parts from o towards c and c which are the centres for the ends of the ellipse. Make equilateral triangles on c c, when ee will be the centres for the sides of the ellipse. If the long axis is more than twice the short one, this construction will not make a good ellipse. 5. To construct an ellipse. Given the two axes, set off half the long axis from c to ff', which will be the two focuses in the ellipse. Divide the long axis into any num- |ber of parts, say a to be a division point. Take A a as radius and f as centre and describe a circle arc about b, take a B as radius and f as centre describe another circle arc about b, then the intersection b is a point in the ellipse, and so the whole ellipse can be constructed. 6. To draw an ellipse that will tangent two parallel lines in A and B. Draw a semicircle on A B, draw ordinates in the circle at right angle to A B, the corre- sponding and equal ordinates for the ellipse to be drawn parallel to the lines, and thus the jº curve is obtained as shown by the gure. CONSTRUCTIONS. 47 7. To construct a cycloid. The circumference C=3-14 D. I.)ivide the Tolling circle and base line C into a number of equal parts, draw through the division point the ordinates and abscissas, make a aſ = 1 d, b bºs-2/ e, c cº-3' f, then a ly and c are D9ints in the cycloid. In the Epicycloid and Hypocycloid the abscissas are circles and the Ordinates are radii to onc common centre. 8. Evolute of a circle. Given the pitch p, the angle v, and radius r. Divide the angle v into a number of equal parts, draw the radii and tangents for each part, divide the pitch p into an equal number of equal parts, then the first tangent will be one part, second two parts, third three parts, &c., and so the Evolute is traced. 9. To construct a spiral with compasses and four centres. Given the pitch of the spiral, construct a square about the centre, with the four sides together equal to the pitch. Prolong the sides in one direction as shown by the figure, the corners are the centres for each arc of the external angles. 10. To construct a Parabola. Given the vertex A, axis a , and a point P. Draw. A B at right angle to ac, and B P parallel to a, divide A B and B P into an equal num- ber of equal parts. From the vertex A draw lines to the divisions on B J’, from the divi- sions on A B draw the Ordinates parallel to ar, the corresponding intersections are points in the parabola. 11. To construct a Parabola. Given the axis of ordinate B, and vertex A. Take A as a centre and describe a semicircle from B which gives the focus of the parabola at Draw any ordinate y at right angle to the abscissa A ac, take a as radius and the focus f as a centre, then intersect the ordinate y, by a circle-arc in P which will be a point in the parabola. In the same manner the whole Parabola is constructed. 12. To draw an arithmetic spiral. Given the pitch p and angle v, divide them into an equal number of equal parts say 6. |make 01:01, 0.2–02, 0.3=0 3, 0.4=04, 05=05, and 0.6=the pitch p, then join the points 1, 2, 3, # 5, and 6, which will form the spiral re- Quired. - THE CIRCLE. T H E CIR C L E. Notation of Letters. d = diameter of the circle. r = radius of the circle. = periphery or circumfer- €10 Ce. a = area of a circle or part thereof. b = length of a circle-arc. All measures must be expressed by the same unit. c = chord of a segment, length of. h = height of a segment. s = side of a regular polygon. v = centre angle. w = polygon angle. IFormulas for the Circle. Liameter and Radius. Periphery or Circum- Area of the Circle. p = º 3.14d. | *- º - iſ (l, E : = 0.785d?. p = 2it r = 6.28r. r=# =ºs. a = t r* = 3.14,”. -*. 72 – - 2 2 p = 2/t º yā d= 2\}=1128/. =#=1. p=# =#. r=\;=osºvº. a=#=#. t =3.14159265358979828846264838821950288410/169809 27t=6.283185 #7T-0.785398 T #T }=0.318810 | *=114,5915 3tr=9,424 #T=1. 4T-12.566370 37–1,570796 Fyº-yº A ºr 2 3 T=1.772453 5T=15.707,963 | # .-0.954929 ". T=15.707963 |#t=0.392699 *=1278239 W. =0.564,189 67–18.849556|ºr=0.523599 |" ºr 7t=21.991.148 ºt-0.261799 s Q =2,54647 Wł-078788, 84–25,132,41|in=2004894 |* | Nº 12 Log. T= =3,819718 Log. T-E 9t=28.274334|sºr=0.008726. T * 0.49714987 *- LONGIMETRY. 49 13. The periphery of a Circle is commonly expressed by the Greek letter a = 3-14 when the diameter d = 1 or the unit. For any other value of the di. ameter d, we will denote the periphery by the let- ter p, r = radius, and a = area of the circle. The periphery of a circle is equal to 314 times its diam- eter, c = chord. lº 14. = ** – 0.0175, * ~ 1:5 0.0175rv, b v- 180° - 57.296% TZr 7" to = 18 –4, w = 2(180° — wy. 16. 2T, N. cº-45 °. R- r = -81 - Hâj, sº c = 2 x/2hr — h". 7, 17. (IC p = —7===. * — ſº Tº Lº N* *V *-(−x,−) 18. a , a 4-bº — cºa b (I" - T25T, a+b+c I.ONGIMETRY. 19. * = 0, 10 = w, w -- w = 180°, w > v. 20. D = B + C, A’ + B' + C = 180°, B = D — C, A + B + C = 180°, A’ = A, B = B'. 21. A + B + C = 180°, A’ = A, B" = B. 22. E + C = A + D = 180°, D = B + c, E = A + B. 23. ab & - * (a + b)* = a +2ab + bº. a lab * * 24. lºſ f *# (a-b)* = a” – 2ab + bº. (a-)*|A – f\ LONGIMETRY 54 25. ---> *s (a + b) (a - b) = a – bº. 2 6. 27. 29. z N 24 ,” * ,’ * ~ Y SN SJ ,” * 22' N X \ A : B = a + 8. 2’ N tº \\ a' 62. \\ * % * t 30. a : a = a - a - it, \\ --V-(+)-; 52 º LONGIMETRY. 31. cº - g” + b°, a* = c^ — bº, bº - cº º a”. c” = a” +b^ — 2bd, h = y/aº Tºº, _a” + b” - c”. d––– 33. c” = a + b + 2bd, h h” –Va”-dº, ! d a-t-º-º: : ——g- a : b = h : c, _dº ad –7---, c” ch d = ? -- a : c = b : d, ad re-r bc. LONGIMETRY. - 53 37. a : t = t t b, t” = ab. 38. *-(a + b) (a-b), t = Vaº – 5°. 39. aR - - Q = #2 R – *, R – r" V + ( r) tº -*- t t = V a”- (R = r)”, sin.v = a- ;’ | 40. %\s 3) t = Vă-(RTF)*, a SV% a = v tº + (RTF). Sł |41. S” V - r -\/ .” -4 l =2r — V, S = 2 v FR(FV). r14 (1 + V). 42. (2 P = †—tº dº, l = n v. Twº d” + P”, l 7& = A/ 77° d” + P. ~ 54 - LONGIMETRY. * To find the length of a Spiral. 2S yı ra l r th \ l = ºrn- TE' *- :-E" P -º- L. P = Pitch. l n 44. to find the length of a Spiral. º 1 = 7 m ( R + r.), Šº/ 1-? (R-r). 45. Periphery of an Ellipse. p = 2 v D**FT4674d. 46. To construct a screw Heliºr. _--> s A 47. To square a Circumference. A R. Kº ) 4 |R = 0:555355 d – 1-1107 r = 0.7071 S. S = 0-785398 d = 1-57079 r = 1.4142 R., v-s--→ v d = 1.27322 S = I-79740 R = 2 r. 48. To square a Circleplane. R = 0.626657 d = 1,253314 r = 0.7071 S. S = 0-886226 d = 1.77245 r = 1.4142 R d = 1.12838 S = 1.5367 R = 2 r. POLYBEDRON8. B5 49. - Tetrahedron. r = 0.20413 s. R = 0.61237 s. a = 1.73205 s”. c = 0:11785 sa. 50. Hexahedron. r = 0-50000s. R = 0.86602 s. a = 6:00000 S2. C = 1.00000 Ss. 51. Octahedron. r = 0.40721 s. R = 0.70710 s. a = 3.46410 S2. C = 0-47140 s- 52. Dodecahedron. r = 1-11350 s. R = 1-40122 s. a = 20-6457 S2. C = 7-66312 s”. 53. Icosahedron. r = 0-7558 s. R = 0-95.10 s. a = 8-66025 s”. C = 2-18169 s”. % N Sº r = Radius of an inscribed Sphere. R = Radius of circumscribed Sphere. a = Area of the Polyhedrons. c = Cubic contents of the Polyhedrons. s — Side or edge of the Polyhedrons. 56. a = 0.5d.” Rectangle. a = a b, a = b V dº — 52. Triangle. a-..."- A, a-; V - (º 2 \ 2 b Triangle. a - #!, h, ... " … = d2 E ºw * - V. – ( #: ). 58. Quadrangle. a = #h(a + b). 59. Quadrangle. a- (a [h + h^l + b h’ + c A). . PLANEMEtkr. 57 — 60. . Circle Plane. a = 77 ra = 0-785 d", 3B = ‘. = 0.0796 P2. 61. - Circle Ring. a-tº-r) = a(R+r)(R—r), a = 0-785(D* – d”). 62. Sector. a = #5 r, Tº r" v ra v * 350-III:5; sº 63. Segment. a = }{b r – c (r—h)], Tº 7-2 ty C * a = ** T :(r-h). 64. Quadrant. a = 0.785% - 0.3927 e. E. z = I. c.” 4- * § & 65. a -0.215 r -01075 e. PLANEMETRY. Ellipse. a – it R r- 0-785 D d. Barabola. a = # b h = b”, a -á hy? R. Irregular Figure. a = b(h+ h’ + h"). Ellipsoid. a = 8-88 r V R*-Er”, a = 2.22 d v D3-H dº, 70. Cylinder. a = 2 ºr h = t d h, 71.The road to Extremity of Space. . 1st. Draw a Circle and inscribe a Square, and in that Square a Circle, &c, &c., &c. . . . . . . The last figure that can be drawn, is one extremity of Space Required if the last one is a Circle or a Square? 2nd. Draw a Circle and circumscribe a Square, and around that Square a Circle, &c., &c., &c. . . . . . The last one that can be circumscribed is the other extremity of space. Required if the last figure is | a Circle or a Square 2 SURFACE of SoLIDS. "TT2. 59 Sphere. a = 4 7t r* = 12:56 r" – 7t d”. 73. Forus. 3.34784. a = 4 77° R r = 39.44 R r, a = 9.86 D d. 4.20% (o o 74, Sphere Sector. a--#444 c). Circle Zone. &ti –2+ r A -4(*#4 #) Cone. a = t R s, al, Fº 7. R ^y R2 + 73. Cone. -- d s R=s++,+4. *--tº- TD – d. a- +(D+d), , 180 D_1800D d). T TR - S STEREOMETRY. Sphere. c -º- 4.189 r, C = * -0.5% d”. Forus. c - 2 it, Rrº – 1974 R r", c -2,463 D d". Sphere Sector. - c = #7 r" h = 2,0944 r" h, # It r*(r T V r" – 4 cº. Zone. c - 7 h"(r – # h), C = 7 *** – h). w Cone. 7, 7” ) - c = **, *= 1,047 rºa, U} C = 0.2618 d” h. Comic Frustum. c - it h(R" R r + r"). * - c = 1, 7 h(D* + D d + d”) STEREOMETRY. 61 84. Cylinder. 7/ C = #: -001962 A. 85. Ellipsoid. c = 0.424 tº R r" = 4.1847 R r", C = 0.053 ſt” D d” = 0.5231 D d” 86. Paraboloid. c = $t r* h = 1.5707 r" h. 87. Pyramid. C = # a h, n sh -ā- C = *V/ rº-3- 88. Pyramidic Frustum. C -#4 + a + VA a). 89. Wedge Frustwm. (E = #(a + b). 62 STEREOMETRY. 90. Cash." - c-10453 (0.4 D-402 D at 0154), ! Tº . - e º - Gallon-gºſ4D-42 Da 11:54). 91. Cylinder Sections. c = "t r*(l + l' – #r), c - it r(, ; 1)–21 r. 92. Circular Spindle. c=t(; cº–02dſcłżvºdăjVdKTG)| Example 1. Fig. 56. The base of a Triangle is b = 8 feet, 8 inches, and the l height, h = 5 feet, 6 inches. What is the area, a = ? - . . .” 30, E * := sº = 22.6875 square feet. Example 2. Fig. 62. A Circle Sector having an angle v = 39° and the radius r = 674 inches. What is the area of the sector a = ? . . _7trº v_3-14 × 67-752 × 39° t- a = -āj--—sº- = 1562-1 square feet. Eacample 3. Fig. 75. A Spherical Zone having its diameter c = 18% inches and height h = 7# inches. What is the convex surface of the Zone 7 7t/ 3-14 - := *- º = nºs • 5% •º Fº, = & T \% + la) 4. (18 52 + 7.75 ) 315 96 Square inches. Example 4. Fig. 52. Require the radius R of a Sphere that will circumscribe a Dodecahedron with the side s = 9 inches. - - R = 1.36428 × 9 = 12:27852 inches, the answer. Example 5. Fig. 89. A Frustrum of a Cone having its bottom diameter D = 13 inches, the top diameter d = 5% inches, and the height h = 25 inches. What is the cubic contents c = ? - c = i, th(D. H. Da 4-dº)-- 0:2618×25 (13 + 13 × 5.25 + 5.25%)=20995 | cubic inches. - - Example 6. Fig. 90. A Cask having its bung diameter D = 36 inches, head diameter d = 28 inches, and length l = 56 inches, (inside measurement) how | many gallons of liquid can be contained in the cask? (The gallon = 281 cub. ii.) | Gallon = #(º X 302 + 2 × 36 X 28 + 1.5 X 2s.)= 214 gallons. GEOMETRY-TABLE of Polygons. - 63 Example 7. Fig. 14. Require the length of the circle-arc b, when the angle w = 42°, and the radius r = 4 feet, 3 inches? _* r *_ 3.14×425X42 _ 3. b = is = –156– = 3-113 feet. Example 8. Fig. 16. Require the radius of a circle-arc, whose chord is 9 feet, 4 inches, and height, h = 1 foot, 8 inches? _ cº-H4H2 9:332-H4×1-66° 98-0711 - ~ 3, T TSXI-66 T ~ T13 25 Example 9. Fig. 32. The three sides in a triangle being, a = 6'42, b = 7.75, and c = 8*66 feet. How high is the triangle over the base b? ‘. . _a2+b^–c4 6'42"--7-75°– 8.66% 1. - the height h = }/a2-d2 = V6ſ2?–T51752 = 6.24 feet, the answer. - Example 10. Fig. 41. The radius of a walking beam is, r = 8.36 feet, the stroke S = 5.5 feet. How much is the vibration. W= ? - - - •52 Vibration, P = ?' – Vra— S* = 886– 8:303 _º . . . 4 V =0.471 feet = 5-65 inches = 5; the answer. = 7-384 feet. TABLE OF POLYGONS. Centre | Polygon | Side Area Apotem Side Area Number Angle w. Angle v. = k R. la A Sº. = k R. = h r. = k r?. of sides - - ------ - . tº: (9 (L) (º (G) . 4. T.Tāli;0&T 603TTT32 O-4350 O-5000 || 3:Töäi Tetragon. 4| 90° 90° 1.4142|| 1:0000 || 0-7071 ||2-0000 Pentagon. 5, 72° 108° 1:1755] 17205 || 0-8090 | 1.4536 | Hexagon. 6' 60° 120° 1-0000 2'5980 || 0-8660 I-1547 Heptagon. 7| 51043/128°17' | 0-8677 3-6339|| 0-9009 || 0-9631 Octagon. 8] 45° 135°. 0-7653 4-8284 || 0-9238 0-82S4 || 3 Nonagon. 9| 40° 140° 0-6840 | 6’1820 || 0-9396 || 0-7279 Decagon. 10| 369 1440 0.6180 7-6942 0-9510 || 0-649S Undecagon, 11. 32°13' ||147°47' | 0-5634 9-3656 0-9595 0-5872 Dodecagon. 12| 30° 1509 | 0-5176 11-196 || 0-9659 || 0-5359 14|| 25°43' 154°17' | 0-4450 15:334 || 0-97.62|0:4562 15| 24o 1569 0.4158 |17-642 || 0-9781 || 0-4250 16 220307|1570307 0-3900 |20-128 0.9807 0-406.8 18| 200 1609 0-3472 25-534 || 0-9848 || 0.3526 20| 189 |1629. 0-3130 40-634 0-9877 0-3166 24 15 o 1659 0-2610 45-593 0-9914|| 0-2632 Explanation of the Table for Polygons. The number of sides in the polygon is noted in the first column. k = tabular coefficient, to be multiplied as noted on the top of the columns. Example 1. How long is the side of an inscribed Pentagon, when the radius of the circle is 3 feet, and 4 inches? (4 inches = 0-333 feet) - - 3.333X1-1755 = 3-9.179 feet, the answer. - | Example 2. What is the area of a Heptagon when one of its sides is 1375 inches 13752×3-6339=687-02 square inches. 64 CIRCUMFERENCE AND AREA OF CIRCLES. Circum, Area. Circum. Area. Circum. / Area. Diam- O || Diam O Diann- O eter. eter. eter. 1 || 3:1416 || 0-7854 51 | 160-22 || 2042.8 || 101 || 317-30 || 8011.9 2 || 6’2832 3.1416 52 163-36 2123.7 || 102 || 320-44 8171-3. 3 || 9-4248 || 7-0686 53 166-50 2206.2 || 103 || 323-58 8332.3 4 || 12:566 12:5664 || 54 || 169-65 2290-2 || 104 || 326-73 | 8494-9 5 || 15-708 || 19.63.50 || 55 172-79 2375-8 || 105 || 329.87 8659.0 6 | 18-850 28.2743 || 56 175-93 2463-0 || 106 || 333-01 || 8824-7 7 || 21-991 || 38.4S45 || 57 179-07 || 2551.8 || 107 || 336-15 | 8992-0 8 25' 133 50.2655 || 58 182°21 2642-1 || 108 || 3:39:29 || 9,160-9 9 28-274 63-61.73 || 59 || 185-35 2734-0 || 109 || 342-43 | 9331-3 10 || 31.416 || 7S-54 60 | 188'50 2827.4 || 110 || 345-58 || 95.03-3 11 || 34°558 95-03 61 || 191-64 2922.5 || 111 || 34S-72 96.76-9 12 || 37-699 || 113-10 62 19478 || 3019. 1 || 112 || 351-86 9852.0 13 || 40.841 132-73 63 || 197'92 || 3117.2 || 1 13 || 355.00 10028.8 14 || 43-982 || 153-94 64 || 201-06 || 3:217-0 || 114 || 358-14 || 10207-0 15 47-124 || 176-71 65 | 204-20 || 3318.3 || 115 || 361-28 || 10386-9 16 || 50°265 201.06 66 207-35 | 3421.2 || 116 364:42 10568.3 17 | 53.407 226.98 67 210.49 || 3525.7 || 117 || 367.57 10751-3 18 56°549 || 254'47 68 || 213.63 3631-7 || 118 370-71 || 10935.9 19 || 59-690 283.53 69 || 216-77 || 3739-3 || 119 || 373-85 11122.0 20 || 62.832 314-16. 70 219.91 || 3848.5 || 120 || 376-99 || 11310 21 || 65'973 || 346.36 71 223-05 || 3959.2 || 121 || 380.13 || 11499 22 || 69°115 || 3S0-13 72 || 226-19 || 4071.5 || 122 383-91 || 11690 23 || 72°257 || 415.4S 73 || 229-34 4.185.4 || 123 || 386-42 11882 24 || 75°39'S 452.39 74 232-48 || 4300-3 || 124 389-56 || 12076 25 || 78°540 || 490-87 75 235.62 4417.9 || 125 392.70 || 12272 26 81'681 || 530-93 76 238.76 4536-5 || 126 395'84 12469 27 | 84.823 572-56 77 || 241.90 4656-6 || 127 398-98 || 12668 28 87-965 615.75 78 245-04 || 4778.4 || 128 I 402-12 | 12868 29 || 91-106 | 660-52 79 || 24S-19 || 490I-7 || 129 || 405-27 | 13070 30 || 94'248 || 706.86 80 || 251.33 || 5026-6 || 130 40S-41 || 13273 31 || 97,389 754-77 81 254'47 || 5153-0 || 131 || 411:55 13478 32 100°53' | S04-25 82 257-61 5281-0 || 132 || 414.69 13685 33 || 103.67 || 855-30 83 260-75 5410-6 || 133 417-83 || 13893 34 106-81 907-92 || 84 263.89 || 5541-8 || 134 420-97 | 1.4103 35 | 109-96 962-1]. 85 || 267.04 || 5674-5 || 135 || 424-12 || 14314 36 113-10 || 1017.88 || 86 270.18 || 580S-8 || 136 || 427.26 14527 37 116-24 1075-21 || 87 273.32 5944-7 || 137 || 4:30-40 || 14741 38 || 119°38 || 1134-11 || 88 || 276-46 60S2-1 || 138 || 433-54 || 14957 39 122°52 1194-59 || 89 || 279-60 | 6221-1 || 139 || 436-68 15175 40 || 125'66 1256-63 || 90 2S2-74 6361-7 || 140 || 439-82 | 15394 4l 128-81 || 1320-25 || 91 || 285-88 6503.9 || 141 442-96 || 15615 42 || 131-95 || 13S5-44 || 92 || 289-03 | 6647-6 || 142 446-11 || 15837 43 || 135-09 || 1452:20 || 93 292-17 | 6792.9 || 143 || 449-25 | 16061 44 || 138°23 || 1520-52 || 94 || 295-31 || 6939.8 || 144 || 452°39 || J 6286 45 141-37 1590-43 || 95 || 298:45 7088-2 || 145 || 455-53 | 16513 46 || 144-51 | 1661.90 || 96 || 301:59 || 7238-2 || 146 458-67 16742 47 147-65 1734.94 || 97 || 304.73 | 7389.8 || 147 || 461.8]. 16972 48 150-80 | 1809:55 || 98 || 307-88 7543-0 || 148 464.96 || 17203 49 || 153,94 | ISS5-74 || 99 || 311-02 || 7607-7 || 149 || 468-10 || 17437 50 | 157-08 || 1963.5 || 100 l 314-16 || 7854-0 || 150 l 471.24 17671 CIRCUMFERENCE AND AREA of CIRCLEs. 65 Diam- Cter. I51 200 | Circum. 474.38 477-52 480-66 483-81 486-95 490-09 493-23 496-37 . 499-51 502-65. 505'80 508-94. 512-08 515-22 518:36 521.50 524-65 527-79 530-93 534-07 537-21 540-35 543.50 546-64 549-78 552-92 556°06 559-20 562-35 565-49 | 568-63 571-77 574-91 578-05 581-19 584-34 587-48 590-62 593-76 596-90 600-04 603-19 606-33 609-47 | 612-61 615.75 618-89 622-04 625-18 628°32 Area. O | 201 17908, I814.6 18385 18627 18869 19113 19359 19607 19856 20106 20358 20612 20S67 21124 21.382 21642 21904. 22167 22432 22698 22966 23235 23506, 23779 24053 24328 24606 24.885 25165 25447 25730 26016 26302 26590 26880 27172 27.465 27759 28055 28353 28652 28953 29255 29559 29865. 30|72 30481 30791. 3.1103 3.1416 Diam-l etor, Circum. | 631.46 '634-60 637.74 640°89 644:03 647-17 650-31 653-45 656°59 659-73 662-88 666-02 669-16 672.30 675-44 3| 678-58 7 | 681-73 684-87 .688-0I. 691-15 694-29 697.43 700-58 703-72 | 706-86 710-00 713-14 716-28 719.42 722-57 725-71 728.85 731.99 735-13 738-27 741-42 744-56 747-70 750-84 753-98 757-12 760-27 763-41 766'55 769-69 772-83 775-97 779-12 782-26 785-40 Area. 31731 32047 32365 • 32685 33006 33329 33654 33979 34307 34636. 34967 35299 35633 35968 36305 36644 | 36984. 37325 376.68 38013 38360 387.08 3905'ſ 39408 39761 401.15 40471 40828. 41187 41548 41910 42273 42638 43005 43374 43744 44115 44488 44863 45239 45617 45996 46377 46759 | 47144 47529 47916 48305 | 48695 49087 Diam- €ter. 251 252 | 264 272 275 286 293 300 Circum. 788-54 791.68 794-82 797.96 801-11 804-25. 807-39 810:53 813-67 816-81 _819.96 823-10 826-24 829-38 832-52 835-66 838°81 841-95 845-09 848-23 . 851-37 854-51 857-66 860-80 863-94 867-08 870-22 873-36 876-50 879-65. 882-79 885-93 889-07 892-21 895-35 89S-50 90.1-64 904-78 907-92 9II'06 914-20 917-35 920-49 923.63 926-77 929-91 933-05 936-19. 939°34. 942-48 49481 49876 50273 50671 51071 51472 51875 52279 52685 53093 53502 53913 543.25 54739 55155 55572 55990 56.410 56S32 57256 57680 58107 58535 58965 59396 5982S 60263 60699 61136 61575 62016 62458 62902 63347 63794 64242 64692 65144 65597 66052. 66508. 66966 674.26 67887 68349 68813 69279 69747 70215. 70686 CIRCUMFERENCE AND AREA OF CIRCLES. Diam- eter. 301 302 303 304 305 306 307 308 309 310 3.11 312 313 314 315 316 317 3.18 3.19 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 | 349 350 Circum. 945-62 | 948.76 951-90 955-04 958-19 9.61°33 964°47 967-61 970-75 973-89 977-04 980-18 9S3-32 986°46 989-60 99.2°74 995-88 999-03 1002-17 I 005:31 100.8°45 I011°59 101473 1017-88 1021.02 I024-16 1027-30 1030-44 1033°58 1036-73 1039-87 1043.01 1046-15 1049. 29 1052°43 1055-58 1058-72 1061-86 1065-00 1068-14 1071-28 1074-42 1077-57 1080-71 1083-85 1086'99 1090. 13 1093-27 1096.42 1099-56 Area. O 71158 71631 72107 72583 73062 '73542 74023 74506 74991 75477 75964 76454. 769.45 77437 779.31 78427 78924 79423 7992.3 804.25 809.28 81433 S1940 82448 82958 83469 S3982 84.496 850I 2 855.30 86049 86570 87092 87616 88141 88668 891.97 897.27 90259 90792 91327 91863 92401 92941 93482 94.025 94569 951.15 95662 962 II Diam- eter. 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 3.71 372 373 374 375 376 377 378 379 380 381 382 3S3 384 385 386 387 388 389 390 391 392 393 394 395 396 397 399 400 Circum. II 02-70 1105-84 1108.98 I 112-12 1115-27 1118°41 1121-55 1124°69 1127-83 1130-97 1134°11 1137-26 1140°40 1143’54. 1146.68 1149-82 1152.96 II56-11 1159:25 1162-39 1165-53 1168.67 1171.81 1174.96 1178.10 1181°24 I 184°38 1187-52 1190.66 1193.81 1196.95 1200-09 1203-23 1206-37 I.209-51 1212.65 1215-80 1218-94 1222-08 1225-22 1228-36 1231-50 1234.65 1237-79 1240.93 1244-07 1247-21 I250-35 1253-50 1256-64 Area. 96.762 97.314 97.868 9S423 98.980 99538 100098. 100660 101.223 101788 102354 102922 103491 104.062 104635 105.209 105785 106362 106941 107521 108103 1086S7 109272 109858 110447 T11036 T 11628 II 2221 II 2815 113411 114009 114608 .115209 II5812 116416 117021 117628 1IS237 I 18847 119459 1200.72 I 20687 121304 121922 122542 123,163 123786 1244.10 I25036 125664 Diam- eter, 401 402 403 404 405 406 407 408 409 410 41], 412 413 414 415 | 416 417 4.18 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 4.38 439 440 441 442 443 444 445 446 447 448 449 450 Circum. 1259-78 1262-92 1266-06 1269-20 T 272-35 1275-49 1278-63 1281-77 1284-91 1288-05 1291-19 1294°34. 1297-48 1300-62 1303-76 1306-90 1310-04 1313-19 1316.33 I319-47 1322.61 1325-75 1328.89 1332° 04 1335.18 1338-32 I341-46 1344-60 1347.74 1350-88 1354'03 1357-17 1360-31 1363-45 1366-59 1369-73 1372-88 1376-02 1379-16 1382.30 1385-44 1388-58 1391-73 I394.87 1398-01 1401-15 1404-29 1407-43 1410°58 1413-72 Area. O 126293 126923 127.556 128}90 12S825 1294.62 130 100 130741 181382 132025 132670 133317 133965 134614. 135265 135918 136572 E37,228 137.885 138544 139205 139867 140531 141196 141863 14253.1 14320.1 143872 144545 145220 14,5896 146574 147254. J47934 I4S617 149301 149987 150674 151363 152053 152745 153439 154134 154830 1555.28 156228 156930 157633 158337 159043 CIRCUMFERENCE AND AREA OF CIRCLES. 67 eter. Diam- 451 Circum. 1416.86 1420-00 1423-14 1426-28 I429-42 1432-57 1435'71 143S-85 1441-99 1445-13 1448-27 1451'42 1454-56 1457-70 I460-84 1463-98 1467-12 1470-27 1473°41 | 1476-55 1479-69 1482-83 14S5-97 1489-11 1492-26 1495-40 1498'54 | 1501.68 1504-82 1507-96 1511-11 1514-25 1517-39 1520-53 1523-67 1526-81 1529.96 1533-10 1536-24 1539°38 1542-52 1545-66 1548-81 1551-95 1555-09 1558-23 1561-37 I564-51 1567-65 1570-80 Area. O 159751 160466) 161171 I61883 162597 163313 164030 164748 165468 166190 166914. I67639 168365 169093 169S23 170554 171287 172021 172757 173494 174234 174974 175716 176460 177205 177952 1787.01 17945.1 180203 180956 181711 182467 183225 183984 184745 1855.08 1S6272 187038 187805 188574 189345 1901.17 190S90 191665 192442 193221 194000 194782 195565 1963.50 501 502 503 | 504. 505 506 507 508 509 510 5II 512 513 514 515 516 517 518 519 520 .52] 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 Diam- | eter. Circum. || I573-94 1577-08 1580-22 1583-36 1586-50 1589-65 1592-79 1595-93 1599-07 1602-21 1605-35 1608'50 1611-64 1614-78 1617-92 1621'06 1624-20 1627-35 1630-49 1633-63 1636-77 |. 1639-91 1643.05 1646-20 1649-34 1652-48 1655-62 1658-76 1661-90 1665'04 1668-19 1671.33 1674-47 1677-61 1680-75 1683-89 I687-04 1690-18 1693-32 1696°46 I699-60 1702.74 I'705-88 1709-03 1712-17 1715-31 1718-45 1721-59 1724-73 1727-88 229022 229871 230722 231574. 232428 2332S3 234140 234998 235858 236720 237.583 Diam- €1,021”. 55]. 552. 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 57.1 572 573 574 575 576 577 578 579 580 581 .582 583 584 5S5 586 587 588 589 590 59]. 592 593 594 595 596 597 59S 599 600 Circum. 1731-02 1734-16 1737-30 1740-44 1743-58 1746-73 1749.87 1753-01 1756-15 1759-29 1762-43 1765-58 1768-72 1771-86 1775-00 1778-14 1781-28 1784-42 1787.57 1790-71 1793.85 1796-99 1800-13 1803-27 ISO6-42 1809-56 IS12-70 1815-84 ISIS-9S 1822-12 1825-27 1828°41 1831'55. 1834-69 1837-83 IS40-97 1844°11 I 847-26 1850'40 1853-54 IS56.68 1859-82 1862-96 1866-11 1S69-25 1872-39 I875-53 1878-67 1SS1-SI 1884-96 Area. 2384.48 239,314 240182 241051 24.1922 24.2795 243669 244.545 245422 246301 247 ISI 248063 248947 249832 250719 251607 252497 25.3388 254.281 2551.76 2560.72 256970 257869 258770 2596.72 260,576 261.482 2623S9 263298 264.208 265120 266033 266948 267865 2687S3 2697.02 270624 27.1547 2724.71 273397 274325 275.254 276184. 277.117 278051 27 S986 279923 280S62 2S1802 2827.43 58 CIRCUMFERENCE AND AREA OF CIRCLES. Diam- Čter. 601 602. 603 604 605 606 607 60S 609 610 611 612 613 614 615 616 617 61S. 619 (320 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 63 638 639 640 641 642 643 | 645 646 647 ($48 649 650 644. | Circum. 1888-10 1891-24 1894°38 1897-52 1900-66 1903-81 l906-95 1910-09 I913°23 1916-37 1919-51 1922-65 1925-80 1928-94 1932-08 1935-22 I938-36 1941-50 1944-65 1947-79 1950-93 1954-07 1957-21 1960-35 1963-50 1966-64 1969-78 1972-92 1976-06 1979-20 1982-35 1985-49 1988-63 1991-77 1994-91 1998-05 2001-19 2004:34 2007-48 2010-62 2013-67 2016-90 2020-04 2023-19 2026-33 2029.47 2032-61 2035.75 2038.89 2042.04 Area. O 2836S7 284.631. 285578 286526 287.475 288426 2893.79 290333 29.1289 292247 29.3206 294166 29512S 296092 297.057 298024 29.8992 2999.62 300.934. 301907 302SS2 303858 304836 305815 306796 307.779 308763 309748 310736 311725 312715 || 3.13707 3.14700 3.15696 3] 6692 3.17690 318690 319692 320695 321699 3227.05 323713 324722 325733 326745 3277.59 3287.75 329792 330810 33.1831 Diam- etCr. 651 666 668 679 | 684. | 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 Circum. 2045-18 204S-32 | 2051.46 2054.60 2057-74. 2060-88 2064-03 2067-17 2070-31 2073°45 2076'59 2079-73 2082-88 2086:02 | 2089-16 2092-30 2095-44 2098-58 2101-73 2104.87 210S-01 2111-15 2114-29 2117.43 2120-58 2123-72 2126-86 21.30-00 21.33:14 2136°28 2139-42 2142.57 2145-71 2148'85 2151-99 2155-13 215S-27 2161-42 2164°56 2167-70 2179-84 2173-98 2177-12 2180-27 2183-41 2186.55 2.189-69 219.2°83 2195-97 2199°II. Area. O 33.2853 33.3876 334901 335927 336955 337985 339016 340049. 341083 342119 343157 344-196 345237 346279 34.7323 348368 3494.15 350464 351514 352565 353618 354673 355730 356788 35.7847 358908 359971 361035 362101 3631.68 364237 365.308 366380 367453 368528 369605 370684 37.1764 37284.5 373928 37.5013 376099 377187 37.8276 379367 380459 381554 382649 383746 38484.5 Diam- eter. 701 702 703 704 705 706 707 708 709 710 711 712 713 714. 715 716 717 718 719 720 721 722 723 724 725 726 | 727 728. | 729 730 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 731. | 732 Circum. 2202-26 2205-40 2208-54 2211.68 2214-82 2217-96 2221-11 2224-25 2227-39 2230-53 2233-67 2236-81 2239.96 2243-10 2246-24 2249-38 2252-52 2255-66 2258-81 2261-95 2265-09 | 2268-23 22:71-37 2274-51 2277-65 22.80-80 2283-94 2287-0S 2290-22 2293-36 2296.50 2299-65 2302-79 2305.93 2309-07 2312-21 2315-35 2318.50 2321-64 2324-78 2327-92 2331-06 2334-30 2337-34 2340-49 2343-63 2346-77 2349-91 2353.05 2356-19 Area. 385945 || 387,047 | 38815I 389.256 | 390363 | 39.1471 392580 3.93692 | 394.805 395919 397.035 398153 399272 400393 4015.15 40,2639 403765 4.04892 406020 407150 408282 409416 4.10550 4.1.1687 412825 413965 415106 4.16248 4.17393 4.18539 419686 420.835 42.1986 . 423139 424,292 425447 426604 4277.62 42S922 430084. 431247 432412 43.3578 434746 43.5916 437087 438259 439433 440609 44.1786 CIRCUMIFERENCE AND AREA of CIRCLEs. 7.59 765 789 792 793 794 795 796 797 798 799 | S00 Circum. 2359-34 2362-48 2365-62 2368-76 : 2371-90 | 2375-04 2378-19 2381.33 2384-47 2387.61 2390.75 2393-89 2397-04 2400-1S 2403-32 2406-46 2409.60 24.12-74. 24.15" SS 2419' 03 2422-17 24.25-31 2428°45 2431°59 2434-73 2437-88 2441.02 2444°16 2447-30 2450-44 2453-58 2456-73 2459.87 2463-01 2466-15 24.69-29 2472-43 2475-58 2478-72 2481-86 24S5'00 2488-14 2491-28 2494-42 2497.57 2500-71 2503-85 2506-99 2510-13 2513-27 Area. O 44.2965 444146 4453.28 446511 447697 448883 450072 45.1262 452453 453646 454841 456037 457234 458434 4.59.63.5 460837 462041 46.3247 464.454 465663 466873 468085 469.298 470513 471730. 472948 474,168 4.75389 476612 477836 479062 480290 481519 48.2750 483982 485216 486451 487688 48.8927 490 167 491.409 492652 493S97 495.143 496.391 497641 498892 5001.45 50.1399. 502655 Diam- eter. 801 802 803 804 805 806 807 808 809 8.10 81 I SI2 813 814 815 816 817 818 S19, S20 S21 822 823 824 825 826 827 829 840 S41 S42 S43 84.4 S45 846 847 848 849 S50 Circum. 2516.42 25.19.56 2522-70 2525-S4 2528.98 2532-12 2535-27 2538-41 2541-55 2544-69 2547-83 2550-97 2554-1} 2557.26 2560-40 2563-54 2566-68 2569-82 2572.96 2576-11 2579-25 2582-39 2585-53 2588-67 2591-81 2594-96 2598-10 260 I-24 2604’38 2607-52 2610-66 2613-81 2616-95 2620-09 2623-23 2626-37 2629-51 2632-65 2635'80 263S-94 264.2-08 2645-22 2648-36 2651-50 2654-65 2657-79 2660-93 2664-07 2667:21 2670-35 Area. 503912 505171 506432 507694 508958 510223 511490 512758 514028 515300 516573 517848 519124 520402 521681. 52.2962 524.245 52.5529 526S14 528 102 529391 530681 531973 53.3267 534562 53585S 537.157 538.456 53975S 541061 542365 5436'71 5449.79 54.6288 5475.99 548912 550226 551541 55.2858 554.177 555497 556819 55SI42 5594.67 560794 56.2122 5634.52 564.783 566]. 16 567 450 872 873 874 S75 S76 S77 S7S 879 S80 SSI SS2 SS3 884. S85 886 888 S89 890 891 892 S93 S94. S95 896 897 S98 899 900. Circum. 2673.50 2676-64 2679-78 2682-92 2686-06 2689-20 2692-34 2695-49 2698-63 2701-77 2704-91 27.08-05 2711-19 2714:34 2717-48 2720-62 2723-76 2726-90 27.30-04 2733-19 2736-33 2739-47 2742-61 2745-75 274S-89 27 52-04 2755-18 275S-32 276.1°46 2764-60 2767-74 2770-88 2774-03 2777-17 2780-31 2783-45 2786-59 2789-73 2792-88 2796-02 2799-16 2S02-30 2805-44 280S-5S 28.11-73 2814-87 281S-01 2S2I-15 2824-29 2827-43 568786 57()124 57.1463 572803 574.146 575490 576835 578182 579530 580880 582232 58.3585 5.84940 586297 587.655 589014 590.375 591 738 593] 02 59.446S 595.835 597.204 5985.75 5.9994.7 60.1320 602696 604073 605451 60.6831 608212 609595 (310980 612366 61.3754 615143 6.16534 617927 619321 620717 622114 6235.13 6249.13 62631.5 6277.18 629] 24 6305.30 63 1938 633348 634760 63617.3 CIRCUMFERENCE AND AREA OF CIRCLES. Diam- eter. 901 902 903 904. | 906 907 90S 909 910 9II 912 913 914 915 916 917 918 919 920 921 922 923 924 925 | 926 927 92S 929 930 931 932 933 Circum. 2830.58 2833°72 2836-86 284.0-00 2843-14 2846-28 2849°42 2852-57 2S55-71 2858-85. 286L-99 2865-13 2868-27 2871-42 2874.56 2877-70 2880-84 28S3-98 2887-12 2890-27 2893°41 2.896-55 2899.69 2902-83 2905-97 2909-11 2912°26 2915-40 2918'54. 2921°68 2924'82 2927-96 2931-11 Area. O 637587 639003 641840 643261 644683 646107 647533 648960 650388 651818 653250 654684 656118 657.555 658993 660433 661874 663317 664761 666.207 667.654 669103 670554 672006 674.915 676372 677831 679291 680.752 682216 683680 640421 673460. Diam- eter. 934 935. 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 '958 959 960 961 962 963 964 965 966. Circum. 2934-25 2937-39 2940°53 2943-67. 2946-81 2949-96 2953-10 2956'24. 2959°38 2962-52 2965-66 2968-81 2971-95 2975-09 2978-23 298I-37 2984-51 2987.65 2990°S0 2993-94 2997-0S 3000-22 3003-36 3006'50 3009-65 30.12-79 3015.93 3019-07 3022-21 3025-35 3028'50 3031°64. 3034°78 Area. O 685147 6866.15 688084 689555 691028 692502 6939.78 695.455 696934 6984.15 699897 701:380 702865 704352 705840 707330 70S822 710315 711809 713307 7.14803 716303 717804 719306 720S10 722.316 7238.23 725332 726842 728354. 729867 731382 732899 Diam- | eter.' 967 968 969 970 971 972 973 974. 975 97.6 977 978 979 980 981 982 983 984. 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 Circum. O 3037-92 3041:06. 3044-20 3047°34 3050°49 3053-63 3056-77 3059-91 3063-05 3066-19 3069°34. 3072°48 3075-62 3078.76 3081°90 3085'04 3088-19 309.1-33 3094-47 3097-61 3100-75 3103-89 3107-04 3110-18 3113-32 31.16°46 3119-60. 3122.74 3125-88 3129-03 3132-17 3135-31 3138°45 Area. O 734417 73.5937 737458. 738981 740506 742032 7.43559 745088 746619 748151 749685 75.1221 752758 Y54.296 755837 757378 758922 760466 762013 763561 765111 766.662 768215 769769 771325 772S82 774441 776002 777.564 779.128 780693 78.2260 783828 Explanation of the Preceding Table. When the diameter is expressed in more or less units than in the table, add or subtract so many figures more in the circumference; add or subtract twice as Imany in the area. Piameter. 9370 93.7 9.37 0.937 Examples. Circumference. 29486.7 294,367 29,4367 2.94.367 Area. 68955500 (3895.55. 68.9555 0.689555 NYSTROM’s CALCULATOR. 7L NY S T R O M 'S C A L CU L A T0 R. Wi Tº a sº º &āſūll \\ Will - sºs Hºnºlº º º. 3 º ºº: "º £º & º ALL calculations in this PocKFT Book have been computed by this Instrument. || It eonsists of a silvered brass plate on which are fixed two moveable arms, ex- tending frong the centre to the periphery. On the plate are engraved a number of curved lines in such form and divisions, that by their intersection with the arms, numbers are read and problems solved. - The arrangement for trigonometrical ealculations is such that it is not neces- sary to notice sine, cosine, tan, &c., &c., operating only by the angles them- selves expressed in degrees and Ininutes. This makes trigonometrical solutions so easy, that any one who understands Simple Arithmetic, will be able to solve trigonometrieal questions. Calculations are performed by it almost instantly, uo matter how egmplicated they may be, while there is nothing intricate or difieult in its use. The author of this book, who is the inventor, has thoroughly tested its practical utility. Without this instrument not one-tenth of the calculations and tables which he is eontinually bringing out, could be produced. A b W E R T IS E M E N T . THE attention of Engineers, Ship builders, and all whose business requires frequent and extensive calculations, is called to Nystrom's Caleulalor. Price $30. To be obtained with description by applying to John W. NYSTROM, Phila- delphia. - Communieations will be promptly attended to. This Calculator received the First Premium at the Franklin Instituto Exhibition. WM. J. YOUNG, Mathematical, Optical, and Calculating Machine Manufacturer, $ 43 N. 7th St., Philadelphia. Circumferences AND AREAs of Circles. - . :#—3.3– i|ſ # | { * 3-141 3-534 3-927 |H4-319 4-712 5-105 5-497 5-890 6°283 6-675 7-06S 7-461 7-854 8-246 8-639 9-032 9°424 9-817 10-21 10-60 10-99 H1-38 11-78 12-17 12-56 12-95 13:35 13-'74 14°13 14°52 14-92 issi º ! ! Area. •00076 .00306 -04908 *1104 “1503 •1963 •24S5 •3067 •3712 “4417 •5184 •6013 •6902 •7854 •9940 1-227 I-484 1-767 2-073 2°405 2°761 3-141 3°546 3-976 4-430 4-908 5*411 5-939 6°491 7-068 7-669 8-295 8-946 9-621 10-320 11-044 11-793 12-566 13'364 14-186 15-033 15-904 16-800 17.720 18-665 Q . | 01227 | •02761 •07669. Circ. Area : Diame- ſº % ter. 5—l 115-70 19-635 H16-10 |20-629 # -H16.49 21-647 H16-88 22-690 *—#17-27 |23-758 H 17-67 24-850 # -H18-06 |25,967 H 18:45 27-108 6—H 18-84 |28-274 H 19-24 || 29°464 3 -H 19-63 || 30-679 H20:02 |31-919 *—H20:42 || 33-183 H20-81 || 34°471 # -H21-20 35-784 - H21-57 37-122 7–H 21-99 : 38-484, †: 39.871 # —H22-77 |41.282 H23-16 |42-718 3—H23-56 |44-178 H23-95 |45-663 # -H24-34 |47-173 H24-74 48-707 s+ 25:13 50°265 * j; 51:848 # -H25-91 i 53-456 H26-31 |55-088 *—-26-70 56-745 - H2709 |58.42% # —H27-48 |60-132 | #: 61-862 | 9–H282. 63-617 H28.66 |65:396 3. —H29-05 67-200 429.45 69-029 *—H29-84 || 70-882 i. 72-759 —H30-63 || 74°662 H31-02 || 76-588 10–H31-41 78°539 31-80 |80-515 + -H32-20 |82-516 H32-59 | 84-540 #–432-98 86-590 • H33-37 88-664 # H33-17 |90-162 - #3416 |92.885 Diame- ter. Circ. .11 —r 34.55 H34.95 3 —H35-34 - ſº #—H 36-12 H36-52. # -H36.9 \|37-30 12–H37-69 H 38-09 # —#38-48 H338. # - H39-27 H39-66 : # -|-|40-05 H40-44 13–H40-84 - 41-23 # -H41-62 H42.01 #–H42-41 H42-80 # -H43-19 - 43-58 14—H43-98 H44-37 # + 44-76 H45-16 #–H45-55 H45.94 # -H46-33 H46-73 15–H47-12 H 47-51 # —H47-90 H48-30 *—H48-69 - 49-08 # -H49-48 H49-87 16—H 50-26 H50-65 # –H51-05 H 51-44 3–H51.83 H 52-22 a 433.6% Häi Óð 95-033 97-205 99-402 103-86 106-13 108°43 1 10-75 '[I3-09 115-46 117-85 120°27 122-71 I25-18 127-67 130-19 132-73 135°29 137-88 140°50 143°13 T45°80 148°48 151-20 , 153-93 156-69 159°48 162°29 165-13 167-98 170-87 173-78 176-71 179-67 182-65 185-66 188-69 191-74 194-82 197-93 201:06 204-21 207-39 210-59 213-82 217-07 220-35 223-65 101-62 . CIRCUMFERENCES AND AREAS OF CIRCLES. 73 Circ. I Area. Circ. Area. Circ. Area. } Diame- Diame- Diame- : ter. - ter. ter. N - 17—n 53-40 |226'98 || 23—II'72-25 || 415°47 29–T191-10 | 660-52 ; : ; H 53°79 |230°33 || - Higº 420-00 |º 666-22 3 -H 54-19 |233.70 || 3 + 73-04 |424-55 # -fi. 671-95 H54.58 |237:10 H73.43 |429-13 92-28 |677-71 *H ; ;| 3 ||##|;| Ti; ; H 55-37 243-97 H 74-21 |438°30 H93-06 | 689-29 # -H 55-76 |247-45 # -H 74-61 |443-01 3 +|93:46 |695-12 H 56-16 |250-94 {{}..., |44. | 93-85 700-98 18–H 56-54 |254-46 24–H 75°39 |452:39 || 30–H 94.24 |706'86 H 56-94 258'01 || H75-79 |457-11 H 94-64 || 712-76 + -H 57.33 |261.58 # -H 76-18 |461-86 3 -HQ5-03 |718-69 H 57.72 265-18 Hié-57 |466:63 H 95.42 | 724-64 #—H 58-II 268-80 4–H 76-96 |471-43 3-H95.81 |730-61 H 58-51 272°44 H77-36 |476-25 H96-21 | 736-61 --|58-90 276-11 # -H77-75 |481-10 3 -H 96.60 |742-64 H 59-29 279-81 H78-14 || 485-97 96-99 |748-69 19–H59-69 |283:52 || 25–H18:54 |490-87 || 31-#97-38 |754-76 || 60-08 || 287-27 H'78-93 |495'79 H 97-78 760-86 +-H60.47 |291-03 # -H79-32 |500-74 3 -HQ8-17 |766.99 |J60-86 |294.83 H79-71 |505-71 H98:56 |773-14 * —H61-26 |298-64 *—H80-10 || 510-70 1–H98-96 |779-31 +61-65 |302.48 HS0-50 515-72 H 99-35 | 785-51 # -H62-04 |306:35 # -H80-89 520-70 : +ji 791-73 |-| 62°43 || 310°24 H.81-28 |525.83 H 100-1 || 797-97 20–H 62-83 314-16 28-H; 539.93| 32-H100: 80424 ||63-22 |318.0% H82-07 |536'04 | H100-9 810°54 # -H63-61 |322-06 3 -H82.46 541-18 || 4 -H101.3 |816.86 – 64-01 || 326-05 –82-85 546°35 H101-7 S23-21 #—H64.40 || 330-06 *—H83-25 |551'54 3-H102.1 |829-57 H64-79 |334:10 H33-64 556-76 H102.4 S35-97 * +651: 333:16 # -H84:03 |562:00 3 -H102.8 |842-39 H65-58 |342:25 H84-43 |567.26 H103.2 |848-83 21-Hº 346.36 27—H84-82 572-55 33—H103.6 |855-30 H66.36 |350-49 H35-21 577-87 |#. 861-79 a -lº 354.65 # -H85-60 583-20 4 -H104.4 |868:30 H67-15 358-84 H.86° 588-57 H104.8 |874-84 }–H 67-54 |363-05 *—H86-39 593-95 3-H105.2 |881:41 |67-93 ||367-28 †: 599-37 |; ; # -||68-32 |371.54|| 3 +87-17 |604.80 3 -H106 |894.61 H68-72 |375-82 H.87.57 610-26 H106.4 901:25 22–H69-11 |380-13 || 28–H87-96 ||615-75 || 34–H106.8 |907-92 H69-50 || 384-46 H$8°35 | 621-26 || H107.2 |914-61 # -H69-90 388-82 3 -HS$75 |626.79 3 -H107.5 |921:32 || 70-29 393-20 -89° 14 632-35 HI07.9 |928-06 #—H70-68 |397.60 #–H89:53 |637.94 4–H108.3 |934-82 471-07 |402-03 HS9-92 643-54 H108-7 |941-60 # HY1.47 |406-49 # H90.32 649-18 * H109-1 |948:41 |#3; 410-97 # 654'83 | 109.5 |955’25 1.96 I& f. QSI3 0.9/IZ ..!-fºló, j.jg|IZ I.jf Ig 6-88.IZ 1.8%I& 9.8 Ilić, 8-80 I& 3.3606 0-880% 6-31.0% 6-390% 8-39.0% 8.2 f()Z 8-3803 8-3303 8-3 I0% 6-300% 0-866. I I-886.I 3.81.6 I 9-396 I 9-896 I 6-876 I I-736L j.jø6 I 1-#I6 I 0.906 I 3.g6SI 1.988.I I-91.81 9.998.I 6-998 I | 7-1.jSI 6-188 I j.8%SI : 6-8 ISI 9.60SI | I,00SI 1.061, I | 8. IS/I || 0-311|| 1-391. I | 7-891, I I-ji II | 6,781. I vo.IV I-99 I T A-99 I H- 9.99 IH 6.j9IH-f Q. fºll H I.j9 IH- # A-89 IH 8-89 I H-39 6-39 I - g-Z9IH- # fºil - 1-I9IH-? 9. [9][ - ..I.9 IH- ? 9.09LH 3.09 IHIQ. S-69 I j.69|| 1. # •69 I – 9.8giH–? 3.89 IH 8-19 IH- { # - • 191 H-09 9.99 I – Z-991|+ # 6-gg I T g-gg IH-f I-gg IH 1-pg|IH- # 8-fºg IF 5.8%ill–67 9.99 I - I-8g|I|-|- # M-39 IH 3.3g I |-|- # gigi. g-IGI |-|- { I-09 IH 1-09 IH-87 j-09 IH •09 IH- # , 9.6%I - Z-6fl H—% 8.8%I - .#.Sf IH- # º: 9-1 fºll-l— lift *Ioq - -ouTuſCI - J.9% II 9.9 IAI 8-101, I 3.869'I I.689 I 0.089 [ 6-019 [. G.I.99.I 8-399 I 8.879 I 6-#89 I 6.9%9I 0.1. L91 I-809 I 3.669 I f.06GI 9. [89 I 8.61 GI 0.79g.I Z.99g I g.97g I 8-139T I.6%g I g.0%g I 6. IIGI 9.80%I I-76; I [.987 I 9-11; I I.69%I 9.09 FI Z.Zgi, I 2.87%I 9.g.8f I 6.9%f I 9. SIf I 3.0 If I 6. IOPI 1-868. I fº. 988.I 3-118 I 0.698 [ 8.098.I 9.398 I G.fºfg.I F.988.I 8.838 I 3.0%8 I ‘volv | ‘olſo Z. If I T S.9f IH- # #.9f IH .9f IH–% 9. Qi IH Z-97LH- # 6.jf IH Q-77 IH-97 iſſil M-87 IH- # §til 6-37 IH-ſº 9.37 IH I-37 IH- # A-If I - 3. If IH-97 6-07 IE G.07I | # I.0 fºlſ – 8.68|IH-? 7.68 I- .68 Il-j- # 9.881} . z-sgill—FW 8. 18L – f. 181|-|- # 9.93 Il-j-ž 3.99. I- - 8.98.IH- # j. G8I - .ggill—gy 9.f3T – 8.78 IH- # 6-88. IL g.88 I 4–% I-88T - M-38 [|- # 8-38 L - - 6-I9 IH-37 g.Táil I. IGI H-. § A-08 I - 8-08 I 4–% 6.6% I - g-6&IH # I-6&IH 8’Sº, Ill-Ij, *Ioq -ouTuſCI Z.ZI8I 3.708 I Z.96%I 3.88%I 3.08%I 8-31&I G.j9%I 9.99%I M.87%I 6.07%I I.88%I f. GZZI 9.1 IZI 6.60%I 3.30%I G.f6 II 6.98TI 8.6/II 1.Ill I I.f4) II 9.99 II 0.67 II g. If II I.jpg|II | 9.9%II 3.6LII S.T.III f.j,0II I-160ſ 1.680T j-Z80T 3.910 I 6-190I | 1.090T G.890.T 3.970T I-680 I 0.380 I 6. f60L S. l IOI 8.0 [0T A-800I 81.966 0S.686 j,8.386 06.9/6 66-896 II. Ž96 “go.IV rºll .SZIHF # 9.1%IF Z.1zIH-# 8.9%II j.9%IH- # •98, IF 9.gg|IH-07 3.9%IH 8.f.g. IH- # j.jøII- .#z IHY I-8&T F g.gzIH- { # g-ZZIH-69 I.3% IF J. L3 I H # 3:13; 6.0&IH-? g.0%IF I.0%IH- # *::IIH - 3.6LIH-88 6.8 IIFT 9.8TI T # Z-SITF 8.1 IIH-f j.1T IH'. ..IIH- # 9.9 IIH Z.9 II H- 18 S. GII - #.gII + # . GII H 9.5II ||—# Z.FIIH 8.8II+ # #.9 IIH .g.II Høg J. & II || g.g|II H. : 6-III H g. III H-? I. III F 1.0LIH- # 8.0 II I 6.60 L --gg "Iaq -ouTuſCI "oi (O “SATOMIC) RO SW3(HW QINV S3ONGIH3AWDOHIO j/, CIRCUMFERENCES AND AREA8 OF CIRCLES. 75 Circ. Area. Circ, Area. Circ. Area. Plºt O O Diame- Diº O ©Ea e tº I’. - ter & W 53—n 166°5 |2206'1 59 — 1185-3 || 2733-9 || 65—F1204-2 || 3318-3 H 166-8 2216-6 - Hiß, 2745-5 - . # 3331.0 3 -iù.3 |3%. , #: 2757-1 || 3 -H2Oſ39 |3343-8 - - H16.6 2237.5 - 186°5 |2768-8 #205-3 ||3356-7 4–H 168 |2248-0 3-H186-9 |2780-5 #—#205-7 ||3369-5 || 168-4 2258°5 || H187-3 |2792.2 - 206-1 || 3332-4 § -H 168-8 |2269:0 # -H187-7 |2803-9 # -- 206.5 |3395-3 | 169-2 2279-6 HI88-1 |2815-6 |2003 ||34082 54–||169.6 |22002 || 60—H1884 |2827.4| 66–H207-3 |3421.2 | | ||70° 2300-8 HIS8-8 |2839-2 207-7 ||3434-1 + +|170-4 |2311'4 + + 1892 |2851.0 # —H 208-1 |344.7-1 Hºſº |}}}}} H189-6 |2862-8 H208.5 |346.0:1 3—H 1712 |2332-8 3–H190: 2874-7 # –H208-9 |3473-2 H 171-6 || 2343’5 H 190°4 || 28S6-6 H209-3 ||34S6-3 # -H172 |2354.2 # -H190.8 |2898.5 # -H209-7 ||3499.3 - 172-3 || 2365-0 - 191-2 : 2910-5 H 210° 3512-5 55–H172-7 ||2375'8 61—H 191-6 |2922-4 || 67—H 210°4 |3525-6 - 173-1 || 2386-6 H 19.2° 2934-4 + 210-8 3538-8 3 -H173°5 23974 # -fig2.4 |2946-4 # -H 211-2 |3552-0 HII.3.9 |24083 Higg.8 |2958-5 |Āiº 3565-2 #—H1743 |24.192 #–H1932 |2970.5 #–H 212" |3578-4 -|174-7 |2430.1 |||}|...}} 212°4 || 3591-7 | # -H175°1 |2441-0 # -H193.9 |2994-7 # -H212-8 |3605-0 ||175.5 2452-0 _j}; 3006-9 – 213-2 || 36|S-3 56–H1759 |24630 || 62-H1947 |3019-0 | 68–H213.6 |3631-6 ||176-3 |2474-0 l. Hº 303.1-2 H 214° 3645-0 + -H176-7 |2485-0 # -H195-5 3043-4 # -H214-4 365S-4 - 177-1 || 2496-1 H1959 |3055-7 H214.8 |3671-8 #—H177.5 |2507-1 || 3–H1963 |3067.9 #-H215.1 |3685-2 H177-8 |2518-2 H196-7 |3080-2 H215.5 3698-7 # -H1782 |2529.4|| 4 -H1971 |3092.5 * Tº ſº; – 178-6 || 2540-5 H 197°5 || 3104-8 216-3 ||3725-7 57–H., |}}}} || 33-#21.9 |3|12| 69-Hålº 3.3% |-|179°4 || 2562-9 HI98-3 3129.6 - 217-1 |3752-8 # -H179-8 |2574.1 3 -H1987 |3142.0 # -H217.5 |3766.4 H1802 |2585.4 H199° 3154.4 H 217-9 |37S0-0 #—H180.6 2596-7 #-Flº 3166-9 #–H 218-3 ||3793-6 isi. 2608-0 +199.8 |3179.4 H218-7 ||3807-3 # -H181°4 2619-3 # 4|2003 |#iº # -H 219.1 |3821-0 H181-8 2630-7 H200-6 || 3204.4 H219.5 |3834-7 58–4–182°2 2642-0 || 64-H201" | 3216-9 || 70—H2] 9.9 |384S-4 – 182-6 || 2653-4 |#4 3229-5 H220-3 ||3862-2 # -H182.9 2664-9 # -H201’S 3242-1 # -H220-6 3875-9 ||133-3 |2676.3 H202-2 3254'S H221 3SS9.8 #—#183.7 2687-8 }–H202.6 |3267.4 #—H221.4 |3903-6 |-|IS4"I 2699-3 |f|..., |...}. H221-S 3917.4 # -H 184.5 2710-8 # -H203-4 3292-8 # + 222-2 3931-3 Hist; |2722.4 H2088 3305-5 #: 39.45'2 CIRCUMFERENCES AND AREAS OF CIRCLES. 76 Circ. | Area. Circ. Area. Cire. Area. - / * |/? - Diame- Diame- gº Ż, Diame- ter. e O O ter. O ter. O O 71-T13%. , |}}}} 77–1241.9 |4656-6 || 83–T1260-7 5410-6 | #: 3973-1 H242-2 |4671-7 H.2611 |5426.9 + -H%-8 3987-1 # +242.6 |4686-9 3 —#261.5 |5443.2 H224-2 |4001:1 H243. 4702-1 #261.9 |5459-6 }—H 224-6 |4015-1 }-H243°4 |4717.3 *—#262-3 |5476-0 | - 225. 4029-2 H243-8 || 4732-5 | H262-7 || 5492-4 # -- 225-4 |4043-2 # -H244-2 |4747-7 # +|2831||55083 |225.8 |4067-3 H244-6 |4763-0 - 263°5 5525-3 72–1226.1 |49.1%| 78–H345, 47.8%| 84-H283.3 |5541. H226.5 |4085-6 | H 245-4 |4793-7 || H.264-2 || 5558-2 # -H226.9 |4099.8 # -H 245-8 |4809-0 3 -H264.6 |5574-8 H227-3 |41.14:0 H246-2 |4824.4 265° 559.1-3 #—4227-7 |4128-2 #—H246.6 |4839-8 }-H265-4 5607.9 H228-1 || 4142.5 H 247° 4855-2 H.265.8 |5624.5 # --|228.5 || 4156-7 # -H247.4 |4870-7 # -F|266-2 |5641-1 – 228.9 |417.1-0 j}. 4886-1 H 266-6 5657-8 73 — 229°3 || 4.185.3 79—H.248-1 || 4901-6 || 85–H 267 5674-5 4229-7 || 41997 || H.248-5 || 4917.2 - 267-4 5691-2 # -H230-1 |4214-1 # - 2489 |40327 + -H 267-8 5707.9 ,}J 230°5 |4228-5 1249.3 |49.48% H268-2 |5724.6 # —H230-9 |4242-9 #—H249-7 |4963-9 3-H269-6 || 5741-4 |-|231.3 |4257.3 |}}} 4979-5 1268.9 |5758.2 # -H231-6 |4271-8 # -H250-5 |4995-1 # -H269-3 |5775-0 |232° 4286-3 H250-9 5010-8 H269-7 || 5791-9 74-1-232°4 |4300-8 80—H251-3 || 5026.5 86—H270*1 5808-8 - 232-8 4315-3 || H251.7 5042-2 H270-5 |5825.7 # -H233-2 |4329.9 || 4 -H252-1 || 5058-0 3 -H270.9 |5842-6 —233°6 4344°5 H252-5 5073-7 H27I-3 || 5859-5 #—H234" |4359-1 *—H252.8 5089.5 3-H271-7 || 5876.5 - 234°4 |4373-8 +253-2 || 5105-4 H272-1 5893-5 # —#1234:8 |4388.4 # +253.6 |51212 § -H272.5 5910-5 H235-2 |4403.1 H254. 5137-1 H272-9 |5927.6 75–14235-6 |4417-8 || 81—H254.4 5153-0 | 87—H273-3 || 5944-6 |-|236° |4432-6 H254.8 || 51 68-9 H273.7 5961-7 # -H236'4 |4447-3 # -H255-2 || 5184-8 3 -H274.1 |5978-9 H236-7 |4462.1 H255-6 || 5200-8 H 274°4 5996-0 *—H237-1 |4476-9 #—H256- || 5216-8 3–H274.8 |6013-2 4237.5 |4491.8 -256°4 || 5232.8 4275-2 |60304 # †. 4506.6 a -lºg 3.3% # -H275-6 |604-6 || - 238-3 || 4521-5 H257-2 || 5264.9 | H276- . 6064-8 76–4238-7 |4536°4 82—H257.6 5281-0 || 88—H276-4 | 6682-1 Tiš 4551-4 H258- 5297-1 H276-8 |6099.4 } -H239-5 |4566-3 # -H258-3 || 5313-2 3 -H277.2 6116-7 — 239°9 4581-3 H258-7 5329-4 H277-6 61.34'0 #—||240-3 |4596.3 *—H259.1 5345-6 }-H278. 6151.4 – 240-7 || 4611-3 H259-5 5361-8 | H 278-4 6168-8. * +24.1.1 |4626-4 # -H259.9 5378-0 # H278.8 61862 241°5 4641'5 #: 5394-3 |2792 |62036 CIRCUMFERENCES AND AREA8 of CIRCLEs. Circ. | Area. Circ. Area. ijiane- Diame- % Diame- ter. O ter. ſº ter. § w SS 89–T1279-6 |6221:1 93—11292'l 6792.9 97– - H.2799 || 6238-6 292-5 | 6811-1 º † —H280-3 || 6256-1 3 #292.9 6829.4 # e H 280°7 6273-6 |-|293-3 | 6847-8 º }–H 281"I 6291-2 3—H293-7 |6866-1 #— sº H281°5 6308.8 H 294°1 | 6884-5 * 3 +|281.9 |6326.4 # -H294.5 |6902-9 # - . . º. | 282-3 || 6344-0 – 294-9 || 6921-3, || 307-4 || 7523.7 90—H282-7 || 6361-7 94—H295-3 |6939.7 98—|| 307-8 75.42-9 H283-1 6379-4 - 295-7 || 6958-2 308-2 7562-2 # +|283.5 |6397-1 3 -H296 |6976-7 + --|308.6 |7581-5 - 283.9 |6414-8 – 296.4 || 6995-2 |-|309-0 || 7600-8 *—H284.3 |6432.6 3—H296.8 || 7013-8 3 - || 309.4 |7620-1 – 284-7 || 6450-4 – 297-2 || 7032-3 H309.8 || 7639:4 # -H285:1 |6468-2 3 +|297.6 |7050.9 # -||310-2 || 7658-8 |-|285*4 || 6486.0 H298 || 7069-5 L|310-6 || 7678°2 91-### 6503.8 || 95—H298.4 || 7088.2 99–4 311-0 || 7697.7 286-2 || 6521-7 – 298.8 || 7106.9 | 311-4 || 7717-1 # -H286-6 |6539:6 # -H299-2 |7125.5 # -H311-8 || 7736-6 H287. 6557.6 H299-6 || 7144.3 - 312-1 || 7756-1 } –H287.4 6575-5 3–H300. 7163-0 #–H312-5 7775-6 – 287-8 || 6593-5 H300.4 |7181-8 |312-9 |7795-2 # -H288-2 6611.5 # -H300.8 || 7200-5 # —H313-3 || 7814-7 _f." 6629-5 |-|301-2 || 7219.4 |-|3|13-7 || 7834-3 92–H289° | 6647-6 96—H301-5 | 7238:2 || 100–H314-1 || 7853-9 – 289°4 | 6665-7 H; 7257.1 |-|314.5 || 7853-6 # -|-|289-8 | 6683-8 3 -H302.3 |7275-9 # –H314.9 || 78.93.3 H290 2 | 6701-9 |-|302-7 || 7294-9 |-|315-3 || 7913-1 #—H290.5 |6720-0 3—H303-1 |7313.8 #–H315-7 || 7932.7 H290.9 |673S-2 -|303.5 | 7332-8 |-|3|16-0 || 7942-4 # —H291.3 6756.4 § –H303.9 |7351.7 # —H316-4 7972-2 H291.7 |67764 |304.3 |##0% TÉ's #16 To find the angle of an arc of a circle. To find the radius of an arc of a circles EXPLANATION OF THE TABLE FOR SEGMENTs, &c. The chord divided by the height is the gauge in the Table, the quotient in the first column. k = tabular coefficient, always to be multiplied by the chord. RULE. Divide the base (chord) of the arc by its height, (sine verse) and find the quotient in the first column. The corresponding number in the second column is the angle of the arc in degrees of the circle. - RULE. Divide the chord of the arc by its height, and find the quotient in the first column. The corresponding number in the third column, multiplied by the chord, is the radius of the arc. | TABLE FOR SEGMENTS &c., of A CIRCLE. I t 78 - Chord div. Centre Radius, Cir. Arc, Area Seg. Surface Solidity Chord lºy beight. Angle v. r = k c. b = k c. |a = k cº. a = k cº, c= h cº. c = k r. 2- ) \ \s «ºs 22-f--- STN §§ - N/ ~~~ * * ^^ Nº *~2’ 458'08 IS 57.296 || 1:0000 || 01091 || 78539 || 00085 || 01744 229-18 2 28.649 || 1:0000 || 00218 || 78549 || “.00172 '03490 152-77 3 19:101 || 1:0000 | -00327 | 78462 || 00255 || 05234 114-57 4 14-327 | 1.0000 || -00436 || 78574 || 00310 || 06978 84-747 5 11.462 | 1.0001 | .00647 '78586 || “00401 || 087.22 76.375 6 9:55.30 | 1.0003 | "00741 || 78599 || “00514 || "10466 65°943 7 8-1902 | 1.0004 || 00910 || 78621 | "00592 | "12208 57.273 8 7-1678 1.0006 || -01089 || 78630 || “00686 13950 50-902 9 || 6-3728 | 1.0008 || 01254 || 78665 || 00772 | 15690 45-807 10 5'7368 1.0011 || 01407 || 78695 || “00857 || 17430 4.1-203 II. 5°2167 | 1.0013 || -0.1552 •78730 || 00964 19168 38-133 12 4,7834 || 1:0016 || 01695 || 78725 || 01031 20904 35-221 13 || 4-4168 || 1-0019 || 01S4] | 78794 || 01.114 | *22640 32-742 14 4-1027. 1-0023 •02000 || 78.832 •01199 2437.2 30°514 15 3-8307 | 1.0027 | *02157 || 788S9 '0.1288 26.104 28-601 16 3.5927 | 1.0029 || 02269 || -78909 || 01375 27834 26-915 17 3-3827 | 1.0034 •02434 •78969 -01462 29560 25.412 18 3-1962 | 1.0039 || -02592 •79028 || 01542 | "31286 24-06S 19 3.0293 1-0044 || -02744 || 79084 || 01635 | 33008 22.860 20 2-8793 1-0048 •0287S •79I40 -01722 | "34728 21°760 21 2-7440 | 1.0054 •03040 -79234 || 01802 || 36446 20-777 22 2.6222 | 1.0059 || -03178 •79300 || 01897 || 38160 19.862 23 2-5080 || 1:0066 •03343 -79.340 || 01984 -39872 19-028 || 24 2-4050 || 1:0072 || 03493 -79416 || 02072 || 41582 18°261 25 2-3101 || 1:0078 || -03639 || 79.486 || 02159 || 43286 17:553 26 2-2233 || 1:0084 •03784 •79530. •02248 || 44990 16-970 27 2-1418 || 1:0091 -03970 -79639 || 02315 46688 . 16-2S8 28 2.0673 || 1:01.01 || 04115 •79748 || 02424 °483S4 15-721 29 1:9969 || 1:01.05 || 04230 -79811 || -02511 || 500.76 15-191 30 1-9319 | 1.0113 || -0.4385 •79907 || 02600 •51762 14°970 31 I-87.10 1-0121 || 04476 •78530 || 02692 | "53445 14:230 32 1.8140 | 1.01.29 || 04710 •80098 || -0.2778 •55126 13-796 33 1.7605 || 1:01:38 •04S42 •SOISI •02866 •56802 13.382 34 1:7102 || 1:01.46 || 04989 || -80300 || 02956 .584.79 12:994, 35 I-6628 1-0155 || 05137 || "S0405 || 03046 || 60140 12-733 36 1.6184 | 1.0167 || 053II -80531 •03137 || -61802 12-473 37 I-5758 || 1:01.74 || 05401 || -80622 || 03226 63460 I 1-931 38 || 1:5358 | 1.0.184 || 05628 || "S0713 •03328 || 65||112 I 1-621 39 I-4979 || 1-0194 •05755 •80850 || 03418 || 66760 11°342 40 1.4619 | 1.0204 || 05899 || '80987 || -0.3506 | 68404 11.060 41 I-4266 | 1.0207 || -06001 || -81046 || -03589 || 70040 10-791 42 I-3952 || 1-0226 -06196 || “.81240 || -03680 •71672 10-534 43 I-3643 | 1.0237 .06359 || “81377 || 03773 || 73300 I0-289 44 || 1-3347 | 1.0248 || -06574 -81505 || -03864 || 74920 10-043 45 1.3066 | 1.0260 •06628 -81756 •03890 '76536 9.8303 46 12797 | 1.0272 | -06826 |.81795 || 04050 || 78146 9-6153 47 1-2539 1.0290 || 06998 || 81939 . .04143 || 79748 9°4092 | 48 12289 | 1.0297 || 09 138 •82064 || -042.47 '81346 TABLE Foſt SEGMENTs &c., of A CIRCLE. 79 Chord div Centre Radius Cir. Arc. Area Sag. Surface Solidity Chord | by height. Angle v. r = k c. b = k c. |a = k e?, a = k cº. c = k c*. c = k r. - º ,--T--> 2T. 2^ <> <> & 52 *------> * *>, > es <-> Yvº * S.J.,’ `-J2’ ~~ N/ ~~ 9.2113 49° | 1.2057 | 1.0309 || 07290 | 82244 || 04330 | 82938 9-0214 50 1-1831 | 1.0323 •07453 | "82384 || "04424 -84522 8-8387 51 1-1614 | 1.0336 .07611 | 82562 || 04519 -86102 8-6629 52 I •1406 || 1-0349 || -0.7758 || “82729 || 04614 || '87674 -8-4462:4. 53 I-1206 | 1.0364 •07959 || “83363 || 04685 '89238 8-3306 54 I-1014 | 1.0378 •08083 | "83072 | "04805 •90798 8 1733 55 | 1-0828 1-0393 |-08246 '83249 •0490I •92348 8, 0215 56 1-0650 1-0407 || -08400 || "S3422 || 05002 || “93894 7.8750 57 1-0478 1-0422 •08579 •83602 || 05098 '95430 7.7334-F 58 1.0313 || 1:0431 || -08680 ‘83796 || 05191 ‘96960 7-5895 59 1-0 154 || 1-0454 •08S91 | "S4064 05299 || '984S4 7-4565 60 1-0000 1-0470 -09106 || “84266 || 05400 || 1:0000 7.3358 61 •98515 1-0486 || 09209 || “843S0 || “.05466 || 1-0150 7.2118 62 •97080 1-0503 || -09375 -84581 •05583 1-0300 7,0914 63 •95694 | 1.0520 .09540 -84791 || 05684 1-0450 6'9748 64 •94352 1-0537 || -09697 || “S4996 || “.05784 || 1-0598 6'8616 65 •93058 1-0555 || 09865 -85.215 "05885 1-0746 6.7512 66 •91804 || 1.0573 || “10036 •85441 ‘05987 1-0892 6-6453 67 •90590 | 1.0591 || -10201 || -85.640 -06088 || 1:1038 6'5469 68 •89415 | 1.0610 | 10367 -85815 •06181 1-1184 6-4902 69 •88276 | 1.0629 10520 | 85.464 •06201 || 1:1328 6'3431 70 •87172 1-0648 || 10710 || -86.350 | *06396 || 1-1471 6-2400 71 -86102 || 1:0668 10887 | S6699 || 06515 1-1614 6-1553 72 •85065 | 1.0687 || -11046 || “86834 || 06604 || 1:1755 6-0652 73 | 84058 || 1:0708 || -11225 | S70S1 || 06709 || 1:1896 5-9773 74 -8.3082 1-()728 •11385 '87935 || 06815 || 1:2036 5-8918 75 •82134 1.0749 •11563 •87590 •06921 I-21.75 5-8084 76 •81213 || 1:07.70 || 11736 | 87853 || 07037 || 1:2313 5-7271 77 •803.19 1-0792 || 11910 || “SS120 -07136 | 1.2450 5-6478 78 •79449 | 1.0814 || -12072 | "S83S9 || 07244 1.2586 5-5704 79 •78606 | 1.0836 || -12281 •88677 -07352 | 1.2721 5*4949 80 •77786 | 1.0859 1244.1 | "S$949 || -07462 | 1.2855 5°4254 81 || -76988 || 1:08.82 | "12660 | *S916.1 •07512 I-2989 5.3492 82 || -76.212 | 1.0905 || "12793 -89.520 -07683 || 1:3] 21 5-2705 S3 •75458 1-0920 -12958 -89958 •07S19 || 1:32.5% 5-2101 84 •74724 | 1.0953 || 13157 || "90095 || -07907 1-33S3 5-1429 85 •74009 || 1-0977 •].3330 || -90420 •07960 I-3512 5-0.772 .86. •73314 || 1:1012 || 13546 '90734 || 08102 1-3639 5-0.134 87 •72637 1-1027 | 13704 || -91036 || 08340 I-3767 4-9501 88 •71978 || 1-1054 •13893 | "91363 | -08436 | 1.389.3 4-88S6 89 •71336 || 1:1079 14078 91696 -08530 | 1.4818 4-8216 90 •70710 | 1-1105 || 14279 || -92210 || -0862I | 1.4142 4-7694 91 •70.101 || 1:1132 •14449 || -92352 | "0S716 1-4265 4-7117 92 •69508 || 1:1159 •14643 •924.76 •0S798 || 1:4387 4°6615 93 •68930 | 1.1186 •14S17 | *92914 || -0S932 1-4567 4-5999 94 •68366 || 1-1211 | "I 5009 || -93385 | -09.076 I-4627 4-5453 95 •67817 1-1242 •15211 •93746. •09.197 | 1.4745 4°4845 96 •672S2 1-1271 | "15375 •94272 •09348 1-4863 #I ‘ā'ichio v ×o “oy SINSIVogS Hair a'iqvi, IZ06. 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S, • 2^ : \–’ 9) Kºź <> <=}= ~ y = 5 |º y = 3 | "… y = n | Coºrs o Tºy F | Toy F.J. Tº sº | Tālāq Kh ploua Alipuos | boujins | 33s waiv ‘oivºlio snipuh enuso . Aip plotſ; ) 08 TABLE FOR SEGMENTS &c., of A CIRCLE, Sy the square of the chord, is the area of the segment. - 81 cºord div. centre T'Radius cir arc. Areases. surface | Solidity Chorl } by height. Augle v. = & c. b = # c. . . a = k c*..] a = k c*. c = k c*. c = k r. SN,” - YJr. *~~~~ . * SJ," e ^* `---2” 2-7276 145 •52426 1-3265 •26889 || 1:2077 | 16965 1-9074 2-7002 || 146 •52284 || 1.3320 •27.196 || 1:2166 || 17209 || 1-91.26 2:6816 147 •52147 | 1.3377 .27449 1'2219 || "17205 || 1917.6 2-6533 148 •52015 | 1.3433 .27772 1-2318 17605 || 1-9225 2-6301 || 149 •51887 | 1.3491 -28168 || 1:2396 || “17809 || 1:9272 2-6064 150 •51764 || 1.3549 •28369 || 1:2476 "18023 1-93.18 2°5830 151 -51645 || 1:3608 28674 || 1:2563 | 18666 19363 2-5598 || 152 •51530 | 1.3668 .28983 || 1:2648 18751 | 1.9406 2-5239 || 153 .51420 I-3729 || 29397 || 1:2801 | "18845 19447 2-5143 154 .51315 | 1.3790 .29607 || 1:2824 || '18913 I-9487 2°4919 155 •51214 || 1:3852 •29928 I-2914 | *19147 I '9526 2-4699 || 156 •51117 | 1.3919 || -30259 || 1:3004 || “19374 I-9563 2-4478 . 157 -51014 | 1.3973 || 30560 | 1:3094 | • 19607 1-9598 2-4262 158 •50936 | 1.4043 •30905 || 1-3191 || “20029 I-9632 2-4.047 159 •50851 | 1.4.109 || 31239 || 1:3287 "20095 19663 2-3835 160 •50771 | 1.4175 -3.1575 | 1.3368 20342 19696 2-3613 161 •50695 || 1:4243 •31931 || 1:3490 | *20609 || I-9725 2-3417 162 .50623 || 1:4311 || 32263 | 1.3583 | "20847 | 1.9753 2-3211 163 .50555 | }•4380 . .32618 1-3682 “21105 || 1-9780 2°3004 || 164 •50491 | 1.4450 -32969 || 1:37.91 21371 1-98.05 2-2805 165 •50431 | 1.4520, -33327 | 1.3895 || 21634 19829 2-2605 166 •50374 || 1.4592 || 33684 | 1.4021 || 2:1904 || 1-9851 2-2408 167 •50323 1-4665 •34048 1-4111 || "22177 | 1.9871 2.2212 H 68 •50275 | 1.4739 •34422 || 1-4222 21946 I-9890 2-2013 I69 -50231 | 1.4813 || 34802 || 1:43.44 22766 I-9908 2-1826 170 •50191 1-4889 || 35230 1-4476 23028 1-9924 2-1636 171 •50154 | 1.4966 •35563 I-4565 23266 | 19938 2-1447 | 172 •50.122 || 1:5044 •35953 || 1:4684 || “23650 1-9951 2-1271 173 •500.93 | 1.5123 |-36337 || 1:4797 || 23900 | 19962 2-1075 174 450068 1.5202 || 36747 | 1.4927 | -24225 1-99.72 2-0892 175 -50047 | 1.5283 | "37152 || 1:5052 •24537 || 19981 2-0710 176 •50030 L-5365 •37562 | 1:51.79 •24856 I'9988 2-0.5.30 177 •50017 | 1.5448 || -37974 || 1:5308 || 251.79 1-9993 2-0352 178 •50007 || 1:5533 •38401 || 1:5439 •25531 1.9996 2:01.75 179 •50002 || I-5618 •38828 || 1:55.73 •25840 1-9999 2-0000 180 -50000 | 1.5707 •39269 || 1:5708 26.179 || 2:0000 To find the length of an arc of a circles RULE. 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().967%96-3 96.10660-g 91.9/I 91.9 9% 0000000;0- 1110f 36.3 0000000-g g399 I Q39 93, J.99999 If 0. I66??88-3 96.16868-j, f{89 I 91.9 #2, I93S17870. 0.198858-3 g|I38961.7 1.91%I 6%g £3, gfgfgfg}(). $630%08-2, 89 [?069.7 8:590'ſ #85 33, 8706 IQ ºf 0. $53,689 I-3 J,919 &SQ-7 I936 If? L3 000000090. J. J. If?IA-Z 093IZ1%.7 0008 007 O3, 619. I89%90. 9IOf 899.3 6868899.j, 69.89 IQ3 6H 99.9999990. iſ IF LO39-3 1.0%373.5% 3883 fºg 8I 63.93%8890° 9ISZI/g-g 990 Igº, I.i. 9I67 68% AI 0000093.90. IZF86 IQ.C. 0000000-f 9607 993 9I 199999990. IZIZ997.3 88.86%/8.3 | g 133 gºº, 9 I Ilg8&fl10. &WIOI7-Z #199 If 1.3 fºlz 96I #I 11,03369.10- Jiř98L93.3 || 31999.09.3 16I& 69I 3I 388839880. 98%f 68%-3, 9IOIf 97.3 831. I #I ŽI I60606060- I086833.3 8%99 Ig-3 I88I IZI II 00000000I. 1jºgfj9 [.g 111339 I-8 000I 00I 0I IIII IIIII. 1880080-3 || 0000000.8 631 I8 6 00000093I. 0000000.3 II,378%S-2, %IQ , #9 8 £f IAGSZip I. ZI863. I6. I 3Ig 1979.8% 959 67 1. 19999999.I. 90%III8.I 16876??-? 9I& 99. 9 00000000%. 69 1660/...I 0890.98%-3 gº I 9% g 000000093. IIOf 1.SG.I 0000000-3, #9 9I f £8888.8889. 9673%ff. I 809038/.'ſ 13 6 g 000000009. OIZ66%.I 98.IZVIf...I 8 f Ž 00000000"I 0000000-I | 0000000-I ſ I I ‘stuoordioasſ shootſ / ‘shootſ /* 'sogno | "serunbs requin'N | &S TABLE OF Squares, CUBEs, Squarig AND CUBE ROOTs. 83 Number. Squares. | Cubes. V Roots. */ Roots. Reciprocals, 53. 2S09 148877 7-280.1099 3-7562858 01886.7925 54 2916 157464 7-3484692 3-7797631 •018518519 55 3025 166375 7-4161985 3-8029.525 •018181818 56 3136 175616 7-4833148 || 3-8.258624 •01785,7143 57 3249 I851.93 7°5498.344 3-8485011 •017543.860 58. 3364 195112 7-6157731 3-8708766 •01724.1379 59 3481 205379 7-6811457 3.8929965 •016949.153 60 3600 216000 7-7459667 3-91.48676 •016666667 61 3721 226981 || 7-8102497 3-9304972 •0163934:43 62 3844 238328 || 7-8740079 || 3-957S915 •016129032 63 3969 250047 || 7-937.2539 3-9790571 •0158730.J.6 64 4096 262144 8-0000000 4-0000000 •015625000 65 4225 274625 8-0622577 4-02.07.256 •015384615 66. 4356 || 287496 8-1240384 4:04.12401 •015151515 67 4489 3.00763 8-1853528 4-0615480 •014925373 68 4624 || 314432 8°2462113 4-0816551 •014.705882 69 4761 328509 8:3066239 4-1015661 •0+4492754 70 4900 343000 8-3666003 4-1212853 •014285,714 71 5041 3.57911 8°426,1498 4°1408178 •014084.517 72 5184 373248 S-4852814 4°1601676 •013SSSSS9 73 5329 3890.17 | 8'5440037 4°17.93390 •0136986.30 74 5476 405224 8-6023253 4-1983364 •013513514 75 5625 421875 8-6602540 4°2171633 •013333333 76 5776 || 438.976 8-7177979 4-2358236 •013157.895 77 5929 456533 || 8-774.9644 4-2543.210 •012987.013 78 6084 474.552 8-8317609 4-2726586 •012820513 79 6241 49.3039 || 8-8881944 4-290S404 || -0.1265.8228 80 6400 512000 8-944.2719 4-3088695 •012500000 81 6561 || 531441 9-0000000 4.3267.487 012345679 S2 6724 551368 9-0553851 4-344.4815 •012.195122 83 6SS9 571787 9-1 104336 4:362070'ſ •01204S193 84 7056 59.2704 9-1651514 4-3795.191 •011904762 | 85 7225 614125 9°2195445 4-3968296 •011764706 86 7396 636056 9°2736,185 4-414.0049 || 011627.907 87 7569 || 65S503 9-327.3791 4'43104.76 •011494.253 88 7744 681.472 9-3808315 4°44'79692 •011363636 || 89 7921 704969 9-4339S11 || 4-4647451 -01 1235.955 90 8100 729000 || 9-4868330 4:48.14047 •011111111 91 82SI 753.571 9°53939.20 4-4979414 •010989.011 92 | 8464 778688 9-5916630 4-5.143574 •010869565 93 8649 804357 9-64.36508 4°5306549 •01075.26SS 94 8836 8305S4 9-695.3597 4°5468359 •010638.298 95 9025 || 857.375 9-74,67943 4-5629026 •0105.26316 96 92.16 8S4736 9-7979590 4-5788570 •0104.16667 97 94.09 9.12673 9-8488578 || 4-5947009 •010309278 98 96.04 941.192 9-8994949 4-61043.63 •010204082 99 || 9801 97.0299 || 9-949S744 4-626.0650 •010101010 100 10000 || 1000000 || 10-0000000 4-6415888 •01000000ſ) 101 10201 || 1030301 || 10-0498756 4-65700.95 •009900990 102 I0404 || 1061208 || 10-0995.049 4-67.23287 0.09803922 103 I0609 || 1092727 | 10-148S916 4-6875482 •009'70873S 104 || 10816 10-1980390 4'7025694 •009615385 1124864 * TABLE of Squares, CURES, SQUARE AND CUDE Roots 84 Number. Squares. | Cubes. v Roots. WTRoots. Reciprocals. 105 11025 1157.625 | 10.2469508 || 4-7 176940 •009523810 I06 || 11236 1191016 || 10°2956.301 || 4-7326235 •009.433962 I07. 11449 || 1225043 || 10-3440804 4*7474594 •00934.5794 108 11664 | 1259.712 || 10-3923048 4-7622032 •009.259259 109 11881 1295029 10°4'403065 4.7768562 •009174312 110 || 12100 1331000 || 10:4880885 4-79.14199 || “009090909 111 12321 || 1367631 10-5356538 4'8058995 || “009009009. 112 I2544, 1404928 || 10°5830052 4.8202845 . . *0089285.71 113 12769 || 1442897 || 10-6301.458 4'8345881 •0088.49558 114 12996 || 1481544 10-6770783 4'8488076 •0087.71930 115 13225 1520875 10-7238053 || 4-8629442 •008695652 116 13456 1560896 || 10-7703296 4-8769990 •008620690 117 136S9 | 1601613 || 10-8166538 4'890.97.32 •008547009 118 13924 1643032 | 10-8627805 4-904 S681 •0084.74576 119 14161 | 1685159 || 10-9087121 4-9186847 | *00S403361 120 14400 1728000 | 10-9544512 4°9324242 •008333333 121 I4641 1771561 | 11:0000000 4.9460874 •008264463 122 14884 1815848 || 11-0453610 4-9596757 •0081967:21 123 15129 | 1860867 || 11-0905365 4-973.1898 | *008130081 124 15376 | 1906624 || 11-1355.287 4'9866310 •008064516 125 15625 | 1953125 | 11-1803399 5-0000000 •008000000 126 15876 2000376 || 11-22497.22 5-0.132979 •007936508 127 16129 2048383 || II-2694277 | 5-0265257 •007874016 128 16384 || 2097 H52 || 11.3137085 5-0396842 •0078.12500 -129 16641 2146689 || 11-3578167 5-05277.43 •007.751938 130 16900 2197000 || 11-4017543 5°06'57970 •007692308 131 17161 || 2248091 | 11:44.55231 5-0787531 •007633588 132 17424 2299968 || 11-489.1253 5-0916434 || “.007575758 133 17689 || 2352637 II •5325626 5-1044687 •007518797 I34 17956 || 2406104 || II-5758369 5-1172.299 •007462687 135 18225 2460375 11-6189500 5-1299278 •007407407 136 18496 || 2515456 | 1.I. 6619038 5-1425632 •007352941 137 18769 || 2571353 | II-7046999 5-155.1367 •007299270 138 19044 2628072 | 11-7473401 5-1676493 •00724.6377 139 19321 2685619 II-7898.261 5-1801.015 •007194245 140 19600 27.44000 | 11-832.1596 || 5-1924.941 •007 142857 141 19881 2803221 11-8743421 5-204S279 •007092199 142 20164 2863.288 || 11-9163753 5-2171034 •007042254 143 20449 292.4207 || II-9582607 5-2293.215 •006993007 . 144 20736 || 29.85984 || 12:0000000 5°241.4828 *0069.44444 145 21025 3048625 | 12:04.15946 5-2535879 •006S96552 146 21316 || 3112136 | 12:08.30460 5°2656374 •006S49315 147 21609 || 3176523 12-1243557 5-277.6321 •006802721 L48 || 21904 || 3241792 | 12-1655251 5°2895725 •0067.56757 149 22201 || 3307949 | 12-2065556 5°301.4592 •0067.11409 150 22500 3375000 | 12-2474487 5-3132928 •006666667 151 22801 || 3442951 12-2882057 5°3250740 •006622517 152 23104 || 3511008 || 12:3288280 5-336.8033 •006578947 153 23409 || 3581577 12-3693169 5°3484.812 •0065.359.48 I54 23716 || 3652264 | 12:4096736 || 5-3601084 •00649.3506 155 24025 | 3723875 12'4498996 5-3716S54 || -0.0645-1613 156 24336 37.96416 || 12:48.99960 5°3832126 •006410256 TABLE OF SQUARES, CUBEs, SQUARE AND CULe Roots. 85 . Number. Squares. | Cubes. V Roots. &/ Roots. Reciprocals. 157 || 24649 || 3869893 12-5299641 5-39.46907 •006369427 158 24964 || 394.4312 || 12:5698051 5-4061202 •006329114 1.59 25281 | 4019679 12.6095202 5*4.175,015 •006289.308 160 25600 | 4096000 | 12-6491106 5-428S352 | *006250000 l61 25921 || 4173281 12-6885775 || || 5-4401218 •006.211180 162 26244 || 4251528 12:7279221 5*4513618 •006172840 l63 || 26569 || 4330747 | 12-7671453 5-4625556 || “006134969 164 || 26896 || 4410944 | 12-8062485 5-4737037 •006097561 165 || 27.225 || 4492.125 | 12-8452326 5°4848066 •006060606 166 || 27.556 || 45.74296 | 12-88.40987 5*4958647 •006024096 I67 - || 27889 || 4657463 | 12.92.28480 5-5068784 •005.98SO24 I68 2.8224 || 4741632 12-96.14814 5'51784.84 •005952381 169 28561 || 4826.809 13-0000000 5-5287.748 •005917160 l'ſ 0 28900 || 4913000 || 13-03840.48 5:5396.583 •005882353 171 29241 || 5000211 || 13-0766968 5-5504991 •005S47953 I72 29584 5088.448 13-1148770 5'5612978 •0058.13953 173 29929 || 51777.17 13-1529464 5-5720546 •0057.80347 174 30276 || 5268024 || 13-1909060 5'5827702 •005747 126 175 80625 5359375 | 18-2287566 5-5934447 •005714286 176 30976 || 545.1776 I3-2664992 5-6040787. •005681818 177 31329 || 5545233 13-304.1347 5-6146724 •00564971S 178 3.1684 || 5639752 13°3416641 5-6252263 •005617978 179 32041 || 5735339 13-3790SS2 5:6357408 •0055S6592 180 32400 || 5832000 | 13-4164079 5.6462.162 *00555,5556 ISI 32761 5929741 13°4536240 5-6566528 •005524S62 182 33 124 6028568 13°490.7376 5-6670511 *005494505 183 33489 || 6128487 || 13:52.77493 5-67.74114 *0054644S1 184 33856 || 62295.04 || 13-5646600 5-6877.340 *0054347S3 185 34225 | 6331625 I3-60.14705 5-6980.192 *0054.05405 186 34596 || 6434.856 13-6381817 5-70S.2675 *005.3'ſ 6344 187 34969 || 6539.203 13-6747943 5-71S4791 *00534.7594 188 35344 | 6644672 I3-7113092 5-72S6543 *005319149 189 35721 | 6751269 13-747.7271 5-7387936 •005291005 190 36100 | 6859000 13-7840.488 5-748897.1 •005.26315S 191 36481 | 6967871 13-S202750 5-75S9652 •005235602 192 36864 || 7077888 13-8564065 5-7689982 •005.20S333 193 37249 || 7189517 | 13-89.24400 5-778.9966 •005 ISI347 194 37636 | 7301384 || 13-928.3883 5-78S9604 *005154639 195 38025 74.14875 13-964.2400 5-79.SS900 •005128205 196 3S416 7529536 14:0000000 5-8087.857 •005102041 197 38809 || 7645373 14-0356.688 5-81S6479 •005076142 198 39.204 || 7762392 I4-0712473 5-8284867 •005050505 199 || 39601 || 7880599 14-1067360 5-8382725 -005025 126 200 40000 || 8000000 || 14-1421356 5-8480355 •005000000 201 40401 8120601 || 14-1774469 5.8577660 •004975124 202 40804 || 824.2408 || 14-2126.704 5-S674673 •00495.0495 203 41209 || 8365427 | 14-247S068 5-S771307 •004926.108 204 41616 | 8489664 || 14-2S28569 5-8867653 -004901961 205 42025 | 8615125 14-3178211 5-89636S5 •004878049 206 - || 42436 || 8741816 || 14-3527001 5-905.9406 •004S5.4369 207 42849 || 8869743 || 14-3S74946 || 5.9154817 ‘004830918 208 43264 | 89.98912 || 14-4222051 5°924992.1 •00480769? 86 . TABLE of SQUARES, CUBEs, SQUARE AND CUBE ROOTS. Number, Squares | Cubes. V Roots. N/ Roots. Reciprocals. 209. 43681 || 9,1293.29 || 14-4568323 5°934,4721 •004784689 210 || 44100 926.1000 || 14-4913767 5°9439220 •004761.905 || 211 44521 | 9393931 || 14.5258.390 5°95334.18 •004.739336 212 44944 9528.128 14:5602198 5-9627320 •0047169S ſ 213 || 45369 || 9663597 14:5945.195 5-97.20926 •004694836 214 45.796 || 98.00344 || 14-6287388 5°9814240 •004672897 | 215 46225 | 9938.375 || 14-66.28783 5°990.7264 •004651.163 216 46656 10077696 || 14.69693.85 6:0000000 -004629630 217 47089 10218313 || 14-7309.199 || 6-00924.50 *004608295 218 47524 10360232 || 14-7648231 6-01846.17 •004587156 219 || 47961 | 10503459 || 14-7986486 || 6-0276502 || 004566210 | 220 4S400 10648000 || 14-83.23970 6-0368107 •004.5454.55 221 48841 || 107.93861 14.8660687 6-0.459435. •004524887 | 222 49.284 ||1094.1048 || 14-8996644 6-0550489 •004504505 223 49729 11089567 14-9331845 6-0641270 •004484305 || 224 501.76 11239424 || 14-9666.295 6-0731779 •004.464286 || 225 50625 | 11390625 | 15-0000000 6-0824020 *004444444 | 226 51076 || 1154.3176 | 1.5-0.332964 6'099] 994 | *004424779 227 51529 || 11697083 | 15-0665.192 6-1001702 •004405286 228 51984 || 11852352 | 15-0996689 6-1091.147 •004385965 229 52441 12008989 15-1327460 6°1180332 •0043.668] 2 230 52900 | 12167000 | 15-1657509 6-1269257 •004.347826 231 53361 | 12326391 || 15-1986842 6-1357924 •004.329004 232 53824 12487.168 15:2315462 6-1446337 •0043.10345 233 54289 || 12649337 15-2643375 6-1534495 •004291845 234 B4756 | 12812904 || 15-29.70585 6°] 622401 •004273504 235 55225 | 1297,7875 | 15-3297097 6-17100.58 •004.255.319 236 55696 || 1314.4256 15.362.2915 6-1797466 || -004237288 237 56169 |1331.2053 | 15:394S043 6°1884.628 •004219409 238 56644 || 1348.1272 | 15.4272486 6-1971.544 •0042016S1 239 57121 J.365.1919 || 15.4596.248 6-2058.218 •004184100 240 57600 | 13824000 15-4919334 6°2144650 -004166667 | 241. 58081 | 13997.521 15:5241747 6°2230843 •0041493.78 242 58564 || 14172488 15-5563492 6-231.6797 •004132231 243 59049 14348907 || 15:5884573 6*24025.15 -004115226 | 244 59536 | I 4526784 || 15-6204994 || 6’2487998 •004.098361 245 60025 || 14706125 | 15.6524758 6°25'73248 •004081633 | 246 60516 14886936 | 15.6S43871 6:2658266 •00406504.1 | 24.7 61009 || 15069223 || 15.7162336 6°2743054 •0040485.83 | 248 61504 || 15252992 | 15-7480.157 6-2827613 - || 00403.2258 249 62001 || 15438249 || 15.7797338 6-291.1946 •004016064 | 250 62500 15625,000 || 15-8113883 6°2996053 •004000000 25I 63001 || 15813251 | 15-84297.95 6-3079935 •003984064 | 252 63504 || 16003008 || 15-8745079 6-3163596 •003968254 253 64009 | 16194277 15-9059.737 6-324.7035. •00395.2569 254 64516 16387064 15-937.3775 || 6′3330256 •003937008 255 65025 | 16581375 | 15-96.87.194 6-34.13257 •0039.21569 | 256 65536 | 16777216 | 16.0000000 6-3496.042 •0039.06250 257 66049 |1697.4593 | 16.0312195 6°35'78611 •003891051 258 66564 17173512 | 16-0623784 6°3660968 •003S75969 259 67081 |1737.3979 | 16.0934769 6-374.3111 •00386.1004 |. 260 67600, 1757.6000 | 16-1245155 6°38'25'043 •00384615.4 1 ABLE of SQUAREs. CUBES, SQUARE AND CUBE, Roots. 87 -- Number. Squares. | Cubes. V Roots. &/ Roots. Reciprocals. | 261 68121 17779581 | 16:1554944 || 6-3906.765 | -003831418 262 | 68644 17984.728 16-1864141 || 6-3988279 •003816794 || - 263 69169 | 18191447 | 16:2172747 || 6-4069585 •003802281 264 69696 | 18399744 16-2480768 6-4.150687 •003787879 265 70225 18609625 | 16-2788206 6°4231583 •003773585 266 70756 | 1882.1096 || 16-3095064 || 6-4312.276 •003759398. 267 71289 19034163 | 16-3401346 6-43927.67 •003745318 268 71824 | 19248832 | 16-3707055 6-4.473057 •003731343 269 72361 19465109 | 16-4012195 6°4553148 •003717472 270 72900 | 1968.3000 | 16.4316767 || 6-463304] -003703704 271 73441 | 1990.2511 | 16-4620776 || 6-47.12736 •003690037 272 73984 || 20123643 | 16-49.24225 6-4792236 •0036764.71 273 74529 2034.6417 | 16-5227116 6-4871541 •003663004 274 75076 20570824 | 16.5529454 6-495.0653 •00364963.5 275 75625 20796875 | 16.5831240 6-5029.572 •003636364 276. 76176 21024576 | 16-6132477 6-5] 08300 •003623.188 277 76729 21253933 16-6433170 6.5186839 || 003610108 278 77284 21484952 | 16.678.3320 6'52651.89 •003597122 279 7784I 23717639 || 16-7032931 6-5343351 •0035S4229 280 78.400 21952000 | 16-7332005 6'542] 326 •003571429 281 78961 22.188041 | 16.7630546 6-54991.16 •003558719 282. 795.24 224.25768 16-7928.556 6-5576722 •003546099 283 || 80089 22665187 | 16.8226038 6'5654144 •003533569 | 284 80656 22906304 || 16-8522995 6-5731385 •0035.21127 285 81225 || 23149] 25 16-88L9430 6-5808443 •003508772 286 81796 || 23393.656 16-9115345 6-5885323 •0034.96503 | 287 82369 || 23639903 || 16.94107.43 6-596.2023 •0034.84321 288 82944 23887872 | 16-97.05627 6.60385.45 •003472222 || 289 835.2L | 24.137569 || 17-0000000 6-61 14890 -0034.60208 | 290 84.100 24389000 || 17-0293864 6.6191060 .003448276 | 291 84681 24642.171 || 17-0587221 6-626.7054 •003436426 292 85264 24897088 I 7-08.80075 6-634287.4 •003424658 293 85849 25.153757 | 17-1172428 6-641S522 •0034.12969 294 86436 25412184 I7-1464282 6-6493998 •003401.361 | 295 87025 || 2567.2375 17-1755640 6-6569302 •003389831 296 87616 || 25934836 || 17-2046505 6-6644437 •003378378 || 297 88209 26198073 || 17-2336879 || 6-6719403 •003367003 || 298 88804 || 26463592 || 17-26.26765 6.6794200 •003355705 || 299 | 894.01 || 26730899 || 17-2916.165 6.686883I. •00334,4482 300 90000 || 27000000 || 17-3205081 6.6943295 •003333333 | 30I 90601 || 27270901 || 17-34935L6 || 6.7017593 •003322259 302 || 91204 || 27543608 || || 7-378l472 6-7091729 •00331.1258 303 91809 27818127 | 17-4068952 6-7 165700 •00330.1330 304 92416 28094464 || 17-4355958 6-72395.08 •0032S9474 305 93025 28372625 17.4642492 6-73.13155 -0032.78689 i. 306 93636 28652616 || 17-4928557 6-73S6641 •003267974 || 307 94.249 || 28934443 17-5214,155 6-7459967 -003257329 | • 308 94864 29218112 || 17-54992S8 6-7533134 •003:24.6753 || . 309 95481 2950.3609 || 17-5783958 6-7606143 •003236246 | 310 96.100 || 29791000 || 17-6068169 6-7678995 •003225S06 3] 1 96.721 30080231 17.6351921 6-7751690 •00321.5434 . 312 97344 30371328 || 17-663.5217 6-7824229 •003205128 $9% ºf 1,300- IZ879.1400- I8 f{9/300- 380011300- -81.11/1300- g|IggS/300. 963861,300- 0&II08300- 6868.08%00- W 069 IS300- 69 SiºS300- I983.8 S300- 6060f 8300- 3006?8300- $f IAgS300- 0889.98%00. $9981.S300- fjSIS83,00- § 1 IQ6SZ00- 19986SZ00- 11,6906300- Žgiºg I6300- 1168Z6200. 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"Ioquin'N 68 ‘sootſ auno GNV asynt's ‘satino “saavnbs ao attvl, 90 TABLE OF SQUARES, CUBES, SQUARE AND CUBE R00Ts. N umber. Squares. Cubès. V Roots. W foots. Reciprocals. 417 173889 || 725.11713 || 20-4205779 7°470999.1 •002398082 418 || 174724 | 73034632 || 20-4450483 || 7°4769664 •002392344 | 419 || 175561 | 735.60059 20-4694.895 7'4829242. •002386635 | 420 176400 || 74088000 || 20.4939.015 || 7-4888724 •0023.80952 | 421 177241 746.18461 | 20-5182845 7'49481.13 •002375297 | 422 || 178084 7515.1448 20-5426386 7-500 7406 •002369668 || 423 178929 || 75.686967 20-5669638 || 7-5066607 | *002364066 | 424 179776 76225.024 || 20-591.2603 7-51257H5 •002358491 | 425 | 180625 || 767656.25 | 20-61552S1 7-5184730 •002352941 || 426 | 181476 || 77308776|| 20-6397674 7°5243652 •00234.7418 || 427 | 182329 || 77854.483| 20.6639783 7:5302482 •00234.1920 | 428 183184 7840.2752 | 20-688-1609 7-5361.221 •002336449 | 429 18404I 78953589 20-7123152 7-54.19867 •00233.1002 | 430 | 184900 || 79507000 | 20.7364414 7-5478423 •002325581 431 185761 80062991 20-7605.395 7~5536888 •002320186 432 186624 80621568 20.7846097 7.5595.263 •00231.4815 433 187489 81182737 || 20-8086520 7-5653548 •00230.9469 434 || 188356 81746504, 20-8326667 7-571 1743 •002304147 | 435 | 189225 82312875 | 20-8566536 7°5769849 •00229S851 436 190096 || 82S81856| 20-8806130 || 7-5827865 •002293578 || 437 | 190969 83453453| 20-9045450 7-5885793 •002288330 | 438 191844 | 84027672| 20.9284.495 7-5943633 •002283105 439 192721 | 84604519 20.9523268 || 7-6001385 -002.277904 | 440 I93600 | 85184000 || 20-9761770 7-6059049 •0022.72727 | 44.1 | 194481 | 85766121| 21-0000000 7-6116626 •002267574. 442 195364 || 86350888 21-0237960 7-61741.16 •002262443 | 443 |1962.49 86938307| 21-0475652 7.6231519 •002257336 || 444 || 197136 || 875283S4| 21:07.13075 7-628S837 •0022.52252 || 445 | 198025 88121125 | 21-0.950231 7-6346067 •002247] 91 446 | 1989.16 || 88716536| 21.1.187121 7-6403213 •00.224.2152 | 447 | 199809 | 893I4623| 21-1423745 7-6460272 •002237136 448 |200704 | 899.15392 || 21-1660105 7-65.17247 •002232143 449 |201601 || 90518849| 21-1896201 7-6574.138 •0022271.71 450 |202500 91.125000 || 21-2132034 7-6630943 •002222222 451 | 203401 || 917.33851 21-2367606 7-6687665 •002217295 452 204304 || 92345408 || 21-2602916 7-6744303 •002212389 | 453 205209 || 92.9596.77| 21.2837.967 7-6800857 •002.207506 || 454 206116 || 9357.6664| 21-30727.58 7-6857.328 •002.202643 | 455 207025 | 94.196375|| 21-3307290 7-6913717 •002197802 456 |207936 94818816 21-354.1565 7-6970023 •002192982 457 | 208849 || 95443993 || 21.3775.583 7-7026246 •002188184| 458 209764 9607 1912 21:4009346 7-7082388 •002183406 459 || 210681 96702579 21-424.2853 7-71884.48 -002178649 | 460 |211600 | 97336000 21-4476106 7.7194426 •002173.913 || 461 212521 | 97.972181| 21-4709106 7-7250325 -002169197 | 462 || 213444 || 98.611128 || 21.494.1853 7-73061.41 •002164502 463 |214369 || 99.252847 21-5174348 7-7361877 -002159827 464 |215296 || 99.897344| 21-5406592 7-74.17.532 •002155.172 465 216225 10054.4625 || 21-5638587 7.7473.109 •002150.538 466 || 217156 || 101.194696 || 21.5870331 7-752S606 •002145923 467 218089 || 101847563| 21-6101828 || 7-7584023 •002141328 468 || 219024 || 1025.03232 21-6333077 7-7639361 •002136752 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTs. 91 Number squares. Cubes. VRoots. &/ Roots. Reciprocals. 469 219961 103161709 || 21-6564.078 7-7694620 •0021321.96 470 220900 | 103823000 || 21-6794834 7-7749.801 •002127660 471 221841 |104487111 || 21-7025344 7-7804904 •002123142 472 222784 || 105154048 || 21-7255610 7-7859928 •002118644 473 223729 || 1058288.17 | 21-7485.632 || 7-7914875 •002114,165 474 224676 | 106496424 || 21-7715411 || 7-79697.45 -0021097.05 476 2256.25 | 107171875 21-7944947 7-8024538 •002105.263 476 226576: 107850I '76 21.8174242 7-8079254 •002100840 477 227529|108531333| 21.8403297 7-8133892 •002096436 478 228484 || 1092.15352 21-8632111 7-8188456 •002092050 479 229.441 10990.2239| 21-8860686 7-824.294.2 •002087683 480 230400||110592000 || 21-9089023 7-8297.353 •0020S3333 481 231361||111284641 || 21.9317122 7-835.1688. •002079002 482 || 232324 |III980.168 21°9544984 7-8405949 •002074689 483 233289 112678587 21-9772610 7-84601.34 •002070393 484 234256 || 113379904 22:0000000 7.8514244 •002066116 485 23.5225 114084125 22:02:27.155 7.85682S1 •002061856 486 2361.96 || 11479.1256 22:0454077 7-8622242 •0020576I3 487 237169 | 1.1550.1303 || 22-0680765 7-8676130 •002053388 488 238144 11621.4272 22°0907220 7-8729944 •002049180 489 23912I | 116930.169 || 22-1133444 7-878.3684 •002044990 490 240100||117649000 22-1359.436 7-8837352 •002040816 || 491 241081 || 1837.0771 22-1585.198. 7-8890.946 •002036660 492 242064 || 1190.95488 22:1810730 7-894.4468 •002032520 | 493 2430.49 119823157 22-2036033 7-8997.917 •002028398 494 244036|| 120553784 22-2261108 7-9051294 •002024.291 || 495 245025 | 121287.375 22-2485955 7-9104599 •002020202 496 246016 || 122023936 22-27.105.75 7-9157832 •002016129 | 497 247009 | 122763473 22-2934968 7-9210994 •002012072 498 248004 || 123505992 22-3159136 || 7-9264085 •002008032 499 249001 | 12425.1499 || 22.3383079 || 7-9317104 -002004008 500 250000 | 125000000 22.3606798 7-937.0053. •002000000 501 251001 || 125751501 22-3830.293 7-94.22931 •001996008 502 252004 || J26506008 22°4053565 7-9475739 •00199.2032 503 253009 127263527 22.4276615 || 7-9528477 •0019.88072 504 254016 128024064. 22.4499443 7-958II.44 •001984.127 j. 505 255025 | 128787625 22:4722051 7-9633743 •001980198 || 506 256036 129554216 22.494.4438 7.96S6271 •0019762S5 | 507 257049 130323843 22-5166605 7-9.738731 •0019723S7 | 508 258064 |1310965.12 22-538.8553 7-97.91122 •0019.68504 509 259081 131872229 22°5610283 7-9843444 •001964.637 510 260100 132651000 || 22:5831796 7-9895697 •001960784 511 26112I 1334.32831|| 22-6053091 7-9947883 •001956947 512 262144 134217728 22-6274.170 8-0000000 •001953125 | 513 263.169 || 13500.5697 || 22°6495033 8-0052049 •001949318 514 264,196 || 135796744 22-6715681 S-0104032 •0019455.25 515 265225 |136590875 22.6936,114 8-0 155946 •00194-1748 516 266256 13738.8096 || 22-7156334 8-0207794 •001937984 517 267.289 1381.884] 3 || 22-737.6341 8-02595'74 •001934.236 518 26S324 || 138991832 22-7596,134 S.0311287 •001930.502 519 269361 139798359 22.7815715 || 8.0362935 •001926782 520 270400 140608000 22.8035085 S-0414515 •001923077 TABLE of SQUARRS, CUBES, SQUARE AND CUBE ROOTS. Number, 521 522 524. 526 . 527 532 539 540 541 544 Squares. 271441 2724S4 273.529 27457.6 27.5625 276676 | 2777.29 27S784 279S41 280900 2S1961 2S3024 2S40S9 2S51.56 | 2S6225 2S7296 2SS369 2894.44 290521 291600 292681 2937.64 2948-49 295936 297025 29S116 299209 300304 301401 302500 303601 304704 30.5S09. 306916 308025 309||136 310249 311364 3.12481 313600 314721 315844 316969. 318096 319225 320356 321489 322624 323761 324900 326041 3.27184: Cubes. 141420.761 142236648 143055667 143S77S24 144703125 145531576 146363183 I4; 197952 148035889 14.SS77001 1497.21291 15056S768 151419.437 i52273304 153130375 1539.90656 15.4854,153 155720S72 156590819 15746.4000 1583.40421 1592200SS 160103007 1609S91S4 16187S625 1627.71336 163667323 I64566592 165469149 16637 5000 167284151 168196608 169112377 170031464 170953875 17IS79616 17280S693 173741112 174676S79 175616000 176558.481 177504328 17 S453547 1794.06144 18036.2125 181321496 182284263 183250.432 IS4220009 IS5193000 186169.411 187149248 V Roots. 22°8254244 22°8473193 22-869.1933 22-S910463 ... 22.912S785 22-9346899 22-956.4806 22-9782506 23-0000000 23-0217.289 23-043437.2 23-0651252 23-0867928 23-10S4400 23-1300670 23-1516.738 23-1732605 23-1948.270 23-2163735 23-2379001 23-25.94067 23-2808935 23-3023604 23-3238076 23-3452351 23-3666.429 23-3SS0311 23-4093998 23-4307.490 23°45207 SS 23°4733S92 23:49.46802 23:5159520 23-5372046 23-55843S0 23:57.96522 23-6008474 23-6220236 23-6431808 23-6643.191 23-6S54386 23.7065392 23-7276210 23-74S6S42 23-7697.286 23-79.07545 23-8117618 23-8327506 23-853.7209 23-8746728 23-S956063 23-91652.15 A- * Roots. 466030 517479 •056SS62 & : : *0671.432 •0722620 •0773743 •0824S00 •0875794 8-0926723 8-0977589 8-102S390 8-107912S 8-1129803 8-1180414 8-1230962 8-12S1447 S-1331870 i 8-1382.230 8-14.32529 S-1482765 S-1532939 .8°1583051 S-1633102 8-16S3092 8-1733020 8-1782SS8 8-1832695 8-1882441 S-1932.127 8-1981753 8-2031319 8-208082.5 S-2130271 8'21796.57 8-22.28985 8-2278254 8-2327463 8-2376614 8-2425706 8-24.74740 8-2523715 S-257.2635 8-26.21492 8-2670294 8-27-19039 8-27677.26 S-2816255 8°2864928 8°2913444 8-296.1903 8-3010304 620IS0 Reciprocals. •0019193S6 •001915709 •0019 12046 •001908397 •001904762 *001901141 •001897533 •001S93939 •001890.359 •00ISS6792 •00ISS3239 •001S79699 •00IS'ſ 617.3 -00187.2659 *001869.159 •001S65672 •001862197 *001S58736 *0018552SS *001851852 *001S4S429 *00184501S *001841621 •00IS3.8235 *001834862 •001831502 *001S28154 *001S248.18 •001821494 •001S181S2 •001814882 •00181.1594 •0018083.18 *001805054 •001801802 •00179S561 •001795332 •0017921.15 •001788909 •001785,714 •001782531 •001779.359 •001776199 •001773050 -001769912 •001766784 •001.76366S •001760563 •001757469 •001'ſ 543S6 •0017513].3 •001'ſ 48.252 TABLE OF SQUARES, CUDES, SQUARE AND CUBE Roors: 93 Number. Squares. Cubes. Aſ loots. V Roots. Reciprocals. 573 328329 188132517 23-9374.184 8-305S651 •001745201 574 3294.76 | 1891.19224 23°95S2971 8-3106941 •0017.42160 575 330625 | 1901.09375 23-97.91576 8-31551.75 •001739.130 576 331776 191102976 24-0000000 8-3203353 •001:736111 577 332927 192100033 24-0208.243 8°3251475 •001733102 578 334084 193100552 || 24°0416306 8-32995.42 •001730104 579 335241 194104539 24-0624188 8-3347553 •001.727116 5S0 336400 195112000 24-0S3.1891 8-339550.9 •001.724.138 581 337561 | 1961.22941 24-1039416 8-3443410 •001721170 5S2 33.8724 197137368 24-1246762 8-349.1256 •001718213 583 339SS9 198155287| 24-1453929 8-3539047 •001715266 584 34.1056 | 199176704 || 24-1660919 8.3586784 •001712329 5S5 342225 200201625 24-1867732 8-3634.466 •00I '709.402 586 34.3396 201230056 24-2074369 8-3682095 •0017064S5 587 344569|202262003 || 24:2280829 8:37.29668 •001703578 588 345744 || 203297472 | 24'2487I13 8-37.77.188 •0017.00680 589 346921 | 204336469 24-2693222 8-3S24653 •001697.793 590 34S100 | 205379000 || 24-2S99156 8-3872065 •00169491.5 591 349.281 2064.2507I | 24-3104996 8-3919428 •001692047 592 || 350464| 20747.4688 || 24:3310501 8-3966729 •0016S91S9 593 || 351649 20852.7857 || 24°3515913 8-40.13981 •0016S6341 594 352S36 209584584 24-3721 152 8’4061180 •0016835.02 595 35.4025 210644875 24:39.2621S 8-4108326 •00168067.2 596 355216| 21170S736 24-4131112 8-4155419 •001677852 597 356409| 212776173| 24'4335834 8-4202460 •001675042 598 357.604 || 213847.192 || 24'4540385 8-4249448 •001672241 599 35S801 || 214921799 || 24'4744765 8-4296.383 •0016694.49 600 360000|216000000 || 24°4948.974 8-434.3267 •001666667 601 361.201 2170S1801 24-5] 53013 8-439009S •001663S94 602 362404 218167208 || 24°5356883 8-4436877 •00I 661130 603 36.3609 || 219256227 24-5560.583 8-44S3605 •00165S375 604 364S16 220348864 24-5764115 8-4530281 •0016556.29 605 366025 2214451.25 24°5967.478 8-4576906 •001652S93 606 367236 222545016 || 24-6170673 8.46234.79 •001650165 607 368449 2236.48543 || 24.637 3700 8-4670001 •001647446 608 369664|224755712 24-6576560 S-4716471 •001644.737 609 370SSI 2258665.29 || 24'67792.54 S-47.62892 •00164.2036 610 372100 226981000 24-69S1781 S-4809.261 •001639344 611 373321 22.809913] | 24-7IS4] 42 S-4855579 •00I 636661 6] 2 374544. 2292.20928 24-73S633S 8-4901848 •001633987 613 375769|230346397 24-7588368 8-494.8065 •001631321 614 376996 || 231475544 24*7790234 S-499.4233 •00162S664 615 878225|232608375 24,799.1935 8-5040350 •001626016 616 379456 233744896 24.8193473 8-50864.17 •001623377 617 380689 || 234885,113 24°S394847 S-5132435 •001620746 618 3S1924 236029032 24-8596058 8-5178403 •001618.123 619 383.161 237176659 24-8797 106 8-5224331 •001.615509 620 384400|238328000 24-8997.992 8-52701S9 •001612903 621 3S5641 239.483.061 24-9198.716 S-5316009 •001610306 622 386884|240641848 24-9399278 8:536.1780 •0016077.17 623 388129 || 241804367 24-95996.79 S-5407501 •0016ſ, 51.36 624 389376] 242970624 24-97.99920 8-5453173 0.01602564 94 TABLE OF SquaRES, CUBES, SQUARE AND CUPE Roots. Number. Squares. | Cubes. | V Roots. & Roots. Reciprocals. 625 || 39.0625 |244140625 25.0000000 | 8-54987.97 •001600000 626 || 39.1876 || 245134376 25-0199920 8°5544372 •Ü01597444 627 | 393129 246491883 || 25-0399.681 8°5589899 •00159.4896 628 394384 247673152 25-0.599282 8°5635377 •001592357 629 || 3956.41 248858.189 25-0798724 || 8°5680807 •001589825 630 || 396900 250047000 || 25-0998008 || 8°57'26189 || "001587302 631 || 398161|251239591 || 25-1197134 8°57'71523 •001584.786 632 399424|252435968 || 25-1396102 || 8°58'16809 •001582.278 633 400689| 253636137 25-1594913 8°5862247 •001579779 | 634 401956 254840104 || 25°1793566 8-5907238 •001577.287 635 | 403225 256047S75|| 25-1992063 || 8°5952380 -001574803 636 | 404496 || 257259456 || 25-2190404 || 8-599.7476 •001572327 637 || 405769 258474853 25-238.8589 8-6042525 -001569859 638 |407044|259694072| 25.2586619 8.6087526 •001567398 || 639 || 408321||260917119 25-27S4493 8-61324.80 *001564945 640 | 409600 262144000 || 25-2982213 || 8:6177388 •001562500 641 || 410881 263374,721 || 25°347977S 8.6222248 •001560.062 642 412164| 264609288 || 25-3377189 8-626.7063 •001557.632 643 413449 2658477.07 || 25-3574447 | 8-6311830 -001555210, 644 || 41.4736|267089984 || 25°3771551 8°6356551 •001552.795 645 || 416025 268336125 || 25°3968502 || 8.640.1226 *00155038S 646 417316, 269585.136|| 25-4165302 || 8-64.45855 *00 1547988 647 418609| 270840023 25-4361947 | 8-6490437 *001545595 648 || 419904 27209.7792 || 25-4558441 8-6534974 *001543.210 649 || 421.201 || 273359449 || 25-4754784 || 8-657.9465 *001540832 650 || 422500|274625000 || 25-4950976 | 8-6623.911 *001538.462 651 || 423801 || 275894.451 25-5147013 | S-666.8310 *0015.36098 652 425.104, 27716.7808 || 25-5342.907 || 8-6712665 •001533742 653 || 426409 278445.077 25-553S647 | 8-6756974 *0015.3i 394 654 || 4277.16 2797.26264 25-5734237 | 8-6S01237 •001529052 655 || 429025 28101.1375 25'5929678 || 8-684.5456. "001526718 656 430336|282300416 || 25-6124969 S-68896.30 *001524390 657 431649 283593393 || 25-6320H12 8-69.337.59 •001522070 658 432964. 284890312|| 25-6515107 || 8-6977.843 •001519757 659 || 434281 286191179 || 25-6709953 S-7021882 "00151745.1 660 || 435600|2874.96000 || 25-6904652 8.7065877 *00151515.2 661 || 436921|288804781 25.7099.203 || 8-7109827 *001512859 662 || 438244|2901.17528 25-7293607 || 8-7153734 •001510574 663 439569| 291434.247 || 25.7487864 8-71975.96 •00150.8296 664 440896|292754944|| 25-7681975 8-724.1414 || 001506024 665 44.2225 29.4079625 || 25.7S75939 8-7285187 •0015037.59 666 || 443556|295408296 || 25.8069758 8-7.328918 •001501502 667 || 4448S9| 296740963 25-8263431 || 8-7372604 •001499.250 668 || 446224| 298077632 || 25-8456960 8-74.16246 •001497.006 669 || 447561 299418309 || 25-8650343 || 8-745.9846 •001.494.768 670 448900 |300763000 || 25-88.43582 8-7503401 •001492537 671 || 450241 |302111711 || 25-9036677 | 8-7546913 •001 4903.13 672 || 451584 |303464448 || 25.9229628 8-759.0383 •001488.095 673 || 452929 |3048.21217 25-94.22435 | 8.7633.809 •001485.884 674 154276|306182024, 25.9615100 8-7677192 •001483680 675 455625 || 307546875|| 25-98.07621 8,7720532 •001481481 676 || 456976) 308.915776] 26.0000000 || 8-1763830 •001479290 TABLE or SquarES, CUBEs, SQUARE AND CUBE Roots. 95 Number. Squares. | Cubes. V Roots. */ Roots. Reciprocals. 677 |458.329 |310288733| 26-0192237 || 8-7807084 || -001477.105 678 459684 || 311665752 26-0384331 8°7850296 •00 1474926 679 || 461041 3130468.39| 26-0576284 8°7893466 •001.4727.64 680 462400 314432000 || 26-0768.096 8-79.36593 || -001470588 681 |463761 |315821241 26-09597.67 8-79796.79 •001468.429 682 465,124 317214568 26-1151297 8-8022721 •001466.276 683 |466489 31861.1987 26-1342687 8-8065722 •001464.129 684 |467856 |320013504| 26-1533937 8-8108681 •001461988 685 4692.25 321419125 26-1725,047 8°815.1598 •001.459854 686 470596 || 322828.856 26-1916017 8’S194474 •001.457726 687 |471969 |324242703| 26-2106848 8-823.7307 •001.455604 688 || 473344 325660672 26°2297541 8-828.0099 •001453488 689 |474721 |3270S2769 26-2488.095 8-S322850 -001451379 690 476100 || 328509000 26°26785II 8-8365559 •001449275 691 |4774S1 |320939371 26-28.68789 8-8408227 •001.4471.78 692 478864 33137.388S 26-3058929 8.8450854 •0014.45087 693 480249 || 33281.2557 26-3248932 8-8493440 •0014.43001 694 || 4S1636 334255384 26°34387.97 8-S535985 •0014.40922 695 || 48.3025 | 335702375 26-3628.527 8-857S489 •00I 43S849 696 |4S44.16||337.153536 26:38.18119 8-8620952 •001:1367S2 697 485809 || 33860SS73 || 26-4.007576 8-8663375 -001434.720 698 4S7204 |34006S392 || 26-4196896 8-8705757 •001432665 699 || 4SS601 || 341532099 || 26°438.6081 8-8748099 •001430615 700 490000 || 343000000 || 26°45'75131 8-S790.400 •00142S571 701 |491401 |344472101| 26-4764046 8-SS32661 •001426534 702 || 492804 |345948408 || 26°4952S26 S-SS74SS2 •001424501 703 || 494209 |3474.28927 | 26°51414.72 8-8917063 -0014224.75 704 || 4956.16||348913664 26-5329983 8-S959204 •001420.455 705 - |497025 350402.625 | 26°5518361 S-9001:304 •001418440 706 || 498436||351895816 || 26°5706605 8-9043366 •001416431 707 || 49.9849 |353393243 26°589.4716 8-90S5387 •001414427 708 || 50.1264 |354894.912 26-6082694 S-91.27369 •001412429 709 || 5026S1 |356400S29 26-6270539 S-91693.11 •0014] 0437 710 || 504100 |35791.1000 || 26-6458252 8-92.11214 •001408451 711 || 505521 |359.425431 26-6645833 8-925.3078 •001406470 712 506944 |360944,128 26-6S33281 8-929.4902 •001404494 * 18 || 508369 |36246.7097 || 26-7020598 8-933.6687 •001402525 714 || 509796||363994344 26-72077.84 8-937S433 •001400560 715 51.1225 365525875 26-739,4839 8-9420140 •001398601 716 || 512656|367061696 || 26-7581763 S-9461809 •001396648 717 || 514089 |368601813 || 26-7768557 8-95.03438 •001394700 718 || 515524 |370146232 || 26-7955220 8-95.45029 •001392758 719 || 516961 |371694959 || 26°SI 4.1754 8-9586581 •001390821 720 || 518400|373248000| 26-8328157 8-96.28095 •0013SSSS3 721 51984.1 |374805361| 26-8514.432 8'966.9570 •001386963 722 || 521284 3.76367048 26-87.005% 7 8-9'ſ 11007 •001.385042 723 522729 37.7933067 26-8886.593 8.975.2406 •0013S3-126 724 524.176 37950.3424 26-90.724.81 8-9793.766 •0013S1215 725 525625 |38107S125 26.925S240 8-9S35089 •001379,310 726 52.7076 |3S2657176 26.94.43872 8.9876373 •001377.410 727 528529 |3S4240583 26-96.29375 8-9917620 •0013755] 6 728 529984 || 385828352 || 26°9814751 8-9958.899 •001373626 TABLE of SQUARES, CUBEs, SQUARE AND CUDE ROOTS. q6 Number. Squares. Cubes. v Roots. & Roots. | Reciprocals. . 729 531441 3874.20489 || 27-0000000 9-0000000 •001371742 730 532900 || 389017000 || 27-0185122 9-0041134 •001369863 731 534.361 || 3906.17891 27-03701.17 9:0082229 •001367989 732 535824 || 3922.23168 || 27-0554985 9:0123288 •001.366.120 733 537289 |3938.32837 27-0739727 9° 0164309 •001364.256 734 538756 || 395446904 27-0924344 9-0205293 || 001362398 735 540225 | 39.7065.375 | 27-110SS34 9°0246239 •001360544 736 541696 || 398688256 || 27-1293199 9°0287.149 •001358696. 737 543169 |400315553 27-1477149 9-0328021 •001356852 738 544644 |401947272 || 27-1661554 || 9-0368S57 •001355014. 739 546121 |4035.83419 || 27-1845544 9-0409655 •001353180 740 || 547600 |405224000 27-2029140 9-0450419 •001351351 74.1 5490S1 |406869021 27-22.1315.2 9-0491142 •001349528 742 550564 | 408518488 27-2396.769 9-0531831 •001347709 743 55.2049 || 410172407 || 27-2580.263 9-0572482 •001345895, 744 553536|41.1830784| 27-2763634 9-0613098 •00] 34.4086 745 555025 413493625 27°29.46881 9-0653677 •00134.2282 746 5565.16 || 415160936 27-3130006 9-0694220 | •001340483 747 55S009 |4|16832723 27-3313007 9-0734726 •00 13386SS 74S 5595.04 || 41850.8992 || 27-349588.7 9-0775197 •001336898 749 561001 || 420 189749 || 27-3678644 9-0815631 •001.335] 13 750 562500 |4218.75000 || 27-3S61279 9-0856.030 •001.333333 751. 564001 |423564751 27.4043792 9-0896352 •001.331558 752 565504 || 425259008 27-4226184. 9-0936719 •001.3297 S7 753 567009 || 426957777 27.440.8455 9-0977010 •001.32802.1 754. 5685.16 || 42866 1064 27-4590604. 9-1017265 •001.326260 755 57.0025 |430368.875 27-477.2633 9°10574S5 •001.324503 756 571536 432081216 || 27-495454.2 9-1097669 •001322751 757 573049 |4337,98093 27°5136330 9-1137818 •001321004 7.58 574564 435519512 27°5317998 9-11779.31 •001319.261 7.59 57.6081437245479| 27-5499546 9-1218010 •001317523, 760 57.7600 438.976000 27-5680975 9-1258053 •001315789 . 761 579121 |440711081 27°5862284 9-1298061 •001314060 762 580644 442450728 27.6043475 9-1338.034 •001312336, 763 58.2169 || 44.419494.7 27-6224.546 9-1377971 •001310616 764 583696 || 44594.3744 27-6405499 9°1417874 •001.308901 765 585225 || 447697125 27-6586334 9-1457.742 •001307190 766 586756 4494.55096 || 27-6767,050 9-1497.576 •001305483 767 588.289 |451217663 27.6947648 9-1537.375 •001.303781 768 589824|452984832 27-7128129 9-1577139 •001302083 769 591361 454756609 || 27-7308492 || 9-1616869 •00.1300390 770 592900 4565.33000 27-7488739 9-1656565 •001298.701 771 594441 458314011 || 27-7668868 9-1696225 •001297.017. 772 595.984|460099648 27-7848880 9-1735852 •001295337 773 59.7529 |4618S9917 | 27-8028775 9-1775445 •001293661 774 5990.76 463684.824 27-8208555 9-1815003 •00129.1990 775 600625 465484375 27-8388218 9-1854527 •U0129.0323 776. 602176 467288576 27.8567766 9-1894.018 •001288660 777 603729 |469097433 27-8747197 9-1933474 •00128.7001 778 605284 |470910952 27-8926514 || 9-1972.897 •001285347 779. 606841 4.72729139| 27.9105715 9-2012286 •001283697: 780 608400 47455.2000 l 27,9284801 9-2051641 •001282051. TaBLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTs. 97 Number. Squares. | Cubes. VRoots. &/ Roots. Regiprocals. 781 || 609961 || 476379541 || 27.946.3772 9-2090962 •001280410 782 611524 || 4782.11768 27-9642629 9°2130250 •00127.8772 783 613089 || 48004.8687 / 27-98.21372 9°2169505 •001277139 784 614656 || 48.1890304 28-0000000 9-2208726 •00] 27.5510 785 616225 || 4837.36625 | 28-01785.15 9°2247914 •001.273.885 786 617796 || 485587656| 28.0356915 || 9.228.7068 •0012.72265 787 619369 || 487443403 28-0535203 9°2326189 •00I 270648 788 620944 || 489303872 28-071337.7 9-2365277 •001.269.036 789 622521 || 491.1690.69 28-0801438 9°2404.333 •001.267.427 790 624 100 493039000 || 28-1069386 9°2443355 •001265823 791 625681 || 494913671 || 28-1247222 9°2482344 •001.264.223 792 627264 || 496793038 28-1424.946 9-2521.300 •00I 262626 || 793 628849 || 498677.257 || 28°1602557 9°2560224 •001.261034 794 || 630436 500566184 || 28°1780056 9°2599.114 •001:2594.46 795 63.2025 502459875 28-1957.444 9-2637973 •001.257862 796 633616 504358336|| 28-2134720 9-2676798 •001.256281 || 797 635209 || 50626.1573 || 28°231.1884 9°27.15592 •00°254705 798 | 636804 || 508169592 28-2488938 9-275.4352. •001.253.133 799 63S401 || 510082399 || 28°2665881 9-2793031 •001:25.1564 800 640000 || 512000000 | 28-2842712 9-2831777 •001:250000 801 641601 || 51392.2401 || 28-3019.434 9-287.0444 •001248439 | 802 643204 || 515849608 28°3196045 9-2909072 •001:246SS3 803 644.809 || 517781627 | 28.3372546 9-294.7671 •0012.45330 804 || 646416 519718464 || 28-3548938 9-2986239 •001243.781 805 648025 521660.125 28-3725219 || 9:3024775 •001.242236 806 649636 5236.06616 || 28-3901391 9-3063278 •0012.40695 807 651249 || 525557943 28°4077.454 9-3101750 •001.239157 808 652864. 527514112| 28:4253408 9-31401.90 •001.237624 809 654481 | 5294.75129 28-4429253 9-3178599 || 001236094 | 810 656100 531441000 || 28°4604989 9-3216975 || 001234568 811 657721 533411731| 28.4780617 9-3255320 •001.233046 812 659344 || 535387328 28°4956137 9°3293634. •00I 231527 S13 660969| 537367797| 28.5131549 9-3331916 •001.230012 814 662596 539353144 28°5306852, 9-337.0167 •001.22S501 815 664225 54.1343375 28-548.2048 9:34.08386 •001226994 816 665856; 543338496 || 28.5657137 9-34465.75 •001.225499 817 | 667489: 545338513| 28:5832119 9-3484731 •001.223990 818 669124 547343432 28-6006993 9-3522857 •001222494 | S19 6'70761 || 549353.259 28-6181760 9°3560952 •001.221001 820 672400 || 55.1368000 || 28-63.56.421 9-3599.016 •0012.19512 821 674041 553.3876.6L 28-6530976 9-3637049 •0012.18027 822 675684. 5554.12248 || 28-6705424 9°36'75051 •001216545 823 677329 55744.1767 28.6879716 9-37.13022 •001:215067 824 678976 559476224, 28-7054002 9-3750963 •001.213592 825 68.0625 || 561515625 28-7228132 9-3788873 || 0012.12121 826 682276|| 563559976| 28-7402157 || 9:3826752 •001:210654 S27 68.3929 || 5656092.83 || 28-7576077 9°3864600 •001209190 | 828 685584 567663552 28-7749.891 9-3902419 •001207729 829 687.241 || 56.9722789 28-7923601 9-39.40206. •001206273 | 830 6SS900 || 57.1787.000 || 28-809.7206 9-3977964 •001204819 831 690561 || 573856191} 28-82.70706 9-4015691 •001203369 832 692224| 575930368] 28-84.44102 9-40533.87 •001201923 7 98. TABt.; of Squares, CUBES, SQUARE AND CUBE Roots. Number. Squares. Cubes. V Roots. J/ Roots. Reciprocals. 833 6938S9 || 57800.9537 28-861.7394 9°4091054 •001200480 834 695556 580093704 || 28-8790582 9°4128690 •001}99041 835 697.225 || 582182875 28-8963666 9°4166297 •001.197605 S36. 69.8896 || 584277.056 28-9136646 9°4203873 •001.1961.72 837 7.00569 586376253| 28-9309523 9°4241420 •001.1947.43 838 702244 588480472 28-9482297 9°427.8936 •001.193317 839 703921|590589719| 28.9654967 9°4316423 •0011.91895 840 || 705600|592.70.4000 || 28-9827.535 9°4353.800 •001.190476 841 707281 594823321 29-0000000 9-4391307 •001189061 842 708964. 5969476SS 29-0172363 9°4428704 •001187648 843 710649 |5990.77107 || 29-0344623 9°4466072 •001186240 844 712336|601211584] 29-0516781 9°4503410 •001184834 845 714025|603351125| 29-0688837 9°4540719 •0011834.32 846 715716 605495736 29-08607.91 9°45′77999 •001182033 847 717409 6076454.23 29-1032644 9°4615249 •001180638 848 719104|609800192 29-1204396 || 9:4652470 •0011792.45 849 720801 ||611960049| 29-1376046 9°4689661 •001177856 850 722500|614125000 29-1547595 9°4726824 •00117647.1 85.1 724.201 ||616295051 29-1719043 || 9:47.63957 •001175088 852 725904|61847.0208 || 29-1890390 9°4801.061 •001173709 853 727609 || 620650477 29-2061637 9°4838136 •001172333 854 729316|622835864| 29-2232784 9°4875182 •001170960 S55 731025 625026375 29-2403830 9°4912200 •001169591 856 732736 627222016 || 29-2574777 9°4949188 •001168224 857 734449 |6294227.93| 29-2745623 9°4986147 •001166861 858 736,164|6316287|12 || 29-2916370 9°5023078 •001165501 859 7378.81| 6338.39779 29-3087018 9°5059980 •001164144 860 739600|636056000 29-3257566 9°5096854 •0011627.91 861 74.1321 | 63S2773S1 || 29-34280.15 9'5133699 •001161440 S62 743044 640503928| 29-359.8365 9°5170515 •001160093 863 744769| 642735647| 29-3768616 9°5207303 •001158749 864. 746496 || 64497.2544 29-3938769 9°5244063 •001157.407 865 748225|647214625 | 29-4108823 9°5280794 •001156069 866 749956 64946.1896 || 29.4278779 9°5317497 •001154734 867 751689| 65|1714363 29°4448637 9°5354,172 •001153.403 868 753424 65397.2032 || 29°4618397 9°5390SIS •001152074 869 755161|656234909| 29:4788059 9°5427437 •001150748 870 756900 || 658503000 29-495'7624 9°5464027 •001}494.25 871 758641|660776311| 29.5127091 9°55'00589 •001148106 872 760384|663054848 29-5296461 9°5537.123 •001146789 873 762129 6653.38617 | 29-5465734 9°5573630 •001145475 874 || 763876|667627624, 29.5634910. 9°5610.108 •001144.165 875 7656.25 | 669921875 29'5803989 9°5646559 •001142857 876 767376|672221376 29-5972972 9:5682782 •00114 1553 877 769129 674526133 29-6141858 9°57'19377 •001140251 878 770884|676836152 29-6310648 9°5755745 •001138952 879 772641 67915.1439 29-64.793.42 9°579.2085 •001137656 880 774400|681472000] 29-6647939 9-582S397 •001136364 881. 776161|683797841| 29-6816442 9°5864682 •001135074 882 777924 | 686128968 || 29-6984848 9°5900937 •001133787 883 779689| 688465387 29-7153159 9°5937169 •001132503 884 781456 || 690807104 || 29-732.1375 9°597.3373 •001131222 TABLE of Squares, CUBEs, SQUARE AND CUBE ROOTs. -001068376 | & -h. 99 Number. Squares. | Cubes. V Roots. &/Roots. Reciprocals. 885 783225 | 693154.125 29°74894.96 9-60.09548 •001.129944 886 78.4996 || 695506456 29-7.657521 9-6045696 •001 128668 887 || 786.769 || 69786.4103 || 29-782.5452 9-6081817 •001.127396 | 888 || 788544 70022.7072 29-7993289 9-61.17911 •001.126126 889 790321 |702595369| 29-8161030 9.6153977 •001.124859 896 792100 || 704969000 29-8328678 9-6.190017 •001123596 89. 793881 | 707347971 29-849.6231 9-6226030 •001122334 892 || 795664 || 707932288 29-8663690 9-626.2016 •001121076 893 797449 |7|12121957 29-883.1056 9-6297975 •0011198.21 894 || 799.236 || 714516984| 29-8998328 9-6333907 •001118568 895 |801025 716917375|| 29-9165506 9:6369812 •001117818 896 || 802816 |7|19323136|| 29-9332591 9-6405690 •001116071 897 || 804609 || 72.1734273| 29-94.99583 9-644.1542 •001114827 898 || 806404 || 724150792 29.9666481 9-64.77367 "001113586 899 || 808201 | 7265.72699 29-9.833287 9-65.13166 | *00111234.7 900 810000 || 729000000 || 30-0000000 9-654893S •001111111 901 || 811801 || 731432701 || 30-01666.21 9-6584684 •0011098.78 902 | 813604 || 7338.70808 || 30-0333148 9-6620403 •001108647 903 || 815409 || 736314327 30-0499584 9-6656096 •001107420 904 817216 || 738763264 || 30-0665928 9-669.1762 •001106195 905 S19025 741217625 30-0832.179 9-6727403 •001104972 906. 820836 7436.77416 || 30-0998339 9-6.7630.17 •001103.753 907 |822649 746142643| 30-1164407 9-679S604 •001102536 908 824464 7486.13312 20-13303S3 9-6834,166 •001101322 909 8262S1 7510S9429 || 30-1496.269 9-68697.01 •001100110 910 82S100 75357 1000 || 30-1662063 9-6905.211 •001098.901 911 || 829921 756058031 || 30-1827765 9-6.940694 •001097.695 912 831744 758550828 || 30-1993377 9-6976151 •001096491 913 | 833569 || 761048497 || 30-2158899 9-7011583 •00I 09:5290 914 || 835396 || 763551944|| 30-2324329 9-7046989 •001094,092 915 837.225 766060875 || 30-2489.669 9-70S.2369 •001092896 916 |839056 |768575296 || 30-2654919 9-71 17723 •001091703 917 | 840889 || 7710952.13 || 30-2820079 9-7153051 •001090513 918 842724 773620632 30°2985148 9-71883.54 •0010893.25 919 |844561 776151559 || 30-315012S 9-7223631 •001088] 39 920 846.400 778688000 || 30-3315018 9-72.58883 •001086957 921 848241 || 781229.961 || 30-347981S 9.7294.109. •0010857.76 922 850084 || 7S3777.448 || 30-3644529 9-7329.309 •001084599 923 |851929 786330467 || 30-380915.1 9-73644S4 •001083423 924 |853.776 78.8889024 || 30-397.36S3 9-73.99634 •0010S2251 925 855.625 79.1453.125 || 30-413S127 9-74347.58 •00108108.1 926 857476 |794022776|| 30-4302481 9-7469.857 •001079914 927 | 859329 || 79.6597983 30-44667.47 9.7504930 •001078.749 928 861184 799.178752 || 30-4630924 ‘9-75399.79 •00107.7586 929 863041 801765089 || 30-479.5013 9-7575.002 || 001076426 930 |864900 80.435.7000 30°4959014 9-7610001 •001075.269 931 866761 |806954491 30°5122926 9-7644974 •001074114 || 932 868624 |809557568|| 30-5286750 9-76.79922 •001072961 933 870489 || 812166237 || 30-54504.87 9-7714845 •00107 1811 934 |872.356 |814780504 || 30-5614136 9-7749743 •001070664 935 874225 817400375 || 30-5777697 9-7784616 •001069519 | 936 || 87.6096 || 820025856 || 30-5941171 9-7819466 tº s f- tºr º : 100 TABLE of Souattes, CUBEs, SQUARE AND CUBE Roors. Number. Squares. | Cubes. v Roots. J Roots. Reciprocals. 937 87.7969 |$22656953 : 30-6104557 9-7854288 . .001067236 938 6798.44 825293672 30-6267.857 || 9-7889087 -001066098 939 881721 827.936019 30.6431069 9:7923861 •001064963 940 | 883600 |8305.84000 || 30-6594.194 || 9°7958611 || -001063830 941 885481 |833237621| 30-6757233 9:7993336 -001062699 942 SS7364 835896888 || 30-6920185 9-8028036 || -00106157 i 943 |889.249 |838561807 || 30-7083051 9-8062711 || -001060445 944 | 891136|841232384 || 30-7245830 9'8097362 -001059322 945 | 893.025 | 843908625 30,7408523 9-8131989 -00105820I 946 | 89.4916 |846590536 || 30-7571130 9°8166591 •001057082 947 | 896809 |8492.78123| 30-7733651 9:8201169 -001055966. 948 |898.704|851971392 || 30-7896086 9.8235723 || -001054852 949 900601 |854670.349 || 30-8058436 9-8270252 •001053741 950 |902500 857375000 || 30-8220700 9.8304757 -00105.2632 951 |904401 |860085351 || 30-8382879 9-8339238 •001051525 952 906304 |86280.1408 30-85.44972 9.837.3695 -001050420 953 908209 |865523177|| 30-8706981 9-8408127 | -0010493.18 954 |910116 |868250664 || 30-8868904 || 9-84.42536 •00Iſ)48.218 955 |912025 |870983875|| 30-9030743 || 9-8476920 •00104.7 120 . 956 |913936 |873.722816 || 30-91924.77 9-851.1280 •001046025 957 |915849 |876.467493 || 30-9354,166 9'85456.17 •001044932 958 |917.764 |879217912 || 30-9515751 | 9-8579.929 •00104384.1 959 |919681 |881974079 30.9677.251 | 9-8614218 •00] 04:2753 960 92.1600 |884736000 || 30-98.38668 9:8648483 || -00104.1667 961 |923521|887503681 31-0000000 9.8682724 -001040583 | 962 |925444 | 890277128 || 31.0161248 || 9°8716941 •0010395.01 963 |927.369 |893056347 31.03224.13 9-875,1135 | -001038422 964 929.296 | 89584.1344 31-0483494 9'8785305 •001 037344 965 93.1225 |898632125 31.0644491 || 9-881945.1 •001 036269 966 | 933156|901428696 || 31.0805405 || 9.8853574 || 0010.35197 967 935089 |904231063 31-0966:236 || 9-8887673 -001034126 968 || 937024 |907039232 || 31-1126984 || 9-892.1749 -001033058 969 |938961 |909853209 || 31-1287648 9-895.5801 •001081992 970 9.40900 91.2673000 || 31.1448230 9-89898.30 •001030928 971 942841 |915.4986] 1 || 31-1608729 || 9.9023835 •001029866 || 972 94.4784 |918330048 || 31-1769,145 || 9-9057817 •001028807 973 |946729 |921167317| 31-1929.479 || 9.9091776 || -001027749 974 |948676 |924010424 31-20897.31 9-91.25712 •001026694 975 95.0625 |926859375 31.2249900 || 9-91596.24 •001025641 976 952576 |929714.176 31-2409987 9-91935.13 •001024590 977 |954529 |932574833 || 31-256.9992 || 9,92273.79 •001023541 978 |956484 |935441352 || 31-2729915 9-926,1222 || -001022495 979 |95844.1 |9383137.39|| 31-2889757 9-9295042 •001021450 980 |960400 941192000 || 31-30.49517 9-9328839 •001020408 981 | 962361 94.4076141 31-3209195 || 9.9362613 •001019-168 982 |964324946966168|| 31-3368792 9-9396363 •001 018330 983 |966.289 949862087 31.3528.308 || 9-9430092 •001017294 984 |9682561952763904| 31:3687743 || 9-94.63797 •001016260 985 |970.225 |955671625 31°3847.097 || 9-94.97479 •001015228 986 |972.196 |958585256 || 31.4006.369 9.9531138 •00101.4199 987 | 974169 96.1504803 || 31-4165561 9.9564775 •00101.3171 988 |976144.964430272. 31.4324673 9.95983.89 ‘U01012146 Tants of Squares, CUDES, SQUARE AND CUPE Roors. 101 | Number. Squares. Cubes. VRoots. & Roots. Reciprocals. 989 978121| 967361669 31.4483704 || 9.9631981 || 001011122 990 980.100 97.0299000 31.4642654 9.9665549 || 001010101 991 | 98.2081 || 97.3242271 31.4801525 || 9.9699055 || 001009082 ... 992 || 984064 97.6191488 31°49.60315 9-9732619 || 001008065 993 || 986.049 || 979146657 || 31.5119025 || 9-9766120 -001007049 994 | 988036 982.107784 || 31°5277.655 9.9799599 || 001006036 995 || 990025 | 985.074875 || 31°5436206 || 9-9.833055 || 001005025 996 || 99.2016 || 988047936 || 31.5594677 | 9-9866488 || "001004016 997 || 99.4009 || 99.1026973 || 31-5753068 9-9899900 •001003009 998 || 996004 || 99.4011992 || 31-5911380 9-9933289 || 001002004 999 || 99.8001| 997.002999 || 31-606.9613 9.9966656 -001001001 1000 || 1000000|1000000000 || 31-6227766 || 10-0000000 || 001000000 1001 || 1002001 || 1003003001 || 31-6385840 | 10-0033222 || 0009990010 1002 |1004004|10060.12008 || 31-6543.866 | 10:0066622 || 0009980040. 1003 |1006009|100.9027027 31-6701752 | 10.0099899 || 00099700.90 1004 ||1008016|101.2048064|| 31-6859590 | 10-0133155 || 000.9960159 1005 || 10100.25: 1015075,125 || 31.7017349 || 10:01.66389 || 00099.50249 1006 || 1012036|| 1018108216 || 31-7175030 || 10-0199601 || 0009940358 1007 || 1014049|1021147343| 31.7332633 || 10:0232791 || 0009930487 1008 || 1016064 1024.192512 31-7490.157 | 10-0265958 || 000.9920635 I009 || 1018081| 1027243729 31-7647.603 || 10-0299.104 || -0.0099.10803 1010 || 1020100 || 1030301 000 31°7804972 10-0332228 •0009900990 1011 || 102.2121 | 1033364331 || 31-7962.262 || 10-0365830 || 00098.91197 1012 || 1024144 || 1036433728 31-8119474 10-0398.410 | *0009881423 1013 || 1026169 || 103950919.7 31.8276609 || 10-0431469 || “.0009871668 1014 || 10281964 104259.0744|| 31.8433666 || 10-0464506 || 0009861933 1015 |1030225|1045678.375 31.8590646 || 10:0497.521 || 0009852217 1016 || 1032256 || 104.8772096 || 31-8747549 || 10-0530514 || 0009842520 1017 | 1034289 || 1051871913 || 31-890.4374 || 10-0563485 •0009832.842 1018 || 1036324 1054977S32 || 31-906I123 10-0596435 | 00098.231.83 1019 |1038361|1058089859| 31.9217794 || 10.0629364 || 0009813543 1020 | 1040400 1061208000 || 31.9374388 || 10-0662271 •0009803922 1021 | 1042441 | 1064332261 31.9530906 || 10-0695156 || 000.9794319 1022 1044,484 || 1067462648. 31.968.7347 10-0728020 •0009.784736 1023 || 046529 || 1070599.167 31-98.43712 || 10-0760S63 •000.9775171 1024 || 1048576|107374.1824| 32.0000000 || 10-0798684 •000.97656.25 1025 | 1050625 1076890625 32-0156212 10-0826484 || 000.9756098 1026 105.2676|1080045576|| 32.0312348 || 10-0859262 || 000.9746589 1027 | 1054729 1083206683 || 32-0468407 || 10-0892019 || 0009.737.098 1028 |1056784|1086373952 32-0624391 || 10-0924755 || 00097.27626 1029 |1058841|1089547389 32-0780298 || 10-0957469 || 000.9718173 1030 ||1060900||1092727000 || 32.0936.131 | 10-0990.163 .0009708738 1031 ||1062961||109591.2791 32-1091887 | 10:1022835 | 0009699321 1032 ||1065024|1099104768] 32-1247568 10-1055487 || 000.9689922 1033 |1067089 1102302937| 32-1403173 || 10-10881.17 | .000968.0542 1034 |1069156||1105507304; 32-1558704 || 10-1120726 || 000.967.1180 1035 |1071225||11087.17875 32-1714159 || 10-1153314 || 0009661836 1036 || 1073296 || 111.1934656 || 32-1869539 || 10-1185882 || 000.9652510 I037 1075369||1115157653| 32-2024.844 || 10-1218428 || 000.9643202 1038 || 1077.444 111838.6872 32-21800.74 10-1250.953 •000.9633911 1039 || 1079521 1121622319 || 32-2335.229 || 10-12S3457 | *0U0962.4639 1040 | 1081600l 11248.64000 || 32°24903.10 || 10-1315941 || “0009615385 TABLE OF SQUAREs, CUBEs, SquarE AND CUBE Roots. 102 Number, Squares. | Cubes. M Roots. $/ Roots. Reciprocals. 1041 1083681 1128111921 32°2645316 || 10-1348403 | "00096061.48 1042 1085764 1131366088 || 32°2800248 10-1380845 | *000.9596929 1043 || 1087849 1134626507 || 32°2955105 || 10-1413266 | *0009587.728 1044 || 1089936||1137893184 || 32°3109888 || 10-1445667 | *0009578544 1045 ||1092025 |1141166125 | 32-3264598 || 10-1478047 | *0009569378. 1046 || 1094116 11444.45336|| 32°34.19233 || 10-1510406 || “.0009560229 1047 1096209 || 1147730823| 32°35'73794 | 10-1542744 || 0009551098. 1048 1098304 || 1151022592 || 32°3728281 | 10-1575062 | "000954.1985 1049 1100401|1154.320649| 32°3882695 || 10-1607359 || 0009532888. 1050 1102500 1157.625000 || 32°4037035 || 10-1639636 "0009523810 1051 1104601 || 1160935651 32°4191301 || 10-1671893 °000.9514748 1052 1106704 || 1164252608 || 32°4345495 || 10-1704129 "0009505703. 1053 |II08809 || 1167575877 32°44996.15 10-1736344 | *00094.96676. 1054 1110916||1170905464| 32°4653662 | 10:1768539 "0009487666 1055 1113025 |117424.1375|| 32-4807635 | 10-1800714 || 000.9478673. 1056 1115136 1177583616 32°4361536 || 10-1832868 || "0009469697 1057 |1117249|1180932.193| 32:5115364 10-1865.002 || 0009460738 1058 III9364 1184287112 || 32°5269119 || 10-1897] 16 | *000945.1796 1059 |1121481 1187648379| 32:5422802 || 10-1929209 || 0009442871 1060 11236.00 1191016000 || 32°5576412 || 10-1961283 | "0009.433962. 1061 1125721 || 194389981 || 32°5729949 || 10-1993.336 °00094.25071 1062 1127844. 1197770328 32°5883415 || 10-2025369 || 0009416196 1063 1129969 1201.157047 32°6035807 || 10-2057,382 | "00094.07338 1064 || 1132096 || 1204550144|| 32°61901.29 || 10-208.9375 °0009398496 1065 1134225|1207949625 | 32-6343377 | 10-2121347 || 0009389671 1066 |1136356 | 1211355496 || 32°6496554 || 10-2153300 | *0009380863 1067 |1138489 || 1214767763 32°6649659 || 10-2185233 | *0009372071 1068 1140624|1218.186432 32’6802693. 10-2217146 ‘0009363296. 1069 |1142761 | 1221611509 || 32-6955654 10-2249039 || “.0009354537 1070 1144900 1225043000 || 32-7108544 || 10-2280912 ‘0009345794 1071 1147041 1228480911| 32.7261363 10-2312766 || 0009337068 1072 |1149184|1231925248| 32-74.14111 || 10-2344599 || 0009328358. 1073 |1151329|1235376017| 32-7566787 | 10:2376413 || 0009319664. 1074 II 53476 | 1238833224|| 32-7719392 || 10-2408207 | *0009310987 1075 1155625 1242296875 32-787.1926 || 10.2439981 | *0009302326 1076 1157776||1245766976|| 32-8024398 || 10-2471735 | *0009293680. 1077 | 11599.29 1249.243533 || 32-8176782 | 10-25034.70 | *00092.85051 1078 1162084|1252726552| 32-83.29.103 || 10-2535186 || 0009276438. 1079 1164241|1256216039 || 32-8481354 || 10-2566881 | "0009267841. 1080 1166400||1259712000 || 32-8633535 | 10-2598557 || 0009259259. 1081 1168561|1263214441. 32.8785644 || 10-2630213 || 0009250694 1082 1170724 }}}}}}}}| 32-8937.684 || 10°266.1850 | *0009242.144 1083 |1172889|1270238787; 32.9089653 | 10-2693467 || 0009233610 1084 1175056|1273760704| 32-924.1553 || 10-2725065 ‘O009225092 1085 |1177225|1277289.125 32-9393382 | 10.2756644 || 00092.16590. 1086 |1179396 |1280824056| 32.9545141 || 10-2788203 || 0009208103 1087 | 1181569 | 1284365503 || 32-9696830 | 10-2819743 | "0009199632 1088 1183744|1287913472| 32.9848450 | 10-285.1264 || 0009191176 1089 || 11859.21 1291467969 || 33-0000000 || 10-2882765 °0009182736 1090 || 1188100 1295029000 || 33-015.1480 | 10-2914247 | "0009174312 1091 11902S1 |1298596571 33-0302891 || 10-294.5709 || “0009165903 1092 | 1192464) IS02170688 || 33-0454233 l 10-2977153 | "0009157509 —"— I6908%8.98 69&If 1,8000. 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Z886788-0T | II07997.88 || 000836.707I |00ff ga.I | 0&II 0999868000. g968I88-0I £197195.88 |69.189TIO7III913%I 6III jºgy;68000. 08.088/8-0I 0.109987-88 Z809 (7.1681 jø66%I | 8 III Iggzgó8000. 91,01918-0|| | 6679.137-88 |819899868 I | 68917&I IIII £91,0968000. 301931.8-0I Z989,907-88 |9688&66881 99.797&I 9III 0I98968000. £II9698-0|| || 1919 I63.88 g1896 IQ881 |gzó87%I | g III I9991,68000. g0I#998-0T g889918.88 |fi/969%88I |9660%I | #III 93.1.jS68000. 91.03898-0T | 9799 (98.88 1686718191 69188&I £III 90SZ668000. 6303093-0|| || 07999;8.98 |8%69809181 |##998&I ZIII 0060006000. 79601,98-0T 9999.I.98.88 I£9088 II.3T | IZgłężI | H II 6006006000. 0886998.0L g3999.I.3.88 |000I89/98 I 00IZg&I | 0III £8 IIIO6000. 8118098-0T | 919,9109-98 ||660886898 I | [8866&I 60II IL393,06006. 1991/78-0T | 6839986-88 |&Iligº,093I |799.13&I. 80 II 7�. 2199;78-0I g609113-28 |g;0&19998.I 67%gZZI 20II I691706000. 89.8g.IF8-0I £819.9%-88 |910668%ggl]9868461| 90II #116;06000. 1817888.01 || 80%9E%-38 |gg9%ggð781|gz01zz1| g0II I161,906000. g863g23-0T ggó993%-8% iſ 98.3199.78L 9'I88IZI | #0II 38L9906000. 01/1688-0T | 89.771.I.G.38 1616 (61.78.I 6099.I&T £0II OFW 106000. 1890638.01 | 8988.961.89 |80%81%$881 |70771&I z0II Zgg3806000. 78%6939-0|| || 003318T.88 I038.39783.I | TOZZIZI I0II 6060606000. ZI08%g.0L | 61.7%99ſ-88 |000000Igg|00001&I | 00II I8I6606000. IZ19618-0I | 689HI9 I-28 ||66%£1823&I I0810&H | 660I 897.1016000. II#9916-01 || 0280981-88 |&6Igg1%g|70900ZI 8601 0.11g II6000. £8078]3.0L | 80660&I-88 31968I0%&T | 60780&I 260i 800%I6000. g81%013-0T | 106890I-88 |981%gg9131|91&10&I 9601 03%3I6000. 898 L103-0|| || Zf 81,060.88 |g|1838.6%I8I g3,066 II g60 I 891.07I6000. z866806-0T 80199.10-38 |789888.6081 |92896II | #601 I31.6%I6000. 1, 198008-0T | 9099,090.8% 19819A908I 67976[I £60 I ‘SIRoo.Idioatſ ‘shootſ /? sloog#/. ‘seqmo serumbs sequn N. 30ſ '81.00%I guno grºw auwabs ‘satioo ‘Saawnbs ad RTGVL 104 TABLE OF Squarks, CURES, SQUARS AND CUBE ROOTs. Number. Squares. Cubes. V Roots. § Roots. | Reciprocals. | il45 |1311025 1501.123625 || 33-8378.486 || 10-4616896 || 000$733624 | 1146 |1313316 15050601.36 || 33-85262.18 10-4647343 | "000$726003 1147 |1315609 || 1509003523 || 33-S673SS4 10-4677773 || 0008718396 1148 |1317904|151295.3792 || 33-8821487 | 10-4708158 || 0008710801 1149 132020I 1516910949 33-896.9025 || 10:4738579 || 00087.03.220 1150 |1322500 [1520875000 || 33-9116499 || 10:4768955 || 0008695652 1151 || 1324801 1524845951 | 33-9263909 || 10-4799314 | *{\00S688097 1152 |1327104 1528823S08 || 33-94.11255 | 10:4829656 -0008680556 | 1153 |1329409|1532808577|| 33-9558537 || 10:4859980 || 0008673027 1154 1331716 1536800264 || 33-97.05755 | 10:48902S6 -000S6655II 1155 1334025 1540798875 || 33-9852910 || 10-4920575 || -0008658009 1156 |1336336 1544804416 || 34-0000000 || 10-4950847 -000S650519 1157 |1338649|1548816893 34.0147027 | 10:49S1101 || 0008643042 1158 1340964 1552836312| 34°0293990 || 10-5011337 •0008635579 1159 1343281 | 1556862679 34-0440890 | 10-504.1556 || 000S628.128 1160 1345600 1560896000 || 34-05877.27 | 10:5071757 || 0008620690 1161 1347921 15649362S1 || 34-0734501 || 10:5101942 || 0008613264 1162 1350244 1568983528 || 34.0881211 || 10:5132109 || 0008605852 1163 |1352569||1573037747| 34-0127858 || 10:5162259 || 0008598452 1164 || 1354896 15770989.44|| 34°1174.442 10-5192391 || 0008591065 1165 1357.225 1581167125 | 34-1320963 | 10-5222506 •000S583691 1166 1359556 1585242296 || 34-1467.422 || 10:5252604 || -0008576329 1167 1361889; 1589324.463 || 34-1613S17 | 10:52.826S5 •000S568980 1168 || 1364.224 1593413632 34-1760.150 10-5312749 || 0008561644 1169 |1366561; 1597509809| 34-1906420 | 10:5342795 || 0008554320 1170 136S900 1601613000 || 34-2052627 | 10-5372S25 || 000$547009 1171 1371.241 1605723211 || 34°2198773 || 10:5402837 •0008539710 1172 | 1373584; 1609840448 || 34-2344S55 | 10:5432S32 •0008532423 1173 1375929 1613964717 | 34-2490S75 || 10-5462810 -0.00852.5149 1174 1378276; 1618096024, 34-2636834 || 10:549.2771 || -0008517888 1175 || 1380625 1622234375 || 34-2782730 || 10-5522715 •0008510638 1176 || 1382976 1626379776|| 34-29.28564 || 10-5552642 •0008503401 1177 1385329 1630532233 34'307.4336 10-558.2552 -0008496.177 1178 |1387684|1634691752|34'3220046 || 10:56.12445 || 0008488964 | 1179 |1390041|1638858339| 34-3365694 | 10-564.2322 || 000848.1764 1180 || 1392400 1643032000 || 34-3511281 | 10-5672181 | 0008471576 1181 1394.761 | 1647212741 34-3656805 || 10-5702024 •000846.7401 1182 | 1397.124 1651400568 34°3802268 || 10-5731849 || 0008460237 1183 || 1399489 1655595487 34°39476.70 || 10-5761658 •000.8453085 1184 1401856 | 1659.797504 || 34.4093011 || 10.5791449 || 0008445946 1185 1404225 | 1664006625 | 34°423S289 || 10-5821225 || 00084388.19 11S6 | 1.406596 1668222856 34°4383.507 || 10-5850983 || 000843.1703 || 1187 || 1408969 16724.46203 || 34-4528663 | 10:5880725 || 00084.24600 1188 1411344|1676676672 34-4673759 || 10-5910450 | 00084.17508 1189 1413721 1680.914629 34°4818793 || 10-5940.158 || 00084.10429 | 1190 1416100 1685159000 || 34°4963766 10-5969.850 •000S403361 1191 1418481 | 16894.1087.1 34°5108678 10-59995.25 || -0008396.306 1192 || 1420S64 1693669888 || 34°5253530 10-6029.184 || 0008389.262 II93 |1423249 1697.936057 || 34°5398321 | 10-6058826 •0008382320 il 94 || 1425636 1702.209384 34°5543051 | 10-60SS451 •000837.5209 TI95 |}; 17064S9875 34°5687.720 | 10-61.18060 •000836820.1 1195 1430416' 1719777536! 34°5832329 10-6147652 -0008361204 IZ8%IOS000. 97.36T0S000. ŽS9960S000. 6&I380S000. g89S80S000. 390970S000. 0.89 Ig0S000. 8I0Sg0S000. 9Igf30S000. 930 [10S000. fºg 1,10S000. f 1,0780S000. 9I90608000. 99 IA608000. 83/80 IS000. 0080LIS000. 88S9 IISO00. 1158&IS000. I8008 ISO00. 96998IS000. 33887 ISO00. 69667 ISO 00. 1,0999 ISOO.0. 99%89 ISO00. 98669 ISO00. 9I991 ISO()0. 9088SIS000. 8000618000. I3196L8000. 97f803S000. ISIOI&S000. 1369|I3S000. f$983&S000. £9 f{}838000. 3331,838000. 330ff38000. 93S093S000. 8891.938000. 39779&8000. 663 [138000. 9718138000. f009SZ8000. $718I63S000. 99.1863S000. 87990.38000. 399&I88000. 8976188000. 96.89%38000. 88.88898000. 7860788000. 97% ºf £8000. 6L679.88000. ‘SIGoondyoakſ &gój991-0T SSfºg.994-0T 801,909!-0T §I61, 19 J-0T $0.1679.1-0 I 1130391-0T 93FI6??-0T 61,939f 1-0I 101.88f 1-0I 6 ISFOf 1-0 I 9I691,81-0T 1669.781-0T 390SI31-0T &II6831-0T 97IO931-0T 99TI931-0T 89 I3031-0T Q9I31.Il-0T A&Iffl 1-0T £80g III-0I 8309801-0T 176990.1-0 I 99S 1301-0T Siſ 18669-0T 93.96969-0T 9Si. Of 69-0I I88II69-0I 09 IZSS9-0T 81.6%g89-0I Il 18389-0T 399 f619-0I AI899.19-0T 99.0931.9-0T 661,901.9-0T 9Ig 1199-0T IIGS799-0T 306SI99-0T 01.96899-0I £33,0999-0T 09SOgg 0.0I 0Sf I099-0T 9S03179.0T 31.9%ff 9.0T fifág If 9.0L 6618839.0I 8887939.0T 09Sf 389.0L 1939.6%9-0T 1989.939.0I I989839.0L 88.19039-0T 833.11. I9.0I ‘shoot /* g3?0129.99 ZASSZI8.98 396/S63.98 Q 1997.83.98 Žiž880.1%.98 T90%993.98 f0%037%.g3 6638/33.98 19898 Ig.98 8I8566 I.G8 3f6%gSI.gg SOIOIAI.G8 AI6199 I.Q8 89.99%f I.Q8 I9888&I.98 1660? II.g3 91.9S660.98 960.99$0.99 89.g.8II,0.99 89601,90.98 6038�.gg 8699 SZ0.93 S&S37T0.98 0000000.98 fºLI1986.f3 69 I?II.G.f3 99TI/96.f3 #0IS.f3 #S6f SZ6.f3 Q0SIf I6.f.8 1998668.f3 II &ggSS.f3 gT6II 18.78 I09SQ98.j9 88.09.f3 g6f IS38.78 7061818.78 393 f661.f3 £f g09S1.f3 $1190/1.f3 jj6%991.f3 gg06 If 1.f3 10 IQ 131.78 660 IgE 1.f3 I801869.78 #06Y.j'8 9II,8699.78 69ff:gg9.j9 39TOI79.73 #61,9939.j9 993 IZIQ.f3 61,891.69.78 ‘shooti/* 366591,856T $33,960.686 I 986787f86.I 93IIS1636I jS178IgzóI MO696;0&61 SSff08g I6I IZg0%II6I 000%.9906 I 6L6?I0306 I & 138Lif/68I 89.06T836SI 99.338&S881 91839.98SSI j,06080618. I 1989 Igf/SI 89 I696698.I I68607998T 000.19809SI 686.Igggg.9I 39.850SIGSI gSOfSzlfSI 91.III/37SI g3999&S98T Afāj91.98SI 1999.13638I Sj,086/j38T I989 I?0&SI 000SfSg ISI 69.5989 IISI 38639690ST §IggSjø08I 969gif|OS61. I 91.88I9861, I fi'8SSI6SJI 1690,11781 I. 83 IO980SLI I869g 69 11. I 000I99 [1]. I 63831. I/91, I &I6061,391, I £f 191jSg|I 9IS650fg|I g&IOG9671. I {#99.1339; II 1์11 807799981. I I09863681. I 0000008& II 669889831. I 368; 18611. I $2,332,09 IAI *Sogno #0g/ggſ 600999 I 9Igggg. I G3.0099 I 989 lifº I 670gjº I .#99%pg|I ISOOfg|I 0.09139T IZIggg I jpg&gg I 691089 I 969], Zg I g339&QI 99.13%g I 683.0%g I #381. IgE I98g IGI 006&Ig|I Tiff0IGI #861,09 I 6%gg09 I 913.809 I g39009 I 91 IS6?I 631,965 I #8386;I If806 FI 00:58SPI I969Sf I #3938; I 680 ISFI 99.9S lip I Q33917 I 96.18 li; I 698 [1571 ##6S9%I IZg.99%I 00If 9; I I89I9 FI j9369; I 6;899; I 98 fºg #I Q30%gif| 9I96##I 60% ºf FI #0Siftºff I IOjziſ; I 0000fpl. #0%ggſ, I 60868?I sorumbs I091.8FI | SF&I JiřáI * 9%I . gif&I ##&I 8;&I ŽižI TjøI 0%I 69&T 883T 13&I 93&I 93&I #8&I £831 383 I I£& I 03&I 63%I 833. I 13&I 933L 93% I fg3 I 833. I 33&I T3&I 033 I 6I&I SI&I AI&I 9I&I 9 [&I #IZI. £IZI ZIZI IIZI 0IZI 60&I 80%I 10&I 90% I 90% i 70%I 80&I &0& I I0&I 00&T 66 II 86 TT 16II ‘IoqūlūN G0I ‘goo?I agno qMW sawmbS ‘Sagno ‘SIAvat S so sigvu, 105 TABLE of SQUARES, CUBEs, SQUARE AND CUBE Roots Number. Squares. | Cubes. VRoots, $/ Roots. Reciprocals. 1249 || 1560.001 1948441249 || 35-3411941 || 10-7693001 || 0008006405 1250 1562500 1953125000 || 35°3553391 || 10-7721735 | 0008000000 1251 1565001 1957816251 35-3694784 || 10-7750453 || 0007993605 1252 1567504|1962515008 || 35-3836120 10-777.9156 |-0007987.220 1253 |1570009 || 1967:221277 35-397.7400 || 10-780.7843 •000798.0846 1254 15725.16|| 1971935064 || 35-411S624 10-78365.16 || -0007974482 1255 |1575025 |1976656375|| 35-4259792 || 10-7865.173 || 0007968127 1256 1577536 | 1981385216 35-4400903 || 10-7893815 -000796.1783 1257 | 1580.049 | 1986121593 || 35°4541958 || 10-7922441 || -0007955.449 1258 || 1582564 || 1990865512| 35-4682957 10-7951053 || 0007949.126 1259 1585081 1995616979 || 35°4823900 || 10-7979649 || 0007942812 1260 | 1587600 |2000376000 || 35°4964787 | 10-8008230 -0007936508 1261 159012I 2005.142581 || 35-5105618 || 10-8036797 •0007930214 1262 1592644 2009916728 35°5246393 || 10-8065348. •0007923930 1263 1595169 20146984.47 || 35-5387113 || 10-8093884 •0007917656 1264 1597.696 || 2019487744 35-5527777 | 10-8122404 •0007911392 1265 | 1600225 2024284625 35-5668385 | 10-8150909 || -0007905138 I266 | 1602756, 2029089096 || 35°5808937 || 10-8179400 || 0007898.894 1267 | 1605289 203390.1163 35°5949.434 || 10-8207876 •0007892660 1268 1607824 2038720.832 35-6089876 || 10-8236336 || -0007886435 1269 | 1610361 2043548109 || 35-6230262 10-8264782 •0007880221 1270 1612900 2048383000 || 35-6370593 || 10-82932.13 •000.7874016. 1271 1615441 2053225511 || 35°6510869 || 10-8321629 || 0007867821 I272 1617984 2058075648 35-6651090 || 10-835.0030 -000786.1635 1273 1620529 2062933417 | 35-6791.255 || 10-83784.16 -0007855460 1274 1623076 206779S$24 35°6931366 || 10-84067S8 •0007849294 1275 1625.625 2072671875 35-707 1421 | 10-8435144 -0007843.137 1276 1628176 2077.552576|| 35-7211422 || 10-8463485 •0007836991 1277 | 1630729 |2082440933 35'735.1367 10-8491812 -0007830854 1278 1633284|2087336952| 35-7491258 || 10-8520125 -0007824726 1279 | 1635.841 2092240639 35-7631095 || 10-8548422 || -00078.18608 1280 | 1638400|2097.152000 || 35-7770876 10-8576704 || -00078.12500 1281 1640961 |210207 1841 35-7910603 || 10-8604972 || -0007806401 1282 | 1643524 2106997768 35-8050276 10-8633225 || -0007800312 1283 | 16460.89 |211.1932187| 35°8189894 || 10-8661454 •0007794232 1284 |1648656|2116874304| 35.83294.57 | 10-8689687 | 0007788.162 1285 | 1651225 |2121824.125 35'8468966 10-8717897 || -0007782101 1286 1653.796 || 2126781656 || 35-8608421 10-8746091 -0007776050 1287 | 1656369|2131746903 || 35-8747822 || 10-877.4271 •0007770008 1288 |1658944|2136719872| 35-8887169 || 10-8802436 |-0007763975 1289 1661521 21417.00569 || 35-9026461 10-8830587 •000.7757952. 1290 1664100 |2146689000 || 35°916.5699 || 10-8858723 •0007751938 1291 1666681|2151685171| 35-9304884 || 10-8886845 || 00077.45933. 1292 | 1669264 2156689088 35°9444015 || 10-8914952 | -0007739938 1293 1671849 |2161700757 35-9583092 || 10-8943044 -0007733952 1294 1674436 2166720.184 || 35-97.22115 10-897 1123 •0007727975 1295 | 1677025 217 17473.75 35°986.1084 || 10-8999186 •0007722008 1296 16796.16 2176782336|| 36-0000000 || 10-90272.35 -0007716049 1297 | 1682209 || 2181825073 || 36-0.138852 | 10-9055269 || -00077.10100 1298 || 1684.804|2186875592. 36.0277671 | 10-9083290 || 0007704160. 1299 || 1687401 || 2191933899 || 36-0.416426 || 10-91,11296 || -0007698229 1690000 l 2197000000 36-0555,128 ' 10-91392.87 || -0007692308 1300 TABLE of SQUAREs, CUBEs, SQUARE AND CUBS Ro Number. 1301 1302 1303 1304 1305 1306 1307 I308 1309 1310 1311 1312 1313 1314 1315 1316 1317 13|8 1319 1320 I 321 1322 1323 I 324 I 325 1326 1327 I 328 1329 1330 1331 1332 | 333 1334 I 335 1336 1337 I 338 I 339 1340 1341 1342 1343 1344 1345. 1346 1347. T348 1349 I350 1351 1352 Squares. 1692601 I695.204 1697809 1700416 1703025 1705636 1708249 1710864 1713481 17 | 6100 | 17 18721 1721344 1723969 H 7.26596 1729225 | 731856 1734,489 1737 124 1739761 I 742400 1745041 1747684 1750329 1752976 1755625 1758276 1760929 1763.584 1766241 1768900 1771561 1774.224 1776889 1779556 I 782225 I784896 1787.569 1790244 1792.921 I795600 1798.281 1800964 I 803649 1806336 1809025. 1811716 1814.409 1817104 1ST 9801 1822500 1825201 1827904 Cubes. 2202073901 2207155608 2212.245127 2217342464 222244.7625 2227560616 2232681443 2237810II2 22429.46629 2248091000 2253243231 2258403328 22635.71297 226874.7144 2273930875 2279.122496 2284.322013 22S9529432 229.47447.59 22999.68000 230519916.1 2310438248 2315685.267 23.20940.224 2326203125 2331473976 233675.2783 23420395.52 234.7334289 2352637000 235794.7691 2363266368 2368593037 2373927.704 23.79270375 238.462.1056 238997.9753 2395.346472 2400.721219 2406104000 241.1494821 2416893.688 2422300607 24-27715584 24331386.25 243.85697.36 24-44008923 2449456.192 245491.1549 2.46037 5000 2465846551 24.71326.208 Roots. 36°06937.76 36-0S323.71 36-0970913. 36"I 109.402 36°1247837 36°1386.220 36°1524550 36°1662826 36*I 801 050 36-1939221 36°20'77340 36°22 15406 36°2353419 36-249 1379 36°2626287 36°27671.43 36°2904.246 36-304.2697 36°3 180396 36°33 18042 36°3455637 36°35.93179 36°37.30670 36°386SI 08 36°4005494 36°4142829 36°42801 12 36°44.17343 36.4554523 36.469 1650 36-4828727 36°4965752 36-5102725 36°5239647 36-5376518 36-5513388 36'5650 106 36-5786S23 36*5923489 36*6060104 36°6’196668 36-6333181 36-6469144 36°6606056 36-67424] 6 36-6878726 36°701 4986 36-715] 195 36-72S7353 36-7423.461 36-7559519 36.7695526 § Roots. I 0.9167265 I 0-9195228 I 0-9223177 I 0.925IIII I 0-9279031 10°9306937 10°9334829 10-9362706 10°9390.569 10°9418418 10-9446.253 I 0-9475074 I 0°950IS80 I 0°9529673 I 0°955.7451 10°9585.215 I 0-96 ſ 2965 I 0-9640701 I 0°9668423 I 0°9696131 10°9723.825 10°975.1505 10-977.9171 I 0°9806823 I 0°9834.462 10-9862086 I 0°9889696 I 0-99.17293 I 0°9944876 I0°997.2445 I 1-0000000 II*0027541 II -0055069 1 1 0082583 II () 11:0082 1 1 0137569 11:01.65041 II •0192500 } | *0219945 I 1-0247377 11-0274.795 11-0302199 I 1-0329590 11 0356967 II*0384330 I 1-04 || | 680 11-043901.7 11-0466339 1 1 0.493649 I 1-05.20945 I 1.0548227 . 11-0575497 OTR 107 Reciprocals. •0007686.395 •0007680492 •0007674579 •0007668712 •0007662.835 •0007656968 •0007651 109 •0007645260 •00076394.19 •0007633588 •0007627765 •000762.1951 •0007616446 •000761 0350 •0007604.563 •00075987.84 •00075930 14 •0007587253 •000758 1501 •0007575758 -000757 0023 •0007564.297 •0007558579 •00075.52870 -0007547 170 •000754 |478 •0007535795 •000.7530 120 •00075.244.54 •0007518797 •00075,13148 •000750750S •000750 1875 . •0007496.252 •000.7.190637 •000.7485030 •0007479432 •0007473S42 •000.7468260 •000.7462687 •0007457122 •000745 1565 •000.744.6016 •0007.440,476 •0007434944 •000.7429421 •000.7423.905 •000.74 18398 •000.74 12S98 •0007407407 •000.740 1924 •000: 396.450 szooa auno aNv suvnos ssuno sauvnos ao TITVI M096 Il,000- " Il 08S669if. lg f98A8g l928, 9IZIl 6I I 70f I i78g lº II,000. | Gif 1,876 I-II | 9lif999 7-1,9 | l68 l l9I94% | 60f S96I | 30f I 89948Il,000- | 68IZZ6I-II | 70688 fi - l'8 | 80S91 l.gg28 | f09g96I | 07I 6g l'1,8Il,000. | 86996SI-II | Gf8668 f. l 8 | I0% f88671,8 | I0S696I | I0 fI l98Zi Ll,000. 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II II91S [3. II 0&II9 IZ. II 1. I978.I.Z.II I0ISOIZ. II 819 ISO3. II 330gg03. II 61 f$30%.II 8 I6 I003-II shoot /* 39 IILC8. II 1,970.96%. II. Jiř0 If 1.3.II 99.808.9%. II jºg0053%. II I899.1QI-83 36977; I-88 6Igg [8 I-88 II,83SII-88 81 IIQ0E-88 6866. I60-88 99.9881,0.8% 93.8.1990-88 3969Zg 0.88 Z89?680.88 1908.9%0.S3. 999 Ig [0.88 0000000-88 S68S986. 1.8 I9198.1.6.18 890g.096-18 6 I981.f6, 13 g89 If 36.18 j,01.60%6.1.8 838.11.06-18 9069 #68-19 8869 ISS. 1.8 536 IS9S. 1.8 f086F98.18 69 11. If S.1.8 9099 S&S. 18 80f 39 IS. 1.8 89 IIZ08. 1.8 £1,8SS81-18 98.999 || || 1.3 . 39 IP391-18 331. 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ZI8603 IG/& §f [993.98.1% 9If Igj61, 13 8&I9093/13 f$39860 IO8 "sequo 9866TT3 g301 II3. 9 IIf II & 60%III.3, #0880T& T07g () [8, 0.09%013 I096603 #01.960% 60S$60% 9 [6060% G30S803 98 IGS03 | 6f 33.80% #986/.0% ISf92.0% 0092/03 I310.10% fjS/90% 696f 90% 96.0%90% g3369.0% 99.999.0% 68; 8.90% fº,909.0% I91. If 0% 006??08, If 0%f.0% jSI630% 63893.0% 91.f3.80% g39080& 9/11.30% 636f 60% jS0Z30Z Iſø610Z 00f 9 [0% I998IO3, j%A0IO3, 68S 1.00% 99.09.00% Q& 3.00% 969666. I 699.966 I fj/866 I I36066 I 00ISS6 I ISZGS6 I #97&S6I 6796.16I 99.8926. I G30f 1.6 L "serenbs i "IGQUIn N zgſt Igift I #If I 99 FI ggſ I #giffſ £g?I. Zgift I Igi'ſ 0g?I 6?? I Słf I Jiff I 95 fl. giff. I #ff I £ff I Ziff I If? I Off I 69'ſ I Sgiſ I Agf I 937 I ggſ I jgif|L 83; I 09:#1 6Zip I 8 1&WI 9&#I gz; † #zVI 9&#I ZZyl IZ#I O&#I 6If I SIf I AIf I 9If I gTip I £If I ŽIf I IIf I 0If I 60f I S07E. 10’ſ I 90f I 907I =- ‘SIOOH ROIng) (INV awabs 'sauno 'ssuvabs so atavi I- 110 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS. Number, Squares. Cubes. V Roots. S/T Roots, Reciprocals, 1457 2122849 309299.0993 || 38-1706693 || 11-3366964 || 0006863412 1458 |2125764 |3099363912 || 38-1837662 | 11-3392894 || "0006S587] 1 1459 || 2128681 3.105745.579 38-196S585 11-34.18813 °00068.54010 1460 |2|131600 3112136000 || 38-2099.463 11-3444719 | *0006849315 1461 2134521 || 3118535181 38-2230297 || 11-347,0614 || 000634.4627 1462 2137444 31249,43128 38-2361085 | 11-3496497 *00068399.45 1463 |2140369 3131359.847 | 38-249.1829 || 11:3522368 | *0006835270 1464 |2143296,3137785344; 33.2622529 11:3548227 | 0006830601 1465 |2146225 314.42 19625; 38-2753184 11:3574075 °0006825939 1466 214.9156; 3150662696 || 38-2853.794 || 11'3599.911 | "0006821282 1467 2152089 3157 114563 || 38-301.4360 | 11°3625735 | *00068.16633 1468 2155024, 3163575232 || 38-3144881 | 11-365.1547 | *00068.11989 1469 2157961 3170044709 || 38-327.5358 || 11-3677347 || "0006807352 1470 |2160900 |3176523000 || 38-3405790 | 11:3703136 ‘0006802721 1471 |2163841. 31830.10111 || 38-3536.178 || 11:3728914 || 0006798097 | 1472 |2166784; 31.89506048 || 38-3666522 || 11-3754679 || “.0006793478 1473 |2169729 |3196010817| 38-3796821 | 11.3780433 | "0006788866 1474 |2172676 3202524424, 38-392.7076 11-3806175 ‘0006784261 1475 2175625 3209046875 || 38-4057287 || 11-383.1906 || “0006779661 1476 2178576; 32.155781.76 || 38-4187454 11-3857625 "0006775068 1477 2181529 32221.18333| 38-4317577 11-38.83332 ‘0006770481 1478 |2184484|3228667352 || 38-4447656 11.390.9028 || 0006765900 1479 21874.41 3235225239 || 38-457.7691 || 11-39347] 2 | "0006761325 1480 2190400 3241792.000 || 38-4707681 || 11:3960384 || “.0006756757 1481 |2193361 32483.67641 || 38-4837627 11:3986045 || "0006752.194 1482 |2196324|3254952.168] 38-4967530 | 11-4011695 || 0006747638 1483 2199289; 3261545587 38-5097390 11.4037332 || “.0006743088 1484 - |2202256|3268147904| 38.5227206 | 11:4062959 || 0006738544 1485 2205225 || 3274759.125 || 38-5356977 | 11.4088574 || “.0006734007 1486 |2208196||3281379256|| 38-5486705 | 11.4114177 || 0006729474 1487 2211169 || 32880.08303 || 3S-5616389 || 11-4 1397.69 ‘00067.24950 1488 |2214144|3294.646272| 38-57.46030 | 11-4165349 || 00067.20430 I 489 2217.121 33012931.69 || 38-5875627 11-4190918 - "00067.15917 1490 |2220100 3307949000 || 38.6005181 11-4206476 || 00067.11409 1491 22230S1 || 3314613771 || 38-6134691 11-424.2022 | *0006706908 | 1492 2226064 || 3321287488 || 38.6264158 || 11-4267556 || “.0006702413 1493 2229049 332797 0157 || 38-6393582 | 11-4293079 °0006697924 1494 2232036 333466 1784 || 38-6522962 11-4318591 ‘0006693440 1495 2235.025 334.1362375 || 38-6652.299 11-434.4092 | "0000688963 J 496 |2238016 || 334807 1936|| 38-6781593 11-4369581 °0006684492 1497 |224|1009; 3354790473 || 38-6910843 11-4395059 | *0006680027 1498 |2244004; 3361517992 || 38-7040050 11.4420525 "0006675567 | 1499 |2247001 || 336825.4499 || 38-7169214 || 11-44.45980 °000667] 114 1500 |2250000 3375000000 || 38-729.8335 | 11-4471424 ‘0006666667 1501 |2253001 || 3381754501 || 38-7427412 || 11-4496857 || “.000666.2225 1502 |2256004 |3388518008 || 38-7556447 11-4522278 || “.0006657790 1503 |2259.009 339.5290527 38-7685.439 11-4547688 || “.0006553360 1504. 2262016 || 340.2072064 || 38-7814389 11-4573087 - "0006648936 1505 |2265025 |3408862625 38-7943294 | 11:45.98476 ‘0006644518 || 1506 2268036||3415662216 || 38-S072158 11.4623850 || “.0006640.106 1507 || 227.1049 || 34224.70843 38.8200978 || 11-4649.215 °0006635700 1508 2274064. 34.292885.12 38-8329757 Il-4674568 *0006631309 | TABLE of SQUARES, CUBEs, SQUARE AND CUBE Roors. 111 Number. 1509 1510 1511 1512 I 513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 I 526 1527 1528 1529 1530 1531 1532. 1533 1534 1535 1536 1537 1538 1539 1540 J541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 2430481 2433600 Squaros. 2277081 2280.100 2283121 228.6144 22891.69 2292.196 2295225 2298256 2301289 2304324 2307361 2310.400 2313441 23.16484 2319529 2322576 2325625 2328676 2331729 2334784 2337841 2340900 2343961 234.7024 235.0089 2353156 2356225 23.59296 2362369 2365.444 2368521 2371600 2374681 23.77764 2880849 2383936 2387025 239.01.16 2393209 2396.304 2399.401 2402500 2405601 2408704. 2411809 2414916 24.18025 242} 136. 2424.249 2427364 Cubes. 34.36115229 3442.95.1000 | 3449.795831 3456649,728 3463512697 34.70384.744 3477.265875 3484156096 3491055413 359796.3832 3504881359 351.1808000 3518743761 3525688648 3532642667 3539605824 35465781.25 35.53559576 3560558183 3567549.552 3574558889 3581577,000 35886.04291 3595.640.768 3602686437 3609741304 36.16805375 36238.78656 363096.1153 3638052872 3645.1538.19 365226.4000 365.7983421 | 3666512088 3673650007 3680797.184 3687953625 3695.119336 3702294323 3709478592 3716672149 3723875000 3731087.151 373830.8608 3745539377 375.2779464 3760028875 3767.287.616 | 3774555693 3781833112 37891-19879 37964.16000 Roots. 38°8458491 38-8587184 38-8715834 38-88.44442 38.8973006 38-9101529 38-9230009 38.9358447 33-94.86S4] 38-96.15.194 38-9743505 38.9871774 39-0000000 39.0128.184 39°0256326 39-0384426 39-0512483 39°0640499 39.0768.473 39-0896406 39-1024296 39°1152144 39°127995] 39-1407716 39°1535439 39-1663.120 39-1790.760 39°1918359 39-2045915 39°2173431 39°2300.905 39-2428337 39-2555728 39-2683078 39-2810387 39-2937.654 39-3064880 39-31920.65 39.3319208 39°3446311 39-3573373 39°3700394 39:38.27373 39°3954.312 39-4081210 39-4208067 39°4334883 39°4461658 39°4588393 39°4715087 39-4841740 39°49.68353 J Roots. 11.46999.11 1I’4725242 11°4750562 11:477.5871 11-480.1169 II*4826455 11-485.1731 I 1-4876995 II-49022.49 11°4927.491 II-4952722 11:49.77942 11-50031.51 11'5028348 11-505.3535 11-5078711 11:5103876 II-5129030 11.515417.3 11-5.1793.05 11:5204425 11.5229535 11:5254634 11.5279,722 11-5304799 11 °5329865 11-535.4920 11:53.79965 1:1-5404998 11.5430,021 11.5455033 II -548.0034 I I-5505025 11-5530004 11-5554972 II-5579931 11-5604878 11-5629815 II*5654740 11:5679655 11:57.04559 11-572.9453 11-5754336 11-5779208 11-5S04069 II -5S28919 11-5853759 11-5878588 11:59.03407 II •5928215 II-595.3013 11:5977799 Reciprocals. •0006626905 •0006622517 •0006618134 •0006613757 •0006609385 •000660.5020 •()006600660 •0006596.306 •000659 1958 •00065S7615 •0006583278 •0006578947 •0006574622 •0006570302 •0006565988 •00065.61680 •0006557377 •000655.3080 •0006548788 •0006544503 •00065.40222 •0006535.948 •000653] 679 •0006527415 •0006523157 •0006518905 || •0006514658 •0006510417 •0006506181 •0006501951 •00064977.26 •000649.3506 •00064892.93 •0006485084 •0006480SS1 •000647.66S4 •00064.724.92 •0006468.305 •0006464] 24 •0006459948 •0006455.778 •000645-1613 •00064.47453 •0006443.299 •0006439150 •0006435006 •0006430868 •00064.26735 •00064.22608 •ſ)006418485 •000t,414368 •0006410256 112 TABLE OF SQUARSS, CUBES, SQUARE AND CUDR Roots. Number Squares. | Cubes. VRoots. &/ Roots. Reciprocals. 1561 |2436721 |38037214S1 || 39°5094925 11-6002576 •00064.06150 1562 2439S44; 3S1103632S | 39°522.1457 11-6027.342 •0006402049 1563 |2442969|3S18360547; 39.5347948 || 11-6052097 || 0006397953 1564 |2446096 |38.2564,1444, 39°5474399 || 11-6076841 || 0006393862 1565 24492.25 3S33037125 || 39-5600809 || 11-610.1575 •0006389776 1566 2452356 |3840389496 || 39'5727 179 11-6126.299 || 00063856.96 1567 2455489 |3847751263. 39'585350S 11.6151012 || 0006381621 1568 245S624|3855123432 || 39°59.79797 11-6175715 •0006377.551 1569 |246.1761 |386.2503009 || 39-6106046 Il-6200407 || -0006373486 1570 2464900 3869883000 || 39.6232255 Il-6225088 || 0006369.427 1571 2468041 3877.292411 || 39°6358424 11-62497.59 •0006365372 1572 |2471184|3884701.248 39-64.84552 11-6274420 .0006361323 1573 |2474329 |38.921.19157| 39.6610640 11-6299070 || 0006357.279 1574 |2477476||3899547224| 39-6736688 || 11-6323710 || 0006353240 1575 2480625 |3906984375 39-6862696 || 11-6348339 •0006349206 1576 2483776||3914430976 39-6988665 11-6372957 •0006345178 1577 |2486929| 3921887033 39:7114593 11-6397566 .000634.1154 I578 2490084|392935.2552 39°7240481 11-6422164 || 0006337.136 1579 |2493241|3936.827.539 || 39:7366329 11-6446751 •0006333122 | 1580 |2496400|39443.12000 || 39:7492.138 11-64.71329 || 0006329114 1581 2499561|3951805941 39-7617907 || 11-6495895 •0006325.111 1582 2502724|395930936S | 39°7743636 11-6520452 •0006321113 1583 |25058S9|3966822287| 39.7869325 11-6544998 || 0006317119 1584 |2509056 397434.4704 || 39-7994976 II-6569534 •0006313131 1585 |2512225 |39.81876625 39-8120585 || 11-6594059 || 00063091.48 1586 |2515396 || 398.9418056 39-82.46155 Il-6618574 || 00063051.70 1587 |2518569|3996969.003 || 39.837.1686 11.6643079 •000630.1197 1588 |2521744|4004529472 39-8497.177 | 11-6667574 .0006297.229 | 1589 |2524921 |4012099.469 39-S622628 || 11-6692058 0006298.266 1590 |2528100 |4014679000 || 39-8748040 | 11-6716532 || 0006289.308 1591 2531281 |4027268071 39.8S73413 11-6740996 || 0006285355 1592 |2534464|4034.866688 39-8998.747 | 11.6765449 || 0006281407 1593 |25376.49 4042474857 39-9124041 | 11-6789892 || 0006277.464 1594 2540836|40500.925S4| 39-9249295 || 11-6814325 | 0006273526 1595 2544025 |4057719875|| 39-9374511 II-6838748 || 0006269592 1596 |2547216|4065356736. 39.9499687 | 11.6S63161 || 0006265664 1597 2550409 |4073003173 || 39-9624824 11-6887563 •0006261741 1598 || 2553604 | 408065919.2 39.9749922 11-691.1955 || 0006257822 1599 || 2556801 | 4088324799 || 39-9S74980 11.6936337 || 0006253909 1600 2560000 | 4096000000 | 40-0000000 | II-6960709 •0006250000 ~~~ —ur-ºu- To find the Square Root of Numbers exceeding 1600, Example 4. Require the Square loot of 34698. In the column of Squares you will find, +34969 = 1872, +34969 = 1873, —34596 = 1868 - 373 = 000.727, Y34698 = 186'27 nearly. --34698 = 186°,+... 271 divided by Evolunow, 118 when the number contains Integer and Decimals. Example 5. Required the Square Root of 7845°45° In the column of Squares you will find, - +7849-96 = 88.63, +7849-96 = 88-62, –7845-45 = 88'53", —7832.25 = 88 5s, 451 divided by 1771 = 000256. - W7855 – 885256 nearly. *When the number of ciphers in the integer is even; the number of figures taken in the Square Column must also be even; but when the number i..." the integer is odd, the number taken in the Square column must To find we cºe Root of Numbers exceeding 1600. - . * 6. Required the Cube Root of 5694958? In the Cube column you will º fin +5735339 = 1798 +5735339 = 1793. –5694958 = 1788-- —5639752 = 1788. T40581 divided by T955& = 000-4225, J/5694058 = 1784225 nearly. When the number contains Integer and Decimals. ſº 7. Required the Cube Root of 4186'586? In the column of Cubes you will find, - # +4251-528 = 16:28 4251-528 = 1623 –4186'585 = 16*18° 4173.281 = 161s 64942 - 78.247 = 00-088 $/4186586 = 16-183 nearly. sa-The following notice must be particularly attended wº, when extracting &he Root of numbers with decimals. - - º 2 ciphers in the integer must be 5, 8, or 11 ciphers in the Cube column. is && - * - & 3, 6, Qr 9 & 4& - : .6% &é 6& 4, or 7 “ &&. 5 * & &&. 5, or 8 “ {{ 6 & &&. & 6, or 9 “ {é 7 & & &e 7, or 10 “ &&. Example 8. Required the Cube Root of 6135875? In the Cube column and 8 | figures you will find, +61629.875 = 395. +61629875 = 39-5's -61358-750 = 3948*** –61162984 = 39-43 TTFIZ5 divided by 466891 = 00:05807 & GT35875 = 39.45807. To find the Fourth Root. . Rule. Extract the Square Root of the number as before described, and of that root extract the Square Root again, then the last is the Fourth root of the number. - - Jºaºple 9. Required the fourth root of 2469781? -*- 3/246978T = V Văşīšī- VISTAT63 =39,6467, the answer. To find the Sixth Root. Rule. Find the Cube Root of the number as before described, and of that root extract the Square Root, and then the last is the Sixth root of the number. . - - - * 114 IRREGULAR FIGUREs. To find the Area and Solidity of Irregular Figures. Chapman's rule in the construction of ships, Stockholm, 1775. * Fig. $93. º # gº Wºr– Tº ſº T.”I -3 Jº-- 3 Divide the base A B into any even number of equal parts. 3 = distance between |the ordinates; Q = area of the projecting figure. . 2={(a +4842,44442.44/40). . . . i. • 2 ºn uſ- Suppose this area to revolve around the axis A B and form a solid figure like a |handle, an urn or a gun; then the solidity C of the figure will be— - § o= º(a+4+2a+44'42-44/.429. . . . 2. The practical calculation of these formulas is set up Ordinates. Mult. Product. as in table for Formula 1. Suppose a = 1.25, b = 1.15, a | 1.25 | 1 1.25 c=1.52, d =1.86, e = 2, f= 1.77, and g = 1.20. à | 1.15 4 4.50 | The distance between the ordinates being 5 = 2, | 1.53 | 3 3.01 then the area will be, Q=}x 28.51 = 19 square of d; I.86 4 7.44 3 e 2. 2 4.00 |whatever measure used. . f | 1.77 4 7.08 The convex surface S of the figure will be, S=2ir Q | g | 1.20 | 1 1.20 = 2X 3.14 X19 = 119.3 square. Q 9,506 74 28.5l. This rule can also be employed in calculating the ordinates. | Mult. | Product. cubic contents of earth-work in excavations and . embankments, in which the ordinates are expressed 33 I 36 in areas of the sections. * b 30 4 120 . Suppose a = 36 square feet, yards, metres, or 9. 42 2 84 whatever unit of measure, b = 30, c = 42, d = 56, d 56 4. 224 e = 84, f= 72, and g = 50, the distance between | * $4 2 168 the sections being, say, 50 feet. The calculation is f 72 4 288 Set up as in the preceding table, namely: g 50 1 50 Volume G= 50 × 323.3 = 1616.5 cubics of what- c | 323.3 || 3% 970 ever unit of measure used. This rule is universally employed for calculating the areas of water-lines, cross- sections and cubic contents of displacement in ships (known as Simpson's rule). When the cubic content is required between each section, calculate it as ex- plained in Excavation and Embankment. Surface of Revolution. The surface generated by a line revolving around an axis, is equal to the length of the line multiplied by the circumference of its centre of gravity. N. B. The line, whether straight or curved, must be in the same plane as the axis. - Solidity of Revolution. The solidity generated by a plane revolving around an axis, is equal to the area of the plane multiplied by the circumference of its centre of gravity. N. B. The revolving plane must be in the same plane as the axis. .” Table of 8th Ordinates, for Railroad Curves. 115 {A 77 gle C W Ordinates. | Angle. Ordinates. 1. r. I a.º.º. 5. 4. h. Iºw"|1, r. a.º.º. 5. 4. h. .00084|:00164|-00193|-00218, 53°-05313|-08932|-IT063|-11773 •00191 |-00327 •00409 || -00436 5 4 |*05422 || -09130 |-11318|-12003 •00299 || -0.0522 •00561|-00659 55 °05531 |-09308 || 11510 | 12235 •00382 |-00654|-00818 -00872, 56 |'05646|-09487 | 11731 | "12466 -00437|-00818|.01023|-01091 57 |:05760|-09673|-11950ſ-12698 •00573|-00928|-01228-01309|| 58 05875|-09853|-12170 12932 •00675 -0.1173 -01432 -01527. 59 °05989 || 10037 12393 || 13162 | .00764|-01309|-01639|-01746. 60 |-06094 -10220 | 12612|-13397 | •00845 || 01474|-01842|-01964 61 -06261| 10427|-12840 13631 •00955 || 01637 || 02047 |-02183. 62 |-06331 -10593 || 13054 || 13866 •01053 |-01801 |-02250 | 02402 63 06451 || 1078] }•13281 14101 •01146|-01965 |-02456|-02620. 64 |-06570 | 10964 || 13505 || 14337 | •01245 -02129 |-02662|-02839. 6.5 °06681|-11101| 13765 | 1.4573 •01284 || 02271 •02861|-0305S 66 |-06805 || 11342 | 13956 || 14810 •01438|-02461 •03081 -03282. 67 || 06914 -11532 | "14181 | "15048 •01535 | -02625 |-032.77|-03496. 68 -07040 -11721 14409 || 15286 •01630 |-02789 |-03484|-03715. 69 |-07168|-11912 | "I4637 |-15526 |-01730 |-02956|-036.93|-03935 7 0 |-07284|-12103|-14864]-15765 ſ -01858 |-03125 |-03996 || 04154|| 7 || |-07407 || '12294 15087 | 16005 -01922|-03286|-04103|-04374| 72 07535|-12485|-15323|-16245 || •02022 |-03453|-04309|| 04594 73 |-07656|-12685 15555 16487 -02119|-03619|-04522|-04814|| 74 -0.7784|-12877|-15785|-16729 -02215|-03787|-04720|-05034 7.5 |07912|-13078|-16016|-16972 •02311|-03934|-04930|-05255] 76 || 08040|-13292-16247|-17216 | •02413|-04117 |-05138|| 05475 ºf 7 |-08168|-13472 | 16482 17460 •02508 |-04283) 05346|-05696; 78 |-08297 13670 16716|-17706 •02610 || 04457 -05552 - 79 |-08426 13868|16951 | "I'7951 •02708 |-04621 |-05761 • 8 0 |-08560 |-14070 -17187 | 18198 •02813 |-04793 •059.70 || - 8 1 |-08695|-14274] 17423|-18445 | •02911 || -04970 || -06188 || - 82 |-08829|-14477|-17660 | 18694 •03005 || -0.5.125 |-06386 • 83 -08944 -14681 17901 || '18943 •03107 || -05298 || 06596 || - 84 |-09105|-14888 -18140|| 19193 •03.191 || -0.5464 •06806 || - 85 09235|-15120 | 18379|-19444 •03310|-05637 |-07016 || - 86 -09377|-15304, 18622}-19695 | •03412 || 05804 || -07424 S 7 || 09518|-15509 18865 | 19946 | •03515 -0.5992 •07452 || - 88 |-09660 lbſ/56 19108 20201 || •03616|-06147|-07646|. 89 |-09780|-15931|*19350 20555 -03718 || 06327 |-07858 || - 9 0 |-09944|-16144|-19597 20710 •03821 || 06492 |-08069 • 91 -100981-16359; 19842|-20966 | •03905 |-06631|-08243 |. 92 || 10240|-16575 20092 |-21223 •04030 || 06836 || 08494 || - 93 -10384|-16787|-20338|-21481 •04.133 -07012|-08707 || - 94 -10537|-17005|-20589|-21740 •04241 •07182|-08920 | - 95 -10692|-17224|-20837|-22000 •04363 -07353 |-09130 • 96 ||10851-17444|-21091|-22262 | •04522 •07531 || 09346 • 97 -10997|-17666|-21342|-22523 •04556 || 07706 || -09562 || - 98 ||11150|-17888|-21596|-22786 ..]-04682|-07894|-09790|. 99 ||11310|| 18111|-22800 |-23050 -04833|-08059|-09991|-10627|100 ||11468 18354|-22107 || 23315 -04879|-08236|-00207|-108561 0 1 11626, 18500 22364|-23596 -04982|-08413|-00422|-110851 02 ||11791-18793|-22623|-23848 |-05096]-08593|-10639|-11314|103 |11959, 19021|-22876|-24107 •05204-08768|-10855; 11543,104 || 12116, 19256|23147] .24386 116 RAIL Road Curves. - * - - - º - * - R A I L R O A D G U R W E S. WHEN Railroads are to be connected by curves, we commonly have given the distance (chord c,) between the two ends o o of the tracks, and the tangential angle c, By these the curve is to be constructed. - Fa'ample 1. Fig. 94. The chord C = 168 feet, and the tangential angle w = 19930'. Required the centre angle w =, and the radius R = ? - w = 2(19930) = 39°. R = ask c = 1:4979×168 = 251:647 feet. k = See Table for Segments, &c., of a circle. . By Tangential Angles. The curve to be laid out by the three tangential angles ror, ron, and noo, each angle = }v = 6° 30'. Required the chord r = ? The centre angle for the chord r is 2×(6°30') = 139, and r = 12k R = 0.2264×251:647 = 56.974 feet. By Angles of Deflexion. Divide the centre angle w into an even number of parts = z. Set off at o the angle z = r on, and bisectitinto ro r and ron-find the chord r, and sub-chord a, and continue as shown by Figure. , Example 2. Fig. 94. The tangential angle v = 78°, and the chord G = 638 feet. Required the centre-angle w = ? Radius R = ? Chord r = ? and the sub- chord a = ? - - w = 2×789 = 1569. R = 156k c = 0-51117X638 = 326-126 feet. Let the curve belaid out by 6 angles of deflexion, and z =#X156°= 20°, and r = ask R = 0.44992326’126 = 146-73 feet. w : a = ask r == 0:4495×14673 = 66.012 feet. By Ordinates, - Example 3. Fig. 95. The chord C = 368 feet, and v = 36°. Required the height h = ? - h = #C(cosec.v — cot.v.). From - - - - - - - cosec.36° = 170130 Subtract - - - - - - - cot.36° = 1.37638. The height h = 0.32492×184 = 59-785 feet. At a = 92 feet from h. Required the ordinate y? * O * sinz = ** = 0.2938926 – sinuſo 6. O - y = x868(#– **)- 45°9448 feet. - By Sub-Chords- * Example 4. Fig. 96. The ends o and o of the tracks form different angles w | and W to the chord C, and therefore must be connected by two curves of differ- ent radii, R and r. The chord C = 869 feet, w = 38°, and W+ 86°. Required the distance from 0 to the height h; n = ? sub-chord b = ? sub-chord a =? | radii R and r = ? - - - w = #X38° = 199, and V = }X86° = 43°. 869 tan.199 b = 234-35 sec.43° = 320'42 feet. R = 88ka = 1.5358×671-21 = 1030-2 ft. a = sec.190(869–234-35) = 671-21 ft. r = sek b = 0-73314×320:42=23491 ft. By Eight Ordinates. Example 5. Fig. 100. Required 8 ordinates for a curve of chord C = 710 feet | and the centre angle w = 69°7 (See Table on the preceding page.) 1st and 7th Ordinates 0.07168×710 = 50-8928 feet. 2nd ** 6th &&. 0.11912×710 = 84.5752 “ 3rd ** 5th “ 0.14637x710 = 103-9227 “ 4th or height h 0.15526×710 = II0.2346 “ FAILROAD OURyss. . . . 117. w = 2w, R = "k C = #C cosec.v. |r = "k R, a = "k r = 2r sin.ºz. 95. - By Ordinates. h = #C(cosec.v — cot.w). gy = #C *; — cot.w ), 3. sin. Jº * 96. By Sub-chords. 72 = #y. h = n tan.V, b = n sec.V, * - #. a - sec.v(C — n), 97. |Parallel tracks by a reverse curve. Formulas same as above. The length o o = 2C, length of |a circle arc l = 0.035v R. Tºg, The greatest radius in a reverse - Cºrve. |w = }(V+3v), W = w ł-W – v, a = w k R, b = "k R, wº - - F-F-A. R = C sec.w(sin.V-ysin.” V – cos.ºw}. 100. Curve by 8 Ordinates. The ordinates are calculated in the accompanying Table, the chord C = 1 or the unit. If the angle w is large, or there be some obstacle on the chord C, find the height h and lay out the curve by two or more sets | of 8 ordinates. 118. BY ORDINATES AND SUBcHonds. IBy Ordinates and Subchords. Eacample 6. Fig. 101. The tangents t being prolonged to where they meet at a, divide that angle into two equal parts, say W-75°. Required the tangents t- ? external secant S= ? chords C= ? and the angle we ? Radius of the curve R=1500 feet. - t—R cot.T59–1500XO-26794=401.91 feet. " Centre angle wago—75°=15° for half the curve. S=R (sec.159–1) =1500 (1*0352–1) =52.8 feet. The chords C=k R=0:26104×1500=391-56 feet. Measure off from a the tangents and the external secant. Draw the chords C C, and divide them each into eight equal parts. In the table of ordinates under w=15° will be found the 1st. 7th. 0-01438×391-56=5:631, 3rd. 5th. 0.03081×391.56=12-063, 2nd. 6th. 0.02461X391-56=9-636, 4th. , '0:03282×39-56=12-851, Thus by only four multiplications, 16 ordinates in the curve is obtained. Should there be any obstacles for the chords C. C. as is often the case in excavations and on embankments, a line can be drawn further in on the track parallel to the chord and the ordinates obtained by subtraction, readily understood by the Engineer. Ellipse by Ordinates. By this arrangement ellipses can be constructed of any proportions. One of the two axes is divided into 16 equal parts. The ordinates drawn and calculated as shown by the figure 102. • Parallel Tracks by a semi-Ellipse, Ea’ample 7. Fig. 103. The instrument placed at , b and b', divide the angles W and w each into two equal parts, prolong the chords which will meet at a, a point in the curve. Divide the chords each into eight equal parts, and draw the Ordinates parallel to the tracks as shown in the figure. The grand chord C is the unit for calculating the ordinates, which latter are alike on both the chords c', c”. 1st, 2nd. 3rd. 4th. 5th. 6th. 7th. 0-1795G 0-2058C 0.2029C 0-1830C 0-1477C 0-1091C 0.0586C. Suppose the grand chord to be C=2050 feet. Required the length of the 6th ordinate? 0-1091X2050=223-655 feet. Tracks mot Parallel by Elliptic, arc, Ea’ample 8. Fig. 104. , Divide the angles W and w each into two equal parts, prolong the subchords until they intersect one another at a, which is a point in the curve. Divide the chord Cinto eight equal parts, join a with the 4th division and draw the other ordinates paraſlel thereto. Suppose the angles are W-18° and w=12°, the centre angle will be 30° for which the ordinates are to be calculated from the table. The chord C=125 feet. Required the 3rd and 5th ordinates? 0.06188X125=7-335 feet. Springing of Rails. - - Eacample 9. Fig. 105. A rail of L=21 feet is to be curved to a radius of R=1250 feet. Required the spring S=4 in sixteenths of an inch. 24X212 e - S = Tºjo" = 8'47 Sixteenths. Super Elevation of the Extermal Rail. Example 10. Fig. 106. A train running M-80 miles per hour on a curve of R=1550 feet radii, the gauge of the track is G=5 feet. Required i #nºe of inclination v-4 and the super elevation of the external rail h-? . - -*==0.0387-tan.2°13. 15X1550 h=G sin.1921'=5X0.02356=0.1178 feet, or nearly 13, inches. . It is practically impossible to lay the super elevation to suit the dif- ferent speeds of trains. . If a mean speed is taken, the faster passenger trains will wear the outer rail, and the slow or freight train will wear the inner rail, - toº.19 = RAILRoad CURVEs. - 119 1101. By ordinates and subchords. - - t = R cot. W= R tan.w, W=90—w, S= R (sec.w–1)= R (cosec. W-1) C=k R. For k, see table of segments. 102. - - - Ellipse by ordinates. 1 = 0.4840C 5 = 0-92040 2 = 0.6616C 6 = 0-9682 O' 3 = 0.7808 C 7 = 0-9922C 4 = 0-8660 C 8 = 0 the unit. 3. - - Parallel tracks by elliptic curve: h=# C. w = 2 v. W = 2 W, C. sin. W C. sin.w c' – –— » c’’= −, 2 sin.v 2 sin. V See example for ordinates. |104. Tracks not parallel by elliptic arc. Angle of the arc = W-- w. Ordinates to be calculated from the table. , 109. — - - - - - Spring of Rails. _1-5 Da T R - - 2.4 L = spring in inches. Inclination of tracks in curves: M* º tanº= I5 R' h = G sin.v. Meaning of letters, see example. 120 LAYING OUT RAILwax CURVEs. Explanation of the Figures on the Following Page. The most correct and positive ways of laying out railway curves are by external secant or by sinus-versus either to be employed, as the ground permits. The operation is well understood by the figures 107 and 108. The natural secant and sinus-versus are found in the trigonometrical tables. Subtract 1 from the natural secant, and the remainder will be the external secant. Multiply the external secant by the assumed radius, and the product is the external secant s in the same unit of measure as the radius. - v. The centre angle is divided by 2 and 2 as many times as may be required for setting out the curve. - ... • - - Fig.107 is used when there are obstacles inside the curve, and Fig. 108 when the outside is inaccessible. The sinus-versus in the tables, multiplied by the assumed radius, will be the height of the curve above the chord. When the inside of the curve is obstructed, and the point T of intersection is also inaccessible, then the curve can be laid out as illustrated by Fig. 109. Fig. 110 illustrates how to lay out a curve by chords of 100 feet. Tangential angles for a chord of c = 100 feet, and different radii. R from 500 feet to . . 3 miles (fig. 110). - . Ič. tan. angle. || R. tan. angle. || R. tan, angle. , Feet. O & 2 * Feet. O v. a f Miles. O & * * 500 5 43 46 3000 0 57 18 0.125 4 20 26 600 4 46 29 3500 0 49 6 0.25 2 10 13 700 4 5 33 4000 0 42 58 0.5 1 5 6 800 3 34 52 || 4500 0 38 12 0.75 0 43 25 900 3 10 59 5000 0 34 23 1 mile. 0 32 33 1000 2 51 53 5500 O 31 15 1.25 0 26 2 1100 2 36 16 6000 0 28 39 1.5 0 21 42 1200 2 23 15 7000 0 24 34 1.75 0 18 42 1500 1 54 35 8000 0 21 30 2 0 16 17 2000 1 25 56 9000 0 19 6 2% 0. 13 1 2500 I 8 46 10000 0 IT 12 3 0 10 51 Fig. 116 illustrates a section of a cut or embankment through sloping ground. The meaning of letters is the same as that on the following pages on excavation and embankment. Fig. 117. Sidings for parallel tracks.-D = distance over tangent points; W= i. between centres of tracks, and R = radius of curvature; v = angle of frog- plates. - The different operations of laying out the curves are so well understood by railroad engineers that it is considered unnecessary to enter into detailed description. The formulas and figures are | |intended only as a memorandum. - RAILROAD CURVEs. . . . . . 121 107. By external secants. External wants = R(secan – 1). W = 90-w; w = 90 —W, w = 2w. - tangent t =R cotW-R tan, w. 108. By sinus-versus. ... • w = 180 — W. c = 2E sin.w. *g sº-º-º: –. c = 2E sin.v. 2 sin.w . sinus-versus h = R sin.w. 109. When the point T is inaccessible. w = 90 — v. b = 2d cot.w. a + d = R sec.w. d = }b tan.v. - a = R sec.w—#5 tan.v. 110. Tangential angle for a chord of c = 100 feet, and different radii B from || 500 feet to 3 miles. w- 2v. sinºw-4. 2R JR = —*— tº -- in.ºw. 2 sinº' " 2R sin.ºw 111. Railway cut or embankment through . side slopes. ( s) _ T , , b sin.(90+22—s =;+dians. ,c- sin.(90–2—s) a = 90 + 2–s. v =90—z—s." e= b cos.s - sin.(90+2 —s) a = d(d sin.s see.2 + r.). Sidings of parallel tracks. D=2V W(R-3W). 2R, * D in. P. R=#44% SlT1,0) t -, tº 122 Excavation AND EMBANEMENT. EXCAVATION AND EMBANKMENT. Example 1. The Roadway of an excavated channel is r = 15 feet, the depth D ==9 feet, and the breadth at the top b = 46% feet. Require the slope S = ? 46-5 – 15 - Formula 6. S = : 2 X 9 =175 or 13 to 1. Example 2. The Road way is to be r = 15, D = 18, and the slope S. = 1}, = ? Require the breadth b = ? and the cross-section A Formuta 4. b=2x18x125+15–60 feet. Jormula 7. = #(º + 15 )= 675 square feet. Eacample 3. The Road-way is to be r=16 feet, the slope S=1}, and the depth D = 11 feet. Required the area of Cross-section A =? Formula 9. A=11 (11 × 13 + r) = 357-5 square feet. Example 4. The Road-way r=18 feet, slope S= 14, d = 14 feet 6inches, and the length from o is l = 55 feet. Required the cubic contents c =? - Formula 11. c = 55 X 1s(**** + #)- 11995.676 cubic feet, divided by 27 = 444.28 cubic yards. Example 5. The Road-way is r = 16 feet, slope S = 14 feet, D = 17°5, d = 7.4 and the length L = 100 feet. Required the cubic content C =? - 17.52 + 7.42 + 17.5 × 7-4 16 = 14 - ** = Aſ T. "I tº • Formula 12, c 100ſ ( 3 )+; (18+1 4)] <= 44445 cubic feet, or 1645-4 cubic yards. The computation is executed thus. 17.5 17.5 7.4 - - 7.4 700 24.9 W 1225 8 - 129-50 1992 17-5° 306-25 | From table 7-4* = 54-76 ſ of Squares. 199-2 - - × 100 = 44445. cubic feet. ! $ EXCAVATION AND EMBANKMENT. 123 113. wº :: §s #:::::::::: | * sº - i * : %Ø # sº *; #tººº Letters in the Formulas correspond with the Figure, - D - S = cot. v, - - 1. A = #(º + r), - 7. | d - a = DS, ſº 2. a-; (5 + r.), tºº 8. a = D cot. v, - tº 3. A = D(D S + r.), 9. a = d(d S + r.), - 10. b = 2 D S + r, º 4. - - d S r | s= }; - - 5. D * + d” + Dal c-1. [s(*::::::Pº) s-ºſ, - - 6. r 2 p. + 3 (D+ d)]. - 12. Letters Denote, — A and a = Cross-Sections in square feet, of the excavated channel or embankment. - D and d = a depth in feet, of the Sections. r = width in feet of the Road-Way. b = Base in feet of the embankment, or top breadth of the channel. L = length in feet, between the two Sections A and a. . . ! = length in feet, from the Section a to the point owhere the ground is level with the road. - C = cubic contents in feet, between A and a. c = cubic contents in feet, between a and o. - S = slope of the sides. The slope is commonly given in proportions, thus: “Slope = i+ to 1,” which means, that the side slopes 13 feet horizontally for i foot vertical. * - t = angle of the slope. 124 RAILROADS, T R A C T I O N O N R O A. D. S. Letters demote. - F= tractive force in pound avoir., necessary to overcome the rolling friction, and ascending inclined plains. M-miles per hour of the train or force F. - T = weight of the load in tons, including the weight of the carriages. On rail-roads T includes the weight of the locomotive and tender. t = weight of the locomotive resting on the driving wheels in tons. h = vertical rise in feet per 100 of inclined roads. b = base in feet #. 100 of the inclined road or plain. *: k = tractive coefficient in pound per ton of the load T, as noted in the accompanying Table, under the different conditions of the road. A = area of one of the two cylinder pistons in a locomotive, in sq. in. P = mean pressure of steam in lbs. per sq. in, on cylinder pistons. S= stroke of pistons in feet. gº D = diameter of driving wheel in feet. - H= actual horse power of a locomotive or the power necessary for the load. About 25 per cent. is allowed for friction and working pumps. j= adherence coefficient of the driving wheels to the rails, in pounds per ton of the weight t. n = revolutions per minute of driving wheels. d = continued working hours of a horse. v = velocity in feet per second, t'= weight of a horse in pounds. Eacample 11. Fig. 114. The area of one of the two cylinder pistons in a locomotive is A=314 square, inches, stroke of piston P=2 feet, mean- pressure P=80 lbs, per square inch. Driving wheels D=4 feet diameter. Required the tractive force F=% of a locomotive. F= * - 12560 lbs. the answer. The adhesive force of the driving wheels to the rails, ft, must always be greater than the retractive force of the locomotive, otherwise the wheels will slip on the track. Eaxample 12. Fig. 115. A locomotive of t=15 tons on an inclined plain rising h-10 feet, and the base b=99.5 feet per 100. f=560, other dimen- sions being the same as in the preceding example. Required the tractive, retractive and adhesive forces? Tractive, F= nºxº –22:4X15X10=9200 lbs. Retractive, F=22.4×15X10=3360 lbs. Adhesive, F=*.*-sassius. Consequently the locomotive can ascend the inclined plain with a tractive force of 8358–3360=4998 lbs., without slip in the driving wheels. Ea’ample 13. Fig. 116. A train of T-200 tons is to be drawn M=20 miles per hour on a horizontal track in good condition, k=4. Required retractive force F=? F = 200 (4+}/20) = 1694.4 lbs. the answer. Eacample 14. Fig. 117. A train of T−150 tons is to be drawn up an in- clined plain of hi-9 feet in 100, with a speed of M-16 miles per hour, k=4. Required the necessary horse power of the locomotive H=? _16×150 . . - TT375 - - Eaſample 15. Fig. 118. , Required the tractive ability F=? of a horse, running M=7 miles per-hour, in d-4 continued hours. -: 375 = 26.8 lbs. the answer, 7/4 (22.4×94-4+W 16) = 1342-144 horses. RAILWAYS AND COMMON ROADS. - 125 114. Adhesive force = ft. 115. A S P - D n Adhesive, #~24. retractive. 116. . - | F-T (k+v/M). 1. - - - Traction Coefficient at very Slow Speed. On railroads in good condition, carriage axles well lubricated, . 4 On railroads under ordinary, not very good condition, • * . . . 8 On very smooth stone pavement, . . . . . ſº * º º . ~ 12 On ordinary street pavements in good condition, e * º • 20 P On street pavements and turnpikes, . • º 30 On turnpikes new laid with coarse gravel and broken stones, . . 50 On common roads in bad condition, . • * e º e • 150 On natural loose ground or sand. . . . . . . . . . 560 Adherence Coefficients f On rails of maximum dryness, • . . . . . . 672 &6 very dry, e - e º º e g º e . 560 &é under ordinary circumstances, . tº º º . . 450 & 4 in wet weather, • - & cº e • º & . 315 * {{ with Snow or frost, . . . . . . . . . 224 In railway curves the retractive force is augmented so many per cent, as the whole train occupies degrees in the curve. Railway Gauges. Gauge feet. in. The most general gauge in coal mines, . º e & © © 2 6 Denver and Rio Grande railway, . º º º e a - a . 3 Rio Grande and Texas, º e º º e • - e e 3 6 The most general gauge in the United States, England, France, Prus- sia, Sweden, Mexico, Chili and Peru, . . . . . . 4 83 The compromised gauge, . e ę is e & º © e 4 9 Camden and Amboy, e e º © e º, º o º - 4 10 In the Southern States and in Russia, . & e g •. .. 5 Irish railways, º a . e. o © º ſº • * . 5 3 Louisiana and Texas, also in Canada and India, • - • s 5 6 Great Western in England, - - - - - - - - 7 Rain-fall in Inches at Different Seasons of the Year. Locations. Year. Spring. Summer. Fall. Winter. Nishny, Taguilsk, Russia, . . 18.26 3.35 9.28 || 3.70 1.93 Tobolsk, Siberia, . . . . . . 17.76 || 2.29 9.05 4.02 2.40 NertChinsk, Asia, . . . . . 18.13 2.32 10.5 4.96 0.35 Yakoutsk, East Siberia, . . . I0.25 1.46 3.35 3.59 1.85 Peking, China, . . . . . . 23.88 2.17 17.7 3.50 0.51 Macao, Quang-tong, . . . . . 67.81 18.8 28.0 17.7 3.31 Saigon, India, - - - - - - 62.80 5.86 || 28.9 28.0 0.04 Yokohama, Japan, . . . . . 35.02 7.52 12.0 | 15.2 || 0.295 Manilla, Philip. Islands, . . 71.31 4,77 34.1 || 25.6 || 4,84 For rain-fall, see page 359. - NAVIGATION. 127 TRAVERSE SAILING AND SURVEYING. To navigate a vessel upon the supposition that the earth is a level plane, on which the meridians are drawn north and south, parallel with each other; and || the parallels east and west, at right-angles to the former. - * * NT *-C. W. Z S These four quantities bear th d = l l', distance from l to l'in miles. C = N l l', course, or points from the meridian. iſ = la, departure or difference in longitudes, in miles. w = a l', difference in latitudes, in miles. ! = latitude in degrees. L = difference in longitude, in degrees or time. Traverse Formulas. distance is measured by the log and time; and The line N S represents a meridian north and south; the line WE represents a parallel east and West. A ship in l sailing in the direction of ll, and having reached l', it is required to know her position to the point l, which is measured by the line ll', and the angle Nll’; and imagined by the lines l a and a l’ While the vessel is running from 1 to l', the the course Nll' is measured by the compass commonly expressed in points. e following names. b = d sin. C, - T = w tan.C, - u = 60 cos.l. L., T! -: v d”—w”, w = d cos. C, - w = U cot. C, - tº E 60L cos.l tan.C w = Vala – tº, _ U wº *-āo º d = cos. C, d = 60L cos.l Tsin. CT' d = V bºwº, 1--"..- COS. 60L’ _ d sin. C cos.l = T60LT.” l y as u tan.C ; cos.l = TGOLT.” 15, 4 - y - Ty - tº 3. L = Ocos.T.’ 16, _d sin.C. . . . . 7, T 60cos.l.” 17, 8, */ tan.c Tº 60cos.7’ - 18, 9, cos, C = | • * • 19, 10, - sing - - - 20, 11, - - iſ 12, tanc- 3. - wº 21, 13, sin. C = 60 L jº, 22, 14, tan.c-904 °osº. 23, - te - 128 LAND SURVEYING. . Example, 1. A vessel sails east-north-east (6 points) 236 miles. Required her departure b, and difference in latitude w. - - . | Formula 1, b = d sin. G-236 × sin.6 points=218 miles departure, and w=d cos. c. = 236 X cos. 6 points = 90.3 miles difference in latitude. - . . Example 2. A ship sails in north latitude in a course C= ESEłE = 63 points; at a distance of 132 miles she made a difference in longitude of L = 3° 34'. What Hatitude is she in f : - - - - .* Formula 14. cos. l = or 1=53°15' the latitude. . - In high latitudes and very long distances, the preceding formulas will not give such correct results as may be desired, because they are set up with the supposi- tion that the earth is a level plane; but by the aid of spherical trigonometry we are enabled to ascertain courses and distances correctly from and between any known points on the earth. (See Spherical Trigonometry.) LAND SURVEYING. Application of formulas on the preceding page. N AN - A 2. gºing 1892 singt-ogºsº - 60L 60 × 3 + 34 3. - Cºsº, 63 - >Nº. $/º ºc, * A tº -C $ > . ...” -- . . .º., * 223 ſ Tº % - The operation is readily understood by the illustration. When only an azimuth compass is used, the course C at each station is measured from the magnetic needle or meridian to the direction of the survey. When a theodolite is employed, the course C is read as carefully as possible from the compass at the first station, but at the second station the angle v between the distances is measured, from which subtract the first course, and the remainder will be the second course. At the third station subtract the second course from the angle between the distances, and the remainder will be the third course, and so on. The calculated course is compared with that shown by the connpass at each station; if a difference is ob- served, there may be some errors in the subtraction or angle measurement, or Some local attraction of the magnetic needle, which is sometimes the case near great deposits of iron ores. The angles and courses are measured by the theodo- lite because they cannot be read so delicately on the compass. At the 5th station, where the 4th and 6th stations are on the same side of the meridian and both north of 5, add the 4th course to the angle 4, 5, 6, and the sum is the new course. On return to the 1st station, where the 7th and 2d sta- tions are both on the same side of the meridian, and one north and the other south, add the anglo 2, 1, 7, to the 7th course, subtract the sum from 180°, and the remainder should be the 1st course, which shows the accuracy of the survey. If the measurements are not correct, there will be errors on return to the first station, as seen at the foot of the traverse table. The correction for varia- tion of the compass is made on the map. ! * TRAVERSE TABLE. . 129 tion. C. Traverse Table for the Survey. º Sta- Course Sin. : cos. Dist. ‘. . . Latitude...'. Departure. 200|162.42|... 116.70 | . . . 2 185 | . . . . 81.68 |165.98 | . . . 3 - |263| 95.81 | . . . .244.90 |... | 4 |S. 42 25 E., |{:}#}|228 |... [16831||153.78|... 5 - 6 | 1 |N.35°42' E, (; 2 S. 63.48 E., |{:}; N.68 38 E., |{:}; N. 85 51 W., {:};}|223| 16.12| . . . . . . . .222.42 | 6 |S. 72 18 W., {:}|321] . . . . 97.58|. . . .305.78 | 7 ||N. 64 27 W., {:}|170] 73.32|. . . . . . . 15337 |Sum of N.S.E. and W., . .347,67|347,57681.36|681.57 | | Subtract the smallest, . . . 1347.57 681.36 | Errors in the measurement, . . 0.10 0.21 | Find the natural sines and cosines in the trigonometrical tables. . The distance, d, multiplied by the cosine for the course C, will be the difference! in ‘latitude formula 5. - ... — ‘ - ! . . The distance, d, multiplied by the sine for the course C, will be the departure] formula 1. ... - & The formulas and traverse table will answer for any unit of measure, but if the | above traverse had been made in miles, whether on land or sea, each departure should be divided by cosine for the mean latitude between each two stations, formula 16, in order to obtain the true difference in longitude. To divide by cosine is the . same as to multiply by the secant for the same angle. Haength of a Degree in Parallel of Latitude. Multiply the length of a degree at the equator (60 sea-miles = 69.03 statute miles = 110.83 kilometres) by cosine for the latitude, and the product will be the length of a degree in parallel of latitude. - - - The length of a minute or second at the equator, multiplied by the cosine for the latitude, will be the corresponding length in the parallel of that latitude. Measurement over Sloping Ground. d = Sloping dts- b = Base, or hort- I k = Difference & v = Angle of the tance. zontal distance. height. slopes. 3. d=h. cosec. v. b = d cos. v. h = d sin. v. sin. =} d=b SęC. J. b = h cot. v. h = 6 tan. v. .. tan, v=} 130 Mariners? Compasse. North, South. |Points.Degrees. sineC. Cos.C. tan.C. N | 4 || 2° 49' j .0491 9988 || -0492 # 8 26 . 1544 9880 1982 N. by E. S. by E. 1. 11 15 •1936 || -981] | 1989 and and - # 14 4 || .2430 -9700 || 2505 N. by W. S. by W. 1% 16 52 •2001 || -9570 •3032 1}_l 1941 || 3368 || 9416 || 3577 N. N. S. S. E. 2 22 30 .3827 | -9239 •4142 Nº. and # 25 19 || 4276 || -9039 || 4730 N. N. W. S. S. W. 2 28 7 || 4713 | "S820 5343 - J_2}_' 30 56 || 5140 || 8577 || 5993 N. E. by N. S. E. by S. 3 33 45 5555 | -8314 | 6883 and and } 36 44 -5981 -8014 || 7463 N. W. by N. S. W. by S. 3# 39 22 | "G343 •7731 8204 3} 42 11 || 6715 || 7410 || 9002_ - \| 4 || 45 o T-707) || 7071 iſ 1,000 *.*. 5. 4} || 47 49 -74.10 || 6715 I-103 N. W. S. W # 50 37 -7731 6345 | 1.218 -- ~. º |_43_|_53 26 || 8014 || 5981 || 1348 N. E. by E. S. E. by E. 5 56 15 -83.14 5555 I*496 - * º | # 59 4 º #: # N. W. by W. S. W. by W. 61 52 | * •47 - w, v, w y 5}___64 41 || 9039_|_4276 || 2:114 . E. N. E. JE. S. E. 6 67 30 -0239 || 3827 || 2:414 and and 6# 70 19 || -9416 || 3368 2-795 W. N. W. W. S. W. 63 73 7 || -9570 2901 || 3:295 63 75 56 || 0700 2430 || 3:991 - 7 78 45 -9811 || 1936 5.027 * .N. *... s. * | 81 34 9880 | 1544 6'744 w". N w"; s 73 | 84 22 || 9952 || 0979 || 11:14 ... by N. • Oy S. 73 87 Il -9988 || -0491 2032 East or West . º 8 90.9 1-000 0.000 CO | | NAVIGATION. 181 -º-º: --> 5:33 ºf º zº-ºº: º §º * SS º Ji ºn 4-- * Eº 2. ' º - Esº's £ººs #####ſº FSºiſſºs #############:S Distance and Dip of Horizon, from different heights above the surface of the ocean. Height. Distance. Dip. Height. Distance. Dip. Height. Distance. Dip. I'eet. Miles. * f * I'eet. Miles. .., ze Feet, Miles. o f * 0.582 1 mile. || 0 59 16 5.29 3 56 || 150 16.22 || 0 14 07 1% 1.31 0 59 17 5.45 4 03 200 18.72 |0 16 18 2 1.87 1 24 18 5.61 4 11 300 22.91 || 0 19 56 3 2.29 1 42 19 5.77 4 17 400 26.46 |0 23 03 4. 2.63 1 58 20 5.92 4 24 b00 29.58 |0 25 46 5 2.96 2 12 25 6.61 4 55' || 1000 32.41 || 0 28 18 6 3.24 2 25 || 30 7.25 5 23 || 2000 59.20 |O 51 42 7 3.49 2 36 35 7.83 5 49 || 3000 '72.50 1 3 24 8 3.73 2 47 40 8.37 6 14 || 4000 83.70 |114 15 9 3.96 2 57 45 8.67 6 36 || 5000 93.50 |1 21 54 10 4.18 3 07 50 9.35 6 58 || 1 mile. 96.10 1 24 01 11 4,39 3 16 60 10.25 7 37 || 13. “ 108.96 || 35 40 12 4.58 3 25 70 11.07 8 14 || 2 “ 123.23 |148 20 13 4.77 3 33 80 11.83 8 48 || 2:# “. 140,64 |2 3 50 14 4.95 3 41 90 12,55 9 20 || 3 “ 154.10 |2 15 50 15 5.12. 3 49 100 13.23 9 51 || 5 “ 199.15 |2 57 15 * For smaller heights, see Curvature of the Earth. The refraction is included in the dip of horizon. - The distance being the tangent a b in statute miles, at the elevation a c, in feet. Example 1. The lighthouse at a is 100 feet above the level of the sea. Required * - the distance a b. Height 100 feet = 13.23 miles. Example 2. The flag of a ship is seen from a in d. Required the distance a d, when the flag is known to be 50 feet above the level d' of the sea? Height of the light 100 = 13.23 miles a b, Height of the flag 50 = 9.35 “ b d, Distance to the ship = 22.58 miles a d. Ecdimple 3. A steamer is seen at e, the horizon b seen in the masts is assumed to be 16 feet above the level e'. Required the distance to the ship? Height of the light 100 = 13.23 miles a b, The assumed height 16 = 5.29 “ e b, Distance to the ship = 7.94 miles a e, 182 CURVATURE of THE EARTH, |CORRECTION FOR OURVATURE OF THE EARTH - IN LEVELING. . . . . . Notation of letters. D = distance in miles from the level to the stave or other object, and d = the same distance in feet. - - C = correction for curvature in feet at the stave; always negative. c = the same correction in inches. - =22. D=1.2247 Vo. C C == e := 3486543. d = 1867.3/ c. The accompanying table gives the curvature for distances from 100 feet to 20 miles. For greater distances see table of Distances and Dip of Horizon. • Difference of Apparent and True Level or. Curvature of the Earth, with and without Refraction. Sº- Distance. Curvature. Curv. and refill Distance. Curvature. Curv. and ref. Feet. inches. . Feet. Miles. Feet. - "Feet. 100 .0028 .0002 l 0.666 . . . 0.575 200 .01.15 .0008 2 2,666 2.283 300 .0258 .0018 - 3 6.000 5.141 400 - .0489 .0033 4 10.675 9.150 500 .0717 .0051 5 . 16.675 14,291 600 .1032 .0073 6 24.083 20,583 700 .1405 .0100 7 32.683 28.16'ſ 800 ..1835 . . . .0130 8 42.691 36.591 900 .2223 .0158 9 w, 54.025. 46.031 1000 .2868 .0204 10 66,700 57.175 1500 .6453 .0459 11 80.708 69.175 2000 1.147 0817 - 12 96.050 82.325 2500 1792 .1276 13 112.716 96.616 3000 2.581 1836 14 130.732 112.058 3500 3.513 .2500 15 150.075 126.633 4000 4.589 372 16 170.750 147.191 4500 5,557 396 17 I92.766 165.225 5000 7.170 ,5110 18 216.108 185,233 5500 8.676 6.185 19 240.783 206.391 6000 10,324 .7360 20 266.800 228.683 º DIVERGENCY OF THE PARALLEL. 133 T0 FIND THE DIVERGENCY OF THE PARALLEL . . . FROM THE PRIME VERTICAL. - - - - - Notation of letters. - l = latitude of the parallel in degrees. - w = distance on the prime vertical, expressed in angle of the great circle from the base-meridian. - - c = divergency in feet of the parallel at the angle v. • . c = 729000 sin.” #v × l. e. The divergency is calculated in the accompanying table for distances from one second to one degree, also expressed in feet and miles on the prime vertical. The coefficient c = 729000 sin.” #v, which, multiplied by the latitude of the parallel in degrees, gives the divergency in feet. . . - - Example 1. Suppose the distance on the prime vertical to be v = 6' = 6 miles and 4770 feet, the latitude of the parallel being 48°. Required the divergency. From the table, 0.5551 × 48° = 26.6448 feet, the divergency required. Divergency of the Parallel from the Prime vertical. |Distance on prime vertical. Coefficient. ||Distance on prime vertical. Coefficient. ; Seconds v. Feet. C. Minutes v. | Miles. Feet. C. - - 1 101.25 0.00000434 | l 1 795 0.0154213 2 202.5 0.00001735 l; 1 3832.5 0.0346979 3 303.75 0.00003855 2. 2 1590 0.061685 4. 405 0.00006916 2#. 2 4627.5 0.0964 5 506.25 | 0.0001071 3 3 2585 0.1387.917 F 6 607.5 0.0001542 4. 4 3180 0.24674 7 708.75 0.0002099 5 5 3975 0.3855 8 810 0.00027665 6 6 4770 0.55516 9 911.25 0.0003470 7 8 285 0.75564 10 1012.5 0.0004284 | 8 9 IO80 0.986.96 11 1113.75 0.00051833 || 9 10 1875 1.249.1253 12 1215 0.0006168 I:0 11 2670 1,5420 13 1316.25 0.00072394 11 12 3465 1.865820 14 1417.50 0.0008396 12 13 4260 2.220604 15 1518.75 0.0009638 13 14 5055 2,6062 16 1620 0.0010966 14 16 570 3.02.256 17. 1721.25 0.0012380 15 17 1365 3.4696 18 1822.5 0.0013879 16 18 2160 3.94.783 19 1923,75 0.0015464 18 20 3750 4.9965012 20 2025 0.0017135 20 23 60 6.1680 25 2531.25 0.002677 25 28 4035 9.63.7500 30 3037.5 0.0038553 30 34. 2730 13.8785 35 3543.75 0.0052475 35 40 1425 18.8895 40 4050 0.0068539 40 45. 120 . 24.6720 45 4556.25 0.0086742 45 51 4095 31.22815 50 5062.5 0.010709 Ö0 57 2790 38.5500 55 5568.75 0.012958 55 63. 1485 46.6455 60 6075 0.1542.13 60 69 180 55.5151 - These calculations are necessary The length of minutes and seconds on the parallel is equal to that in the table, multiplied by cosine for the latitude. * in running a parallel of latitude by fore and back sighting, and also for laying out the parallels and meridians on a map. 134 - TRIGONOMETRY. * TRIG O N 0 M ETR Y. TRIGONOMETRY is that part of Geometry which treats of triangles. It is divided into two parts—viz., plane and spherical. - Plane Trigonometry treats of triangles which are drawn (or imagined to be) on | a plane. Spherical Trigonometry treats of the triangles which are drawn (or imagined to be) on a sphere. - - ... • - : A triangle contains seven quantities—namely, three sides, three angles and the surface. When any three of these quantities are given, the four remaining ones can by them be ascertained (one side or the area must be one of the given quanti-l ties), and the operation is called solving the triangle, which is only an application of arithmetic on .#. objects. For the foundation of the above-mentioned solution, there are assumed eight help quantities, which are called Trigonometrical functions, and are here denoted with their names and number, corresponding with Figure.126. Example 1. Fig. 121. An inclined plane a = 150 feet long, and c = 27 feet, the height over its base. What is the angle of inclination C=? Formula 14. sin.c= * = *-- 0.18000. - a 150 - - - - Find 9.18000% in the table of sines, which will be found at 10°30', which is the l angle C nearly. - * , -- - . " . . . . ; ! Example 2. Fig. 122. An oblique-àngled triangle has the sides c = 27.6 feet, the | angle C = 34°10', and the angle A = 47° 40'. How long is the side a =? c sin. A 27.6 × 5in. 47o 40/ = 36.33 feet, the answer. sin.C. Rin, 34° 10' Formula I. a = By Logarithms. log.a = log.c + log.sin. A — log.sin.C. c + log. 27.6 = 1.44090 A + log. sin. 47° 40' = 9.86878 * 1.300683, C – log. sin. 34° 10' = 9.74942 log. 36.4 = 1,56026, or a = 36.4 feet. : Frample 3. Two ships of war notice a strong firing from a castle. In order to be safe, they keep themselves at a distance beyond the reach of the balls from the castle. To measure the distance from the castle, they place the vessels 800 yards from each other, and observe the angles between the castle and the vessels to be | A = 63045, B = 75°50'. What will be the two distances from the castle? C = 180 – 63°45' —75°50' = 40°25'. To A the distance will be, b = % sin.B. 800 × sin. 75°50' 1195.75 yards - sin.C. sin. 40° 25' - To B the distance will be, - csin. A = 800 × sin: 63045/ sin.C. sin. 40° 25' Q. = 1106.6 yards. * The index of a logarithm for a fraction is negative; but in the logarithms for the trigonometrical functions, 10 is added to the index, for which it appears so much less than 10 as the real negative index. Therefore, when trigonometrical logarithms are added, 10's must be rejected from the sum of the index, which will be understood by the examples. - 185 120. \, . . . - 6” - *|v 2. ' 4. _2^ \ >k 8 7 - 1 Sinus abbreviated sin.C. 2 Cosimus - & cos.C. ' 3 Sinus-versus 16 sinu.C. 4 Cosinus-versus & cosu.C. ' 5 Tangent . &&. tan.C. 6 Cotangent « . cot.C. 7 Secant sa- 8. & sec.C. . ,” 8 Cosecant sr. / + *N &c. coSec:6% r = Radius of the circle, which is the unit by which the functions are mea- | sured. r" – sin."C+cos’C. 1 . - - sec. C = −r. * cos.C.' - sin. C - . . tan. C = ** ... - , - cos. C' cosec. C = sin. C. ' I sinv.C =1 -- cos. C, tan.C * -º cot, C' - , COU. cosy. C = 1 — sin. C, os.C. . – 2 sin cons. cot. c = *% * sin.2C = 2 sin. C cos. C, sin. C . – : * ~ * . . sin.AC = }V(sin.”C+sinv.*C), - 1 º ~ : cot. C T Tan. G'. sin.(C+B)=sin.C cos. B+ | - Positive and Negative Signs. - Angles. Tsin. Tcos. Tsiav. Too Tian. cot. sec. coset. +0° +0 | +1 | +0 | +1 | +0 | +oo +1 +oo +90° +1 --0 | +1 | +0 | Foo || 4-0 || 4-co + 1 . +180° #0 | –1 || +2 | +1 || To || Too –1 | too | +270° | – || To +1 | +2 +oo || 4 || Too –1 +360° | +" | +1 | 40 | + | #9 l =2 | ++ l =2 | When a quantity has reached 0 or Co, it has ceased not be increased or diminished. to exist, because it tan | Example. What is the length of the secant for an angle of 74o 1871. Secant C = cos. 7 #Tº = 8.095. - 1 136 . RIGHT-ANGLED-EBIANGLE. . roRMULA FOR RIGHT-ANGLED TRIANGLES. Q = vº l, Q - r #26 , 10, *-āo ?. 2-, *.c, il, . 5 - Q = 3 cº cot.C. > 12, . o-, evgºro is, •-2\/º. 4 in C-4, 14.) b = a cos.C, 5, cos. C = #, 15, *- : cot.C., 2 16, 2 5 6 b= a sin,B, 7 - 8 4Q b = c tan.B, 3, sin.2c-ºſ, 17, b = */ {ar,C’ 9, an.c- ba - 18, ! Say the angle to be 0 = 60°. In the first column of the table of sines, 60° ; corresponds with 0.86602 in the next column, which is the length of sin.60°, when the radius of the circle is one, or the unit, and the expression sin. 60°X36 ; means 0-86602×36 = 31-17672, and likewise with all the other Trigonometrical expressions. . - - - - - . . . . . . In a triangle the functions for an angle have a certain relation to the oppo- | site side; it is this relationship which enables us to solve the triangle by the ap- plication of Simple Arithmetic. . . . . . . . . . . . . . . . : In triangles the sides are denoted by the letters a, b, and e, their respective | opposite angles are denoted by A, B, and C, and the area by Q. . . . . . . . . Example 1. Fig. 136 The side c in a right angled Triangle being 865 feet, and || the angle G = 39°20'. How long is the side a =? . . . . . . . ; Formula 2 a-ºº-Hºº-ºº-ººoºet, the answer. sin.O. T sim. 390,207 T 0-63383 . º & º * * --, . a ºr *- • * * * * * * , e, *, * 40 º • * , - -oblique-andrew Tatangiº: tº ‘ i37. —ºx tº: * ---> Formula For OBLIQUE-ANGLED TRIANGLEs. A . a : b = sin.A. : sin.B, and : c - sin. B. ; sin.c. a ; c = sin. A : sin.C, and Q : ab = sin.C.: 2. c sin. A d = sin. Cº., – c sin. A sin.(A+B)' 22 **7 sin.C., b = C sin.B b = • 29 & sin. C c sin.A.’ c sin. B = —b- 9 y c sin. A &=º sº . :# 29 Sin. Aë = -7- , . . 4. bc.” . . a sinic o-yº COSA, , , /T20 sin.A.T V sin. Bsin.(A+B) 10, 11, |sinº B-V g E; =2). 14, | i. o- e sin. A sin B 19, |b = Vº; 3TC), all S= (a+b+c) T12, *A-V *(s-4), 15, y bc ; : - , *Tº sin(AEE)” “ V(SFa)(SFB)(SFC Tsin. A sin. CT’ /T20 sinc T 2. • sin. A sin(A+C) 138 ; SPHERICAL TRIGONOMETRY. SPHERICAL TRIG0NOMETRY. “Spherical Trigonometry treats of triangles which are drawn (or im- | agined to be) on the surface of a sphere. Their sides are arcs of the great circle of the sphere, and measure by the angle of the arc. Therefore the trigonometrical functions bear quite a different relation to the sides. - Every section of a sphere cut by a plane is a circle. A line drawn through the centre and at right angles to the sectional circle is TN called an axis, and the two points where the axis meets the surface of the sphere are called the poles of the sectional circle. When the cutting plane goes through the centre of the sphere, it will pass through the great circle, and is then called the Equator for the poles. Axis = N.S. Equator — G.E.T. W. Three great circle-planes, a aſa/a", b b/b”, and c c'c", cutting a sphere, NESW, will form a solid angle at the centre O, and a triangle ABC on the surface of the sphere, in which the arcs a, b, c, are the sides. The angles formed by each two planes are congruent to each of the appertinent angles A, B and C. Spherical Distances. For the spherical distances, letters will denote, l = lower latitude, in degrees from the equator. {{ l/ = highest latitude, {{ $ C = course, from the latitude l to l'. C/ = course, from “ l/ to l. d = shortest distance between l and l’ in degrees of the great circle. L = difference in longitude between l and l', in degrees, or time. tan. m = cot. V cos...L. n = 90 =F l — m. — l, when l and l’ are on one side of the equator. + l, when l is on one side, and l’ on the other. Then sin. l’ cos. m. cos. d = −, . tº COS.770, º sin. L. cos.l/ sin. C = —, g G sin.d. sin. L cos.l sin.CW = + sin.d 5 tº * > pool. o aoſ N & - *###"...} New York. W – R2O 90/ & & ***ś..} Liverpool. L = 71°8' difference in longitude. tan. m = cot. 53° 22′ × cos. 71°8' = 13°31'. n = 900 — 13° 31' – 40°42' = 35°47'. sin. 530 22′ × cos. 359.47/ = 479.58/. CoS. 13931// Shortest distance = 470 × 60 + 58 = 2878 geographical miles. * * ‘. . & Formula 1... cos. d = sin.C == sin. 71°8'X cºs. 53°22' = 490 23 – 4; points, ". - sin. 479 58/ course from New York NEłE. . 3. Example. Required the shortest distance and course from New York to Liver- Right-ANGLED Spherical Triandra - 139 RIGHT-ANGLED SPHERICAL TRIANGLE. 125. b . sin.b == sin.a sin.B, 1, tº . sin.b tan.c = tan.a cos. B, 2, sin. B = sin.a' 12, cot. C = cos. a tan.B, 3, b tan.c = sin.8 tan.C, 4, sin. C = #. 13, cos. a = cos.b cos.c, 5, g cos. B = cos.ö sin. C, 6, tan. C = tan.c 14 tail.a = tan.b 7 & sin.5’ y “– in C, "| tan.h tan. B = tant y 15, sin. c = tan. B’ 8, SII). C sinº _ cos:0 sin.a = #. 9 9, cos.” - sin.B. - 16, º cos. B cos.b = cos. B sin. C :- cos. 5’ - 10, - sin. C’ 17, - COS. a . cot. C cos." - cos.; 11, cos.a = tan. B’ 18. The sum of the three angles in a spherical triangle is greater than two right angles, and less than six right angles. - . . . - - : By Spherical Trigonometry we ascertain distances and courses on the surface || of the earth; positions and motions of the heavenly bodies, &c, &c. Examples | will be furnished in Geography and Astronomy. – ". Example 1. Fig. 140 In a right-angled spherical triangle the slde or hypothe- nuse a = 36°20', the angle B = 68° 50'. How long is the side b = ? - Formula 1. sin.b = sin.a.sin. B = sin.36°20'xsin.68960'. Q, logisin. 36°20' = 9:77267 B - log.sin. 68° 50' = 9:96966 The answer, log.sin. 33°32' = 9:74233 or b = 33932. –º- z--> 126. sin.a : sin.b = sin. A : sin.B, sin.a = tº:", - - - sin.B. sin.b.; sin.c = sin.B : sin.C, sin.ö – sinºsinº, - - sin. C. cos.;(A–B) cos. #(A+B) ' sin.}(A – B) sin. (ATB). cos.;(b— c) cos.{(b+c)? tan.(a+b) = tan.ºc tan.(a — b) = tan.c tan.*(A+B) = cot.; A. w sin. #(5–c sin. #(b+c) 9 sin. :(b+c) sin. (5-c) tanke – tank(a-b) ####, cotta-taniſh–c) 20. tanºſa–B)-cotiaº - - 26, 140 ontºux Awarm seminica Taussia. d OBLIQUE-ANGLED SPHERICAL TRIANGLE. - 19 t - 21, 22, 23, 24, 25, gº. - How long is the side b = ?. - sin.c sin. B sin.72° 30′X′sin.17° 30' Formula 20. Sinº- +...+ =#### C + logsin. 72° 30′ = 9:97942 JB + log.sin. 17° 30' = 9:47812 + . - = TI-45754 c +log. sin. 79°50′ = 9.90312 - The answer log.sin. 16956 – 9:16ſſ2 orb– 16° 56' Example2. Fig.141 Oblique angled spherical triangle. c = 72°30'. B = 11°so. oriºus Aneirº Spanical Tarangie. 14, - obLIQUE-ANGLED SPHERICAL TRIANGLE. . tanºonººnºon - 72 - anºanſa —c) - . . . . . . . . tan.m = tan.c cos.A, wº- gº º º 27, C =sinº tan:4, - - - - - 28, sin.(b-m) . cos. .ſb —m COS. (Z = sosocos(5-m) * gº * * { … 29, c o cos.m. . . cos.n = ** **, - - - - - 30, COS.C. . . . . . . . cos. c tan. A - s - cot.ſm = — 3) * ~ * s * . . gº gº 31, - tan.a - s = a +b+c S = A+B+C, sin. # A += V* c) sin.(s –b), - *...* gº 32, sin.b sin.c - tº - cos.Sºcos.(S EA) . . sin...a – V* sin. B sin. C * - - 33, . . . ." To Find the Area of a Spherical Triangle. * Let Q be the area of the triangle in square degrees; if R = radius of the sphere, the length of one degree will be, - 27 R __ wn aerº--—#– = #F, or one square degree-s:#5. 360 cot.;c cot.%a+cos. B . - cot.*Q = % # , - - - - 1, ſº sin. #c sin. #a sin. B into -*:::: * - - - - 2, g 142 • ANALYTICAL GEOMETRY. ANALYTICALGEOMETRY AND CONIGSECTIONS. An equation of a line is generally re- & ferred to rectangular lines, A B = axis A. JºZºZ. 128. of , ordinate and C D = axis of abscissa, - º, The position of any point P in the curved line PI Q is defined by the rectangular distances, y the Ordinate and a the abs- cissa; a. and y are variables, depending on one another. Any change in either of them will produce a change in the other, in accordance with the formulae for the line. The position of a number of points can be determined, located and joined into the required line of the equation. The ordinate y generally constitutes the first member of the equation, and its value is determined by assumed values of the abscissa ar. g * The junction of the two axes is called origin, and denoted by 0. The line will not pass through the origin when the equation has a constant term. Properties of Lines Referred to Rectangular Co-ordinates. The tangent of any curve, . . . . HP=v Wi-F#. • 1.l $/ - da: . The subtangent of any curve, . . . .H G = 9. g O 2. $/ - - dy” The normal of any curve, . . . . PIE = y A/1 + 㺠• 3. 30 - du. The subnormal of any curve, . ſe e G E = y; e G . . Q 4. 2C The point of inflection, I, where convex and concave curves tangent, or where a curve re- d”y verses, is when . . . . . . . . . . = 0, or oo. . • 5. - da,” . Let 2 denote the length of any curve, then d2 = W da,” + dy”. Ö 6. , ºr º - - - d23 The radius of curvature of any curve is . R = - g • 7. da, d”y . . . * & © dy - - The ordinate y is a maximum or minimum when 7, = 0, . . . . º 8. Q} (See Maxima and Minima.) A curve is convea, to the axis of abscissa when the ordinate and second differen- tial coefficient have the same sign, but concave when either of them is positive and the other negative IQ is convex, and PI concave, to the abscissa C. D. . A Comic Section is the section obtained when a plane cuts a cone. The comic sections are of five different kinds, namely: 1st. Thiangle. When the plane cuts the cone through its axis. 2d. Circle. When the plane cuts the cone at right angles to its axis. 3d. Ellipse. When the plane cuts the cone obliquely, passing through the two sides. 4th. Parabola. When the plane cuts the cone parallel to one side. 5th. Hyperbola. When the plane cuts the cone at an angle to the axis less than the angle of the axis and the side of the cone. - & BYPERBOLIC LOGARITHMS. - 143 HYPERBOLIC LOGARITHMS. Hyperbolic logarithms are used in formulas derived from the calculus when the | differential cannot be integrated without the aid of hyp. logarithms. The common logarithm multiplied by 2.30258509 will be the hyperbolic logarithm, and the hyp. log. multiplied by 0.43429448 will be the common logarithm. 2/ - / ſº = hyp.logiz' – hyp.logao – hyplog+. 2, 20 200 O Hyperbolic Logarithms. OTo 9. § 0.0 || 0,1 \ 0.2 0.3 || 0.4 0.5 || 0.6 0.7 0.8 0.9 — co - 00000|8.390.568.7960219.08371 |9,306859.48918|9.64332|9.77685}9,89463 0.00000|0.09530|0.18213|0.262340,33646|0.40505}0.46998||0.53063|0.58776}0.64181 0.69315|0.74190|0.78843|0.83287|0.87544|0.91629|0.95548||0.99323||1.02962|1.06473 1.09861|1.13140|1.16314|1.195941.22373|1,25276.1.28090|1.3083441.33406 1.36099 1.386.29 |1.41096 || 1.43505|1.458591.48161||1.504081.52603]1,54753|1.56859|1.58922 1.60944|1,62922 | 1.64865.11,66770. 1,68633|1.70475||1.72276; 1.7404611,757851.77495 1.79.175||1.80827 | 1.82545||1,84055) 1.85629}1.871801,88658||1.90218.1.91689|1.93149 1.94591|1.96006||1.97406|1.98787|2.001492,01490|2.028162,04115|2,05415|2.06690 2,07944|2.091902.104,182.11632.2.12830 2.14007|2.15082í2.16338||2.174822.18615. 2.1972212.20837 l.2,21932|2.23014 2.24085 2.25129.12.2619112.27228 12.28255.12.29.171 Hyperbolic Logarithms. O. to 35 9. | No. O I 2 3 4. 5 6 7 8 9 10 2.30258|2,395892.484.91 |2,56494.2.63906|2.7080512.77259|2,83321||2.89037 2.94.444 20 | .99573|3,04452|3.091043.13549;3.17805}3.21888|3.25810|3.295843.33220 |3.36780 30|3.40120 .43399 .46574] .49651 .52636] .55535 .58352 .61092] .63759| .66356 40 .68888 .71357 .73767 .76120 .78419 .80066. ,82864 .85015 .87120 | .89.182 50 | .91202 .93183 .95124 .97029 .98898|4.00733;4.02535|4.043054.06044|4,07754 60 |4,09434|4.11087.14.127134.14313|4.15888 .17439] .18965 .20469 .21951] .23411 70 .24849] .26268| .27667 .29046 .30406| .31749; .33073 .34380 .35671 .36945 80 .38203| .394.45 .40672| .41884 .43082 .442651 .45435 | .46591; .47734 .48864 90 || 49981 .51086 52.179 .53260 .54329 .55388 .56435| .5747 1 .58497 | .59512 100|4,60517|4.615124,624974,6347314.64439|4,65396/4.66343|4,67283|4682134,69135 110 || 70048] .70953 .71849| .72739 .73619| 74493 .75359| 76217 | 77068| 77912 120 . .78749 .79579 .80402 .81218 .82028 .82831|| .83628| .84418 .85203| .85981 130 | .86753| .87519 .88280 .89035. .89784 .90527 .91265| .91998 || .92725] .93447 140 || 94164 94876 .95583 .96284 .96981| .97673 .98360 .99043 .997.215.00394 1505.01063|5.017285,02388||5.03044|5.03695|5,043425,04985|5,056245.06259|5.06890 160 07517 .08140} .08760/ .09375] .09986 .10594 .11199 .11799] .12396 .12990 170 | .13580. .14166 .14749| 153291.15905 .16478] .17048 .17615 .18178 .18738 180 .19205| .19850 .20400|, .2094.S. .214.93| .22035 .22574 .23111 | .23644. .241.75 190 | .247.02 .25227 . .25750] .26269 .26786] .27300|| .278.11 .28320|| .2S826 .29330 200|5.298325.303305,3082645,31320|5.31812}5.323015.32787|5.332725.337545.34233 210 | .34711 .351861 .35658] .36129 .36597 | .37064; .37528 .37989 | .38450| .38907 220 .39363| 39816 .40268| .40717 .41164: .41610; .42053| .42495 || .42934] .43372 230 .43808) .44242] .44674} .45104. .45532 .45958 .463S3] .46806. .47227 . .47646 240 || 48064 .484.79 .488.93| .49306 .497.16|| .50126; .50533} .50939| .51343| .51745 250 ič.52146|5.525.455,52943|5.53339||5.53733|5.541265.54517 |5.54907 5.55296|5.55683. 260] .56068| 56452 .56834 .57215] .57595|.57973] .58349 .58725] .59098 .594.71 270 . .59842| .60212 .60580|.60947] .61313| .61677 .62040 .62402 .62762] .63121 280 || 63479| .63835 ,64191| .64544: .64897 .65249| .65599 . .65948 .66296 .66642 290 ) .66988] .67332) .67675] .68017| .68358 .68697 .69036] .69373} .69709] .70044 300 H5.70378; 5.707 1115,71043|5.713735.71703|5.72031 |5.7235S 5.72685 (5.73010|5.73834 310 .73657 | .73979 74300|| .74620 .74939| .75257 | .75574| .75890 || 76205H 76519 320|| 76832] .771441 .77455 .77765 .78074 .78382 .78690 .78996 || 79301 .79606 330 || 79909| 80212 .80513| .80814] .81.114: .81413 | .81711] .82008 || .82304 .82600 340 .82894 .83188] .83481| .83773: .84064| .84354 .84644 .84932 . .S5220 | .85507 350 15,8579315,8607815,8636315.86647 15.86929.15.8721215,8749315,87773 |5,8805315.88332 144 Conio Secross. v = v2rº-rº, - y’-twº - r=-a-, a = r+ v r" — y”. Circle. y = v r” — a 4, . +a;”, - •- Vºy. * = TſI31. Circle Arc. - y – Vºcº-º —a, - &–4hº a = -si-. Circle Arc. cº-i-hº T8% cº–4h” 8.T.’ 132. y = \/ ( )—e —- & -= 133. Ellipse. y – “v2my->. I31. Elº. T gy = m” #) CONIC SECTIONs. 145 135. - Ellipse. e = ſmº-nº, _2n” p- ºr 136. Ellipse. R = 2m -- r, r = 2n — R. 137. - JParabola. 2) = V pr, p–4m, r = x/y-Gº-m)*= x +m, y = V tº — 44°.* y–2-ſtºrm)-r, t = V4 rºyº, t = 2 vſ r(x+m). : 138. Hyperbola. . R — r = 2m, © = e 7)2 y= # v(e"—1)(c. — m”), 1.16 - Loganizaws. . 106 ARITH Ms. A Logarithm is an exponent of a power to which 10 must be raised to give. a certain number, which will be understood by this tº; tºl ă ş § 2; E go § 3 ; : : ; J: Er : # 5 # # log. 100 = 2 because 103 = 106. log. 10000 = 4 ** 10% = 10006) log. 5012 = 3-7 “ #03.7 = 5012. The etrºit of the logarithm is ealled the characteristie or indes, snd the deeşmead. * * part is called the mantissa, the sum of the characteristic and wantissa is the Loga- ; The invariable number 10 is the base for the system of Ługa- Tlt]]. IſºS. - - - . It is not necessary that the bººse should be 10, it can be any number, but all i. tables of Bogarithms now in common use, are calculated with 16 to the use. - - - . . . The nature of logarithms in connection with their numbers are sueh, that the | indea; cf the logarithm is always one less than the number of figures. in the number, (when the base of the logarithm is 10,) as, index 5012 = 3 mantissa 5012 = 0-7 logarithra 5012 = 37. Let 10 be raised to any power as, and the power of 10* = a or log. g = 2, the power of 10° = b or log. b = 2. Let the product of ab = c and the quotient. = C, 10*X10* = 10&^* = ab = e or log. c = 2+s. EO2: - Q: - # = 10− =#=d ©r Rog. d = 2–2. Og = m.º. - - or log. * = 2×log. a. Wa = n or log. a = }og. a 3. - Any number represented by the letters a, b, e, or d, can be a power of 10, which | exponent is the logarithm for the mumber. Logarithms are calculated for every number with three figures in the accompanying Table, by which any operation in |Multiplication, Division, Involution and Evolution can be performed by simple Addition or Subtraction of Logarithms. Tables of Logarithms are commonly more | extensive, and ealculated for any mumber of four or five figures, which would | occupy too much room in this book; but by the proportional parts, the logarithm can be found by this Table, to four or five figures. The index of the logarithms do not appear in the Table, only the mantissa. It is easily remembered that the $ndez is one less, than the mºvember of figures in the number; then when the nup- | ber is only one figure, the index is 9 ; and when the number is a fraction, the index is negative. * - - When the logarithm is to be jound for a fraction, we commonly have the fraction expressed in a decimal; and then the negative index is equal to the number of ciphers before the first figure, and commonly marked after the marº- tissa; thus explained in whole numbers and fractions: log. 365 = 2-56229. log. 0-365 = 56229—1. Bog. 46-7 = 1-66931 log. 0-0467 = 66931—2. - log. 7-59 = 0-88024 Hog. 0.00759 = 88024—3. - In the accompanying Table of Logarithms, for the trigonometrical kines the negative index is marked thus, - . log. sin. 35° 40′ = log. 0.58396 = 1:76572. Logarithms. - 147 To find the Logarum of Numbers. To 45 in the first column of the Table, answers 65321 in the next oolumn, which is the mantissa; index = 1 because 45 is two figures. - Then, log. 45 = 1°65321, the answer. Example 2. Find the logarithm of 768? - - - - | Opposite 76 in the first-column, answers 88536 in the column marked 8 on the | top or bottom. Index = 2 because 768 is three figures. Then, log. 768 == 2-88536. Example 3. Find the Logarithm of 6846? log. 6840 = 3.83505 Proportional part, 64X0-6 = 384 ... • log. 6846 = 3-835434 the answer. - To find the number for a given Logarithm. Example 1. What number answers to the logarithm 3-871577 In the Table you will find in the column of logarithms, that log. 7440 = 3-87157. Example 2. What number answers to the logarithm 3-801884? Given logarithm 3.801884, Subt, nearest table log, 380.1400 = log. 6330, Divided by proportional part, 69|484 - - - - - - 7, - . 6337 the req. numb. Multiplication by Logarithms. Rule. Add together the logarithms of the factors, and the sum is the loga- rithm of the product. * - Example 1. Multiply 425 by 48. To log. 425 = 2-62839, Add . log. 48 = 1.68124, The product, log. 20400 = 4:30963. Example 2. Multiply 79600 by 0-435. To log. 79600 = 4.90091, Add, log. 0-435 = 63848—1, The product log. 34690 = 4:53939. Division by Logarithms. Rule. From the logarithm of the dividend subtract the logarithm of the di- visor, and the difference is the logarithm of the quotient. Example 1. Divide 43800 by 368. From log. 43800 = 4.64147, Subtract log. 368 = 2.56584, The quotient log, 119 = 2,07563. Example 2. Divide 36 by 0.625. From log. 36 = 1-55636, Subtract, log. 0-625 = -79588-1. The quotient, log. 576 = 176048. ... A negative indea: follows an opposite operation of its mantissa, as if the man- tissa is subtracted, add the negative index, and vice versa. - Envolution by Logarithms. Rule. Multiply the logarithm of the number by its exponent, and the pro- duct is the logarithm of the power of the number. Involution by Logarithms. Rule. Divide the logarithm of the number by the index of the root, and the quotient is the logarithm of the root of the number. - * - - - -- 143 LOGARITHMS OF NUMBERs. No. 100 to 1600. Logarithms. 00000 to 20412. NO 1 2 3 4 5 6 8 9 43 100 || 00000 || 00043 || 00087 || 00130 |00173 00217 || 00260 || 00303 |00346 00389 || 1 || 4 101 || 0432 0475 || 0518 || 0561 || 0604 || 0647 || 0689 || 0732 || 0775 || 0817 || 2 | 9 102 0860 | 0903 || 0945 || 0988 || 1030 1072 1115 1157 || 1199 || 1242 |3|13 103 || 1284 1326|| 1368 || 1410 || 1452 1494 || 1536 1578 || 1620 1662 |4|17 104 || 1703 || 1745 || 1787 | 1828 1870 | 1912 || 1953 | 1995 || 2036 | 2078 || 5 || 22 105 || 02119 || 02160|02202 || 02243 |02284 || 02325 || 02366 02407 || 02449 || 02490 || 6 || 26 106 || 2531 2572 2612 2653| 2694 2735 | 2776 2816 || 2857 2898 || 7 || 30 107 2938 2979 || 3019 3060 3100 3141 || 3181 3222 || 3262 || 3302 || 8 || 34 108 || 3342 3383 || 3423 3463 3503 || 3543 || 3583 || 3623 || 3663 3703 || 9 || 39 109 || 3743 3782 || 3822 || 3862 || 3902 || 3941 || 3981 | 4021 | 4060 4100 - iſo loſió9 || 04179| 04218 || 04258 |04297 || 04336|04376 || 0415 || 04454 014.g3! ...+! iii || 4532 || 4571 || 4610 || 4656 || 4689 || 4727 || 4766 || 4805 || 4844 4333 || 4 ii.2 4932 || 4961 || 4999 || 5038 || 5077 || 5115|| 5154 || 5192 533i 5269 |3| 8 ii.3 || 530s 5346|| 5385 5423 546i 5500 || 5538 5576|| 5614 || 5652 |3|12 iſ 5650 5729 || 5767 5805 || 5843 || 5881 5918 || 5956|| 5934 6032|#| || ii.5 |06070 || 06ios logiq6 || 06183 || 06221 | 06258 loé296 || 06333| 06371 || 06403 ||3| ?! ii.6 || 6446 || 64.83 || 6521 || 6553 || 6595 || 6633| 66.70 || 6′07 || 6744 || 678i |}|3: iii 6sij | 6856|| 6393 || 6930 | 6667 || 7004 || 704i | 7078 || Wii.5 tiši || 33 Tiš | 7188 || 7225 | 7262 || 7298 || 7335 | 7372| 7403 || 74.45| 7432 || 7518; ; ; Tig | 7555 7591 || 762s 7664 || 7700 | 737 || 7773 | 1809 || 7846 | 7382|9| 37 120 |07918 07954 07990 || 08027 08063 08099 || 08135 | 08171 || 08207 || 08243 39 121 || 8279 8314|| 8350 8386 8422 8458 || 8493 || 8529 8565 | 8600 || 1 || 4 122 8636 8672 8707 || 8743 || 8778 || 8814 || 8849 || 8884 || 8920 | 8955 |2| 8 123 S991 || 9026 || 9061 | 9096 || 9132 || 9,167 || 9202 9237 9272 | 9307|3| 12 124 || 9342 | 9377 94.12 9447 9482 9517 | 9552 9587 9621 | 9656 |4|16 125 |09691 || 09726 || 09760 | 09795 || 09830 09864 || 09899 || 09934 09968 10003 || 5 | 20 126 10037 || 10072 | 10106 || 10140 || 10175 10209 || 10243 10278 || 10312 || 0346 || 6 || 23 | 127 | 0380 0415 || 0449 || 0483 || 0517 | 0551 || 0585 || 0619 || 0653 || 0687 || 7 || 27 128 || 0721 || 0755 || 0789 || 0823 0857 || 0890 || 0924 || 0958 || 0992 || 1025 || 8 || 31 129 1059 || 1093 || 1126 || 1160 || 1193 | 1227 | 1261 1294 1327 1361||| 9 || 35 130 11394 | 11428 |11461 || 11494 || 11528 11561 || 11594 | 11628 || 11661 | 11694 – †: 131 || 1737 || 1760|7|1793 || 1826||1360 | 1893 || 1926 |Ti959 || igg2 2024 | . 37. 132 | 2057 | 2000 || 2:23 2156| 2išg 2222 2254 2287| 2320 || 2:52 || 4 133 || 2335 2418 2150 | 2183 || 2516 || 2543 || 2531 || 2613| 2646 || 2678 |}|...? 134 2716 2743| 2775 2803 || 2340 || 2872 | 2005 || 2537 || 2565 || 3001 || 3 || iš5||13633 | 13066||13098 || 13136||13162 |13194|13326||13258||13250 |1332214 # 136 || 3354 || 3336||73jiš || 3450|T34S1 || 3513| 3545 || 357 || 3605 || 3640|5|| 137 3672 | 3704; 3735 | 3767 || 3799 || 3830 || 3362 | 3393 || 3335 | 3556 || ?? 138|| 338s | 4019 4051 || 4082| 4ii.4 || 4145 || 4f76 || 420s | 4339 4270 || 3% 135|| 4301 || 4333| 4364 || 4395| 4126 4457 4486 || 4520 || 455i 4582 |}| 39 140 ||14613 14644|14675 || 14706|14737 14768|14799 || 14829 |14360 14391 |9| 33 141 4922 || 4953 4983 5014 || 5045 5076 || 5106 || 5137 5168 || 5198 || 35 142 || 5229 5259 5290 5320 5351 5381 || 54.12 5442 5473 5503 || 1 || 4 143 5534 5564 || 5594 || 5625 5655 5685 5715 5746 || 5776 5806 |2| 7 144 5836 || 5866 || 5897 || 5927 || 5957 5987 6017 6047 6077 6107 || 3 || 11 145 || 16137 | 16167 | 1619.7 | 16227 | 16256 16286 16316 | 16346. 16376 | 16406 || 4 || 14 I46 6435 | 6465 || 6495 || 6524 6554 || 6584 6613 | 6643 | 6673 || 6702 || 5 | 18 147 || 6732 6761 || 6791 6820 | 6850 | 6879 || 6909 || 6938 || 6967 || 6997 || 6 || 21 148 || 7026 7056 || 7085 7114 7143 || 7173 || 7202 || 7231 || 7260 | 7289 || 7 || 25 149 || 7319 | 7348 || 7377 7406 || 7435 | 7464 || 7493 || 7522 || 7551 7580 || 8 || 28 150 17609 || 17638 17667 17696 17725 || 17754 || 17782 || 17811 || 17840 || 17869 || 9 || 32 151 || 7898 || 7926 7955 7984 || 8013 8041 i 8070 8099 || 8127 8] 56 g 152 $134 || 8313| $2.1 8270 $398 || 8327 $355 | 8334|| 3412 || 34.41 || 3: 153 || $469 || $463 || $526 $554|| $583 $61i $636 $667| $696 || 3724 || 3 iší || $753 | 8730| $805 || $837| 8865 | 88.93| 333i $gſgl §§77 || 3005 i3 .. 155 |19033 1906i |13039 || 15117 |13145||13173||1320i | 16229 |1325i 19235|}}} 156 || 3313| Tö346|| 33.63 || 9396 || 3424 || 3451 || 94.70 || 9507|Tö535 | 3562 |*|}} iš7 || 3560 | 66is 3645 3673| 9700 | 9728| 3756 giš gºſi || 3338||5|| J58 || 9866 98.93||| 99.21 | 99.48 || 9976 20003 || 20030 || 20058 || 20085 20112 º 20 159 |20140 |20167|20194 |20222|20249 || 0276|| 0303 || 0830) 0358 0385|g : No. 0 1 2 3 4 5 6 7 8 9 || 9| 30 #| |&t 20% lºst; gºlf |&If |z1|, |yorſ § 5 |930ſ 900. 98% 99% ºg | 93% |30% if |G|9388 |9088 |981& 99.8 |9?!? 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II:#9 || 1889 |#999 || 0:589 |9|89 9 |3| IZZ9 |8619 |#LI9 || 0g I9 |9&I9 |z019 |6109 I QS69 6969 |986G ZI69 || 88SQ #989 || OfSg ;|3|iº |iº |zīz; †: ; ; ;is o: || |$10g | 866 |696; tº |0%ft g68; Ilsº if |S|gllſ | 8:lï fºllº | 669; fl.9?_ 099p || 939t Gº |3|1%gró |30gtó llìfà |&gº, 86% goñº. 81.81% Ží |}|6|ft| | #3%f |& | #03; 108Iſ |ggiº |Q&I. § |8 || 080p || 900; 985& 9969 |0868 g063 |QSS3 § || ||61||8 || Pºlº 6318 || 018 ||6198 tº 98 |6398 § || |8393 || 30% lºg | Zºg_{9% 1088 |9|88 86 ||6 || 9913 | 1913 || ZIP3 | 9892, 0993 | #393 || 8092, 9% |8 || Q0gz, 615% ggiº, 1złż, IO-52, 91% Oggz, 33 || || 9 FZZ | 0323 #6TZ | 1913 | If IZ | g IIZ | 680% 6L | 9 || GS6 LZ | 896LZ Z96L6 || 906LZ || 08SIZ | #gSIZ | L3SL3 9 L | 9 || ZZLI 969 I 699 I ºf 91 || LIQI | 069| | #99T 3I f | 89%I Ig; L gOFI 81.3L | Zg&I gzgI | 66%I 6 |8 || Z6LI g9II | 68LI ZIII | Q30I 690L |Z90I 9 |6 |gz60 | 8680 | I].80 | ##80 || LISO || 9610 | 8910 8 |I| 99.90% 63903; 30.90% g1903 || Spg03 || 0Zg0Z || 86.50% I3 6 8 || || 9 G # | 8 9950% 69503 I I91 09.I “ON 'gīzīg o! ZI;0& *Suru] Irešorſ '00% O] 009.I ‘ON 6FI sagawa N Io SWHLIXIvoo'I 6L |6 350 LoGARITHMs of NUMBERs. No. 2200 to 2800. Logarithms. 34242 to 44,716. No. 0 1. 2 3 4 5 6 7 : 8 9 20 | 220 || 34242 34262 34282 34301 || 3.4321 || 34341 34361 || 34380 .34400 34420 II 2 221 || 4439 4459 || 4479 4498 || 4518 4537 4557 || 4577 4596 || 4616 || 2 || 4 222 || 4635 || 4655 4674 4694 4713. 4733| 4753 4772 4792 || 4811 || 3 || 6 223 4830 || 4850 4869 || 4889 || 4908 || 4928 4947 4967 4986 5005 || 4 || 8 224 5025 5044 5064 5083 || 5102 || 5122 || 5141 || 5160 || 5180 || 5199 || 5 || 10 225 35218 35288|35257 |35276 |35295 35315 35334 35353 |35372 || 35392 || 6 || 13 220 5411 5430 5449 || 5468 5488 5507 || 5526 5545 5564 5583 || 7 || 14 227 | 5603 || 5622 || 5641 5660 || 5679 5698 || 5717 | 5736 5755 5774 || 8 || 16 228 5793 5813 || 5832 5851 || 5870 5889 || 5908 5927 || 5946 5965 9| 18 229 || 5984 || 6003 : 6021 || 6040 || 6059 || 6078 || 6097 || 6116 || 6135 | 6154 230|| 36||73 ||36192|36211 36229 |36248 || 36267 ||36286 || 36305 |36324 || 36342|- . 19 23i || 6361 || 6380 || 6399 || 6418 || 6436 || 6455 || 6474 || 6493 || 6511 || 6530 || ? 232 || 6519 || 6568 || 6586 | 6605 | 6624 | 6642) 6661 | 6680 | 6698 || 6717 |2| 4 233| 6736 | 6754| 6773 | 6791 | 6810 | 6829 | 6847 | 6866 | 6884 || 6903 ||3| 6 234| 6922 || 6940; 6959 || 6977. 6996 || 7014 || 7033 || 7051 || 7070 || 70s& |4| 8 235|37;07 ||37125 |37144 37162|37181 || 37199 |3721s 37236||37254 37273|5||19 236 || 735i | 7310|| 7328 || 7346||7365 | 7383 || 7401 || 7420 || 7438 || 7457 |S| 11 237 iſ is 7493 || 7511 || 7536|| 7548 || 7566 || 75$5 || 7603 || 762i 7639||13 238|| 7658 igić ió94| Tiz, Tiš1 || 7149) iſ 67 || 7785| 7803 1822|}|}} 239| 7340 || 7858 || 7876 7894| 7912 || 7931 || 7949 || 7967 || 7985 8003 |9| 17 240 || 38021 || 38039 || 38057 ||38076 38093 ||38112 || 38130 38148 ||38166 || 38.184 || 18 24l 8202 || 8220 | 8238 8256 || 8274 8292 || 8310 || 8328 || 8346 8364 || 1 || 2 242 || 8382 8399 || 8417 | 8435 | 8453 | 8471 | 8489 8507 || 8525 | 8543 |2| 4 243 || 8561 8578 8596 8614 || 8632 | 8650 | 8668 8686 8703 || 87.21 |3| 5 244 || 8739 8757 8775 8792 8810 | 8828 8846 8863 || 8881 8899 || 4 || 7 245 38917 | 38934 3S952 38970 38987 39005 || 39023 39041 |39058 39076 || 5 || 9 246 9094 || 91.11 9129 || 9146|| 9164 || 9,182 9.199 || 92.17 | 9235 | 9252 G| 11 247 || 92.70 | 9.287 || 9305 || 9322 || 9340 | 9358, 9.375 | 9393 || 9410 9428 || 7 || 13 248 || 9445 9463| 94.80 9498 || 9515 9533 9550 9568 || 9585 9602 || 8 || 14 249 || 9620 9637 9655 9672 9690 9707 || 9724 9742| 9759 9777 || 9 || 16 250 .39794 39811 39829 39S46 || 39863 39S81 || 39898 || 39915 39933 39.950 25i || 9967 || 968514.0002 || 40019 |40037 | 400.54|40071 | 40088|40166 | 40123 || - 17 252] 40i40 | 40157 loſić | Töig2|Tú209 || 0226 || 0243 || 0261 || 0278 || 0295 || || ? 253) offiz || 0329| 0346 || 0364| 0381 | 0398 || 0:15 || 04:32| 0449 || 0466||3| 3 254|| 0183 || 0500 || 0518 off.35| 0552 6569 || 0586 Ö603 || 0620 | 0637 |}| 3 2551.40654 |4067.1140688 |40705|40722 |40739|40756 | 40773|4ó790 | 40807 |#| || 256} 0324 || 0841 || 0858 | Tö875|| Toš92 || 0909 | Tö926 | Tö943 || 0960|| 0976|*| .9 257 ogg3 idiol io27 | 1044) ió61 iO78 || 1095 | ii.11 | ii.28 || 1145||9||19 25s jié2 | 1179 || Tigg | 1212| 1229 | 1246 1263 | 1280; 1296 || 1313| 7 || 12 259 || 1330 1347 | 1363 | 13sol isºft | 1:14|| 1430 i347 || 1464 1481 |S| 14 260 || 41497 || 41514|41531 || 41547 |41564 || 41581 |41597 || 41614|41631 || 41647 |9| 12 261 | 1664 1681 | 1697 1714 || 1731 1747 || 1764 || 1780 || 1797 | 1814 | 16 262 1830 1847 1863 | 1880 | 1896 || 1913 1929 1946 || 1963 | 1979 || 1 || 2 263 || 1996 || 2012 || 2029 2045 2062 | 2078 || 2095 || 2111 || 2127 | 2144 2} 3 264 || 2160 2177 || 2193 2210 2226 2243 2259 2275 2292 2308 || 3 || 5 265 |42325 42341 |42357 || 42374 || 42390 || 42406 || 42423 || 42439|42455 || 42472 || 4 || 6 266 || 2488 2504 || 2521 2537 || 2553 || 25.70 || 2586 2602 || 2619 || 2635 || 5 || 8 267 || 2651 2667 || 2684 2700 2716 2732 || 2749 2765 || 2781 2797 || 6 || 10 268 || 2813 || 2S30 2846 2S62 2878 2894 || 2911 || 2927 2943 2959 || 7 || 11 269 2975 2991 3008 || 3024 || 3040 3056 3072 3088 || 3104 3120 || 8 || 13 270 || 43136 || 43152 |43169 || 4318.5 4320.1 43217 43233 43249 || 43265 43281 || 9 || 14 : 339. ºl;| 33% 3343| 36|| 3 || 33% 34%| 3425 | #|- 15 272 3457 || 3473 || 3489 || 3505 || 3521 || 3537 || 3553 3569 || 3584 || 3600 * 273 || 36.j6 || 3632. 3648 || 3664 3630 || 3696 || 3712 3727 | 3743 || 3759 || 3 | 274 || 3775 3791 || 3S07 || 3S23 || 3838 || 3854 || 38.70 || 3886 || 3902 || 3917 : : Žiš.[4%|43942|43%; 439s, 43.96 |4}}|44%. 44.44|44.9 |44. || || 276 || 4091 || 4107 || 4,122 || 4138 || 4154 41.70 || 4185 || 4201 || 4217 | 4232 - 277.| 4318 4264 || 4279 || 4255 || 4311 || 4326|| 4342 4358| 4373 || 4389 || 3 278 I 4404 || 4430 4136 445il 4.467 || 4.483| 4498 || 4514. 4529 || 3545, 9 ! 279 || 4560 4576 4592 4607 || 4623 || 4638 4654 || 4669 ; 4685 4700 { # No. 0 1 2 3 4 5 6 7 8 9 || 9 | 1.4 LoGARITHMs of NUMBERs. - - - - 154. No. 2800 to 3400. Logarithms. 44716 to 53148. No. () 1 2 3 4 5 6 7 || 8 9 16 280 44.716 || 44731 44747 44762 || 44778 || 44793 44809 || 44824 44840 || 44.855 1| 2 281 || 4871 || 4886 49(32 || 4917 || 4932 || 4948 4963 || 4979 || 4994 || 5010 |2| 3 282 5025 5040 5056 5071 5986 || 5102 || 5117 || 5133 || 5148 || 5163 ||3| 5 283 || 51.79 || 5194 i 5209 || 5225 5240 5255 5271 5286 || 5301 || 5317 || 4 || 6 284 5332 5347 || 5362 5378 5393 || 5408 || 5423 5439 || 5454 5469 || 5 || 8 285 |45484 || 45500 |45515 45530 |45545 || 45561 || 45576 || 45591 || 45606 || 45621 || 6 || 10 | 286 5637 || 5652 5667 | 6682 5697 || 5712 5728 5743 || 5758 5773 || 7 || 11 287 5788 || 5803 || 5818 || 5834 5849 || 5864 5879 5894 || 5909 || 5924 || 8 || 13 288 5939 5954|| 5969 5984 || 6000 | 6015 6030 6045| 6060 | 6075||9|| 14 289 || 6990 || 6105 || 6120 | 6135 | 6i50 || 6165 6180 | 6195|| 6210 || 6225 290 |46240 || 46255 || 462.70 || 46285 || 46300 || 46315 || 46330 || 46345 |46359 |46374 291 || 6389 || 6404 || 6419 6434 || 6449 6464 || 64.79 6494 || 6509 || 6523 - 292 || 6538 || 6553 || 6568 6583 || 6598 || 6613 | 6627 | 6642 6657 | 6672 || 15 293 || 6687 | 6702 || 6716 6731 || 6746 6761 || 6776 || 6790 || 6805 | 6820 || 1 || 2 : 294 | 6835 | 6850 | 6864 6S79 | 6894 || 6909 6923 || 6938 || 6953 | 6967 || 2 3 295 || 46982 46997 || 47012 || 47026 47041 47056 470.70 || 47085 || 47100 || 47114 || 3 || 5 | 296 || 7129 || 7144 || 7159 || 7173 || 7188 7202 || 72.17 7232 || 7246 || 7261 || 4 || 6, 297 || 1276 || 7290 || 7805 | 7319 || 7334 || 7349 || 7363 | 7378 || 7392 || 7407 || 5 || 8 298 || 7422 || 7436 || 7451 || 7465 7480 7494 || 7509 || 7524 || 7538 7553 || 6 || 9. 299 || 7567 || 7582 || 7596 || 7611 || 7.625. 7640 || 7654 || 7669 || 7683 || 7698 || 7 || 11 300 47712 47727 |47741 47756 || 47770 || 47784 || 47799 || 47S13 47828 || 47842 || 8 || 12 301 || 7857 || 7S71 || 7885 7900 7914 || 7929 || 7943 || 795S 7972 || 7986 || 9 || 14 302 || 8001 || 8015 || 8029 || 8044 8058 || 8073 || 8087 | 8101 || 8116 8130 303 || 8+44 8159 || 8173 || 8.187 8202 || 8216 || 8230 8244 || 8259 8273 304 || 8287 | 8302 || 83.16 || 8330 || 8344 8359 || 8373 8387 | 8401 | 84.16 305 || 48430 || 484.44 || 48458 || 48473 || 48487 48501 || 48515 48530 4854.4 || 48558 || 14- 306 || 85.72 8586 8601 || 8615 || 8629 || 8643 8657 | 8671 8686 || 8700 || 1 || 1 307 || 8714 8728 8742 | 8756 S770 8785 8799 || 8813 || 8827 | 8841 || 2 || 3 308 || 8855 || 8869 || 8883 || 8897 | 8911 | 8926 | 8940 | 8954 896S | 8982 || 3 || 4 309 || 8996 || 9010 || 9024 9038 9052 9066 || 9080 9094 || 9108 || 9122 |4| 6 310 || 49136 || 49150 || 49164 || 49178 || 4919.2 49206 || 49220 |49234 |49248 || 49262 || 5 || 7 311 || 9276 9290 || 9304 93.18 9332 | 9346 || 9360 93.74 || 9388 6|| 8 312 || 94.15 9429 94.43 94.57 94.71 94.85 || 9499 9513 9527 95.41 || 7 || 10 313 || 95.54 9568 9582 || 9596 || 9610 9624 96.38 || 9651 || 9665 96.79 || 8 || 11 314 || 9693 9707 || 97.21 | 9734 97.48 9762. 9776 || 9790 9803 || 98.7 || 9 || 13 315 # 49831 || 49845 49859 49872 49886 49900 || 49914 || 49927 || 4994.1 || 49.955 316 9969 || 99.82 99.96 50010 || 50024 50037 50051 || 50065 50079 || 50092 317 | 50106 || 50120 H5O133 || 0147 O161 || 0174 || 0188 || 0202 || 0215 || 02:29 318 || 0243 | 0256 || 0270 | 0284 || 0297 || 0311 || 0325 | 0338 || 0352 | 0365 || 13 319 || 0379 || 0393 || 0406 || 0420 || 0433 || 0447 || 0461 | G474 | 0488 || 0501 || 1 || 1 320 |50515 50529 |50542 50556 |50569 50583 |50596 || 50610|50623 50637 ||3| 3 321 || 0651 || 0664 || 0678 || 0691 || 0705 || 0718 || 0732 0745 || 0759 || 0772 || 3 || 4 322 || 0786 0799 || 0813 0826 || 0840 || 0853 0866 || 0880 || 0893 || 0907 |4| 5 | 323 || 0920 | 0934 || 0947 || 0961 || 0974 || 0987 || 1001 || 1014 || 1028 1041 || 5 || 7 324 || 1 106S | 1081 | 1095 || 1108 || 1121 | 1135 | 1148 || 1162 | 1175 || 6 || 8 325 || 51188 51202 || 51215 || 51228 51242 || 51255 || 51268 || 51282 || 51295 || 51308 || 7 || 9 326 || 1322 || 1335 | 1348 || 1362 | 1875 | 1388 || 1402 || 1415 || 1428 1441 || 8 || 10 327 | 1455 1468 || 14S1 || 1495 || 1508 || 1521 | 1534 || 1548 || 1561 | 1574 |9| 12 328 || 1587 | 1601 || 1614 | 1627 | 1640 | 1654 || 1667 | 1680 | 1693 1706 329 1720 || 1733 1746 || 1759 || 1772 || 1786 || 1799 || 1812 || 1825 | 1838 330 |51851 || 51865 |51878 || 51891 |51904 || 51917 |51930 51943 || 51957 51970 | 12 331 || 1983 || 1996 || 2009 || 2022 || 2035 | 2048 2061 2075 | 2088 || 2101|1|T1 332 2114 2127 | 2140 2153| 2166 2179 || 2.192 2205 2218 2231 |2| 2 333 2244 225; 2270 2284 2297 2310 || 2323 2336|| 2349 2362 |3| 4 334 || 2375 2388 || 2401 || 2414| 2427 | 2440 2453 2466 || 2479 2492 |4| 5 335 |52504 || 52517 |52530 52543 |52556 52569||52582 52595 |52608 || 52621 |5|| 6 336 || 2634 2647 2660 2673 2686 2699 || 2711 || 2724| 2787 || 2750 || 6 || 7 337 || 2763 2776|| 2789 2802 || 2815 2827 2840 2853 2866 2879 |7| 8 338 || 2892|| 2905 || 2917 2930| 2943 2956 2969 || 2982] 2994 || 3007 ||3|10 339 || 3020 3033 || 3046 3058 3071 3084 3097 || 3110 || 3122 || 3135 |g| II No. 0 1 2 3 4 5 6 7 8 9 152 No. 3400 to 4000. LOGARITIIMS OF NUMBERS. Logarithms. Log. 53148 to 60206. NO. 340 34] 342 343 344 345 346 347 348 349 350 351 0 53148 3275 3403 3529 3656 537.82 3908 4033 4158 4283 54.407 4531 55023 5145 5267 5388 5509 1 53161 328S 3415 3542 3668 53794 3920 4045 4170 4295 54419 4543 2 53173 3301 3428 3 || 4 5 53186 53199 || 53212 3314 || 3326 || 3339 3441 || 3453 || 3466 * / 53237 3364 3491 3618 3744 5387 () 3995 .4120 4245 4370 54494 4617 4741 4864 4986 55108 5230 5352 5473 5594 55.715 5.835 5955 6074 6.194 56312 6431 6549 666'ſ 8 53250 3377 3504 3631 3757 53882 4008 I ł 8 9i i i º -l T I LOGARITIIMS OF NUMBERS. - 153 No. 4000 to 4600. Logarithms. Log. 60206 to 66276. No. 0 1 2 3 || 4 5 || 6 7 | 8 9 11 400 60206 60217 60228 60239 || 60249 60260 60271 60282 60293 60304 || 1 || 1 401 || 0314 || 0325 || 0336 || 0347 || 0358 0369 0379 || 0390 0401 || 0412 || 2 || 2 402 || 0423 || 0433 || 0444 || 0455 || 0466 0477 || 0487 0498 || 0509 || 0520 || 3 || 3 403 || 0531 0541 || 0552 || 0563 || 0574 || 0584 || 0595 0606 || 0617 | 0627 || 4 || 4 404 || 0638 || 0649 || 0660 || 0670 || 0681 || 0692 || 0703 || 0713 || 0724 || 0735 || 5 || 6 405 |60746 |60756 |60767 60778 |60788 60799 || 60810 | 60821 |60831 60842|| 6 || 7 406 || 0853 || 0863 || 0874 0885 0895 || 0906 || 0917 | 0927 | 0938 0949 || 7 || 8 407 || 0959 || 0970 || 0981 0991 || 1002 || 1013 || 1023 || 1034 || 1045 1055 || 8 || 9 408 || 1066 1077 || 1087 1098 || 1109 || 1119 || 1130 || 1140|| II51 1162 || 9 || 10 409 || 1172 1183 1194 | 1204 || 1215 1225 | 1236 1247 | 1257 | 1268 410 || 61278 61289 || 61300 6131() || 61321 61331 || 61342 | 61352 61363 61374 411 || 1384 1395 || 1405 || 1416 || 1426 1437 1448 1458 || 1469 1479 412 || 1490 1500 || 1511 1521 || 1532 || 1542 || 1553 || 1563 1574 1584 413 ; 1595 || 1606 || 1616 1627 | 1637. 1648 || 165S | 1669 || 1679 || 1690 414 || 1700 1711 || 1721 || 1731 1742 1752 || 1763 || 1773 || 1784 || 1794 415 || 61805 61815 || 61826 61836 || 61847 || 61857 || 61868 61878 61888 || 61899 416 || 1909 || 1920 1930 | 1941 || 1951 | 1962 || 1972 1982 || 1993 || 2003 417| 2014 || 2024 || 2034 2045 2055 2066 2076 2086| 2007 || 2107 41S 2118 2128 || 2138 2149 || 2159 21.70 || 2180 || 2190 2201 2211 424 2137 2147 2757 2701 || 2778 2788 2.98 || 2:08 2818 2829 || 10 433 |62839 62849 |62859 |62879 |62880 62892|62900 62910|62921 | 62931|1}^{ 426|| 2941 2951 2961 | 2012| 2982 2992 || 3002 || 3012| 3022 || 3033 ||3| 3 427 3043 3053 || 3003 || 3973 || 3083 || 3994| 3104 || 3114|| 3124 || 3:4) || 3 428 3144 3153 || 31% | 31|| 3183 || 3 ||3| 3205 || 325|| 3225 || 3339|| || || 429 || 3246 || 3256 || 3:06 .3276 || 3286 || 3296 || 3306 || 3317 | 3327 | 3337 || || 430|6334||6335||6336|| 63311||6338|| | Gº |63407 || 6341||63428 63438|| || 431 || 3448 || 3458 34.8 3478] 3488 3498 || 3:08 || 3318|| 3528 3538|}| } 433 3348 || 3358 || 3368 || 3:19 || 3389 || 3339| 2009 || 36||9|| 3629 || 3639 $| 3 433 3649 || 3059 || 3669 || 3679 || 3689 || 3599 || $199 || 3119 || 3729 || 37.39|| || 434 || 3749 3759 || 3769 3779 || 3789 || 3799 || 3809 || 3819 || 3829 || 3839 - 435 | 63849 63859 63869 || 63879 || 638S9 || 63899 63909 63919 63929 || 63939 436 || 3949 || 3959 3969 3979 || 3.98S 3998 || 4008 4018 4028 || 4038 437 | 4048 4058 || 4068 4078 || 4088 4098 || 4108 || 4118 4128 || 4:137 438 || 4147 || 4157 || 4167 || 4177 || 4187 || 4197 || 4207 || 4217 || 4227 | 42:37 439 || 4246 || 4256 || 4266 4276 || 4286 4296 || 4306 4316 || 4326 || 4835 440 || 64.345 || 64355 || 64365 64375 64385 64395 64.404 64414 || 64,424 || 64434 441 4444 || 44 4464 4473 || 4483 4493 || 4503 || 4513 || 4523 || 4532 442 || 4542 || 4552 || 4562 4572 4582 4591 || 4601 | 4611 || 4621 4631 443 || 4640 || 4650 || 4660 || 46.70 || 4680 || 4689 || 4699 || 4709 || 4719 || 4729 444 || 4738 || 4748 || 4758 || 4768 || 4777 || 4787 || 4797 || 4807 || 4816 || 4826 445 64836 || 64846 || 64.856 64865 || 64875 64885 || 64895 || 64904 || 64914 || 64.92 446 || 4933 || 4943 4953 || 4963 4972 || 4982 || 4992 || 5002 || 5011 || 5021 447 5031 || 5040 || 5050 5060 5070 5079 || 5089 5099 || 5108 || 51.IS 448 || 5128 || 5137 || 5147 || 5157 || 5167 || 5176 || 51.86 || 5196 || 5205 || 52.15 449 5225 || 5234 5244 || 5254 5263 || 5273 || 5283 || 5292 || 5302 || 5312 450 6532] | 65331 || 65341 || 65350 || 65360 || 65369 || 65379 65389 || 65398 || 65408 9 451 || 5418 5427 | 5437 5447 5456 || 5466 || 5475 || 5485| 5495 5504 || 1 || 1 452, 5514 5523 5533 5543 5552 5562 5571 5581 || 5591 5600 |2| 3 453 || 5610 || 5619 || 5629 || 5639 5648 || 5658 || 5667 5677 || 5686 || 5696 || 3| 3 454 || 5706 || 5715 || 5725 | 5734 || 5744 5753 || 5763 5772 || 5782 || 5792 |4| 4 455 || 65801 || 65811 || 65820 65830 || 65839 65849 || 65858 || 65868 65877 | 66887 || 5 || 5 456 || 5896 5906 || 5916 5925 || 5935 | 5944 5954 5963 || 5973 || 5982 || 3 457 || 5992 || 6001 || 6011 || 6020 | 6030 | 6039 || 6049 || 6058 6068 6077 || 7 || 6 458 || 6087 || 6096 || 6106 || 6115 6124 || 6134 6143 || 6153 || 6162 || 6172 ; . 459 || 6181 | 6.191 6200 6210 | 6219 6229 6238 6247| 6257 6266 154 LoGARITHMS OF NUMBERs. No. 4600 to 5200. Logarithms. Log 86276 to 71600. NO. 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 4S2 4S3 484 4S5 486 487 488 4S9 490 491 492 493 494 495 496 497 498 499 500 50L 0 66276 2 66295 3 4 66.304 || 66314. 6398 || 6408 6492 || 6502 65S6 || 6596 66SO | 6689 66773 || 66.783 6867 6876 pse O 66323 6417 6511 6605 6699 66792 6SS5 6978 7071 7164 67256 '7348 '7440 7532 7624 6 66332 7 66342 6436 8 66351 9 66.361 6455 6549 66.42 6736 66829 6922 7015 7108 0321 7(1406 0492 0578 I LOGARITHMS OF NUMBERS. 155 No. 5200 to 5800. Logarithms. Log. 71600 to 76343. No. 0 1 2 3 || 4 5 || 6 7 || 8 9 9 520 T1600 || 71609 || 71617 | 71625 71634 71642 || 71650 | 71659 || 71667 71675 || 1 || 1 521 | 1684 || 1692 || 1700 1709 || 1717 | 1725 || 1734 1742 1750 1759 |2| 2 | 522; 1767 1775 || 1784 1792 1800 | 1809 || 1817 | 1825 | 1834 || 1842|3| 3 523 1850 | 1858 || 1867 | 1875 1883 |. 1892 || 1900 1908 || 1917 | 1925 || 4 || 4 524 1933 || 1941 || 1950 | 1958 1966 1975 | 1983 || 1991 || 1999 || 2008 || 5 || 5 525 | 72016 || 72024 || 72032 | 72041 || 72049 || 72057 | 72066 | 72074 || 72082 | 72090 || 6 || 5 526 2099 || 2107 || 2115 2123| 2132 2140 || 2148 || 2156 2165 2173 || 7 || 6 527 2181 || 2189 || 2198 || 2206 || 2214 2222 2230 2239 2247 2255 || 8 || 7 528 2263 2272 || 2280 2288 || 2296 2304 || 2313 || 2321 || 2329 2337 || 9 || 8 529 2346 || 2354 || 2362 2370 2378 2387 || 239 2403 || 2411 || 2419 530 || 72428 || 72436 72444 72452 || 72460 | 72469: 72477 | 72485 || 72493 || 72501 531 || 2509 || 2518 2526 2534 2542 i 2550 || 2558 || 2567 2575 2583 532 || 2591 || 2599 || 2607 2616 2624 2632 2640 2648 || 2656 2665 533 2673 2681 2689 2697 2705 || 2713 2722 || 2730 || 2738 2746 534 || 2754 || 2762 || 27.70 || 2779 || 27S7 2795 || 2803 || 2811 || 2819 2827 535 | 72835 | 72843 || 72852 72S60 || 72868 || 72876 || 72S84 || 72892|| 72900 | 7290S 536 || 2916 2925 || 2933 2941 || 2949 2957 2965 2973 || 2981 || 2989 637 || 2997 || 3006 || 3014 : 3022 || 3030 || 303S .3046 3054 3062 3070 538 || 3078 || 3086 3094 || 3102 || 3111 || 3119 || 3127 | 3135 || 3143 || 3151 539 || 3159 || 3167 || 3175 3183 3191 || 3199 || 3207 || 3215 || 3223 3231 540 | 73239 73247 73255 73263 || 73272 73280 || 73288 || 73296 || 73304 || 73312 541 || 3320 || 3328 || 3336 || 3344 3352 3360 3368 || 3376 || 33S4 || 3392 542 || 3400 || 3408 || 3416 || 3424 || 3432 | 8440 || 3448 || 3456 || 3464 || 3472 543 || 3480 || 3488 3496 || 3504 || 3512 || 3520 i 3528 || 3536 || 3544 || 3552 544 || 3560 || 3568 || 3576 || 35S4 || 3592 3600 || 3608 || 36.16|| 3624 || 3632 8 545 || 73640 || 7364S | 73656 | 73664 || 73672 | 73679 || 73687 | 73695 || 73703 || 737.11 1 || 1 546 || 3719 3727 i 3735 3743 || 3751 3759 || 3767 3775 || 3783 || 3791 2| 2 547 || 3799 || 3807 || 3815 3823 3830 3838 || 3846 || 3854 || 3862 3870 3 2 548 || 3878 || 3886 || 3894 || 3902 || 3910 3918 || 3926 3933 3941 || 3949 4|| 3 549 || 3957 || 3965 3973 3981 || 3989 || 3997 || 4005 || 4013 || 4020 | 402S 5| 4 550 || 74036 || 74044 || 74052 74060 || 74068 || 74076 || 74084 74092 || 74099 || 74107 6|| 5 551 || 4115 || 4123 || 4131 || 4139 4147 4155 || 4162 || 4170 || 4178 || 4186 T | 6 552 || 4194 4202 || 4210 || 4218 || 4225 4233 || 4241 || 4249 || 4257 4265 8| 6 553 || 4273 || 42SO || 4288 || 4296 || 4304 4312 || 4320 4327 || 4335 | 4343 9| 7 554 || 4351 || 4359 4367 || 437.4 || 4382 || 4390 || 4398 || 4406 || 4414 || 4421 555 || 74429 || 74437 || 74.445 74453 || 74461 || 74468 || 74476 || 74484 || 74492 || 74500 556 || 4507 || 4515 || 4523 4531 4539 || 4547 4554 || 4562 45.70 || 4578 557 || 4586 4593 || 4601 || 4609 || 4617 4624 || 4632 4640 || 4648 || 4656 558 || 4663 || 4671 || 4679 || 4687 || 4695 702 || 471() || 4718 || 4726 4733 559 4741 4749 || 4757 4764 || 4772 || 4780 || 4788 4796 || 4803 || 4811 560 || 74S19 74S27 || 74S34 || 74842 || 74850 | 74S58 74865 74873 || 74881 || 74S89 561 || 4896 || 4904 || 4912 || 4920 || 4927 | 4935 || 4943 || 4950 || 4958 || 4966 562 || 4974 498L || 4989 4997 || 5005 || 5012 5020 5028 || 5035 5043 563 || 5051 5059 || 5066 5074 || 50S2 5089 || 5097 || 5105 || 5113 || 5120 564 5128 5136 || 51.43 || 5151 || 5159 || 5166 5174 || 5182 || 5189 || 5197 565 75205 || 75213 j'75220 75228 || 75236 || 75.243 || 75251 75259 || 75266 || 75274 566 || 5282 |- 5289 || 5297 || 5305 5312 5320 5328 5335 | 5343 5351 567 || 5358 || 5366 5374 || 53S1 || 5389 5397 || 5404 || 54.12 5420 5427 568 || 5435 | 5442 5450 || 5458 || 5465 5473 || 54S1 5488 || 5496 || 5504 569 || 5511 5519 || 5526 5534 5542 5549 5557 5565 5572 || 5580 570 || 75587 75595 || 75603 || 75610 || 75618 75626 75633 75641 || 75648 75656 2. 571 5664 5671 || 5679 || 5686 || 5694 || 5702 || 5709 || 5717 | 5724 || 5732 || 1 | 1 572 5740 || 5747 || 5755 5762 || 5770 || 5778 5785 5793 5800 || 5808 || 2 | 1 573 || 5815 || 5823 || 5831 || 5838 || 5846 5853 || 5861 | 5868 || 5876 5884|3| 2 574 || 5891 5899 || 5906 || 5914 || 5921 5929 || 5937 || 5944 || 5952 | 5959 || 4 3 575 | #5967 || 75974 || 75982 75989 || 75997 || 76005 || 76012 || 76020 || 76027 76035 5 4 576 || 6042 | 6050 | 6057 | 6065 | 6072 | 6080 || 6087 | 6095 6103 || 6110 || 6 4 577 || 6118 || 6125 | 6133 || 6140 6148 || 6155 | 6163 || 6170 || 6178 6185 || 7 || 5 578 || 6.193 || 6200 || 6208 || 6215 6223 6230 || 6238 6245 || 6253 || 6260 |S| 6 579 || 6268 || 6275 | 6283 6290 6298 : 6305 || 6313| 6320 6328 6335||9|| 6 No. 0 1 2 3 || 4 5 | 6 7 8 9 6 8 [ A. 9 g # 9 Ž I 0 Fox II90 #090 S690 I690 |#Sg() 1,190 01.g.0 #990 | 1QCO Ogg.0 689 9; g0 99.90 || 08:30 ££g0 || 9 IGO 6090 z0g0 96#0 6S50 ZS$0 | Sea grifo S9;0 3950 ggiò . 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I 8&ll | LINI : 0 III f{}}L L69 I 0691 999 $89 IS 1.19 US II.9LS #99 IS 1991S IQ9IS frºTS | 199TS : [8.91S fºlS QQQ 1.I9 I } Il 9T | #09 I S69 I I69 I isg| | SºGI | I].g.I #99 I SQGI jjº UQQ [ fºg I Søgt Iggi GZg|I SIGI I IIgſ | Q09L | S6#[ I6 FI £g.9 GSFL | Sºft | Lºſ L goñL | SQFI IGFI gif|L | SęfL ISFI gzi I &Q STFT | IIFI QOf I S69 [ ] [69L | QSQL | Slºt I18N gSQI SQ9I IQ9 IgºlS i gif&IS SQQIS Igg|S| Q&IS | SIQIS II9 IS Q09 IS S6&IS I6&IS 099 9 || 6 |gs&I S1&L | IL&I go&L | SgáI Iggſ | GFGL | S&L | I&L | FCCI 6f) 9 || 8 || SI&L IIGI iOZI S6II | I6II fSII | 81LL | Iºll #9TL | Sg|II Sfº g | | | Igll #II | 181T | ISII #&II IIII I IIII | #0II 160I 060L ºf{} # | 9 |f$0L | 1,10I |010I foot | 1901 || 090L |$for 180L | 080I gCOI |9ts # | g | LIOIS OIOlS £00IS 96608 || 0660S £S60S |9160S 6960S $9608 9960S gig g | f | 6f 60 | SF60 |9860 | 6360 3360 9I60 | 6060 3060 g6S0 6SS0 #9 z | g | ZSSO glSO | SQSO 39S0 |ggSO SFSO | If S0 gSS0 S&SO I&SO ºf 9 I z |#ISO | SOSO IOSO f610 || 1810 | 1810 |fl.10 1910 || 0910 #gt0 | I I lifl, Of 10 # 8810 || 93.10 || 03:10 £ITO 90.20 6690 Q690 9SQ0 Iłº & 61,90S Z190S 99.908 6990S 3990S Sf90S S$90S z890S Qº0S SIQ0 Qis 6 8 9 | g | f | 8 || 3 | I | 0 '0N º • S gº * * 0Igf8 on SI908 So I 'surugiaruso'I ()001, Ol 00:#9 “ON Eºſ 'ssagwn'N Jo SwitHRIvoo'I. 258 * LoGARITIIMs of NUMBERs. No. 7000 to 7600. Logarithms. Log. 84510 to 88081. No. 0 1. 2 3 4 5 6 7 8 : : 9 700 | 84510 | 84516 || 84522 | 84528 |84535 | 8454.1 | 84547 | 84553 |84559 | 84566 | . , 7 701 || 4572 4578 || 4584 || 4590 4597 4603 || 4609 || 4615 || 4621 4628 || 1 | 1 702 || 4634 || 4640 || 4646 4652 || 4658 I 4665 || 4671 4677 46S3 4689 || 2 | 1 703 || 4696 || 4702 || 4708 || 4714 || 4720 || 4726 || 4733 || 4739 || 4745 4751 || 3 || 2 704 || 4757 || 4763 || 4770 4776 || 4782 || 4788 || 4794 || 4800 || 4807 || 4813 || 4 || 3 705 | 84S19 | 84.825 | 84831 | 84837 S4S44 | 84850 | 84S56 | 84862 | 84868 | 84874 || 5 || 4 706 || 4880 || 4887 || 4893 || 4899 || 4905 || 4911 || 4917 | 4924 || 4930 || 4936 || 6 || 4 707 || 4942 4948 I 4954 || 4960 || 4967 || 4973 || 4979 || 4985| 4991 || 4997 || 7 || 5 708 || 5003 || 5009 || 5016 || 5022 || 5028 5034 || 5040 5046|| 5052 5058 || 8 || 6 709 || 5065 5071 || 5077 || 5083 5089 || 5005 || 5101 || 5107 || 5114 || 5120 | 9 || 6 710 85.126 85132 85.138 85.144|85150 | 85.156 85163 || 85169 || 85175 .85181 711 || 51S7 || 5193 || 5199 || 5205 || 5211 || 52.17 | 5224 || 5230 || 5236 || 5242 712 5248 || 5254 || 5260 || 5260 || 5272 5278 || 52S5 5291 || 5297 5303 713 || 5309 || 5315 5321 5327 || 5333 5339 || 5345 5352 5358 5364 714 || 5370 || 5376 || 53S2 || 53SS 5394 || 5400 || 5406 || 54.12 || 5418 || 5425 715 85431 85437 || 85443 || 85.449 |85455 | 85.461 || 85467 85.473 || 85.479 || 85485 716 5491 || 5497 || 5503 || 5509 || 5516 || 5522 || 5528 5534 || 5540 || 5546 717 | 5552 || 5558 || 5564 5570 || 5576 || 5582 || 55SS 5594 || 5600 || 5606 71S 5612 || 5618 5625 5631 || 5637 5643 || 5649 || 5655 5661 5667 719 || 5673 || 5679 || 56S5 || 5691 || 5697 || 5703 || 5709 || 5735 | 5721 || 5727 720 857.33 85739 85745 $5751 S5757 85763 || 85769 || 85775 85.781 8578S 721 || 5794 | 5800 || 5806 || 5812 581S 5824 || 5830 || 5836 || 5842 5848 722 || 5854 5860 || 5866 5872 || 5878 5884 || 5890 5896 || 5902 || 5908 723 || 5914 5920 5926 5932 5938 5944 5950 5956 || 5962 596S 724 || 5974 5980 || 5986 5992 || 5998 || 6004 || 6010 || 6016 || 6022 6028 6 725 |80034 86040 |86046 86052|86058 |86064|86010 | 86016|86082 |800SS| 1 || || 726 6094 | 6100 6106 || 5112 || 3118 G124| 6130 G136|| 6141 6147|| 3 || 1 727 | 6133 6159 || 6165 6171 6177 6183 6189 || 6195 || 6201 || 6207 || 3 || 2 728 6213 6219 || 6225 | 6231 6237 6243 || 6249 6255 6261 6267 || || 2 729 || 6273 || 6279 || 6285 6291 || 6297 6303 || 6308 || 6314|| 6320 | 6326|| 5 || 3 730|86332 |86338|86344 |86350 |86356 |86362|86368 |86374|86380 | 86.386 || || || I31 6392 || 6398 || 6404 || 6410 | 6415 || 6421 | 6427 | 6433 || 0439 6445 || 7 || 1 732 6451 6457 6463 || 6469 || 6415 || 6481 6487 6493 || 6499 || 6504 || 3 || 5 733| 0510 || 65.16 || 6522 652S 6534 6540 || 6546 6552 || 6558 6564 || 3 || || 734 || 65.70 || 6576 || 65SI 65S7 || 6593 6599 || 6605 | 6611 | 6617 | 6623 735 | 86629 || S6635 | 86641 86646 86652 86658 || 86664 || 866.70 || S6676 || 86682 '736 || 6688 || 6694 6700 6705 || 6711 || 67.17 6723 6729 6735 6744 737 || 6747 | 6753 || 6759 6764 || 6770 6776 || 6782 | 6788 6794 | 6800 738 | 6806 | 6812 | 6817 | 6823 | 6829 | 6835 | 6841. | 6847 | 6853 | 6859 739 || 6864 | 68.70 || 6S76 | 68S2 | 6888 | 6894 || 6900 || 6906 || 6911 || 6917 740 || 86923 || 86929 || S6935 | 86941 || 86947 | SG953 || 86958 S6964. |86970 86976 74.1 6982 6988 || 6994 || 6999 || 7005 || 7011 || 70]." | 7023 || 7029 || 7035 742 || 7040 7046 || 7052 || 705S | 7064 || 7070 || 7075 70S1 || 70S7 70.93 743 || 7099 || 7105 || 7111 || 7116 || 7122 || 712S | 7134 || 7140 || 7146 || 7151 744 7157 7163 7169 || 7175 || 7IS1 || 71S6 || 7192 || 7198 || 7204 || 7210 745 87.216 || S7221 S7227 | 87233 S7239 87245 |87251 | 87256 87.262 87268 746 7274 || 7280 || 7286 7291 || 7297 | 7303 || 7309 | 7315 7320 | 7326 747 | 7332 || 73.38 || 7344 || 7349 || 7355 7361 | 7367 || 7373 || 7379 || 7384 748 || 7390 | 7396 || 7402 || 7408 || 7413 || 7419 || 7425 || 7431 || 7437 || 7442 749 || 7448 || 7454 || 7460 7466 || 7471 7477 || 7483 || 7489 || 7495 || 7500 750 |87506 87512 || 87518 87523 87529 || 87535 | ST541 || 87547 || 87552 | S7558 5 751 7564 || 75.70 || 7576 || 7581 || 7587 || 7593 || 7599 || 7604 || 7610 || 7616 || 1 || 1 752 7622 || 7628 || 7633 7639 || 7645. 7651 || 7656 7662 7668 7674 || 2 | 1 753 || 7679 7685 || 7691 || 7697 || 7703 || 770S 7714 || 7720 77.26 7731 || 3 || 2 754 || 7737 7743 || 7749 || 7754 || 7760 7766 7772 7777 7783 7789 || 4 || 2 755 |87795 || 87800 | S7806 || 87812 || 87818 87823 87829 | STS35 | STS41 87846 || 5 || 3 756 7S52 7858 || 7864 || 7869 || 7875 7881 7887 || 7892 7898 || 7904 || 6 || 3 757 iT910 || 7915 || 7921 || 7927 || 7933 7938 - 7944 || 7950 || 7955 7961 || 7 || 4 758 || 7967 7973 || 7978 || 79S4 7990 || 7996 || 8001 || 8007 || 8013 || 8018 || 8 || 4 759 || 8024 8030 || 8036 | 8041 i 8047 8053 8058 8064 8070 8076 || 9 || 5 No. 0 1 2 3 4 5 6. 7 8 9 º LOGARITHMS OF NUMBERs. 159 No. 7600 to 8200. Logarithms. Tog. 88081 to 91381. No. 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 6 '760 |88081 || SS087 88093 88098 || 88104 || 8S110 || 88.116 S8121 || 88127 88133 1 || 1 761 || 8138 8144 || 8150 || 8156 || S161 | S167 8173 8178 || 8184 8190 2 | 1 762 8195 || 8201 || 8207 8213 || 8218 || 8224|| 8230 S235 | 8241 8247 3 || 2 763 || 8252 S258 || 8264 || 8270 S275 82S1 || 8287 8292 || 8298 || 8304 4 2 | 764 || 8309 || 83.15 || 8321 | 8326|| S332 | 8338 8343 S349 || 8355 | S360 5 3 765 S8366 | 88372 |88377 | 88383 8S389 || 88395 || 88400 | 88406 || 88412 || 88417 6 || 4 766 | 8423 | 8429 || 8434 | 8440 | 84.46 | 8451 | 8457 | 8463 | 8468 | 84.74 7 || 4 767 84.80 | S485 | 8491 | 8497 || S502 || 8508 || S513 | S519 || 8525 | 8530 8 || 5 768 || 8536 8542 | 8547 8553 8559 8564 857 8576 S581 8587 9 5 769 8593 || 8598 || 8604 || 8610 || 8615 8621 || 8627 S632| 8638 8643 – 1 – 770 88649 || 88655 |88660 | 88666 || SS672 | 88677 || 88683 || SS6S9 || 88694 | 88700 771 8705 || 8711 || 8717 | 8722 || S728 8734 8739 || S745 || S750 | 8756 772 || 8762 | 8767 | ST73 8779 S784 || 8790 || 8795 || SS01 || 8807 || S812. 773 || 88.18 8824 || 8829 || S835 | 8840 8846 || 8852 || $857 || 8863 || 8868 774 8874 SS80 8885 8891 || 8897 | 8902 || 8908 | 8913 | 8919 | 8925 775 88930 || 88.936 || 8894.1 88947 || 8S953 88958 88964 || 8S969 || SS975 8S981 776 | 89S6 | 8992 | 8997 || 9003 || 9009 || 9014 i 9020 | 9025 | 9031 | 9037 777 9042 9048 9053 | 9059 9064 9070 || 9076 9081 9087 9092 778 || 9098 || 9,104 || 9109 || 9,115 9120 || 9,126 || 9131 || 9137 || 9,143 91.48 779 || 9154 9159 || 9,165 91.70 || 917G | 9182| 91S7 || 9193 || 9198 || 9204 780 | 89209 | 892.15 # 89.221 | S9226 | 89232"| 89237 89.243 | 89248 || 89.254 | 89260 781 9265 9271 9276 92S2 || 92S7 9293 9298 || 9304 || 93.10 | 9315 782 || 9321 | 9326 || 9332 9337 || 9343 || 9348 || 9354 9360 || 9365 | 9371 783 || 93.76 93.82 || 9387 | 9393 || 939S 9404 || 9409 || 9415 || 9421 9.426 784 9432 94.37 || 9443 | 9448 9454 9459 94.65 94.70 || 94.76 9481 785 89.487 | 89492 || S949S | 89504 || S9509 | 89515 | 89520 | 89526 || S9531 | 89537 786 || 9542 | 954S 9553 9559 || 9564 95.70 || 9575 9581 9586 || 9592 787 || 95.97 9603 || 9609 || 9614 || 9620 9625 || 9631 || 9636 || 9642 | 9647 788 9653 9658 || 9664 9669 || 96.75 | 9680 9686 9691 || 96.97 97.02 789 || 9708 || 9713 || 9719 || 9724 || 9730 9735 | 9741 9746 97.52 | 97.57 790 | 89763 | 897.68 || 897.74 | 897.79 |89785 | 89790 | 89796 | 89.801 || 89.807 | 89812 791 || 98.18 98.23 - 98.29 9834 9S40 9845 9851 9856 || 98.62 9867 792 || 98.73 || 987S 9883 || 9889 9S94 | 9900 9905 99.11 || 9916 9922 793 9927 99.33 9938 9944 99.49 9955 9960 | 99.66 || 9971 || 9977 794 99.82 | 9988 9993 || 9998 || 90004 || 90009 || 90015 90020 90026 90031 795 || 90037 90042 || 90048 || 90053 || 90059 90064 90069 || 90075 900S0 | 900S6 796 || 0091 || 0097 || 0102 || 0108 || 01.13 || 0119 || 0124 || 0129 || 01:35 || 0140 797 || 0146 || 015I O157 0162 || 0168 || 0173 || 0179 || 0184|| 0189 Q195 798 || 0200 | 0206 ()211 || 0217 || 0222 || 0227 || 0233 || 0238 || 0244 || 0249 799 || 0255 || 0260 || 0266 || 0271 || 0276 || 02S2 || 0287 || 0293 || 0298 || 0304 800 |90309 || 90314 |90320 | 90325 190331 90336 90342 $)0347 |90352 9035S 801 || 0363 || 0369 || 0374 || 03S0 || 03S5 || 0390 || 0896 || 0401 || (1407 || 0:412 802 || 0417 | 04:23 l ()42S 0434 || 0439 0445 || 0450 0455 0461 0466 803 || 0472 0477 || 0482 0.488 || 0493 || 0499 || 0504 || 0509 || 0515 || 0520 804 || 0526 || 0531 || 0536 || 0542 || 0547 || 0553 || 0558 || 0563 i O569 || 0574 805 |90580 || 90585 |90590 90596 |90601 || 90607 |90612 90617 |90623 90628 5 806 || 0634 || 0639 || 0644 || 0650 || 0655 | 0660 || 0666 || 0671 0677 | 0682 || 1 || 1 807 || 0687 || 0693 || 0698 || 0703 || 0709 || 0714 || 0720 || 0725 || 0730 || 0736|| 2 || 1 808 || 0741 0747 0752 || 0757 || 0763 O76S 0773 ()779 || 07S4 || 07S9 || 3 || 2 809 || 0795 || 0800 || 0806 || 0811 || 0816 (1822 || 0827 | 0832 || 0838 || 0843 || 4 || 2 810 || 90S49 |90854 |90859 |90865 |90S70 | 90S75 |90881 90SS6 || 90S91 || 90897 i 5 3 811 || 0902 || 0907 || 0913 || 0918 || 0924 || 0929 || 0934 || 0940 || 0945 0950 || 6 || 3 812 || 0956 0961 || 0966 || 0972 || 0977 0982 || 0988 || 0993 || 0998 || 1004 || 7 || 4 813 || 1009 || 1014 || 1020 | 1025 || 1030 | 1036 || 1041 IO46 || 1052 1057 | 8 || 4 814 || 1062 | 1068 || 1073 || 1078 1084 1089 || 1094 1100 || 1105 || 1110 || 9 || 5 815||9|1116 91121 191126 91132 91137 91142||91148 91153 |91158 || 91164 816 || 1169 || 1174 1180 | 11S5 || 1190 | 1196 || 1201 || 1206 || 1212 || 1217 817 | 1222 | 1228 || 1233 | 1238 || 1243 | 1249 || 1254 | 1259 | 1265 | 1270 818 || 1275 | 12S1 || 12S6 1291 || 1297 || 1302 || 1307 || 1312|| 1318 || 1323 819 || 1328 1334 || 1339 || 1344 || 1350 | 1355 1360 | 1365 || 1371 || 1376 No. 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 160 LOGARITHMS OF NUMBERS. No. 8200 to 8800. Logarithms. Log.91381 to 94.448. 820 821 822 S23 824 | 825 # 826 827 828 829 : 83C # 831 | 832 | 833 834 S35 S36 J 837 838 839 | 84() 841 842 843 844 | 845 846 847 848 849 | 850 S51 : 852 853 854 855 856 857 | S58 859 860 | 861 862 |NO. O 91381 I 91387 l440 1492 1545 I59S 91651 1703 1756 I808 2 91392 1445 1498 3 || 4 91397 91.403 1450 i 1455 1503 || 1508 1556 1561 1609 || 1614 91661 91666 1714 || 1719 1766 || 1772 1819 || 1824 7 91418 1471 1524 1577 | I630 8 91424 9 91429 1482 1535 1587 1640 91693 1745 1798 1850 1903 l— 91955 2007 2059 2111 2163 92215 LOGARITHMS OF NUMBERS. 161 No. 8800 to 9400. NO, 880 SSI 882 SS3 0 94448 4498 4547 4596 4645 94694 1 9.4453 4503 4552 4601 4650 94699 4748 4797 4846 96853 6900 ||. 039 970S6 7271 94458 Logarithms. Log.94448 to 97.313. 3 || 4 || 5 || 6 7 || 8 9 || 5 94463 |94468 |94473 |94478 |94483|94488 94493 || 1 || 1 4512 || 4517 | 4522 4527 || 4532 4537 4542|| 2 | 1 4562 || 4567 4571 4576 || 4581 || 4586 || 4591 || 3 || 2 4611 || 4616 || 4621 || 4626 || 4630 || 4635 | 4640 || 4 || 2 4660 4665 |. 4670 || 4675 4680 4685 4689 || 5 || 3 94709 |94714 || 947.19 |94724 |94729 |94734 || 94.738 || 6 || 3 4758 || 4763 4768 || 4773 || 4778 || 4783 || 4787 || 7 || 4 4807 || 4812 || 4817 || 4822 || 4827 4832 || 4836 || 8 || 1 4856 4861 || 4866 || 4871 || 4876 || 4880 || 4885 || 0 || 5 4905 || 4910 || 4915 || 4919 || 4924; 4929 || 4934 94954 94959 |94963 |94968 || 94973 |94978 || 94983 5002 || 5007 || 5012 || 5017 5022 || 5027 | 5032 5051 || 5056 || 5061 5066 || 5071 || 5075 5080 5100 || 5105 || 5109 || 5114 || 5119 || 5124 || 5129 5148 || 5153 5158 || 5163 || 5168 || 5173 5177 95.197 || 95.202 || 95207 || 952.11 952.16||95221 | 95.226 5245 5250 5255 || 5260 5265 || 52.70 || 5274 5294 || 5299 || 5303 || 5308. 5313 || 5318 5323 5342 5347 5352 || 5357 5361 || 5366 5371 5390 || 5395 || 5400 || 5405 5410 5415 || 5419 95.439 |95444 |95448 || 95.453 | 95.458 |95463 | 95468 54S7 || 5492 || 5497 || 5501 || 5506 5511 || 5516 5535 | 5540 5545 || 5550 | 5554 || 5559 || 5564 5583 55SS 5593 || 5598 || 5602 || 5607 |. 5612 5631 5636 || 5641 5646 5650 || 5655 5660 95.679 |95684 || 956S9 95694 || 95698 |95703 || 95708 5727 || 5732 5737 || 5742 5746 5751 5756 5775 5780 [ 5785 || 5789 || 5794 5799 || 5804 5823 5828 || 5832 || 5837 5842 5847 5852 5S71 5875 5880 5885 5890 5895 || 5899 95918 95923 || 95928 |95933 || 95938 95942 95947 5966 5971 5976 5980 || 5985 5990 || 5995 6014 || 6019 || 6023 6028 || 6033 || 6038 6042 6061 || 6066 6071 || 6076 || 6080 || 6085 6090 6109 || 61.14 || 6LIS 6123 6128 || 6133 || 6137 96.156|96161 96.166|96171 96.175||96180 | 96.185 G204 || 6209 || 6213 || 6218 || 6223 6227 | 6232 6251 6256 6261 || 6265 62.70 || 6275 6280 6298 || 6303 || 6308 || 6313 || 6317 | 6322 6327 6346 || 6350 | 6355 || 6360 || 6365 || 6369 || 6374 96.393 || 96.398 || 96402 || 96407 || 96412 96417 | 96421 6440 || 6445 6450 || 6454 6459 || 6464 || 6468 6487 6492 6497 || 6501 6506 || 6511 6515 6534 6539 || 6544 || 6548 || 6553 6558 6562 65S1 || 6586 6591 || 6595 | 6600 6605 || 6609 96628 96.633 || 96638 96642 || 06647 || 96652 96656 4. 6675 | 6680 | 6685 | 6689 | 6694 | 6699 || 6703 || 1 || 0 6722 || 6727 | 6731 6736 || 6741 6745 675() || 2 || 1 6769 || 6774 || 6778 || 6783 || 6788 || 6792 || 6797 || 3 || 1 6816 || 6820 | 6825 | 6830 6834 6839 6S44 || 4 || 2 96S62 96867 96.872 96S76 96.881 || 96S86 || 96.890 || 5 || 2 6909 || 6914 6918 || 6923 || 6928 || 6932 6937 || 6 || 2 6956 || 6960 6965 6970 6974 || 6979 || 69S4 || 7 || 3 7002 || 7007 || 7011 || 7016 || 7021 || 7025 7030 || 8 || 3 '7049 || 7053 || 7058 || 7063 || 7067 || 7072 '7077 || 9 || 4 97.095 || 97100 97104 || 97109 || 97114 || 97,118 97.123 'WI42 || 7146 715I T155 || 7160 7165 || 7I69 7188 7.192 7197 || 7202 || 7206 || 7211 || 7216 7234 7239 || 7243 || 724S 7253 || 7257 | 7262 72SO | 72S5 | 7290 || 7294 | 7299 || 7304 7308 3 4 5 | 6 || 7 || 8 9 162 LOGARITHMS of NUMBERS. No. 9400 to 10000. . Logarithms. Log.97.313 to 99996. Woj_0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 8 940 97.313 | 97.317 | 97.322 || 97.327 | 97331 97336 97340 97345 97350 97354 1 || 1 941 || 7359 7364 || 7368 || 7373 || 7377 | 7382 | 7387 | 7391 || 7396 || 7400 2 || 1 , 942 7405 || 7410 || 7414 || 7419 || 7-124 742S | 7433 7437 || 7442 || 7447 3 2 943 || 7451 7456 || 7460 | 7465| 7470 7474| 7479 || 7483| 7488 7493 || 4 || 3 944 || 7497 || 7502 || 7506 || 7511 || 7516 || 7520 || 7525 || 7529 || 7534 7539 5 3 945 || 97543 || 97.548 |97552 | 97557 97.562 || 97566 |97571 97575 |97580 || 97585 6 || 3 946 || 7589 7594 || 7598 || 7603 || 7607 || 7612 || 7617 7621 7626 7630 7 || 4 947 || 7635 7640 || 7644 || 7649 || 7653 || 7658 7663 || 7667 || 7672 || 7676 8 || 4 948 7681 7685 || 7690 7695 || 7699 || 77041 7708 || 7713 || 7717 | 7722 9 || 5 949 || 7727 | 7731 || 7736 || 7740 || 7745 7749 || 7754 || 7759 || 7763 || 7768 950 97.772 | 97,777 97.782 |977S6 || 97.791; 97.795 || 97800 || 97804 || 97809 || 97813 951 7818 7823| 7827 | 7832 || 7836 || 7841 || 7845 || 7850 || 7855 7859 952 7864 7868 || 7873 7877 7882 7886 7891 || 7896 || 7900 || 7905 953 || 7909 || 7914 || 7918 7.923 || 7928 7932 || 7937 || 7941 || 7946 7950 954 || 7955 || 7959 7964 7968 || 7973 || 7978 || 7982 7987 || 7991 7996 955 98000 | 98005 || 98009 || 98014 || 98019 |98023 || 98.028 |98032 9SO37 98041 956 || 8046 || 8050 || 8055 8059 || 8064 || S068 || 8073 || 8078 || 8082 8087 957 8091 8096 || 8100 8105 || 8109 81.14 || 8118 8123 8127 | 8132 958 || 8137 || 8141 i 8146 8150 | 8155 8159 8164 || 8168 || 8173 8.177 959 || 8182 | 81.86 8191 8195 || 8200 | 8204 || 8209 || 82.14 || 8218. 8223 960 |98227 | 98.232 || 98.236 |98241 |98245 || 98250 | 98254 98259 |98263 || 98.268 961 8272 | 8277 8281 8286 S290 8295 || 8299 || 83 8308 8313 962 8318 8322 || 8327 | 8331 || 8336 || 834() || S345 8349 || 8354 || 8358 963 || 8363 || 8367 || 8372 8376 || 8381 8385 || 839() $394 || 8399 || 8403 964 8408 || 84.12 | 8417 | 8421 | 8426 | 8430 | 8435 | S439 || 8 84.48 965 || 98453 98.457 || 9S462 | 98466 || 98471 98475 9S480 | 9S484 |98489 |98493 966 | 8498 || 8502 || 8507 || 8511 || 8516 || 8520 | 8525 | 8529 || 8534 8538 967 || 8543 | 8547 || 8552 | 8556 || 8561 i 8505 || 8570 8574 || 8579 8583 968 || 8588 || 8592 || 8597 || 8601 || 8605 || 86.10 || 86.14 || 8619 || 8623 || 8628 969 || S632 || 8637 || 8641 || 8646 || 8650 | 8655 || S659 || S664 || 8668 || 8673 970 |98677 || 98682 98686 98.691 9S695 || 98.700 98.704 || 98.709 |98713 || 987.17 971 || 8722 || 8726 || 8731 || S735 | 8740 || 8744 || 8749 || 8753 || 875S | 8762 972 8767 | 8771 8776 8780 || 8784 || 8789 || 8793 8798 || S802 || 8807 973 || 88.11 || 8816 || 8820 8825 | 8829 8834 || 8838 || 8S43 || 8847 || 8851 974 || 8856 8S6() || 8865 8869 || 88.74 || 8S78 || 8883 || 8887 || 8892 || 8896 975 |989.00 | 98.905 || 98.909 || 98.914 |98918 98923 || 98927 | 98.932 || 98.936 98.941 976 | 8945 | 8949 | 8954 | 8958 || 8963 | 8967 | 8972 | 8976 || 89S1 | 8985 977 | 8989 || 8994 | 8998 || 9003 || 9007 || 9012 || 9010 || 9021 || 9025 9029 978 || 9034 || 9038 9043 9047 9052 9056 || 9061 || 9065 9069 || 9074 979 || 9078 || 9083 || 9087 | 9092 || 9096 || 9100 9105 || 9,109 || 9,114 || 91.18 980 99.123 99127 99131 || 99136 99.140 || 99.145 || 99149 || 99.154 |99158 99.162 981 || 9,167 || 91.71 || 9,176 || 9,180 || 9185 9189 || 9,193 9198 || 9.202 || 0207 982 92.11 || 92.16 || 9220 92.24 || 92.29 9233 || 9238 9242 || 924.7 | 9251 983 || 9255 9260 || 9264 || 9269 || 9273 || 927.7 || 92S2 || 9286 || 9291 9295 984 || 9300 || 9304 || 9308 || 9313||| 93.17 | 9322 || 9326 9330 || 93.35 | 9339 985 |99344 99.348 || 99352 99357 |99361 |99366 199370 99.374 |993.79 |99383 4. 986 || 9388 || 9392 || 9396 || 9401 || 9405 || 9410 94.14 94.19 || 9423 9427 || 1 || 0 987 9432 || 9436 || 94.41 9445 9449 || 9454 || 9458 || 9463 || 9467, 947 l. 2 || 1 988 || 9476 94.80 9484 || 9489 || 9493 9498 || 9502 || 9506 || 9511 || 9515 || 3 || 1 989 || 9520 | 95.24 9528 9533 || 9537 || 9542 9546 9550 | 9555 | 9559 || 4 || 2 990 |99564 99568|995.72 99.577 199581 | 99585 |99590 |99594 |99599 || 90603 || 5 || 2 991 || 9607 || 96.12 96.16 || 9621 9625 | 9629 || 9634 || 96.38 9642 | 9647 || 6 || 2 . 992 || 9651 | 9656 9660 9664. 9669 9673 || 9677 9682| 9686 9691 || 7 || 3 993 || 9695 9699 || 9704 || 9708 || 97.12 97.17 | 97.21 97.26 9730 | 9734 || 8 || 3 994 || 97.39 9743 || 97.47 97.52 || 9756 9760 || 9765 97.69 || 97.74 || 97.78 || 9 || 4 995 |99782 99.787 |99791 |997.95 |99800 99804|99808 99813|998.17 | 99.822. 996 || 98.26 98.30 || 9835 | 9839 || 98.43 9848 || 9852 9856 || 9861 | 9865 997 || 98.70 || 9874 || 98.78 9883 || 98.87 9891 || 9896 || 9900 9904 || 99.09 998 || 9913 99.17 | 9922 9926 9930 9935 99.39 9944 9948 9952 999 || 9957 9961 9965 99.70 || 99.74 997S 9983 9987 99.91 9996 No. 0 | 1 || 2 | 3 || 4 || 5 || 6 || 7 || 8 || 9 LoGARITHMS TRIGONOMETRIC. 163 Oh 0°. Logarithms. 179°111.h M.S. M Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. l'Aſ. S. 00 || 0 || Inf.Neg. Infinite. Inf-Neg. Infinite. 10.00000 || 10.00000 || 60 | 60 4 || 1 || 6.46373 13.53627 | 6,46373 || 13.53627 00000 00000 || 59 || 56 8 2 76476 23524 '76476 23524 00000 00000 || 58 52 12 3 94085 05915 94085 05915 00000 00000 || 57 || 48 16 4 || 7.06579 || 12.93421 || 7,06579 || 12.93421 00000 00000 || 56 || 44 20 5 || 7.16270 | 12,83730 || 7.16270 | 12.83730 || 10.00000 || 10.00000 || 55 | 40 24 6 || 24.188 75812 24188 75812 00000 00000 || 54 || 36 28 7 30882 69118 30882 69118 00000 00000 || 53 || 32 32 S 36682 63318 36682 63318 00000 00000 || 52 || 2 30 9 || 41797 58.203 4.1797 58203 00000 00000 || 51 || 24 40 10 || 7.46373 12,53627 || 7.46373 || 12,53627 | 10.00000 || 10.00000 || 50 20 44 || 11 50512 49488 50512 49488 00000 00000 || 49 || 16 48 || 12 54291 45709 54.291 45709 00000 00000 || 48 || 12 52 || 13 57767 | \ 42233 57767 42233 00000 00000 || 47 8 56 || 14 60985 39015 6098 39014 00000 00000 || 46 || 4 1 || 15 7.63982 | 12,360.18 || 7.63982 | 12,360.18 10.00000 || 10.00000 || 45 || 59 4 || 16 66784 33216 66785 33215 00000 00000 || 44 56 8 || 17 694.17 30583 69418 30582 00001 9.99999 || 43 || 52 I2 | 18 '71900 2S100 71900 2S100 00001 99999 || 42 || 48 16 || 19 '74248 25752 742.48 25752 00001 99999 || 41 || 44 20 | 20 || 7.764.75, 12,23525 || 7,76476 | 12.23524 || 10.00001 9.99999 || 40 || 40 24 || 21 78594 21406 78595 21405 ()0001 99999 || 39 || 36 28 || 22 80615 19385 80615 193S5 00001 99999 || 3S 32 32 || 23 82545 17455 S2546 17454 00001 99.999 || 37 || 28 36 || 24 84393 15607 84394 15606 ()0001 99999 || 36 24 40 || 25 || 7.86.166 | 12:13834 || 7.86167 || 12.13833 || 10.00001 9.99999 || 35 | 20 44 || 26 87870 12130 87871 121:29 00001 99999 || 34 || 16 48 || 27 89509 10491 | 895.10 10490 00001 99999 || 33 || 12 52 || 28 91088 08912 || 91089 08911 00001 99999 || 32 || 8 56 || 29 92612 07388 926.13 07:387 00002 99.998 || 31 || 4 2 || 30 || 7.04084 || 12.05916 || 7.940.86 12,05914 || 10.00002 9.99998 || 30 58 4 i 31 95508 04:492 95510 04:490 00002 99.998 || 29 || 56 8 32 96.887 03113 96S89 03111 00002 99.998 || 2S 52 12 || 33 98223 O1777 9S225 O1775 00002 99.998 || 27 || 48 16 || 34 99.520 0.0480 99522 00478 00002 99.99$ 20 || 44 20 || 35 | 8.00779 || 11.99221 i 8.00781 || 11.992.19 || 10.00002 9.99998 || 25 40 24 || 36 02002 97998 || 02004 97.996 00002 99.998 || 24 || 36 28 || 37 03.192 96.808 03.194 96.806 00003 99.997 23 || 32 32 || 38 {}4350 95650 04:353 95647 00003 99.997 || 22 || 28 36 || 39 05478 94522 054S1 94519 00003 99.997 || 21 || 24 40 | 40 S.06578 || 11.93422 || 8,06581 11.93419 || 10.00003 9.99997 || 20 | 20 44 || 41 07650 92350 || 07653 92347 00003 99.997 || 19 || 16 48 42 0S696 91.304 || 0S700 91.300 00003 99.997 || 18 || 12 52 : 43 097.18 902S2 09722 9027S 00003 99.997 || 17 8 56 44 10717 892S3 10720 89280 00004 99.996 || 16 || 4 3 || 45 8.11693 11.88307 || S.11096 || 11.88304 || 10.00004 9.99996 || 15 57 4 || 46 12647 87353 12651 87349 00004 99.996 || 14 || 56 8 || 47 I3581 86419 I3585 864.15 00004 99.996 || 13 || 52 12 || 48 14495 85.505 14500 85.500 00004 99996 || 12 || 48 16 49 15391 84609 || 15395 S4605 00004 99.996 || 11 || 44 20 || 50 || 8.1626S 11,83732 || 8.16273 || 11.837.27 | 10.00005 9.99995 || 10 | 40 24 || 51 17128 82872 17133 82867 00005 99.995 9 : 36 28 52 17971 82029 17976 82024 00005 99.995 8 || 32 32 53 18798 8.1202 || 18804 S1196 00005 99.995 7 || 28 36 54 19610 80390 19616 80384 00005 99.995 6 || 24 40 || 55 8.20407 || 11.79593 8.20413 11.79587 || 10.00006 9.99994 || 5 || 20 44 || 56 21189 78S11 || 21195 78805 00006 99.994 || 4 || 16 48 57 21958 78042 21964 78036 00006 99.994 || 3 || 12 52 58 22713 77287 22720 '77280 00006 99.994 || 2 8 56 || 59 23456 76544 23462 76.53S 00006 99.994 || 1 4 4 || 60 241S6 75814 || 24.192 75808 00007 99.993 0 || 56 . M. S. M I Cosino. Secant. Cotangent Tangent. I Cosecant. Sine. M. M. S. 6h 1909 S9°| 5h 164 LOGARITHMS TRIGONOMETRIC. On 11° Logarithms. 1789111.h M.S. M Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. M. S. 4. 0 | 8.24186 | 11.75S14 || 8.24.192 || 11.75808 || 10.00007 9.99993 || 60 || 56 4 l 24903 '75097 24910 75090 00007 99.993 || 59 || 56 8 2 25609 '74391 25616 74384 00007 99.993 || 58 52 12 3 26304 73096 26312 '73688 00007 99.993 || 57 || 48 16 4 26988 '73012 26996 73004 00008 99992 || 56 || 44 20 5 || 8.27661 11.72339 8.27669 II.7.2331 || 10.00008 9.99992 || 55 | 40 24 6 2S324 71676 28332 71668 00008 99.992 || 54 || 36 28 7 28977 '71023 289S6 71014 00008 99992 || 53 || 32 32 8 2.9621 70379 29629 70371 0000S 99992 || 52 28 36 9 30255 697.45 30263 69737 00009 99.991 || 51 24 40 || 10 || 8.30879 11.69121 8.308S8 11.69112 || 10.00009 9.99991 || 50 20 44 || 11 31495 6S505 31505 68495 00009 99.991 || 49 || 16 48 || 12 32103 67897 32112 67SSS 00010 99990 || 48 || 12 52 || 13 32702 67298 32711 67289 00010 99990 47 8 56 || 14 33292 66708 33302 66698 00010 99990 || 46 4 5 15 8.33S75 11.66125 || 8.33886 11.661.14 || 10.00010 || 9.99990 || 45 || 55 4 i 16 34450 65550 34461 65539 00011 99.989 44 || 56 8 : 17 35018 (34982 35029 . G4971 00011 99.989 || 43 52 12 18 35578 64422 35590 64.410 00011 99.989 || 42 || 48 16 || 19 36131 63S69 36143 63857 00011 99.989 || 41 || 44 20 || 20 || 8.36678 11.63322 || 8.366S9 11,63311 || 10.00012 || 9.99988 || 40 || 40 24 21 37217 62.783 37229 62771 00012 99988 || 39 || 36 28 22 37750 62250 37762 62.238 00012 99988 || 3S 32 32 || 23 38276 G1724 38.289 61711 00013 99.987 || 37 || 28 36 24 387.96 61204 38809 61.191 00013 999S7 || 36 || 24 40 25 || 8.39310 11.60690 || 8.39323 II.60677 || 10.00013 9.9998T | 35 ; 20 44 || 26 39S18 60.182 39S32 60168 00014 99986 || 34 || 16 48 || 27 40320 59680 40334 59666 00014 99986 || 33 || 12 52 28 40816 5918.4 40830 59170 00014 . 99986 || 32 8 56 29 41307 58693 41321 586.79 00015 99.985 || 31 4 6 30 8,41792 11.58208 || 8.41807 II.5.8.193 10.00015 9.99985 || 30 || 54. 4 : 31 42272 5772S 4.2287 57713 00015 99.985 || 29 || 56 8 || 32 42746 57254 42762 57.238 00016 99.984 28 || 52 12 || 33 43216 56784 43232 56768 00016 99.984 || 27 || 48 16 || 34 436SO 56320 || 43696 56304 00016 99984 || 26 || 44 20 || 35 | 8.44139 11,55861 || 8,44156 11.55844 || 10.00017 9.99983 || 25 | 40 24 || 36 44594 55406 44611 55389 00017 99983 || 24 || 36 2S 37 45044 54956 45061 54939 00017 99983 || 23 || 32 32 || 38 45489 b4511 45507 54493 00018 99.982 || 22 || 28 36 || 39 45930 54070 45948 54052 00018 99.982 || 21 || 24 40 40 8.46366 11.53634 || 8.46385 11,53615 || 10.00018 9,999.82 || 20 20 44 || 41 46799 5320.1 46817 53183 00019 99981 || 19 || 16 48 || 42 47226 52.774 47245 52755 00019 99981 | 18 || 12 52 || 43 47650 52350 47669 52331 00019 99981 || 17 8 56 || 44 48069 51931 48089 51911 00020 99.980 16 4 7 || 45 8.4S485 11.51515 S.48505 || 11.51495 || 10.00020 9.99980 15 || 53 4 || 46 48S96 51104 48917 51083 00021 99.979 || 14 || 56 8 || 47 49304 50696 49325 50675 00021 99.979 13 || 52 12 || 48 49708 50292 497.29 50271 00021 99.979 || 12 || 48 16 || 49 501()8 49.892 501.30 49870 00022 99.978 || 11 || 44 20 || 50 8.50504 II.49496 || 8.50527 II.49473 || 10.00022 9.99978 || 10 || 40 24 || 51 50897 49.103 50920 490.80 00023 99977 9 || 3 28 52 51287 48713 51310 48690 00023 99977 8 || 32 32 53 51673 4S327 51696 48304 00023 99.977 7 || 28 36 || 54 52055 47945 52079 47921 00024 99976 6 24 40 || 55 8,52434 11,47566 S.52459 || 11.47541 || 10.00024 9.99976 5 20 44 56 52S10 4.7.190 52835 47165 00025 99.975 4 16 48 || 57 53183 40817 53208 46792 00025 99.975 3 || 12 52 58 53552 46448 53578 46422 00026 99.974 2 8 56 59 53919 46081 539.45 46055 00026 99.974 l 4 8 || 60 54282 45718 54308 45692 00026 99.974 || 0 || 52 M. S. M Cosine. Secant. iCotangent Tangent. || Cosecant. Sine. M. M. S. 6h 1910 88° 5h LoGARITHM's TRIGONoMETRIC. Oh 20 Logarithms. 1779|11h M.S. M ; Sine. Cosecant. | Tangent. || Cotangent. Secant. Cosime. J. M. M. S. 8 0 || 8.54282 11.45718 || 8.54308 11.45602 || 10.00026 9,999.74 60 52 4. 1. 54.642 45358 54669 45331 00027 99.973 || 59 || 56 8 2 54999 45001 55027 44973 00027 99.973 || 58 || 52 12 3 55354 44646 || 55382 44618 00028 99.972 || 57 || 48 16 4. 55705 44.295 || 55734 44266 00028 99.972 56 || 44 20 5 || 8,56054 || 11,43946 || 8,56083 11.43917 | 10.00029 || 9.99971 || 55 40 24 ... 6 56400 43600 56429 435.71 00029 99.971 || 54 36 28 7 56743 4.3257 56773 43227 00030 99.970 || 53 32 32 8 57084 42916 57114 42886 00030 99.970 52 || 28 36 9 || 57421 42579 57452 42548 00031 99969 51 || 24 40 || 10 || 8.57757 11,42243 8.57788 11.42212 10.00031 9.99969 50 20 44 || 11 58089 | \, 41911 58121 41879 00032 99.968 || 49 || 16 48 || 12 584.19 41583. 58451 41549 00032 99.968 I 48 || 12 52 || 13 58747 41253 58779 41221 00033 99967 || 47 8 56 14 590.72 40928 59105 40895 00033 99967 || 46 4 9 || 15 || 8.59395 11.40605 || 8,59428 11.40572 || 10.00033 9.99967 45 || 51 4 16 59715 40285 59749 40251 O0034 99966 || 44 56 8 || 17 60033 39967 60068 39932 00034 99966 || 43 || 52 12 18 60349 39651 60384 39616 00035 99965 42 || 48 16 19 60662. 39338 60698 39.302 00036 99964 41 || 44 20 || 20 || 8.60973 11.39027 || 8.61009 11.38991 || 10.00036 9.99964 40 || 40 24 || 21 61282 38718 61319 38681 00037 99963 || 39 || 36 2S 22 61589 384.11 61626 38374 00037 99963 || 38 || 32 32 || 23 61894 3S106 61931 38069 00038 99962 || 37 || 28 36 || 24 621.96 37804 62234 37766 00038 99962 || 36 || 24 40 || 25 || 8.62497 11.37503 || S.62535 11.37465 || 10.00039 9.9996.1 || 35 | 20 44 || 26 62795 37205 62834 37166 00039 99961 || 34 || 16 48 || 27 63091 36909 63131 36869 00040 99960 || 33 12 52 28 63385 366.15 63426 36574 00040 99960 32 8 56 || 29 63678 36322 63718 36282 0004:1 99959 || 31 4 10 || 30 || 8.63968 11.36032 || 8.64009 11.35991 || 10.0004:1 9.99959 || 30 || 50 4 || 31 64256 35744 64298 35'702 00042 99958 29 || 56 8 || 32 64543 35457 6.4585 35415 00042 99958 28 || 52 12 33 64827 35.173 64870 35.130 000:43 99.957 || 27 || 48 16 || 34 651.10 34890 65154 34S46 00044 99.956 26 || 44 20 || 35 | 8.65391 11.34609 || 8,65435 11.34565 10.00044 9.99956 25 || 40 24 || 36 65670 34330 65715 34285 00045 99955 || 24 36 28 || 37 65947 34053 G5993 34007 00045 99955 || 23 32 32 || 38 66223 337.77 66269 33731 00046 99.954 22 28 36 || 39 66497 33503 66543 33457 000:46 99.954 21 24 40 40 || 8.66769 11.33231 || 8.66816 || 11,33184 || 10.00047 9.99953 20 20 44 || 41 67039 32961 67087 32.913 00048 99.952 | 19 16 48 I 42 67308 32692 67356 32644 00048 99.952 || IS 12 52 || 43 67575 32425 67624 32376 00049 99951 || 17 S 56 || 44 6784.1 32.159 67890 32110 00049 99951 || 16 4 11 || 45 8.68104 || 11.31896 || 8.68154 || 11.31S46 || 10.00050 9.99950 || 15 || 49 4 || 46 68367 3.1633 6S417 31583 00051 99949 || 14 || 56 8 || 47 68627 31373 68678 31322 00051 99949 || 13 || 52 12 || 48 68886 31114 68938 31062 00052 99948 || 12 || 48 16 || 49 69144 30856 69.196 30804 00052 99948 11 || 44 20 || 50 || 8.69400 || 11.30600. 8.69453 11.30547 || 10.00053 9.9994.7 10 40 24 || 51 69654 30346 69708 30292 00054 99.946 9 || 36 28 52 69907 300.93 69962 30038 00054 99.946 8 || 32 32 || 53 TO159 2984.1 70214 297 S6 00055 99.945 'I 28 36 || 54 70409 29591 70465 295.35 00056 99.944 6 || 24 40 || 55 || 8.70658 II.29342 8.70714 || II.29286 || 10.0005 9.99944 5 || 20 44 56 70905 29.095 '70962 29038 00057 99.943 4 || 16 48 || 57 71151 28849 71208 2S792 0005S 99942 3 || 12 52 58 '71395 28605 71453 285.47 00058 99942 2 8 56 || 59 71638 28362 71697 28303 00059 99.941 l 4 12 || 60 71880 281.20 71940 28060 00000 99.940 0 48 M. S. M_i Cosine. Secant. "Cotangentl Tangent. Cosecant, Sine. M M. S. 6h 1929 S79 5h LoGARITHMS TRIGONOMETRIC. Oh 39 Logarithms. 1769 |11h M. S. M. Sine. Cosecant, I Tangent. | Cotangent. Secant. Cosine. M. M. S. 12 || 0 || 8,71880 11.28120 || 8,71940 11,28060 || 10.00060 9.99940 || 60 || 48 4 1. 72120 27SSO '72181 27819 00000 999.40 || 59 || 56 8 2 72359 27641 72420 27580 • 00061 99.939 58 || 52 12 3 72597 27.403 72659 27341 00062 99938 || 57 || 48 16. 4 72S34 27166 '72S96 27104 00062 99.93S | 56 || 44 20 5 || S.73069 11.26931 8.73132 11.26868 || 10.00063 9.999.37 || 55 40 24 6 73303 26697 73366 26634 ()0064. 99936 54 36 28 7 '73535 26465 73600 26400 00064. 99936 || 53 || 32 32 S 737.67 26233 '73S32 26168 00065 99.935 || 52 || 28 36 9 '73997 26003 74063 25937 00066 99.934 || 51 || 24 40 || 10 || 8.74226 11.257'74 || 8,74292 || 11.25708 || 10.00066 9.99934 || 50 || 20 44 || 11 74.454 25546 74521 25479 00067 99933 || 49 16 48 || 12 74680 25320 '74748 25252 •00068 99932 || 48 12 52 || 13 74906 25094. 74974 25026 00008 99932 || 47 8 56 || 14 '751.30 24870 75199 24801 00069 99.931 || 46 4 13 || 15 8.75353 11,24647 || 8.75423 || 11.24577 || 10.00070 9.99930 || 45 || 4-7 4 16 75575 24.425 75645 24355 00071 99929 || 44 || 56 8 17 75795 24.205 75867 24.133 00071 99929 || 43 52 12 18 76015 23985 '760ST 23.913 00072 99928 42 || 48 16 19 76234 23766 76306 23694 0007.3 99927 41 || 44 20 i 20 || 8.76451 II.23549 || 8.76525 11.23.475 10.00074 9.99926 || 40 || 40 24 || 21 76667 23333 76742 23258 00074 99926 39 || 36 2S 22 76SS3 23117 76958 23042 00075 99925 || 3S 32 32 || 23 77097 22903 77173 22S27 00076 99924 || 37 28 36 24 '77310 22600 77387 22613 00077 99.923 || 36 24 40 || 25 || 8.77522 II.22478 S.77 600 || II.22400 || 10.00077 9.99923 || 35 20 44 || 26 '77733 22267 77S11 221S9 00078 99922 || 34 || 16 4S 27 77943 22057 '78022 21978 00079 9992.1 |33 || 12 52 || 2S 7S152 21848 7S232 21768 00080 99920 || 32 8 55 29 78.360 21640 78441 21559 00080 99920 i 31 4 I4 || 30 || 8.78568 11.21432 || 8.78649 11.21351 || 10.000S1 9,999.19 || 30 || 4-6, 4 || 31 78774 21226 78.855 21145 000S2 99.918 29 || 56 8 32 '78979 21021 79001 20939 000S3 99.917 | 28 52 12 || 33 '79183 20817 79.266 20734 00083 99.917 | 27 || 48 16 || 34 793S6 20614 '79470 20530 00084 99916 || 26 || 44 20 || 35 | 8.795SS 11.20412 || 8.79673 11,20327 || 10.00085 9,999.15 # 25 | 40 24 || 36 79789 2021.I 79875 2012.5 00086 99.914 || 24 || 36 28 || 37 7.9990 20010 800'76 19924 000S7 99.913 || 23 32 32 || 38 80189 19811 S0277 19723 000ST 99.913 22 || 2S 36 || 39 80388 I9612 804.76 19524 00088. 99.912 21 || 24 40 40. 8.80585 11.19415 || 8,80674 || 11,19326 10.000S9 9.99911 20 20 44 || 41 80782 19218 80872 1912S 00090 99.910 19 16 48 || 42 80978 19022 81068 18932 00091 99909 || 18 12 52 43 81173 18827 81264 18736 00091 99.909 || 17 8 56 || 44 81367 18633 81.459 18541 00092 99908 || 16 4 15 45 || 8.81560 11.18440 || 8.81653 11.18347 || 10.00093 9.99907 || 15 || 4-5 4 || 46 S1752 18248 S1846 18154 00094. ‘99906 || 14 56 8 || 47 81944. 18056 82038 17962 00095 99905 || 13 || 52 J2 || 48 82.134. 17S66 82230 17770 00006 99904 || 12 || 48 16 || 49 82324. 17676 82420 17580 000.96 99904 || 11 || 44 20 || 50 || 8.825.13 || 11.17487 || 8.82610 || 11.17390 10,000.97 9.99903 || 10 | 40 24 || 51 82.701 17299 S2799 1720.1 00098 99.902 9 36 28 || 52 82888 I7I12 82987 17013 O0099 99901 8 32 32 || 53 83075 16925 83175 16S25 00.100 99900 7 || 28 36 || 54 83261 16739 S3361 16639 00101 99S99 6 || 24 40 || 55 || 8.83446 11.16554 |8.83547 11.16453, 10.00102 9.998.98 5 i 20 44 || 56 83630 16370 83732 1626S 00102 99898 4 || || 6 48 || 57 S3813 I6187 83916 16084 00.103 99.897 3 || 12 52 58 83996 16004 84.100 15900 00.104 998.96 2 S 56 || 59 84.177 15S23 84.282 1571S 00105 99.895 I 4 16 60 84358 15642 84.464 15536 00100 998.94 () || 44 M. S. M Cosine. Secant, iCotangentl Tangent. || Cosecant. Sine. M. M. S. 6h 1939 86° 5h LoGARITHM's TRIGONOMETR1c. Oh 49 Logarithms. 1759|11 h M.S. M Sine. Cosecant. | Tangent. Cotangent. Secant. Cosine. M. M. S. 16 0 || 8.84.358 11.15642 || 8,84464 11.15536 || 10.00105. 9.99894 || 60 || 44 4 I 84539 15461 84646 15354 00107 99.893 || 59 || 56 8 2 847.18 15282 84826 15174 00108 99.892 || 58 52 12 3 S4S97 15103 85006 14994 0.0109 99.891 || 57 || 48 16 4. 85075 14925 85.185 14815 0.0109 998.91 || 56 || 44 20 5 || 8,85252 | 11.14748 || 8.85363 11.14637 || 10.00110 9.99890 || 55 40 - 24 6 85.429 1457.1 85540 14460 00111 998S9 || 54 || 36 28 7 || -85.505 14395 85.717 14283 00.112 99.888 || 53 || 32 32 8 857.80 14220 85893 14107 00113 99.887 || 52 || 28 36 9 85955 14045 S6009 13931 00114. 99886 || 51 || 24 40 # 10 || 8.86128 11.13872 || 8.86243 11.13757 || 10.00115 9.99885 .50 20 44 || 11 S6301 13699 S64.17 135S3 00II6 99.884 49 || 16 48 || 12 86474 | \ 13526 86591 13409 00117 998S3 || 48 12 52 13 86645 13355 86763 13237 00.118 99882 47 8 56 || 14 86816 13184 86935 13065 00II9 99SS1 || 46 4 17 | 15 8.80987 11.13013 || 8.87106 11.12894 || 10.00120 9.99.880 || 45 || 43 4 || 16 87.156 12844 87.277 12723 00.121 99S79 || 44 || 56 8 || 17 873.25 12675 87.447 12553 00.121 99S79 || 43 52 12 || 18 S7494 12506 87616 I2384 00.122 99.878 || 42 || 48 16 || 19 87661 12339 || 87785 12215 00.123 99.877 || 41 || 44 20 | 20 | 8.87829 11.121.71 8.S7953 11.12047 || 10.0012.4 9.998.76 40 || 40 24 || 21 87995 12005 88.120 11880 00.125 99S75 39 || 36 28 22 88.16.1 11839 882ST 11713 00.126 99S74 || 3S 32 32 || 23 8S326 11674 S8453 11547 00.127 995.73 || 37 || 28 36 || 24 8S490 II510 8S618 II.382 00128 99S72 || 36 24 40 || 25 | 8,88654 || 11.11346 |8.88783 II.11217 | 10.00129 9.99$71 || 35 | 20 44 26 88S17 11183 8S948 11052 00.130 998.70 || 34 || 16 48 27 88980 11020 891.11 10SS9 00131 99S69 || 33 || 12 52 28 89142 10858 892.74 10726 00.132 99S68 || 32 8 56 29 89304 10696 89.437 10563 00.133 99.867 || 31 4 18 || 3() || 8.80464 11,10536 || 8.S9598 11.10402 || 10.00134 9.99866 || 30 42 4 || 31 89625 10375 S9760 10240 00.135 99.865 29 || 56 8 32 89784 I0216 899.20 10080 00.136 99S64 || 28 || 52 12 || 33 899.43 10057 900S0 0.9920 00.137 99S63 || 27 || 48 16 || 34 90102 09S98 90240 09760 00.138 99SG2 26 || 44 20 || 35 | 8.90260 II.09740 8.90.399 11.09601 || 10.0013 9.99861 || 25 40 24 l 36 904.17 095S3 90.55'ſ 09443 00140 99860 || 24 || 36 28 || 37 90574 0.9426 90.715 09.285 00141 99S59 || 23 || 32 32 || 3S 90730 09270 90S72 09128 00142 99.858 22 || 28 36 || 39 90SS5 09115 91029 OS971 O0143 99S57 || 21 || 24 40 || 40 || 8.91040 11.08960 | 8.91.185 11.08815 i 10.001.44 9.99856 || 20 20 44 || 41 91195 08S05 91340 08660 001.45 99855 19 16 48 || 42 91349 08651 91495 08505 00146 99.854 || 18 || 12 52 43 9.1502 08498 91650 08:350 001.47 99853 17 8 56 || 44 91655 08:345 91803 OS197 00148 998.52 16 4. 19 || 45 8.91807 II.08.193 8.91957 11,08043 || 10.001:49 9.99851 || 15 || 41. 4 || 46 91959 OSO4] 92110 07890 0.0150 99.850 || 14 56 8 47 92110 O7890 92.262 0773S ()0152 99.848 || 13 || 52 I2 || 48 92.261 07739 92414 07586 0.0153 99.84'ſ I2 || 48 16 || 49 924.11 07589 92.565 07435 00154 99.846 11 || 44 20 50 8.92561 11.07439 || 8.92.716 11,07284 || 10.00155 9.99$45 || 10 || 40 24 51 92710 ()7290 92866 07.134 0.0156 998.44 9 || 36 28 || 52 928.59 0.7141 93016 069S4 00.157 99S43 8 32 32 53 93007 O6993 93165 06S35 00.158 99S42 7 2S 36 || 54 93154 06S46 93313 || 06687 ()0159 99.841 6 24 40 55 || 8.93301 11.06699 || 8.93462 II.0653S 10.00160 9.99$40 5 20 44 || 56 93448 06552 93609 06.391 0.0161 99839 4 || 16 48 || 57 935.94 0.6406 93756 06244 ()0162 99S3S 3 : 12 52 58 93740 06260 93.903 06097 00163 99837 2 8 56 59 93885 06115 9.4049 05951 00164. 99S36 l 4 20 || 60 94030 05970 94.195 0.5805 00166 99$34 0 || 4-0 M. 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S. 28 || 0 || 9.0S589 10.91411 || 9,08914 || 10.910S6 10.00325 9.99675 60 || 32 4 l OS692 91.30S 09019 909S1 00326 99674 || 59 || 56 8 2 0S795 91205 09123 90S77 00328 99672 58 || 52 2 3 QSS97 91.103 09.227 90773 00330 9967O 57 || 48 16 4 OS999 91001 09330 90670 00331 99669 || 56 || 44 20 5 || 9,0910] 10.90899 || 9.09.434 || 10.90566 || 10.00333 9.99667 || 55 | 40 24 6 09.202 90708 09537 90463 00334 99666 || 54 || 36 2S 7 09304 90696 09640 90360 00336 996.64 53 || 32 32 8 0.9405 90595 0.97.42 9025S 00337 99.663 52 28 36 9 09:506 90494 09S45 90.155 00339 99.661 || 51 || 24 40 || 10 || 9,09606 || 10.90394 || 9,099.47 | 10.90053 || 10.00341 9.99659 50 | 20 44 11 09707 90293 10049 S9951 00342 99658 || 49 || 16 4S | 12 09807 90.193 10150 89850 00344 99656 || 48 || 12 52 || 13 09907 90093 10252 8974S 00345 90655 47 8 56 || 14 10006 S9994 10353 89647 0.0347 99653 || 46 4 29 || 15 9.10.106 10.89894 |9,10454 || 10.89546 || 10.00349 9.99651 || 45 || 31 4 || 16 10205 89.795 10555 894.45 00350 99650 || 44 || 56 8 || 17 10304 89696 I0656 89344 00352 9964S || 43 52 12 18 10402 8959S 10756 89244 00353 99647 || 42 || 48 16 || 19 10501 S9499 10S56 89.144 00355 99645 || 41 || 44 20 | 20 9.10599 || 10.89.401 |9,10956 || 10.89044 || 10.00357 9.99643 || 40 || 40 24 21 10697 89.303 11056 8S944 0035S 99642 || 39 || 36 2S 22 10795 89205 11155 88.845 00360 99640 || 38 32 32 || 23 10893 89.107 11254 8S746 00362 99638 || 37 28 36 || 24 10990 S9010 11353 8S647 00363 99637 || 36 24 40 || 25 || 9,11087 10,88913 |9.11452 | 10.8S548 || 10.00365 9.99635 | 35 | 20 44 26 11184 SSSI6 11551 SS449 00367 99633 34 16 4S 27 11281 8S719 11649 88.351 0036S 99632 || 33 || 12 52 28 11377 8S623 11747 S8253 00370 99630 || 32 8 56 || 29 11474 88526 11845 88.155 00371 99629 || 31 4 30 || 30 || 9,11570 10.SS430 || 9,11943 || 10.SS057 || 10.00373 9.99627 || 30 30 4 || 31 11666 SS334 12040 S7960 00:375 99625 29 56 8 || 32 11761 S8239 || 12138 S7862 00376 99624 || 28 || 52 12 || 33 IIS57 8S143 12235 S7765 00378 99622 || 27 || 48 16 || 34 11952 SS048 I2332 S7668 003S0 99620 26 || 44 20 i 35 | 9,12047 || 10.S7953 || 9,12428 || 10.S7572 || 10.00382 9.99618 25 | 40 24 || 36 12142 87858 12525 S7475 0.0383 996.17 24 || 36 2S 37 12236 87764 12621 8.7379 003S5 996.15 || 23 || 32 32 || 3S 12331 87669 I2717 872S3 0.0387 996.13 22 || 28 36 || 39 12425 87575 12S13 871S7 003SS 996.12 21 || 24 40 || 40 9.12519 10.S74S1 || 9,12909 IO.S7091 || 10.00890 9.996.10 || 20 20 44 || 41 12612 87388 13004 S6996 00392 99608 || 19 || 16 4S 42 12706 87294 I3099 86901 00393 99607 18 || 12 52 || 43 12799 872O1 13194 86806 00395 99605 || 17 8 56 || 44 12892 87.108 13289 | S6711 00897 99.603 || 16 4 31 || 45 || 9,12985 || 10.S7015 || 9,13384 || 10.86616 10.00399 8.99601 || 15 || 29 4 || 46 13078 86922 13478 86522 00:400 99600 || 14 || 56 8 || 47 13171 86829 13573 86427 00402 99.598 || 13 || 52 12 || 48 13263 86737 13667 S6333 00404 99596 || 12 || 48 16 49 13355 86645 13761 8623) 004()5 99595 i 11 || 44 20 || 50 || 9,1344.7 10,86553 |9.13854 10,861.46 | 10.00407 9,99593 || 10 | 40 24 || 51 13539 86461 13948 86052 U()409 99591 9 || 36 2S 52 13630 86370 14041 85959 004:11 99589 8 || 32 32 || 53 13722 86278 14134 858(36 00412 99588 7 || 28 36 54 3813 86.187 14227 85773 00414 99586 6 || 2 40 55 || 9,13904 || 10.86096 || 9.14320 10.856S0 || 10.00416 9.99584 || 5 || 20 44 56 13994 86006 14412 S55SS 004:18 99582 4 16 48 57 14085 85915 14504 85496 004:19 99381 3 || 12 B2 5S 14175 85S25 14597 85403 004:21 995.79 2 8 56 || 59 14266 85734 146S8 85.312 0.0423 99577 l 4 32 || 60 14356 S5644 14780 85220 004:25 995.75 () ; 28 M. S. M. Cosine. Secant. Cotangent Tangent. Cosecant. Sine. M. M. S. 6h 1979 82° 5h LoGARITHM's TRIGONoMETRIC. Oh 89 Logarithms. 1719, 11 h M.S. M. Sime. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. M. S. 32 || 0 || 9,14356 || 10.85644 |9.14780 | 10.85220 || 10,00425 9.995.75 60 || 28 4 || 1 14445 85555 || 14S72 8512S 0.0426 995.74 || 59 || 56 8 || 2 14535 85465 14963 85037 004:28 995.72 58 52 12 3 14624 85376 15054 84946 004:30 995.70 || 57 || 48 16 4 14714 85286 || 15145 84855 004:32 99568 || 56 || 44 20 || 5 || 9.14803 || 10.85.197 9.15236 || 10,84764 10.00434 9.99566 || 55 40 24 || 6 14891 85109 || 15327 84673 004:35 99565 || 54 || 36 28 'I 14980 85020 || 15417 S4583 00437 99563 || 53 || 32 32 || 8 15069 84931 || 1550S 84.492 00439 99.561 || 52 || 28 36 || 9 15157 84843 || 15598 84.402 00441 99559 || 51 24 40 || 10 || 9,15245 || 10.84755 9.15688 10.84312 || 10.00443 9,995.57 || 50 | 20 44 || 11 15333 i 84667 15777 84223 00444 99.556 || 49 || 16 48 || 12 15421 84579 || 15867 84.133 00446 99554 || 48 || 12 52 || 13 15508 84492 15956 84044 00448 99552 || 4-7 || 8 56 14 15596 84.404 || 16046 83954 00450 99.550 || 46 l 4 33 || 15 || 9,15683 || 10.84.317 | 9.16135 | 10.83865 || 10.00452 9.99548 || 45 27 4 16 15770 84230 | 16224 S3776 00454. 99.546 44 || 56 8 || 17 15S57 84.143 16312 836SS 00455 99.545 43 || 52 12 18 15944 84056 || 10401 83599 00457 99543 || 42 48 16 || 19 16030 839.70 || 16489 S3511 0.0459 99.541 || 41 || 44 20 20 | 9.16116 10.83884 |9.16577 | 10.83423 || 10.00461 9.99539 || 40 || 40 24 21 16203 S3797 16665 83.335 00463 995.37 39 || 36 28 22 16289 837.11 16753 83247 00465 99.535 3S 32 32 || 23 16374 83626 16S41 83159 00467 99533 37 || 2S 36 || 24 16460 S354 16928 83072 00468 99.532 || 36 24 40 || 25 || 9,16545 || 10.83455 |9.17016 || 10.829S4 || 10.00470 9,995.30 || 35 | 20 44 26 16631. 83369 17103 S2S97 00472 99.528 34 || 16 48 27 16716 83284 || 17190 S2810 00474 99526 || 33 12 52 || 28 16801 S3199 || 17277 82723 00476 99524 || 32 || 8 56 29 16S86 831.14 || 17363 $2637 00:478 99522 || 31 || 4 34 || 30 || 9.16970 || 10.83030 |9.17450 || 10.8255() || 10.00480 9.99520 i 30 || 26 4 || 31 17055 82.945 17536 82464 00482 99518 29 56 8 || 32 17139 82861 17622 S2378 004S3 99517 | 28 52 12 || 33 I7223 82777 ; 17708 82292 00485 99515 || 27 || 48 16 || 34, I -17307 82693 || 17794 82206 00487 99513 26 || 44 20 || 35 | 9.17391 || 10.82609 || 9.17SS0 | 10.8.2120 || 10.00489 9.99511 25 | 40 24 || 36 17474 S2526 17965 S2035 ()0491 99509 || 24 || 36 28 || 37 I'7558 82442 | 18051 81949 00493 99507 || 23 || 32 32 38 17641 82.359 || 18136 81864 00495 99505 22 || 28 36 || 39 17724. 82.276 18221 81779 00497 99503 || 21 || 24 40 || 40 9.17807 || 10.82193 || 9,18306 || 10.81694 || 10.00499 0.995.01 || 20 20 44 l 41 17890 82110 | 18391 81609 00501 99.499 || 19 || 16 48 || 42 17973 82027 | 18475 81525 00503 99.497 || 18 12 52 || 43 1S055 81945 18560 81440 00505 99.495 || 17 || 8 56 || 44 18137 81S63 18644 SI356 00506 99.494 | 16 || 4 35 || 45 || 9.18220 | 10.81780 || 9.18728 || 10.SI272 10.00508 9.994.92 || 15 25 4 / 46 18302 S1698 || 18812 811S8 00510 99.490 || 14 || 56 8 || 47 IS383 81617 18896 81104 00512 9948S | 13 || 52 I2 48 18465 81535 | 18979 81021 00514 99.486 || 12 || 48 16 || 49 I8547 81.453 19063 SO937 00516 99484 || 11 || 44 20 || 50 || 9.18628 || 10.81372 || 9,19146 || 10.80854 || 10,00518 9.99482 || 10 || 40 24 51 I8709 81291 19229 80771 0.0520 99.480 || 9 || 36 28 52 IS790 S12.10 | 19312 806SS 00522 99.478 || 8 || 32 32 || 53 18871 81129 || 19395 80605 005.24 99476 || 7 || 28 36 || 54 18952 81048 || 19478 80522 005.26 99.474 || 6 || 24 40 55 9.19033 || 10.S0967 || 9.19561 10.80439 || 10.00528 9,994.72 || 5 || 20 44 || 56 19113 80SS7 || 19643 S0357 00530 994.70 || 4 || 16 48 || 57 I9193 80S07 I9725 80275 00532 99.468 || 3 || 12 52 58 19273 S0727 19807 80193 00534 99.466 || 2 || 8 56 || 59 19353 SOG47 || 19889 80.111 00536 99.464 || 1 || 4 36 || 60 19433 80567 || 19971 80029 00538 99.462 || 0 || 24 M. S. M I Cosine. Secant. Cotangent| Tangent. Cosecant. Sine. M M. S. | 6h 1989 S1° 5h LOGARITHMS TRIGONOMETRIC. 172 . - Oh 9° Logarithms, 1709|11.h M.S. M Sine. Cosecant. Tangent. Cotangent, Secant. Cosime. M. M. S. 36 || 0 || 9,19433 10.80567 || 9,19971 || 10.80029 || 10.00538 9.99462 || 60 || 24 4 || 1 1951: S0487 || 20053 '79947 00540 99.460 || 59 56 8 2 19592 804.08 20134 79866 00542 99.458 || 58 || 52 12 || 3 19672 80328 20216 79784 '00544 99.456 57 || 48 16 || 4 19751 80249 || 20297 79703 0.0546 99.454 || 56 || 44 20 || 5 || 9,19830 10,801.70 || 9,20378 || 10,79622 || 10.00548. 9.09452 || 55 | 40 24 || 6 1990.9 80091 || 20459 79541 00550 99.450 54 || 36 28 || 7 19988 80012 || 20540 7.9460 00552 99448 || 53 || 32 3 8 20067 799.33 20621 793.79 00554 99446 52 || 28 36 || 9 20145 79S55 20701 79.299 00556 99.444 || 51 || 24 40 || 10 || 9.20223 || 10.79777 || 9.20782 10.79218 || 10.00558 9.99442 50 20 44 11 20302 79698 || 20862 '[9138 00560 99440 49 || 16 48 || 12 20380 '79620 20942 79058 00562 99.438 || 48 || 12 52 || 13 20458 79542 21022 78978 00564 99.436 || 4-7 || 8 56 || 14 20535 79465 21102 7SS98 00566 99.434 || 46 || 4 37 || 15 || 9,20613 || 10.79387 |9.21182 10,78818 || 10.00568 9.99432 || 45 || 23 4 || 16 20691 79309 || 21261 78739 005'71 99.429 || 44 || 56 8 || 17 20768 '79232 21341 78.659 00573 994.27 || 43 52 12 | 18 20845 79155 21420 78580 005.75 99.425 || 42 || 48 16 || 19 20922 790.78 || 21499 '78501 00577 99.423 41 || 44 20 | 20 9.20999 || 10.79001 |9.21578 || 10.78422 || 10.00579 9.99421 40 || 40 24 || 21 21076 78924 || 21657 78343 00581 994.19 || 39 || 36 28 22 21153 '78847 21736 '78264 0.0583 994.17 || 38 || 32 32 || 23 21229 78771 || 21814 'YS186 005S5 994.15 # 37 || 28 36 24 || 21306 78694 || 21893 78.107 0.0587 99.413 || 36 || 24 40 || 25 || 9.21382 | 10,786.18 || 9.21971. 10.78029 || 10.005S9 9.994.11 || 35 | 20 44 || 26 21458 78542 22049 77951 00591 99.409 34 || 16 48 27 21534 78466 22127 77S73 00593 99.407 33 || 12 52 || 28 21610. 78.390 || 22205 77795 00596 99.404 || 32 || 8 56 29 21685 '78315 222S3 77717 00598 99402 || 31 || 4 38 30 || 9.2.1761 10.78239 |9.22361 || 10.77639 || 10.00600 9.99400 || 30 22 4 3L 21836 78.164 || 22438 77562 00602 99398 || 29 56 8 32 21912 78088 22516 77484 U0604 993.96 || 28 52 12 || 33 21987 '78013 22593 '77.407 00606 99394 || 27 || 48 16 || 34 22062 77938 || 22670 '77330 00608 99392 || 26 || 44 20 i 35 | 9.22137 || 10.77863 |9.22747 | 10.77253 || 10.00610 9.993.90 °25 || 40 24 || 36 22211 777S9 22824. 77176 00612 99388 || 24 36 28 37 || 22286 '77714 22901 '77099 00615 99385 || 23 || 32 32 38 22.361 '77639 22977 77023 00617 99383 || 22 28 36 || 39 22435 77565 || 23054 76946 00019 99381 || 21 || 24 40 | 40 || 9.22509 || 10.77491 || 9.23130 | 10.768.70 || 10.00621 9.993.79 20 20 44 || 41 22583 '77417 2.3206 76794 00623 99.377 19 || 16 48 || 42 22657 77343 || 232S3 76717 0.0625 993.75 18 || 12 52 43 22731 77269 23359 76641 00628 993.72 || 17 || 8 56 44 i 22805 77195 || 23435 7 6565 006.30 99370 16 || 4 | 39 45 9.22878 || 10.77122 |9.23510 || 10.76490 | 10.00632 9,99368 15 # 21 4 || 46 22952 77048 || 23586 76414 00634 99366 14 56 8 || 47 23025 76975 23601 76339 00636 99364 13 52 12 48 23098 76902 || 23.737 76263 00638 99362 12 48 16 || 49 231.71 76829 || 23812 76.188 00641 99359 II || 44 20 || 50 9,23244 || 10.76756 |9.23887 || 10.76113 || 10.00643 9.09357 || 10 40 24 || 51 23317 76683 || 23962 76038 006:45 99355 9 36 28 52 23390 76610 || 24037 75963 00647 99353 8 32 32 53 23462 76538 || 24,112 '75888 00649 99351 || 7 || 28 36 || 54 23535 76465 || 24186 75S14 00652 993.48 || 6 || 24 40 || 55 || 9,23607 || 10.76393 || 9,24261 10.75739 || 10.00654 9.99346 || 5 || 20 44 || 56 23679 76321 24335 75665 00656 99344 || 4 || 16 48 || 57 23752 76248 || 24.410 75590 0065S 99.342 || 3 || 12 52 || 58 23.823 76177 24484 75516 00660. 993.40 || 2 || 8 56 || 59 23.895 76105 || 24558 754.42 0.0663 99337 || 1 || 4 40 60 23967 76033 || 24.632 75368 00665 99.335 || 0 || 20 M. S. M I Cosſne. Secant. Cotangent| Tangent. Cosecant. Sine. M IM. S. 6h 1999 80° 5h LOGARITHM's TRIGONoMETRIC. 173 Oh 100 Logarithms. 1699||11h M. S. M. Sine Cosecant. | Tangent. Cotangent. Secamt. Cosine. M. M. S. 4-0 || 0 || 9.23967 10.76033 9.24632 10.75368 || 10.00665 9.99335 | 60 20 4 || 1 24039 '75961 .24706 '75294 00667, 99333 59 || 56 8 2 241.10 75890 2 [779 75221 O0669 99331 58 || 52 12 3 24181 75S19 24S53 75147 00672 9932S 57 || 4S 16 || 4 24253 75747 24926 75074 00674 99326 56 || 44 20 5 9.24324 || 10.75676 |9.25000 | 10.75000 || 10.00676 9.99324 55 | 40 24 6 24395 T5605 25073 74927 00678 99322 || 54 || 36 28 7 24466 75534 || 25.146 74854 00681 99319 || 53 || 32 32 8 24536 754.64 25219 74.781 00683 99.317 || 52 || 28 36 9 24607 '75393 25292 74708 00685 99315 || 51 || 24 40 || 10 || 9,24677 10.75323 9.25365 | 10.74635 | 10.00687 9.99.313 || 50 || 20 44 || 11 24748 7 5252 || 25437. 74563 00690 993.10 || 49 16 48 || 12 24818 | 75182 25510 74490 00692 99308 || 48 || 12 52 13 24SS8 751.12 25582 74418 00694 99306 || 47 8 56 || 14 24958 75042 || 25655 74345 00696 99304 || 46 4 41 || 15 || 9,25028 || 10.74972 |9.25727 | 10,74273 || 10.00699 9.99301 || 45 19 4 || 16 2509S 74902 25799 742.01 00701 99.299 || 44 56 8 || 17 2516S 74832 25871. 74.129 00703 99.297 || 43 || 52 12 18 25237 74763 25943 74057 . 00706 99.294 || 42 || 48 16 || 19 25307 74693 || 26015 73985 00708 99.292 || 41 || 44 20 | 20 9.25376 || 10.74624 19.26086 10.73914 || 10.00710 9.99200 40 || 40 24 || 21 25445 74555 26158 73842 007 12 99288 || 39 || 36 28 22 25514 74486 || 26229 73771 007 15 99.285 38 || 32 32 || 23 25583 74417 26301 73699 00717 992S3 || 3 28 ! 36 24 25652 74348 26372 73628 00719 99.281 i 36 || 24 40 |25 || 9.25721 || 10.74279 |9.26443 | 10.73557 || 10.00722 9,99278 || 35 | 20 44 26 25790 74210 26514 73486 00724 99276 || 34 || 16 48 || 27 25858 74.142 26585 '73415 007.26 99274 || 33 12 52 28 25927 74073 26655 '73345 007 29 99.271 || 32 || 8 56 || 29 25995 74005 267.26 T3274 00731 99.269 || 31 || 4 4-2 || 30 || 9.26063 || 10.73937 - 9.26797 || 10.73203 || 10.00733 9.99267 30 18 4 || 31 26131 73869 26867 73T33 00736 99.264 || 29 || 56 8 || 32 26199 73S01 26937 73063 00738 99.262 || 28 52 12 || 33 26.267 73733 27008 72992 0.0740 99.260 7 || 48 16 || 34 || 26335 '73665 27078 | 72922 00743 99257 || 26 || 44 20 || 35 | 9.26403 || 10.73597 |9.27148 || 10.72852 || 10.00745 9.99255 || 25 | 40 24 || 36 26470 73530 27218 72782. 00748 99.252 || 24 || 36 28 37 26538 73462 27.288 72712 00750 99.250 || 23 || 32 32 || 38 26605 733.95 27357 72643 00752 99.248 22 || 28 36 39 26672 73328 27427 72573 00755 99.245 21 || 24 40 | 40 9.26739 10.73261 |9.27496 || 10.72504 || 10.00757 9.99.243 || 20 || 20 44 || 41 26806 73194 27566 72434 00759 99.241 19 || 16 48 # 42 26873 73127 27635 72365 007 62 99.23S | 18 || 12 52 || 43 26940 '73060 27704 '72:296 00764 99.236 || 17 8 56 || 44 27007 72993 27773 '72227 00767 99.233 16 4 43 || 45 9.27073 10.72927 9.27842 | 10.72158 10.00769 9.99.231 || 15 ſº 4 46 27140 72.860 27911 72089 007'71 99.229 || 14 || 56 8 || 47 27206 72794 27.980 '72020 0077.4 99.226 || 13 || 52 12 || 48 27.273 72727 28049 71951 ()()776 99.224 12 || 48 16 || 49 27339 72661 28117 '71883 00779 99221 || 11 || 44 20 || 50 || 9.27405 || 10.72595 || 9.2S]S6 || 10.71814 || 10.00781 9.992.19 || 10 || 40 24 || 51 274.71 72529 || 28254 71746 00783 99.217 9 || 36 28 52 27537 '72463 28323 71677 00786 99.214 || 8 || 32 32 || 53 27602 72398 is 28391 ’71609 00788 99.212 7 2S 36 54 27668 72332 28459 71541 007.91 99.209 6 24 40 || 55 || 9,27734 || 10.72266 |9.28527 | 10.71473 || 10.00793 9.99207 || 5 || 20 44 56 7799 72201 28595 71405 00796 99.204, 4 || 16 48 || 57 27864 72136 28662 71338 00798 99202 || 3 || 12 b2 || 58 27.930 72070 28730 71270 00800 99.200 2 8 56 || 59 27995 72005 28798 71202 00803 99.197 1 4 44 || 60 28060 71940 28S65 71135 00805 99.195 () || 16 M. S. M. Cosime. Secant. Cotangentl Tangent. Cosecaut. Sine. M. M. S. 6a |100 79° 5h LOGARITHMS TRIGONOMETRIC. 11o Logarithms. 1689 M Sine. Cogecant. I Tangent. | Cotangent. Secant. Cosine. M. M. S. 0 || 9.2S060 I0.71940 || 9.2SS65 I0.71135 | 10,00S05 9.99.195 || 60 | 16 l 281.25 71S75 2S933 71067 0080S 99.192 || 59 || 56 2 2S190 71810 : 29000 71000 00S10 99.190 58 || 52 3 2S254 71746 20067 '70933 008:13 991S7 || 57 || 48 4 2S319 71681 291.34 70S66 00S15 99185 || 56 || 44 5 9.2S3S4 10,71616 || 9.2920 l 10.70799 || 10,00S18 9.99.182 || 55 40 6 284.48 '71552 29:26S 70732 00S20 99.180 || 54 || 36 7 2851.2 714SS 2.9335 70665 00823 9917.7 || 53 || 32 S 2S577 71423 29.402 70598 00S25 99175 52 2S 9 2S641 '71359 2946S 70532 00S2S 99172 || 51 24 10 || 9,2S705 10.71295 |9.29535 10.70465 10.00S30 9.99.170 50 20 11 2S769 71231 29601 70.399 00S33 99.167 49 || 16 12 2SS33 7.1167 29668 . 70332 00835 99.165 || 48 || 12 13 28S96 '71104 29734 70266 00S38 99.162 47 8 14 2S960 71040 298.00. 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If | 3 || 8 ‘.99 || 69 I6;S6 60QIO fi-II.g 99 Sãf QQJSG y | 1.f3 If I f O9 || 09 || $6f86'6 90910'0I Q6 Il.9 OT || 90S&# 6 || 001.SG'OI 009.If"6 || 0 || 0 *S*JV] W. *auſsoo “quuoos ºuaºutºloo | "JuaşuuJ, ºucoosoo "augS JW ‘S’ W. u0II of 9T "suuppau Sorſ o9'Il ul ‘OIMISINONO91&II, SINHLIHV50T 81: LOGARITHMS TRIGONOMETRIC. 169 Logarithms. 163°10'h M Sine. Cosecant. | Tangent. Gotangent. Secant. Cosime. M. M. S. 0 || 9,4403 10.55966 || 9.45750 10.54250 || 10.01716 9.9S284 || 60 56 1 44078 55922 45797 54203 01719 98.281 59 || 56 2 44.122 55S78 45845 54155 Ol'723 98.277 || 58 52 3 44.166 55834 45892 54108 01727 98.273 || 57 || 48 4. 44.210 55790 45.940 54060 O1730 9S270 56 || 44 5 9.44253 10,55747 || 9,45987 10.54013 || 10.01734 9,98266 || 55 40 6 44.297 55703 46035 5396 0.1738 98262 || 54 || 36 7 4.1341 55659 46082 539 S 0.1741 98.259 || 53 || 32 8 44385 55615 46130 53870 0.1745 98.255 || 52 || 28 9 44428 55572 4617'ſ 53S23 0.1749 98.251 || 51 || 24 10 9.444.72 10.55528 9.46224 10.53776 || 10.01752 9.98248 || 50 | 20 11 44516 55484 46271 58729 0.1756 98244 49 || 16 12 44559 55441 46319 53681 O1760 98240 || 48 || 12 13 44602 55898 46366 53634 0.1763 98237 || 47 8 14 44646 55354 46413 53587 017 o'7 98.233 || 46 4. 15 9.44689 10.55311 || 9.4646() 10.53540 || 10.01771 9,98229 || 45 55 16 44733 55.267 46507 53493 0.1774 9S226 || 44 56 17 44776 55224 46554 53446 0.1778 98222 || 43 || 52 18 44S19 55181 46601 53.399 01782 98.21S 42 || 48 19 44862 5513S 46648 53352 0.1785 98.215 || 41 || 44 2.) | 9,44905 10.55095 || 9,46694 10.53306 || 10,01789 9,982.11 || 40 40 21 44948 55052 46741 53259 0.1793 98.207 || 39 || 36 22 44992 5500S 46783 53212 0.1796 98.204 || 38 || 32 23 450.35 54965 468.35 53165 OIS00 9S200 || 37 28 24 45077 5.4923 46SSl 53119 01804 9Slgó | 36 24 25 || 9.45120 10.54SS0 || 9,4692S 10,53072 || 10.01808 9.98192 || 35 | 20 26 45163 54837 46975 53025 01811 98.189 || 34 || 16 27 45206 54794 47021 52979 01815 98.185 || 33 || 12 2S 45249 54751 47068 52931 01S19 9S1S1 || 32 S 29 45292 54.708 47I14 52SS6 01S23 9SI'77 || 31 4 3() 9.45334 10.54666 | 9.47160 10.52840 || 10.01826 9.9Sl'74 || 30 || 54. 31 45377 54623 47207 52793 01S30 9S1.70 || 29 || 56 32 454.19 54581 47253 52747 01S34 9S166 2S 52 33 45.462 5453S 47299 52701 0183S 9S162 27 || 48 34 45504 5.4496 47346 52654 01841 9S159 || 26 || 44 35 | 9.45547 10.54453 |9.47392 10.52608 || 10.01845 9.9S155 ||25 | 40 36 455S9 544ll 47.43S 52562 0.1849 9S151 24 || 36 37 45632 5436S 47484 52516 0.1853 9S147 || 23 32 38 45674 54326 7 530 52470 01856 98144 22 || 28 39 45.716 542S4 47576 52424 01860 98.140 || 21 24 40 || 9.45758 10.54242 || 0.47622 10 52378 || 10.01864 9.9S136 || 20 20 41 45801 54199 4766S 52332 01S6S 9SI32 19 16 42 45S43 54157 47714 52.286 01871 9.S129 || 18 || 12 43 45SS5 54115 47760 52240 01875 9S125 17 8 44 459.27 54073 47806 5219.4 01879 9S121 16 4 45 9.45969 10.54031 9.47852 10,52148 || 10.01883 9,98117 | 15 53 46 4601.1 53989 47897 52103 01SS7 98113 || 14 || 56 47 46053 53047 || 47943 6205.7 01S90 98.110 || 13 || 52 48 46095 53905 479S9 52011 Ol S94 9S106 || 12 || 48 49 46136 53864 4S035 51965 0.189S 9S102 || 11 || 44 50 9.461.78 10.53S22 || 9.48080 10,51920 || 10,01902 9.9S098 || 10 40 51 46220 53780 4S126 51874 0.1906 9S094 9 || 36 52 46262 5373S 4S171 51S29 0.1910 98090 8 || 32 53 46303 53697 4S217 517S3 01913 9SOS7 7 || 28 54 46345 53655 48262 5173S U1917 9SOS3 6 || 24 55 || 9,46386 1053614 |9.4S307 10,51693 10,01921 9.9S079 5 || 20 56 4642S 53.572 4S353 51647 ()1925 9S()75 4 16 57 46469 53531 4$398 51602 0.1929 QS071 3 12 58 465.11 534S9 48443 51557 0.1933 98067 2 8 59 46552 53448 4S489 51511 01937 98063 l 4 60 465.94 53406 48534 51466 0.1940 98060 0 || 52 M Cosime. Secant. ICotangentl Tangent. Cosecant. Sume. M AI. S. 1069 73° 4h 180 IOGARITHMS TRIGONOMETRIC. 1h 170 Logarithms. 162°10b |M.S. M Sine. Cosecant. | Tangent, Gotangent. Secant. Cosime. M. M. S. 8 0 || 9.46594 || 10.53406 |9.4S534 T0.51466 || 10.01940 9,98060 || 60 || 52 4. 1. 46635 63365 48579 51421 01944 98056 || 59 || 56 8 2 46676 53324 48624 51376 0.1948 98052 58 || 52 12 3 46717 53283 48669 b1331 O1952 98.043 || 57 || 48 16 4 46758 53242 487.14. 51286 0.1956 98044 56 | 44 20 5 9,46800 10.53200 || 9.48759 10.51241 || 10.01960 9,98040 || 55 | 40 24 6 46841 53159 || $804 51196 0.1964 9S036 || 54 || 36 28 7 46SS2 53118 48849 51151 0.1968 98.032 || 53 || 32 32 8 46923 53077 48894. 51106 0.1971 98.029 || 52 28 36 9 46964 53036 48939 51061 0.1975 98.025 || 51 || 24 40 10 || 9.47005 10.52995 || 9.48984 || 10.51016 || 10.0.1979 9,9802.1 50 || 20 44 i 11 47045 52955 49029 50971 O1983 98017 49 16 48 || 12 470S6 52914 49073 50927 O1987 98013 || 48 || 12 52 13 47127 52873 491.18 50882 O1991 98.009 || 47 8 56 14 47168 52S32 49163 50837 01995 9S005 || 46 4 9 || 15 || 9,47209 10.52791 9.49207 10.50793 || 10.01999 9.98001 || 45 || 51 4 || 16 47249 52751 49252 50748 02003 97.997 || 44 || 56 8 || 17 47290 52710 49296 50704 02007 97993 || 43 52 12 18 47330 52670 4934.1 60659 02011 97.989 || 42 || 48 16 19 47371 52629 49385 50615 02014 97.986 || 41 || 44 20 | 20 9.47411 10.52589 |9.494.30 10.50570 || 10,02018 9.97982 40 || 40 24 21 47452 52548 49474 50526 02022 97.97S 39 || 36 28 22 47492 b2508 49519 50481 02026 97.974 || 3S 32 32 || 23 47533 52467 49563 50437 02030 97.970 || 37 28 36 || 24 47573 52427 49607 50393 02034 97966 || 36 || 24 40 || 25 9.47613 10.52387 9.49652 10.50348 || 10.02038 9.97962 35 | 20 44 || 26 47654 62346 49696 50304 02042 97.958 34 || 16 48 27 47694 52306 49740 50260 02046 97.954 || 33 || 12 52 28 47734 52266 497.84 50216 02050 97.950 32 8 56 29 47774 52.226 49828 50172 02054 97.946 || 31 4. 10 || 3() 9.47814 || 10.52186 || 9,49872 10.50128 || 10,02058 9.97.942 30 50 4 || 31 47854 B2146 49916 50084 02062 97.938 29 || 56 8 || 32 47894 52106 49960 50040 02066 97.934 || 28 52 12 || 33 47934 Ö2066 50004 499.96 02070 97.930 27 || 48 16 || 34 47974 52026 500.48 49952 02074 97.926 || 26 || 44 20 || 35 | 9.48014 || 10,51986 || 9,500.92 || 10.499.08 || 10.02078 9.97922 || 25 || 40 24 || 36 48054 51946 50136 49864 02082 97918 24 || 36 28 || 37 48094 51906 50180 49820 02086 97914 || 23 32 32 || 38 48.133 51867 b0223 49777 0.2090 97910 || 22 || 28 36 || 39 481.73 518 b0267 49733 0.2094 97.906 21 || 24 40 | 40 || 9.482.13 10.51787 9.50311 10.49689 || 10.02098 9.97.902 20 20 44 || 41 48252 Ö1748 50355 49645 02102 97898 || 19 16 48 42 48292 51708 50398 49602 02106 9'ſ S94 || 18 12 52 || 43 48332 51668 50442 49558 02110 97890 || 17 8 56 || 44 48371 51629 50485 49515 02114 97886 || 16 4 11 || 45 || 9.484.11 10.51589 || 9.50529 10.49471 || 10.02.118 9.97882 15 49 4 || 46 48450 51550 50572 49428 , 02122 978.78 || 14 || 56 8 || 47 484.90 61510 50616 49384 0.2126 97.874 || 13 || 52 12 || 48 48529 51471 50659 4934.1 02130 978.70 || 12 || 48 16 49 48568 51432 50703 49297 02.134. 97.866 11 || 44 20 || 5() 9.48607 10.51393 || 9,50746 10.492.54 || 10.02.139 9.97.861 10 || 40 24 || 51 48647 51353 50789 4921 02143 97.857 9 || 36 28 || 52 48686 51314|| || 50833 4916 021.47 97853 8 || 32 32 53 48725 61275 b0876 49124 0.2151 97849 7 28 36 || 54 48764. 5] 236 50919 49081 02155 97.845 6 24 40 || 55 || 9.48803 10.51.197 9,50962 10.49038 || 10.02.159 9,97841 5 i 20 44 56 48.842 51158 51005 4S995 02163 97.837 4 16 48 || 57 48881 51119 51048 48952 02167 97.833 3 12 62 || 58 48920 51080 5.1092 48908 02171 97829 2 8 56 || 59 48959 51041 51135 48865 021.75 97825 I 4 12 || 60 48998 51002 511.78 48822 0217.9 97821 0 || 48 M. S. M Cosine. Secant. Cotangent| Tangent. || Cosecant. Sine. M. M. S. 7h 107 o 72° 4th LOGARITHMS TRIGONOMETRIC, Ih 180 Logarithms. 161° 10h M. S. M. Sine. Cosecant. I Tangent. Cotangent. Secant. Cosine. M M. S. I2 0 9.48998 10.51002 || 9.511.78 10.48822 || 10.0217.9 9,97821 60 || 48 4 1 49037 50963 51221 487.79 021S3 97817 59 56 8 2 49076 50024 51264 48736 0218.8 97812 || 58 52 12 3 491.15 50885 51306 486.94 02.192 97808 57 48 I6 4 49153 50847 51349 48651 02:196 97804 || 56 || 44 20 5 9,49192 || 10.50808 9.51392 || 10.48608 || 10.02.200 9.97800 55 || 40 24 6 49231 50769 51435 4.8565 0.2204 97.796 || 54 || 36 28 7 49269 50731 61478 48522 0.2208 97.792 53 || 32 32 8 49.308 50692 51520 48480 02212 97788 || 52 28 36 9 49347 50653 51563 48437 02216 97.784 || 51 || 24 40 || 10 || 9,49385 10.50615 || 9,51606 || 10.48394 || 10,02221 9.97.779 || 50 20 44 || 11 49424 50576 51648 48352 02:225 97.775 49 16 48 || 12 494.62 50538 51691 48309 0.2229 97.771 || 48 12 52 13 495.00 50500 5.1734 48266 02233 97.767 || 47 8 56 14 49539 50461 51776 48224 02237 97.763 || 46 4 IL3 15 || 9,49577 10,50423 || 9,51819 10.48181 i 10,02241 9,97759 || 45 47 4 || 16 496.15 | * 50385 51861 48.139 0.2246 97754 || 44 || 56 8 : 17 49654 50346 51903 48097 02:250 97750 || 43 52 12 18 49692 50308 51946 48054 02254 97746 42 || 48 16 || 19 49730 50270 51988 48012 0.2258 97.742 || 41 || 44 20 | 20 || 9,497.68 10.50232 9.52031 10.47969 || 10.02262 9.9773 40 || 40 24 || 21 49806 50.194 52073 47927 0.2266 97.734 || 39 || 36 28 22 498.44 50156 52115 47885 0.2271 97729 38 || 32 32 || 23 49882 50118 52.157 47843 02275 97.725 || 3 28 36 || 24 49920 50080 52200 47800 02279 97721 || 36 || 24 40 || 25 9.49958 10,50042 || 9,52242 | 10,47758 || 10,02283 9,97717 || 35 | 20 44 || 26 499.96 50004 52284 47716 02287 97713 || 34 16 48 || 27 50034 49966 52326 47674 02292 97'70S 33 || 12 52 28 60072 49928 52368 47.632 02:296 97.704 || 32 || 8 56 29 501.10 49890 52410 47590 02300 97.700 31 4 I4 30 || 9,50148 10.49852 9.52452 10.47548 || 10.02304 9.97696 || 30 |46 4 31 | < 50185 49815 52494 47506 02309 97.691 || 29 56 8 || 32 50223 49777 52536 47464 02313 97.687 28 52 12 || 33 50261 49739 52.578 47422 02317 97.683 27 || 48 16 || 34. 50.298 49702 526:20 47380 02321 97.679 26 44 20 || 35 || 9,50336 10.49664 || 9,52661 10.47339 || 10.02326 9.97 674 || 25 40 24 || 36 50374 49626 52.703 47297 02330 97.670 24 || 36 28 || 37 50411 49589 52745 47255 02334 97.666 || 23 32 32 || 3S 50449 49551 52787 47213 02338 97662 22 || 28 36 || 39 504.86 49514 52829 47III 02343 97657 21 24 40 || 40 || 9.50523 || 10.494.77 9.52870 || 10.47130 || 10.02347 9.97653 20 || 20 44 || 41 50561. 49439 52912 47088 02351 97.649 || 19 || 16 48 || 42 50598 49.402 52953 47047 02355 97645 18 12 52 || 43 50635 49365 52995 47005 02360 97640 17 8 56 || 44 50673 49327 53037 46963 02364 97636 16 4 15 45 || 9.50710 | 10.49290 9.53078 10.46922 || 10.02368 9.97632 || 15 45 4 || 46 50747 49253 53120 46880 02372 97.628 14 || 56 S 47 50784 49216 53161 46839 02377 97.623 13 52 12 || 48 60821 49179 53202 46798 ()2381 97619 || 12 48 16 || 49 50858 49142 53244 46756 023S5 97.615 # 11 || 44 20 || 50 || 9.50896 || 10.4910.4 || 9,53285 10.46715 10.02390 9.97610 || 10 40 24 || 51 50933 49067 53327 46673 02394 97606 9 || 36 28 52 50970 49030 53368 46632 02398 97602 8 || 32 32 53 5.1007 48993 53409 46591 ()2403 97597 7 2S 36 54 51043 48957 53450 46550 02407 97.593 6 || 24 40 || 55 || 9.51080 10.4S920 9.53492 10.46508 || 10.02411 9.97589 6 20 44 56 51117 48883 53533 464.67 024.16 97.584 4 16 48 57 51154 48846 53574 46426 02420 97580 3 || 12 52 58 51.191 48809 53615 46385 02424 97.576 2 8 56 59 51227 4S'773 53658 46344 02.429 97.571 l 4 16 || 60 51264 48736 || 53697 46303 02433 97567 || 0 || 44 M. S. M. Cosine. Seeant. Cotangent| Tangent. Cosecant. Sine. M (M. S. 7h 108° 71° 4h LOGARITHMS TRIGONOMETRIC, 1h 190 Logarithms. 1609 |IOh M.S. M. Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. M. S. 16 || 0 || 9,51264 || 10.48736 || 9.53697 || 10.46303 || 10.02433 9,97567 60 || 44 4 I 51301 4S699 5373S 46262 0.2437 97563 || 59 || 56 8 2 51338 48662 || 53779 46221 02442 97558 || 58 || 52 12 || 3 51374 4S626 || 53820 46180 02:446 97554 57 || 48 16 4 51411 48589 53861 46139 02450 97.550 || 56 || 44 20 || 5 || 9.51447 | 10.4S553 9.53902 || 10.46098 || 10.02455 9.97545 55 40 24 6 ë1484 48516 53943 46057 02:459 97.541 || 54 || 36 28 || 7 51520 4S480 || 53984 46016 02464 97536 || 53 32 32 8 51557 48443 54025 45.975 O2468 97.532 || 52 28 36 9 51593 48407 54065 45935 024.72 97528 || 51 || 24 40 || 10 || 9.51629 || 10.48371 |9.54106 || 10,45894 || 10.02477 9.97523 || 50 20 44 || 11 51666 48,334 54147 45853 02:481 97519 || 49 || 16 4S 12 51702 48298 || 54,187 45813 024S5 97515 || 48 || 12 52 || 13 51738 48262 54.228 45772 02490 97510 || 47 || 8 56 || 14 51774 48226 54269 45731 02494 97506 || 46 4 I? | 15 || 9,51811 || 10.48189 || 9.54309 || 10.45691 || 10.02499 9.97501 || 45 43 4 16 51847 48153 54350 45650 02503 97497 44 || 56 8 || 17 51883 4S117 54390 45610 02508 97492 || 43 || 52 12 18 51919 4SOS1 || 54431 45569 02512 974.SS 42 || 48 16 19 51955 48045 54471 4.5529 02516 97.484 || 41 || 44 20 20 9,51991 || 10.48009 || 9.54512 || 10.454SS | 10.02521 9.97479 i 40 || 40 24 21 52027 47973 || 54552 45 02525 97.475 39 || 36 28 22 52063 47937 || 54593 45.407 02530 974.70 || 38 || 32 32 23 52099 47901 54633 45367 02534 97466 || 37 || 28 36 24 52135 47 S65 54.673 45327 02539 97461 || 36 24 40 || 25 || 9.52.171 || 10.47829 || 9,54714 10,45286 || 10.02543 9.97.457 || 35 | 20 44 || 26 52207 47793 54754 45246 02547 97.453 ||34 || 16 48 27 52242 47758 54794 45206 0.255.2 97448 || 33 || 12 52 28 52278 47722 54835 45165 0.2556 974.44 || 32 8 56 29 52314 476.86 54.875 451.25 02561 97.439 || 31 4 18 || 30 || 9,52350 10,47650 9.54915 10.450SS 10.02565 9.97435 || 30 || 4-2 4 || 31 52385 4T 615 54955 45045 02570 97.430, 29 || 56 8 32 52421 47579 54995 45005 02574 974.26 28 52 12 || 33 52456 47544 550'35 44965 0.2579 97.421 27 || 48 16 || 34 52492 47508 55075 44925 02383 97.417 26 || 44 20 || 35 | 9.52527 | 10.474.73 || 9,55115 | 10.44SS5 || 10.02588 9,97412 25 | 40 24 || 36 52563 47.437 55 155 44845 02592 97.408 24 36 2S 37 52598 47402 || 55,195 44$05 02597 97.403 || 23 || 32 32 || 3S 52634 47366 55235 44765 0.2601 97.399 || 22 || 28 36 || 39 52669 47331 55275 7 0.2606 97.394 21 24 40 40 || 9.52705 10.47295 9.55315 10,446S5 10.02610 9.97390 20 20 44 || 41 52740 47260 55355 44645 02615 97.385 || 19 16 48 || 42 527.75 4.7225 55395 44605 0.2619 97.381 18 12 52 || 43 52811 47.189 55.434 44566 0.2624 9.7376 || 17 || 8 56 || 44 52846 47154 55474 44526 0.2628 97.372 | 16 || 4 19 || 45 9.5288.1 || 10,47119 |9.55514 || 10,44486 || 10.02633 9.97367 || 15 || 4:1 4 || 46 52916 47084 55554 44446 02637 97363 14 || 56 8 || 47 52951 47049 55593 44407 0.2642 97.358 13 || 52 12 || 48 52986 47014 || 556.33 44367 0.2647 97.353 || 12 || 48 16 49 63021 46979 55673 44327 0265] 97349 || 11 || 44 20 50 || 9.53056 10.46944 9.557 12 10.44288 || 10,02656 9.97344 || 10 || 40 24 || 51 53092 46908 55752 442.48 0.2660 97.340 9 || 36 28 52 53126 46874 557.91 44,209 0.2665 97335 8 || 32 32 53 53161 46839 55831 44,169 02669 9733L || 7 || 28 36 || 54 531.96 46804 || 55870 44.130 0.267.4 97.326 || 6 || 24 40 || 55 9.53231 10.46769 || 9.559 || 0 || 10.44090 || 10.02678 9,97322 || 5 || 20 44 || 56 53266 46734 || 55949 44051 U26S3 97.317 || 4 || || 6 48 || 57 53301 46699 55989 44011 0.2688 97312 3 || 12 52 58 58336 46664 5602S 43972 0.2692 97.308 2 8 56 59 5337() 46630 56007 4:3933 0.2697 97.303 l 4 20 || 60 53405 46595 || 56.107 43S93 O2701 97.299 0 || 4-0 M. S. M. Cosine. Secant. Cotangent Taugent. Cusecant. Sine. M. M. S. 7h ||109 70° 4h LOGARITHMS TRIGONOMETRIC. 1b | 20° Logarithms. 1590 M.S. M Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosime. 'M 20 || 0 || 9,534)5 10.46595 || 9.56107 10.43S93 || 10.027Ol 9.972.99 || 60 4 1 53440 46560 56146 43854 ()2706 97.294 || 59 8 2 53475 46525 56.185 43815 02711 97.289 58 12 3 53509 46491 56224 437.76 02715 97.285 57 16 4. 53544 46456 56264 43736 02720 972S0 || 56 20 5 || 9,53578 10.46422 i9.56.303 || 10.43697 || 10.02724 9.97276 || 55 24 6 53613 46387 56342 43658 02729 97.271 54 28 7 53647 46353 56381 43619 02734 97.206 || 53 32 8 536S2 46318 56420 43580 02738 97.262 52 36 9 53716 46284 56459 43541 02743 97.257 || 51 40 || 10 || 9,53751 || 10.46249 |9.5649S 10.43502 || 10.02748 9.97252 50 44 11 537 S5 46215 56537 43463 02752 97248 I 49 48 || 12 63819 46181 56576 43424 02757 97243 || 48 52 || 13 53854 46 tº 6 566.15 433.85 02762 97238 || 47 56 | 14 53SS8 46112 56654 43346 02766 97234 46 21 || 15 || 9,53922 || 10.4607S 9.56693 10.43307 || 10.02771 9.97229 || 45 4 || 16 53957 46043 56732 43268 02776 97224 || 44 8 || IT 53991 46009 56771 43229 02780 97.220 || 43 12 18 54.025 45975 56810 43190 02785 97.215 || 42 16 || 19 54059 45.941 56849 43151 0.2790 97.210 || 41 20 | 20 | 9.54093 10.45907 9.568.87 10.4.3113 || 10.02794 9.97206 || 40 24 21 54127 45S73 56926 43074 0.2799 97.201 || 39 2S 22 54161 458.39 56965 43035 02804 97.196 || 3S 32 || 23 54195 45S05 57004 4.2996 O280S 97.192 || 37 36 || 24 54229 4577 l 57042 42958 028:13 97.187 || 3 40 || 25 || 9.54263 I0.45737 || 9.57081 10.42919 || 10.02818 9,97182 || 35 44 || 26 54.297 45703 5712() 42S80 028:22 97.178 34 48 || 27 54331 45669 5715S 42842 02827 97.173 || 33 52 28 54365 456.35 57197 42803 02S32 97.168 || 32 56 || 29 54399 45601 57235 42765 Q2837 97.163 || 31 22 || 30 || 9.54.433 10.45567 || 9.57274 || 10.42726 || 10.02841 9.97.159 || 30 4 || 31 54466 45534 57.312 42688 02S46 97.154 || 29 8 || 32 54500 45500 57351 42649 02851 971.49 2S 12 || 33 54534 45466 573S9 42611 02855 97145 || 27 16 || 34 54567 45433 57428 42572 02S60 9714 26 20 || 35 9.54601 10.45399 || 9,57466 10.43834 || 10.02S65 9.97135 || 25 24 || 36 54635 45365 57.504 42-496 02S70 97.130 || 24 2S 37 54.668 45332 57543 42457 0287.4 97.126 23 32 || 38 54.702 45298 57581 42419 02879 97.121 22 36 || 39 54735 45265 57619 42381 028S4 97.116 21 40 || 40 || 9,54769 10.45231 || 9,57658 1042342 || IC.02889 9.97 111 || 20 44 || 41 54802 45198 57696 42304 02893 97.107 || 19 48 || 42 54836 45164 57734 4.2266 02898 97.102 || 18 52 || 43 54869 45131 57772 42228 02903 97097 || 17 56 44 54903 45097 57810 42.190 0290S 97.092 || 16 23 || 45 9.54936 10.45004 |9.57849 || 10,42151 || 10.02913 9.97087 || 15 4 || 46 54969 45031 57SS7 42113 0.2917 97083 || 14 8 || 47 55003 44997 57,925 42075 02922 97.078 || 13 12 || 48 55036 44964 57.963 42037 0.2927 97073 || 12 l6 49 55.069 44931 580t)] 41999 02932 97.068 || 11 20 || 5() || 9,55102 || 10.44898 || 9.5S039 10.4.1961 || 10.02937 9.97003 || 10 24 51 55136 44864 58077 4.1923 02941 97.059 9 28 || 52 55169 44831 58115 4.1885 02946 97.054 8 32 53 55202 44798 58153 4.1847 02951 97049 7 36 || 54 55235 44765 5819 L 41809 0.2956 97044 6 40 55 || 9,55268 10.44732 || 9,58229 10.41771 || 10,02961 9.97039 5 44 56 55301 44699 58.267 41733 0.2965 97.035 4 48 # 57 55334 44666 5S304 41696 02970 97.030 3 52 58 55367 44633 5S342 41658 02975 97.025 2 56 || 59 55400 44600 58380 41620 029S0 97020 l 24 || 60 55433 44567 5S418 41582 029S5 97.015 0 M. S. M I Cosine. Secant. Cotangent} Tangent. || Cosecant. Sine. M 7h 1109 699 184 LOGARITHMS TRIGONOMETRIC, Ih 210 Logarithms. 1589 M.S. M Sino. Cosecant. | Tangent. Cotangent. || Secant. Cosime. M 24 || 0 || 9,55433 10.44567 9.584.18 10.415S2 || 10,02985 9.97015 60 4 I 55466 44534 5.8455 41545 0.2990 7010 || 59 8 2 55499 44501 58493 41507 ()2995 97.005 || 58 12 3 55532 44,468 58531 4.1469 0.2999 97.001 || 57 16 4 55564 44436 58569 4.1431 0300.4 96996 || 56 20 5 || 9,55597 10.44403 |9.58606 10.41394 || 10.03009 9,96991 || 55 24 6 55630 44370 586.44 41356 03014 96.986 || 54 28 7 65663 4.4337 58681 41319 03019 969S1 || 53 32 8 55695 44305 58719 41281 03024 96.976 || 52 36 9 55728 4.4272 58757 41243 03029 96971 || 51 40 || 10 || 9,5576i 10.44239 |9.58794 || 10.4120 10.03034 9.96966 50 44 II 65.793 44207 58832 41168 0.3038 96962 || 49 48 || 12 55826 44,174 5SS69 41131 03043 96.957 || 48 52 || 13 55858 4.4142 58907 4.1093 03048 96952 || 47 56 || 14 55891 44.109 58944 41056 03053 96.947 || 46 25 | 15 9.55923 10.41077 || 9,58981 10.41019 || 10.03058 9,96942 || 45 4 || 16 55956 44044 590.19 40981 03063 96937 || 44 8 || 17 55988 44012 59056 40944 03068 96932 : 43 12 18 56021 43979 59094 40906 03073 96927 || 42 16 || 19 56053 43947 5913.1 40869 03078 96.922 || 41 20 20 9.560S5 10.43915 9.59].68 10.40832 || 10.03083 9.96917 | 40 24 || 21 56.118 43SS2 59205 40795 03088 969.12 || 39 28 22 56150 43850 59.243 40757. 03093 96907 || 38 32 23 56182 43.818 59280 40.720 03097 96.903 ſ 37 "36 || 24 56215 43785 593.17 40683 03.102 96898 || 36 40 || 25 || 9,5624'ſ 10.43753 || 9.59354 10.4064 10.03107 9.96893 : 35 44 26 56279 43721 5939]. 40609 03112 96888 || 34 48 || 27 56311 43089 59.429 40571 03117 96$83 || 33 52 || 28 56343 43657 59.466 40534 03122 96.878 || 32 56 || 29 56375 43625 59503 40.497 03127 96.873 || 31 26 30 9.56.408 10.43592 |9.59540 10.40460 || 10.03132 9.96868 || 30 4 : 31 56440 43560 59577 40423 (1313 96863 || 29 8 || 32 564.72 43528 59614 40386 03142 96858 || 28 12 || 33 56504 43496 5965} 40349 03147 96853 || 27 16 || 34 56536 43.464 59688 4.0312 03.15.2 96.848 || 26 20 || 35 || 9.5656S 10.43432 |9.59% 10.40275 || 10.03157 9.96843' || 25 24 || 36 56599 43401 59762 40238 03.162 96838 24 28 37 56631 43369 59799 4020.1 03167 96S33 23 32 || 38 56663 43337 59S35 40165 03172 96828 22 36 || 39 56695 43305 598.72 40128 03177 96.823 21 40 || 40 || 9,56727 10.43273 j9.59909 10.40091 || 10.03182 9.96818 || 20 44 || 41 567.59 43241 59946 40054 Q318 96813 || 19 48 || 42 56700 43210 59983 40017 03.192 96808 || 18 52 || 43 56822 431.78 60019 39.981 03197 96.803 || 17 56 || 44 56854 43.146 60056 39944 03202 96798 || 16 27 || 45 || 9,56886 10,43114 || 9,60093 10.39907 || 10.03207 9.96793 || 15 4 || 46 56917 43083 60130 39870 03212 96788 14 8 47 56949 48051 G0166 39834 03217 96783 || 13 12 || 48 56.980 43020 60203 39797 03222 96.778 12 16 49 57.012 42988 60240 39760 03228 96.772 11 20 || 5() || 9,57044 10,42956 || 9,60276 10.39724 || 10.03233 9.96767 || 10 24 || 51 57075 42925 60313 $9687 03238 96.762 9 28 52 57107 42893 60349 39651 03243 96.757 8 32 || 53 5713 42862 60386 39.614 03248 96752 7 36 54 à7169 42831 60422 39578 03253 967.47 6 40 55 || 9.57 201 10.42799 || 9.60459 10,395.41 || 10 0S258 9.967 42 5 44 56 57.232 42.768 60495 39505 03263 96.737 4 48 || 57 57264 42'ſ 36 60532 39468 O3268 96732 3 52 58 57295 42705 60568 394.32 ()3273 96.727 2 56 || 59 57326 42674 60605 39.395 03278 96.722 I 28 || 60 57358 42642 60641 39.359 O32S3 96717 () M. S. M I Cosime. Secant. Cotangent| Tangent. || Cosecant. Sme. M 7h [1110 689 LOGARITHMS TRIGONOMETRIO, 18.5 1h 1229 Logarithms. 1570 10h M.S. M., Sine. Cosecant. Tangent. | Cotangent. Secant. Cosine. M. M. S. 28 || 0 || 9,57358 10.42642 || 9.60641 10.39359 || 10.03283 9.96717 | 60 || 32 4 l 57389 42611 60677 39323 0.3289 96711 59 || 56 8 2 57420 42580 60714 39.286 03294. 96.706 || 58 52 12 3 57451 42549 60750 39250 03299 96.701 || 57 || 48 16 4 57482 425.18 60786 39214 3304 96696 || 56 || 44 20 5 || 9.57514 || 10.42486 || 9.60823 10.391.77 || 10.03309 9.96691 55 || 40 24 6 57545 42455 60859 3914.1 03314 96686 54 || 36 28 7 57576 42424 60895 39105 03319 96681 53 || 32 32 8 57607 42393 60931 39069 03324 96676 || 52 28 36 9 57.638 42362 60967 39033 O3330 96670 || 51 24 40 || 10 || 9,57669 10.42331 || 9,61004 10,38996 || 10.03335 9.96665 || 50 20 44 || 11 57700 42300 61040 38960 0.3340 96660 || 49 || 16 48 || 12 57.731 42269 61076 38924 03345 96655 || 48 || 12 52 || 13 57.762 42.238) 61112 38888 03350 96650 || 47 8 56 || 14 57.793 42207 61148 38852 03355 96645 || 46 4 29 || 15 9.57824 10.42176 |9.61184 10.38816 || 10.03360 9.9664() }. 45 || 31 4 16 57855 42145 61220 38780 03366 96634 44 || 56 8 I'ſ 578S5 42115 61.256 38744 03371 96629 || 43 || 52 12 18 57916 42084 61292 3S708 03376 96624 || 42 || 48 16 || 19 57947 42053 61328 38672 08381 96619 || 41 || 44 20 || 20 9.57978 10.42022 9.6.1364 10.38630 || 10.03386 9.96614 || 40 || 40 24 21 58008 4.1992 61400 38600 03392 96608 || 39 || 36 28 22 58039 41961 61436 38564 03397 96603 || 38 32 32 || 23 58070 41930 61472 38528 03402 96598 || 37 28 36 || 24 58101 4.1899 61508 38492 0.9407 96593 || 36 || 24 40 || 25 9,5S131 10.4.1869 ; 9.6.1544 10,38456 || 10.03412 9.965S8 || 35 20 44 || 26 58,162 41838 61579 38421 03418 96582 || 34 16 48 || 27 58,192 4.1808 61615 38385 03423 96.577 || 33 12 52 || 28 58223 41777 6.1651 38349 03428 96572 32 8 56 || 29 58253 41747 61687 38313 03433 96.567 || 31 4 30 || 30 || 9,58284 10,41716 |9.61722 10,38278 || 10.03438 9.96562 || 30 30 4 || 31 58314 41686 61758 38242 03444 96.556 || 29 56 8 || 32 58345 41655 61794 38206 034.49 96551 28 || 52 12 || 33 58,375 4.1625 61830 38170 03454 96546 27 || 48 16 || 34 58.406 41594 6 1865 3S135 03459 96.541 26 || 44 20 + 35 || 9,58436 10.41564 || 9.6.1901 10.38099 || 10.03465 9.96535 || 25 | 40 24 || 36 58467 41533 61936 38064 03470 96530 || 24 || 36 28 || 37 58497 41503 61972 3S028 03:475 96525 || 23 || 32 32 || 38 58527 4147: 62008 37992 03480 96520 22 || 28 36 39 58557 41443 62043 37957 034.86 96.514 || 21 24 40 40 || 9.5S588 I0.41412 || 9.62(179 10,37921 || 10.03491 9.96509 20 || 20 44 || 41 586.18 41.382 62114 37886 03496 96504 || 19 16 48 || 42 58648 41352 G2150 37.850 0.3502 96.498 || 18 || 12 52 || 43 58678 41322 62.185 37.815 0.3507 96493 17 8 56 || 44 58709 41291 6222] 37779 03512 96.488 || 16 4 31 45 || 9,58739 10.41261 9.62256 10,37744 10.03517 9.964S3 || 15 29 4 || 46 58769 41231 62292 37'708 03523 96477 || 14 || 56 8 || 47 58799 41201. 62327 37673 03528 96472 || 13 || 52 12 || 48 58829 41171 62362 37638 03533 96.467 || 12 || 48 16 || 49 58859 41.141 62398 37602 03539 96461 || 11 || 44 20 50 || 9.58889 10.4.1111 || 9.62433 10,37567 || 10.03544 9,96456 || 10 || 40 24 || 51 58919 41081 62468 37532 0.3549 96.451 9 || 36 28 52 58949 41051 62504 37496 03555 964.45 8 || 32 32 53 58979 41021 62.539 37461 03560 9644() 7 28 36 || 54 59009 40991 6257.4 37.426 03565 96435 6 || 24 40 || 55 || 9,59039 10.40961 || 9.62609 10.37391 || 10.03571 9.96429 5 i 20 44 || 56 59069 40931 62645 37355 0.3576 964.24 4 16 48 || 57 590.98 40902 626.80 37320 03581 96419 3 12 52 || 58 591.28 40872 62715 37285 03587 964.13 2 8 56 || 59 59158 40842 62750 37250 0.3592 96408 I 4. * 32 || 60 5918.8 40812 62785 37215 03597 96403 () 28 M. S. M. Cosine. Secant. Colangent! Tangent. Cosecant. Sine. M. M. S. 7b. 1129 67° 4h LOGARITHMS TRIGONOMETRIC. 1h 1239 Logarithms. 1569 |10h M. S. M. Sine. Cosecant. | Tnugent. Cotangent. Secant. Cosine. M M.S. 32 || 0 || 9,59188 10.40812 || 9.62785 10,37215 || 10.03597 9.96403 || 60 || 28 4 I 5921S 4()7S2 62820 37180 0.3603 96.397 59 || 56 8 2 59.247 40, 53 62855 37145 0360S 96.392 || 58 || 52 12 3 59.277 40723 62890 37110 03613 96387 || 57 || 48 16 4 59307 40693 62926 37074 03619 963S1 || 56 44 20 5 || 9.59336 10.40664. 19.62961 10.3703 10.03624 9.96376 || 55 || 40 24 6 59366 40634 62996 37004 03630 96370 || 54 || 36 2S 7 59396 40604 (53031 36969 03635 96365 53 || 32 32 8 59.425 40575 63066 36934 03640 96360 52 28 36 9 5945.5 40545 63101 36899 03646 96.354 || 51 24 40 || 10 || 9,59484 10.40516 || 9.63135 10.36865 10.03651 9.96349 || 50 20 44 || 11 59514 40486 6317() 36830 0.3657 90343 || 49 16 48 || 12 59.543 40457 63205 36.795 03662 96.338 || 48 || 12 52 13 59573 40427 63240 36760 0.36b'ſ 96.333 || 47 8 56 || 14 59.602 40398 G327.5 36725 03673 96327 46 4 33 || 15 9,59632 10.40368 || 9.63310 10,36690 10.03678 9.96322 || 45 27 4 16 59661 40339 63345 36655 0.3684 96316 44 || 56 8 || 17 59.690 40310 63379 J6621 0.3689 96311 43 52 12 || 18 59.720 40280 63414. 36586 03695 96305 || 42 || 48 16 || 19 59749 40251 634.49 30551 03700 96300 || 41 44 20 || 20 || 9,597.78 10.40222 || 9,634S4 || 10.365.16 || 10.0.3706 9.96294 40 || 40 24 || 21 59S08 40192 6.3519 36481 03711 || > 96.289 || 39 || 36 28 || 22 59837 4016.3 63553 36447 03716 96.284 || 38 || 32 32 || 23 59866 40134 63588 36412 (13722 9627S 37 28 36 || 24 59895 40105 63623 36.377 09727 96273 || 36 24 40 |25 || 9.59924 10.40076 || 9.6365'ſ 10.36.343 || 10.03733 9.96267 || 35 20 44 || 26 59954 40046 63692 36308 0373S 96262 || 34 || 16 48 || 27 59983 40017 63726 3627.4 0.3744 96.256 || 33 || 12 52 || 28 60012 39988 637 (31 36239 03749 96251 || 32 8 56 29 600 11 39959 63796 36.204 037:55 96.245 || 31 4 34 || 3 9.6007() 10.39930 9.63S30 10.:}(31.70 || 10.03700 9.96240 || 30 26 4 : 31 60099 3990]. 63S65 36135 ()3766 962.34 29 || 56 8 || 32 60.128 39872 63899 36101 03771 96229 28 || 52 12 || 33 60.157 398.43 63934 36066 03777 96223 27 || 48 16 || 34 6()186 39814 63968 36032 037.82 96218 || 26 || 44 20 || 35 | 9.60215 10,39785 || 9.64003 10.35997 || 10.03788 9.96212 || 25 | 40 24 36 60244 39756 64037 35963 03793 962()7 || 24 36 28 || 37 60273 39727 64()72 35928 03799 96.201 || 23 32 32 || 38 6()302 39698 64100 35894 U3S()4 96.196 || 22 || 28 36 || 39 60331 39669 64140 35860 03810 96.190 21 || 24 40 || 40 || 9.6()359 10.39641 || 9.64175 10.35825 || 10.03815 9.96.185 || 20 20 44 || 41 60388 3961.2 64209 3.5791 0.3821 96.179 19 || 16 48 || 42 60417 395.83 64243 35757 03826 96.174 18 || 12 52 || 43 60446 3955.4 64278 3.5722 03832 96.168 17 8 56 | 44 60474 395.26 64312 356SS O3S38 96.162 16 4 35 || 45 || 9.60503 10.39497 || 9.64340 10.35654 || 10.03843 9.96157. 15 || 25 4 || 46 605:32 39468 64381 35,019 03849 96.151 || 14 || 56 8 I 47 60561 39439 644.15 35585 03854 96.146 || 13 52 12 || 48 60589 394.11 64449 3555.1 03860 96140 || 12 || 48 16 || 49 60618 39.382 6+483 3551.7 03S65 96.135 | 11 || 44 20 || 50 9.60646 10,39354 - 9.645 l'7 10.35483 || 10.03871 9.961.29 || 10 || 40 24 || 51 60675 39325 (34.552 35448 (3S77 96.123 9 || 36 28 || 52 60704 39296 G4586 35414 03882 96.118 8 || 32 32 || 53 60732 3926S 64620 353SU) 03888 96.112 7 || 28 36 54 60761 39239 64654 35346 03S93 96.107 6 || 24 40 55 9.6()789 10,392.11 || 9.6468S 10.35312 || 10.03899 9,96101 6 || 20 44 || 56 60818 391.82 6 º'722 35278 03905 96.005 4 16 48 || 57 60846 39154 64756 35244 03910 96090 3 || 12 52 || 58 60875 39.125 (54790 35210 0.3916 96084 2 8 56 || 59 60903 39097 04824 3517 6 03921 96079 I 4 36 60 60931 39069 64S58 35142 ()3927 96073 0 24- M. S. M Cosine. Secant. Cotangentl Tangent. Cosecant. Sine. M. M. S. 7h ||1139 66° 4h LOGARITHMS TRIGONOMETRIC. 1h |24° Logarithms. 155° 10h M.S. M Sine. Cosecant. | Tangent. Cotangent. Secant. Cosine. M. M. S. 36 0 9.60931 10,39069 || 9.64S58 10,35142 | 10.03927 9.96073 || 60 || 24 4 l 60960 390.40 64892 35108 03933 96.067 || 59 || 56 8 2 60988 39()12 64926 35074 03938 96062 || 58 52 12 3 61016 38984 64960 35040 03944 96.056 57 || 48 16 4 61045 38.955 64994 35006 O3950 96.050 || 56 || 44 20 5 || 9,61073 10,38927 || 9.65028 10,34972 || 10.03955 9.96045 || 55 40 24 6 61101 38899 65062 34938 0.3961 96.039 || 54 36 28 7 61129 38871 65096 34904 03966 96034 || 53 : 32 32 8 6115S 38842 65130 34870 0.3972 96.028 || 52 i 28 36 9 61186 38S14 65164 3 1836 03978 96022 || 51 || 24 40 || 10 || 9.61214 10.38786 9.65197 10.34803 || 10.03983 9.96017 | 50 i 20 44 || 11 61242 3875S 65231 34769 03989 96.011 || 49 || 16 48 12 61270 387.30 65265 3 1735 03995 96005 || 48 12 52 || 13 61298 38.702 65.299 34.701 0.4000 96000 || 47 8 56 || 14 61326 38674 65.333 31667 04:006 95.994 || 46 4 37 || 15 9.61354 10.38646 9.65.366 10.34634 || 10,04012 9.95988 || 45 || 23 4 || 16 61.382 386.18 (55400 34600 04018 95.982 44 || 56 8 17 G1411 38589 65434 34566 04023 95.977 || 43 || 52 I2 18 6143S 38562 65467 34533 04029 95971 || 42 || 48 I6 || 19 61466 38534 65501 34499 04035 95965 || 41 || 44 20 | 20 || 9.61494 10.3S506 9.65535 10.34465 10,04040 9.95960 40 | 40 24 || 21 61522 3S478 t;5568 34.432 0.4046 95954 || 39 || 36 2S 22 61550 3.8450 65602 34398 04052 95.948 || 38 || 32 32 || 23 G1578 3S422 65636 34364 0405S 95042 || 37 i 28 36 || 24 ($1606 3839-4 65669 3433 1 ()4063 95937 || 36 ſ 24 40 || 25 9.6.1634 10.38366 || 9.65703 10.34297 || 10.04069 9.95931 || 35 | 20 44 || 26 6] 662 3833S 65736 34264 0407.5 95925 || 34 || 16 48 27 61689 38311 65770 34-23U) 04080 95.920 i 33 || 12 52 28 61717 38283 65803 3419.7 04086 95914 || 32 8 56 29 61745 3S255 65837 34163 04092 9590S 31 4. 38 || 3() || 9.61773 10.3S227 | 9.65S70 10.34130 || 10.04098 9.95902 || 30 || 2'2 4 || 31 61800 3S200 b5904 34096 04:103 95S97 29 56 8 || 32 61828 3817.2 659,37 34063 0.4109 95891 28 52 12 || 33 61856 38144 65971 34029 04115 95885 27 || 48 16 || 34. 61883 38117 6600.4 33996 04121 95879 || 26 || 44 20 || 35 ; 9,01911 10.38089 || 9,66038 10.33962 || 10.04127 9.95873 || 25 40 24 || 36 61939 38061 (36071 33929 04132 95S6S 24 || 36 28 || 37 61966 38034 66104 33896 04138 95862 || 23 || 32 32 || 38 61994 3S006 661.38 33862 04144 95S56 22 || 28 36 || 39 62021 37979 G6171 33829 04 50 95850 21 || 24 40 || 40 9.62049 10.37951 9.66204 10.33796 || 10.04.156 9.95844 20 | 20 44 || 41 62070 37.924 6623S 3.37.62 04:161 95839 19 || 16 48 || 42 62104 37896 66271 33729 04167 95833 18 || 12 52 43 62.131 37869 66304 33696 04:173 95827 17 8 50 || 44 62.159 37841 66337 33663 04179 9582] 16 4 39 || 45 9.62186 10,37814 || 9,66371 10,33629 || 10.04.185 9.958.15 || 15 21 4 || 46 G2214 37786 66.404 33596 04:190 95810 || 14 || 56 8 || 47 62241 37759 (36437 33563 04:196 958.04 || 13 || 52 T2 || 48 62268 37732 6647() 335.30 04202 9579S | 12 || 48 16 || 49 62296 7704 66503 33497 0420S 95792 || 11 || 44 20 || 5() 9.6232. 10.37677 |9.66537 10.33463 || 10.04214 9.95786 10 || 40 24 || 51 02350 37.650 (36570 33430 04220 95780 9 || 36 28 || 52 62377 37623 66003 33397 ()4225 95775 8 32 32 || 53 624ſ)5 37595 666.36 33364 04231 95.769 7 28 36 54 62432 37568 66669 33,331 04237 95.763 6 || 24 40 || 55 9.62459 10.37541 |9.66702 10.33298 || 10 ()4243 9.95757 5 || 20 44 || 56 62486 37.514 (36735 33265 04249 95.751 4 || 16 48 || 57 62513 3.7487 GG76S 33232 04:255 95.745 3 || 12 52 || 58 62541 37459 66801 33199 04:261 95.739 2 8 56 59 62568 37.432 66S34 3.316t, 04:267 9573. l 4. 40 60 62595 37405 (,6867 33.133 04:272 95.728 0 || 20 M. S. Cosine. Secant. ICotangent Tangent. Cosecant. Sune. M. M. S. 7h 11 65° 4h 188 LOGARITHMS TRIGONOMETRIC. 1h |250 Logarithms. 1549|10h M.S. M Sine. Cosecant. Tangent. | Cotangent. Secant. Cosime. M. M. S. 4-0 || 0 || 9.62595 10,37405 || 9.66867 10.33133 || 10.04272 9.95.728 60 || 20 4. I 62622 37378 669()0 33100 04:278 95.722 || 59 || 56 8 2 62649 3T351 66.933 33067 04:28.4 95716 58 || 52 12 3 62676 37324 66966 33034 04290 957 10 || 57 || 48 16 4 62703 37.297 66999 33001 04:296 95704 || 56 || 44 20 5 || 9.62730 10,372.70 || 9.67032 10.32968 || 10.04302 9.95698 || 55 40 24 6 62757 37243 67065 33935 04308 95692 || 54 || 36 28 7 62784 37216 67098 32902 04314 95686 53 || 32 32 8 62S11 371S9 67131 32S69 0.4320 956S0 52 || 28 36 9 62838 37162 67163 32837 ()4826 95674 || 51 24 40 || 10 || 9.62865 10,37135 || 9,67196 10.32804 || 10.04832 9.95668 || 50 20 44 || 11 62892 3710S 67229 327.71 04837 95663 49 || 16 48 || 12 62918 37082 67.262 32738 04:343 95657 || 48 12 52 13 62945 37055 67295 32705 04349 95651 || 47 8 56 || 14 62972 37028 67327 32673 0.4355 95.645 || 46 4 41 15 || 9.62999 10.37001 |9.67360 10.32640 || 10.04361 9.95639 || 45 19 4 || 16 63026 36974 67393 32007 0.4367 95633 I 44 56 8 || 17 63052 36948 ($74.26 32574 04:373 95627 || 43 || 52 12 || IS G3079 36921 67458 32542 04379 95621 || 42 || 48 16 || 19 63106 36894 67491 32509 04:385 95615 || 41 || 44 20 | 20 || 9,63133 10,36867 || 9,67524 10.32476 || 10.04391 9.95609 || 40 || 40 24 || 21 G3159 36841 b'7556 32444 04397 95603 || 39 || 36 28 22 63186 36814 67 589 32411 04403 95597 || 38 || 32 32 || 23 632.13 36787 67622 32378 0.4409 95591 || 37 || 28 36 || 24 63.239 36761 67654 32346 04415 955S5 || 36 || 24 40 || 25 || 9.63266 10.36734 || 9,676S7 10.32313 || 10.04421 9.95579 || 35 | 20 44 || 26 63292 36708 67719 32281 0.4427 95573 || 34 || 16 48 || 27 63319 36681 67,752 3224S 04:433 95567 || 33 || 12 52 2S 63345 36655 67785 32215 04:439 95561 32 8 56 29 63372 56628 67S17 32183 04445 95555 31 4 4-2 || 30 || 9.63398 10,36602 || 9.67850 10,32150 || 10.04451 9.95549 30 18 4 || 31 63425 36575 67882 32.118 04:457 95543 29 || 56 8 || 32 63451 36549 67915 32085 04463 95537 2S 52 12 || 33 63.478 36522 67.947 32053 04:469 95531 || 27 || 48 16 || 34 G3504 36496 67.980 32020 04475 95525 26 || 44 20 || 35 | 9.63531 10.36469 || 9.68012 10.31988 || 10.0448l 9,95519 || 25 | 40 24 || 36 63557 36443 6S044 31956 044ST 95513 || 24 36 28 37 63583 36417 680.77 3.1923 04493 95.507 || 23 || 32 32 || 38 63610 36390 68109 31891 04500 95500 22 || 28 36 || 39 63636 36364 68142 3185S 04506 95494 | 21 24 40 | 40 || 9.63662 10.3633S 9.68174 10 31826 || 10.04512 9.95488 || 20 | 20 44 i 41 63689 36311 68206 31794 04518 95.482 19 || 16 48 || 42 63715 36285 68239 31761 04524 95476 18 || 12 52 || 43 63741 36259 6S271 31729 04530 954.70 || 17 8 56 || 44 63767 36233 6S303 3.1697 04:536 95.464 || 16 4. 43 || 45 9.63794 10.36206 || 9,68336 10.31664 10.04542 9.95458 15 17 4 || 46 63820 361SO 6836S 31632 04548 95.452 || 14 || 56 S || 47 63S46 36154 68400 3.1600 04:554 95446 || 13 || 52 T2 || 48 63872 36128 68432 3.1568 04:560 95440 12 || 48 16 49 63898 36102 68465 31535 04566 95.434 || 11 || 44 20 50 9.63924 || 10.36076 |9.68497 10.31503 || 10.04573 9.95427 | 10 | 40 24 || 51 63950 36050 68529 3.1471 04579 95.421 9 || 36 28 || 52 63976 36024 68561 3.1439 0.4585 95.415 8 32 32 || 53 64002 35998 68593 31.407 04:591 95.409 7 || 28 36 || 54 6402S 35972 (58626 31374 04597 95.403 6 || 24 40 || 55 || 9.64054 10.35946 9.6S658 10.31342 || 10.04603 9.95397 5 i 20 : 44 || 56 64080 35920 68690 31310 04609 95.391 4 || 16 48 57 64 106 35894 687.22 31278 0.4616 95.384 3 || 12 52 || 5S 64 132 35868 68.754 31246 04622 95.378 2 8 50 || 59 64.158 35842 GS786 31214 04628 953.72 l 4. 44 || 60 6418.4 35816 6SS18 31182 0.4634 95366 () | 16 M. S. M I Cosine. Secant. ICotangent| Tangent. || Gosecant. Sine. M AI. S. 7h 1159 64° 41i LOGARITHMS TRIGONOMETRIC, 1 h 269 Logarithms. 1539 M.S M Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M 44 || 0 || 9,64184 10,35816 || 9,688.18 10.31182 | 10.04634. 9,95366 || 60 • 4 1 64210 35790 68850 3.1150 0.4640 95360 || 59 S 2 64236 357.64 68S82 31.118 04646 95.354 || 58 12 3 64262 35.738 68914 31086 04652 95.348 57 16 4 6428S 35.712 68.946 31054 04659 95341 || 56 20 5 9,64313 10.35687 |9.68978 10.31022 || 10.04665 9.95335 55 24 6 64.339 35661 69010 30990 04671 95.329 54 28 7 64.365 35635 69042 30958 0.4677 95323 53 32 8 64.391 35609 69074 30926 04683 95.317 | 52 36 9 ($44.17 35.583 69106 30894 04690 95.310 || 51 40 || 10 || 9.644.42 10,35558 || 9.69138 10.30862 || 10.04696 9,95304 || 50 44 || 11 64468 35.532 69170 30830 04702 9529S 49 48 || 12 64494 35506 69202 30798 0470S 95.292 || 48 52 || 13 64519 354S1 || 69234 30766 04714 95286 || 47 56 || 14 645.45 35455 69266 3073 04721 95279 || 46 45 || 15 9,64571 10.35429 |9.69298 10,30702 || 10,04727 9.95273 || 45 4 || 16 64596 35404 69329 3067.1 04733 95267 || 44 8 || 17 64622 35378 69361 30639 04739 95261 || 43 12 || 18 64647 35353 69393 30607 04746 95.254 || 42 16 || 19 64673 35327 69425 30575 04752 95.248 || 41 20 | 20 || 9,64698 I0.35.302 || 9.69.457 10.30543 || 10.047.58 9.95.242 | 40 24 21 64724 35276 69488 30512 04764 95236 || 39 28 22 64749 35:251 69520 30480 04771 95.229 || 3S 32 || 23 647.75 35225 69552 3044S 04777 95223 || 37 36 24 64800 35200 69584 30416 04783 952.17 || 36 40 # 25 9.64S26 10.35174 || 9,696.15 10.30385 || 10.04789 9,952.11 || 35 44 26 6485.1 35149 69647 30353. 04796 95204 || 34 4S 27 64877 35123 69679 30321 04S02 95198 || 33 52 28 64902 35098 69710 30290 0480S 95192 || 32 56 || 29 64927 35073 69'742 30258 04815 95185 31 46 || 30 || 9,64953 10,35047 || 9.69774 || 10.30226 10.04821 9.95.179 || 30 4 || 31 64978 35022 69805 30195 O4S27 95173 || 29 8 || 32 65003 34997 69837 30163 04833 95167 || 2S 12 || 33 65029 34971 69868 301.32 04840 95.160 27 16 || 34 65054 34946 69900 30.100 04S46 95154 || 26 20 || 35 | 9.650.79 I0.34921 || 9.69932 10,3006S 10.04852 9.95148 || 25 24 || 36 65104. 34896 69963 3003 04859 951.41 24 28 || 37 65130 34870 60995 30005 04865 95135 23 32 || 3S 65155 34845 '70026 299.74 04871 951.29 || 22 36 39 651S0 34820 7005S 29942 048.78 95.122 || 21 40 || 40 || 9,65205 10.34795 || 9,700S9 10.29911 || 10,04SS4 9,951.16 20 44 || 41 65230 34770 701.21 29S79 ()4890 95110 || 19 48 || 42 65255 34745 70152 29848 0.4897 95.103 || 18 52 || 43 652S1 34719 701S4 29816 04903 95097 17 56 | 44 65306 34694 70215 29785 04910 95090 | 16 4-7 || 45 9.65331 10.34669 || 9,70247 10.297.53 || 10.04916 9.950S4 15 4 || 46 65356 34 70278 297.22 04922 95078 || 14 8 || 47 65381 34619 '70309 29691 04929 95071 || 13 12 || 48 65406 34594 70341 296.59 04935 95005 || 12 16 || 49 65431 34569 70372 29628 0.4941 95059 || 11 20 || 50 || 9.65456 || 10,34544 9.70404 || 10.29596 || 10,04948 9.95052 || 10 24 || 51 654S1 3451 70435 29565 04954 950.46 9 28 52 65506 34494 70466 295.34 04961 95039 8 32 53 65531 34469 70498 29502 04967 95033 7 36 54 65556 34444 70529 2.94.71 0.4973 95027 6 10 || 55 9.65580 10.34420 || 9.70560 10.294-40 || 10,04980 9.95020 5 44 || 56 65605 34.395 70592 2940S 0.4986 95014 4. 48 57 6563 34370 70623 2937.7 ()4993 95007 3 52 || 5S 65655 34.345 70654 29346 0.4999 95001 2 56 || 59 65680 34320 70685 29315 05()(); 94995 l 48 || 60 65705 34295 70717 292S3 05012 94988 0 M. S. M Cosine. Secant. ICotangentl Tangent. CoscCant. Sine. M 7h 116° 639 190 LOGARITHMS TRIGONOMETRIO, 1h 1270 Logarithms. 1529 |10h M.S. M. Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosime. M. M. S. 4-8 || 0 || 9.65705 10,34295 || 9.70717 10.29283 || 10.05012 9.94988 || 60 || 12 4 l 65729 34271 70748 29252 05.018 94982 59 || 56 8 2 65754 34246 70779 29221 0.5025 94975 58 || 52 12 3 65779 34221 7()810 29.190 05031 94909 || 57 || 48 16 4. 65804 34,196 ||. 70841 291.59 05038 94.962 56 || 44 20 5 || 9.65828 10,341.72 9.70873 10.29127 | 10.05044 9.94.956 || 55 40 24 6 65853 34147 '70904 29096 05051 94949 || 54 || 36 28 7 65878 34122 70935 29065 05.057 94943 || 53 || 32 32 8 659()2 34098 '70966 29034 05064. 9.4936 || 52 28 36 9 65927 34073 70997 29003 05070 94930 || 51 24 40 || 10 || 0.65952 10,340.48 || 9,71028 10.28972 || 10.05077 9.94923 || 50 20 44 || 11 65976 34024 71059 2S941 05083 94917 || 49 || 16 48 || 12 6600 33999 71090 2S910 05089 94911 || 48 || 12 52 || 13 66025 33975 7112 28879 05096 94904 || 47 8 56 || 14 66050 33950 71153 28S47 05102 94898 || 46 4 49 || 15 9.66075 10,33925 || 9.7.1184 || 10.28810 || 10.05109 9.94891 || 45 || 11 4 || 16 66099 33901 '71215 28785 05] 15 94885 || 44 || 56 8 || IT 66124 33S76 71246 28754 05122 94878 || 43 || 52 12 || 18 66148 3.3852 71277 28.723 05129 94871 42 || 48 16 || 19 66173 33S27 7|1308 28692 05135 94.865 || 41 || 44 20 20 9.66197 10.33S03 || 9.71339 10.28661 || 10.05142 9.94858 || 40 40 24 21 66221 337.79 71370 28630 05148 94.852 || 39 || 36 2S 22 66246 33754 71401 28599 05155 9.4845 || 38 || 32 32 23 66270 33730 71431 2S569 05161 9.4839 || 37 28 36 || 24 66295 33705 71462 2S538 05168 9.4832 || 36 24 40 || 25 || 9.66319 10,33681 |9.7 1493 10.28507 || 10.05174 9.94826 || 35 | 20 44 || 26 66343 3.3657 71524 284,76 05181 94819 || 34 16 4S 27 66368 33632 71555 284.45 05187 94813 || 33 || 12 52 28 66392 33608 'ſ 1586 28414 05.194 94.806 || 32 8 56 20 66416 335S4 71617 2S383 0.5201 94790 i 31 4 50 || 30 || 9.66441 10,33559 || 9.71648 10.28352 || 10.05207 9.94.793 || 30 || 10 4 31 66.465 33535 716.79 28321 05214 94.786 || 29 || 56 8 || 32 66489 33511 71709 28291 05:220 94780 28 || 52 12 || 33 66513 33487 71740 2S260 05:227 94.773 27 || 48 16 || 34 66537 33.463 71771 28229 05233 94.767 26 || 44 20 || 35 | 9.66562 10.33438 || 9.71802 10.28198 || 10.05240 9.94.760 25 || 40 24 36 665S6 33414 7.1833 2S167 05247 94.753 24 || 36 28 37 G6610 33390 '71863 28137 05253 94747 || 23 32 32 || 3S 66634 33.366 71894 28106 05260 94740 22 28 36 39 66658 33342 71925 28075 05266 94.734 21 || 24 40 40 || 9.66682 10.33318 || 9,71955 10.28()45 || 10.05273 9.94727 | 20 20 44 || 41 66706 3.3294 71.9S6 2S014 05:280 94.720 | 19 || 16 48 42 66731 33269 7.2017 27983 05286 94.714 18 || 12 2 : 43 66755 33245 72048 27952 05:293 94.707 || 17 8 56 || 44 66779 3.3221 72078 27922 05300 94700 || 16 4. 51 || 45 || 9,66803 10,33197 || 9,72109 10.27891 || 10.05306 9.94694 || 15 || 9 4 46 66827 33173 72140 27860 053.13 946.87 14 || 56 S 47 66851. 33140 72170 27830 05:320 94680 || 13 52 12 || 48 66875 33125 'W220I. 27799 05326 94674 || 12 || 48 16 || 49 66S99 331()1 72231 27769 0.5333 94.667 || 11 || 44 20 50 || 9.66922 10,33078 9.72262 10.27738 || 10.05340 9.94660 || 10 || 40 24 || 51 66946 33054 72293 27707 05346 94.654 9 || 36 28 || 52 6697() 33030 72323 27677 05353 94647 8 || 32 32 || 53 66994 33006 72354 27646 05360 94640 7 || 28 36 || 54 6701S 32982 72384 27.616 05366 94.634 6 24 10 || 55 9.67042 10.32958 || 9.724.15 10.27585 || 10.05373 9.94627 5 || 20 44 || 56 67066 32934 72445 27555 05380 94620 4 || 16 48 57 07090 3.291() '72476 27524 05386 94614 3 || 12 52 5S 67113 32887 72506 27494 05393 94.607 2 8 56 59 67137 32863 7.2537 27463 05400 94600 l 4 52 || 60 6716] 32839 7256'ſ 27433 05407 94593 0 || S M. S. M. Cosine. Secunt. Cotangent Tangent. Cosecant. Sine. M. M. S. 7h [1179 62° 4b. LOGARITHMS TRIGONOMETRIC. 191 1h |289 Logarithms. 151° 10h M.S. M Sine. Cosecant. | Tangent. Cotangent. Secant. Cosine. M M. S. 52 || 0 || 9.67161 10.32839 || 9.72567 10,27433 || 10.05407 9.94593 || 60 || 8 4 l 67185 32S15 72598 27402 05413 94587 || 59 56 S 2 67208 32792 72628 273.72 05420 94580 || 58 || 52 12 3 67232 32.768 72659 27341 05427 94573 || 67 || 48 16 4 67256 32744 72689 27311 05433 94567 || 56 || 44 20 5 || 9.67280 10,32720 9.72720 10.27280 || 10.0544() 9.94560 55 || 40 24 6 67303 32697 72750 27250 05447 94553 54 || 36 28 7 67327 32673 72780 27.220 05454 94546 || 53 || 32 32 8 67350 32650 72811 27 189 05460 94540 || 52 || 28 36 9 67374 32626 72841 27159 05467 94.533 51 || 24 40 || 10 || 9.67398 10.32602 || 9,72872 10.27128 || 10.05474 9.94526 50 20 44 11 67421 32579 72902 27098 054Sl 945.19 || 49 16 48 || 12 67.445 32555 72932 27068 05487 94513 || 48 || 12 52 || 13 67468 32532 72963 27037 05494 94506 || 47 8 56 14 G7492 32508/I '72993 27007 05501 94499 || 46 4 53 15 || 9.675.15 10,32485 9.73023 10.26977 || 10.05508 9.94492 || 45 || 7 4 || 16 67539 32461 73054 269 16 05515 94485 || 44 || 56 8 || 17 67.562 32438 73084 26916 ().5521 94479 43 52 12 || 18 67586 32414 73114 26886 05528 9-1472 || 42 || 48 16 || 19 67609 32391 73144 26856 055.35 94465 || 41 || 44 20 | 20 || 9.67633 10.32367 || 9.73175 10.26825 || 10.05542 9.94.458 || 40 | 40 24 || 21 67650 $2344 73205 26795 055-19 94451 || 39 || 36 28 22 67680 32320 73235 20765 05555 94.445 || 38 || 32 32 || 23 67703 32297 73265 26735 05562 94438 || 37 || 28 36 | 24 67726 32274 73295 26705 05569 94431 || 36 || 24 40 || 25 9.67750 10.32250 || 9,73326 10.26674 || 10.05576 9.94:424 || 35 | 20 44 || 26 67773 32227 73356 26644 055S3 94417 | 34 || 16 48 27 67.796 32204 73386 26614 05:590 9.4410 || 33 12 52 || 28 67S20 32.180 73416 26584 0.5596 94.404 || 32 8 56 || 29 67843 32157 73446 26554 ()5603 94397 || 31 4 54 || 30 || 9.67866 10,32134 || 9.73476 10.26524 || 10.05610 9.9439 30 || 6 4 || 31 67890 32110 7.3507 26493 05617 943S3 || 29 || 56 8 || 32 67.913 32087 7353 26463 0.5624 9.4376 || 28 || 52 12 || 33 66936 32064 73567 26433 0.5631 94369 || 27 || 48 16 || 34 67.959 3204l 73597 26403 05638 94362 26 || 44 20 || 35 9.67982 10.32018 || 9.73627 10.26373 || 10.05645 9.94355 || 25 | 40 24 36 G8006 31994 73657 263.43 05651 94349 || 24 || 36 * 28 || 37 68029 3.1971 736S7 26313 05658 943.42 | 23 32 32 || 38 68052 31948 737.17 26.283 0.5665 943.35 | 22 28 36 || 39 680.75 31925 '73747 26253 05672 94328 || 21 || 24 4() | 40 || 9.68098 10.31902 9.737.77 10.26223 || 10.05679 9,94321 20 | 20 44 || 41 681.21 3.1879 73S07 26193 056S6 943.14 19 16 48 I 42 68144 3.1856 73837 26163 05693 94307 || 18 12 52 || 43 6S167 3.1833 73S67 26133 0570() 94300 || 17 S 50 | 44 68190 31810 73897 26103 05707 94.293 16 4. 55 || 45 || 9.68213 10.31787 || 9.73927 10.26073 || 10.05714 9.942S6 || 15 || 5 4 || 46 6S237 31763 73957 26043 05721 94.279 || 14 || 56 8 || 47 6S260 31740 73987 26013 057 27 94.273 || 13 || 52 T2 || 48 6S2S3 31717 74017 25983 Q5734 94.266 | 12 || 48 16 || 49 68305 31695 '74047 25953 0574l 94.259 || 11 | 44 20 || 5() 9.68328 10.31672 || 9,74077 10.25923 || 10.05748 9.94.252 || 1() 40 24 || 51 68351 31649 74.107 25893 05755 94.245 9 || 36 2S 52 68374 31626 74.137 25863 05762 94238 8 || 32 32 53 6S397 31603 74166 25$34 05769 94231 7 28 36 || 54 68420 31580 '74,196 25804 (J5776 94.224 6 || 24 40 || 55 || 9.68443 10.31557 |9.74226 10.25774 || 10,057.83 9.94217 5 i 20 44 || 56 68406 31534 74256 25744 05790 94.210 4 | 16 48 57 684$9 31511 7.1286 25714 05797 94203 3 || 12 52 || 58 68512 3.1488 7.4316 25684 05804 94.196 2 8 56 || 59 68534 31466 74345 25655 05811 94189 1. 4 56 || 60 6S557 31443 74375 25625 05S18 94182 0 || 4: M. S. M Cosimo. Secant. IGotangent| Tangent. || Cosecant. Sine. M. M. S. 7h 118 61° 4h LOGARITHMS TRIGONOMETRIC. 1b | 290 Logarithms. 150° 10m M.S. M Sine. Cosecant. | Taugent. | Cotangent. Secant. Cosine. M. XI. S. 56 || 0 || 9.68557 | 10.31443 || 9,74375 || 10,25625 || 10.05818 9.94.182 || 60 || 4 4 || 1 685S0 3142() || 74405 25595 05S25 94.175 59 || 56 8 2 68603 31397 74435 255.65 058.32 94168 58 || 52 12 3 68625 3.1375 74465 25535 05839 94161 || 57 || 48 16 || 4 6$648 31352 74494 25506 05846 94154 || 56 || 44 20 || 5 || 9.68671 || 10.31329 |9.74524 || 10.25476 || 10.05853 9.94.147 || 55 | 40 24 || 6 68694 31306 74554 .254.46 05S60 94140 || 54 || 36 28 7 68716 31284 || 7.45S3 25417 O5867 94.133 || 53 || 32 32 8 68.739 31261 746.13 25387 05874 94.126 || 52 l 28 36 9 68762 3123 7.4643 25.357 05SS1 94119 || 51 || 24 40 || 10 || 9.68784 10.31216 || 9,74673 10,253.27 | 10.05888 9.94.112 || 50 | 20 44 || 11 68S()7 31193 74.702 25298 05895 94105 || 49 16 48 || 12 6SS29 3.1171 74732 25268 05902 94098 || 48 || 12 52 13 68852 31148 74762 25238 05910 9.4090 || 47 8 56 14 6SS75 31.125 747.91 25209 05917 94083 || 46 || 4 57 || 15 9.68.897 10.31103 || 9.74821 | 10.25.179 || 10.05924 9.94076 45 || 3 4 16 6SQ20 31080 74851 25149 05931 94069 44 || 56 8 || 17 68942 31058 '74S80 25120 05938 94062 || 43 52 12 || 18 68965 31035 74910 25090 05945 94055 || 42 || 48 16 || 10 680ST 31013 74939 25001 O5952 94048 || 41 || 44 20 | 20 9.69010 || 10.30990 |9.74969 10.25031 || 10.05959 9.94041 40 | 40 24 || 21 6903.2 30968 74998 25002 05966 94034 || 39 || 36 28 22 69055 30945 '75028 24972 05973 9.4027 38 32 32 23 69.077 30923 7505S 24942 059.80 94020 || 37 || 28 36 || 24 69100 30900 75087 24913 05988 94012 || 36 || 24 40 25 9.69122 || 10.30878 9.75117 | 10.24883 || 10,05995 9.94005 || 35 20 44 || 26 69144 30856 75.146 2.4854 06002 93.998 || 34 || 16 48 || 27 691.67 30S33 751.76 28,824 06009 93991 || 33 || 12 52 || 28 691.89 30811 '75205 24.795 06016 939S4 || 32 || 8 56 || 29 69212 30788 '75235 24765 06023 93.977 || 31 || 4 58 || 30 || 9.69234 || 10,30766 || 9,75264 || 10.24736 10.06030 9.93970 || 30 2 4 || 31 69256 30744 75294 24.706 06(37 93.963 29 56 8 32 69279 30721 75323 24677 06045 93955 28 || 52 12 || 33 69301 30699 '75353 24647 06052 93948 || 27 || 48 16 || 34. 69323 30677 75382 24618 06059 93941 || 26 || 44 20 35 9.69345 10,30655 9.75411 || 10.24589 10,06066 9.93934 || 25 | 40 24 || 36 69368 30632 75441 24559 06073 93927 24 || 36. 28 37 693.90 30610 75470 24530 06080 93.920 23 32 32 || 38 69412 30588 75500 24500 06088 93.912 || 22 || 28 36 39 69434 30566 7.5529 24471 06095 93.905 21 || 24 4() | 40 9.69456 || 10.30544 9.75558 10.24442 || 10.06102 9,93898 || 20 20 44 41 694.79 30521 75588 24412 06109 9.3891 19 || 16 48 || 42 60501 30499 75617 243S3 06116 93884 18 || 12 52 43 69523 30477 75647 24353 06124 93S7 17 8 56 || 44 69.545 30455 75676 24324 0.6131 93869 || 16 || 4 59 || 45 9,69567 || 10.30433 |9.75705 || 10.24295 || 10.06133 9.93862 15 1 4 || 46 469589 30411 75735 24265 06145 93.855 ; 14 56 8 || 47 G9611 30389 75764 24236 06153 938.47 || 13 52 12 || 48 696.33 30367 '15793 24.207 06160 93840 i 12 || 48 16 || 49 69655 30345 '75822 24,178 06167 93833 || 11 || 44 20 5() || 9.69677 || 10.30323 9.75852 | 10.24,148 || 10.06174 9.93826 10 || 40 24 || 51 69699 30301 75881 241.19 06181 93S19 9 : 36 28 || 52 69721 30279 75910 24090 06189 93811 8 32 32 53 69743 30257 '75939 24061 O6196 93S04 || 7 || 28 36 54 69765 30235 '75969 24031 06203 93.797 6 || 24 40 55 || 9.697.87 || 10.30213 || 9.75998 || 10.24002 || 10.06211 9.93789 5 i 20 44 || 56 69809 30.191 76027 23973 06218 93.782 || 4 || 16 48 || 57 69831 30169 76056 23944 06225 93775 || 3 || 12 52 || 58 69853 301.47 76086 23914 06.232 93768 || 2 || 8 56 59 69875 301.25 '76I15 23SS5 06240 93.760 || 1 || 4 60 || 60 69897 30.103 76.144 23856 06247 93.753 (j || 0 M. S. M I Cosine. Secunt, 10otangent Tangent. Cosecant. Sine. M M. S. 7th III99 60° 4h LoGARITHMS TRIGONOMETRIC. 2h 30° Logarithms. 1499 || 9h M.S. M. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. M. I.M. S. O 0 9.69897 $ 10.30103 || 9.76144 10.23856 || 10.06247 9.93753 60 || 60 4 Y. 69919 30081 76173 23827 06254 93746 59 || 56 8 2 6994.1 30059 76202 2379S 06262 93.738 58 52 12 3 69963 30037 76231 23769 06269 93.731 || 57 48 16 4 69984 30016 76261 2.3739 ()6276 93724 || 56 || 44 20 5 || 9.70006 10.29994 || 9.76290 10.23710 || 10.06283 9.93717 || 55 || 40 24 6 T0028 299'12 '76319 236Sl 06291 93709 || 54 || 36 28 7 '70050 29950 '76348 3652 O6298 93702 || 53 32 32 S 70072 29928 76377 23623 06.305 93695 52 28 36 9 70093 29907 '76406 23594 06.313 936.87 || 51 || 24 40 || 10 || 9,701.15 10.29885 || 9,76435 10.23565 || 10.06320 9.936.80 || 50 | 20 44 11 701.37 29863 76464 23536 06327 93673 || 49 || 16 48 || 12 70159 29S41 .76493 23507 O6335 93665 || 48 || 12 52 13 701S0 298.20 } 76522 234.78 06342 93658 I 47 8 56 || 14 702O2 2979S '76551 23449 06350 93650 46 4 1 15 || 9.70224 10.297.76 9.76580 10.23420 10.06357 9.93643 || 45 || 59 4 || 16 70245 297.55 76609 23391 06364 93636 || 44 56 8 17 | . TO267 29733 76639 23361 06372 93628 43 || 52 12 || 18 70288 29712 76668 23332 06379 93621 || 42 || 48 16 19 '70310 29690 76697 233()3 063S6 93614 || 41 || 44 20 20 || 9.70332 10.29668 9.76725 10.23275 || 10.06394 9.93606 || 40 || 40 24 || 21 '70353 29647 76754 23246 06401 93599 || 39 36 28 || 22 70375 29625 76783 23217 06:409 93591 || 38 || 32 32 23 7()396 29604 76812 23.188 06416 93584 || 37 || 28 36 || 24 70418 295S2 76841 23159 06423 93577 || 36 || 24 40 || 25 9.70439 10.2956I 9.76S70 10.23130 || 10.06431 9.93569 || 35 | 20 44 26 70461 29539 76899 23101 06:438 93562 || 34 || 16 48 || 27 7()482 295].8 '76928 23072 O6446 93554 || 33 12 52 28 70504 29.496 76957 23043 06:453 93547 32 8 56 29 70525 29.475 76986 23014 06:461 93539 || 31 || 4 2 30 || 9.70547 10,29453 || 9.77015 10.22985 || 10.06468 9.93532 || 30 58 4 31 7()568 29432 T7044 22956 06:475 93525 || 29 56 8 || 32 70590 29.410 7'7073 22927 06483 93517 || 28 52 12 33 70611 29389 77101 22S99 06490 93510 || 27 || 4S 16 || 34 70633 29367 77130 22S70 06498 93502 || 26 i 44 20 35 9.70654 10.29346 9.77159 10.22841 || 10.06505 9.93495 || 25 40 24 36 70675 29325 TT188 2281.2 06513 934S7 24 || 36 2S | 37 70697 29303 772.17 227S3 06:520 934.80 23 32 32 38 70718 2.9282 772.46 22754 U6528 93472 22 || 28 36 39 70739 29.261 77274 22726 06535 93.465 || 21 24 40 40 9.7076 10,29239 9.77303 10.22697 || 10.06543 9.93457 || 20 | 20 44 41 70782 29.218 77.332 22668 06550 93450 || 19 || 16 48 || 42 70S()3 291.97 'I'7361 22639 06558 93442 || 18 || 12 52 || 43 70824 291.76 77390 22610 06565 93435 | 17 8 56 || 44 '70S46 2915.4 '77418 22582 06573 93427 | 16 4 3 || 45 || 9,70867 10.29133 9.77447 I0.22553 || 10.06580 9,934.20 15 || 57 4 46 70888 2.9112 TT476 22524 06588 93412 || 14 || 56 8 || 47 '70909 290.91 775()5 22495 06595 93.405 || 13 || 52 i2 || 48 7()931 29069 77533 22467 (J6603 93397 || 12 || 48 16 || 49 70952 29048 77.562 2243S 06610 93390 || 11 || 44 20 50 9.70973 10.29027 9.77591 I0.22409 || 10.06618 9,93382 || 10 || 40 24 || 51 70994 29006 77619 223S1 06625 93.375 9 : 36 28 || 52 71015 28985 '77648 22352 06633 93367 8 || 32 32 53 71036 28964 77677 22323 06640 93360 7 || 28 36 || 54 71058 28942 77706 22294. 06648 93352 6 || 24 40 || 55 || 9,71079 10.28921 ; 9.77734 10.22266 || 10.06656 9.93344 5 || 20 71100 28900 77763 22237 06663 93337 4 16 71121 28879 777.91 22209 0.6671 93329 3 || 12 71142 28858 77.820 221S0 O6678 93322 2 8 71.163 28837 77.849 22151 06686 93314 I 4. 71184 28816 || 77877 22123 06693 93307 - 0 || 56 | Cosine. Secant. Cotangenti Tangent. || Cosecant. Sine. M M. S. D * 59° 3h LOGARITHMS TRIGONOMETRIC. 310 Logarithms. 1480 9h M Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. M. S. 0 9.7.1184 10.288.16 9.77877 | 10.22123 || 10.06693 ) . 9.93.307 || 60 56 1 T}205 28795 || 77906 220.94 06701 93.239 59 || 56 2 71226 2S774 || 77935 22065 06700 93.291 || 58 || 52 3 '71247 2S753 77963 22037 067 16 93.284 || 57 || 48 4 71268 2ST32 || 77992 2200S 06724 9.3276 l 56 || 44 5 9.71289 || 10.28711 |9.78020 | 10.21980 || 10.06731 9.93269 55 40 6 '71310 2S69() || 78049 21951 06739 93261 54 || 36 7 '71331 28669 || 78077 21923 0674'ſ 93.253 53 || 32 8 '71352 2864S '78106 21894 06754 93246 52 28 9 '71373 2S627 '18135 21865 06762 9.3238 51 24 10 || 9.71393 || 10.28607 || 9.TS163 || 10,21837 || 10.06770 9,932.3() || 50 | 20 11 71414 28586 || 78.192 21808 O6777 93223 || 49 || 16 12 71435 28565 78220 21780 06785 93.215 || 48 || 12 13 71456 28544 78249 21751 06793 93.207 || 47 8 14 71477 28523 78277 21723 06800 93.200 || 46 || 4 15 9.71498 || 10.2S502 || 9.7S306 || 10.21694 || 10-06S08 9.93.192 || 45 || 55 16 71519 284S1 78.334 21656 06816 93.184 || 44 || 56 17 71539 2S461 7S363 21637 06S23 9317T || 43 || 52 J.8 71560 28440 || 7S391 21609 06831 93.169 42 || 48 19 71581 2S419 || 78419 21581 06839 93.161 41 i 44 20 || 9.71602 || 10.2S398 || 9.78448 || 10.21552 || 10.06846 9.93154 | 40 40 21 I 1622 23378 78476 21524 06S54 93.146 3 36 22 T1643 28.357 TS505 21495 06862 93.138 || 33 32 23 71664 28336 7853.3 21467 06869 93.131 || 37 28 24 7 1685 28315 78562 21 06877 93123 || 36 24 25 || 9,71705 || 10.28295 || 9,78590 10.21410 || 10.06885 9.931.15 || 35 | 20 26 T1726 2S274 || 786.18 21382 06S92 93.108 || 34 16 27 71747 28.253 78647 21353 06900 93.100 33 || 12 28 71767 28233 || 78675 21325 06908 93092 || 32 || 8 29 717 S8 28212 '78704 21296 06916 93084 || 31 4 30 || 9,71809 10.2S191 || 9.7873.2 10.21268 10,06923 9.93077 || 30 || 54 31 71829 28171 78760 21240 06931 93069 || 29 || 56 32 71850 28150 || 78.789 21.211 06939 93061 28 52 33 7187 () 28130 || TSS17 21183 06947 93()53 27 || 48 34 71891 28109 78845 21155 06954 93046 || 26 || 44 35 | 9.71911 || 10.28089 |9.78874 10.21126 || 10.06962 9.93038 || 25 | 40 36 71932 28068 || TS902 2109S 0697 () 9:3030 || 24 || 36 37 7 1952 2SO48 || TS930 2.1070 06978 93022 || 23 || 32 38 T1973 28027 T8959 21041 0.6986 93014 || 22 || 28 39 71994 2S()06 7S987 21013 06993 9300T 21 24 40 9.72014 || 10.27986 9.79015 10.20985 || 10.07001 9,92999 || 20 20 41 72034 27966 || 79043 20957 07.009 92991 || 19 || 16 42 72055 27945 79()72 209.28 07U17 92.983 || 18 || 12 43 7207.5 27925 || 7 9100 20900 07024 92976 || 17 || 8 44 72096 27904 || 79.128 20S72 07032 92.968 || 16 || 4 45 || 9.72116 || 10.27884 || 9.79156 || 10.20S44 || 10.07040 9,92960 15 # 53 46 72137 2.7863 || 70185 20815 07048 92952 || 14 56 47 72157 27843 || 792.13 20787 07.056 93944 || 13 || 52 48 72177 27823 T9241 20759 07064 93.936 || 12 || 48 49 72198 27802 || 79.269 20731 07.071 92.929 || 11 || 44 50 || 9,72218 || 10.27782 || 9,79297 || 10.207()3 || 10.07079 9,92921 10 || 40 51 72238 277.62 79326 20674 07087 92.913 9 || 36 52 '72259 27741 || 79.354 20646 07.095 92.905 8 || 32 53 72279 27721 '7938.2 20618 07 103 92897 7 || 28 54 72299 27701 T9410 20590 07.111 92889 || 6 || 24 55 || 9.72320 || 10.27680 || 9.79438 || 10.20562 || 10,07119 9.92881 5 : 20 56 '72340 27660 || 79466 20534 07 126 928.74 || 4 || 16 57 72360 27640 79495 20505 ()7134 92.866 || 3 || 12 58 723S1 27619 79523 20477 07.142 92.858 || 2 || 8 59 72401 27599 || 79551 20449 07150 92850 || 1 || 4 60 72421 27579 795.79 20421 0.7158 92.842 () 52 M Cosing. Secant, Cotangent Tangent. || Cosecant. Sine. M M.S., 1210 * 58° 3h LOGARITHMS TRIGONOMETRIC. 2b | 329 Logarithms. 1479| 9h M.S. M Sine Cosecant. Tangent. || Cotangent. Secant. Cosine. M M. S 8 0 9.724.21 || 10,27579 |9.79579 10.20421 || 10,07158 9.92842 60 H52 4. 1 72441 27559 T9607 20393 07.166 92.834 || 59 || 56 8 2 72-161 27539 79635 20365 07.174 92826 || 58 || 52 12 3 '724S2 27518 '79663 20337 07.182 928.18 57 || 48 16 4 72502 27.498 79691 20309 07.190 928.10 || 56 || 44 20 5 || 9.72522 10.274.78 9,797.19 10.20281 i 10.0719.7 9.92803 || 55 | 40 24 6 72542 2745S 7.9747 2()253 07:205 92795 || 54 || 36 28 7 72562 27,438 79776 20224 07:213 927S7 || 53 || 32 32 8 72582 274.18 79804 20196 07221 92.779 || 52 || 28 36 9 72602 27398 T9832 20168 07:229 92.771 51 24 40 || 10 9.72622 || 10.27378 |9.79860 | 10.20140 || 10.07237 9.92763 || 50 || 20 44 || 11 72643 27357 79888 20112 07245 92.755 49 || 16 48 12 72663 27337 79916 2008.4 07253 927.47 || 48 || 12 52 13 T2683 27.317 79944 20056 07:261 92.739 || 4'ſ S 56 || 14 727.03 27297 799.72 20028 07:269 92731 || 46 || 4 9 || 15 9.72723 10.27277 |9.8000 10.20000 || 10.07.277 9.92723 || 45 || 51 4 || 16 727.43 27257 80028 19972 07:285 92.715 || 44 || 56 8 : 17 T2763 27237 80056 19944 07:293 92.707 || 43 52 12 18 72783 27217 80084 19916 07.301 92699 || 42 || 48 16 || 19 72803 2719.7 8()112 19888 07.309 92691 || 41 || 44 20 20 || 9.72823 10.27177 |9.80140 | 10.10S60 || 10.07317 9,92683 || 40 | 40 24 || 21 72843 27.157 80.168 19832 07:325 92675 || 39 36 28 22 72868 27137 80.195 19805 07333 92667 || 3S 32 32 23 72S83 27117 802.23 19777 07341 92659 || 37 28 36 24 72902 27098 80251 19749 07349 92651 || 36 || 24 40 25 9,72922 10.27078 9.80279 10.19721 || 10.07357 9.92643 || 35 | 20 44 || 26 72942 270.58 80307 19693 07.365 92635 | 34 16 48 || 27 72962 27038 80335 19665 07:373 92.627 33 || 12 62 28 72982 2701S 80363 I9637 07381 92619 || 32 8 56 29 73002 26998 80391 19609 073S9 92611 || 31 4 10 || 30 || 9,73022 10.26978 ; 9,80419 || 10.19581 || 10.07397 9.92603 || 30 50 4 || 31 73041 26959 80447 19553 (J7405 92595 29 56 8 || 32 73061 26939 S()474 19526 O7413 92.587 || 28 || 52 12 || 33 73081 26919 80502 19498 ()7421 92579 || 27 || 48 16 || 34 731()1 26899 80530 19470 07:429 92571 || 26 || 44 20 || 35 || 9,73121 10,26879 || 9.8()558 10.19442 | 10.07437 9.92563 25 | 40 24 || 36 73140 26860 805S6 19414 07445 92555 || 24 || 36 28 || 37 73.160 26S40 80614 19386 07:454 92546 || 23 || 32 32 || 38 73.180 26820 806.42 19358 0.7462 92538 22 28 36 || 39 73200 26S00 80669 19331 07470 92530 21 l 24 40 || 40 || 9,732.19 10.26781 || 9.80697 10.19303 || 10.07478 9.92522 || 20 | 20 44 || 41 73239 26761 S()725 19275 074S6 92514 || 19 || 16 48 I 42 73259 26741 807 53 19247 0.7494 92.506 || 18 || 12 52 || 43 73278 26722 80781 I92.19 07502 92.49S 17 S 56 || 44 73298 26702 80808 I9192 07510 9249() 16 4 II || 45 || 9.733 18 10.26682 9.80836 10, 19164 || 10.0751 9.92482 | 15 49 4 46 73337 26663 80864 19136 075.27 92473 || 14 || 56 8 || 47 73357 26643 80892 19108 07535 92465 | 13 l 52 12 || 48 73377 26623 80919 19081 07543 92457 | 12 || 48 I6 || 49 73396 26604 80947 19053 07:551 92449 || 11 || 44 20 || 5() 9.73416 || 10.26584 9.80975 10.19025 || 10.07559 9,92441 10 40 24 || 51 73435 26565 81003 18997 O7567 92433 9 || 36 28 || 52 '73455 26545 81030 1897 () 07 575 92.425 8 || 32 32 || 53 73474 26526 81058 18942 07584 924.16 7 || 28 36 || 54 73494 26506 81086 18914 07592 92408 6 || 24 40 || 55 9.73513 || 10.26487 9,81113 || 10.18887 || 10.07600 9.92400 5 i 20 44 56 73533 26467 81141 18859 07608 92392 4 || 16 48 || 57 73552 26448 81169 1883]. 07616 92.384 3 || 12 52 || 58 73572 26428 81196 18804 07624 92376 2 8 56 || 59 73591 264.09 81224 18776 07633 92367 1 4 H2 60 73611 263S9 81.252 18748 07641 92359 0 || 4-8 Al. S. M Cosine. Secant. Cotangent| Tangent. || Cosecant. Sine. M A1. S. 8h 1220 57° 3h 196 LOGARITHMS TRIGONOMETRIC, 2h 339 Logarithms. 146° 9th M.S. M Sine, Cosecant. | Tangent. | Cotangent, Secant. Cosine. M. M. S. 12 0 || 9,73611 10.26389 || 9,S1252 10.1874S | 10.07641 9.9235Q 60 || 48 4 l 73630 26370 S1279 18721 07649 92351 59 || 56 S 2 73650 26350 81307 18693 O7657 92343 || 58 || 52 I2 3 '73669 26331 81335 IS665 07665 92.335 | 57 4S 16 4 T36S9 263.11 81362 IS638 07674 92326 56 || 44 20 5 9.73708 10.26292 || 9,81390 10.1S610 || 10.07682 9.92318 55 || 40 24 6 737.27 26273 81418 IS5S2 07690 92310 || 54 || 36 2S 7 T3747 26253 81.445 IS555 07698 92302 || 53 || 32 32 S 73766 26234 81473 IS527 O7707 92.293 || 52 28 36 9 737S5 26215 81500 1S500 07'ſ 15 92.285 || 51 || 24 40 || 10 || 9,73805 10.26195 || 9.S1528 10.18472 || 10.07723 9,92277 || 50 20 44 || 11 73S24 261.76 S1556 18444 07731 92.269 49 16 4S 12 738.43 26157 81583 18417 07740 922.60 || 48 || 12 52 || 13 73863 26137 SIGII 183S9 07748 92.252 || 47 8 56 || 14 73SS2 26118 8163S I8362 O7756 92.244 46 4. 13 || 15 || 9,73901 10.260.99 || 9.S1666 10.1S334 10.07765 9.92235 || 45 || 4-7 4 || 16 73921 26079 81693 18307 07773 92.227 44 56 8 17 73940 26060 81721 18279 077SL 92.219 || 43 || 52 12 || IS 73959 26041 81748 18252 077S9 92211 || 42 48 16 19 7397S 26022 81776 1S224 0779S 9.2202 || 41 || 44 20 | 20 || 9.73997 10.26003 9.8.1803 10.18197 || 10.07806 9.92194 | 40 40 24 21 74017 259.83 8] S31 IS169 07814 92186 || 39 || 36 2S 22 74(36 25964. S1S5S 18142 07 S23 92.177 || 38 || 32 32 || 23 74055 25945 81SS6 IS114 07831 92.169 37 || 28 36 || 24 74074 259:26 81913 ISOS7 07S39 92.161 || 36 24. 40 || 25 || 9,74093 10,25907 || 9.81941 10.18059 || 10.07848 9.92.152 35 | 20 44 || 26 74113 25SS7 S1968 18032 07856 92144 || 34 || 16 48 || 27 74.132 25S6S 81996 ISOſ)4 07864 92.136 || 33 || 12 52 2S '74151 25849 82023 17977 07S73 92.127 32 8 56 29 T4170 25S30 82051 17949 07SS1 92119 || 31 4 I4 3() || 9.74189 10.25S11 || 9,82078 10,17922 || 10.07889 9.92111 || 30 || 46 4 || 31 74208 25792 82106 17894 07S98 92102 || 29 || 56 8 || 32 74227 25773 S2133 17867 07906 9.2094 28 || 52 12 || 33 742.46 25754. 82161 17839 07914 92086 27 || 48 16 34 74.265 25735 821SS I7S12 O7923 92077 || 26 || 44 20 i 35 | 9.74284 10.25716 || 9.S2215 10.17785 10.07931 9.92069 25 | 40 24 || 36 74303 25697 82243 17757 O7940 92060 || 24 36 28 || 37 74322 2567S 82270 17730 07948 92052 || 23 || 32 32 3S 74341 35659 82298 17702 07.956 92044 || 22 || 28 36 39 7.4360 25640 82325 17675 07965 92035 | 21 24 40 || 40 || 9,74379 10,25621 9.82352 10.17648 10 07973 9.92027 20 20 44 || 41 7,4398 25602 823.80 17620 079S2 920IS 19 || 16 48 || 42 '74417 25583 S2407 17593 07990 92010 | 18 12 52 || 43 74436 25564 S2435 17565 07.09S - 92002 || 17 8 56 || 44 7445.5 25545 82462 17538 08007 91993 || 16 4 15 45 || 9,74474 10.255.26 || 9,824.89 10,17511 || 10.08015 9 91985 15 || 4-5 4 46 74493 25507 82517 17483 08024 91976 || 14 || 56 8 || 47 74512 254SS 82544 17456 OS032 91968 || 13 || 52 12 || 4S 74531 25469 82571 17429 0S041 91959 || 12 || 48 16 || 49 '74549 2545.1 S2599 17401 08049 91951 || 11 || 44 20 || 50 || 9.74568 10,25432 || 9.82626 10.17374 || 10.0S058 9,91942 10 | 40 24 || 51 74587 25413 82653 17347 08066 91934 9 || 36 28 || 52 '74606 25394 82681 17319 08075 91925 8 || 32 32 |-53 74625 25.375 82.708 17292 OSOS3 91917 7 28 36 || 54 7 4644 *25356 82735 17265 OS092 91908 6 || 24 40 55 || 9.74662 10.25338 || 9.827.62 10.17238 10,08100 9.91900 5 20 44 || 56 74681 25319 82790 17210 0ST09 91891 4 || 16 48 || 57 || 74700 25.300 S2817 IT183 08117 9.1883 3 || 12 52 5S 74719 252S1 82S44 17156 08.126 91874 2 8 56 || 59 74.737 25263 82S71 I'7129 08134 91S66 l 4. 16 || 60 7475 25244 82S99 17101 0S143 91857 0 | 44 M. 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S. 20 || 0 || 9.75859 10.24141 9.84523 10.15477 i 10.08664 9,91336 || 60 | 40 4 1. 75877 24.123 845.50 15450 0S672 91328 || 59 || 56 8 2 75S95 24105 84576 15424 086Sl 91.319 || 58 52 12 3 75913 24087 84603 15397 08690 91310 || 57 || 48 16 || 4 75931 24069 S4630 I5370 O8699 91301 || 56 || 44 20 5 9.75949 10,24051 || 9.84657 10.15343 || 10.087.08 9.91292 || 55 || 40 24 6 75967 24033 84684 15316 087.17 91283 || 54 || 36 2S 7 75985 24(115 S47I1 15289 08726 91274 53 || 32 32 8 76003 23997 84738 15262 08734 91266 52 28 36 9 '76021 23979 S4764 15236 08743 9.1257 || 51 24 40 || 10 ; 9.7603 10.23961 |9.84.791 10.15209 || 10.08752 9.91248 || 50 20 44 || 11 76057 239.43 84S18 15182 OS761 91239 || 49 16 48 || 12 76075 239.25 S4S45 15155 08770 91230 || 48 || 12 52 || 13 7.6093 23907 84872 15128 08779 91221 || 47 8 56 || 14 76111 23SS9 84899 15101 08788 91212 || 46 4 21 || 15 || 9,76129 || 10.23871 || 9.84925 || 10.15075 || 10.08797 9.91203 || 45 39 4 16 76.146 23S54 84952 1504.8 08806 91194 || 44 56 8 || 17 76.164 23836 849.79 1502I 08815 91185 || 43 || 52 12 18 76.182 23818 85006 14994 08824 911.76 42 || 48 16 || 19 76200 23S()0 85033 14967 OS833 91167 || 41 || 44 20 20 | 9.76218 10.23782 || 9.85059 || 10.1494.1 || 10.08842 9.91158 || 40 40 24 || 21 76236 23764 85086 14914 08851 91149 || 39 || 36 28 || 22 76253 2374.7 851.13 I4887 08859 91141 || 38 32 32 23 76271 237.29 85.140 14860 US868 91132 || 37 || 28 36 || 24 7.6289 23711 85166 14834 08877 91123 36 || 24 40 || 25 || 9,76307 || 10.23693 || 9.85193 10.14807 || 10.08886 9.91.114 || 35 | 20 44 || 26 76324 23676 85.220 14780 08895 91105 || 34 || 16 4S | 27 76342 2365S 85.247 14753 08904 91096 || 33 || 12 52 28 76360 2364() 852.73 14727 08913 91087 || 32 8 56 29 76378 23622 85.300 14700 08922 91078 || 31 4 22 30 9.76395 || 10,23605 || 9.85327 10.14673 || 10.08931 9.91069 || 30 || 38 4 : 31 76413 23587 85354 14646 OS94() 91060 || 29 56 8 || 32 76431 23569 85.380 1462() 08949 91051 || 28 52 12 || 33 76448 23552 85407 14593 0S958 91042 27 || 48 16 || 34 '76466 23534 85.434 14566 OS967 91033 26 || 44 20 ! 35 ; 9.76484 || 10.235.16 9.8546ſ) || 10.14540 || 10,08977 9.91023 || 25 | 40 24 || 36 76501 23499 854.87 14513 08986 91014 || 24 || 36 28 37 '76519 23481 85514 14486 OS995 91005 || 23 || 32 32 38 76537 23463 85540 14460 09004 90.996 22 || 28 36 39 76554 23446 85567 14433 09013 90987 || 21 24 40 || 40 || 9,76572 || 10.23428 |9.85594 || 10.14406 || 10.09022 9.90978 20 || 20 44 || 41 '76590 234.10 85620 14380 09031 90969 ; 19 || 16 4S 42 76607 •23393 85.647 14353 09040 90960 18 || 12 52 || 43 76625 23375 85674 14326 09049 90951 || 17 8 56 || 44 76642 23358 85700 143t)0 09058 90942 16 4 23 || 45 || 9,76660 || 10.23340 |9.85727 10.14273 || 10.09067 9.90933 15 || 37 4 46 76677 23323 85754 14246 09076 90924 || 14 || 56 8 || 47 '76695 23303 85.780 14220 0.9085 90915 13 || 52 12 || 48 76712 23288 S5807 14193 09094 90906 || 12 || 48 1ſ; 49 76730 23270 85834 14166 09104 90896 || 11 || 44 20 || 50 || 9,76747 || 10.23253 || 9,85860 | 10.14140 || 10,09113 9.90887 || 10 40 24 || 51 76765 23235 85887 14113 0.9122 90S78 9 36 28 || 52 76782 23218 85913 14087 09131 90869 8 || 32 32 || 53 76800 23200 85940 14060 0.914() 90860 7 || 28 36 || 54 76817 23183 || 85967 14033 0.9149 90851 6 || 24 40 55 || 9,76835 10.231.65 9,85993 || 10.14007 || 10.09158 9.90842 || 5 || 20 44 || 56 '76852 23148 86()20 13980 09168 90832 4 || || 6 48 || 57 76870 23130 86046 13954 ()9.177 90823 3 || 12 62 || 58 76887 23113 86073 13927 091S6 90814 || 2 8 J 56 59 76904 23096 86100 1390() 0.919.5 90805 I 4 24 60 76922 23078 86126 13874 09204 90796 0 || 36 M. S. M Cosine. Secant. Cotangentl Tangent. Cosecunt. Sine. M M.S. 8h j125° 54° 3h LOGARITHMS TRIGONOMETRIC. 2h 369 Logarithms. 1439 9h M.S. M Sine. Cosecant. | Tangent. Cotangent. Secant. Cosine. M. M. S. 24- || 0 || 9,76922 10.2307S 9.86126 1(). 13874 10.09.204 || 9.90796 || 60 || 36 + Y. 76939 23061 86153 13847 09:213 907S7 || 59 || 56 8 2 76957 23043 86179 13821 09.223 9()777 || 58 || 52 12 3 76974 23026 86206 13794 09232 90.768 57 || 48 16 4 76991 23009 86232 1376S 09.241 90759 || 56 || 44 20 5 || 9.77009 10.22991 9.86259 10.1374L 10.09:250 9.90750 || 55 40 24 6 ‘77026 22974 862S5 13715 09:259 90741 54 36 28 7 77.043 22.357 863L2 13688 09.269 90731 53 32 32 8 77061 22939 8633S 13662 09278 90722 || 52 || 28 36 9 77.078 22922 S6365 13635 09.287 90713 || 51 24 40 || || 0 || 9,77095 10.22905 || 9.86392 10.13608 || 10.09:296 9,90704 || 50 || 20 44 11 771t 2 22888 || 86418 13582 09306 9(K694 || 49 || 16 48 || 12 77130 22S7() $6445 13555 09:315 90685 || 48 || 12 52 || 13 77147 22853 864.71 13529 09324 90676 || 47 8 56 14 7.7164 22S36 86498 13502 09333 90667 46 4 25 | 15 9.77181 10.22819 || 9.86524 || 10.13476 || IO.09343 9.90657 || 45 || 35 4 16 77199 22801 86551 13449 09352 90648 I 44 || 56 8 : 17 77216 22784 86577 13423 O9361 90639 || 43 || 52 12 18 77233 22767 86603 13397 0937() 90630 || 42 || 48 16 19 77.250 2275() S6630 13370 093S() 90620 || 41 || 44 20 | 20 9.77.268 10.22782 9.86656 10.13344 || 10.09389 9.90611 || 40 || 40 24 2. 772S5 22715 866S3 13317 09398 90602 || 39 36 28 22 77302 22698 867 09 13291 09:408 90592 38 || 32 32 || 23 77319 22681 86736 13264 09417 905S3 || 37 28 36 || 24 773.36 22664 86762 1323S 0.9426 90574 || 36 24 40 || 25 || 9.77.353 10.22647 || 9.86789 10.13211 || 10.09435 9.90565 || 35 20 44 26 77.370 22630 86S15 131S5 09445 90555 || 34 16 48 || 27 77.387 22613 86842 13158 09:454 90546 || 33 || 12 52 28 '77405 22595 8686S 13132 09:46.3 90537 || 32 8 56 || 29 77422 2257S 86894 13106 09473 90527 || 31 4 26 30 9.77439 10.22561 || 9.8692}. 10.13079 || 10.09482 9.90518 || 30 || 34 4 : 31 77456 2. 86947 13053 ()94)1 90509 || 29 || 56 8 32 77473 22527 86974 13026 O950+ 90499 || 28 || 52 12 || 33 '77.490 22510 87000 13000 095.10 90490 27 || 48 16 || 34 77507 22493 8702'ſ 12973 09520 90480 || 26 || 44 20 || 35 ; 9.77524 || 10.22476 9.8705 10.12947 || 10.09529 9,90471 || 25 40 24 || 36 77541 22:459 87079 12921 09:538 90.462 || 24 || 36 28 37 77558 22442 87 106 12894 ().9548 90452 || 23 || 32 32 || 38 77575 22425 87.132 12S6S 09:557 90.443 || 22 2S 36 || 39 77592 22408 87.158 12842 09566 90434 21 || 24 40 || 40 9.77609 10.22391 || 9.871S5 10.12815 10.0957.6 9.904:24 20 || 20 44 41 '77626 22:37.4 S7211 12789 0.95S5 90.415 || 19 || 16 48 || 42 '77643 22357 87.238 12762 Q9595 90.405 || 18 || 12 52 || 43 7766() 22340 87264 12736 09604 90.396 || 17 8 56 l 44 77677 22323 87290 T27L() 09614 903S6 || 16 4 27 || 45 9.77694 || 10.22306 || 9.87317 10.12683 || 10.09623 || 9.90377 | 15 || 33 4 46 777.ll 22.289 87343 12657 09032 90.368 || 14 || 56 8 47 77728 22272 87369 12631 09642 90.358 || 13 || 52 12 || 48 7774 22256 87396 12604 ()965 l 90.349 || 12 || 4S I6 || 49 77761 22239 87422 12578 09661 90339 || 11 || 44 20 || 50 || 9.77778 10.22222 || 9.874.48 10.12552 10.09670 9.90330 || 10 || 40 24 || 51 7779.5 22205 874.75 12525 Ü9680 90.320 9 || 36 28 || 52 77812 22:188 875O1 12499 09689 90.311 8 || 32 32 || 53 77829 22171 875.27 12473 09699 90.301 7 || 28 36 54 77S46 22.154 87554. 12446 0970S 90292 6 || 24 10 || 55 9.77862 10.22138 || 9.8758() 10.12420 || 10.097.18 9.90282 5 i 20 44 || 56 77879 22121 87606 12394 0.9727 90.273 4 || || 6 48 57 77896 22104 87633 12367 (9.737 90263 3 || 12 52 || 58 77913 220ST 87659 12341 097.46 90254. 2 S 56 || 59 77930 22070 876S5 12315 09756 90244 I 4 28 60 77.946 22054 877 11 122S9 09765 90235 () || 32 M. 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S. 0 || 9,78934 || 10.21006 || 9,89281 10.10719 || 10.10347 9.89653 || 60 28 1 78950 21050 89.307 10693 10357 89643 || 59 || 56 2 78967 21033 89.333 10667 10367 S9633 || 58 || 52 3 789S3 21017 S9359 10641 I0376 89624 57 || 48 4 TS999 21001 89.385 10615 10386 896.14 || 56 || 44 5 9.79015 10.20985 9,894.11 10.10589 || 10.10396 9,89604 || 55 || 4() 6 79031 20969 89.437 10563 10406 89594 || 54 || 36 7 79(147 20953 89.463 10537 10416 S95S4 || 53 32 8 79063 209:37 89.489 105.11 10426 89574 || 52 28 9 79079 20921 S9515 104S5 10436 89564 || 51 || 24 10 || 9,79095 || 10.20905 || 9,89541 10.10459 || 10.10446 9.SS)554 || 50 | 20 11 791.11 20889 S9567 10433 10.456 89544 || 49 || 16 12 79°128 20872 89593 10407 10466 895.34 || 48 || 12 13 7.9144 20856 896.19 I0381 104.76 80524 || 47 8 14 '79160 2084Q 89645 10355 104S6 89.514 || 46 4 15 9.791.76 10.20S24 || 9.89671 10.10329 10.10496 9.89504 || 45 || 27 16 7.9192 26S08 89697 10303 10505 S9495 44 i 56 17 79208 20792 89723 10277 105.15 89.485 43 52 18 79.224 T 20776 897.49 10251 10525 89.475 || 42 || 48 19 79240 20760 897.75 10225 10535 $9465 41 || 44 20 || 9.79256 10.29744 9.80801 10.10199 || 10.105.45 9.88455 | 40 || 40 21 79272 20728 89S27 10173 10555 894.45 || 39 || 36 22 79288 20712 89853 10147 10565 89435 | 3S 32 23 79304 20096 89879 10121 10575 89.425 || 37 f 28 24 793.19 206S1 89905 I0095 I0585 89.415 || 36 || 24 25 9.793.35 10.20665 9.89931 10.10069 || 10.10595 9.SS405 || 35 | 20 26 79351 20649 S995.7 10043 10605 89395 || 34 || 16 27 79367 20633 S9983 10017 106.15 89.385 33 || 12 28 793S3 2001.7 90009 ()9991 10625 89.375 || 32 8 29 79399 20601 90035 0.9965 10636 89364 31 4 3() 9,794.15 10.20585 || 9,90061 10.09939 || 10.10646 9.89354 || 30 || 26 31 7943i 20569 90086 09914 10650 89344 || 29 || 56 32 79.447 20553 90112 09888 10666 89.334 || 28 || 52 33 '79463 20537 9013S 09862 10676 89.324 27 || 48 34 79478 20522 90164 09S36 106S6 893.14 || 26 || 44 35 | 9.79494 || 10.20506 || 9.90190 10.09810 || 10.10696 9.SS304 || 25 | 40 36 795.10 20490 90216 09784 10706 892.94 || 24 || 36 37 79526 20474 90242 09758 10716 89.284 || 23 || 32 38 79542 20458 9026S 097.32 10726 80274 || 22 || 28 39 79558 20442 90294 09706 I0736 S9264 || 21 || 24 40 || 9.79573 || 10.20427 | 9.90320 | 10.09680 || 10 10746 9.892.54 || 20 | 20 41 79589 20411 90346 09654 10756 892.44 || 19 16 42 79605 20395 90371 096.29 10767 89233 18 || 12 43 79621 20379 90.397 09003 107.77 S9223 17 8 44 79636 20364 90.423 (J9577 10787 892.13 | 16 4 45 9.79652 10.20348 || 9.90449 10.09551 || 10.10797 9 S9203 || 15 25 46 T9668 20332 90475 09525 1080'ſ 89193 || 14 || 56 47 '79684 20316 90501 09499 10817 89.183 || 13 || 52 48 79699 20301 90527 09473 I0827 89.173 || 12 || 48 49 79715 20285 90553 09447 10838 89.162 || 11 || 44 50 9.79731 I0.20269 || 9.90578 10.09422 || 10.10848 || 9,89152 || 10 | 40 51 79746 20254 90604 09396 I()S5S 89142 9 || 36 82 797 62 20238 90630 09:370 10868 89132 8 || 32 53 79778 20222 90656 09344 10S78 89.122 7 2S 54 79793 20207 906S2 093.18 10SS8 89112 6 24 55 9.79809 || 10.20191 9.90708 10.09292 || 10 10899 9.89101 5 i 20 56 79S25 20175 90734 09.256 10909 89.091 4 || 16 57 79S40 20160 907 59 09.241 10919 89081 3 || 12 58 79856 20144 907S5 09:215 10929 89.071 2 8 59 79872 2012S 90SII 0.9189 10940 S9060 l 4 60 7.98ST 20113 90837 0916.3 10950 S9050 0 || 24- M Cosime. Secant. " Cotangenti Tangent. Cosecant. Sune. M Al. S. 128 51° 35 LOGARITHMS TRIGONOMETRIC. 399 Logarithms. 140° 9h M Sirae. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. M. S. 0 || 9.79887 10.20113 || 9.908:37 10,0916.3 || 10.10950 9.89050 || 60 || 24- 1 '799)3 20097 90S63 091.37 10960 890-40 || 59 || 56 2 79918 20082 90889 09111 I097 () 89.030 5S 52 3 '7993.4 20006 90914 09086 109SO 89020 57 || 48 4 7995() 2005() 90940 ()9()60 T099 L 89009 || 56 || 44 5 || 9.79965 10.20035 | 9.90966 10.09034 || 10.11001 9.8S999 || 55 40 6 7. 9981 20019 90992 09008 11011 88989 || 54 36 7 7.9996 20004 91018 08982 11022 8S978 53 32 8 80012 19988 91043 08957 II032 S8968 || 52 28 9 SO(27 1997:3 91069 08931 11042 88958 || 51 || 24 I() 9.80043 10, 19957 9.91095 || 10.08905 || 10.11052 | 9.88948 || 50 | 20 11 S0058 19942 91121 OS879 11063 88937 || 49 || 16 12 8007.4 19926 91.147 OSS53 11073 8S927 || 48 || 12 13 S00S9 1991]. 91172 08S28 11083 88917 || 47 8 14 801(\5 1989.5 911.98 08802 1109 | 88906 || 46 4 15 || 9,8012) 10.198.80 || 9.91224 10.0S776 10.11104 9.88896 || 45 23 16 80136 1986 || 9125() 08750 11114 88S86 || 44 || 56 17 80151 19849 91276 087 24 11125 8S875 || 43 || 52 1S 80.166 19834. 91301 08699 T1135 88S65 || 42 48 19 SO182 19818 91327 08673 11143 $8855 || 41 || 44 20 || 9.8U197 10.19803 9.91.353 10.0S647 || 10.11156 9.88844 || 40 4() 21 802.13 19787 91379 0S621 11166 88834 || 39 || 36 22 80228 19772 9.1404 OS506 1 1176 88.824 || 38 || 32 23 80244 19756 91430 08570 11187 88813 37 || 28 24 80259 1974L 91.456 08544 III.97 88803 || 36 || 24 25 9.80274 10.19726 9.91482 10.0°518 || 10.11207 9.88793 35 | 20 26 80290 19710 91507 OS493 11218 887S2 || 34 16 27 8():305 19695 9153.3 08:467 II 228 88772 33 12 28 8032() 19680 9.1559 08441 11239 S8761 || 32 8 29 80336 19004 91585 ()8415 11249 88751 || 31 4 30 || 9.80351 10.19649 || 9.9161() 10,0S390 || 10.11259 9.887.41 || 30 22 31 80366 I9634 91636 0S364 11270 88730 29 56 32 8(3S2 19618 91662 083:38 II2SO 88720 || 28 || 52 33 80397 1960.3 91688 08312 11291 S8709 || 27 || 48 34 80412 19588 91713 082S7 11301 88699 || 26 || 44 35 | 9.80428 10.19572 || 0.01739 10,08261 || 10.11312 9.88688 25 | 40 36 80443 19557 9.1765 0.8235 11322 88678 || 24 || 36 37 80458 19542 917.91 08209 11332 88668 || 23 32 3S 80473 19527 91816 08.184 11343 88657 22 || 28 3 80489 1951]. 91.842 08.158 11353 88647 || 21 24 40 || 9.SU504 10.19496 || 9.91S68 10 ()8132 || 10 11364 9.88636 || 20 i 20 41 80519 19481 91893 08.107 I1374 88626 || 19 16 42 80534 19460 91919 08081 I 1385 S8615 18 12 43 80550 19450 91945 0S055 11395 88605 || 17 8 44 80565 19435 9.1971 OSU29 11406 88594 | 16 4 45 || 9.80580 I0.19420 9.91996 10,08004 || 10.11416 9 88584 || 15 l 21 46 80595 19405 92022 O7978 11427 88573 || 14 || 56 47 80610 19390 92048 ()7952 Tlá37 88563 || 13 || 52 48 80625 19375 92073 07927 11448 88552 | 12 || 48 49 80641 I9359 92099 (J7901 11458 885.42 || 11 || 44 5() 9,80656 10.19344 || 9.92.125 10.07875 || 10.11469 9.88531 i 10 | 40 51 8067.1 19329 92.150 ()7850 11479 88521 9 || 36 52 80686 19314 9217.6 07824 11490 88510 8 || 32 53 S{)701 |9299 92.202 O7798 115()1 SS499 7 || 28 54 80716 19284 92227 07773 11511 88489 6 || 24 55 9.80731 10 19269 || 9.92253 10.07747 || 10 11522 9.88478 5 20 56 80746 19254 9:2279 ()7721 11532 88468 4 || 16 57 8()762 1923S 92304 ()7 696 11543 88457 3 || 12 58 8()777 T9223 923.30 ()7(;7() 11553 884.47 2 8 59 80792 19208 92.356 07(44 11564 884.36 1 4 60 8080'ſ 19193 923S1 07619 1157.5 8S425 || 0 || 20 M Cosine. Secant. Cotangentl Tangent. Cosecant. Sine. M. Al. S. |1299 50° 3h. LOGARITHMS TRIGONOMETRIC. 2h 400 Logarithms. 1399 || 9h M. S. M. Sine. Cosconnt, Thngent. | Cotangent. Secant. Cosine. M iM. S. 40 || 0 || 9,80807 10.19193 || 9.92381 10.07619 || 10.11575 9.8% A25 || 60 | 20 4 I 80822 19178 92407 ()7593 11585 88415 || 59 56 S 2 80S37 1916.3 92433 07567 11596 884.04 || 58 || 52 I2 3 80852 19148 92.458 ()7542 11606 88394 57 || 48 16 4 80867 19133 92484 ()7516 11617 88383 || 56 || 44 20 5 || 9,80882 10.19118 || 9.9251() 10.0749() || 10.11628 9.88372 55 | 40 24 6 80897 19103 92535 07:465 1163S 88362 54 || 36 28 7 80912 19088 92561 07439 11649 88.351 || 53 || 32 32 8 80927 1907.3 92587 07413 11660 88.340 52 28 36 9 80942 19058 92612 07.388 11670 8833() || 51 || 24 40 || 10 || 9,80957 10.19043 || 9.926.38 10.07362 || 10.11681 9.883.19 || 50 20 44 || 11 809.72 19028 92663 07:337 11692 88.308 || 49 || 16 48 # 12 80987 19013 92689 07311 11702 88298 || 48 || 12 62 || 13 81002 18998 92.715 (J7285 IT'ſ 13 882S7 || 47 S 56 14 81017 18983 92740 07260 11724 88276 || 46 4 41 || 15 || 9,81032 10.18968 || 9.92766 10.07234 || 10.11734 9.88266 || 45 || 19 4 || 16 8104.7 18953 92.792 ()7208 11745 88.255 || 44 || 56 8 17 81061 18939 92817 07.183 11756 SS244 43 || 52 12 || 18 81076 18924 92843 O7157 L1766 88234 || 42 || 48 16 || 19 81091 18909 928.68 07132 11777 88223 || 41 || 44 20 20 9.81106 10.18894 || 9.92894 || 10,07106 || 10.11788 9.882.12 || 40 || 4) 24 || 21 81121 18879 92920 07080 11799 88201 || 39 || 36 2S 22 81136 18864 929.45 0.7055 T1809 88.191 38 || 32 32 23 81151 18849 92971 O7()29 11820 88.180 || 37 || 28 36 || 24 81166 18834 92996 07004 11831 88.169 || 36 || 24 40 || 25 || 9.81180 10.18820 || 9.93022 10.06978 || 10.11842 9.88.158 || 35 20 44 || 26 81195 18805 93048 ()6952 11852 8814S 34 16 48 || 27 81210 18790 93073 06927 11863 S8137 || 33 12 52 || 2S 81225 18775 930.99 06901 I1874 88126 || 32 8 56 29 81240 1876() 93.124 06S76 11885 88.115 || 31 4 4-2 || 30 || 9.8.1254 10.18746 || 9,93150 10,06850 || 10.11895 9.88105 || 30 18 4 || 31 81269 18731 931.75 06825 11906 8S094 || 29 56 8 || 32 81284 18716 93.201 (16799 11917 S8083 || 28 || 52 12 || 3:3 81299 1870L 93227 06773 11928 88072 27 || 48 I6 || 34 81314 18686 93252 06748 11939 88061 26 || 44 20 35 || 9,81328 10.18672 || 9.93278 10.06722 || 10.11949 9.88051 || 25 || 40 24 || 36 81343 18657 93303 06697 II960 88040 || 24 || 36 2S 37 81358 18642 93329 06671 11971 88029 || 23 || 32 32 || 38 81372 18628 93354 06646 II982 88018 22 || 28 36 || 39 81387 18613 93380 ()6620 11993 88007 || 21 || 24 40 40 9.81402 10.18598 || 9.93406 10.06594 || 10.12004 9.87996 || 20 || 20 44 41 81417 18383 93431 06569 12015 87985 19 16 48 || 42 81431 18569 93457 ()6543 12025 7975 | 18 || 12 52 || 43 81446 T855-4 93.482 06518 I2036 87.964 || 17 8 56 i 44 81461 18539 935.08 ()6492 12047 87953 || 16 4 43 || 45 || 9.81475 10.18525 9.93533 10.06467 10.12058 9.87942 15 || 17 4 || 46 81.490 18510 93559 06441 12069 87931 || 14 || 56 8 || 4'ſ 81505 18495 93.584 06416 12080 87920 || 13 || 52 12 || 48 81519 18481 93610 0639() I2091 879(99 || 12 || 4S I6 || 49 81534 18466 93636 06364 12102 87898 || 11 || 44 20 || 50 || 9.8.1549 10,18451 9.93661 10.06339 || 10.12113 9.87887 || 10 || 40 24 51 8]503 18437 936.87 06313 I2123 87877 9 || 36 28 52 81578 18422 93712 06.288 12134 87866 S 32 32 || 53 81592 18408 93.738 ()6262 I2145 87855 7 28 36 || 54 81607 I8393 93.763 (36237 12156 87844 6 || 24 40 || 55 || 9,81622 10.18378 || 9.93780 10.06211 || 10.12167 9.S7833 5 || 20 44 || 56 81636 18364 93814 ()6186 1217.8 S7822 4 i 16 4S 57 81651 18349 93840 06160 T2189 87811 3 || 12 52 5S 81665 18335 93865 .06135 12200 87800 2 8 56 59 81680 18320 93891 (16109 12211 87.789 l 4 44 60 81694 18306 93916 06084 12222 87778 () | 16 M. S. Aſ Cosime. Secant. || Cotangenti Tangent. || Coaccumt. Sine. M M.S. 8|| |1309 499 || 3h 204 LOGARITHMS TRIGONOMETRIC. 2h 419 Logarithms. 1389 || 9h M.S. M. Sine. Cosecant. | Tangent. | Cotangent. Secant. Cosine. M. M. S 4-4. 0 || 9.81694 I0.18306 || 9.93916 10.06084 10,12222 9,8777S 60 16 4 1 81709 JS291 93942 ()605S l2233 87.767 59 56 8 2 81723 18277 93.967 06033 12244 877.56 58 || 52 12 3 81738 18262 93.993 06007 12255 87.745 57 || 48 16 4 81752 18248 94.018 05982 12266 87.734 56 || 44 20 5 || 9,81767 10.18233 l 9.94044 10.05956 || 10.12.277 9.87723 55 40 24 6 81781 IS219 94.069 05931 12288 8,712 54 || 36 28 7 81796 18204 94095 05905 12299 87.701 53 || 32 32 8 81810 18190 94.120 05880 12310 87690 || 52 2S 36 9 81825 18175 94.146 05854 12321 87679 51 24 40 || 10 || 9.81839 10.1S161 || 9.94.171 10.05829 || 10.12332 9.87668 50 20 44 11 81854 18146 94.197 05803 12343 87657 || 49 16 48 || 12 81S6S 18132 94.222 05778 12354 87646 || 48 || 12 52 13 SISS2 18118 94248 05572 12365 87635 || 47 8 56 || 14 81897 18103 94.273 057 27 12376 87624 || 46 4. 45 15 9.8.1911 10,180S9 9.94.299 10.05701 || 10.12387 9,876 13 45 15 4 || 16 S1926 lS()74 943.24. 05676 I2399 87.601 || 44 || 56 8 17 81940 18060 94:350 05650 12410 87.590 43 || 52 12 18 81955 18045 94375 05625 1242.1 - 87579 || 42 || 48 I6 || 19 81969 18031 94.401 05:599 12432 S7568 || 41 || 44 20 || 20 9.81983 10.18U17 9.94.426 10,05574 || 10.12443 9.87557 40 || 40 24 || 21 81998 18002 94452 05548 1245.4 87.546 || 39 || 36 28 || 22 82012 17988 94477 05523 12465 8753 38 32 32 23 82026 17974 94503 05497 12476 875.24 || 37 28 36 24 8204] I7959 94528 05472 ‘I:2487 87513 || 36 || 24 40 25 9.82055 10,17945 i 9.94554 10.05446 || 10.12499 9.87501 || 35 20 44 || 26 82069 17931 94.579 05421 12510 S7490 || 34 || 16 48 || 27 82084 17916 94604 05396 12521 87.479 || 33 || 12 52 28 82098 17902 946.30 05:370 12532 87468 || 32 8 56 || 29 82112 ITSSS 94655 05:345 12543 87457 : 31 4 46 || 30 9.82126 10,17874 || 9,9468l 10,05319 || 10,12554 9,87446 || 30 14. 4 || 31 82I41 I'ſ 859 94.706 05294 12566 87434 || 29 || 56 8 || 32 82155 17845 94.732 ()5268 I2577 S7423 || 2S 52 12 33 82169 17S31 94.737 05243 I2588 87412 7 || 48 16 || 34 82184 I'7816 94.783 05217 12599 ST401 || 26 || 44 20 i 35 | 9.82198 10.17802 9.94.808 I0,05192 || 10,12610 9,8739() 25 || 40 24 || 36 82212 17788 94834 05166 12622 87378 24 || 36 28 || 37 82226 I7774 94S59 05141 I2633 S7367 23 || 32 32 || 38 82240 1776) 94884 05116 12644 87356 22 28 36 || 39 8.2255 17745 94910 ()5090 12655 87345 || 21 24 40 40 || 9,82269 10,17731 || 9.94935 10,05065 || 10,12666 9.87334 20 | 20 44 || 41 82.283 17717 94061 05039 12678 873-22 || 19 || || 6 48 || 42 82.297 17703 94986 05014 12689 87311 18 || 12 52 || 43 82311 17689 95012 04988 12700 8.7300 17 8 56 || 44 82326 I7674 95037 04963 12712 87.288 || 16 4 47 || 45 || 9,82340 10.17660 9.95062 10.04938 || 10.12723 9.87.277 ; 15 13 4 |. 46 82354 17646 950SS 04912 12734 87.266 || 14 || 56 8 || 47 82368 17632 95113 04887 12745 87255 || 13 || 52 12 48 82.382 17618 95139 04861 12757 87243 | 12 || 48 16 || 49 82396 17604 95164. 04836 12768 87.232 || 11 || 44 20 50 || 9.82410 10, 17590 || 9,95190 10,04810 || 10,12779 9.87221 || 10 | 40 24 || 51 824.24 17576 95.215 04785 12791 87.209 9 || 36 28 || 52 82.439 17561 95.240 04760 12802 87.198 8 || 32 32 || 53 82.453 I'7547 95.266 04734 12.813 87187 7 2S 36 || 54 82.467 I7533 95.291 04709 12825 S7175 6 || 24 30 55 || 9.82481 10.17519 || 9.95317 10,046834; 10.12836 9,87164 5 i 20 44 || 56 82495 17505 95.342 04658 12847 S7153 4 || || 6 48 57 82509 17491 95368 0.4632 12859 87.141 3 || 12 52 58 82523 17477 953)3 04607 12S70 87.130 2 S 56 59 82537 I7463 95.418 04582 I288.1 87119 l 4 4-8 60 82551 17:449 95444 04556 12893 87107 () 12 M. S. M Conine. Secant. Cotangent Tangent. Cosecant. Sine. Aſ M.S. 8h ||131 48° 3h LOGARITHMS TRIGONOMETRIC. 2h 429 Logarithms. 1370 9h M.S. M Sime, Cosecant. Tangent. | Cotangent. Secant. Cosine. M. M. S. 4-8 0 || 9.82551 10.17449 || 9.95444 10.04556 || 10.12893 9.87107 || 60 | 12 4 I 82565 I7435 95.469 0.4531 12904 87096 || 59 || 56 8 2 82579 I'7421 95495 04.505 12915 87085 || 58 52 12 3 82593 IT407 95520 04480 12927 87073 ſ 57 || 48 16 4 82607 17393 955.45 04:455 12938 87062 || 56 || 44 20 5 9.82621 10,17379 || 9.955'71 10.04429 || 10,12950 9.87050 || 55 40 24 6 82635 I'7365 95596 04.404 12961 87 ()39 || 54 36 28 7 8264 17351 95622 04:378 12972 8.7028 53 || 32 32 8 82663 17337 95647 04:353 12984 87016 || 52 || 28 36 9 82677 17323 95672 04328 12995 87005 || 51 24 40 10 || 9,82691 10.1730 9.95698 10.04302 || 10.13007 9.86993 || 50 | 20 44 11 82705 17295 95723 04:277 13018 86982 || 49 16 48 12 82.719 17281 95.748 0.4252 13030 86970 || 48 || I2 52 || 13 82.733 17267 * 95774 04226 13041 86959 || 47 8 56 || 14 827.47 17253 95799 04201 13053 86947 || 46 4 49 || 15 || 9,82761 10.17239 || 9.958.25 10,0417 10.13064 9.86936 || 45 || Ill 4 || 16 82775 17225 95850 04:150 13076 86924 || 44 || 56 8 || 17 82788 17212 95875 04:125 13087 86013 || 43 || 52 12 18 82802 J'ſ 198 95901 04U99 13098 86902 42 || 48 16 || 19 82816 17184 95926 04074 13110 86$90 4l || 44 20 20 || 9,82830 10,17170 || 9.95952 10,04048 || 10.13121 9.86879 40 40 24 21 8284.4 17156 95.977 04023 13133 86867 || 39 36 28 || 22 82858 I'7142 96.002 03998 13145 86855 || 38 || 32 32 || 23 82872 17128 96.028 0.3972 13156 86844 || 3 2S 36 || 24 828S5 I'7115 96053 03947 13168 86832 || 36 || 24 40 || 25 || 9,82899 10,17101 || 9.96078 10,03922 || 10.13179 9.86821 || 35 | 20 44 26 82.913 17087 96.104 (3896 13.191 86809 || 34 16 48 || 27 8.2927 17073 96.129 03871 13202 86798 || 33 12 52 28 82941 17059 96.155 03845 13214 867S6 || 32 8 56 29 82955 17045 96.180 03820 13225 86775 31 4 50 || 30 || 9.82968 10,17032 || 9.96205 10,03795 || 10.13237 9,86763 || 30 || 10 4 31 82982 17018 96231 03769 13248 86752 29 || 56 8 || 32 82996 17004 96.256 03744 13260 86740 28 || 52 12 || 33 83010 16990 962S1 03719 13272 86728 27 || 48 16 || 34 83023 16977 96307 03693 13283 86717 | 26 || 44 20 || 35 ; 9.836)37 10,16963 9.96332 10,03668 10.13295 9.86705 || 25 40 24 || 36 83051 16949 96.357 03643 13306 86694 || 24 || 36 28 || 37 83065 16935 96383 036.17 I3318 86682 23 || 32 32 || 38 83078 16922 96.408 03592 13330 S667() 22 || 28 36 || 39 83092 16908 96433 03567 1334.1 S6659 || 21 24 40 # 40 9.83}06 10,16894 || 9.96459 10.03541 || 10.13353 9.86647 20 20 44 41 83.120 16880 96.484 03516 13365 86635 | 19 || 16 48 I 42 83133 16867 9651() 03490 13376 86624 i 18 || 12 52 43 83147 16853 96535 03465 13388 86612 || 17 8 56 || 44 83.16.1 16839 96560 03440 13400 86600 16 4 51 || 45 9.83,174 10.16826 || 9.96586 10.03414 || 10.13411 9.86589 15 9 4 || 46 83.188 16812 96611 ()3389 13423 86577 || 14 56 8 || 47 83202 16798 96636 03364 13435 86565 || 13 || 52 12 || 48 83215 I6785 9666.2 03338 13446 86554 || 12 || 48 16 49 83229 16771 96687 0.3313 13458 86542 || 11 || 44 20 || 50 || 9.83242 10.16758 || 9.96712 10,03288 || 10.13470 9.86530 || 10 || 40 24 || 51 83256 16744 96.738 03262 13482 865.18 9 || 36 28 || 52 83270 16730 967G3 03237 13493 86507 8 32 32 53 83283 16717 96788 03212 13505 86495 7 28 36 || 54 83297 16703 96814 03186 1351.7 864S3 6 24 $0 55 9.83310 10.16690s 9,96839 10.03161 || 10.13528 9.864.72 5 20 44 || 56 8332.4 16676 96.864 ()3136 13540 86460 4 || 16 48 || 57 83338 1666.2 96.890 03110 13552 864.48 3 12 52 || 58 83351 16649 96915 03085 13564 864.36 2 8 66 || 59 83365 16635 96.940 0.3060 13575 864.25 I 4 52 || 60 83378 16622 96966 03034 13587 864.13 () || S M. 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S. 0 || 9.84177 10.15823 || 9,98484 || 10.01516 | 10.14307 9.85693 || 60 || 4 1. 84190 1581() 98.509 0.1491 14319 85681 || 59 || 56 2 84203 15797 98534 O1466 ſ 14331 85669 || 58 || 52 3 84216 J5784 985.60 0.1440 14343 $5657 57 || 48 4 84229 15771 985S5 0.1415 14355 85645 || 56 || 44 5 || 9.84242 10.1575S 9.98610 || 10.0139() { 10.14368 9.85632 || 55 | 40 6 84255 15745 9S635 0.1365 14380 85620 54 || 36 7 S4269 I5731 98661 0.1339 14392 85.608 || 53 || 32 8 84.282 I5718 9S6S 6 0.1314 14.404 85596 || 52 28 9 84295 15705 98.711 0.1259 14417 S55S3 || 51 || 24 10 9.84308 10.15692 || 9.98737 10.0.1263 || 10.14429 9.8557 | || 50 20 11 84321 15679 98762 01.238 14441 $5559 || 49 || 16 12 84334 I 5066 98.787 012.13 14453 85547 || 48 || 12 13 84347 15653 98812 01.188 14466 85534 || 47 8 14 84360 15640 98838 0.1162 14.478 85522 || 46 4 15 9.84373 10.15627 | 9.98SG3 10.01137 || 10.14490 9.85510 || 45 || 3 16 S4385 15615 98S 88 01112 1450.3 85497 || 44 || 56 17 84398 I5602 9S913 ()1087 14515 854S5 || 43 || 52 18 84411 15589 98.939 U1061 14527 85.473 || 42 48 19 84424 15576 9S964 01.036 1454) S5460 || 41 || 44 20 || 9.84437 10.15563 9.98989 10.01011 || 10.14552 9.8544S 40 || 4() 21 84.450 15550 99.015 ()0985 14564 8.5436 || 39 || 36 22 84.463 15537 9904() 00960 1457.7 85.423 38 || 32 23 84476 15524 99.065 00935 14589 854.11 || 37 || 28 24 84489 15511 99000 00910 14601 85399 || 36 24 25 || 9.84502 || 10.15498 || 9.99.116 10.00884 || 10.14614 || 9.85386 || 35 | 20 26 845.15 15485 99.141 008:59 14626 85374 || 34 16 27 84528 15472 99166 008:34 14639 85.361 || 33 || 12 2S 84540 15460 9919.1 00809 14651 85349 || 32 8 29 8+553 15447 99.217 U0783 14663 85.337 || 31 4 30 || 9.84566 10.15434 || 9.99.242 10.00758 || 10.14676 | 9.85324 30 || 2 31 845.79 15421 99267 0()733 146SS 85.312 || 29 || 56 32 S4592 15408 99.293 00707 14.701 85299 || 28 52 33 84605 15895 99.318 006S2 14713 852S7 27 || 48 34 84618 15382 99.343 00657 14726 85274 26 || 44 35 | 9.84630 || 10.15370 9.99368 10.00632 || 10.14738 9.S$262 || 25 | 40 36 846.13 15357 99.394 00606 14750 85.250 || 24 || 36 37 84636 15344 994.19 0.0581 14763 85.237 || 23 || 32 38 84669 15:331 99.444 00556 14775 85.225 || 22 28 39 84682 1531S 99.469 00531 14788 85212 21 24 40 || 9,84694 10,15306 || 9.994.95 10 00505 || 10 14800 9.85200 | 20 | 20 41 S47 ()7 15293 99.520 00480 14813 85.187 | 19 || 16 42 84720 15280 995.45 0.0455 148.25 S5,175 | 18 12 43 S4733 15267 99570 004:30 14838 85162 17 S 44 S4745 15255 99596 00404 14850 85150 | 16 4. 45 9.847.58 10.15242 9.99621 10.00379 10.14863 9 85.137 || 15 1 46 84771 15229 99646 00354 14875 85.125 || 14 56 47 84784 15216 99672 00328 14888 85112 || 13 || 52 48 84796 15204 99.697 00:303 14900 85100 | 12 || 48 49 84809 15.191 9972.2 ()0278 14913 85087 | 11 || 44 5() 9.84822 || 10.15178 || 9.997.47 10.00253 10.14926 || 9.85074 || 10 | 40 51 84835 15] 65 99773 U0227 14938 85062 9 : 36 52 84S47 15153 99798 00202 14951 85049 8 || 32 53 84860 15140 998.23 00.177 14963 85037 7 2S 51 84873 15127 99S48 00152 14976 85024 6 || 24 55 || 9.84885 || 10.15115 || 9.998.74 || 10.00126 || 10.14988 9.85012 5 || 20 56 84898 15102 99899 00101 I5001 849.99 4 || 16 57 84911 I5()89 99924 00076 15014 849S6 3 || 12 58 84923 15077 99949 00051 15026 84974 2 S 59 8.4936 15064 99.975 00025 15039 84961 l 4 60 84949 15051 10,000U0 00000 15051 84949 0 || 0 M Cosime. Secant. ICotangent. | Tangent. || Cosceant. Sune. M. M. S. 1349 45° 3h 208 ExPLANATION OF TIIE TABLES. EXPLAN ATION OF THE TAB L E S. The outer columns in the trigonometrical tables contain the angle in time of hours, minutes and seconds, corresponding to the same angle in degrees and min- utes in the next columns. The hour is noted at the top and bottom, the minutes in black, and the seconds in ordinary figures. To find the Logarithm and Natural Lime for Seconds exceeding Minutes of a Degree. Example 1. Find the logarithm for sin. 38° 47' 55". or si O 48' = 9.7 * * From table, {**:::::::::::::::} diff. 15. Correction, 15 × 55 : 60 = + 14 nearly. The required log. sin. 38° 47' 55" = 9.79698 In practice, the difference is subtracted direct from the tables. Example 2. Find the natural cos. 43°29' 19°. From table, cos. 430 29' = 0.72557 Correction, 20 × 19: 60 = — 6 nearly. The required cos. 43°29' 197 = 0.72551 The correction is added when the function is increasing, and subtracted when decreasing. To find the Angle corresponding to a given Logarithm or Nat- ural Haine. Example 3. Log. sin. = 9.56429. Required the angle. or si O / - From table, log. sin. 21° 31 = 9.56.440 “ “ 210 307 = 9.56.408 {{ 4) The angle required, “ “ 21°30'29"= 9.56.429 21. Correction, 21 × 60: 32 = 29 seconds nearly. Example 4. Cosine = 0.35254. Required the angle. From table, ſº 69° 22' = 0.85239) diff, 27. “ 690 21/ = 0.35266 The required angle, “ 69° 21' 27’’= 0.35254 & Correction, 12 × 60: 27 = 27 secouds, nearly. diff. 32. Conversion of Minutes and Seconds into Decimals.of a Degree or of an IMour. M. Decimal. M. Decimal | M. Decimal. S. Decimal. S. Decimal. S. Decimal. 1 .016666 || 21 | .350000 || 41 .6S3333 1 | .000277 21 .005833 41 | .011388 2 | .033333 22 | .366666 || 42 | .700000 2 . .000555 22 | .006111 || 42 || 011666 3 .050000 || 23 .383.333 43 | .716666 3 || 000833 || 23 .006388 || 43 .01.1944 4 || 066666 || 24 .400000 || 44 | .733333 4 | .001111 || 24 ,006666 44 .012222 5 .083333 || 25 | .416666 || 45 | .750000 5 | .001388 25 | |00694-4 45 .012500 6 | .100000 || 26 | .433333 || 46 .766666 6 | .001666 .007 222 || 46 .012777 7 | .116666 27 .450000 || 47 | .783.333 7 | .001944 .007500 || 47 || 01:3055 8 . .133333 || 28 .466666 || 48 .800000 8 . .002222 .007.777 || 48 || 013333 9 | .150000 || 29 .483.333 || 49 | .816666 9 .002500 .008055 49 || 01:3611 10 .166666 || 30 .500000 || 50 8.33333 || 10 | .002777 .00S333 50 | 013888 11 .183333 ||31 | .516')00 || 51 | .850000 || 11 .003055 || 31 | .008611 || 51 || 014166 12 .200000 32 .533333 52 | .866666 || 12 | .003333 || 32 .008888 || 52 .01.4444 13 .216666 |33 | .550000 || 53 | .88.3333 || 13 .003611 || 33 .009166 || 53 || 014722 14 .233333 ||34 || .566666 || 54 .900000 || 14 | .003888 34 .009.444 || 54 ,015000 15 .250000 || 35 | .583333 55 .916666 || 15 .004,166 || 35 .0097.22 55 | .015277 16 | .266666 || 36 | .600000 || 56 .933333 || 16 | .004.444 || 36 .010000 || 56 .015555 17 | .283333 ||37 | .616666 57 .950000 || 17 | .004.722 || 37 .010277 || 57 || 015833 1S .300000 || 38 .633333 58 .966666 || 18 .005000 || 38 .010555 58 .016111 19 .316666 || 3 .65000ſ) || 59 983333 || 19 .005277 || 39 | .010S33 || 59 . .016388 20 | .333333 || 40 | .666666 || 60 (1.000000 || 20 | .005555 || 40 .01.1111 || 60 || 016666 i :6 NATURAL I/INES. 09 Natural Trigonometrical Functions. 1790 M. Sine. Vrs. Cos. Cosec'nte Tang. | Cotang. Secante.IVrs.Sin. Cosine. M. 0 | .00000 | 1.0000 Infinitel .00000 Infinitel 1.0000 |.00000 | 1.0000 || 60 1 | . 0029 . .99971 || 3437.7 | . 0029 || 3437.7 | .0000 || 0000 | .0000 || 59 2 0.058 . 9942 1718.9 0058 || 1718.9 .0000 | . 000() .0000 || 58 3 | . 0087 . 9913 || 1145.9 ()087 || 1145.9 .0000 i. (1000 | .0000 || 57 4 Q116 . 9884 859.44 0.116 || 859.44 | .0000 . 0000 | .0000 : 56 5 | .00145 |.99854 | 687.55 | .00145 | 687.55 | 1.0000 .00000 | 1.0000 55 6. 0.174 . 9825 572.96 | . 0.174 572.96 | .0000 | . 0000 | .0000 || 54 7 | . 0204 |. 9796 || 491.11 . . 0204 || 491.11 . .0000 | . 0000 | .0000 || 53 8 0233 . 97.67 429.72 0233 || 429.72 | .0000 | . ()()00 | .0000 || 52 9 | . 0262 |. 9738 || 3S1.97 . 0262 || 381.97 || 0 , 0000 | .0000 || 51 10 .00291 || 99.709 || 34377 | .00291 || 343.77 | 1.00(K) i .00000 .99999 || 50 11 | . 0320 |. 9680 312.52 0320 312.52 .0000 || ()000 | . 9999 || 49 12 | . 0349 |. 9651 286.48 |. 0349 || 286 48 | .0000 | . 0001 | . 9999 || 48 13 | . 0378 . 9622 || 64.44 |. 0378 || 64.44 .0000 | . 0001 | . 9999 || 47 14 | . 0407 | . 9593 || 45.55 0407 || 45 55 .0000 | . 0001 | . 9999 || 46 15 || 00436 .99564 229.18 .00436 229.18 1.0000 || 00001 | .99999 || 45 16 | . 9534 14.86 . 0465 || 14.86 .0000 | . ()001 | . 9999 || 44 17 | . 0494 | . 9505 || 02:22 |. 0494 | 02.22 | .0000 | . 0001 | . 9999 || 43 18 . 0524 |. 9476 | 190.99 |. ()524 || 190.98 || 0000 || 0001 |. 9999 || 42 19 . O553 |. 94.47 | 180.93 0553 | 180.93 | .0000 | . 0001 | . 9998 || 41 20 | .00582 .99418 || 171.89 | .005S2 || 171.88 | 1.0000 | .00002 | .99998 || 40 21 | . 0611 . . 9389 63.7() 0611 63.70 | .0000 | . 0002 | . 9998 || 39 22 . 0640 | . 9360 || 56.26 06:40 || 56.26 .0000 | . 0002 | . 9998 || 38 23 . O669 |. 9331 || 49.47 O669 || 49.46 .0000 | . ()002 | . 9998 || 37 24 . . 0698 | . 9302 || 43.24 0698 || 43.24 .0000 | . 0002 | . 9997 || 36 25 | .00727 .99273 | {37.5% .00727 | 137.5L | 1.0000 || 00003 | .99997 || 35 26 . U766 |. 9244 || 32.22 0756 32.22 .0000 . ()003 | . 9997 || 34 27 | . 0785 |. 92.15 27.32 | . O785 27.32 | .0000 | . 0003 |. 9997 || 33 28 | . 0814 | . 9185 22.78 . 0814 || 22.77 | .0000 | . 0003 | . 9997 || 32 29 . Q843 . 9156 | 18.54 |. 18.54 || 0000 | . ()003 . 99.96 || 31 30 || 00873 || 99127 | 114,59 |.00S73 114.59 | 1.0000 || 00004 | .99996 || 30 31 | . O902 | . 9098 || 10.90 0902 || 10.89 | .0000 | . 0004 | . 9996 || 29 32 | . Q931 |. 9069 || 07.43 0931 || 07.43 .0000 |. 0004 | . 9996 || 28 33 || . Q960 | . 9040 || 04.17 096() 04.17 | .0000 | . 0005 | . 9995 || 27 34 . O989 | . 9011 || 01.11 0989 || 01.11 . .0000 | . 0005 | . 9995 || 25 35 | |01018 .98982 98.223 || 01018 98.218 1.0000 || 00005 | .99995 || 25 36 1047 | . 8953 || 5.495 1047 5.489 | .0000 i. ()005 |. 9994 || 24 37 1076 |. 8924 2.914 | . 1076 2,908 || 0000 , 0006 | . 9994 || 23 38 1105 . 8895 0.469 1105 0.463 | .0001 . . 0006 | . 9994 22 39 1134 . 8865 || 88.149 |. 1134 || 88.143 | .0001 0000 | . 9993 || 21 40 || 01163 .98836 || 85.946 .01164 || 85.940 | 1.0001 || 00007 |.79993 || 20 41 E.193 | . 8807 || 3.849 1193 3.843 | U001 |. 0007 | . 9998 || 19 42 1222 . 8778 1.853 | . 1222 | 1.S47 .0001 0007 | . 9992 | 18 43 1251 . 8749 79.950 1251 || '19.943 .0001 | . ()008 | . 9992 || 17 44 1280 | . 8720 || 78.133 1280 || 78.126 .0001 |. 0008 |. 9992 | 16 45 .01300 .98691 || 76.306 |.01309 || 76.390 | 1.0001 || 00008 | .99991 || 15 46 . 1338 . 8662 || 4.736 |. 1338|| 4.729 .0001 0009 | . 9994 || 14 47 1367 | . 8633 || 3.146 | . 1367 || 3.139 | .0001 |. ()009 |. 99.91 || 13 48 1396 . 8604 || 1.622 1396 || 1.6]5 | .0001 0010 | . 99.90 12 49 1425 | . 8575 0.160 1425 || 0.153 .0001 0.010 | . 9990 || 11 50 || 01454 .985.46 68.757 |.01454 | 68.750 | 1,0001 ||.00010 | .99989 || 10 51 1483 | . 8516 || 7.409 1484 || 7.402 | .0001 |. 0011 | . 99.89 || 9 52 | . 1512 |. 8487 || 6.113 |. 1513| 6.105 | .0001 00.11 . . 9988 || 8 53 . 1542 | . 8458 4.866 1542 || 4.85S | .0001 0012 | . 9988 || 7 54 . 1571 . 8429 || 3.664 |. 1571 3.657 | .0001 | . 0012 | . 9988 || 6 55 .01600 | .98400 62.507 |.01600 || 62.499 || 1.0001 | .00013 | .99987 5 56 | . 1629 |. S371 | 1.391 1629 1,383 | .0001 . . 0013 | . 99S7 || 4 57 . 1658 . . 8342 | 0.314 1658 || 0.306 || 0001 || , ()014 | . 99S7 || 3 58 | . 1687 | . 8313 || 59.274 1687 59.266 ,000l j . 0014 |. 9986 2 59 . 1716 . 82S4 S.270 1716 || S 261 .0001 |. OU15 | . 9985 || 1 60 . 1745 | . 8255 7.299 1745 || 7.290 . .0001 | . ()015 | . 9985 || 0 M. Cosime. Wrs. Sin. I Secante. Cotang. Tangent. Cosec'n't Wrs. Cos. Sine. M. 900 Natural. 899 NATURAL LINES. 1o Natural Trigonometrical Functions. 1780 M Sine. Vrs.Gos. Cosec'nte Tang. | Cotung. Secante.}Vrs. Sin Cosine. Aſ 0 1 01745 .98255 57.299 .01745 57.290 | 1.0001 | .00015 .99985 60 1 | . 1774 |. 8226 || 56.359 |. 1775 56.350 | .0001 |. 0016 |. 9984 || 59 3 |. 1803 |. 8196 55.450 |. 1804 || 55 441 .0002 |. 0016 |. 9984 || 58 3 | . 1832 . 8167 || 54.570 |. 1833 || 54.561 | .0002 |. 0017 |. 9983 || 57 4 | . 1861 |. 8138 53.718 . 1S62 | 53.708 .0002 |. 0017 |. 9983 || 56 5 .01891 | .98109 || 52.891 |.01S91 || 52.882 | 1.0002 | .00018 .99982 55 6 | . 1920 |. 8080 l 2.090 |. 1920 | 2,081 .0002 |. 0018 |. 9981 || 54 7 | . 1949 }, 8051 | 1.313 . 1949 | 1.303 | .0002 |. 0019 |. 9981 || 53 8 |. 1978 |. 8022 || 0.558 |. 1978 || 0.548 .0002 |. 0019 |. 99.80 || 52 9 : . 2007 |. 7993 || 49.826 |. 2007 || 49.816 | .0002 |. 0020 |. 9980 || 51 10 .02036 | .97964 || 49.114 | .02036 | 40.104 | 1.0002 |.00021 .9997.9 || 50 11 | . 2065 |. 7935 | 8.422 |. 2066 || 8.412 .0002 |. 0021 |. 9979 || 49 12 |. 2094 |. 7906 || 7.750 |. 2095 || 7.739 .0002 |. 0022 |. 9978 || 48 13 | . 2123 |. 7877 || 7.096 |. 2124 || 7.085 | .0002 |. 0022 |. 9977 || 47 14 |. 2152 |. 7847 || 6.460 | . 2153 || 6.449 .0002 |. 0023 |. 9977 || 46 15 | .02181 |.97818 45.840 | .02182 45.829 | 1.0002 |.00024 .99976 : 45 16 |. 2210 |. 7789 5.237 |. 2211 || 5.226 .0002 |. 0024 |. 9975 || 44 17 | . 2240 |. 7760 4.650 |. 2240 || 4.638 .0002 |. 0025 |. 9975 || 43 18 |. 2269 |. 7731 || 4.077 |. 2269 || 4,066 .0002 |. 0026 . 9974 || 42 19 |. 2298 |. 7702 || 3.520 | . 2298 || 3.508 | .0003 |. 0026 . 9974 || 41 20 | .02327 | .97673 || 42.976 |.02327 || 42.964 | 1.0003 |.00027 . .99973 || 40 21 |. 2356 |. 7644 || 2.445 |. 2357 || 2.433 | .0003 |. 0028 |. 9972 || 39 2: . . 2385 |. 7615 1928 |. 2386 1916 | .0003 |. 0028 |. 9971 || 38 23 . 2414 |. 7586 | 1.423 | . 2415 | 1.410 | .0003 |. 0029 |. 9971 || 37 24 . 2443 |. 7557 0.930 | . 2444 0.917 | .0003 |. 0030 |. 9970 || 36 25 | .02472 .97528 40448 |.02473 || 40.436 | 1.0003 |.00030 .99969 35 26 . 2501 |. 7499 || 39,978 |. 2502 || 39.965 | .0003 |. 0031 |. 9969 || 34 27 25.30 | . T.;69 || 9.518 |. 2531 || 9.506 | .0003 |. 0.032 | . 9968 || 33 2S 2559 | . T440 || 9.069 256)| 9.057 | .0003 |. 0.033 . 9967 32 29 |. 2589 |. 7411 || 8.631 |. 2589 8.618 .0003 |. O(|33 |. 9966 || 31 30 | .02618 |.973S2 || 38.201 |.02618 || 38.188 | 1.0003 |.00034 .99966 || 30 31 | . 2647 |. 7353 || 7.782 |. 2648 || 7.769 | .0003 |. 0035 | . 9965 29 32 . 2676 |. 7324 || 7.371 | . 2677 || 7.358 .0003 |. 0.036 |. 9964 28 33 . 2705 | . T295 || 6.969 |. 2706 || 6.956 .0004 |. 0.036 . 9963 || 27 34 |. 2734 |. 7266 || 6.576 |. 2735 | 6.563 .0004 |. 0037 . 9963 || 26 35 | .02763 .97237 || 36.191 |.02764 || 36.177 | 1.0004 |.00038 .99962 || 25 36 |. 2792 | . 7208 || 5.814 |. 2793 5.800 | .0004 |. ()039 |. 9961 || 24 37 | . 2821 | . TIT9 || 5.445 |. 2822 || 5.431 | .0004 |. 0040 |. 9960 23 38 |. 2850 |. 7150 5,084 |. 2851 5.069 .0004 |. 0041 |. 9959 22 39 | . 2879 |. 7121 || 4.729 |. 2880 || 4.715 .0004 |. 0041 |. 9958 21 40 || 02908 |.97091 34,382 |.02910 || 34,368 | 1.0004 |.00042 | .99958 20 41 2937 ||. T062 || 4.042 | . 2939 || 4.027 . .UU04 |. 0043 . 9957 19 42 2967 | . TO33 3,708 . 2968 || 3.093 .0004 ||. ()044 | . 9956 || 18 43 2996 ||. T004 || 3.381 | . 2997 || 3.366 .0004 |. 0045 |. 9955 17 44 | . 3025 | . 6975 || 3.060 |. 3026 || 3.045 .0004 |. 0046 | . 9954 || 16 45 | .03054 | .96946 || 32.745 | .03055 || 32.730 | 1.0005 . .00046 | .99953 || 15 46 . 3083 |. 9692 || 2.437 |. 3084 || 2.421 | .0005 |. 0047 . 9952 : 14 47 | . 3112 | . 6888 i 2.134 |. 3113 || 2.118 .0005 |. ()048 |. 9951 || 13 48 | . 31.41 |. 6859 | 1.836 |. 3143 T.820 .0005 |. 0.049 |. 9951 | 12 49 |. 31.70 | . 6830 | 1.544 . 31.72 | 1.52S .0005 |. 0.050 |. 9950 || 11 50 | .03199 || .96801 || 31.257 |.03.201 || 31.241 | 1.0005 |.00051 | .99949 || 10 51 | . 3228 . 6772 (),976 . 3230 || 0.960 .0005 | . 0.052 . 9948 || 9 52 . 3257 | . 6743 0.699 |. 3259 0.683 | .0005 |. 0.053 | . 99.47 8 53 | . 3286 |. 6713 || 0.428 |. 3288 || 0.411 | .0005 i. 0.054 |. 9946 || 7 54 . 3315 . 6684 0.161 | . 3317 | 0.145 | .0005 |. O(|55 |. 9945 || 6 55 .03344 .96655 29.899 |.03346 29.882 | 1.0005 . .00056 | .99944 || 5 56 . 3374 |. 6626 9.641 |. 3375 || 9.624 | .0006 |. 0.057 | . 9943 || 4 57 | . 3403 |. 6597 || 9.3SS | . 3405 || 9,371 | .0006 |. 0058 . 9942 || 3 58 | . 3432 | 6568 || 9,139 |. 3434 || 9,122 | .0006 |. 0059 |. 9941 || 2 59 | . 3461 |. 6539 || 8,894 | . 3463 || , 8.877 | .0006 |. 0.060 | . 99.40 || 1 60 , 3490 | . 6510 || 8.654 |. 3492 8.636 | .0006 |. 0.061 | . 9939 || 0 M l Cosine. Wrs. Sin. Secante. | Cotang. Tangent. Cosec'nt Wrs. Cosi Sine. M 910 Natural. 889 NATURAL LINES. 29 Natural Trigonometrical Functions. 1779 [11h M Sine. Wrs. Cos. Cosec'nte | Tang. Cotang. Secante. Wrs. Sin Cosine. M. M.S. 0 || 03490 | .96510 28.654 |.03492 28 636 | 1.0006 | .00061 | .99939 || 60 52 1 i. 3519 |. 6481 | 8,417 | . 3521 | 8,399 .0006 |. 0062 |. 9938 || 59 || 56 2 . . .3548 . 6452 | 8.184 |. 3550 8.166 .0006 |. 0063 |. 99.37 || 58 || 52 3 | . 3577 |. 6423 || 7.955 |. 3579 || 7.937 .0006 |. 0064 |. 9936 || 57 || 48 4 | . 3606 |. 6394 || 7.730 |. 3608 || 7.712 .0006 |. 0065 |. 9935 | 56 || 44 5 | 03635 | .96365 27.508 |.03638 27.490 | 1.0007 |.00066 .99934 || 55 | 40 6 |. 3664 |. 6336 | 7.290 |. 3667 || 7.271 | .0007 |. 0067 l. 9933 || 54 || 36 7 | . 3693 |. 6306 || 7.075 |. 3696 || 7.056 | .0007 l. 0068 |. 9932 || 53 || 32 8 . . 3722 |. 6277 || 6.864 . 3725 | 6.845 | .0007 |. 0.069 |. 9931 || 52 || 28 9 | . 3751 |. 6248 6.655 |. 3754) 6.637 .0007 |. 0070 |. 9930 || 51 || 24 10 : 03781 | .96219 || 26.450 |.03783 26.432 | 1.0007 |.00071 .99928 50 | 20 11 | . 3810 | . 6190 || 6 249 |. 3812 || 6.230 | .0007 l. 0073 |. 9927 || 49 | 16 12 | . 3839 |. 6161 | 6.050 |. 3842| 6,031 | .0007 |. 0074 |. 9926 || 48 || 12 13 | . 3868 |. 6132 || 5.854 |. 3871 || 5.835 | .0007 |. 0075 |. 9925 || 47 || 8 14 |. 3897 |. 6103 || 5.661 |. 3900 || 5.642 || 0008 |. 0076 |. 99.24|46 || 4 15 .03926 . .96074 || 25.471 |.03929 || 25.452 | 1.0008 |.00077 | .99923 || 45 || 51 16 | . 3955 |. 6045 || 5.284 |. 3958 || 5.264 .0008 |. 0078 |. 9922 || 44 || 56 17 | . 3984 |. 6016 || 5.100 |. 3987 || 5.080 | .0008 |. 0079 |. 9921 || 43 || 52 18 . 4013 |. 5987 || 4.918 . . 4016 || 4.898 .0008 |. 0080 |. 9919 || 42 || 48 19 |. 4042 |. 5958 4.739 . 4045 || 4.718 .000S ). 0082 |. 99.18 || 41 || 44 20 | .04071 |.95929 || 24.562 |.04075 24.542 | 1,0008 |.00083 |.99917 | 40 | 40 21 4100 |. 5900 || 4,388 | . 4104 || 4,367 .0008 |. 0084 |. 9916 || 39 || 36 22 | . 4129 |. 5870 || 4.216 | . 4.133 || 4.196 | .0008 |. 0085 |. 9915 || 38 32 23 | . 4.158 |. 5841 || 4.047 l. 4162 || 4.026 .0009 |. 0086 |. 9913 || 37 | 28 24 . 4.187 |. 5812 || 3 880 |. 4191 || 3.859 .0009 |. 0088 |. 9912 || 36 || 24 25 | .04217 | .95783 23.716 |.04220 || 23.694 | 1.0009 .00089 .999.11 || 35 | 20 26 | . 4246 | . 5754 || 3.553 |. 4249 3,532 .0009 . ()090 |. 99.10 || 34 16 27 | . 4275 |. 5725 || 3.393 |. 4279 || 3.372 | .0009 |. 00.91 |. 9908 || 33 || 12 28 . 4304 |. 5696 || 3.235 | . 4308 || 3.214 | .0009 |. 0093 |. 9907 ||32 || 8 29 | . 4333 |. 5667 || 3.079 |. 4337 3.058 | .0009 |. 0094 | . 9906 || 31 || 4 30 | .04362 .95638 22.925 | .04366 22.904 || 1.0009 |.00095 .99905 || 30 || 50 31 | . 4391 |. 5609 || 2.774 |. 4395 || 2.752 | .0010 |. 0096 |. 99.03 || 29 || 56 32 . 4420 |. 5580 2.624 |. 4424 || 2.602 | .0010 |. 0098 |. 9902 || 28 || 52 33 . 4449 |. 5551 2.476 |. 4453 || 2.454 .0010 |. 0099 |. 9901 || 27 || 48 34 . . 4478 |. 5522 || 2.330 . 4483 || 2.308 || 0010 |. 0100 |. 9900 || 26 || 44 35 | .04507 |.95493 22.186 |.04512 22,164 | 1.0010 |.00102 | .99898 || 25 | 40 36 |. 4536 |. 5464 2.044 |. 4541 || 2.022 | .0010 |. 0103 |. 9897 || 24 || 36 37 | . 4565 |. 5435 | 1.904 |. 4570 | 1.881 | .0010 |. 0104 |. 9806 || 23 || 32 38 | . 4594 | . 5405 | 1.765 . . 4599 || 1.742 | .0010 |. 0100 |. 98.94 22 || 28 39 |. 4623 |. 5376 | 1.629 |. 462S | 1.606 | .0011 |. 0.107 |. 98.93| 21 || 24 40 .04652 |.95347 || 21.494 |.04657 || 21.470 | 1.0011 || 00108 |.90892 || 20 | 20 41 . 4681 |. 5318 || 1.360 |. 4687 | 1.337 | .0011 . 0110 |. 9890 | 19 | 16 42 | . 4711 |. 5289 | 1.228 |. 4716 | 1.205 | .0011 |. 0.111 | . 9889 18 || 12 43 | . 4740 | . 5260 | 1.098 |. 4745 | 1.075 .0011 |. 0112 | . 9888 || 17 | 8 44 |. 4769 |. 5231 || 0.970 774 0.946 .0011 |. 0114 |. 9886 | 16 || 4 45 | .04798 .95202 || 20.843 .04803 20.819 | 1.0011 |.00115 .99885 || 15 49 46 | . 4827 | . 5173 || 0.717 | . 4832 || 0,603 || 0012 . 0116 |. 9SS3 || 14 56 47 . 4856 |. 5144 || 0.593 |. 4862 || 0.569 | .0012 |. ()118 |. 9882 || 13 52 48 | . 4885 |. 5115 || 0.471 |. 4891 || 0.446 | .0012 |. 0119 |. 98.81 | 12 || 48 49 | . 4914 | . 5086 || 0.350 . . 4920 || 0.325 | .0012 |. 0121 | . 987.9 || 11 || 44 50 .04943 | .95057 | 20.230 | .04949 20.205 | 1.0012 .001.22 | .998.78 || 10 || 40 51 | . 4972 | . 5028 || 0.112 | . 4978 || 0.087 .0012 . 0124 . 9876 9 || 36 52 j . 5001 | . 4999 || 19.995 |. 5007 || 19.970 | .0012 |. 0125 | . 9875 || 8 || 32 53 | . 5030 . 4970 9.880 |. 5037 || 9.854 | .0013 | . 0127 | . 9873 || 7 || 28 54 . 5059 | . 4941 || 9.766 |. 5066 9.7 10 | .0013 |. O128 . 98.72 || 6 || 24 55 | .05088 .94.912 || 19.653 | .05095 || 19.627 | 1.0013 |.00129 .998.70 || 5 || 20 56 . 5117 | . 488.3 || 9,541 | . 5124 || 9,515 . .0013 |. 0131 |. 9869 | 4 || 16 57 . 5146 | . 4853 || 9,431 | . 5153 9.405 | .0013 |, 0132 | . 9867 || 3 || 12 58 . , 5175 | . 4824 || 9,322 |. 5182 9.296 | .0013 |. 0134 | . 9866 || 2 || 8 59 . 5204 |. 4795 9.214 |. 5212 || 9,188 .0013 |. O135 | . 98.64 || 1 || 4 60 . 5234 . 4766 9.107 | . 5241 || 9,081 | .0014 |. 0137 |. 9863 || 0 || 48 M Cosine. Wrs. Sin. Secante. Cotang. Tangent. | Cosec'nt IVrs. Cosi Sine. M. M.S. 92 Natural. 87° 5h 212 . NATURAL LINES. Oh 39 Natural Trigonometrical Functions. 1769 |11.h M. S. M | Sine. Wrs. Cos. Cosec'nte Tang. | Cotang. Secante. {Wrs. Sinj Cosime. M M.S. 12 || 0 | .05234 .94766 | 19.107 .05241 || 19.081 | 1.0014 | .00137 | .99863 || 60 || 48 4 || 1 | . 5263 |. 4737 || 9.002 |. 5270 8.975 | .0014 |. 0138 |. 9861 59 || 56 8 | 2 | . 5292 |. 4708 || 8.897 |. 5299 || 8.871 | .0014 |. O140 | . 9860 l 58 52 12 || 3 | . 5321 |. 4679 || 8,794 |. 5328 8,768 . .0014 |. O142 |. 9858 57 || 48 l6 || 4 |. 5350 |. 4650 | 8,692 |. 5357 | 8.665 | .0014 |. O143 |. 9857 || 56 || 44 20 || 5 || 05379 |.94621 | 18.591 .05387 | 18.564 1.0014 .00145 .99855 55 40 24 || 6 | . 5408 |. 4592 || 8.491 |. 5416 || 8.464 | .0015 1. O146 |. 9854 || 54 || 36 28 || 7 |. 5437 |. 4563 || 8.393 |. 5445| 8.365 .0015 |. O148 |. 9852 || 53 || 32 32 || 8 | . 5466 |. 4534 || 8.295 |. 5474 || 8.268 .0015 |. O149 |. 9850 52 || 28 36 || 9 | . 5495 . 4505 || 8.198 |. 5503 || 8.171 .0015 1. O151 . 9849 || 51 24 40 || 10 || 05524 |.94476 | 18.103 |.05532 | 18.075 | 1.0015 |.00153 |.99$47 || 50 | 20 44 || 11 |. 5553 |. 4447 || 8.008 |. 5562 || 7.980 | .0015 |. 0154 |. 9846 || 49 | 16 48 || 12 | . 5582 |. 4418 || 7.914 |. 5591 || 7.SS6 | .0016 |. O156 |. 9844 48 || 12 52 || 13 | . 5611 . . 4389 || 7.821 . 5620 | 7.793 .0016 |. O157 |. 9842 || 47 || 8 5. 14 |. 5640 |. 4360 || 7,730 |. 5649 || 7.701 | .0016 |. O159 |. 98.41 || 46 || 4 13 | 15 | .05669 |.94331 || 17,639 | .05678 || 17.610 | 1.0016 | .00161 .99839 || 45 || 47 4 || 16 | . 5698 |. 4302 || 7.549 |. 5707 || 7.520 | .0016 |. 0162 |. 9837 || 44 || 56 8 || 17 | . 5727 |. 4273 || 7.460 | . 5737 || 7.431 | .0016 |. 0164 |. 9836 || 43 || 52 12 | 18 |. 5756 |. 4244 || 7.372 |. 5766 || 7.343 .0017 |. 0166 |. 9834 || 42 48 16 || 19 | . 5785 |. 4214 || 7.285 | . 5795 || 7.256 | .0017 |. 0167 |. 9832 || 41 || 44 20 | 20 | .05814 | .94.185 || 17.198 |.05824 || 17.169 | 1.0017 | .00169 |.99831 || 40 | 40 24 || 21 |. 5843 |. 4156 || 7.113 |. 5853 || 7.084 | .0017 |. O171 |. 98.29 || 39 || 36 28 22 | . 5872 |. 4127 7.028 |. 5883 || 6.999 | .0017 |. 0172 |. 9827 | 38 || 32 32 || 23 | . 5902 |. 4098 || 6.944 . 5912 || 6.915 .0017 |. O174 |. 98.26 || 37 || 28 36 || 24 . 5931 |. 4069 || 6.861 |. 5941 6.832 .0018 . OI76 |. 9824; 36 24 40 ||25 | .05960 |.94040 | 16.779 .05970 | 16.750 | 1.0018 |.00178 |.99822 || 35 | 20 44 26 |. 5989 |. 4011 || 6.698 |. 5999 || 6.668 .0018 . O179 |. 98.20 || 34 || 16 48 27 | . 6018 |. 3982 | 6.617 | . 6029 || 6.587 .0018 |. 0181 |. 9819 || 33 || 12 52 || 28 6047 |. 3953 || 6.538 . 605S 6,507 | .001S . 0183 | . 98.17 | 32 || 8 56 29 6076 . 3924 || 6.459 |. 6087 6.428 .0018 . ()185 | . 9815 || 31 || 4 14 || 30 | .06105 .93895 | 16.380 . .06116 16.350 | 1.0019 |.00186 .998.13 || 30 46 4 31 6134 |. 3866 || 6.303 | . 6145 || 6.272 | .0019 |. 0188 . 9812 || 29 || 56 8 || 32 | . 6163 |. 3837 6.226 |. 6175 6.195 | .0019 |. O190 | . 9810 || 28 || 52 12 || 33 | . 6192 |. 3808 || 6.150 |. 6204 || 6.119 | .0019 |. 0192 | . 98.08 || 27 || 48 16 || 34 | . 6221 | . 3777 6.075 |. 6233 || 6. .0019 |. 0194 | . 9806 26 || 44 20 35 | .0625() .93750 | 16.000 |.06262 | 15.969 | 1.0019 |.00195 |.99804 || 25 | 40 24 || 36 | . 6279 | . 3721 5.926 . 6291 5.894 .0020 | . ()197 | . 9803 || 24 || 36 28 37 | . 6308 |. 3692 || 5.853 . 6321 || 5.821 .0020 |. 0199 . 9801 || 23 32 32 || 38 | . 6337 |. 3663 || 5.780 . . 6350 5.748 .0020 |. 0201 . 9799 || 22 || 28 36 || 39 . 6366 |. 3634 5.708 . 6379 || 5.676 | .0020 j. 0203 |. 9797 || 21 24 40 | 40 | .06395 |.93605 || 15.637 |.06408 || 15.605 | 1.0020 | .00205 |.997.95 || 20 | 20 44 || 41 | . 6424 |. 3576 5.566 |. 6437 5.534 .0021 |. 0206 |. 9793 19 16 48 || 42 | . 6453 |. 3547 5.496 |. 6467 || 5.464 .0021 |. 0208 |. 9791 18 || 12 52 || 43 | . 6482 |. 3518 5.427 | . 6496 || 5.394 .0021 |. 0210 | . 9790 | 17 | 8 56 44 . 6511 |. 3489 || 5.358 . 6525 || 5.325 | .0021 |. 0212 . 9788 || 16 || 4 15 || 45 .00540 | .93460 | 15.290 . .06554 || 15.257 | 1.0021 .00214 | .997.86 15 || 45 4 || 46 | . 6569 |. 3431 || 4.222 | . 6583 || 5.189 .0022 |. 0216 | . 9784 || 14 || 56 8 || 47 | . 6598 | . 3402 || 5.155 . 6613 || 5.122 .0022 i. ()218 . 9782 || 13 || 52 12 || 48 . 6627 | . 3373 || 5.089 . 6642 5.056 .0022 |. 0220 . 9780 | 12 || 48 16 || 49 | . 6656 . 3343 || 5.023 |. 6671 || 4,990 | .0022 |. 0222 |. 9778 || 11 || 44 20 50 .06685 .93314 || 14.958 |.06700 || 14.924 | 1.0022 .00224 |.99776 10 | 40 24 || 51 | . 6714 | . 3285 || 4.893 |. 6730 4,860 .0023 |. 0226 . 9774 || 9 || 36 28 52 | . 6743 | . 3256 4.829 |. 6759 || 4,795 .0023 |. 0228 . 9772 || 8 || 32 32 || 53 6772 .. 3227 || 4.765 . . 6788 || 4.732 .0023 . 0230 |. 9770 || 7 || 28 36 54 . 6801 | . 3198 || 4.702 |. 6817 || 4,668 .0023 |. 0231 . 9768 || 6 || 24 40 55 | .06830 .93169 14.640 .06846 || 14.606 | 1.0023 |.00233 .99766 || 5 || 20 44 || 56 | . 6859 . 3140 || 4.578 |. 6876 || 4.544 | .0024 | . 0235 | . 9764 || 4 || 16 48 57 6888 . . 3111 || 4,517 | . 6905 || 4.482 .0024 . ()237 . 9762 || 3 || 12 52 || 58 . 6918 . 3082 4.456 |. 6934 || 4.421 .0024 |. 0239 |. 9760 || 2 || 8 56 59 . . 6947 | . 3053 4.395 | . 6963 || 4.361 | .0024 |. 0241 . 9758 || 1 || 4 16 60 . 6976 | . 3024 || 4.335 | . 6993 || 4.301 | .0024 |. 0243 |. 9756 || 0 || 44 M.S. M I Cosine. Vrs. Sin. Secante, | Cotang. Tangent. |Cosec’nt IVrs. Cos| Sine. M M.S. 6h 1939 Natural. 86° 5h NATURAL LINES. 49 Natural Trigonometrical Functions. 1750 M Sine. Wrs. Cos. Cosec'nte | Tang. Cotang. Secante.}Wrs. Sin| Cosine. M 0 | .06976 .93024 14.335 | .06993 || 14.301 | 1.0024 .00243 | .997.56 || 60 1 7005 | . 2995 || 4.276 |. T022 || 4.241 | .0025 |. 0246 |. 9754 || 59 2 . . 7034 . 2966 4.217 | . 7051 4.182 | .0025 | . 024S | . 97.52 58 3 | . 7063 . 2937 || 4.159 |. 7080 || 4,123 .0025 |. G250 | . 9750 || 57 4 |. 7092 |. 2908 || 4.101 |. 7110 || 4.065 | .0025 |. 0252 | . 9748 || 56 5 || 07121 |.92879 || 14,043 .07139 14.008 || 1.0025 | .00254 . .997.46 55 6 | . 7150 | . 2850 || 3.986 . 7168! 3.951 | .0026 |. 0256 |. 9744 || 54 7 | . TIT9 |. 2821 3.930 |. 7197 || 3.894 | .0026 |. 0258 . 9742 53 8 . . 7208 |. 2792 || 3.874 |. 7226 3.838 | .0026 i. 0260 . . 97.40 || 52 9 . 7237 . 2763 || 3.818 . 7256 3.782 .0026 |. 0262 |. 9738 51 10 || 07-66 || 92734 || 13,763 |.07.285 || 13.727 | 1.0026 .00264 |.997.36 || 50 11 | . T295 |. 2705 || 3.708 |. 7314 || 3.672 .0027 . . 0266 | . 9733 || 49 12 . 7324 |. 2676 || 3.654 |. 7343 || 3.617 | .0027 . . 0268 |. 97.31 || 48 13 | . T353 |. 2647 || 3.600 | . T373 || 3.563 .0027 . . 0271 | . 9720 || 47 14 | . T3S2 |. 2618 || 3.547 | . T.402 || 3.510 | .0027 . . 0273 , 97.27 46 I5 | .07411 .92589 || 13.494 | .07431 || 13.457 | 1.0027 .00275 | .997.25 || 45 16 || 440 || 2:30 3.4ſi || 460 3.404 || Jºš I ºf gº || || IT | . T469 . .2531 || 3,389 |. 7490 3.351 .0028 |. 0279 |. 97.21 43 18 | . T498 |. 2502 || 8.337 | . T519 || 3.299 || .0028 |. 0.281 |. 971S 42 19 . 7527 | . 2473 || 3.286 |. 7548 || 3.248 .0028 |. (.284 |. 97.16 || 41 20 | .07556 .92444 || 13.235 | .07577 13.197 | 1.0029 .00286 .99714 40 21 | . 7585 . 2415 || 3.184 . 7607 || 3.146 .0029 |. 0288 . . 97.12 || 39 22 | . 7614 | . 2386 || 3.134 |. 7636 || 3.096 | .0029 |, 0.290 | . 9710 || 38 23 . 7643 |. 2357 3.084 . 7665 3.046 .0029 . . 0292 | . 97.07 || 37 24 | . T672 |. 2328 3.034 |. 7694 || 2.996 | .0029 |. 0.295 | . 9705 || 36 25 | .07701 | .922.99 || 12.985 | .07724 12.947 | 1.0030 .00297 .997.03 ſ 35 26 . 7730 | . 227() || 2.937 . 7753 2.898 .0030 . ()299 | . 97.01 || 34 27 | . 7759 |. 2241 2.888 | . 7782 2.849 .0030 |. 0301 | . 9698 || 33 28 . 7788 || 2212 2.840 | . T812 || 2.801 | .0030 |. 0304 . 9696 || 32 29 | . TS17 | . 2183 || 2.793 . 7841 || 2.754 .0031 . ()306 . 9694 || 31 30 | .()7846 | .92154 12.745 .07870 | 12.706 | 1.0031 .00308 .996.92 || 30 31 | . 7875 . . 2125 2.698 ||. 7899 2,659 .0031 |. 0310 | . 9689 29 32 | . T904 . . 2096 || 2,652 | . T929 || 2.612 .0031 |. 0313 | . 9687 28 33 . 7933 . . 2067 2.606 | . 7958 2.566 | .0032 |. 0315 . 9685 27 34 | . W962 | . 2038 2.560 | . T987 2.520 | .0032 . 0317 | . 9682 || 26 35 | .07991 .92009 || 12.514 .0S 016 || 12.474 | 1.0032 |.00320 | .996S() 25 36 | . 8020 | . 19SO || 2.469 |. S046 || 2.429 .0032 | . ()322 . 967S | 24 37 | . 8049 | . 1951 || 2.424 . 807 5 || 2.384 .0032 . 0324 |. 96.75 23 38 | . 8078 . 1922 2.379 . 8104 || 2.339 .0033 |. 0327 . 96.73 ſ 22 39 || . S107 | . 1893 || 2.335 | . 8134 || 2.295 .0033 |. 0329 |. 9671 || 21 40 | .08136 .91S64 || 12.291 | .0S163 || 12.250 | 1,0033 |.00.,31 | .9966S 20 41 |. 8165 |. 1835 | 2.248 |. 8192 || 2.207 | .0033 |. 0334|. 9666 19 42 | . 8.194 | . 1806 || 2.204 . 8221 2.163 .0034 | . 0336 | . 9664 || 18 43 | . 8223 . 1777 || 2.161 | . 8251 2.120 | .0034 |. 0339 | . 9661 || 17 44 . 8252 . 1748 2.1.18 . 828() 2.077 .0034 |. 0341 . . 9659 16 45 .082S1 | .91719 12.076 .08309 || 12.035 | 1.0034 |.00343 .99656 || 15 46 . 831() |. 1690 || 2.034 |. 8339 || 1.992 | .0035 | . 0346 |. 9654 14 47 . S339 . 1661 | 1.992 | . 8368 1.950 .0035 | . ()348 . 9652 13 4S . 8368 . 1632 || 1.950 | . 8397 | 1.909 | .0035 | . 0351 | x 9649 || 12 49 : 8397 | . I 603 | 1.909 | . 8426 | 1.867 .0035 | . 0353 |. 9647 11 50 .08426 .91574 11 S68 .08456 11.826 | 1.0036 |.00356 .99644 || 10 51 | . 8455 | . 1545 | 1.828 . 8485 | 1.785 .0036 . 035S . 9642 9 52 | . S484 | . I516 || 1.787 | . 8514 | 1.745 .0036 |. 0360 | . 9639 || 8 53 | . 8513 . 1487 | 1.747 . 8544 1.704 .0036 | . 0363 |. 96.37 7 54 . 85.42 . 1458 || 1.707 |. 8573 || 1.664 .0037 |. 0365 | . 9634 || 6 55 | .08571 .91429 || 11.668 .08602 || 11.625 | 1.0037 .00368 .99632 || 5 56 . S600 | . 1400 | 1.628 . 8632 1,585 .003'ſ . 0370 | . 96.29 || 4 57 | . 86.29 |. I371 | 1.589 |. 8661 | 1.546 || 0037 || 0373 |. 9627 || 3 5S . 8658 . . 1342 1,550 | . 8690 | 1.507 | .0038 |. 0375 | . 96.24 || 2 59 | . 86S7 | . 1313 | 1.512 . .8719 | 1.468 .0038 |. 0378 |. 9622 || 1 60 | . 8715 . 1284 || 1.474 |. 8749 || 1.430 .0038 |. 038() |. 96.19 || 0 M Cosine. I Wrs. Sin.I Secante. Cotang. Tangent. Cosec'mt i Vrs. Cosl Sine. i M 949 Natural. 859 214 NATURAL LINES. Oh 5° Natural Trigonometrical Functions. 1749 |I|h M. S. M. Sine, [Wrs. Cos. Cosec'nte | Tang. Cotang, | Secante. Wrs. Sin Cosine, H M M.S 20 || 0 | .087.15 .91284 || 11.474 .08749 || 11.430 | 1.0038 |.003S0 .996.19 60 |40 4 1 S744 | . 1255 1.436 | . 8778 1,392 | .0038 . (.3S3 . 96.17 59 56 S 2 87.73 | . 1226 1.39S . 8807 || 1.354 .0039 || . 0386 | . 9614 || 58 52 12 || 3 8802 . . 1197 | 1.360 | . S837 | 1.316 .0039 || . (388 . 96.12 || 57 || 48 16 || 4 | . 8831 |. 1168 || 1.323 . 8866 | 1.279 .0039 |. 0391 i. 9609 || 56 || 44 20 || 5 | .OS860 j .91139 || 11.286 .08895 11.242 | 1.0039 .00393 | .99607 || 55 | 40 24 || 6 | . 8889 . 1110 | 1.249 |. 8925 | 1.205 .0040 l. 0396 |. 9604 || 54 || 36 28 || 7 | . 8918 . . 1082 | 1.213 |. 8954 | 1.168 .004) . 0398 }. 9601 || 53 || 32 32 8 . . 8947 . 1053 | 1.176 8983 || 1.132 | .0040 |. 0401 | . 9599 || 52 2S 36 9 S976 | . I 024 | 1.140 9013 | 1.095 | .0040 |. 0404 | . 9596 || 51 24 40 || 10 || 09005 | .90995 || 11.10 k . .09042 11.059 1.0041 .00406 .99594 || 50 20 44 i Il 90.34 . 0966 | 1.069 |. 9071 | 1.024 | .0041 |. 0409 | . 9591 || 49 || 16 48 || 12 | . 9063 | . O937 || 1 033 . 9101 || 0.988 | .0041 . 0411 | . 958S 48 || 12 52 || 13 | . 9092 | . 0908 || 0.998 9130 || 0.953 . .0041 | . 0414 . 9586 || 47 8 56 || 14 | . 9121 . 0879 || 0.963 | . 9159 0.918 . .0042 |. 0417 | . 95S3 46 || 4 21 | 15 | .09150 | .90850 || 10.929 .09189 || 10.SS3 | 1.0042 .00419 .99580 || 45 39 4 16 91.79 . OS21 || 0,894 92.18 ().S48 .0042 ſ. 0422 | . 957S 44 || 56 S 17 92.0S . ()792 0.860 9247 0.814 | .0043 |. 04:25 95.75 || 43 || 52 12 18 , 9.237 . ()763 0.826 . 927.7 || 0,780 . .0043 . 0427 95.72 || 42 48 16 || 19 | . 9266 , 0.734 0.792 9306 || 0.746 .0043 . ()430 | . 95.70 || 41 || 44 20 20 | .00295 || 907(5 || 10.75S .09335 | 10.712 | 1.0043 | .00433 .99567 || 40 40 24 21 | . 9324 | . (676 || 0.725 9365 0.678 .004.4 . 0436 . 9564 || 39 || 36 28 || 22 . 9353 | . ()647 0.692 | . 9394 || 0.645 | .004.4 . 0438 |. 9562 | 38 32 32 || 23 | . 93.82 | . 0618 || 0.659 9423 0.612 .0044 |. 04:41 | . 9559 || 37 28 36 || 24 94ll j . 05S9 || 0 626 . 9453 || 0.579 .004.4 |. 0444 . 9556 36 24 40 25 | .09.440 | .9(1560 10.593 | .09482 10.546 | 1.0045 .00446 .99553 35 | 20 44 || 26 9469 . 053I 0.561 9511 || 0.514 | .0045 |. (34.49 | . 9551 || 34 || 16 48 || 27 9498 | . U502 || 0 529 9541 || 0.4S1 | .004.5 | . 0452 | . 9548 || 33 12 52 28 9527 | . (K173 || 0.497 957() 0.449 .0046 . 0455 . 95.45 || 32 || 8 56 || 29 9556 | . (A44 ().465 9599 || 0.417 | .0046 , ()458 95.42 31 || 4 22 || 30 .(;9584 .90415 || 10.433 .09629 || 10.385 | 1.0046 |.00460 .9954() 30 || 38 4. 31 96.13 | . (.386 (1.402 96.58 || 0.354 .0046 |. (2463 . 9537 || 29 || 56 8 || 32 964.2 . . (357 0.371 96.88 || 0.322 .0047 i. ()466 | . 9534 28 52 I2 || 33 9671 | . 0328 || 0.340 97.17 | ().291 .0047 . 0469 | . 9531 || 27 || 48 16 || 3 || 97.00 | . 0300 || 0.309 | . 9746 || 0.260 . .0047 j. 0472 . 9528 26 || 44 20 | 35 | .09729 .90271 || 10.278 .09776 10.229 | 1,0048 .00474 .99525 25 40 24 || 36 97.58 (242 || 0.248 9505 || 0.199 || 00:48 . 0477 9523 24 36 28 37 97S7 | . 0213 || 0.217 98.34 0.168 .004S . 04SO | . 9520 || 23 32 32 38 98.16 | . UIS4 0.187 98.64 0.138 .004S . 0483 . 9517 22 || 28 : 39 9845 01:55 0.157 | . 98.93 || 0.108 .0049 . 0486 | . 9514 || 21 || 2-1 4() 40 | .09874 | .90126 || 10.127 | .09922 || 10.078 | 1.0049 .00489 | .99511 || 20 | 20 44 || 4 | 990.3 0097 || 0.098 | . 9952 || 0.048 | Ul'49 |. ()491 | . 9508 || 19 16 48 || 42 9932 0068 0.06S . 9981 || 0.019 .0050 | . ()494 . 9505 || 18 || 12 52 43 996.1 0.039 || 0.039 || 10011 || 9.9893 | .0050 | . (A497 | . 9503 || 17 8 56 || 44 99.90 | . 0010 || 0.010 | .1004() .9601 | Ul 50 |. 0500 |. 9500 | 16 || 4 23 45 | . 10019 .899S1 || 9.9812 .10009 . .9310 | 1.0050 .00503 | .99497 15 37 4 || 46 0048 . 9952 . .9525 0099 || .9021 | .0051 ('506 | . 94.94 || 14 || 56 8 47 ()()77 | . 9923 | .9239 0.128 ,8734 . .0051 ()509 . 94.91 || || 3 || 52 I2 || 48 01(J6 | . 98.94 i .8955 0.158 .84.48 .0051 0512 | . 94.88 || 12 || 48 16 || 49 O134 . 98.65 . .8672 0187 | .8164 0.052 (1515 . . 9485 11 || 44 20 i 50 .10163 ,89836 || 9.8391 || 10216 || 9.7882 | 1,005.2 .00518 .99482 10 | 40 24 51 0.192 | . 9807 | .8112 | . U246 .7601 | .0052 . 0521 | . 94.79 || 9 || 36 28 || 52 02:21 97.79 .7S34 |. O275 .7322 | .0053 0524 . 94.76 || 8 || 32 32 53 02:50 9750 | .7558 . U305 | .7044 | .0053 ()527 94.73 || 7 || 28 36 || 54 ()279 97.21 .7283 . ()334 .6768 . .0053 (530 | . 94.70 || 6 || 24 40 || 55 || 10308 .89692 || 9.7010 .10363 9.6493 I.0053 | .00533 .99467 || 5 || 20 44 56 0337 9663 i .6739 . U39.3 : 6220 | .005 ! 0536 . 9464 || 4 || 16 48 57 | . 0366 | . 96.34 .6409 | . 0422 || 5949 : .0054 (1539 . 9461 || 3 || 12 52 || 58 . . 0.395 9605 | .6200 | . 0452 5,679 .0054 |. 0542 |. 9458 || 2 8 56 || 59 . . 0424 | . 9576 .5933 . . 04S1 5,411 .0055 |. 0545 . 9455 | 1 || 4 24 60 | . 0453 | . 9547 5668 . 0510 5144 | .0035 | . (548 | . 9452 || 0 || 36 M. S. M Cosine. Wrs. Sin. Secante. Cotang.; Tangent. Cosee'nt (Wrs. Cosi Sine. 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Tuoi.11auourośIII, Turugu N o9| q0 'SSINIT TY&Iſld. WN NATURAL LINES. 7o Natural Trigonometrical Functions. 1729 M | Sine. Wrs. Cos. Cosec'nte | Tang. Cotang. Secante. iVrs. Sin Cosine. M 0 .12.187 .87813 || 8.2055 .12278 8.1443 1.0075 .00745 .99255 60 1 | . 2.216 | . 7787 .1861 . . 2308 .1248 .0075 |. 0749 . 9251 59 2 i - 224.5 7755 .166S . 2337 .1053 | .0076 ()752 . 924.7 58 3 || - 2273 '77.26 .1476 . 2367 .086() .0076 . . 0756 . 9.244 || 57 4 | . 2302 7697 .1285 | . 2396 || 0667 | .0076 0760 | . 9240 || 56 5 | .12331 .87669 || 8.1094 .12426 8.0476 | 1.0077 . .00763 .992.37 || 55 6 | . 2360 | . 764) .0905 2456 .0285 .0077 0767 . 9233 || 54 7 | . 238.) | . 7611 .0717 2485 .0095 .007S 0770 . 9229 || 53 8 . . 24.18 7582 .0529 2515 # 7.9906 | .0078 0774 | . 9226 52 9 2447 7553 .0342 2544 || 7.97.17 .0078 . 0.778 | . 9222 || 51 10 | .12476 | .87524 8.0156 .12574 || 7.95:30 | 1.007.9 .007 S1 | .992.19 || 50 11 2504 7495 || 7.9971 2603 || 9344 .0079 0785 92.15 49 12 2533 74.67 .97S7 2633 . .915S | 0079 0788 9211 || 48 13 2562 743S .9604. 2662 : .8973 .008() 0792 | . 9208 I 47 14 | . 2591 | . T409 .9421 | . 2692} .S789 . .008() O796 9204 || 46 15 .12620 | .87380 || 7.9240 | .12722 || 7.S606 | 1.0080 || 00799 || .99200 || 45 16 2649 |. 7351 .905.) | . 275 l l .8424 | .U081 0803 91.97 || 44 17 | . 2678 . 7322 .8879 |. 2781 .S243 | .00S] (S07 | . 9193 || 43 18 2706 | . T 293 .8700 | . 2810 i .8062 .0082 08.10 | . 9189 || 42 19 2735 | . T265 .8522 | . 2840 .7SS2 | .0082 0814 | . 918.6 || 41 20 | .12764 .87236 || 7.S$44 |.12869 || 7.7703 | 1.00S2 |.00SLS .991S2 | 40 21 2793 7207 .816S . 2899 .7525 | .00S3 . 0822 91.78 || 39 22 2822 717S .7992 | . 292S .734S .00S3 08:25 91.74 || 3 23 285.1 71.49 .7817 | . 295S | .7171 | .0084 ()829 917.1 ſ 37 24 . 2S79 7120 .7642 | . 29SS .6996 .0084 OS33 9167 || 36 25 | .12908 .87091 || 7.7.469 |..] 3017 | 7.6821 | }.0084 || 00837 .9916.3 || 35 20 2937 7 06:3 .7296 | . 3047 | .6646 | .0085 084() 9160 || 34 27 2966 7034 .7 124 |. 3076 . .6473 | .0085 0S44 . 9156 || 33 28 . 2995 7005 6953 | . 3105 6300 OU85 0848 9152 || 32 29 3024 6976 .6783 |. 3136 .6129 .0056 ()852 91.48 || 31 30 J3053 |.86947 || 7.6613 | .13165 7,5957 | 1.00S6 |.0, S55 | .99144 || 30 31 || - 30S 1 G918 .6444 . 3195 5787 | .0087 |. Q859 9141 29 32 31.10 6S90 .6276 | . 3224. 5617 | 0087 0863 91.37 || 28 33 3139) GS61 .6108 . 3254 5449 .0087 0867 9133 || 27 34 3168 6832 .5942 |. 3284 .52S) | .00SS . 0.871 9129 || 26 35 | .13.197 .86803 || 7.5776 .13313 || 7.51.13 | 1.0088 || 00875 | .991.25 || 25 36 3226 b774 .5611 |. 3343 j .4946 .0089 |. 0878 9121 || 24 37 3254 67+5 .5446 |. 3372 .47SO | .0089 0882 9118 || 23 3S 3283 67.17 .52S2 34()2 | .4615 .0089 0886 | . 91.14 22 39 3312 (56.88 .5119 34.32 .4451 | .0090 O890 9110 || 21 40 | .1334l .86659 7.4957 . .13461 7.4287 | 1.0090 ,00894 | .90 106 || 20 41 337() | . 6630 .4795 349 l ; .4124 .0090 0898 9102 || 19 42 3399 || . 6601 .4634 3.320 .396 | | .0091 ()002 | . 9098 || 18 43 3427 | . 6572 .4474 355() .3S()() | .0091 ().905 9094 || 1 || 44 3456 . 6544 .4.315 3580 | .3639 .()092 |. ().909 9()90 | 16 45 .134S5 | .865.15 7.4156 | .13600 || 7.3479 | 1.0092 .00913 | .99086 || 15 46 3514 | . 6486 .3998 3639 .3319 | .0092 |. 0.917 | . 9083 || 14 4'ſ 3543 , 6457 .3S40 3669 .3160 | .0093 0921 9079 || 13 48 3571 . 6428 .3683 3698 .3002 | .0093 ||. 0925 90.75 || 12 49 3600 6400 .3527 3728 .2844 | .0094 | . ().929 907 () 11 50 | .13629 .86371 || 7.3372 .13757 || 7.2687 | 1.0094 .00933 | .99067 10 51 . 3658 6342 .3217 3787 .2531 .0094 U937 9063 || 9 52 36.87 (33.13 .3063 3S17 .2375 | .0095 0941 9059 || 8 53 3716 . 6284 .2909 . 3846 . .2220 | .0095 0945 9055 || 7 54 3744 625 j .2757 3S76 .2066 | .0096 0949 9051 || 6 55 .13773 || 86.227 || 7.2604 || 13006 7.1912 | 1.0096 | .00953 . .99047 5 56 3S()2 6198 .2453 3935 | .1759 .0097 ()957 9043 || 4 57 38.31 6169 .2302 3965 .1607 | .0097 0961 9039 || 3 5S 386() 61-10 .2152 399.5 .1455 | .()097 0.965 90.35 | 2 59 3888 6111 2002 4024 .1304 .0098 O969 90.31 || 1 60 3917 | . 6083 .1853 |. 4054 .1154 .0098 | . 0973 9027 | () M. l Cosine. Wrs. Sin." ScCaute. Cotaug. Tangent. | Cosec'nt IVrs. Cos] Sime. M 97o Natural. 829 u? ol.8 *I*.11nºt; N. oS6 *S*IWI JW A -ºs. 1111,00soo; “anożut,LI'5ub)00 "aludoos I’ugs's 1A “outsoo | IV +52, 0 || 6918 IQ7. 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ST60 58 12 || 3 | . 5730 , 4270 | .3574 |. 5928 .2783 | .0126 l. 1245 . 8755 57 16 || 4 | . 5758 |. 4242 .3458 |. 5958 | .2665 | .0125 |. 1249 i. 8750 56 20 || 5 | .15787 |.84213 || 6,3343 |.15987 | 6.2548 | 1.0127 | .01254 . .987.46 55 24 || 6 | . 5816 | . 4.184 || 3228 |. 6017 | .2432 | .0127 l. 1259 |. S741 54 28 || 7 | . 5844 |. 4155 .3113 |. 6047 | .2316 | .0128 l. 1263 |. 8737 53 32 || 8 | . 5873 . . 4127 .2999 |. 6077 | .2200 | .0128 |. 1268 |. 8732 || 52 36 || 9 | . 5902 . . 4098 | .2885 . 6107 | .2085 || 0129 |. 1272 . 8727 || 51 40 || 10 .15931 |.84069 6.2772 .16137 || 6.1970 | 1.0129 i.01277 |.9S723 50 44 || 11 | . 5959 |. 4041 .2659 |. 6167 .1856 | .0130 l. 1282 |. 8718 || 49 48 || 12 | . 5988 |. 4012 || 2546 |. 6196 | .1742 | .0130 l. 1286 | . 8714 || 48 52 | 13 | . 6017 | . 3983 | .2434 |. 6226 .1628 0131 |. 1291 | . 8709 || 47 56 || 14 | . 6045 |. 3954 .2322 |. 6256 | .1515 || 0131 |. 1296 |. S704 || 46 37 | 15 . .16074 |.83926 || 6.2211 | 16286 || 6.1402 | 1.0132 .01300 | .98700 45 4 | 16 | . 6103 |. 3897 .2100 |. 6316 .1290 | 0132 i. 1305 | . 8695 || 44 8 II . 6132 . 386S | 1990 . 6346 .1178 .0) 33 . . 1310 | . 8690 43 12 18 . 6160 l. 3840 | .1880 i. 6376 .1066 || 0133 |. 1314|. 8685 || 42 16 || 19 | . 6189 |. 3811 .1770 . 6405 | .0955 | 0134 |. 1319 |. SGS1 || 41 20 | 20 ! .16218 |.837.82 | 6.1661 | .16435 | 6,0844 | 1.0.134 |.01324 .98676 | 40 24 21 . 6246 |. 3753 .1552 . 6465 .0734 . .0135 | . 1328 . 8671 39 28 22 . 6275 |. 3725 | .1443 |. 6495 | .0624 || 0.135 |. 1333 . 8667 || 38 32 || 23 | . 6304 |. 3696 | .1335 | . 6525 | .0514 | .0136 i. 1338 |. S662 || 37 36 24 | . 6333 3667 .1227 | . 6555 . .0405 | .0136 |. I343 |. 8657 36 40 || 25 .16361 | .83639 6.1120 | .16585 || 6.0296 | 1.0.136 |.013.47 | .98652 35 44 || 26 . 6390 | . 3610 ! .1013 | . 6615 ! .0188 .0137 l. 1352 i. 864S 34 48 27 | . 6419 |. 3581 | .0906 | . 6644 ,00SU | .0137 l. 1357 | . 8643 || 33 52 28 . 6447 | . 3553 | .0800 | . 6674 5.9972 | .0138 . 1362 | . 86.38 || 32 56 29 . 6476 |. 3524 .0694 | . 6704 || 5.9865 ,0138 . . I367 . 8633 || 31 38 30 .16505 | .83495 6.0588 .16734 5.9758 | 1,0139 , .01371 .9Su28 30 4 || 31 | . 65.33 . 34.66 | .0483 | . 6764 .9651 .0139 . I-376 . 8624 || 29 8 || 32 | . 6562 | . 3438 . .0379 . 6794 | .9545 .0140 i. 1381 |. S619 28 12 || 33 . 6591 |. 3409 | .0274 . 6824 .9439 || 0140 i. 1386 . $614 27 16 34 . 6619 |. 3380 . .0170 |. 6S54 || 9333 || 0141 |. 1391 | . 8609 26 20 || 35 | .16648 .83352 6.0066 .16884 5.9.228 | 1.014L .01395 | .98604 || 25 24 || 36 | . 6677 , 3323 5.9963 | . 6914 | .9123 .0143 i. 1400 | . 8600 24 28 || 37 | . 67(35 | . 3294 | .9860 | . 6944 . .9819 .0142 i. 1405 | . 8595 23 32 38 | . 6734 . . 3266 | .9758 |. 6973 || S915 . .0143 . 1410 |. 8590 22 36 || 39 | . 6763 | . 3237 | .9655 . 7003 | .88.11 . .0143 |. 1411 | . $585 21 40 | 40 | .16791 83208 5.9554 .17033 || 5,870S | 1.01.44 |.014:20 .98580 20 44 || 41 | . 6820 |. 3180 | .9452 ) . 7063 | .8605 | .0144 |. 1425 | . 8575 | 19 48 || 42 | . 6849 |. 3151 | .9351 | . TO93 .85()2 | .0145 |. 1430 | . 8570 18 52 || 43 | . 6878 . 3122 | .9250 . 7123 8400 | .0145 |. 1434 | . 8565 17 56 || 44 . 6906 | . 3094 | .9150 | . 7153 .8298 || 0146 |. I439 | . 8560 16 39 i 45 | .16935 | .83065 || 5.9049 |.17183 5.8.196 | 1.0146 |.01444 | .98556 15 4 || 46 . 6964 |. 3036 .8950 | . T 213 .8095 || 01:47 l. 1449 | . 8551 i 14 8 || 47 | . 6992 | . 3008 || SS50 | . T.243 | .7994 || 0147 |. 1454 | . 8546 || 13 12 || 48 | . TU21 | . 2979 | .8751 | . '7273 | .7894 .0148 . I459 i. 8541 || 12 16 || 49 . '7050 | . 2950 | .8652 | . T303 .7793 .0148 i. 1464 . 8536 ll 20 50 .17078 .82922 || 5.S554 .17333 5,7694 | 1.0149 : .01469 .98531 10 24 51 . 7107 | . 2893 .8456 | . T363 | .7594 | .0150 . . 1474 . 8526 9 28 52 | . 7136 j . 2S64 .8358 | . Tà93 | .7495 | .0150 |. 1479 . 8521 || 8 32 53 | . TI64 . 2836 | .8261 |. 7423 .7396 | .0151 | . 1484 | . 85.16 7 36 54 | . TI33 . 2807 || S163 | . T 453 .7297 | .0151 |. 1489 . 8511 || 6 40 || 55 | .17221 .8277S 5.8067 .17483 || 5,7199 || 1,0152 | .01494 | .98506 || 5 44 || 56 | . TX50 | . 2750 .7970 . 7513 | .7101 | .0152 . 1499 | . 8501 || 4 4S 57 '7279 |. 2721 || 7874 . . 7543 | .7004 .0153 - 1504 | . 8496 || 3 52 58 . '7307 | . 2692 .7778 . 7573 . .6906 .0153 . I509 . 849 I 2 56 59 . . 7336 | . 266 | | .7683 | . 7603 | .6809 || 0154 |. 1514 | . 8486 || 1 40 || 60 . 7365 | . 2635 | .7588 . 7633 . .6713 .G. 54 . 1519 |. 8481 || 0 M. S. M Cosine. Wrs. Sin. Secante. Cotang. Tangent. Cosed mt i Vrs. Cos! Sine. M 6h 999 Natural. 80° NATURAL LINES. 2T9 '100 Natural Trigonometrical Functions. 1699 |I|h M Sime. Wrs. Cos. Cosec'nte | Tang. | Cotang. Secante, Vrs. Siu | Cosine. M. M. S 0 .17365 .82635 | 5,7588 .17633 || 5.6713 | 1.0154 || 01519 | .9848] | 60 20 1 | . T393 | . 2600 .7493 . 7663 .6616 | .0155 . 1524 |. S476 59 || 56 2 | . T 422 : . 2578 .7398 | . T693 .6520 .0155 |. 1529 . 847 1 || 58 || 52 3 | . TA5I | . 2549 .7304 |. 7723 .6425 | .0156 . 1534 . S465 || 57 || 48 4 | . T479 . 2521 ,7210 | . 7753 .6329 | .0156 . 1539 |. 8460 56 || 44 5 | .17508 .82492 || 5.7117 .17783 || 5.6234 | 1.0157 .01544 .98455 55 | 40 6 : . T5:37 2463 .7023 |. 7S13 . .614() || 0157 | . 1550 S450 54 36 7 | . (56.5 2435 .6930 . 7843 | .6045 .0158 . . 1555 84.45 || 53 || 32 8 . . .7594 2406 .683S . 7873 | .595l .0158 |. 156() 84.40 52 || 28 9 | . T822 . 2377 .6745 | . T903 .5857 .0159 |. 1565 8435 || 51 24 10 .17651 .82349 || 5.6653 | .17933 5.5764 1.0159 || 01570 .9S430 50 20 11 | . T6S0 | . 2320 .6561 . 7963 .5670 29160 . 1575 | . 8425 || 49 || 16 I2 | . TTO8 2291 .6470 . 7993 || 5578 || YO160 |, 1580 . 84.19 || 48 12 13 | . T737 . 2263 .6379 |. 8023 .5485 | .0161 . 1585 84.14 || 47 8 14 | . TT66 22:34 .6288 . . 8053 .5393 | .0162 |. 1591 . 8409 || 46 4 15 .17794 | S2206 || 5.6.197 .18083 5.5301 1.(I62 .01596 .98404 || 45 19 16 | . T$23 2177 .6107 | . 8113 | .5209 || 0163 |. 1601 | . 8399 || 44 || 56 17 | . TS52 2I4S .6017 | . 8143 | .5117 | .0163 |. 1606 | . 8394 || 43 || 52 IS . 7880 212() .592S 8173 .5026 .0164 . 1611 |. 8388 || 42 || 48 19 | . 7909 2091 .5838 8203 | .4936 .0164 |. 1617 | . 83S3 || 41 || 44 20 | .17937 .82002 || 5,5749 |.18233 || 5.4845 | 1.016.5 |.01622 | .98378 || 40 4) 21 7966 2034 .5660 8263 .4755 | .016.5 |. 1627 S373 39 || 36 22 7995 2005 ,5572 8293 || 4665 .0166 |. 1632 |. 8368 || 38 || 32 23 8023 1977 .54S4 8323 | .4575 .0166 |. 1638 S362 37 28 24 | . 8052 1948 .5396 8353 .4486 .0167 |. 1643 S357 || 36 24 25 .18080 .81919 5.5308 .18383 5.4396 | 1.0167 |.0164S .98352 # 35 20 20 S109 1891 .5221 S413 || 4308 0168 |. 1653 || - 8347 || 34 || 16 27 SI38 1862 .5134 84.44 || 42.19 (J169 |. 1659 8341 || 33 12 28 S166 1834 .5047 8474 || 41.31 0.169 |. 1664 S336 || 32 8 29 S195 IN05 .4960 S50 [ . .4043 | .017() }. 1669 |. 833l 31 4 30 | .18223 |.S1776 5.4874 | .18534 || 5.3955 | 1.017t) || 01b.74 .98325 || 30 | 18 31 S252 . 1748 .478S 8564 | .3868 .0171 |. 16S0 | . 832) 29 || 56 32 | . 82S1 I719 .4702 S594 | .3780 . .0171 |. 1685 S315 28 52 33 8309 | . I69 [ .4617 8624 .3694 | .0172 |. 1690 8309 || 27 || 48 34 | . 8338 1662 .4532 8654 || 3607 .0172 |. 1696 | . 8304 || 26 || 44 35 | .18366 | .81633 || 5.4447 .18684 || 5,3521 | 1.0173 || 01701 || 98.299 || 25 | 40 36 . 8395 1605 .4362 S714 | .3434 0.174 |. 1706 S293 l 24 || 36 37 84.24 1576 .4278 S745 | .3349 | .0174 1712 . S288 || 23 || 32 38 | . $452 1548 .4.194 87.75 .3263 .0175 1717 | . 82S3 22 || 28 39 . 8481 1519 .4110 8805 .3178 .0175 1722 | . 8277 || 21 24 40 .18509 .81490 || 5.4026 .18835 5.3093 | 1.0176 .01728 .98272 20 20 41 85.38 | . 1462 .3943 8865 : .300S | .0176 17.33 S 267 || 19 16 42 | . 8567 ? - 1433 .3S60 | . 889.5 .2923 || 0177 1739 S261 18 2 43 S595 1405 .3777 8925 .2839 .0177 1744 | . 8256 17 8 44 S624 1376 .3695 8955 .2755 ,0178 1749 . 825ſ) || 16 4 45 .18652 .8134S 5.3612 | .18985 || 5.2671 | 1.0179 |.01755 .98245 || 15 17 46 | . S6Sl 1319 .3530 9016 || .258S .0179 |. 1760 . 82.40 || 14 || 56 47 87 00 1290 .3449 9046 .2505 .01S() 1766 . 8234 || 13 || 52 48 | . S73S | . 1262 .3367 9076 .2422 || 0180 1771 . 8229 || 12 || 48 49 S767 1233 .32S6 9106 || 2339 .01Sl 1777 . 8,223 || 1 || || 44 50 | .18795 |.81205 || 5.32U5 .19136 || 5.2257 | 1.0181 .01782 .982.18 10 | 40 51 . 8824 | . 1176 .3124 9166 | .2174 .01S2 |. 1788 | . 8212 9 || 36 52 $852 1147 .3044 9.197 .2092 || 0182 1793 . S207 S 32 53 . 8SSI . 1 ll.9 .2963 9227 2011 | .0183 |. 1799 || - 820.1 7 2S 54 | . 8909 109() .2S83 . 9257 | .1929 | .018 k . . 1804 |. Sl'96 6 24 55 I8938 . .81062 5.2803 .19287 || 5, 1848 | 1.0.184 |.01810 | .98190 || 5 || 20 56 | . 8967 103. .2724 9317 1767 .0185 |. 1815 SlS5 4 16 57 | . 8995 1,005 .2645 | . 9347 1686 .0185 |. IS21 | . 8179 || 3 || 12 5S l . 9024 0976 .2566 937S .1606 | .0186 . 1826 S174 2 8 59 9052 0948 .2487 940S .1525 | .01S6 |. 1832 | . $168 l 4. 60 | . 9081 . 0.919 .240S | . 9438 .1445 | .0187 |. 1837 i. 8163 || 0 | 16 M Cosine. Wrs. Sin. Secante, Cotang. Tangent. Cosec'nt IVrs. Cosi Sine. M M.S. 100° Natural. 79° 5h NATURAL LINES. 1689 11o Natural Trigomometrical Fumctions. M Sine. Wrs. Cos. Cosec'nte | Tang. | Cotang. Secante. {Wrs. Sinj Cosine. M 0 | .19081 .80919 5.2408 .19438 5.1445 | 1.0187 i.01837 .98163 60 I 9109 | . ()890 | .2330 | . 9468 .1366 | .0188 . 1843 . Slă7 59 2 913S . OS62 .2252 | . 9498 12S6 | .0188 . . 1848 . . 8152 || 58 3 9166 | . 0.833 .217.4 | . 9529 1207 || 0189 |. 1854 | . 8146 57 4 9.195 | . 0805 | .2097 9559 .1128 .018.9 1859 .. 8140 56 5 | .19224 |.80776 || 5.2019 | .19589 || 5.1049 | 1,0190 .01865 | .98135 | 55 6 9.252 . 07:48 .1942 96.19 .09.70 | .019.1 IS71 | . 8129 54 7 9.281 . ()719 .1865 9649 .0892 || 0191 | . 1876 . . $12.4 53 8 . . 9300 | . ()691 | .1788 |. 9680 . .0814 | .0192 1SS2 | . 81.18 || 52 9 9338 . 0662 .1712 97.10 || 0736 || 0192 18S7 .. 8112 || 51 10 .19366 | .80634 || 5.1636 .19740 5.0658 | 1.0193 i.01893 | .98107 || 50 11 | . 9395 . 0605 | .1560 97.70 | .0581 .0193 ) . 1899 || . 8101 || 49 12 | . 9423 . 0576 . .1484 |. 9800 ! .0504 | .0194 i. 1904 |. 8095 || 48 13 9452 | . 0548 || .1409 98.31 .0427 | .0195 | . 1910 . S090 47 14 94S0 | . 0519 | .1333 986] 1 0350 | .0195 | . I916 . . 8084 || 46 15 | .19509 | .80491 || 5.1258 | .19891 5.0273 | 1.0196 | .01921 .98078 || 45 16 953.7 | . 0462 | .11S3 99.21 | .019.7 | .0196 1927 | . 8()73 || 44 17 9566 | . 04:34 .1100 | . 9952 . .0121 | .0197 | . 1933 | . 8067 || 43 18 . 9595 | . 0405 || 1034 99.82 .0015 U198 |. 1938 . . 8061 || 42 19 96.23 0377 .0960 .20012 || 4.9969 .0198 1944 8056 || 41 20 ! .19652 | .80348 || 5.0886 .20042 i 4.98.94 | 1.0199 |.01950 .98050 | 40 21 96.80 | . 0320 ! .0812 U073 .9819 .0199 || . 1956 . 8044 || 39 22 9709 | . 0291 .0739 0.103 | .9744 | .0200 |. 1961 | . 8039 || 38 23 9737 | . 0263 .0666 0.133 | .9669 | .020.1 1967 | . 8033 || 37 24 9766 . 0234 .0593 0.163 .9594 .0201 | . 1973 | . 8027 || 36 25 | .19794 | .80206 || 5.0520 .20194 || 4.9520 | 1.0202 |.01979 .98021 35 26 | . 9823 , 0.177 | .(?447 0.224 .9446 | .0202 | . 1984 | . 8016 || 34 27 | . 9851 | . O149 | .0375 . 025 .9372 | .0203 1990 | . S010 || 33 28 . 9S80 | . 0120 | .0302 0285 .9298 .0204 . 1996 | . 8004 || 32 29 . 9908 . 0092 .0230 0315 .9225 .0204 |. 2002 7998 || 31 30 || 19937 .S0063 || 5.0158 .20345 || 4.9151 | 1.0205 | .02007 .97992 || 30 31 . 9965 | . 0035 .0087 | . ()375 | .907 S .0205 2013 79ST 29 32 | . 9994 | . 0006 .0015 . 0406 | .9006 . .0206 2019 | . '7981 || 28 33 | .20022 |.799.78 || 4.9944 ()436 .8933 .0207 . 2625 | . T975 || 27 34 | .2(1051 | .79949 || 4.98.73 0466 .8860 | .0207 2031 7969 26 35 | .20079 .7992] 4.9802 . .20497 4.S788 | 1.0208 || .02037 .97963 25 36 . 0108 . 9S92 .97.32 Q527 | .8716 || 0208 || . .2042 .. 7957 || 24 37 | . U136 . 98.63 9661 0557 | .8644 | .0209 | . 2048 . 7952 23 38 | . 0165 . 983 95.91 0588 .8573 || 0210 ! .. 2054 | . T946 22 39 . 0193 ) . 9807 .9521 0618 || Sã01 || 0210 |. 2060 | . T940 || 21 40 .20222 | .79778 || 4.9452 .20648 || 4.8430 | 1.02.11 | .02066 .97934 20 41 - U250 | . 9750 .938.2 0679 .833.9 .02.11 . . 2072 7928 || 19 42 | . 0279 | . 97.21 | .9313 07.09 | .82S8 || 0212 | . 2078 , 7.922 18 43 | . (307 | . 9693 .924.3 ()739 .8217 | .02.13 . 2084 | . T916 17 44 | . 0336 | . 9664 .9175 0770 .8147 || 0213 |. 2089 |. 7910 | 16 45 .20364 || 79636 || 4.9106 | .20800 || 4.8077 | 1,0214 .02095 .979C4 15 46 . . 0.393 | . 9607 .9037 0830 | .8007 || 0215 i. 2101 7899 || 14 47 | . 0421 | . 9579 .8969 U861 .7937 || 0215 i. 2107 j. 78.93 || 13 48 . G450 | . 9550 .S901 0891 .7867 .0216 |. 21.13 '7887 12 49 | . 0478 . 9522 .8833 O921 | .7798 || 0216 | . 21.19 . 7881 || 11 50 .20506 .79493 || 4.8765 | .20952 4.7728 1.0217 | .02.125 .978.75 10 51 | . 0535 | . 9465 | .8697 U982 .7659 .0218 . 2131 | . T869 || 9 52 | . 0563 | . 94.36 S630 1012 || ,7591 ,0218 |. 2137 , 7S63 || 8 53 . 0592 | . 94US 8563 1043 .7522 || 0219 |. 2143 , 7857 7 54 | . 0620 | . 9379 | .8496 1073 .7453 .0220 |. 2149 j . 7851 || 6 55 .20649 .79351 4.8429 .21104 4.7385 1.02.20 || 02155 .97.845 5 56 | . (1677 | . 9323 .8362 |. 1134 | .7317 .U221 i. 2161 . 7839 || 4 57 . 0706 | . 9294 .8296 1164 .7249 | .02.21 |. 2167 | . T833 || 3 58 . 0734 . . 9266 | .8229 1195 || 7181 | .0222 |. 2173 |. 7827 | 2 59 Ü7b3 . 9237 | .8163 1225 || 7 114 .0223 1. 2179 . 7821 || 1 60 0791 . 9209 | .8097 | . I256 || 7046 .0223 i. 2185 , 7815 || 0 M | Cosine. Vrs. Sin. Secante. Cotang. Tangent. Cosec'nt l Vrs. Cos! Sime. M 1019 Natural. 780 NATURAL LINES. 129 Natural Trigonometrical Functions. 1670 M | Sine. Wrs. Cos. Cosec'nte Tang. Cotang. Secante. Wrs. Sin Cesire. M 0 | .20791 || 79209 || 4.8097 .21256 || 4.7046 | 1.0223 |.02185 97815 60 1 | . 0820 |. 9180 8032 |. 1286 | .6979 | .0224 |. 2191 |. 7809 || 59 2 | . 0848 |. 9152 | .7966 |. 1316 | .6912 .0225 |. 2197 |. 7803 || 58 3 | . 0876 |. 9123 || 7901 |. 1347 | .6845 .0225 |. 2203 |. 7806 || 57 4 |. 0905 . 9105 || 7835 | . 1377 | .6778 || 0226 |. 2209 |. 7790 56 5 | .20933 |.79066 4.7770 | .21408 || 4.6712 | 1,0226 |.02215 | 97.784 || 55 6 . . 0962 | . 9038 || 7706 |. 1438 . .6646 | .0227 l. 2222 |. 7778 54 7 | . 0990 |. 9010 || 7641 .. 1468 || 6580 | .0228 |. 2228 |. 7772 || 53 8 . . 1019 |. 8981 | .7576 ſ. 1499 || .6514 .0228 |. 2234 |. 7766 || 52 9 |. 1047 | . 8953 || 7512 . 1529 | .6448 .0229 |. 2240 l. 7760 51 10 | .21076 | 78924 || 4,7448 |.21560 || 4.6382 | 1.0230 .02246 |.97754 || 50 11 1104 . 8896 | .73S4 | . 1590 .6317 | .0230 |. 2252 |. 7748 || 49 12 | . 1132 |. 8867 || 7320 |. 1621 | .6252 .0231 |. 2258 |. 7741 || 48 13 | . II61 |. 8839 || 7257 |. 1651 | .6187 .0232 |. 2264 |. 7735 | 47 14 . 1189 |. 8811 || 7193 |. 1682 .6122 .0232 |. 2271 |. 7729 || 46 15 | .21218 .78782 || 4.7130 | .21712 || 4.6057 | 1.0233 |.02.277 .97723 45 16 | . I246 . 8754 || 7067 |. 1742 | .5093 .0234 i. 2283 . 77.17 | 44 17 | . 1275 |. 8725 | .7004 |. ITT3 . .5928 .0234 |. 2289 . 7711 || 43 18 | . 1303 |. 8697 .6942 |. 1803 .5864 .0235 . . 2295 |. 7704 || 42 19 | . 1331 |. 8668 .6S79 |. 1834 .5800 | .0235 . . 2302 |. 7698 || 41 20 | .21360 |.78640 4.6817 | .21864 || 4.5736 1.0236 |.02308 .97692 || 40 21 | . 1388 |. 8612 .6754 |. 1895 .5673 .0237 i. 2314 - 7686 || 39 22 . 1417 | . 8583 .6692 . 1925 | .5609 | .0237 |. 2320 | . T680 || 38 23 . 1445 |. 8555 . .6631 |. 1956 .5546 .0238 |. 2326 |. 7673 || 37 24 | . 1473 |. 8526 .6569 |. 1986 . .5483 .0239 |. 2333 |. 7667 || 36 25 | .21502 || 78508 || 4.6507 .22017 || 4,5420 | 1.0239 |.02339 .97661 || 35 26 . 1530 |. S470 .6446 |. 2047 | .5357 | .0240 |. 2345 | . T655 || 34 27 | . 1559 |. 8441 . .6385 | . 2078 | .5294 | .0241 |. 2351 |. 7648 || 33 28 | . 1587 |. S413 .6324 |. 2108 || .5232 | .024.1 |. 2358 . 7642 || 32 29 . 1615 | . 8384 .6263 |. 2139 .5169 | .0242 |. 2364 . 7636 || 31 30 | .21644 .78356 || 4,6202 | .22169 || 4.5107 | 1.(1243 .023.70 | .97630 || 30 31 | . 1672 |. S32S .6142 |. 2200 | .5045 | .0243 |. 2377 ||. T023 29 32 . 1701 | . 8299 || .6081 | . 2230 | .4983 | .0244 i. 2383 . . '7617 | 28 33 | . 1729 |. 8271 .6021 |. 2261 | .492.1 | .0245 |. 23S9 |. 7611 27 34 . 1757 | . 8242 .5961 |. 2291 .4860 | .0245 i. 2396 | . T604 || 26 35 | .21786 .78214 4,5901 | .22322 4.4799 || 1.0246 |.02402 | .9750S 25 36 | . 1814 , 8.186 .5841 | . 2353 || 4737 | .0247 |. 2408 . 7592 24 37 | . 1843 |. 8154 .5782 . 2383 . .4676 | .0247 |. 2415 . 7585 || 23 38 | . 1871 .. 8129 | .5722 | . 2414 .4615 .024S |. 2421 . 7579 22 39 . 1899 | . 8100 | .5663 |. 2444 .4555 . .0249 |. 2427 | . 7573 21 40 | .21928 .78072 || 4,5604 || .22475 || 4.4494 | 1.0249 |.02434 .97566 || 20 41 | . 1956 | . 8 .5545 | . 2505 | .4434 | .0250 |. 2440 . 7560 | 19 42 | . 1985 | . 8015 .5486 |. 2536 .4373 | .0251 |. 2446 |. 7553 18 43 | . 2013 | . T987 || 5428 . 2566 : 4313 | .0251 |. 2453 |. 7547 17 44 | . 2041 . 7959 | .5369 |. 2597 || 4253 .0252 |. 2459 |. 7541 | 16 45 | .22070 .77930 || 4,5311 | .22628 || 4.4.194 | 1.0253 |.02466 .97534 15 46 | . 209S | . T902 | .5253 |. 2658 . .4134 .0253 |. 2472 |. 7528 || 14 47 . 2126 . 7873 .5195 | . 2689 | .4074 | .0254 . . 2479 |. 7521. 13 48 | . 2155 | . TS45 .5137 | . 2719 .4015 | .0255 . . 2485 |. 7515 i 12 49 |. 2183 | . TS17 | .5079 |. 2750 | .3956 .0255 |. 2491 |. 7508 || 11 50 .22211 | .77788 || 4.5021 | .22781 || 4,3897 | 1.0256 |.02498 .97502 10 51 . 2240 | . Tſ60 || 4964 |. 2811 | .3838 | .0257 °l. 2504 . 7495 || 9 52 | . 2268 . 7732 || 4907 | . 2842 .3779 | .0257 |. 2511 . . .7489 || 8 53 | . 2297 . 7703 || 4850 | . 2872 .3721 | .0258 |. 2517 | . 7483 || 7 54 | . 2325 | . T675 | .4793 |. 2903 .3662 | .0259 |. 2524 |. 7476 || 6 55 | .22353 || 77647 || 4.4736 .22934 || 4.3604 || 1,0260 . .02530 | .97470 5 56 | . 2382 |. 7618 .4679 |. 2964 | .3546 .0260 . . 2537 |. 7463 || 4 57 | . 2410 | . T590 .4623 |. 2995 | .3488 .0261 |. 2543 |. 7457 3 58 . 2438 | . 7561 .4566 |. 3025 .3430 | .0262 |. 2550 | . 7450 || 2 59 | . 2467 | . T533 .4510 |. 3056 | .3372 | .0262 |. 2556 |. 7443 || 1 60 | . 2495 | . 7505 | .4454 |. 30S7 | .3315 .(1263 . 2563 |. 7437 || 0 M Cosime. Wrs. Sin. Segante. Cotang. Tangeut. Cosec'nt Wrs. Così Slne. M 1029 Natural. 77o NATURAL LINES. 130 Natural Trigonometrical Functions. 1660 11h M ). Sine. Wrs. Cos.jQosec'nte | Tang. Cotang. Secante.jWrs. Sin Cosine. M. M. S. 0 .22495 |.77505 || 4.4454 .23087 4,3315 | 1,0263 .02563 .97437 60 || 8 I 2523 | . T476 .4398 31.17 | .3257 | .0264 i. 2569 '[430 59 56 2 2552 | . TA4S .4342 3148 .3200 .0264 |. 2576 |. 7424 58 52 3 2580 | . T.420 .4287 |. 3179 | .3143 .0265 |. 2583 74.17 || 57 || 48 4 2608 | . T391 .4231 3209 || 30S6 | .0266 . 25S9 7411 || 56 || 44 5 | .22637 .77363 || 4.4.176 .23240 4,3029 || 1,0266 .02596 97.404 55 40 6 2665 | . T335 .4121 3270 .2972 .0267 |. 2602 '7398 || 54 || 36 7 2693 7306 .4065 . 3301 | .2916 .0268 |. 2609 | . T391 || 53 || 32 8 2722 7278 .4011 |. 3332 2859 .0268 |. 2616 73S4 || 52 28 9 2750 7250 .3956 | . 3363 .2803 | .0269 |. 2622 . T378 51 24 10 .22778 |.77221 || 4.3901 .23393 || 4,2747 | 1.0270 |.02629 |.973.71 || 50 20 11 2807 | . TIQ3 .3847 |. 3424 .2691 .027 l 2635 7364 || 49 || 16 12 | . 2S35 7165 ,3792 | . 3455 / .2635 | .0271 2642 7358 || 48 h 12 13 2863 '7136 .3738 . 3485 .2579 .0272 2649 7351 47 8 14 2892 7108 3684 |. 3516 .2524 .0273 2655 7344 || 46 4. 15 .22920 | .77080 || 4,3630 | .23547 || 4.2468 | 1.0273 |.02662 | .97.338 || 45 || 7 16 2948 | . T052 .3576 . 3577 | .2413 .0274 2669 7331 || 44 || 56 17 2977 . 7023 .3522 3608 || .2358 .0275 2675 7324 || 43 52 18 3005 | . 6995 .3469 3639 .2303 | .0276 2682 7318 || 42 || 48 19 3033 6967 .3415 3670 | .2248 || 0276 . 2689 7311 || 41 || 44 20 | .23061 .76938 || 4.3362 .23700 || 4,2193 | 1.0277 || 02695 .97.304 || 40 || 40 21 3090 691() .3309 3731 .2139 .0278 i. 2702 7298 || 39 || 36 22 | . 31.18 6882 .3256 . 3762 .2084 .0278 |. 2709 7291 || 38 || 32 23 . 31.46 6853 .3203 3793 | .2030 .0279 |. 2716 . 7284 || 37 || 28 24 3175 | . 6825 ,3150 3823 .1976 .02SO 2722 7277 || 36 || 24 25 .23203 || 76797 || 4,3098 || .23854 4.1921 | 1.0280 || 02729 || 97.271 || 35 | 20 26 3231 6769 .3045 3885 .1867 ()281 i. 2736 7264 || 34 || 16 27 3260 6740 .2993 3916 .1814 .0282 i. 2743 7257 || 33 12 28 3288 6712 .2941 3946 . .1760 | .02S3 |. 2749 7.250 || 32 S 29 3316 6684 .2888 3977 .1706 | .0283 |. 2756 | . 7244 || 31 4 30 .23344 .76655 || 4.2836 .24008 || 4.1653 | 1.0284 |.02763 | .97237 30 || 6 31 3373 . 6627 ,2785 4039 || .1600 .0285 |. 2770 7230 29 || 56 32 | . 3401 | . 6599 .2733 4069 | .1546 | .0285 |. 2777 7223 || 28 || 52 33 | . 3429 | . 6571 .2681 4100 .1493 .0286 I. 2783 7216 || 2 || 48 34 . . 3458 . 6542 .2630 4131 .1440 .0287 |. 2790 7210 26 || 44 35 | .23486 |.76514 || 4,2579 | .24162 4,1388 | 1.0288 . .02797 || 97.203 25 40 36 . 3514 | . 6486 .2527 . 4.192 .1335 | .0288 . . 2804 7.196 || 24 || 36 37 . . .3542 6457 .2476 |. 4223 .1282 .0289 |. 2811 7 189 || 23 || 32 38 3571 . 6429 .2425 | . 4254 . .1230 | .0290 . . 2818 7182 22 || 28 39 . 3599 | . 6401 .2375 . 4285 .1178 .0291 2824 | . TIT5 || 21 24 40 .23627 | .76373 || 4.2324 | .24316 || 4.1126 | 1.0291 || 02831 .97.169 20 | 20 4L | . 3655 6344 .2273 4346 .1073 | .0292 2838 |. 7162 || 19 || 16 42 | . 3684 | . 6316 .2223 j . 4377 .1022 .0293 2845 | . TI55 18 || 12 43 . 3712 6288 21.73 . 4408 . .0970 | .0293 2852 7.148 17 8 44 | . 3740 | . 6260 .2122 | . 4439 || .0918 . .0294 2859 . 7141 || 16 4 45 .23768 . .76231 || 4,2072 .244.70 || 4,0867 | 1.0295 ()2S66 .97134 || 15 5 46 3797 6203 .2022 | . 4501 | .0815 .0296 |. 2873 7127 || 14 || 56 47 38.25 ($175 .1972 | . 4531 | .0764 | .0296 | . 2880 7120 13 || 52 48 3853 6147 .1923 | . 4562 | .0713 .0297 2886 7113 | 12 || 48 49 3881 61.18 .1873 . . 4593] .0662 | .0298 2893 | . TIU6 || 11 || 44 50 | .23910 | .76090 || 4.1824 .24624 4.0611 | 1.0299 |.02900 | .97099 || 10 || 40 51 | . 3938 6062 .1774 |. 4655 .0560 | .0299 2907 7092 || 9 || 36 52 | . 3966 | . 6034 1725 4586 .0509 .0300 2914 | . TÜ86 || 8 || 32 53 3994 | . 6005 .1676 4717 | .0458 .0301 2921 |. T079 || 7 || 28 54 4023 . 5977 .1627 4747 | .0408 | .0302 ||. 2928 7072 || 6 || 24 55 .24051 | .75949 4.1578 . .24778 || 4.0358 || 1,0302 .02935 | .97.065 || 5 || 20 56 4079 |. 592.1 .1529 . 4809 . .0307 | .0303 2942 | . 7058 l 4 || 16 57 | . 4107 . 5892 .1481 | . 4840 | .0257 .0304 2949 . 7051 || 3 || 12 58 . . 4136 |. 5S64 1432 . 4871 | .0207 | .0305 2956 7044 2 8 59 4164 . 5836 .1384 | . 4902 . .()157 | .0305 2963 7037 || 1 4 60 4.192 | . 5808 .1336 . 4933 . .010S / .0306 2970 | . T029 || 0 || 4 M Cosine. Wrs. Sin.) Secante, Cotang. Tangent. Cosec’nt IVrs. Gos Sine. M M.S. 103 Natural. 76° 5h NATURAL LINES. 149 Natural Trigonometrical Functions. I65° 11h M Sine. Wrs. Cos. Cosec'nte | Tang. | Cotang. Secante. {Wrs. Sin| Cosine. M |M.S. 0 .24192 || 75808 || 4,1336 | .24933 || 4.01.08 | 1.0306 |.02970 | .97029 || 60 || 4 1 | . 4220 | . 5779 .12S7 | . 4964 | .0058 .0307 i. 2977 |. 7022 || 59 || 56 2 | . 4249 . 5751 | .1239 |. 4995 .0009 | .0308 |. 2984 |. 7015 || 58 52 3 | . 4277 . 5723 | .1191 |. 5025 | .9959 | .0308 |. 2991 |. 7008 57 || 48 4 . 4305 . 5695 .1144 |. 5056 3.9910 ! .0309 i. 2999 |. 7001 || 56 | 44 5 | .24333 |.75667 || 4.1096 .25087 3.9861 | 1.0310 l.03006 | .96994 || 55 40 6 | . 4361 |. 5638 . .1048 5118 .9812 .0311 |. 3013 |. 6987 || 54 || 36 7 | . 4390 |. 5610 | .1001 |. 5149 | .9763 .0311 |. 3020 |. 6980 || 53 || 32 8 . . 4418 |. 55S2 | .0953 |. 5180 | .9714 .0312 i. 3027 . . 6973 || 52 || 28 9 | . 4446 |. 5554 .0906 |. 5211 . .9665 | .0313 |. 3034 . 6966 || 51 | 24 10 | .24474 | .75526 4.0859 .25242 || 3.9616 | 1.0314 | .030.41 | .96959 || 50 | 20 11 | . 4502 |. 5497 .08.12 |. 5273 | .9568 .0314 |. 3048 |. 6952 49 | 16. 12 | . 4531 |. 5469 .0765 . . 5304 | .9520 ! .0315 |. 3055 |. 6944 || 48 || 12 13 | . 4559 |. 5441 . .0718 |. 5335 | .9471 .0316 i. 3063 |. 6937 47 || 8 14 | . 4587 |. 5413 | .0672 |. 5366 .9423 .0317 |. 3070 |. 6930 || 46 4 15 | .24615 .75385 || 4.0625 | .25397 3.9375 | 1.0317 | .03077 .96923 || 45 || 3 16 | . 4643 |. 535 .0579 . 5428 .9327 | .0318 |. 3084 | . 69.16 || 44 56 17 | . 4672 |. 5328 .0532 |. 5459 . .9279 .0319 |. 3091 |. 6909 || 43 52 18 . 4700 l. 5300 .0486 |. 5490 | .9231 | .0320 l. 3098 |. 6901 || 42 || 48 19 . 4728 |. 5272 .0440 | . 5521 | .9184 .0320 |. 3106 | . 6894 || 41 || 44 20 ! .24756 .75244 || 4,0394 .25552 | 3.9136 | 1.0321 |.03113 | 196887 | 40 | 40 21 | . 4784 |. 5215 .0348 . 5583 | .9089 | .0322 i. 3120 | . 6880 || 39 || 36 22 | . 4813 |. 5187 l .0302 |. 5614 | .9042 .0323 |. 3127 | . 6S73 || 38 || 32 23 . 484.1 |. 5159 .0256 |. 5645 .8994 | .0323 |. 3134 . 6865 || 37 || 28 24 . . 4869 |. 5131 || 0211 . . 5676 .8947 | .0324 |. 3142 i. 6858 || 36 || 24 25 | .24897 | .75103 || 4.0165 .25707 || 3.SS00 | 1,0325 |.03149 .96851 || 35 | 20 26 . 4925 | . 5075 .0120 | . 5738 || 8853 .0326 |. 31.56 | . 6844 || 34 16 27 | . 4953 |. 5046 | .0074 | . 5769 | .8807 | .0327 |. 3163 |. 6836 33 || 12 28 | . 4982 . 5018 .0029 . . 5800 | .8760 || 0327 |. 3171 | . 6829 || 32 || 8 29 | . 5010 | . 4990 || 3.99.84 . 5831 | .8713 .0328 |. 3178 . 6822 || 31 || 4 30 | .2503S .74962 || 3,9939 .25862 || 3.866'ſ 1.0329 |.03185 | .96815 # 30 2 31 | . 5066 | . 4934 .9S94 | . 5893 l S621 | .0330 |. 3192 |. 6807 || 29 || 56 32 | . 5004 | . 4906 | .9850 |. 5924 | .8574 .0330 |. 3200 | . 6800 || 28 || 52 33 . 5122 | . 4877 .9805 |. 5955 .8528 .0331 |. 3207 | . 6793 || 27 || 48 34 . 5151 | . 4849 .9760 |. 5986 .84S2 | .0332 |. 3214 | . 6785 || 26 || 44 35 | .25179 |.74821 3.97.16 .26017 | 3.8436 | 1.0333 |.03222 |.96778 25 | 40 36 | . 5207 | . 4793 .9672 |. 604S .8390 | .0334 |. 3229 |. 6771 24 || 36 37 | . 5235 | . 4765 . .9627 | . 6079 | .8345 .0334 i. 3236 |. 6763 || 23 || 32 38 . 5263 | . 4737 | .958.3 | . 6110 | .8299 || .0335 | . 3244 |. 6756 22 28 39 . 5291 | . 4709 .9539 . 6141 | .8254 . .0336 |. 3251 | . 6749 21 24 40 .25319 .746S0 || 3.9495 .26172 || 3.8208 || 1.0337 |.03.258 |.96741 || 20 | 20 41 3348 | . 4652 | .945.1 | . 6203 | .8163 .0338 |. 3266 |. 6734 19 | 16 42 | . 5376 . 4624 .940S | . 6234 .8118 .0338 |. 3273 | . 6727 | 18 12 43 . 5404 . 4596 | .9364 . 6266 . .8073 .0339 |. 3281 | . 6719 || 17 | 8 44 . 5432 |. 4568 .9320 | . 6297 .8027 . .0340 i. 3288 . . 6712 | 16 || 4 45 - .25460 | .74540 || 3.9277 | .26328 || 3.7983 | 1.0341 . .03295 96704 || 15 | 1 46 | . 5488 . 4512 | .9234 |. 6359 | .7938 . .U341 . . 3303 |. 6697 || 14 || 56 47 | . 5516 | . 4483 .919.0 | . 6390 | .7893 | .0342 |. 3310 | . 6090 13 || 52 48 . 5544 . 4455 .9147 | . 6421 | .7848 .0343 |. 3318 |. 6682 | 12 || 48 49 | . 5573 . . 4427 | .9104 | . 6452 | .7804 | .0344 i. 3325 |. 6675 || 11 || 44 50 | .25601 || 74399 || 3.9061 | .26483 || 3.7759 | 1.0345 |.03332 |.96667 10 | 40 51 | . 5629 | . 4371 .9018 . 65.14 | .77.15 .0345 |. 3340 . 6660 || 0 || 36 52 | . 5657 | . 4344 | .8976 |. 6546 | .7671 | .0346 . 3347 | . 6652 | 8 || 32 53 . 5685 . 4315 | S933 . . 6577 | .7627 | .0347 l. 3355 . 6645 || 7 || 28 54 | . 5713 | . 4287 | .8890 | . 6608 .7583 | .0348 |. 3362 |. 6638 || 6 || 24 55 | .25741 | .74239 || 3.8848 . .26639 || 3.7539 1.0349 .03370 | .96630 || 5 || 20 56 | . 5769 |. 4230 | .8805 | . 66.70 | .7495 | .0349 |. 3377 | . 6623 || 4 || 16 57 5798 | . 4202 | .8763 | . 6701 || 7451 | .0350 . . 33S5 | . 6615 || 3 || 12 58 . 5826 | . 4174 .8721 | . 6732 || 7407 | .0351 |. 3392 |. 6608 || 2 || 8 59 . 5854 | . 4146 | .8679 . 6764 || 7364 .0352 |. 3400 | . 6600 | 1 || 4 60 | . 5882 | . 4118 || 8637 | . 6795 | .7320 .0353 |. 3407 | . 6592 || 0 || 0 M Cosine. Wrs. Sin. Secante. Cotang.jTangent. | Cosec’nt l Wrs. Cos! Sine. M M.S. 104° Natural. 75° 5h MATURAL LINES. Ih 150 Natural Trigonometrical Functions. 1649 |10h M.S. M Sine. Wrs. Cos. Cosec'nte Tang. Cotang. Secante, Vrs. Sin Cosine. M M. S. 0 () .25882 | .74118 || 3.8637 .26795 3.7320 | 1.0353 |.03407 .96592 60 || 60 4 I | . 5910 | . 4090 .8595 | . 6826 .7277 .0353 |. 3415 | . 6585 || 59 || 56 S 2 . . 5938 . . 4062 .8553 | . 6857 .7234 .0354 |. 3422 | . 6577 58 || 52 I2 3 | . 5966 . . 4034 .8512 | . 6888 . .7.191 .0355 |. 3430 | . 6570 || 57 || 48 16 4 5994 |-. 4006 .S470 . 6920 .7147 .0356 |. 3438 . . 6562 || 56 44 20 5 .26022 | .73978 || 3.8428 .26951 || 3.7104 || 1.0357 ,03445 | .96555 55 | 40 24 6 6050 . 3949 .8387 | . 6982 || 7062 .0358 |. 3453 | . 6547 || 54 || 36 28 7 6078 3921 .8346 | . T013 || 7019 .0358 |. 3460 . 6540 53 || 32 32 8 6107 3893 .8304 | . T044 .6976 | .0359 |. 3468 . 6532 52 || 28 36 9 G135 3865 .8263 . 7076 .6933 .0360 |. 34.75 6524 51 24 40 || 10 .26163 .73S37 || 3.8222 | .27107 3.6891 | 1.0361 |.03483 .965.17 || 50 20 44 || 11 6.191 3809 S181 |. 7138 .6848 .0362 3491 6509 || 49 || 16 48 || 12 62.19 37SI 8140 | . W169 .6806 .0362 3498 6502 || 48 # 12 52 || 13 . 6247 | . 3753 .S100 | . T201 | .6764 | .0363 |. 3506 6494 || 47 8 56 || 14 | . 6275 | . 3725 | .8059 |. 7232 | .6722 || 0364 |. 3514 |. 6486 || 46 4 1 || 15 .26303 .73697 || 3.801S .27263 || 3.6679 | 1.0365 |.03521 | .96479 || 45 || 59 4 IG | . 633L 3669 .7978 . 7294 .6637 .0366 3529 | . 6471 || 44 || 56 8 || IT | . 6359 3641 .7937 | . Tº?6 | .6596 .0367 |. 3536 6463 || 43 || 52 12 18 || . 63S7 3613 7897 | . T357 .6554 .0367 3544 6456 || 42 48 16 19 | . 6415 3585 .7857 | . T3S8 | .6512 | .0368 3552 6448 i 41 || 44 20 | 20 .26443 || 73556 3.7816 | .27419 || 3.6470 | 1.0369 |.03560 .96440 | 40 || 40 24 21 | . 6471 . 3528 .7776 | . T451 .6429 .0370 3567 G433 || 39 || 36 28 22 | . 6499 3500 .7736 |. "4S2 63S7 | .0371 3575 6425 || 38 || 32 32 || 23 . 6527 | . 34.72 .7697 . 7513 6346 .0371 3583 6417 || 37 28 36 24 . 6556 |. 3 .7657 | . T544 | .63(5 .0372 3590 6409 || 36 || 24 40 25 .26584 .73416 || 3.7617 .27576 || 3.6263 | 1.0373 .03508 .96402 || 35 20 44 26 | . 6612 . .3388 .7577 | . TG07 .6222 | .0374 3606 6394 || 34 16 48 27 6640 | . 3360 .7538 | . TG3S .6181 .0375 3614 63S6 || 33 || 12 52 28 . 6668 . 3332 .7498 | . '670 | .6140 .0376 36.21 6378 i 32 8 56 29 . 6696 3304 .7459 7701 || 6100 .0376 3629 6371 31 4 2 || 30 .26724 |.73276 || 3.7420 | .27732 3.605.9 | 1.0377 |.03637 .96363 || 30 58 4 || 31 | . 6752 3248 .7380 | . Tſ64 6018 .0378 3645 6355 29 || 56 8 || 32 | . 6780 3220 7341 7795 597.7 | .0379 3652 6347 || 28 || 52 12 || 33 . . 6808 3.192 .7302 7826 .5937 .0380 3660 6340 27 || 48 I6 || 34 6836 3.164 .7263 '7858 .5896 || 03S1 3668 6332 26 44 20 || 35 .26S64 || 73136 || 3.7224 .27889 3.5856 1.03S2 |.03676 .96324 || 25 || 40 24 36 6892 31()8 .7186 7920 .5S16 .0382 3684 6316 24 || 36 28 37 6920 |. 3080 .7147 '7952 | .5776 .0383 . . 3691 | . 630S 23 || 32 32 || 38 6948 3052 .71(JS 7983 | .5736 .0384 3699 . 6301 || 22 28 36 || 39 6976 3024 .7070 8014 | .5696 .0385 37()7 6293 || 21 || 24 40 40 | .27004 .72996 || 3.7031 .28046 || 3.5656 | 1.0386 |.03715 .96285 20 20 44 || 41 | . 7032 2968 .6993 8077 .5616 .0387 3723 |. 6277 || 19 || 16 48 || 42 . 706 2940 6955 | . 8109 .5576 | .0387 373L | . 6269 || 18 || 12 52 || 43 . 7088 |. 2912 6917 | . 8140 .5536 | .0388 3739 . 6261 || 17 8 56 || 44 . 7116 2884 .6878 | . 8171 .5497 .0389 3746 . 6253 | 16 4 3 45 .27144 .72856 || 3.6840 .28203 || 3.5457 | 1.0390 .03754 .96245 || 15 57 4 || 46 | . 7172 2828 6802 8234 . .5418 .0391 3762 | . 6238 || 14 || 56 8 || 4-7 | . Tº()() 2800 .6765 8266 .5378 .0392 3770 ($230 || 13 || 52 12 || 48 || . 7228 2772 .6727 | . 8297 ,5339 .0393 3778 | . 6222 || 12 || 48 16 || 49 . 7256 |. 2744 | .6689 |. 8328 || 5300 .0393 |. 3786 |. 6214 || 11 || 44 20 || 50 .27284 .72716 || 3.6651 | .28360 || 3.5261 | 1.0394 | .03794 | .96206 || 10 || 40 24 || 51 | . T312 2688 .6614 8391 | .5222 | .0395 3802 6198 9 || 36 28 || 52 | . Tºq0 2660 .6576 8423 .5183 .0396 3810 | . 6190 8 || 32 32 53 . 7368 2632 .6539 8454 || 514.4 | .0397 3818 . 6182 7 28 36 || 54 | . T396 | . 2604 ,6502 8486 .5105 | .0398 3826 6174 6 || 24 30 55 .27424 || 72676 || 3.6464. | .28517 | 3.5066 | 1.0399 |.0383 96.166 5 20 44 56 . 7452 2548 .6427 S549 || .502S .0399 3.842 6158 4 16 48 57 || - 7480 252() .639() 8580 .4989 || 0400 | . 3850 Ö150 || 3 || 12 52 58 . '7508 | . 2492 .6.353 8611 .4951 .0401 3858 . . 6142 2 8 56 59 . . 7536 2464 .6316 | . 8643 .1912 | .0402 3866 | . 6134 || 1 4 4. 6() 7564 | . 2436 .6279 $674 .4874 | .0403 3874 | . 6126 0 56 M. S. M I Cosine. Wrs. Sun. ScCante. | Cotang. Tangent, Cosec'ntiVrs. Cos! Sine. M M.S., 7h 1050 Natural. 74° 4h . NATURAL LINES. 169 Natural Trigonometrical Functions. 1639 M | Sine. Wrs. Cos. Cosec'nte | Tang. | Cotang. ScCante. Vrs. Sin Cosime. M 0 .27564 .72436 || 3.6279 .28674 3.4874 | 1.0403 |.03874 .96126 60 1 | . 7592 | . 2408 .62+3 | . 8706 || 4836 .0404 . 3882 | . 6118 || 59 2 . . '7620 | . 2380 | .6206 |. 8737 | .4798 || 0405 i. 3890 |. 6110 58 3 | . T648 |. 2352 .6169 . 8769 .4760 .0406 ||. 3898 |. 6102 || 57 4 | . 7675 | . 2324 .6133 i. 8800 | .4722 || 0406 |. 3906 |. 6094 || 56 5 | .27703 .72296 || 3.6096 | .28832 3.4684 || 1.0407 |.03914 || 96086 || 55 6 | . TT31 | . 2268 .6060 | . 8863 | .4646 .0408 i. 3922 | . 6078 || 54 7 | . TT59 |. 2240 .6024 |. 8S95 .4608 || 0409 |. 3930 | . 6070 || 53 8 . . 7787 . 22.13 .5987 | . 8926 .4570 .04.10 | . 3938 . . 6062 || 52 9 78.15 . 2185 .5951 | . 8958 . .4533 .04.11 . . .3946 | . 6054 || 51 10 | .27843 72157 || 3.5915 . .28990 || 3,4495 i 1.0412 |.03954 .9604.5 50 Il 78T1 . . 2129 .5879 . 9021 .4458 .04.13 . 3962 . 6037 || 49 12 7899 | . 2101 .5843 | . 9053 | .4420 ) ,0413 |. 3971 . . 6029 || 48 M3 . 7927 . 2073 .5807 | . 9084] .4383 || 0414 i. 3979 . 6021 47 14 7.955 | . 2045 .5772 | . 91.16 .4346 .04.15 . . .3987 | . 6013 || 46 15 .27983 || 72017 || 3.5736 .29147 3.4308 || 1.0416 .03995 .96005 || 45 16 . 8011 | . 1989 .5700 |. 9179 .4271 .0417 |. 4003 |. 5997 || 44 17 | . 8039 |. 1961 .5665 |. 9210 .4234 .0418 |. 4011 |. 5989 43 M8 | . 8067 | . 1933 .5629 |. 9242 .4197 .04.19 . 4019 . 5980 || 42 M9 8094 . 1905 .5594 | . 9274 .4160 .0420 . 4628 . 5972 41 20 .28122 | .71S77 || 3.5559 .29305 3.41:24 | 1.0420 |.04036 |.95964 40 21 | . 8150 | . 1849 .5523 . 9337 || 4087 .0421 |. 4044 |. 5956 || 39 22 8178 |. 1822 | .54S8 . 9368 .4050 || 0422 4052 | . 5948 || 38 23 . 8206 |. 1794 | .5453 |. 9400; .4014 J .0423 4060 | . 5940 || 37 24 . 8234 . 1766 .5418 | . 9432 .3977 .0424 |. 4069 |. 5931 || 36 25 | .28262 || 71738 || 3.5383 . .29463 || 3.3941 | 1.0425 i.04077 |.95923 || 35 26 8290 . . 1710 | .5348 . 9495 | .3904 || 04:26 4085 5915 || 34 27 | . 8318 . . 1682 .5813 | . 9526 .3868 .0427 4093 5907 || 33 28 8346 | . 1654 .5279 . 9558 .3832 .0428 4101 , 5898 || 32 29 8374 |. 1626 .5244 | . 9590 || 3795 .0428 41.10, 5890 || 31 30 | .28401 || 71608 F3.5209 | .29621 3.3759 | 1.0429 |.04.118'' .95882 || 30 31 . 84.29 | . 1570 .51.75 . 9653 | .3723 .0430 4.126 | . 5874 || 29 32 . 8457 | . 1543 .5140 | . 9685 .3687 | .0431 4134 |. 5865 28 33 8485 | . 1515 .5106 9T16 .3651 .0432 4143 5857 || 27 34 85.13 | . 1487 .5072 | . 97.48 .3616 .0433 4.151 5849 26 35 .28541 ||.TI459 3.5037 .29780 || 3.3580 | 1.0434 |.04.159 .95840 25 36 8569 | . 1431 .5003 9811 , .3544 .0435 4168 5S32 || 24 37 8597 . 1403 .4969 | . 98.43 .3509 .0436 4176 5824 || 23 38 8.324 . 1375 .4935 | . 9875 . .3473 .043 4.184 |. 5816 22 39 8552 . [347 .4901 | . 99.06 .3438 .0438 4193 5S07 || 21 40 . .28630 | .71320 i 3.4867 | .29938 : 3.3402 | 1.0438 |.04201 | .95799 || 20 41 8708 || . I2.02 . .4833 | . 997() .3367 .0439 4209 | . 5791 || 19 42 | . 8736 | . 1264 .4799 .30001 | .3332 .044() 4218 | . 5782 | 18 43 . 8764. | . 1236 .4766 ()033 .3296 | .0441 4226 5774 ; 17 44 8792 . 1208 .4732 . 0065 .3261 .0442 4234 5765 | 16 45 .28820 | 711S0 i 3.4698 | .30096 || 3.3226 | 1.0443 .04.243 || 95757 || 15 #6 8847 | . 1152 .4665 . ()128 .3]91 .0444 4251 | . 5749 || 1:4 47 | . 8S75 - 1125 .4632 . 0160 | .3156 .0445 |. 4260 5740 || 13 4S , 8.903 | . I (197 .4598 || -- 0.192 .3121 .0446 4268 5732 12 49 8931 | . 1069 .4565 | . 0223 .3087 | .0447 4276 5723 || 11 50 .28959 |.71041 || 3.4532 | .3U255 || 3.3052 | 1.(1448 .04:285 |.05715 || 10 51 | . 8987 | . 1013 | .4498 | . 0287 .3017 | .0448 4.293 5707 || 9 52 . 9014 | . (Y985 .4465 | . 0319 | .2983 ,0449 . 4302 5698 8 53 | . 9042 . 0958 .4432 . 0350 | .2948 .0450 |. 4310 |. 5690 7 54 | . 9070 | . 0930 || 4899 || , 0382 | .2914 | .0451 |. 43}9 |. 5681 || 6 55 .29098 || 70902 || 3.4366 .30414 || 3:2S79 1,0452 .04327 | .95673 || 5 56 . 9.126 . OS74 .4334 | . 0446 .2S45 .0453 |. 4335 | . 5664 4 57 | . 9154 | . 0846 . .4301 | . ()478 .2811 .0454 . 4344 . 5656 3 58 i , 9181 | . 0818 .426S | . 0509 .2777 | .0455 |. 4352 | . 5647 2 59 . 9209 | . 0791 .4236 . 0541 | .2742 .0456 |. 4361 , 5639 || 1 60 | . 9237 | . O'763 .4203 | . ()573 . .2708 || 0457 ºl. 4369 |. 5630 || 0 M Cosine. Wrs. Sin. Secante. Cotang. Tangent. Cosec'nt IVrs. Cosſ Sine. M 1069 g Natural. 730 O 3/. *Iu.Inqu N O IV "ouſS soo’s. Aſ qu,008o0 ſºuq.3uttſ, 5uujoo alumoos ('uſ's '84A "ouphoo 0 |90Ig fgSV gig0 1110 |26FZ 1982 8606 - |z060 I g[[g g885 ' | WI90 1080. 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Iu.Innºunſ "SANI'ſ 'IVºſſ),LVN ! O I NATURAL LINES, 180 Natural Trigonometrical Functions. 1619 M | Sinc. Vrs. Cos.;CoscC'nte T'ang. Cotting. Secuntc. iVrs. Sin Coşine. M. M. S. () .30902 | .69098 || 3.2361 .32492 || 3.0777 | 1.0515 . .04894 | .95106 || 60 || 48 1 | . 0.929 |. 9071 .2332 | . 2524 .0746 .0516 | . 4903 | . 5097 59 || 56 2 . . 0957 | . 9043 .2303 | . 255(; ; ,0716 .0517 | . 4912 | . 5088 || 58 || 52 3 : . ().985 . 9015 .2274 |. 2588 .0686 .0518 |. 4921 |. 5079 || 57 || 48 4 | . T012 | . 8988 .224.5 | . 2621 ,0ſ;55 .0519 | . 4930 . 507 () || 56 || 44 5 .3104() |,68960 3.2216 .32653 || 3.0625 | 1.0520 |.04939 .95061 || 55 40 6 | . 1068 . 8932 .2188 |. 2685. .0595 ,0521 . 4948 ||. 5051 || 54 36 7 | . 1095 | . 8905 .2159 |. 2717 | .0565 .0522 ||. 4957 . 5042 || 53 || 32 8. 1123 |. 8877 .2131 |. 2749 || 0535 | .0523 |. 4966 |. 5033 || 52 28 9 115() 8849 .2102 | . 2782 | .0505 || 0524 |. 4975 . 5024 || 51 || 24 10 || 31178 |.68822 || 3.2074 .32814 || 3.0475 | 1.0625 ||.04985 .950.15 50 20 11 . 1206 | . 8794 | .2045 . 2846 . .()445 ,0526 ||. 4994 | . 5000 || 49 || 16 12 | . 123 8766 .2017 | . 2878 || 0415 | .0527 |. 5003 | . 4997 || 48 || 12 13 | . 1261 8739 .1989 |, 2010 || 0385 .0528 |. 5012 | . 4988 || 47 8 14 | . I 289 87.11 .1960 |. 2943 || 0.356 || 0529 |. 5021 |. 4979 || 46 4 15 ,31316 |.68684 || 3.1932 | .32975-1 3.032G | 1,0530 | .05030 .94970 || 45 |47. 16 | . 1344 8656 .1904 | . 3007 || 0296 | .0531 i. 5039 | . 4961 I 44 56 17 | . 1372 8028 .1876 |. 3039 || .0267 .0532 |. 5048 |. 4952 || 4:3 || 52 18 . 1399 8601 .1848 . . .3072 ,0237 || 0533 |. 5057 | . 4942 || 42 || 48 19 . 1427 8573 .1820 | . 3104 | .0208 || 0534 |. 5066 . 493.3 || 41 || 44 20 | .31454 .68545 || 3.1792 .33136|| 3.0178 || 1,0535 | .05076 .94924 || 40 || 40 21 | . 1482 , 8518 || .1764 |. 3169 .0149 .0536 |. 5085 | . 4915 || 39 || 36 22 | . 1510 | . 8490 .1736 .. 3201 || 0120 .05:37 |. 5094 | . 4906 || 38 || 32 23 . 1537 |. 8463 .1708 |. 3233 .0000 | .0538 |. 5103 |. 4897 || 37 || 28 24 | . 1565 |. 8435 .1681 |. 3265 .0001 | .0539 |. 5112 |. 4888 || 36 24 25 | .31592 | .68407 || 3.1653 .33298 || 3.0032 1,0540 ,05121 .94878 || 35 | 20 26 1620 8380 .1625 | . 3330 || 3,0003 || 0541 |. 5131 |. 4869 || | 16 27 | . 1648 |. 8352 .1598 |. 3362 || 2,0974 | .0542 |. 514() . 486() 33 12 28 . 1675 8325 .1570 |. 3395 .99.15 | .0543 |. 5149 }. 4851 32 8 29 . 1703 8297 .1543 |. 3427 | .9916 ,0544 |. 5158 . 4841 || 31 4 30 | .31730 .68269 || 3.1515 . .33459 || 2.9887 | 1.0545 |.05.168 .94.832 30 46 31 | . 1758 8242 .1488 |. 3492 | .9858 ,0546 |. 5177 |. 4823 29 || 56 32 . 1786 . 8214 .1461 , 3524 .9829 || 0547 |. 5186 |. 4814 || 28 52 33 . 1813 | . 8187 .1433 |. 3557 .9800 | .0548 , 5195 |. 4805 || 27 || 48 34 . 1841 |. 8159 .1406 . 3589 | .9772 .0549 |. 5205 | . 4795 26 || 44 35 | .31868 .68132 || 3.1379 .33621 2.9743 | 1.0550 .05214 .9478ti || 25 4() 36 | . 1896 8104 || .1352 . 3654 .9714 | .0551 |. 5223 . 4777 24 || 36 37 . 1923 |. 8076 1325 | . 3680 .9686 || 0552 |. 5232 | . 4767 23 || 32 38 | . T951 | . 8049 .1298 | . 3718 96.57 | ,0553 |. 5242 |. 4758 22 || 28 39 || . 1978 . 8021 .1271 | . 3751 | .0629 .0554 |. 5251 |. 4749 || 21 24 40 .32006 .67994 || 3.1244 .33783 || 2.9600 | 1.0555 . .0326() .94740 || 20 20 41 . 2034 .. 7966 .1217 | . 3816 .9572 ,0556 52.70 | . 4730 19 16 42 2001 7939 .1190 3848 . .9544 .0557 |. 5279 . 47.21 | 18 || 12 43 : . 2089 7911 .1163 3880 . .9515 . .0558 |. 5288 . . 4712 17 8 44 . 2116 |. 7884 ..] |37 3913 | .9487 | .0.55%) . . 5297 | . 47(12 | 16 4 45 .32144 |.07856 || 3,1110 ! .33045 2.9459 1,0560 |.05:307 .34693 15 || 45 46 | . 217.1 |. 7828 .1083 |. 3978 . .94.31 | .0561 |. 5316 . 4684 || 14 º 66 47 , 2199 |. 7801 .1057 4010 | .9403 .0562 |. 5326 |. 4674 || 13 l 52 48 | . 2226 , 7773 .10:30 | . 4043 | .9375 | .0563 |. 5335 | . 4665 | 12 || 48 49 | . 2254 7746 ..1004 4075 9347 .0565 |. 5344 |. 4655 | 11 || 44 50 | .32282 .07718 3,0977 |,34108 || 2.9319 | 1.0566 |.05:35 | | .04646 || 10 | 40 51 | . 2309 7691 .0951 4140 92.91 .0567 5363 . 4637 || 0 || 36 52 | . 2337 7663 .09.25 417.3 9263 .0508 5373 | . 4627 || 8 || 32 (53 . 2364 , 7636 ,0898 4205 9235 . .0569 b382 | . 4618 || 7 || 28 64 | . 2,392 | . 7 (308 .0872 4238 || .9208 .0570 5391 . . 4608 || 6 || 24 55 | .32419 |,67581 || 3.0846 | .342.70 || 2.9180 | 1.0571 .00401 .94599 || 5 || 20 56 | . 2447 7553 ,0820 43()3 .9152 | .0572 641() . 4590 4 || || 6 57 | . 2474 7526 .0793 4335 | .9125 | .0573 5420 | . 4580 || 3 || 12 68 | . 2502 7498 .0767 4368 90.97 .U574 5429 | . 4571 || 2 8 59 | . 2529 74.71 .0741 |. 4400 9069 .0575 5439 . 4561 || 1 4 60 . 2557 : 7443 ,0715 |. 4433 9042 0576 |. 5448 |. 4552 || 0 || 44 M. I. Cohine. Wrb, Sid. Seoante, | Cotung. Taugent, Cogec'nt IVrs. Cosi Sinc. M M.S. 108° Natural. 71° 4h NATURAL LINES. 1h 190 Natural Trigonometrical Functions. 160°]10, M.S. M | Sine. Wrs. Cos. Cosec'nte Tang. | Cotang. ||Secante. Wrs. Sin Cosime. M M.S 16 || 0 | .32557 | .67443 || 3.0715 .34433| 2.9042 | 1.0576 | 05448 .94552 | 60 || 44 4 1 | . 2584 7416 .0690 | . 4465 | .9015 .0577 |. 5458 |. 4542 || 59 56 S 2 : . 2612 | . 7388 || 0664 |. 4498 | .8987 | .0578 |. 5467 , 4533 || 58 || 52 12 3 | . 2639 |. 7361 | .0638 |. 4530 | .8960 | .0579 |. 5476 |. 4523 || 57 || 48 16 || 4 | . 2667 |. 7333 || 0612 |. 4563 | 8933 . .0580 i. 5486 |. 4514 || 56 || 44 20 5 .32694 | .67306 || 3.0586 .34595 || 2.8905 | 1.0581 I.05495 | .94504 || 55 | 40 24 6 | . 2722 7278 .0561 | . 4628 .8878 .0582 5505 |. 4495 || 54 || 36 2S 7 | . 2749 |. 7251 | .0535 | . 4661 | .8851 | .0584 |. 5515 . . 4485 53 || 32 32 || 8 | . 2777 |. 7223 .0509 |. 4693 .8824 | .0585 |. 5524 |. 4476 || 52 || 28 36 9 | . 2804 |. 7196 | .0484 4726 | .8797 .0586 5534 | . 4466 || 51 || 24 40 || 10 | .32832 |.67.168 || 3.0458 .34758 || 2.8770 | 1,0587 |.05543 |.94.457 || 50 | 20 44 11 2859 7141 . . 4791 | .8743 | .0588 5553 | . 4447 || 49 || 16 48 || 12 28ST 7113 | .0407 | . 4824 | .8716 .0589 5562 | . 4438 || 48 || 12 52 || 13 2914 | . TOS6 | .0382 | . 4856 | .86S9 .0590 5572 | . 4428 || 47 8 56 14 2942 |. T058 | .0357 |. 4889 || 8662 | .0591 |. 5581 | . 4418 || 46 4 17 15 .32969 7031 || 3.0331 .34921 || 2.8636 1.0592 |.05591 | .94409 || 45 || 43 4 || 16 2996 | . T003 || .0306 4954 | .8609 | .0593 |. 5601 | . 4399 || 44 || 56 8 || 17 | . 3024 |. 6976 .0281 | . 4987 | .8582 | .0594 |. 5610 | . 4390 || 43 52 12 | 18 . 3051 6948 || .0256 |. 5019 .8555 . .0595 |. 5620 | . 4380 || 42 || 48 16 || 19 | . 3079 6921 .0231 5052 | .8529 .0596 i. 5629 |. 43.70 || 41 || 44 20 | 20 .33106 | .66894 || 3.0206 | .35085 || 2.8502 | 1.0598 .05639 .94.361 | 40 || 40 24 || 21 | . 3134 6S66 .0181. 5117 | .8476 | .0599 |. 5649 |. 4351 || 39 || 36 28 22 | . 3161 6839 || .0156 5150 | .8449 || 0600 5658 . . 4341 || 38 32 32 || 23 . 3189 |. 681.1 .0131 5183 | .8423 | .0601 5668 . 4332 || 37 28 36 || 24 3216 . 6784 .0106 5215 .8396 || 0602 5678 . 4322 || 36 24 40 || 25 | .33243 .66756 || 3.0081 | .35248 || 2.83.70 | 1.0603 | .05687 .94.313 || 35 20 44 || 26 3271 6729 | .0056 5281 | .8344 | .0604 5697 . 4303 || 34 || 16 48 || 27 | . 3298 . 6701 .0031 5314 || 8318 . .0605 5707 4293 || 33 || 12 52 || 28 . 3326 6674 | .0007 5346 | .8291 .0606 5716 | . 4283 || 32 8 56 || 29 | . 3353 6647 || 2.99.82 5379 | .8265 || 0607 |. 5726 . 4274 || 31 4 18 || 30 || 33381 | 66619 || 2.9957.| .354.12 || 2.8239 | 1.0608 |.05736 .94264 || 30 || 42 4 || 31 | . 3408 6592 9933 5445 | .8213 | .0609 5745 | . 4254 29 || 56 8 || 32 . 3435 | . 6564 99.08 5477 | .8187 | .06.11 5755 | . 4245 || 28 || 52 12 || 33 . 3463 | . 6537 98.84 5510 | .8161 | .0612 5765 4235 | 27 || 48 16 || 34 . 3490 | . 6510 | .9859 5543 .8135 | .0613 57.75 |. 4225 || 26 || 44 20 || 35 | .33518 |.66482 || 2.9835 | .35576 || 2.8109 | 1.0614 |.05784 |.94.215 || 25 40 24 || 36 | . 3545 | . 6455 | .9810 5608 || .8083 .0615 5794 | . 4206 || 24 || 36 28 || 37 . 3572 | . 6427 | .9786 5641 .8057 | .0616 5804 |. 41.96 || 23 32 32 || 38 . 3600 | . 6400 .9762 6674 .8032 .06.17 5814 | . 4186 22 28 36 || 39 . 3627 6373 || 9738 5707 | .8006 | .06.18 5823 | . 4176 || 21 24 40 | 40 | .33655 .66345 || 2.9713 | .35739 || 2.7980 | 1,0619 |.05833 .94167 || 20 | 20 44 || 41 36S2 6318 . .9689 5772 | .7954 .0620 |. 5843 |. 4157 | 19 || 16 4S 42 | . 3709 | . 6290 | .9665 |. 5805 | .7929 || 0622 |. 5853 . 4147 | 18 || 12 52 43 | . 3737 6203 .964l 5838 .7903 || 0623 |. 5863 . 4137 || 17 8 50 || 44 3764 . 6236 .96.17 5871 | .7878 .0624 |. 5872 . 4127 | 16 4 19 || 45 .33792 .66208 || 2.9593 | .35904 || 2.7852 | 1.0625 .05882 .94.118 || 15 || 4-1 4 46 3819 6181 .9569 |. 5936 || 7827 | .0626 |. 5892 . 4108 || 14 || 5, 8 || 47 | . 3846 | . 6153 .9545 | . 5969 .7801 | .0627 |. 5902 . . 4098 || 13 || 52 12 || 48 3874 6126 .9521 | . 6002 || 7776 .0628 |. 5912 | . 4088 || 12 || 48 16 || 49 | . 3901 6099 || .9497 . 6035 | 7751 .0629 |. 5922 . . 4078 || 11 || 44 2 50 .3392S .66071 || 2.9474 .36068 2.7725 | 1.0630 | .05932 .94068 || 10 | 40 24 || 51 | . .3956 6044 | .9450 | . 6101 || 7700 | .0632 . 5941 . 4058 || 9 || 36 28 52 3983 6017 .9426 |. 6134 .7675 || 0633 i. 5951 |. 4049 || 8 || 32 32 || 53 4()]] 5989 | .9402 | . 6167 .765() | .0634 . . 5961 . 4039 || 7 || 28 36 || 54 403 5962 | .9379 |. 6199 .7625 | .0635 |. 5971 . . 4029 || 6 || 24 40 || 55 | .34065 .65935 | 2.9355 | .36232 || 2.7600 | 1.0636 | .05981 .94019 || 5 || 20 44 56 . . 4093 |. 5907 | .9332 . 6265 .7574 | .0637 |. 5991 4009 || 4 || 16 48 || 57 4120 | . 5880 . .9308 . 6298 || 7549 .0638 |. 6001 |. 3999 || 3 || 12 52 || 58 . 4147 5853 .9285 |. 6331 .7524 .0639 . 6011 3989 || 2 8 56 || 59 . . 41.75 5825 | .926.1 | . 6364 | .7500 | .0641 |. 6021 | . 3979 || 1 4 20 | 60 | . 4202 | . 5798 ,9238 . 6397 || 7475 .0642 |. 6031 . 3969 || 0 || 40 M. S. M | Cosine. Wrs. Sin. Secante. Cotang. Tangent. |Cosec'nt l Wrs. Cosl Sine M. M.S. 7h ||1090 Natural. 70° 4h __NATURAL LINEs. 200 Natural Trigomoimetrical Funetions. 1590 M | Sine. Vrs. Cos. Cosec'nte | Tang. Cotang. | Secante.jWrs. Sinj Cosime. M 0 || 34202 .65798 || 2,9238 .36397 2.7475 1.0642 |.06031 |.93969 || 60 1 | . 4229 . 5771 ,9215 | . 6430 .7450 | .0643 i. 6041 |. 3959 || 59 2 . . 4257 | . 5743 .919.1 6463 ) .7425 | .0644 |. 6051 . 3949 || 58 3 | . 4284 . 5716 .9168 |. 6496 || 74()() .0645 . 6061 |. 3939 || 57 4 | . 4311 |. 5689 .9145 | . 65.29 .7376 ()646 . 6071 39% 56 5 | .34339 |.65661 || 2.9122 | .36562 2.7351 | 1.0647 |.06080 .93919 || 55 6 . . 4366 . 5634 .9().98 6595 '7326 .0648 . 6090 3909 || 54. 7 | . 4393 |. 5607 .90.75 6628 .7302 || 0650 ($100 3899 || 53 S | . 4421 - 5579 .9()52 6661 [ .7277 .0651 6110 3889 || 52 9 | . 4448 . 5552 .9029 6694. .7252 .0652 6121 3S79 || 51 10 .34475 |.65525 2.9006 .36727 2.7228 1.0653 |.06131 .93869 || 50 11 . 4502 |. 5497 .8983 6760 | .7204 | .0654 6141 3859 || 49 12 | . 4530 . 5470 .8960 6793 .7179 .0655 6}5} 3849 || 48 13 | . 4557 | . 5443 .8937 | . 6826 .7.155 | .0656 6.161. 3839 || 47 14 . 4584 ||. 5415 .8915 . . 6859 || .7130 .0658 6171 3829 46 15 .34612 |.65388 || 2.8892 .36892 2.7106 | 1.0659 .06181 .93819 || 45 16 . 4639 . 5361 .8869 || - 6925 .7082 | .()660 619.1 ! .. 3S()9 || 44 17 4666 |. 5334 .8846 6958 . .7058 .0661 6201 | . 3799 || 43 18 4693 ) . 5306 .8824 6991 .7033 .0662 621.1 37S9 || 42 19 4721 . 5279 .8801 '7024 .7009 .0663 . 6221 3779 || 41 20 .34748 .65252 2.8778 .37057 2.6985 | 1.0664 |.06231 .93769 || 40 21 | . 4775 | . 5225 .8756 ||. T090 .6961 | .0666 |. 6241 | . 3758 || 39 22 | . 4803 | . 5107 .8733 7123 .6937 .0667 . 6251 | . 374S | 38 23 | . 4830 | . 5170 .S711 '7156 .6913 | .0668 . 6262 373S 37 24 . . 4S57 | . 5143 .8688 | . TIQ() .6SS9 || 0669 |. 6272 3728 || 36 25 | .34884 .65115 2.8666 .37223 2.6865 | 1,0670 .06282 .93718 || 35 26 | . 4912 |. 5088 .8644 '7256 .6841 .0671 . (292 3708 || 34 27 | . 4939 |. 5061 .8621 | . T289 .6817 | .0673 |. 6302 3698 || 33 28 4966 | . 5034 .8599 | . T322 .6794 | .0674 i. 6312 |. 3687 || 32 29 | . 4993 | . 5006 .8577 7355 .6770 .0675, H. 6323 3677 || 31 30 | .35021 .64979 || 2.8554 .37388 2,6740 | 1.0676 |.06333 .93667 30 31 5048 |. 4952 .8532 . 7422 | .6722 || 0677 | . 6343 3657 29 32 . 5075 | . 4925 .8510 | . W455 . .6699 || 0678 i. 6353 |. 3647 || 28 33 . . 5102 | . 4897 .8488 7488 .6675 .0679 |. 6363 3637 || 27 34 . . 5130 | . 4870 .8466 | . T521 .6652 || 0681 6373 3626 26 35 | .35157 .64843 2.84.44 .37554 2.6628 1.0682 .06384 .936.16 25 36 . 5184 |. 4816 .8422 | . T587 | .6604 0683 |. 6394 3606 || 24 37 5211 . . 4789 .8400 7621 .6581 .06S4 |. 6404 3596 || 23 38 . . 5239 4761 .8378 7654 .6558 | .0685 |. 6414 3585 22 39 || - 5266 4734 .8356 7687 .6534 .0686 . 6425 | . 3575 || 21 40 .35293 .64707 || 2.8334 .37720 2,6511 | 1,0688 |.06435 .93565 20 41 | . 5320 |. 4680 .S312 7754 | .6487 || 0689 i. 6445 3555 19 42 5347 | . 4652 .8290 . . '7787 .6464 .0690 . 6456 3544 18 43 : . 5375 | . 4625 .8269 | . TS20 .6441 . .0691 6466 3534 17 44 . 5402 . . 4598 .8247 7853 || 6418 .0692 |. 6476 |. 3524 || 16 45 .35429 .64571 2.8225 .37887 2.6394 | 1,0694 | .06486 | .93513 || 15 46 || ". 5456 |. 4544 .8204 7920 .6371 | .0695 |. 6497 3503 || 14 47 | . 5483 |. 4516 .8182 | . T953 .6348 .0696 6507 3493 || 13 48 || . 5511 |. 4489 .8160 | . T986 .6325 | .0697 |. 65.17 34S2 | 12 49 | . 553S | . 4462 .8139 8()2() .6302 ) .0698 6528 |. 3472 || 11 50 | .35565 .64435 || 2,8117 | .38053 || 2.6279 | 1.0699 ||.06538 .93462 || 10 51 | . 5592 | . 4408 .8096 S086 | .6256 .0701 6548 3451 9 52 | . 5619 | . 4380 .8074 |. 8120 ! .6233 .0702 6559 . .3441 || 8 53 | . 5647 |. 4353 .8053 .. 8153 .6210 .0703 6569 3431 || 7 54 | . 5674 |. 4326 ,8032 | . 81S6 .6.187 | .0704 65.79 3420 || 6 55 .35701 .64299 || 2.8010 .38220 2.6164 | 1.07.05 |.06590 .93410 || 5 56 | . 5728 . 4272 .7989 . 8253 .6142 | .07 07 6600 3400 4 57 | . 5755 4245 .7968 . 82S6 | .6119 .07.08 |. 6611 3389 || 3 58 . 5782 |. 4217 '7947 | . 8320 .6096 .0709 (3621 3379 || 2 Ö9 . . 5S10 | . 4190 .7925 | . 8353 .6073 | .0710 i. 6631 | . 3368 || 1 60 . 5837 | . 4163 7904 | . 83S6 || 6051 | .0711 |. 6642 3358 || 0 M Cosine. Wrs. Sin." Secante. Cotang. Tangent. | Cosec’nt Wrs. Cos | Sime. M 1100 Natural. 699 NATURAL LINES. 21o Natural Trigonometrical Functions. 1589]10h M Sine. Wrs. Cos. Cosco'nte | Tang, | Cotang. Seeante. Wrs. Sinj Cosine. M. M.S. 0 | .35837 .64163 || 2.7904 . .38386 || 2.6051 | 1.0711 |.06642 |.93358 || 60 || 36 1 | . 5864 |. 4136 || 7883 |. 8420 | .6028 .0713 |. 6652 |. 3348 || 59 || 56 2 . . 5891 |. 4109 || 7862 i. 8453 .6006 . .0714 I. 6663 |. 3337 || 58 || 52 3 |. 5918 |. 4082 || 7841 |. 8486 | .5983 || 0715 |. 6673 |. 3327 | 57 || 48 4 | *5945 |. 4055 .7820 |. 8520 | .5960 | .0716 |. 6684 i. 3316 ſ 56 || 44 5 ! .35972 .64027 | 2.7799 .38553 2.5938 | 1.0717 l.06694 93306 || 55 -40 6 6000 | . 4000 .7778 . 8587 | .5916 .0719 |. 6705 |. 3295 || 54 || 36 7 |. 6027 |. 3973 .7757 |. 8620 | .5893 | .0720 |. 6715 1. 3285 || 53 || 32 8 . . 6054 |. 3946 || 7736 |. 8654] .5871 .0721 1. 6726 |. 3274 || 52 28 9 |. 6081 |. 3919 || 7715 |. 8687 | .5848 || 0722 |. 6736 |. 3264 || 51 || 24 10 | .36108 .63892 || 2.7694 | .38720 2.5826 | 1,0723 |.06747 |.93253 || 50 20 11 | . 6135 |. 3865 | .7674 |. 8754 .5S04 | .0725 1. 6757 |. 3243 || 49 | 16 12 | . 6162 |. 3837 . .7653 |. 8787 | .5781 | .0726 |. 6768 |. 3232 || 48 || 12 13 | . 6189 |. 3810 | .7632 |. 8821 | .5759 .07.27 l. 6778 |. 3222 || 47 || 8 14 |. 6217 i. 3783 7611 |. 8854 .5737 || 0728 |. 6789 |. 3211 || 46 || 4 15 .36244 |.63756 2.7591 .38888 2.5715 | 1.0729 || 06799 || 93.201 || 45 || 35 16 |. 6271 |. 3729 || 7570 |. 8921 | .5693 || 0731 |. 6810 |. 3190 || 44 || 56 17 | . 6298 |. 37.02 . .7550 | . 8955 .5671 .0732 I. 6820 |. 3180 || 43 || 52 18 |. 6325 . 3675 ! .7529 |. S988 | .5649 | .0733 |. 6831 |. 3169 || 42 || 48 19 |. 6352 |. 3648 | .7509 |. 9022 .5627 | .0734 |. 6841 - 3158 || 41 || 44 20 | .36379 i.63621 2.7488 .39055 || 2.5605 | 1.0736 |.06S52 | .93148 I 40 | 40 21 . 6406 |. 3593 || 7468 . 9089 | .5583 || 0737 l. 6863 |. 3137 || 39 || 36 22 | . 6433 |. 3566 || 7447 l. 9122 | .5561 | .0738 l. 6873 |. 3127 | 38 || 32 23 . 6460 |. 3539 || 7427 | . 9156 .553 .073 6SS4 . 3116 || 37 28 24 . 6488 |. 3512 || 7406 |. 9189 .5517 | .0740 P. 6894 |. 3105 || 36 || 24 25 .36515 . .634S5 2.7386 .39223 l 2.5495 | 1,0742 ,06905 .93095 || 35 | 20 26 | . 6542 |. 3458 . .7366 |. 9257 | .5473 | .0743 |. 6916 |. 3084 || 34 || 16 27 | . 6569 |. 34.31 | .7346 | . 9290 . .5451 | .0744 I. 6926 3074 || 33 12 28 . 6596 , 3404 || 7325 | . 9324 .5430 .0745 . 6937 | . 3063 || 32 || 8 29 | . 6023 |. 3377 || 7305 |. 9357 | .5408 || 0747 l. 6947 |. 3052 || 3L || 4 30 .30650 |.63350 || 2.72S5 | .39391 || 2.5386 | 1.0748 .06958 . .93042 || 30 || 34 31 . 6677 | . 3323 .7265 . 9425 | .5365 | .0749 |. 6969 |. 3031 29 || 5 32 . 6704 |. 329(3 | .7245 . 9458 .5343 | .0750 |. 6979 |. 3020 || 28 52 33 . 673L |. 3269 .7225 | . 9492 .5322 .0751 |. 6990 |. 3010 27 || 48 # .3158 .3242 || 7205 |...}525 5300 | . .9753 ||. T001 | .2999 || 23 ||34 35 | .36785 .63214 || 2.7185 | .39559 || 2.5278 | 1,0754 |.07.012 | .92988 || 25 | 40 36 . 6SI2 |. 31.87 | .7165 | . 9593 | .5257 | .0755 |. 7022 | . 2978 || 24 36 37 | . 6839 |. 3160 | .7145 |. 9626 .5236 | .0756 |. 7033 |. 2967 || 23 || 32 38 | . 6866 |. 3133 . .7125 | . 9060 | .5214 | .0758 i. 7044 °. 2956 22 || 28 39 | . 6893 |. 3106 .7.105 | . 9694 | .5193 | .0759 |. 7054 |. 2945 21 24 40 .36921 .63079 2.7085 .39727 2.5171 | 1.0760 |.07065 .92.935 | 20 20 41 . 6048 |. 3052 || 7065 |. 9761 | .5150 | .0761 |. 7076 |. 2924 19 | 16 42 | . 6975 |. 3025 .7045 . 9795 .5129 .0763 i. 7087 | . 2913 T8 || 12 43 | . TOO2 . . 2998 || 7026 |. 98.28 .5108 || .0764 i. 7097 |. 2902 || 17 || 8 44 | . T029 . , 2971 . .7006 | . 9S62 | .50S6 .0765 |. 7108 |. 2892|| 16 || 4 45 .37056 |.62944 2.6986 .30896 2.5065 | 1.0766 |.07119 |.92881 || 15 33 46 . 7083 . 2917 | .6967 | . 9930 .5 .0768 . . 7130 | . 28.70 || 14 56 47 | . TIT0 | . 2890 :6947 | . 9963 | .5023 .0769 |. 7141 |. 2859 || 13 || 52 48 | . 7137 i. 2863 .6927 | . 9997 | .5002 | .0770 |. 7151 | . 2848 || 12 || 48 49 . 7164 |. 2836 .6908 || 4003 .49S1 .0771 |. 7162 | . 2838 || 11 || 44 50 .37191 |.62809 || 2.6888 .40065 2.4960 | 1.0773 |.07173 .92827 || 10 || 40 51 | . 7218 . 2782 | .6869 |. 0098 || 4939 .0774 |, 7.184 |. 2816 || 9 || 36 52 . 7245 . 2755 .6849 |. U132 || 4918 . .0775 i. 7195 . 2805 || 8 || 32 53 | . T272 |. 2728 .6830 | . 0166 || 4897 .0776 . 7205 |. 2794 || 7 || 28 54 |. 7299 |. 2701 | .6810 | . 0200 | .4876 . .0778 i. 7216 |. 2784 || 6 || 24 55 .37326 .62674 || 2.6791 .40233 2.4855 1,0779 |.07227 | 927.73 || 5 || 20 56 | . 7353 |. 2047 .6772 | . 0.267 .4834 .0780 . , 7238 |. 2762 || 4 || 16 57 73S0 . 2620 .6752 | . 0301 | .4S13 . .0781 |. 7249 |. 2751 || 3 || 12 58 . 7407 |. 2593 .6733 335 .4792 .0783 |. 7260 . . 2740 || 2 || 8 || 59 | . 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" || 06:10" | 899%. 18396) * | 8L99 || I, " | 899E i; 19 || 98.93, ' | #IgE ' | 6810 | 689;" |}0g) * | *ç99 || 89pg | | Tººl, $ 89 969% ' || 3:03, ' || 8810 || 60liſ' | Off() * | 9999 || 68% #[gſ, " || 3. 69 || 101Z Z621 1820 || 08/9' I figF6) * | g 199" | ZIgz, " | 88%? H 09 || 8LLZ6 Z8310 || $810'I Iglp'º, £0.70%" | 9699", 63;29 1941.9 || 0 JW J ‘auſsoo juſS ‘sa Ai’aºutoos ‘āuru)00 || 3nts), Hayu,22soo’soo’s. A ‘oulS R o/9I "suoyºountſ reogrgououſe31.1.1, relinqu N. oº:3 "S&NIT TVäſliy N. 232 *~ *. ** NATURAL LINEs. * ~. -- * * * * * ~ ** -º- **** -...-- Ih 230 Natural Trigonometicical Functions. 1569||16th M.S. M | Sine. Wrs. Cos. Cosec'nte Tang. | Cotang. Secante, Vrs. Sin Cosime. Mſ M.S. 32 || 0 || 39073 || 60927 | 2.5593 || 42447 || 2.3558 | 1.0864 |.07949 | .92050 || 60 28 4 || 1 | . 9100 . 0900 || 5575 |. 2482 .3539 .0865 i. 7961 | . 2039 || 59 56 8 || 2 | . 91.26 . 0873 || 5558 |. 25.16 || 3520 | .0866 |. 7972 |. 2028 || 58 || 52 12 || 3 | . 9153 | . OS46 .5540 |. 2550 | .3501 | .0868 |. 7984 |. 2016 || 57 || 48 16 || 4 | . 9180 | . 0820 | .5523 |. 2585 .3482 | .0869 |. 7995 . 2005 || 56 || 44 20 || 5 || 39.207 |.60793 2.5506 | .42619 2.3463 | 1.0870 ||.08006 | .91993 || 55 40 24 || 6 | . 9234 . O766 5488 . 2654 .3445 .0872 |. 8018 |. 1982 || 54 || 36 28 || 7 | . 9260 . . ()739 .5471 . 2688 .3426 | .0873 . 8029 |. 1971 || 53 || 32 32 || 8 | . 92S7 | . OT13 || 5453 |. 2722 | .3407 | .0874 |. 8041 |. 1959 || 52 || 28 36 || 9 | . 9314 | . 0686 .5436 . 5757 .3388 .9876 . . 8052 | . 1948 || 51 24 40 || 10 | .3934} | .60659 || 2.54H9 .427.9L 2.3369 1.0877 |.08063 .91936 || 50 20 44 || 11 . 9367 . 0632 | .5402 | . 2826 .3350 | .0878 |. 80.75 | . 1925 || 49 16 48 12 . 9394 . 0606 || 5384 |. 2S60 [ .3332 .0880 H. 8086 |. 1913 || 48 12 52 || 13 | . 942} | . 0579 .5367 | . 2894 .33E3 .0881 . 8098 | . 1902 || 47 || 8 # 56 || 14 | . 9448 |. 0552 || 5350 |. 2929 3294 | .0882 |. 8109 |. 1891 || 46 || 4 33 || 15 | .39474 .60526 2.5333 .42963 || 2.3276 | 1.0884 |.08121 .91879 || 45 || 27 4 || 16 . 950} | . ()499 || .5316 |. 2998 || .3257 ,0885 |. 8132 | . 1868 44 || 56 8 || 17 | . 9528 . 0452 .5299 |. 3032 .3238 | .0886 i. 8+44 , 1856 || 43 52 12 | 18 . 9554 |. 0445 | .5281 |. 3067 .3220 | .0888 |. 8].55 |. 1845 || 42 || 48 16 f{} | . 9581 | . ()419 || 5264 |. 3101 | .3201 .0889 |. 8167 | . 1833 || 41 || 44 20 20 ! .39608 ||.60392 || 2.5247 .43136 2.3183 1.0891 |.08478 .91822 || 40 40 24 21 | . 96.35 | . 0365 .5230 |. 31.70 | .3164 .9892 |. 81.90 | . 1810 || 39 || 36 28 22 | . 9661 . 03:39 || 52.13 | . 3205 .3145 | .6893 |. S.20% | . 1798 || 38 || 32 32 23 . 9688 |. 0312 .5196 |. 3239 .3127 .0895 |. 8313 |. 1787 37 || 28 36 24 | . 9715 | . 0285 .51.79 |. 3274 .3109 .0896 | . 8224 . 1775 || 36 || 24 40 || 25 | .3974} | .60258 || 2.5163 | .43368 || 2:3090 | 1.0897 | .08.236 .91764 || 35 | 20 44 26 | . 9768 . . 0232 || 5146 |. 3343 .3072 .0899 |. 8248 |. 1752 34 16 48 || 27 | . 97.95 | . 0205 || 51.29 |. 3377 .3053 || 0900 |. 8259 |, 1744 || 33 || 12 52 28 || 982} | . 0178 .5112 | . 3412 .3035 .09.02 . . 827} | . I'729 || 32 || 8 56 29 . 9S48 |. O152 | .5095 |. 34.47 | .3017 | .0903 . 8282 . 1718 || 31 || 4 34 || 30 .39875 .60.125 2.50T8 .43481 2.2998 || 1.0904 ||.08294 .91706 || 30 || 26 4 || 31 | . 990} | . 0.098 .5062 |. 35.16 .2980 | .0906 |. 8306 | . 1694 || 29 || 56 8 || 32 . 9928 |. 0072 .5045 |. 3550 | .2962 .0907 |. 8317 | . 1683 || 28 52 12 || 33 . 9955 |. 0045 .5028 |. 3585 .2944 .0908 |. 8329 |. 1674 27 || 48 16 || 34 . 9981 . 0018 .50.11 . . 3620 | .2925 | .0910 i. 8340 | . 1659 || 26 || 44 20 35 | .40008 || 59992 || 2.4995 | .43654 2.2907 | 1.09F1 | .08352 .91648 25 | 40 24 || 36 . . 0035 | . 9965 4978 |. 3689 .2889 .0913 |. 8364 | . 1636 24 || 36 28 37 . 006] | . 9938 . .4961 | . 3723 .287.1 .09:14 |. 8375 | . 1625 || 23 32 32 || 38 | . 00S8 |. 9912 || 4945 | . 3758 .2853 .0915 . . 8387 | . 1613 || 22 || 28 36 || 39 || . 0115 , 9S85 || 4928 . 3793 .2835 .0917 | . 8399 | . 1604 || 3 || || 24 40 | 40 .40141 .59858 || 2.4912 | .43827 | 2.2817 | 1.0918 . .684.10 | .94590 | 20 | 20 44 || 4t | . 0168 |. 9832 .4S95 |. 3862 .2799 || 0920 i. 8422 | . 1578 19 || 16 48 || 42 . O195 | . 980.5 ! .4879 . 3897 .2781 .0921 |. 8434 | . 1566 18 12 52 || 43 | . 0221 | . 9778 .4S62 |. 3932 . .2763 | .0922 i. 8445 | . 1554 || 17 || 8 56 44 . 0248 |. 97.52 || 4846 | . 3966 . .2745 | .09:24 |. 8457 | . I543 | 16 || 4 35 || 45 .40275 59.725 || 2.4829 .44001 2.2727 | 1.09.25 |.08469 |.9].531 || 15 || 25 4 || 46 | . 0301 | . 9699 || 4813 | . 4036 .2709 ,0927 |. 8480 | . 1519 || 14 || 56 8 || 47 | . 3328 |. 9672 | .4797 | . 40.70 | .2691 .0928 |. 8492 | . 1508 || 13 || 52 12 || 48 . 0354 |. 96.45 .4780 . . 4105 .2673 | .0929 |. 8504 || - 1496 || 12 || 48 16 || 49 . 0381 | . 96.19 .4764 . 4140 [ .2655 | .0931 i. 8516 |. 1484 || 11 || 44 20 || 50 | 40408 || 59592 || 2.4748 || 44.175 2.2637 1.0932 || 0852'ſ .91472 10 || 40 24 || 51 . (434 |. 9566 || 4731 |. 4209 .2619 .0934 |. 8539 . 1461 || 9 || 36 28 || 52 : . ()46} | . 9539 || .4715 . 4244 .2602 | .0935 | . 8551 | . 1449 || 8 || 32 32 || 53 | . 0487 . 9512 .4699 . 4279 .2584 .0936 . . 8563 | . 1437 i 7 || 28 36 || 54 | . 0514 | . 9486 | .4683 |. 4314 | .2566 | .0938 . . 8575 | . 1425 || 6 || 24 40 55 .40541 |.59469 2.4666 | .44349 2.2548 || 1.0939 |.08586 | .91.414 || 5 || 20 44 || 56 | . ()567 . 9433 .4650 | . 4383 . .2531 .0941 |. 8598 | . 1402 || 4 || 16 48 57 . 0504 |. 9406 .4634 |. 4418 || .2513 | .6942 i. 86.10 | . 1390 || 3 | }.2 52 58 . . 0620 | . 93.79 | .4618 |. 4453 .2495 .0943 |. 8622 | . 1378 || 2 || 8 56 || 59 . . 0547 | . $353 .4602 |. 4488 . .2478 .0945 |. 8634 |. 1366 || 1 || 4 36 || 60 | . ()674 . 9326 | .45S6 . 4523 .246() ,0946 |. 8645 | . 1354 || 0 |24. M. S. M I Cosime. Wrs.Sin. Secante, Cotang. Tangent, Cosec'nt tWrs. Cosl Sine. M. M.S. 7h ||1139 Natural. 66° 4h *. . . . . . . . . NATURAL LINEs. . . 233 1h 1249 Natural Trigonometrical Functions. 155° 10h M.S. M Sine. Wrs. Cos. Cosec'nte | Tang. Cotang. Secante. Wrs. Sin Cosine. Mſ M.S. 36 || 0 | .40674 |.59326 || 2.4586 | .44523 2.2460 | 1.0946 |.08645 |.91354 60 l 24 4 || 1 | . 0700 |. 9300 .4570 |. 4558 . .2443 | .0948 |. 8657 |. 1343 59 || 56 8 || 2 | . 0727 |. 9273 | .4554 |. 4593] .2425 | .0949 |. 8669 |. 1331 || 58 || 52 12 || 3 | . 0753 |. 9247 | .4538 |. 4627 | .2408 .0951 |. 8681 |. 1319||57 || 48 16 || 4 | . (780 |. 9220 | .4522 |. 4662 .2390 .0952 . 8693 . 1307 || 56 44 20 || 5 || 40806 |.59193 || 2.4506 .44697 2.2373 | 1.0953 |.08705 || 91295 || 55 | 40 24 || 6 | . 0833 |. 9167 | .4490 |. 4732 | .2355 .0955 |. 8716 |. 1283 || 54 || 36 28 || 7 | . 0860 |. 9140 | .4474 |. 4767 | .2338 .0956 i. 8728 |. 1271 || 53 || 32 32 || 8 | . 0886 |. 9114 | .4458 |. 4802 | .2320 | .0958 |. 8740 |. 1260 || 52 || 28 86 9 0.913 | . 9087 .4442 | . 483 .2303 | .0959 |. 8752 . 1248 || 51 || 24 40 || 10 .40939 .59061 2,4426 .44872 || 2.2286 | 1.0961 | .08764 .91236 || 50 | 20 44 || 11 0966 . 9034 .4411 | . 4907 | .2268 .0962 |. 8776 | . I224 || 49 || 16 48 || 12 . 0992 | . 9008 .4395 | . 4942 .2251 | .0963 |. 8788 |. 1212 || 48 || 12 52 || 13 1019 |. S081 .4379 . 4977 | .2234 .0965 |. 8800 | . 1200 || 47 || 8 56 || 14 1045 . 8955 .4363 |. 5012 | .2216 .096 8812 | . 1188 || 46 || 4 37 || 15 .41072 .58928 || 2.4347 | .45047 || 2.2199 || 1.0968 |.08824 | .91176 || 45 || 23 , 4 || 16 1098 . 8901. .4332 |. 5082 .2182 .0969 |. 8836 |. 1164 44 || 56 . 8 || || ||25 |. §§ 4316 || 5 iſ 3155 ºil || $43 |. iii.2|33 || 52 12 || 18 1151 . S848 . .4300 5152 .2147 .0972 . . 8860 | . 1140 || 42 || 48 16 || 19 1178 . 8822 || .4285 5187 | .2130 | .0973 |. 8872 | . 1128 || 41 || 44 20 20 ! .41204 | .58795 || 2.4269 .45222 || 2.2113 | 1.0975 |.08884 | .91116 || 40 | 40 24 || 21 | . 1231 |. 8769 || 4254 5257 .2096 | .0976 | . S896 | . 1104 || 39 || 36 28 22 | . 1257 | . 8742 .4238 5292 .2079 .0978 . S908 . 1092 || 38 || 32 32 || 23 . 1284 . S716 .4222 5327 | .2062 .0979 |. S920 | . 1080 37 || 28 36 24 I310 | . 8689 || .42()7 5362 .2045 .0981 |. S932 |. 1068 || 36 24 40 || 25 | .4.1337 .5SG63 || 2.4.191 .45397 2.2028 1.0982 |.08944 .91056 || 35 | 20 44 || 26 | . 1363 |. S636 .4176 5432 . .2011 | .0984 |. 8956 |. 1044 || 34 || 16 48 || 27 | . 1390 . 861() .4160 5467 .1994 | .0985 |. 8968 , 1032 || 33 || 12 52 || 28 || . 1416 | . 8584 || .4145 5502 | .1977 09S6 . . 8980 | . 1020 || 32 || 8 56 29 1443 | . 8557 .4130 5537 .196() } .0988 |. 8992 | . 1008 || 31 || 4 38 30 .41409 |.5S531 || 2.4114 | .45573 || 2.1943 | 1,0989 |.09004 | .90996 || 30 22 4 || 31 . 1496 . 8504 .4099 5608 || 1926 .0991 |. 9016 | . U984 || 29 56 8 || 32 1522 |. 8478 | .4083 5643 .1909 .0992 |. 9028 . U972 28 || 52 12 || 33 | . 1549 i. 8451 | .4068 5678 .1892 | .0994 | . 9040 | . 0960 27 || 48 16 || 34 1575 | . 8425 | .4053 | . 5713 | .1875 | .0995 |. 9052 | . 0948 26 || 44 20 || 35 | .41602 | .58398 || 2.4037 .45748 || 2.1859 | 1,0997 |.09064 .90936 || 25 | 40 24 || 36 1628 |. 8372 .4022 5783 | .1842 | .0998 |. 9076 . ()924 || 24 || 36 28 || 37 . 1654 |. 8345 | .4007 5819 | .1825 .100() |. 9088 . 0.911 || 23 32 32 38 . 1681 | . 8319 .3992 | . 5854 | .1808 .1001 9101 0899 || 22 || 28 36 || 39 1707 |. 8292 | .3976 5889 | .1792 || 1003 |. 9113 | . 0887 || 21 || 24 40 || 40 | .41734 |.58266 2,3961 | .45924 2.1775 | 1.1004 | .09125 | .90875 | 20 | 20 44 || 41 1760 . 8240 .3946 5960 | .1758 .1005 |. 9137 | . 0.863 || 19 || 16 . 48 # 42 | . 1787 | . 8213 .3931 5995 .1741 | .1007 | . 9149 | . 0851 | 18 || 12 52 || 43 . 1813 | . 8.187 .3916 | . 6030 .1725 | .1008 |. 916.1 | . 0839 || 17 || 8 | 56 || 44 | . 1839 |. 8160 | .3901 6065 .1708 || 1010 i. 91.73 . 0.826 || 16 || 4 39 || 45 | .41866 .58134 2.3886 | .46101 || 2.1692 | 1.1011 | .091.86 | .90814 || 15 21 4 || 46 | . 1892 | . 8108 || .3871 6136 .1675 || 1013 |. 9198. . 0802 || 14 || 56 8 || 47 . 1919 | . 8081 | .3856 | . 6171 .1658 . .1014 |. 9210 | . (790 || 13 || 52 12 || 48 | . 1945 | . 8055 | .3841 | . 6206 | .1642 | .1016 |. 9222 | . 0.778 || 12 || 48 16 || 49 1972 . 8028 .3826 , 6.242 .1625 | 1017 | . 9234 | . ()765 || 11 || 44 20 50 | .41998 || .58002 || 2.3811 .46277 || 2.1609 | 1.1019 |.09247 | .90753 || 10 || 40 24 || 51 2024 | . T975 | .3796 | . 6312 .1592 .192() . 9259 . 07:41 || 9 || 36 . 28 || 52 . 2051 | . T949 .3781 |. 6348 .1576 | .1022 |. 9271 . 0729 || 8 || 32 32 53 | . 2077 . 7923 | .3766 | . 6383 . .1559 1023 |. 92S3 | . 0717 || 7 || 2S 36 || 54 2103 | . T896 || .3751 . 6418 .1 .1025 H. 9296 704 || 6 || 24 40 55 .42130 .578.70 || 2.3736 .46454 2.1527 | 1.1026 .09308 .90692 || 5 || 20 44 || 56 2156 . 7844 .3721 | . 6489 | .1510 | .1028 |. 93.20 | . 0680 || 4 || 16 48 || 57 | . 2183 . 7817 .3706 6524 .1494 | .1929 i. 9332 | . 0668 || 3 || 12 52 || 58 | . 2209 . 7791 .3691 6560 | .1478 || 1031 |. 9345 | . 0655 || 2 || 8 56 59 . . 2235 | . TT64 .3677 | . 6595 .1461 | .1032 |. 9357 | . 0643 || 1 || 4 4-0 || 60 | . 2262 | . TT38 | .3662 | . 6631 .1445 .1934 |. 9369 |. 0631 || 0 || 20 M. S. M I Cosime. Wrs. Sin. Secante. | Cotang. Tangent. |Cosec'mt Wrs. Cos] Sine. M M.S., ..] 7h ||1149 Natural. 65° 4 NATURAT, LINES. 1h |250 Natural Trigonometrical Functions. 1549 || 10h M.S. M Sine. Wrs. Cos. Cosec'nte Tang. Cotang. Secante. Vrs. Sinj Cosine. M. M. S. 40 || 0 || 42262 .57738 || 2.3662 | .46631 2.1445 | 1.1034 .09369 |.90631 || 60 20 4 || 1 | . 228S . 7712 | .3647 . 6666 .1429 .1035 | . 9381 . 0618 59 56 8 || 2 | . 2314 . 7685 .3632 6702 | .1412 || 1037 |. 9394 | . 0606 || 58 52 12 || 3 | . 234] | . T659 . .3618 . 6737 .1396 || 1038 |. 9406 | . 0594 i 57 48 L6 || 4 || . 2367 . 7633 . .3603 6772 .1380 | 1040 l. 9418 . 05S1 || 56 || 44 20 || 5 | .42394 | .57606 || 2.3588 .46808 || 2.1364 | 1.1041 .09.431 | .90569 || 55 || 40 24 || 6 2420 . 7580 .3574 |, 6843 | .1348 || 1043 |. 94.43 | . 0557 || 54 36 28 7 2446 | . T554 .3559 6879 .1331 || 1044 |. 9455 |. 0544 53 || 32 32 || 8 2473 | . T527 | .3544 | . 6914 | .1315 .1046 |. 9468 . 0532 || 52 || 28 36 9 2499 |. 7501 | .3530 6950 | .1299 || 1047 l. 94.80 | . 0520 || 51 24 40 || 10 | .42525 .57475 2.3515 . .46985 || 2.1283 | 1.1049 |,09492 .90507 || 50 || 20 44 || 11 | . 2552 |. 7448 .3501 7021 .1267 .105() |. 9505 | . 0495 || 49 || 16 48 || 12 257S | . 7422 .3486 7056 .1251 | .1052 . 9517 | . 0483 || 48 || 12 52 || 13 2604 | . T396 .3472 7()92 | .1235 | .1053 i. 9530 | . 047() || 4-7 || 8 56 14 263() | . Tà69 .3457 7127 .1219 .1055 | . 9542 . 0458 I 46 4 41 || 15 .42657 .57343 || 2.3443 .47163 2.1203 | 1.1056 .09554 .904:45 || 45 19 4 || 16 26S3 . 7317 .3428 |. 7199 .1187 | .105S . 9567 | . ()433 44 || 56 S 17 2709 | . T290 . .3414 7234 .1171 .1059 . 9579 . 0421 || 43 || 52 12 || 18 2736 . 7264 .3399 72.70 | .1155 | .1061 | . 9592 | . ()408 || 42 || 48 16 || 19 27.62 | 723S .33S5 7303 .1139 .1062 |. 96.04 | . 0396 || 41 || 44 20 | 20 | .42788 |.57212 || 2.3371 .47341| 2.1123 | 1.1064 |.09617 | .90383 || 40 | 40 24 || 21 | . 2815 . 7185 | .3356 7376 .1107 | .1065 |. 9629 |. 0371 39 || 36 28 || 22 | . 2841 . 7159 .3342 7412 | .1092 .1067 |. 9641 |. 0358 38 || 32 32 || 23 2867 7133 .3328 7448 || 1076 | .106S j. 96.54 | . 0346 || 37 28 36 || 24 2893 | . TI()6 | .3313 | . T483 | .1060 .1070 i. 9666 | . 0.333 || 36 || 24 40 25 | .42920 | .57080 || 2.3299 .47519 || 2.1044 1.1072 | .09679 .90.321 || 35 | 20 44 || 26 2946 | . T054 .3285 7555 . .1028 || 1073 |. 9691 . 0308 || 34 16 48 || 27 2972 | . TO2S .3271 7.590 | .1013 | .1075 |. 97.04 | . 0290 || 33 12 52 || 28 . 2998 . 7001 | .3256 7626 .0997 | .1076 i. 9710 | . 0283 || 32 || 8 56 || 29 3025 | . 6975 | .3242 . 7662 .0981 | .1078 i. 9729 |. 0271 || 31 || 4 4-2 || 30 .43051 .56949 || 2.3228 .47697 || 2.0965 | 1.1079 |.097.41 .90258 || 30 18 4 || 31 | . 3077 |. 6923 | .3214 |. 7733| .0950 | .1081 |. 97.54 |. 0246 29 || 56 S 32 3104 . (SQG | .3200 7769 | .0934 .10S2 i. 9766 | . 0233 || 28 || 52 12 || 33 - 3130 . 6870 3.186 | . TS05 .0918 . .1084 i. 97.79 |. 0221 || 27 A48 16 || 34 3.156 |. 6844 .3172 7840 | .0903 | .1085 . 9792 |. 0208 || 26 || 44 20 || 35 | .43182 .56818 || 2.3158 .47876 || 2.0SS7 | 1.1087 .09804 || 90190 25 || 40 24 || 36 320S | . 6791 .3143 7912 .0872 .1()SS | . 98.17 | . 0183 || 24 || 36 28 || 37 3235 | . 6765 3129 7948 || .0856 .1090 . 98.29 | . ()171 || 23 || 32 32 || 38 | . 3261 | . 6739 31.15 7983 .0840 | .1092 |. 9842 | . 0158 22 || 28 36 || 39 . 3287 | . 6713 | .3101 8019 .0825 | .1093 . . 9854 . 0145 || 21 || 24 40 40 | .43313 | .56GS6 || 2.3087 .48055 2.0809 | 1.1095 |.09867 .901.33 || 20 20 44 || 41 | . 3340 ! .. 6660 | .3073 |. 8091. .0794 | .1096 |. 9880 |. 0120 | 19 | 16 48 || 42 | . 3366 | . 6634 .3059 |. 8127 | .0778 .1098 |. 9892 | . 0108 18 || 12 52 43 : . 3392 | . 6608 .3046 8162 | .0763 .1099 ||. 9905 | . 0.095 || 17 || 8 56 || 44 | . 3418 , 6582 .3032 8198 || 0747 .1101 |. 0.917 | . 0082 | 16 || 4 43 45 .43444 .56555 || 2.3018 || 48234; 2.0732 | 1.1102 .09930 | .900.70 | 15 || 17 4 46 3471 . 65.29 .3004 82.70 | .0717 | .1104 |. 99.43 | : 0057 || 14 || 56 S || 4-7 | . 3497 . 6503 .2990 8306 .0701 | .1106 | . 9955 . 0044 || 13 || 52 12 || 48 . 3523 |. 64.77 .2976 | . 8342 . .0686 | .1107 . 99.68 | . 0.032 || 12 || 48 16 || 49 3549 | . 6451 .2962 8378 . .0671 .1109 |. 9981 | . 00I9 || 11 || 44 20 50 | .43575 .56424 || 2.2949 .484.14 2.0655 | 1.1110 |.09993 .90006 || 10 || 40 24 || 51 | . 3602 |. 6398 || .2935 | . 8449 || 0640 | .1112 | .10006 | .89994 || 9 || 36 2S 52 . 3628 . 6372 2921 S485 | .0625 | .1113 |. 0019 | . 9081 || 8 || 32 32 || 53 3654 | . 6346 2907 8521 .0609 | .1115 |. ()031 | . 9968 || 7 || 28 36 || 54 . 3680 | . 6320 ! .2894 8557 . .0594 | .1116 . 004.4 | . 9956 || 6 || 24 40 || 55 .43706 | .56294 || 2.2880 | .4S593 || 2.0579 | 1.1118 .10057 .899.43 || 5 || 20 44 || 56 | . 3732 |. 6267 | .2866 | . 86.29 .0564 .1120 |. 0070 | . 993 4 || 16 48 || 57 | . 3759 . 6241 | .2853 8665 .0548 .I.1.21 |. 00S2 . 9018 || 3 || 12 52 58 | . 3785 |. 6215 .2S39 |. 8701 | .0533 | .1123 . 0096 |. 9905 || 2 || 8 56 59 |. 3S11 |. 6189 | .2S25 |. 8737 | .0518 .1124 |. 0108 |. 9892 || 1 || 4 - 60 3837 | . 6163 .2812 8773 | .0503 | .1126 |. O121 | . 987.9 || 0 | 16 M. S. M I Cosine. Vrs. Sin. Secante. Cotang. Tangent. Cosec'nt IVrs. Cos Sine. M M.S. 7h 1159 Natural. 64° 4h NATURAL LINES. Natural Trigonometrical Functions. 153° . 5036 . 5062 . 5088 . 5114 .45140 . 5166 . 5.191 . 5217 . 5243 .45269 . 5295 . 5321 . 5347 . 5373 . 5399 gosine. Wrs. Cos. ,56163 . 47S2 , 4756 ,54730 . 4705 . 46.79 . 4653 . 4627 . 4601 Wrs. Sin. Cosec'nte 2,2812 .2798 .2784 .2771 .2757 2.2744 .2730 ,2717 .2703 .2690 2.2676 .2663 .2650 .2636 .2623 2.2610 .2596 .25S3 .2570 .2556 2.2543 .2530 .2517 .2503 .2490 2.2477 .2464 .2451 .2438 .2425 2.2411 .2398 .2385 .237.2 2.2153 .2141 .2128 .2115 ,2103 2.2090 .2077 .2065 .2052 .2039 .2027 Sccante. Tang. .4S773 . 8809 . 8845 . S881 . S917 .4895.3 . 8989 , 9025 . 9062 . 909S .49134 . 91.70 . 9206 . 9242 , 927S .49314 . 9351 . 93S7 . 9423 . 9459 .49495 . 9532 . 9568 . 960.4 . 9640 .49677 . 9713 . 9749 . 97S5 .49858 . 9894 . 9931 . 9967 .50003 .500.40 . 0076 . 0113 . 01:49 . 0185 ,50222 . 0258 . ()295 . 0331 . 0368 .50404 . 04:41 . 0477 . 0514 . 055() .50587 . 0623 . 0660 . ()696 . OT33 ,50769 . 0806 . 0843 . OS79 . 0916 . Q952 Oottung. Cotang. 2.0503 .0488 .0263 .0248 .0233 .02.19 2.0204 .0189 .0174 .0159 .014.5 2.013() .01.15 .01.01 .0(186 .0071 2.005.7 .0042 .0028 .()013 1,9998 I.99.84 .9969 .9955 Tangent. Natural. Secante. .1217 .1218 .1220 .1222 .12.23 Cosec'mt Vrs. Sin .10121 . 0133 . 0146 . 0159 . 0172 .10184 . 0.197 . Q210 . ()223 . 0236 .10248 . ()261 . 0274 , ()2S7 0300 .10313 . 0326 . 0338 . 0351 |.. 0.364 10377 . 0390 Cosine. "Sg879 , 9867 . 9854 . 984.1 . 98.28 .S9815 . 9803 . 9790 . 9777 . 9764 .89751 . 97.39 . 97.26 . 9713 . 9700 .89687 6 236 NATURAL LINEs. *kº. -- -º-ºº--- ** 1 h 2 7 O Natural Trigonometrical Functions. 1529 Sine. [Vrs. Cos. Cosec'nte | Tang. Cotang. Secante. Vrs. Sinj Cosime. M ,45399 || 54601 || 2.2027 | .50952 | 1.9626 | 1.1223 |.10899 || 89101 || 60 . 5425 | . 4575 .2014 | . 09SQ | .961.2 . .1225 |. 09.12 | . 9087 || 59 . 5451 | . 4549 .2002 |. 1026 .9598 || .1226 |. 0926 | . 9074 || 58 . 5477 | . 4523 .1989 |. 1062 .9584 .1228 |. 0939 e 9061 || 57 . 5503 | . 4497 .1977 | . 1099 || .9570 .1230 l. (952 . . 9048 56 .45528 || 54.471 || 2.1964 .51136|| 1.9556 | 1.1231 || 10965 | 89034 || 55 . 5554 . 4445 .1952 | . 1172 . .9542 .1233 |. 0979 . 9021 || 54 . 55S0 | . 4420 .1939 | . T209 | .952S | .1235 | . 0992 | . 9008 || 53 . 5606 . . 4394 .1927 | . 1246 .9514 | .1237 . 1005 | . 8995 || 52 . 5632 | . 4368 .1914 | . 1283 .9500 | .1238 |. 1018 |. 8981 || 51 40 # 10 .45658 . .54342 2.1902 | .51319|| 1.9486 || 1.1240 .11032 .88968 || 50 44 || 11 | . 5684 | . .4316 .1889 |. 1356 .9472 .1242 |. 1045 . 8955 || 49 48 || 1:2 . . 5710 |. 4290 .1877 | . 1393 || .9458 | .1243 |. 1058 . 8942 || 48 52 || 13 | . 57.36 . 4264 .1865 | . 1430 | .94.44 | .1245 i. 1072 |. 8928 i47 56 || 14 | . 5761 |. 4238 .1852 | . 1466 .9430 .1247 |. 1085 S915 || 46 49 || 15 | .45787 | .54213 || 2.1840 | .51503 || 1.9416 | 1.1248 4 || 16 . . 5813 . 4187 .1828 . 1540 .9402 | .1250 |. 1112 |. 8888 || 44 T252 & 1253 1. 6 T 8 || 17 | . 5839 . 4161 .1815 . 1577 | .9388 12 || 18 || . 5865 . 4135 .1803 |. 1614 | .9375 16 || 19 . 5891 . 4109 .1791 |. 1651 .9361 | .1255 . 1152 . 8848 || 41 20 20 | .45917 | .540S3 2.1778 .51687 | 1.9347 | 1.1257 .11165 .88835 | 40 24 21 | . 5942 | . 4057 | .1766 |. 1724 . .9333 .1258 |. 1178 |. 8822 || 39 28 22 | . 5968 . 4032 | .1754 . 1761 .9319 .1260 . . TT92 | . 8808 || 38 32 || 23 . 5994 | . 4006 | .1742 . 1798 .9306 .1262 |. 1205 |. 8795 || 37 36 24 | . 6020 |. 3980 .1730 |. 1835 | .9292 | .1264 |. 1218 |. 87S1 || 36 40 || 25 | .46046 .53954 2.1717 | .51872 | 1.9278 1.1265 |.11232 |.88768 || 35 44 26 | . 6072 |. 3928 .1705 | . 1909 . .9264 .1267 i. 1245 . 8755 34 48 || 27 . . 6097 |. 3902 . .1693 . 1946 . .9251 | .1269 |. 1259 |. 8741 || 33 1; is . . §§g|. §§1 | idºl |. 2724) #667 iñº || 1543 || $453||13 16 || 49 . 6664 |. 3336 | .1430 |. 2761 | 8953 .1306 |. 1555 . . 8444 || 11 20 50 | .46690 .53310 || 2.1418 .52798 || 1.8940 I.1308 ||.T1569 |.88431 || 10 -i;ii|; Oh 24 || 51 . 6716 | . 3284 .1406 | . 2836 .8927 | .1310 |. 1583 |. 8417 9 28 52 | . 6741 | . 3258 .1394 | . 2873 . .8913 | .1312 |. 1596 | . 8404 || 8 32 || 53 | . 6767 .. 3233 .1382 . 2010 | .8900 .1313 |. 1610 | . 8390 7 36 || 54 . 6793 |. 3207 .1371 |. 2947 . .8887 | .1315 |. 1623 | . 8376 || 6 40 || 55 .46819 .53181 2.1359 .52984 || 1.8873 | 1.1317 | 11637 .88363 || 5 44 || 56 | . 6844 |. 3156 .1347 |. 3022 | .8860 | .1319 i. 1651 | . 8349 || 4 48 || 57 . . 68.70 | . 3130 .1335 | . 3059 | .8847 | .1320 |. 1664 | . 8336 || 3 52 || 58 . . 6896 |. 3104 .1324 |. 3096 | .8834 .1322 |. 1678 |. 8322 2 56 || 59 . . 6921 |. 3078 .1312 | . 3134 .8820 .1324 i. 1691 | . 8308 || 1 52 || 60 . 6947 | . 3053 .1300 | . 3171 .8807 || 1326 |. 1705 | . 8295 || 0 M. S. M | Cosine. Wrs. Sin. Secante. Cotung. Tangent. Cosec'ntiVrs. Cos] Sime. .* M. S. 7h [1179 Natural. 62° 4h NATURAL LINES. 237 1h 28° Natural Trigonometrical Functions. 151° 10h M. S. M. Sine. Vrs. Cos. Cosec'nte | Tang. Cotang. [Secante. Wrs. Sin Cosine. M. M. S. 52 || 0 | .46947 .53053 2,1300 .53171 | 1.8807 | 1.1326 |.11705 .88295 || 60 | 8 4 1. 6973 .. 3027 .1289 |. 3208 .8794 | .1327 |. 1719 |. 82S1 || 59 56 8 2 . . 6998 | . 3001 .1277 , 32.45 .8781 .1329 |. 1732 8267 58 || 52 12 3 | . 7024 |. 2976 .1266 | . .3283 | .8768 . .1331 |. 1746 S254 57 || 48 16 4 . 7050 2950 .1254 . . 3320 .8754 .1333 1. 1760 8240 56 || 44 20 5 .47075 | .52924 2.1242 | .53358 1.8741 | 1.1334 |.11774 |.88226 || 55 40 24 6 | . TIO1 . 2899 .1231 |. 3395 || .8728 .1336 |. 1787 | . 8213 || 54 || 36 28 7 | . 7127 2S73 .1219 |. 3432 | .8715 .1338 |. 1801 8.199 || 53 || 32 32 8 . 7152 |. 2847 .1208 || , 34.70 | .8702 .1340 |. 1815 8.185 52 28 36 9 . 7178 2822 .1196 || - 3507 .8689 1341 . , 1828 8171 51 24 40 || 10 | .47204 .52796 || 2.1185 .53545 | 1.8676 | 1.1343 |.11842 |.881.58 || 50 20 44 11 7220 2770 .1173 |. 3582 . .8663 1345 |. 1856 | . 8144 || 49 || 16 48 || 12 | . 7255 2745 .1162 |. 3619 .8650 | .1347 |. 1870 8130 || 48 12 52 || 13 | . 7281 |. 2719 .1150 3657 ,8637 .1349 |. 1883 8117 i 47 8 56 || 14 7.306 2694 .1139 . 3694 .8624 1350 |. 1897 |. 8103 || 46 4 53 || 15 .47332 .52668 || 2.1127 .53732 || 1.8611 | 1.1352 |.11911 | .88089 || 45 || 7 4 || 16 | . 7357 2642 ..1116 | . 3769 | .8598 1354 |. 1925 | . 8075 44 || 56 8 || 17 | . 7383 . . 26.17 .1104 |. 3807 .8585 .1356 |. 1938 |. 8061 || 43 || 52 12 18 7409 | . 2591 .1093 |. 3844 S572 1357 1952 8048 I 42 || 48 16 || 19 |. 7434 |. 2565 .1082 3882 .8559 1359 |. 1966 . 8034 || 41 || 44 20 20 | .47460 .525.40 || 2.1070 .53919 | 1.8546 | 1.1361 |.11980 | .88020 | 40 || 40 24 21 | . 7486 |. 2514 .1059 |. 3957 i .S533 1363 . 1994 8006 || 39 || 36 2S 22 . 7511 . . 2489 .1048 |. 3995 || ,8520 1365 |. 2007 '7992 || 38 || 32 32 || 23 || - 7537 2463 .1036 |. 4032 .8507 | .1366 |. 2021 |. 7979 || 37 28 36 || 24 |. 7562 2437 .1025 | . 4070 .8495 1368 |. 2035 7965 36 24 40 25 .47588 .524.12 || 2.1014 ||.54107 || 1.84.82 | 1.1370 | .12049 .87951 || 35 | 20 44 || 26 . 7613 | . 2386 ..1002 |. 4145 | .8469 .1372 |. 2063 7937 || 34 16 48 || 27 7639 2361 .0991 | . 4.183 .8456 1373 i. 2077 7.923 || 33 12 52 || 28 | . T665 2335 .0980 | . 4220 | .8443 1375 |. 2090 7909 || 32 8 56 29 . 7690 2310 .0969 4258 | .8430 1377 |. 2104 7S95 || 31 4. 54. 30 .47716 | .52284 || 2.0957 | .54295 || 1.8418 | 1.1379 || 12118 .87882 || 30 || 6 4 || 31 | . TT 41 |. 225S .0946 |. 4333 .8405 1381 |. 2 (32 |. 7868 || 29 56 8 || 32 . 7767 |. 2233 .0935 | . 4371 | .8392 1382 |. 2146 7854 - 28 52 12 || 33 7792 | . 2207 .0924 |. 4409 .8379 1384 |. 2160 78.40 || 27 || 48 16 || 34 . 7818 2182 .0912 | . 4446 | .8367 1386 || - 2174 '7826 26 || 44 20 || 35 .47844 |.52156 || 2.0901 | .54484 || 1.8354 | 1.1388 |.12188 |.87812 || 25 | 40 24 || 36 7869 |. 2131 .0890 | . 4522 .8341 1390 | . .2202 7798 || 24 || 36 28 37 | . T895 |. 2105 .0879 . 4559 | .8329 1391 22L6 | . 7784 || 23 || 32 32 || 38 | . 7920 . 2080 .0868 |. 4591 | .8316 .1393 |. 2229 |. 77.70 || 22 28 36 || 39 7.946 2054 .0857 |. 4635 | .8303 1395 ||. 2243 77.56 || 21 || 24 40 || 40 .4797 l .52029 || 2.0846 .54673 || 1.8291 1.1397 ||.12257 | .877.42 20 20 44 || 41 | . 7997 2003 .0835 4711 . .8278 1399 ||. 2271 7728 || 19 || 16 48 || 42 | . 8022 |. 1978 .0824 4748 .8265 1401 ||. 22S5 '7715 18 || 12 52 || 43 | . 8048 . 1952 .0812 786 .8253 1402 ||. 22.99 7701 || 17 S 56 || 44 8073 1927 .0801 4824 .8240 1404 ||. 2313 76S7 16 4 55 || 45 .48099 || .51901 || 2.0790 | .54862 | 1.8227 | 1.1406 .12327 | .87673 || 15 5 4 || 46 j . 8124 1876 .0779 4900 .8215 1408 |. 2341 7659 14 || 56 8 || 47 8150 | . 1850 .0768 4937 .S202 1410 |. 2355 | . 7645 13 || 52 12 48 S175 |. 1S25 .0757 4975 .8190 1411 |. 2369 '7631 || 12 || 48 16 || 49 8201 | . 1799 .0746 5013 . .8.177 1413 |. 2383 7617 | 11 || 44 20 || 50 .48226 .51774 2.0735 | .55051 | 1.S165 | 1.1415 ||.T.2397 | .87603 || 10 40 24 || 51 | . 8252 |. 1748 .0725 |. 5089 .S152 | .1417 |. 2411 |. 7588 || 9 || 36 2S 52 | . 8277 . I'723 .0714 5127 | .8140 .1419 |. 2425 7574 8 || 32 32 || 53 | . 8.303 | . 1697 .0703 5,165 .8127 | .1421 |. 2439 |. 7560 7 || 28 36 || 54 . 8328 |. 1672 .0692 | . 5203 | .8115 | .1422 |. 2453 |. 7546 || 6 || 24 40 || 55 .48354 || 51646 || 206S1 | .55241 | 1.8102 | 1.1424 || 12468 .87532 || 5 || 20 44 || 56 | . 8379 |. 1621 ,0670 5279 .8090 .1426 . 2482 75IS 4 || 16 48 57 | . 8405 | . 1595 .0659 5317 | .S078 | .1428 . 2496 7504 || 3 || 12 52 58 . 8430 . 1570 .0648 |. 5355 . .8065 .1430 |. 2510 7490 || 2 S 56 || 59 . . 8455 | . 1544 .0637 5393 .8053 .1432 |. 2524 7476 || 1 4 56 || 60 . 8481 . 1519 .0627 | . 5431 .8040 | . . 253S 7462 || 0 || 4 M. S. M | Cosine. Wrs. Sin. Secaute. Cotang. ITangent. | Cosec'nt.Wrs. Cosſ Sine. M. M. S. 7h 1180 Natural. 61° 4's *t NATURAL LINES. 290 Natural Trigonometrical Functions. 150° M Sine. Wrs. Cos. Cosec'nte | Tang. | Cotang. Secante. Vrs. Sin Cosine. M () || 48481 .51519 || 2,0627 | .55431 | 1.8040 | 1.1433 .12538 .87462 || 60 1 | . $500 | . 1493 .0616 | . 5469 .8028 .1435 | . 2552 . 744S 59 2 . . 8532 . 1468 .0605 | . 5507 | .S016 .1437 l. 2566 | . T434 58 3 | . 8557 | . 1443 .0594 | . 5545 | .8003 .1439 |. 25S0 | . T420 57 4 | . 8583 | . I-417 .0583 | . 5583 | .7991 | .1441 .. 2594 | . T.405 || 56 5 || 4860S .51392 || 2.0573 .55621 1.7979 | 1.1443 .12609 | .87391 || 55 6 | . S633 . . 1366 ,0562 . 5659 .7966 | .1445 | . 2623 | . T377 54 7 . 8659 . 1341 .0551 | . 5697 .7954 .1446 |. 2637 | . T363 53 S 86S4 . 1316 .0540 | . 5735 | .7942 .1448 |. 2651 | . T349' 52 9 . S710 | . 1290 .0530 | . 5774 .7930 | .145() |. 2665 , 7335 || 51 10 || 487.35 | .51265 2.0519 | .55812 | 1.7917 | 1.1452 .12679 | .87320 50 11 S760 | . 1239 .0508 | . 5850 | .7905 | .1454 |. 2694 | . T306 || 49 12 . 8786 . 1214 .0498 ||. 588S .7893 .1456 |. 27.08 . 7292 || 48 13 SS11 | . 1189 .0487 |. 5926 . .7881 .1458 |. 2722 | . T278 i 47 14 | . 8S37 1163 .0476 | . 5964 . .7868 .1459 |. 2736 | . 7264 || 46 15 .48862 .51.138 || 2.0466 .56()03 || 1.7856 | 1.1461 |.12750 .87250 45 16 8887 1112 .0455 |. 6041 .7844 .1463 |. 2765 . . 7235 || 44 17 | . 8913 1087 .0444 . 6079 | .7832 .1465 i. 2779 | . T221 || 43 1S 8938 . . I062 .0434 . 6117 | .7820 | .1467 2793 | . T207 || 42 19 8964. | . 1036 .0423 |. 6156 .7808 .1469 2807 || , 7.193 || 41 20 .48989 || 51011 || 2,0413 | .56194 | 1.7795 | 1.1471 .12821 .87.178 || 40 21 9014 | . 0986 .0402 |. 6232 .7783 | .1473 |. 2836 ) . 7164 || 39 22 9040 | . 0960 .0392 | . 6270 | .7771 .1474 |. 2850 . 7150 || 38 23 9065 | . ()935 .03S1 | . 6309 .7759 .1476 . 2864 | . TI36 || 37 24 9(190 . 0910 ,0370 | . 6347 .7747 | .1478 |. 2879 |. 7121 || 36 25 .491.16 .50884 || 2,0360 .56385 | 1.7735 | 1.1480 |.12893 | .87107 || 35 26 9141 . 0859 .0349 |. 6424 | .7723 .1482 . 2007 | . TO93 || 34 27 9166 | . 0834 .0339 |. 6462 7711 . .1484 |. 2921 |. T078 || 33 28 91.92 | . 0808 .0329 |. 6500 .7699 || .1486 |. 2936 ||. T064 || 32 29 92.17 | . 0.783 .0318 |. 6539 || .7687 .14SS 295() ||. T050 i 31 3() .40242 .50758 2.0308 || .56577 | 1.7675 1.1489 |.12964 |.S7035 || 30 31 9268 . 0732 .0297 6616 .7663 | .1491 2979 |. 7021 || 29 32 9293 ) . 0707 .0287 6654 | .7651 .1493 |. 2993 | . TOU7 || 28 33 931S | . 06S2 .0276 6692 .7639 .1495 3007 | . 6902 || 27 34 9.343 | . 06: .0266 |. 6731 .7627 .1497 |. 3022 . 6978 26 35 | .49369 .50631 || 2.0256 .56769 | 1.7615 | 1.1499 ||.13036 .86964 || 25 36 9394 | . 0606 .0245 |. 6808 || .7603 | .1501 3050 | . 6949 2 37 | . .94.19 . 0580 .0235 | . 6846 . .7591 | .1503 3065 . 6935 || 23 3S 9445 . 0555 .0224 . 6885 . .7579 .1505 |. 3079 |. 6921 || 22 39 947 () | . 0530 .0214 6923 .7567 .1507 . .3094 | . 6906 || 21 40 .40495 .50505 || 2.0.204 .56962 | 1.7555 | 1.1508 ||.T3108 || 86892 20 41 952T | . 0479 (J194 7000 | .7544 .1510 |. 3122 . 6877 | 19 42 95.46 | . 0 0183 7039 || 7532 .1512 3137 | . 6863 18 43 95.71 | . 0429 ()173 7077 .752() | .1514 |. 3151 | . 6849 || 17 44 9596 | . ()404 || .01 63 7116 | .7508 .1516 |. 3166 . . 6S34 || 16 45 .49622 | .50378 || 2.0152 .57155 | 1.7496 | 1.1518 || 13180 .86820 15 46, 9647 . 0353 .0142 7193 || 74S4 .1520 |. 3194 | . 6S05 || 14 47 96.72 . 0328 .01.32 7232 || 7473 | .1522 |. 32(39 . 6791 || || 3 48 96.97 | . 0303 .0122 7270 .7461 .1524 |. 3223 . 6776 || 12 49 97.23 | . 0.277 .01.11 '7309 || 74.49 .1526 3238 , 6762 || 11 50 .497.48 .50252 2.01.01 .57348 1.7437 | 1.1528 .13252 .867.48 || 10 51 9773 . (J227 .00.91 7386 .7426 .1530 |. 3267 . 6733 || 9 52 97.98 | . 0202 | .00S1 7425 | .7414 .1531 |. 3281 | . 6719 || 8 53 98.23 . 0176 / .0071 7464 .7.402 | .1533 |. 3296 | . 6704 || 7 54 9849 | . 0151 .0061 | .. 7502 | .7390 .1535 | . 3310 | . 6600 || 6 55 | .49874 || 50126 2.0050 | .57541 | 1.7379 | 1.1537 | .13325 ,866.75 || 5 56 9899 , 0.101 00:40 758() | .7367 .1539 3339 . 6661 || 4 57 | . 99.24 | . 0076 0030 7619 || 7355 .1541 |. 3354 |. 6646 || 3 58 . 9950 | . 0.050 0020 7657 | .7344 | .1543 |. 3368 |. 6632 2 59 . 99.75 . 0.025 .001() 7696 || 7332 | .1545 3383 . . 6617 | 1 60 | .5()000 | . 0000 .0000 | . TT35 | 7320 .1547 |. 3397 . 6602 || 0 M | Cosine. Wrs. Sin. Secante. Cotang. Tangent. Cosec'nt|Wrs. Così Sine. .* 119° Natural. 60° NATURAL LINES. 300 Natural Trigonometrical Functions. 1499 M Sine. Vrs. Cos. Cosec'nte Tang. Cotang. | Secante. Vrs. Sinj Cosine. Aſ 0 | .50000 || 50000 || 2.0000 .57735 | 1.7320 | 1.1547 l.13397 .86602 || 60 1 | . 0025 49975 | 1.9990 | . 7774 .7309 | .1549 |. 3412 |. 6588 || 59 2 . . 0050 9950 99.80 . 7813 .7297 | .1551 |. 3426 | . 6573 ſ 58 3 | . 0075 9924 .9970 | . T851 .7286 | .1553 |. 34.41 | . 6559 || 57 4 . O101 | . 98.99 .9960 . . TS90 | 7274 .1555°. 3456 . 6544 || 56 5 | .50.126 | .49874 | 1.9950 .57929 | 1.7262 | 1.1557 |.13470 ,86530 || 55 6 0.151 9S49 .9940 | . 7968 .7251 | ..] 559 |. 3485 | . 65.15 || 54 7 0.176 9824 .9930 . 8007 .7239 .1561 |. 3499 . 6500 || 53 8 0201 | . 9799 .9920 | . 8046 . .7228 .1562 |. 3514 | . 6486 || 52 *9 0226 9773 .99.10 |. 8085 | 7216 | .1564 |. 3529 |. 6471 || 51 10 | .50252 || 49748 1.9900 .58.123 | 1.7205 | 1.1566 |.13543 | .864.57 || 50 11 | . 0277 | . 9723 .9890 8162 7.193 .1568 , 355S | . 6442 49 T2 0302 | . 9698 .98SO | . 820I. 71S2 .1570 |. 3572 . 6427 || 48 13 ()327 96.73 | .9870 | . 8240 7170 | .1572 . 3587 || - 6413 || 47 14 0352 96.48 .9860 | . 8279 7159 | .1574 |. 3602 6398 || 46 15 .50377 | .49023 | 1.9850 .58318 || 1.7147 | 1.1576 .13616 | .86.383 || 45 16 . ()402 |. 95.97 .9840 | . 8357 7136 | .1578 |. 3631 6369 || 44 17 0428 . 9572 98.30 8396 || 7124 1580 3646 6354 || 43 18 {}453 954'ſ .9820 |. 8435 | 7113 | .1582 3660 6339 || 42 19 0.478 9522 .9811 $474 | .7101 | .1584 3675 63.25 41 20 | .50503 | .49497 | 1.9801 |.585.13 | 1.7090 | 1.1586 | .13690 .863}() | 40 21 | . 0528 94.72 .9791 . S552 .7079 .1588 37 ()4 6295 || 39 22 | . 0553 9447 .9781 | . 8591 .7067 .1590 3719 ſ281 38 23 | . 0578 |. 9422 .9771 . 8630 | .7 (156 1592 3734 6266 || 37 24 . 0603 9397 .9761 | . 867() .7044 .1594 3749 6251 || 36 25 .50628 493.71 || 1.9752 ,58709 || 1.7033 | 1.1596 ||.13763 .86237 || 35 26 0653 9346 .9742 S748 || .7022 | .1598 3778 |. 6222 || 34 27 0679 9.321 .97.32 8787 | .701() .1600 3793 . 6207 || 33 28 ()704 9296 .9722 S826 .6999 || .1602 3S()7 619.2 || 32 -29 07:29 . 9271 .9713 8865 .6988 | .1604 3$22 6.178 || 31 30 .50754 | .40246 | 1.9703 | .58904 || 1.6977 | 1.1606 ||.T3837 .86163 || 30 31 | . ()779 . 9221 .9693 | . S944 .6965 | .1608 3S52 614S 29 32 . 0804 9.196 .96S3 | . 8983 .6954 .1610 |. 3867 | . 6133 l 28 33 0829 917.1 .9674 9022 | .6943 .1612 3881 611S 27 34 0854 | . 9146 .9664 9061 | .6931 | .1614 3S96 6104 || 26 35 | .50879 || 49121 | 1.9654 |.59100 | 1.6020 | 1.1616 i.139]] | .86089 25 36 0904 . . 9096 .9645 9140 .6909 | .1618 3926 . 6074 24 37 0.929 . 9071 .9635 9.179 .6898 .1620 3941 . .605.9 || 23 38 0954 |. 9046 .9625 9218 .6887 1622 3955 | . 6044 || 22 39 0979 9021 .9616 9258 .6875 . .1624 3970 | . 6030 21 40 || 51004 || 48996 || 1.9606 ||.59297 1.6864 | 1.1626 |.13985 | .86015 || 20 41 1029 . . 8971 .9596 | . 9336 .6S53 I62S 4000 6000 || 19 42 1054 . 8946 . .9587 93.76 | .6842 .1630 4015 5985 | 18 43 1079 892.1 .9577 9415 | .6831 .1632 4030 59.70 || 17 44 1104. 8896 .956S 9.154 | .682() .1634 4044 . 5955 | 16 45 .51.129 | .48871 | 1.9558 || 59494 | 1.6808 1.1636 |.14059 | .85%l 15 46 1154 8846 | .9549 9533 | .6797 | .1638 4074 5926 || 14 47 1179 S821 .9539 . 9572 | .6786 .1640 4089 |. 5911 || 13 48 1204 S796 | .9530 96.12 .6775 .1642 1. 4104 ||. 5896 || 12 49 1229 87.71 .9520 965] | .676 || | .1644 4119 5881 II 50 | .51254 . .487.46 1,9510 | .59691 1.6753 | 1.1646 .14134 .85S66 || 10 51 | . 1279 8721 .9501 97.30 | .6742 .1648 4149 |. 5851 || 9 52 . 1304 8696 | .9491 97.70 | .6731 .1650 4164 . 5836 || 8 53 1329 8671 .9482 9809 6720 1652 417S 5821 || 7 54 | . 1354 . 8646 | .9473 9849 67(){} | .1654 4193 ) . 5806 || 6 55 | .51379 || 48621 | 1,0463 .50888 | 1.6698 || 1.1656 .14208 | .85791 || 5 56 . . 1404 | . 8596 | .9454 9928 6687 .1658 4.223 5777 || 4 57 . 1429 857l .9444 9967 6676 | .1660 4238 57.62 || 3 58 1454 85-16 .9435 | .60007 . .6665 | 1662 4253 5747 || 2 59 1479 . 8521 .9425 | . 0046 . .6654 | .1664 4268 573 l 60 1504 | . 8496 .9416 . 0.086 .6643 | .1666 4283 . 5717 || 0 M l Cosine. Wrs. Sin. Secante. Cotaug. Tangent. ||Gosec'ni Vrs. Cosl Sine. M 120° Natural. e 590 NATURAL LINES. 31o Natural Trigonometrical Functions. 148° 9h M | Sine. Wrs. Cos. Cosec'nte Tang. Cotang. Secante. Wrs. Sin Cosine. M. M. S. 0 .51504 || 48496 || 1.9416 .600S6 || 1.6643 1.1666 1.14283 .85717 | 60 || 56 1 | . 1529 . 8471 .9407 | . 0126 .6632 | .1668 |. 4298 5702 || 59 || 56 2 | . 1554 | . 8446 .9397 | . 0.165 .6621 .1670 . 4313 |. 56S7 58 || 52 3 1578 . 8421 .9388 |. 0205 .6610 | .1672 |. 4328 |. 5672 || 57 || 48 4 1603 | . 8396el .9378 . 0244 .6599 .1674 H. 4343 |. 5657 || 56 44 5 ,51628 .48371 | 1.9369 .60284 || 1.6588 | 1.1676 .14358 .85642 || 55 || 40 6 , 1653 8347 ,9360 |.. 0324 | .6577 .1678 |. 4373 | . 5627 || 54 36 7 1678 8322 .9350 | . 0363 .6566 1681 |. 4388 5612 || 53 || 32 S , 1703 8297 .9341 . . 0403 .6555 1683 . 4403 | . 5597 || 52 28 9 1728 8272 .9332 | . ()443 .65 .16S5 |. 4418 5582 || 51 || 24 10 .51753 .48247 | 1.9322 |.60483 1.6534 | 1.1687 |.14433 .85566 || 50 20, 11 177S 8222 .9313 | . 0522 .6523 .1689 |. 4448 5551 || 49 || 16 12 1803 8197 .9304 | . 0562 .6512 | .1691 |. 4463 |. 5536 || 48 || 12 13 1827 8172 .9295 | . 0602 | .6501 .1693 |. 4479 |. 5521 || 47 8 14 1852 8147 .92S5 | . 0642 | .6490 .1695 |. 4494 | . 5506 || 46 4 15 .51877 .481.23 | 1.9276 .60681 | 1.6479 | 1.1697 |.14509 .85491 || 45 55, 16 | . 1902 809S .9267 | . 0721 .6469 | .1699 |. 4524 . 5476 || 44 56 17 1927 8073 .9258 | . ()761 .6458 | .1701 |. 4539 |. 5461 || 43 || 52 18 . 1952 . 8048 .9248 . 0801 .6447 | .1703 |. 4554 |. 5446 || 42 || 48 19 1977 8023 .9289 |. 0.841 .6436 .1705 |. 4569 |. 5431 41 || 44 20 | .52002 .47998 || 1,9230 .60881 | 1.6425 | 1.1707 |.14584 .85416 || 40 | 40 21 2026 7973 .9221 | . 0920 ! .6415 | .1709 |. 4599 |. 5400 || 39 || 36 22 2051 7049 .92.12 | . 0960| .6404 .1712 |. 4615 5385 || 38 || 32 23 . 2076 .. 7924 .9203 . 100() .6393 | .1714 4630 | . 53.70 || 37 || 28 24 2101 7809 .9193 | . 1040 | .6383 . .1710 |. 4645 i. 5355 || 36 || 24 25 | .52126 .47874 | 1.9184 |.61080 | 1.6372 | 1.1718 |.14660 |.85340 || 35 | 20 26 2151 7S49 .9175 | . 1120 ! .6361 .1720 4675 . 5325 || 34 || 16 27 21.75 7S24 .9166 , 1160 .6350 | .1722 |. 4690 5309 || 33 || 12 28 . 2200 7800 .9157 | . 1200 6340 .1724 4706 | . 5294 || 32 8 29 . 2225 7775 .9148 1240 .6329 | .1726 4721 |. 5279 31 4 30 | .52250 .47750 | 1.9139 .61280 | 1.6318 1.1728 |.14736 .85264 30 || 54, 31 | . 2275 7725 .9130 | . 1320 .630S | .1730 4751 |. 5249 || 29 56 32 . 2299 7700 .9121 1360 . .6207 | .1732 4766 5234 28 || 5% 33 | . 2324 7676 .9112 1400 | .6286 .1734 47S2 |. 5218 27 4 34 . 2319 7651 .9102 | . 1440 .6276 .173 4797 . 5203 || 26 || 44 35 | .52374 |.47626 | 1.9093 .61480 | 1.6265 | 1.1739 .14812 .85188 || 25 | 40 36 | . 2398 7601 .9084 |. 1520 .6255 | .1741 4S27 5173 || 24 || 36 37 2423 | . T577 90.75 1560 | .6244 .1743 4842 |. 5157 || 23 32 38 . . 2448 7552 .9066 1601 | .6233 .1745 4858 5142 22 || 28 39 i , 2473 7527 .9057 1641 .6223 .1747 4873 ||. 5127 || 2 || || 24 40 .52498 | .47502 | 1.9048 .61681 | 1.6212 | 1.1749 .14888 . .85112 || 20 | 20 41 2522 7477 .9039 1721 .62()2 .1751 4904 | . 5096 || 19 16 42 2547 7453 .9030 | . 1761 .619.1 | .1753 4919 50S1 || 18 12 43 2572 7428 .9021 | . 1801 || ,6181 .1756 4934 5066 || 17 8 44 | . 2597 7403 .9013 | . 1842 .6170 | .1758 4949 5050 | 16 4 45 .52621 || 473.79 | 1.9004 |.618S2 | 1.6160 | 1.1760 | .14965 .85035 | 15 || 53 46 2646 7354 .8995 | . 1922 .6149 || 1762 4980 5020 || 14 || 56 47 2671 7329 89S6 . 1962 6130 j .1764 4995 | . 5004 || 13 || 52 48 . 2695 7304 .8977 . 2003 || 6128 .1766 |. 5011 | . 4989 || 12 || 48 49 2720 7280 .8968 2043 .6118 .17 (58 |. 502(; 4974 || 11 || 44, 50 | .52745 .47255 | 1.8959 .62083 1,6107 | 1.1770 .15041 .84959 || 10 || 40 51 | . 277() 723() .8950 | . 2123 .6097 | .1772 5057 4943 i 9 || 36 52 2794 | . T.205 .S941 |. 2164 .6086 | .1775 5072 . 4928 || 8 || 32 53 2819 7181 .8932 : . 22()4 .6076 .1777 5087 | . 4012 || 7 || 28 54 . . 2844 . 7156 .8924 | . 2244 | .6066 .1779 5103 4897 || 6 || 24 55 .52868 || 47131 | 1.8915 .622S5 1,6055 | 1.1781 |.15118 .84882 || 5 || 20 56 | . 2893 7107 .8906 |. 2325 | .6045 | .1783 5133 J866 || 4 || 16 57 | . 29.18 7082 8897 |. 2366] .6034 .1785 5140 4851 || 3 || 12 58 . . 294.2 7057 8888 . . 2406 | .6024 .1787 5164 . 4836 2 8, 59 2967 7033 .8879 . 2446 | .6014 | .1790 518() | . 4820 - 1 4. 60 | . 2992 | . TOO8 .887.1 | . 24S7 || 6003 .1792 |. 5195 | . 4S05 || 0 || 52. M l Cosine. Wrs. Sin. Secante, | Cotang. Tangent, Cosco'nt IWrs, Cos | Sine. M M.S. 1210 Natural. 58° 3h NATURAL LINES. 329 Natural Trigonometrical Functions. 1479| 9h M Sine. Wrs, Cos.;Cosec'nte | Tang. Cotang. ScCante.}Wrs. Sin Cosine. M |M.S. 0 | .52992 .47008 || 1,8871 |.62487 | 1,6003 | 1.1792 |.15.195 | 84805 60 || 52 1 |. 3016 |. 6983 8862 |. 2527 5993 | .1794 i. 5211 . . 4789 || 59 || 56 2 . . .3041 . 6959 . .8853 |. 2568 .5983 .1796 |. 5226 . 4774 || 58 || 52 3 . 3066 |. 6934 || 8844 |. 2608 | .5972 | .1798 |. 5241 . 4758 57 || 48 4 |. 30.90 | . 6909 . .8836 |. 2649 || 5962 | .1800 |. 5257 i. 4743 ſ 56 || 44 5 .531I5 .46885 | 1.8827 | .62689 || 1.5952 | 1.1802 |.15272 | 84728 || 55 | 40 6 . 3140 | . 6860 | 88.18 |. 2730 || 5941 .1805 |. 5288 . . 4712 || 54 || 36 7 | . 3164 |. 6835 | .8809 |. 27.70 || 5931 | 1807 l. 5303 |. 4697 || 53 || 32 8 |. 3189 |. 6811 .8801 | . 2811 .5921 .1809 |. 5319 |. 4681 || 52 28 9 |. 3214 | . 6786 || 8792 |. 2851 || 5910 | 1811 |. 5334 . 4666 || 51 || 24 10 .53238 .46762 | 1.8783 |.62892 | 1.5900 | 1.1813 |.15350 . .84650 50 | 20 11 | . 3263 . 6737 || S775 | . 2933 || 5890 .1815 |. 5365 |. 4635 || 49 16 12 | . 3288 . . 6712 | .8766 . 2973 || 5880 .1818 |. 5381 . 4619 || 48 12 13 | . 3312 | . 6688 . .8757 | . 3014 || 5869 .1820 |. 5396 |. 4604 || 47 8 14 | . 3337 | . 6663 .8749 }. 3055 | .5859 .1822 |. 54.12 | . 4588 46 4 15 .53361 || 46638 | 1.8740 | 63095 | 1.5849 | 1.1824 |.15427 | 84573 || 45 || 51 16 | . 3386 |. 6614 .8731 |. 3136 .5839 .1826 |. 5443 |. 4557 44 || 56 17 34:11 | . ,8723 3.177 | .5829 | .1828 |. 5458 |. 4542 || 43 || 52 18 . 3435 | . 6565 S714 | . 32.17 | 5818 .1831 |. 5474 | . 4526 42 || 48 19 | . 3460 | . 6540 || S706 |. 325S .5808 || .1833 j. 5489 4511 || 41 || 44 20 | .53484 || 46516 || 1.8697 .63299 || 1.5798 || 1.1835 |.15505 ,84495 || 40 || 40 21 3509 , 6491 SG88 |. 3339 5788 .1837 |. 5520 | . 4479 || 39 36 22 3533 |. 6466 S680 |. 33SU 5778 1839 5536 | . 4464 || 38 32 23 3558 | . 6412 8671 |. 3421 | .5768 1841 5552 . 4448 || 37 28 24 3583 | . 6417 8663 | . .3462 .5757 1844 5567 | . 4433 || 36 || 24 25 | .53607 .46393 || 1.SG54 63503 || 1.5747 | 1.1846 . .15583 || 84417 || 35 ; 20 26 3632 | . 6368 .8646 |. 3543 5737 1848 5598 |. 4402 || 34 || 16 27 3656 . 6344 S637 | . 35S4 .5727 185ſ) 5614 | . 4386 || 33 || 12 28 3681 | . 6319 86.29 . 3625 5717 | .1852 5630 7() 32 S 29 3705 | . 6294 i .S620 | . 3666 5707 1855 5645 | . 4355 || 31 4 3() 53730 | .4627() | 1,8611 |.63707 1.5697 | 1.1857 | .15661 .84.339 || 30 50 31 3754 | . 6245 86()3 . 37.4S 5687. 1859 5676 . 4323 29 : 56 32 37.79 |. 6221 S595 | . 3789 | .5677 1861 |. 5692 | . 4308 28 || 52 33 3803 | . 6196 8586 |. 3830 5667 1863 |. 5708 |. 4292 27 || 48 34 3828 . 6172 85.78 |. 3871 | .5657 1866 |. 5723 |. 4276 || 26 || 44 35 | .53852 .46147 1.8569 .63912 || 1.5646 | 1.1868 |.15739 .84261 25 | 40 36 3877 6123. 8561 |. 3953 | .5636 1870 5755 . 4245 24 || 36 37 3901 . 6098 8552 | . 3994 | .5626 1872 57.70 | . 4229 || 23 32 3S 3926 . 6074 8544 |. 4035 | .5616 1S74 . 5786 . 4214 || 22 || 28 39 3950 | . 6049 8535 | . 4076 | .56()6 1877 5802 . . 4198 || 23 24 40 || 53.975 46025 | 1.8527 .64117 | 1.5596 || 1.1879 .15817 | S4182 20 | 20 41 3999 | . 60(K) 85.19 |. 415S 5586 1881 5833 4167 || 19 16 42 4024 |. 5976 | .8510 | . 4199 5577 .ISS3 5849 4151 || 18 || 12 43 4048 . 5951 8502 |. 4240 5567 | .1886 5865 4135 || 17 || 8 44 4073 |. 5927 8493 | . 42S1 5557 | .18S 588() . 4120 | 16 4. 45 .54097 .45902 || 1.8485 .64322 1.5547 | 1.1890 .15896 .84104 || 15 49 46 4122 |. 5878 8477 | . 4363 5537 | .1892 5912 | . 4088 || 14 || 56 47 4146 |. 5854 8468 | . 4.404 5527 1894 |. 5927 | . 4072 || 13 || 52 48 417.1 ! .. 5S29 8460 | . 4446 .5517 1897 5943 | . 4057 | 12 || 48 49 | . 4195 5805 8452 | . 4487 .5507 LS99 5959 |. 4041 || 11 || 44 50 | .54220 .45780 | 1.8443 .64528 || 1.5497 | 1.1901 |.15975 84025 || 10 | 40 51 . 4244 | . 5756 .8435 | . 4569 || 5487 .1903 |. 5991 | . 4009 || 9 || 36 52 426S | . 5731 .8427 | . 4610 | .5477 | .1906 |. 6006 3.993 || 8 || 32 53 . 4293 . 5707 | .8418 |. 4652 .5467 | .1908 |. 6022 |. 3978 || 7 || 2S 54 . 4.317 | . 5682 | .8410 | . 4693 l .5458 .1910 |. 6038 |. 3962 || 6 || 24 55 ,54342 . .45658 || 1.S402 | .64734 || 1.5448 || 1.1912 |.16054 || 83946 || 5 || 20 56 . 4366 |. 5634 .8394 | . 4775 ,5438 .1915 i. 6070 |. 3930 || 4 || 16 57 | . 4391 | . 5609 || S385 | . 4817 | .542S | .1917 | . 6085 |. 3914 || 3 || 12 58 . . 4415 | . 5585 .8377 | . 4858 . .5418 .1919 |. 6101 |. 3899 || 2 8 59 . 4439 |. 5560 .8369 . 4899 || .5408 .1921 . . 6117 | . 3883 || 1 || 4 60 | . 4464 |. 5536 | .8361 | . 4941 | .5399 || .1922 |. 6133 |. 3867 || 0 || 48 ..] M Cosine. Wrs. Sin. Secante. Cotang. Tangent. Cosec'nt IVrs, Così Sine. M |M.S. 1229 - Natural, 579|3|| 16 NATURAL LINEs. 339 Natural Trigonometrical Functions. 1469 9b. M | Sine. Wrs. Cos.; Cosec'nte Tang. Cotang. | Secante. Wrs. Sin Cosine. M M.S. 0 | .54464 .45536 | 1.8361 .64941 1.5399 || 1.1924 .16133 ,83867 || 60 || 48 1 : . 4488 |. 5512 .S352 | . 4982 | .5389 | . .1926 . . 6149 |. 3851 || 59 56 2 . . 4513 | . 54S7 . .8344. |. 5023 .5379 .1928 . 6165 . 3S35 | 58 || 52 3 | . 4537 . 5463 ) .8336 |. 5065 .5369 .1930 i. 6180 |. 3819 || 57 || 48 4 . 4561 |. 54.38 .8328 - 5106 | .5359 .1933 |. 6.196 | . 3804 || 56 || 44. 5. .54586 | .45414 | 1.8320 .65148 || 1.535() i.1935 |.16212 83788 || 55 | 40 6 . 4610 | . 5390 .8311 |. 5.189 || 5340 ) .1937 |. 6228 . 3772 54 36 7 4634 |. 5365 .8303 ||. 5231 .533) | 1939 . . 6244 | . 3756 || 53 || 32 8 . . 4659 |. 534.1 .8295. . 5272 { .5320 | 1942 |. 6260 . . 3740 52 28 9 4683 | . 5317 .8287 | . 5314 .5311 || 1944 6276 - 3724 || 51 24 10 .54708 .45292 | 1.8279 |.65355 | 1,5301 | 1.1946 .16292 .83708 50 | 20 11 . 4732 |. 5268 .8271 . 5397 .5291 .1948 || . 6308 |. 3692 || 49 16 12 | . 4756 . 5244 .8263 | . 5438 .5282 | .1951 |. 6323 |. 3676 || 48 || 12 13 4781 52.19 .S255 5480 .5272 | .1953 . . 6339 |. 3660 || 47 8. 14 | . 4805 | . 5195 .8246 |. 5521 .5262 .1955 . . 6355 | . 3644 || 46 || 4 15 .54829 .45.171 1.8238 |.65563 1.5252 1.1958 i.16371 |.83629 || 45 47 16 | . 4854 |. 5146 .8230 |. 5604 || 5243 | .1960 |. 6387 |. 3613 || 44 || 56 IT | . 4878 |. 5122 S222 5646 .5233 .1962 |. 6403 |. 3597 || 43 || 52 18 . 4902 . . 509S .8214 5688 .5223 .1964. l. 6419 |. 3581 || 42 || 48 19 . 4926 5073 .8206 | . 5729 .5214 | .1967 . . 6435 | - 3565 || 41 || 44 20 .54951 | .45049 || 1.8198 || .65771 | 1.5204 || 1.1969 i.16451 | .83549 40 | 40 21 . 4975 . 5025 .8190 | . 5S13 . .5195 .1971 i. 6467 |. 3533 || 39 || 36 . 22 . 49.99 5000 81S2 | . 5854 .51.85 .1974 H. 6483 . 3517 | 38 || 32 | 23 | . 5024 4976 | .8174 |. 5896 || 5175 .1976 |. 6499 |. 350l ||37 || 28 24 | . 5048 . 4952 .8166 |. 593S 5166 .1978 . . 65.15 |. 3485 || 36 || 24 25 .55072 .4492S 1.81.58 .65980 | 1.5156 1.1980 .16531 .83469 || 35 | 20 26 . 5097 4903 || S150 . 6021 .5147 | .1983 |. 6547 | . 3453 || 34 || 16 27 | . 5121 - 4879 .8l42 | . 6063 .5137 || 1985 . G563 |. 3437 || 33 12 28 . 5145 | . 4855 .8134 6105 .5127 | .1987 |. 6579 |. 34.21 || 32 8 29 |. 5169 |. 4830 -S126 | . 6147 || 5118 .1990 |. 6595 | . 3405 || 31 4 30 || 55.194 .44806 || 1.8118 .6618S | 1.5108 || 1.1992 ||.1661.1 .83388 || 30 || 46 3} | . 5218 47.82 .S110 | . 6230 .5099 || 1994 |. 6627 | . 3372 29 56 32 - 5242 . 4758 -8102 6272 .50S9 .1997 |. 6643 |. 3356 28 || 5. 33 . 5266 |. 4733 .8094 | . 6314 .5080 | 1999 |. 6660 |. 3340 27 || 48 34 . . 5291 |. 4709 .8086 | . 6356 .5070 | 2001 |. 6676 . 3324 || 26 || 44 35 | .55315 .44685 | 1.8078 |.66398 | 1.5061 | 1.2004 |.16692 | .83308 || 25 40 36 | . 5339 |. 4661 .8070 | . 6440 .5051 | 2006 i. 6708 |. 3292 24 36 37 . 5363 |. 4637 .8062 |. 64S2 .5042 | .2008 |. 6724 . . .3276 || 23 || 32 38 - 5.388 . 4612 -8054 || - 6524 .5032 2010 i. 6740 |. 3260 22 | 28 39 . 54.12 |. 4588 .S.047 6566 .5023 2013 |. 6756 |. 32.44 21 24 40 | .55436 | .44564 || 1.8039 |.66608 || 1.5013 | 1.2015 | .16772 |.83228 || 20 20 41 . 5460 . 4540 .8031 6650 .5004 | .201.7 . . 6788 |. 3211 k9 || 16 42 | . 5484 | . 45.15 .8023 . 6692 || 4994 | .2020 . 6804 | . 3195 || |8 || 12 43 : . 5509 |. 4491 .8015 |. 67341 .4985 | .2022 |. 6821 |. 3179 || 17 | 8 44 . 5533 . 446'ſ .8007 6776. .4975 .2024 |... ($837 - 3163 | 16 || 4 45 .55557 | .44443 | 1.7999 || 668+S 1.4966 | 1.2027 | .16853 |.83147 || 15 45 46 | . 55.81 |. 4419 .7992 |. 6860 | .4957 | .2029 |. 6869 |. 3131 || 14 || 56 47 5605 | . 4395 .7984 . 6902; .4947 .2031 |. 6885 |. 31.15 13 || 52 48 .5629 |. 4370 .7976 |. 6944. 4938 . .2034 |. 6901 |. 3098 || 12 || 48 49 5654 |. 4346 .7968 6986. .4928 2036 |. 6918 |. 3082 || 11 || 44 50 .55678 | .44322 | 1.7960 | .6702S 1.4919 | 1.2039 |.1693.4 | .83066 || 10 || 40 5L | . 5702 || - 42.98 .7953 ||. T071 .4910 | .2041 |. 6950 |. 3050 || 9 || 36 52 | . 5726 . 4274 .7945 | . T113 | . .4000 | .2043 |. 6966 - 3034 || 8 || 32 53 5750 | . 4250 .7937 |. Ilā5 .4891 .2046 |. 6982 |. 3017 || 7 || 28 54 || - 5774 |. 4225 .7029 .. 7 197 || 4881 | .2048 || - 6999 || - 3001 || 6 || 24 55 .55799 || 44.201 | 1.7921 | .67239 || 1.4872 | 1.2050 | .17015 .82985 || 5 || 20 56 . 5823 . 417 7914 | . T282 .4863 | .2053 7031 |. 2969 || 4 || 16 57 | . 5,847 . . 4153 .7906 |. 7324 4853 .2055 |. 7047 |. 2952 || 3 || 12 58 . 5S'Il - 4129 .7898 |. 7366 || 4844 | .2057 7064 |. 2936 2 8. 59 | . 5895 |. 4105 .7891 | . .7408 || 4835 | .2060 |. 7080 |. 2920 || 1 || 4 60 5919 . . 4081 .7883 |. 7451 || 4826 . .2062 |. 7096 |. 2904 || 0 || 44 Aſ I Cosine. IVrs. Sin. Secante, I Gotang. Tangent. ||Gosec'nt IVrs.Gosl Sine. I M.I.M.S. 1239 56° 3h Natural. NATURAL LINES. 2h 349 Natural Trigonometrical Functions. 145° M.S. M. Sine. Wrs. Cos. Gosec'nte | Tang. | Cotang. Secante, Vrs. Sim Cosine. M 16 || 0 | .55919 | .44081 | 1.7883 |.67451 | 1.4826 | 1.2062 .17096 |.82904 || 60 4 || 1 | . 5943 |. 4057 .7875 |. 7493 || 4816 | .2064 |. 7112 | . 2887 || 59 8 2 . . 5967 |. 4032 .7867 . 7535 | .4807 | .2067 | . TI29 | . 2871 58 12 3 | . 5992 |. 4008 || 7860 - 7578 .4798 .2069 |. 7145 | . 2855 57 16 || 4 | . 6016 |. 3984 .7852 |. 7620 .4788 .2072 ſ. 7161 . 2839 56 20 || 5 || 56040 .43960 | 1.7844 |.67663 | 1.4779 | 1.2074 .17178 .82S22 || 55 24 || 6 | . 6064 . 3936 | .7S37 77.05 | .4770 .2076 |. 7194 | . 2806 || 54 28 7 | . 6(S8 |. 3912 | .7829 |. 7747 .4761 | .2079 i. 7210 | . 2790 || 53 32 || 8 | . 6112 | . 3.SSS .7821 | . TT90 .4751 .2081 | . T227 . , 2773 52 36 9 | . 6136 |. 3864 .7814 78.32 .474.2 . .2083 7243 | . 2757 || 51 40 || 10 || 56160 | .43840 | 1.7806 | .678.75 | 1.4733 | 1.2086 .17259 || 82741 || 50 44 || 11 | . 6184 |. 3816 | .7798 |. 7917 | .4724 . .2088 . . .7276 2724 || 49 48 || 12 | . 6208 . 3792 || 7791 . 7960 || 4714 2091 7292 | . 2708 || 48 52 || 13 | . 6232 | . 376S .7783 . 8002 .4705 2093 '730S 2692 || 47 56 14 6256 | . 3743 .7776 8045 | .4696 .2095 |. 7325 2675 || 46 17 | 15 | .56280 | .43719 | 1.7768 . .68087 | 1.4687 | 1.2098 i.17341 | .82650 || 45 4 || 16 . 6304 |. 3695 || 7760 8130 | .4678 | .2100 7.357 | . 2643 || 44 8 || 17 | . 6328 |. 3671 .7753 | . 8173 .4669 | .2103 7374 | . 26.26 || 43 12 | 18 . 6353 |. 3647 .7745 | . 8215 .4659 | .2105 |. 7390 | . 2610 || 42 16 || 19 . 6377 |. 3623 || 7738 825S .4650 | .2107 7406 | . 2593 || 41 20 | 20 ! .56401 | .43599 || 1.7730 | .68301 | 1.4641 | 1.2110 ſ.17423 .82577 | 40 24 || 21 | . 6425 |. 3575 | .7723 | . 8343 .4632 .2112 |. 7439 |. 2561 || 39 28 22 . 6449 |. 3551 || 7715 8386 .4623 | .2115 |. 7456 |. 2544 || 38 32 || 23 . 6473 |. 3527 | .7708 $429 .4614 | .2117 7472 2528 || 37 36 24. 6497 | . 3503 .7700 8471 .4605 | .2119 |. 7489 2511 || 36 40 || 25 .56521 .43479 | 1.7693 | 68514 | 1.4595 | 1.2122 |.17505 |.82495 || 35 44 26 | . 6545 . 3455 .7685 8557 .45S6 | .2124 7521 | . 24.78 || 34 48 || 27 6569 . 3431 .7678 . 8600 .4577 .2127 7538 2462 || 33 52 28 . 6593 i. 3407 || 767() | . 86.42 | .4568 .2129 7554 | . 2445 || 32 56 || 29 66.17 33S3 .7663 | . 8685 .4559 | .2132 7571 2429 || 31 18 || 30 | .56641 | .43359 | 1.7655 .6872S | 1.4550 | 1.2134 .17587 |.82413 || 30 4 || 31 6664 . 8335 | .7648 87.71 | .4541 .2136 |. 7604 |. 2396 || 29 8 || 32 6688 , 3311 .7640 | . 8814 | .4532 .2139 |. 7620 2380 28 12 || 33 6712 |. 3287 .7633 . . 8857 .4523 .2141 i. 763 2363 || 27 16 || 34 | . 6736 |. 3263 .7625 8899 || .4514 .2144 7653 | . 2347 || 26 20 || 35 | .56760 | .43239 || 1.7618 .68942 | 1.4505 | 1.2146 |.17670 .82330 25 24 || 36 | . 6784 |. 3216 || 7610 8985 . .4496 | .2149 |. 7686 |. 2314 || 24 28 || 37 | . 6808 | . 8192 || 7603 902S | .4487 | .2151 | . T703 | . 2297 || 23 32 || 38 | . 6832 ||. 316S .7596 |. 907.1 | .4478 .2153 7719 2280 || 22 3 39 6856 || - 3144 | .75SS 9114 | .4469 .2156 |. 7736 2264 21 40 | 40 | .56880 | .4312() | 1.7581 |.69157 | 1.4460 | 1.2158 .17752 .82247 20 44 || 41 . 6904 | . 300G | .7573 . . 9200 | .4451 .2161 |. 7769 | . 2231 || 19 48 || 42 | . 6928 |. 3072 iſ .7566 | . 0243 | .4442 .2163 7786 2214 | 18 52 || 43 6952 |. 3048 .7559 |. 9286 .4433 .2166 7802 2198 || 17 56 44 . 6976 | . 3024 .7551 | . 9329 l .4424 .216S -. 7819 2181 | 16 19 || 45 .57000 || 43000 || I,7544 .69372 | 1.4415 | 1.2171 |.17835 | .82165 || 15 4 || 46 | . T023 2976 .7537 | . 9415 | .4406 | .2173 7852 2148 14 8 || 4-7 | . T047 | . 2952 . .7529 9459 .4397 .2175 | . T868 2131 13 12 || 4S . 7071 2929 .7522 |. 9502 | .4388 .2178 7885 2115 12 16 || 49 . 7095 2905 | .7514 9545 | .4379 .2180 |. 7902 2098 || 11 20 || 50 .57119 .42881 | 1.7507 | .6958S | 1.4370 | 1.2183 .17918 .82082 10 24 || 51 . 7143 2857 .7500 | . 9631 | .4361 | .2185 H. 7935 | . 2065 || 9 2S | 52 | . TI67 283: .7493 9674 | .4352 .2188 i. 7951 |. S 32 || 53 | . T191 2809 || 74S5 97IS | .4343 .2190 |. 7968 |. 2032 || 7 36 || 54 | . T214 | . 2785 .7478 | . 9761 | .4335 | .2193 |. 7985 2015 || 6 40 || 55 .57238 .42761 | 1.747 1 | .69804 || 1.4326 | 1.2195 |.18001 | .8.1998 || 5 44 56 | . 7262 . 2738 . .7463 98.47 | .4317 j .2198 |. 8018 1982 || 4 48 57 | . T286 |. 2714 .7456 9891 | .4308 .2200 |. 8035 1965 : 3 52 58 | . 7310 | . 2690 .7449 9934 | .4299 .2203 |. 8051 1948 2 56 59 . . '7334 . 2666 | .7442 9977 | .4290 | .2205 |. 8068 1932 || 1 20 | 60 | . T358 2642 .7434 .70021 | .42S1 | .2208 |. 8085 . 1915 0 M. S. M Cosine. IVrs. Sin. Secante. Cotaug.j'Tangent, Cosec'ntiVrs. Cos! Sine. M 8h 1249 Natural. 55° NATURAL LINES, 359 Natural Trigonometrical Functions. 1449 M | Sine. Wrs. Cos.;Cosec'nte Tang. Cotang. Secante. Vrs. Sin Cosine. M 0 .5735S | .42642 1,7434 || 70021 1.4281 1,2208 || .18085 .81915 60 1 | . 7381 | . 2618 .7427 | . 0.064 .4273 .2210 |. 8101 | . I898 || 59 | 2 | . 7405 |. 2595 .7420 . 0.107 .4264 .2213 | . 8118 . 1882 58 3 | . T.429 |. 2571 .7413 | . Ulá1 .4255 .2215 8135 | . 1865 || 57 4 . 7453 , 2547 .7405 | . 0194 .4246 | .221S | . 8151 1848 56 5 .57477 .42523 || 1.7398 .70238 || 1.4237 | 1.2220 .18168 .81832 55 6 . 7500 . 2499 .7391 | . U281 | .4228 .2223 |. 8185 . 1815 || 54 7 | . T524 . 2476 .7384 |. 0325 .4220 | .2225 | . 8202 |. 1798 || 53 8 . . 754S | . 2452 .7377 . 0368 .4211 . .2228 8.218 1781 52 9 '7572 . 2428 .7369 | . 0412 4202 | .2230 8235 . . 1765 || 51 10 .57596 | .42404 || 1.7362 .70455 | 1.4.193 | 1.2233 .18252 |.81748 || 50 11 76.19 | . 2380 .7355 | . 0499 || 4185 .2235 8269 IT31 || 49 12 . 7643 |. 2357 .7348 . 0542 .4176 | .2238 82.85 1714 || 48 13 , 7667 . 2333 .7341 . . 0586 || 4167 .2240 S302 |. 1698 || 47 14 | . T691 2309 .7334 . 0629 .4158 .2243 |. 8319 1681 || 46 15 .57714 || 42285 1.7327 | .70673 || 1.4150 | 1.224.5 .18336 .81664 45 I6 | . 773S 2262 .7319 07.17 | .4141 | .2248 . 8353 |. 1647 || 44 17 | . TT62 |. 2238 .7312 0760 | .4132 .225() |. 8369 . 1630 || 43 18 . 7786 |. 2214 .7305 || -- 0804 || .4123 .2253 8386 . 1614 || 42 19 T809 2190 .7298 08:48 || .4115 .2255 8403 1597 || 41 20 .57833 .42167 | 1.7291 .70891 | 1.4106 | 1.2258 .18420 |.81580 | 40 21 | . T857 2143 .7284 ||. Q935 .4097 .2260 . . 8437 1563 || 39 22 | . T881 2119 .7277 . 0979 || 4089 .2263 |. 8453 | . 1546 38 23 . 7904 2096 .7270 | . 1022 .4980 | .2265 |. 8470 1530 37 24 . 7928 . .2072 .7263 |. 1066 .4071 .2268 |. 8487 1513 || 36 25 .57952 . .42048 || 1.7256 .71110 | 1.4063 | 1.2270 .18504 S1496 || 35 26 7975 2024 .7249 1154 .4054 .2273 |. 8521 1479 || 34 27 | . T999 . 2001 .7242 . 1198 || .4045 .2276 |. 8538 1462 33 28 8023 . 1977 .723 1241 .4037 | .2278 8555 . . 1445 || 32 29 | . 8047 1953, .7227 1285 .4028 .2281 |. 85.71 | . 1428 || 31 30 .5S0.70 | .41930 1,7220 .71329 || 1.4019 | 1.22S3 .18588 .81411 || 30 31 | . 8094 | . 1906 .7213 | . 1373 .4011 .2286 8605 1395 29 32 . 8118 |. 1882 .7206 1417 | .4002 | .228S Sö22 1378 || 28 33 8141 1S59 .7199 1461 .3994 | .2291 8639 . 1361 27 34 8165 IS35 .7.192 |. 1505 | .3985 .2293 |. 8656 1344 26 35 .58.189 |,41811 || 1.7185 .71549 || 1.3976 | 1.2296 ||.18673 || S1327 || 25 36 . 8212 . 17SS .717.8 1593 .3968 .2298 8690 | . 1310 24 37 . 8236 . 1764 .7171 1637 | .3959 | .2301 |., S707 | . 1293 23 38 8259 1740 .7164 |. 1681 i .3951 .2304 8724 . . 1276 || 22 39 . 8283 1717 .7157 1725 .3942 .2306 8741 . I259 || 21 40 l. .58307 | .41693 || 1.7151 .71769 || 1.3933 | 1.2309 | .1875S | .81242 20 41 8330 . 1669 .7144 1813 .3925 | .2311 87.75 | . I225 19 42 . 8354 1646 .7137 1857 .3916 | .2314 S792 | . 1208 || 18 43 8378 | . Tö22 .7130 1901 | .3908 .2316 8809 | . 1191 || 17 44 | . 8401 1599. .7123 |. 1945 | .3899 | .2319 8826 . I174 16 45 | .58425 | .41575 | 1.7116 .71990 | 1.3891 | 1.2322 .18843 .81157 || 15 46 | . S448 |. 1551 .7109 2034 | .3882 | .2324 8860 | . 1140 || 14 47 8472 | . 152S .7102 2078 .3874 .2327 8877 | . 1123 13 48 || - S496 1504 .7005 | . 2122 .3865 | .2329 8894 | . 1106 || 12 49 . 8519 1481 .7088 2166 .3S57 .2332 891.1 | . IOS9 || 11 50 .58543 | .41457 | 1.7081 .72211 | 1.3848 1,2335 | 18928 .8.1072 || 10 51 | . 8566 | . 1433 .7075 |. 2255 .3840 | .2337 8945 . 1055 || 9 52 | . 8590 . 1410 .7068 |. 2299 .3831 | .2340 8962 |. 1038 || 8 53 | . 8614 1386 .7061 |. 2344 .3823 .2342 8979 1021 || 7 54 8637 . 1363 .7054 |. 2388 . .3814 .2345 8996 . . 1004 || 6 55 i ,58661 || 41339 || 1.7047 | .72432 | 1.3806 | 1.2348 |.19013 .80987 5 56 . 8684 1316 .7040 | . 24.77 .3797 | .2350 9030 0970 || 4 57 | . 8708 1292 .7033 . 2521 .3789 . .2353 9047 | . ()953 : 3 58 . S731 1268 .7027 . . 2565 .3781 | .2355 9064 . 0.936 || 2 59 . 8755 |. 1245 .702 . 2610 ! .3772 .2358 9081 0.919 1 60 . 8778 | . 1221 .7013 | . 2654 .3764 .2361 | . 9098 0902 || 0 M i Cosine. Wrs. Sin. Secante. Cotang. Tangent. Cosec'nt IVrs. Cos] Sine. M 125° Natural. 549 NATURAL LINES. 369 Natural Trigonometrical Functions. M. : Sine. Vrs. Cos. Cosec'nte Tang. Cotang. ||Secante. Vrs. Sin 0 | .58778 .41221 1.7013 .72654 || 1.3764 I.2361 .19098 I . 8S02 . 1198 .7006 . 2699 .3755 .2363 9115 2 . . 8825 . 1174 .6999 || . 2743 .3747 .2366 9132 3 . 8849 |. 1151 .6993 |. 2788 .3738 .2368 915() 4 | . 8873 | . 1127 .6986 . . 2832 .373U .2371 |. 9167 5 .5SS96 || 41.104 || 1.6979 |.72877 | 1.3722 | 1.2374 |.19184 6 . S920 | . 1080 .6972 |. 2921 .3713 .2376 |. 9201 ‘7 | . 8943 . 1057 .6965 |. 2966 . .3705 | .2379 |. 9218 8 . . 8967 | . 1033 .6959 |. 3010 | .3697 .2382 i. 9235 9 | . 8990 | . 1010 .6952 |. 3055 .3688 .2384 |. 925.2 10 || 59014 | .40986 | 1.6945 |.73100, 1,3680 | 1.2387 |.19270 11 9037 0963 .6938 . . 3144 j .3672 | . .2389 |. 9287 12 9060 | . 0939 .6932 |. 3189 .3663 .2392 |. 9304 13 | . 9084 | . 0916 .6925 3234 .3655 .2395 i. 9321 14 | . 9107 | . 0892 .6918 |, 3278 . .3647 .2397 |. 9338 15 .59131 | 40869 | 1.6912 | .73323 | 1.3638 | 1.2400 | .19355 16 9I54 | . 0845 .6905 ||. 3368 .3630 | .2403 9373 17 | . 91.78 . 0822 6898 |. 3412 || 3622 .2405 93.90 18 . 9201 | . 0799 .6891 3457 | .3613 .240S 9.407 19 92.25 0775 .6885 3502 | .3605 | .2411 i , 9424 20 .59248 .40752 1.6S78 || 73547 i 1.3597 | 1.2413 ; .19442 21 9272 . 0728 .6871 3592 .3588 .2416 [.. 9459 22 | . 92.95 | . 0705 .6865 3637 .35S0 .2419 94.76 23 | . 93.18 | . 0681 .6858 3681 .3572 .2421 9493 24 . 9342 . . 0658 .6851 3726 .3564 .2424 i. 9511 25 | .59365 .40635 | 1.6845 .73771 | 1.3555 | 1.2427 | .19528 26 9.389 | . 0611 .6S38 |. 38.16 .3547 .2429 95.45 27 | . 94.12 | . 0588 .6S31 |. 3861 | .3539 .2432 9562 28 9435 | . 0564 .6825 3906 .3531 | .2435 j. 9580 29 9459 . ()541 .6818 3953 .3522 .2437 95.97 30 .594S2 .40518 || 1.6SI2 .73996 | 1.3514 | 1.2440. .19614 31 . 9506 | . 0494 .6805 | . 4041 .3506 | .2443 |. 9632 32 | . 9529 |. 0471 .6798 4086 .3498 .2445 H. 9649 33 . 9552 . 04:47 .6792 4131 .3489 | .2448 9666 34 95.76 | . 0424 .6785 4176 .3481 | .2451 96.83 35 i .59599 || .40401 | 1.6779 |.742.21 | 1.3473 | 1.2 p53 .1970l 36 9622 | . 0377 .67'72 . 4266 .3465 . .2456 97 18 37 . 9646 | . 0354 .6766 4312 | .3457 | .2459 97.36 38 9669 | . 0331 .675.9 4357 .3449 | .2461 . 9753 39 96.92 | . 0307 .6752 . 4402 . .3440 | .2464 |. 9770 40 .59716 | .402S4 | 1.6746 .74447 | 1.3432 | 1.2467 .19788 41 97.39 . 0261 .6739 | . 4492 | .3424 .2470 9805 42 97.62 . 0237 .6733 4538 .3416 .2472 |. 9822 43 . 97.86 |. 0214 .6726 |. 4583 i .3408 .2475 i. 9S40 44 9S09 . 0.191 .6720 . 4628 .3400 | .2478 9.857 45 .59832 .40167 || 1.6713 | .74673 | 1.3392 | 1.2480 . .19875 46 | . 9856 | . 014.4 .6707 | . 4719 .3383 . .2483 |. 9892 47 9S79 | . 0121 6700 . 4764 .3375 .24S6 99.09 48 | . 99t)2 . 009S .6694 | . 4809 .3367 .248S 9927 49 9926 . 0() .6687 |. 4855* .3359 .2495 9944 50 | .50949 | .40051 | 1.6681 .74900 | 1.3351 | 1.249 p 1.19962 51 | . 99.72 | . 0028 .6674 | . 4946 . .3343 | .2497 99.79 9 52 . 9995 | . 0004 .6668 | . 4901 .3335 | .2499 999'ſ S 53 .60019 .39981 .6661 | . 5037 / .3327 .2502 | .20014 7 54 | . 0.042 | . 995 .6655 | - 5082 .3319 .2505 ()031 6 55 .60065 | .39935 | 1.6648 .75128 1,3311 | 1.2508 || 20049 5 56 | . OUSS . . 99.11 .6642 . 5173 .3303 | .2510 0.066 4 57 . 0112 | . 9888 .6636 | . 5219 3.294 | .2513 ()084 3 58 . . 0.135 | . 9S65 .6629 |. 5264 .3286 .2516 ()101 2 59 | . O158 . 9842 .6623 | . 531() .3278 .2519 O] 19 I 60 | . 01S1 | . 98.18 .6616 | . 5355 .327 () .2521 0.136 0 M Cosime. Wrs. Sim I Secante. Cotang. ITangent. Cosec'nt. Vrs. Cos M 1269 Natural. 39 NATURAL LINEs. 2h 379 Natural Trigonometrical Functions. 1429 M. S. M. Sine. Vrs. Cos. Cosec'nte | Tang. | Cotang. ||Secante.IVrs. Sin| Cosine. M 3. 0 | .6O181 .398.18 || 1.6616 .75355 | 1.3270 | 1.2521 | 20136 |.798.63 60 4 || 1 | . (205 | . .9795 | .6610 | . 5401 | .3262 .2524 |. 0154 |. 9846 || 59 8 || 2 | . 0228 |. 9772 | .6603 |. 5447 | .3254 .2527 l. O171 |. 98.28 || 58 12 || 3 | . 0251 |. 9749 | .6597 |. 5492 .3246 | .2530 l. O189 |. 9811 || 57 16 || 4 | . 0274 |. 97.26 .6591 |. 5538 .3238 .2532 |. 0206 |. 9793 || 56 20 || 5 | .60298 || 397.02 | 1.6584 |.75584 || 1.3230 | 1.2535 | 20224 |.79776 || 55 24 || 6 | . 0320 |. 9679 .6578 |. 5629 | .3222 | .2538 . 0242 |. 9758 || 54 28 || 7 | . (.344 | . 9656 .6572 |. 5675 . .3214 | .2541 l. 0259 |. 9741 || 53 32 || 8 | . 0307 | . 9633 . .6565 |. 5721 .3206 .2543 |. 0277 |. 9723 || 52 36 | 9 | . 0390 9610 | .6559 |. 5767 .3198 | .2546 |. Q294 | . 97.06 || 51 40 || 10 | .60413 | .39586 | 1.6552 | .75812 | 1.3190 | 1.2549 || 20312 |.79688 || 50 44 || 11 . 0437 |. 9563 | .6546 |. 5858 . .3182 | .2552 |. 0329 |. 9670 || 49 48 12 | . 0460 | . 9540 | .6540 l. 5904 . .3174 .2554 i. 0347 l. 9653 l 48 52 || 13 | . 0483 | . 9517 | .6533 |. 5950 | .3166 .2557 l. 0365 | . 9635 i 47 56 || 14 | . 0506 |. 9494 | .6527 |. 5996 | .3159 | .2560 l. 0382 |. 96.18 || 46 29 15 .60529 .39471 | 1.6521 |.76042 | 1.3151 | 1.2563 | .20400 |.79600 45 4 | 16 . . 0552 |. 9447 .6514 |. 6088 | .3143 | .2565 |. 0417 | . 9582 || 44 8 || 17 | . 0576 |. 9424 .6508 . 6134 | .3135 | .2568 |. 0435 | . 9565 43 12 | 18 |. 0599 |. 9401 | .6502 |. 6179 .3127 | .2571 |. 0453 |. 9547 || 42 16 || 19 | . 0622 |. 9378 .6496 |. 6225 | .3119 .2574 |. 0470 | . 9530 || 41 20 | 20 ! .60645 39355 | 1.6489 |.76271 | 1.3111 | 1.2577 l.20488 .79512 40 24 || 21 OGGs . 9332 .6483 |. 6317 .3103 | .2579 |. 0505 |. 9494 | 89 28 22 | . 0691 |. 9309 | .6477 |. 6364 .3095 | .2582 |. 0523 |. 9477 || 38 32 || 23 . 0714 | . 9.285 | .6470 |. 6410 | .3087 .2585 H. (541 |. 9459 || 37 36 24 | . 0737 |. 9262 | .6464 |. 6456 .3079 | .2588 |. 0558 |. 9441 || 36 40 || 25 | .60761 | .39239 || 1.6458 |.76502 | 1.3071 | 1.2591 | .20576 .79424 || 85 44 26 | . 0784 |. 9216 | .6452 | . 6548 | .3064 .2593 |. 0594 |. 9406 || 34 48 || 27 | . 0807 |. 9193 ) .6445 |. 6594 | .3056 .2596 i. 0611 |. 9388 || 33 52 || 28 || . 083 9170 | .6439 |. 6640 | .3048 .2599 |. 0629 |. 9371 || 32 56 || 29 . 0853 |. 9147 | .6433 |. 6686 | .3040 | .2602 |. 0647 |. 9353 || 31 30 || 30 | .60876 | .3912 || | 1.6427 | .76733 | 1.3032 | 1.2605 | .20665 |.79335 || 30 4 || 31 | . 0899 |. 9101 || 6420 |. 6779 | .3024 .2607 |. 0682 |. 93.18 29 8 || 32 | . 0922 |. 9078 || 6414 |. 6825 | .3016 | .2610 |. 0700 |. 9300 28 12 || 33 | . 0945 | . 905 .6408 |.. 6871 | .3009 | .2613 |. 0718 |. 9282 || 27 16 |34 | . 0963 | . 9031 | .6402 |. 69.18 3001 | .2616 I. 0735 | . 9264 || 26 20 || 35 | .60991 | .39008 || 1.6396 .76964 | 1.2993 | 1.2619 | .20753 |.792.47 || 25 24 || 36 |. 1011 | . 8985 | .6389 |. 7010 ! .2985 . .2622 |. 0771 . 9229 || 24 28 || 37 . 1037 | . 8962 .6383 |. 7057 | .2977 .2624 || 0789 |. 92.11 || 23 32 || 38 |. 1061 |. 8939 . .6377 |. 7103 | .2970 .2627 |. 0806 |. 9193 || 22 36 || 39 . 1084 |. S916 . .6371 |. 7149 .2962 .2630 |. 0824 |. 9176 21 40 | 40 | .61107 | .38S93 | 1.6365 .77.196 || 1.2954 | 1.2633 .20842 | .79158 20 44 || 41 |. 1130 | . 8870 | .6359 |. 7242 .2946 .2636 |. 0860 |. 9140 | 19 48 || 42 . I153 |. S847 . .6352 |. 7289 .2938 . .2639 . OS78 |. 9122 || 18 52 || 43 | . 1176 |. 8824 .6346 |. 7335 | .2931 .2641 l. 0895 | . 9104 || 17 56 || 4 | . 1199 || , 8.801 | .6340 l. 7382 | .2923 | .2644 |. 0913 | . 9087 | 16 31 || 45 |.61222 || 3S778 | 1.6334 |.77428 || 1.2915 | 1.2647 | .20931 |.79069 || 15 4 || 46 |. 1245 |. 8755 .6328 |. 7475 . .2007 | .2650 | . 0949 |. 9051 || 14 8 || 47 l. 1268 |. 8732 .6322 |. 7321 | .2900 | .2653 |. 0.967 |. 9033 13 12 || 48 . 1290 | . 8709 | .6316 |. 7568 | .2892 | .2656 |. 09S4 |. 9015 | 12 16 || 49 |. 1314||. 8686 || 6309 |. 7614 2884 .2659 |. 1002 |. 8998 || 11 20 || 50 | .61337 .38663 | 1,6303 || 77661 | 1.2876 | 1.2661 |.21020 | 78980 || 10 24 || 51 | . 1260 | . 8640 | .6297 |. 7708 | .2869 .2664 |. 1038 |. 8962 9 28 52 1383 |. 8617 .6291 |. 7754 | .2861 | .2667 |. 1056 |. 8944 || 8 32 53 | . 1405 |. 8594 | .6285 |. 7801 | .2853 .2670 i. 1074 |. 8926 || 7 36 || 54 |. 1428 |. 8571 | .6279 |. 7848 . .2845 | .2673 |. 1091 |. 8908 || 6 40 55 | .61451 .38548 || 1,6273 .77895 | 1.2838 | 1.2676 | .21109 .78890 5 44 56 |. 1474 |. 8525 | .6267 |. 7941 .2830 | .2679 |. 1127 |. 8873 || 4 48 || 57 | . 1497 |. 8503 | .6261 |. 7988 . .2822 | .2681 |. 1145 |. 8855 || 3 52 || 58 | . 1520 |. 8480 . .6255 |. S035 | .2815 .2684 |. 1163 |. 8837 2 56 || 59 | . Tà43 |. 8457 | .6249 |. 8082 | .2807 | .2687 i. 1181 | . 8819 || 1 32 60 | . 1566 |. 8434 .6243 |. S128 .2799 .2690 i. 1199 |. 8801 || 0 M. S. M Côsine. Wrs. Sin. Secante. Cotang. I ſungent. Cosec'ut. Wrs. Cos! Sine, M 8h 1270 Naturai. 529 NATURAL LINES. 380 Natural Trigonometrical Functions. 1419 9h M || Sine. Vrs. Cos. Cosec'nte Tang. I Cotang. Secante. Vrs. Sin Cosine. M M.S. 0 || 61566 | 38434 || 1.6243 | 78128 || 1.2799 || 1.2690 || 21199 || 78801 | 60 || 28 1 . J589 8411 .6237 || - S175 || 2792 || 2693 1217 . 8783 || 59 || 56 2 . 1612 S388 .6231 . 8222 || 2784 || 2696 I2:35 8765 | 58 || 52 3 . 1635 8365 .6224 || - 8269 || 2776 || 2699 1253 87 47 || 57 || 48 4 . 1658 . 8342 .6218 . 8316 || 2769 || 2702 1271 8729 || 56 || 44 5 | 61681 || 38319 || 1.6212 || 78363 || 1.2761 || 1.2705 || 21288 || 78711 | 55 || 40 6 . 1703 8296 .6206 . 8410 || 2753 || 2707 1306 8693 || 54 || 36 7 , 1726 . 8273 .6200 . S457 || 2746 || 271() 1324 8675 || 53 || 32 8 1749 8251 .694 . 8504 || 273S || 2713 1342 8657 || 52 || 28 - 9 1772 S228 .61S8 8551 || 2730 || 2716 1360 8640 | 51 || 24. 10 | 61795 . .38205 || 1.6182 || 78598 || 1.2723 || 1.2719 || 21378 || 78622 || 50 || 20 11 . 1818 81S2 .6176 . 8645 || 2715 || 2722 . 1396 8604 | 49 || 16 12 1841 8159 .617() S692 || 2708 || 2725 14:14 . 85S6 || 48 || 12 13 1864 , 8136 .6164 . 8739 || 2700 || 2728 1432 | . 8568 || 47 8 14 . 1886 813 .6159 8786 || 2692 || 2731 1450 . 8550 || 46 4 15 || 61909 || 38091 || 1.6153 || 78834 | 1.2685 || 1.2734 || 21468 || 78532 | 45 || 27 16 1932 | . 806S .GT47 8881 || 2677 || 2737 14S6 8514 || 44 | 56 17 1955 8045 .6141 892S | 267) 2739 1504 8496 || 43 | 52 18 . 1978 . 8022 .6135 . 8975 i 2002 || 2742 1522 | . 8478 || 42 || 48 19 2001 7999 .6129 9022 | .2655 || 2745 1540 . S460 | 41 | 44 20 | 62023 | 37976 || 1.6123 . .79070 || 1.2647 || 1.2748 || 21558 || 78441 || 40 || 40 21 . 2046 . 7954 .(S117 9117 || 2639 || 2751 1576 S423 | 39 || 36 22 2069 7931 .6111 9104 || 2632 || 2754 1594 8405 | 38 || 32 23 2092 | . 7908 .6105 9212 || 2624 || 2757 1612 83S7 | 37 || 28 24 . 2115 || . 7885 .6099 9259 | 2017 || 2760 || - 1631 . . S369 || 36 | 24 25 .62137 | 37862 || 1.6093 || 79300 || 1.2609 || 1.2763 || 21649 | 78351 | 35 || 20 26 2160 , 784() .6087 9354 || 2602 || 2766 1667 . 8333 || 34 || 16 27 21S3 . 7817 .6()S . 9401 | 2594 || 2769 1685 8315 || 33 || 12 28 , 2206 7794 G077 94-19 || 2587 772 1703 8297 || 32 8 29 2229 7771 6070 9496 || 2579 || 2775 1721 8279 || 31 4 30 || 62251 | 377.48 || 1.6064 || 79543 || 1.2572 || 1.2778 || 21739 | 78261 || 30 | 26 31 2274 | . 7726 6058 9591 2564 || 2781 1757 8243 || 29 || 56 32 , 2297 7703 6052 9639 || 2557 || 2784 1775 8224 || 28 || 52 33 . 2320 7680 .6(k6 || - 96S6 2549 || 2787 . 1793 8206 || 27 || 48 34 . 2342 7657 .6040 . 9734 || 2542 || 2790 1812 Sl SS || 26 | 44 35 || 62365 .37 635 || 1.6034 || 79781 || 1.2534 || 1.2793 || 2183() || 78170 || 25 || 40 36 . 23S8 7612 .6029 . 9829 || 2527 || 2795 184S S152 i 24 || 36 37 2411 | . 7589 .6023 s. 9876 || 2519 || 2798 JS66 S134 || 23 || 32 38 2433 7566 .(S017 . . 9924 i .2512 | .2S01 ISS4 81.16 || 22 || 2S 3 2456 || . 7544 || 6011 9972 | .2504 || 2S04 1902 S()97 || 2L | 24 40 || 62479 || .37521 || 1.6005 | 80()2() 1.2497 || 1.2807 || 21921 || 78079 || 20 || 20 41 2501 7498 .6000 || .. 0067 || 2489 || 2810 1939 S()61 || 19 || 16 42 2524 . 7476 .5994 . 01:15 || 2482 || 2813 1957 8043 i 18 | 12 43 . 2547 . 7453 .5988 . 0163 || 2475 || 2816 1975 8025 || 17 8 44 2570 7430 || 5982 0211 || 2467 || 2S19 1993 S()()7 || 16 4 45 | 62592 | 37408 || 1.5976 || 80258 || 1.2460 || 1.2822 || 22011 || 77988 || 15 || 25 46 . 2615 || - 7385 .5971 ()306 || 2452 || 2825 2030 , 7970 | 14 | 56 47 2038 7362 5965 0354 || 2445 || 2828 204S 7952 || 13 || 52 48 26G() . 7340 5959 . ()402 | .2437 || 2831 2066 7934 || 12 || 48 49 2083 || . 7317 .5953 ()450 || 2430 || 2834 2084 7915 || 11 || 44 50 || 627(6 | 37294 || 1.5947 | 80498 || 1.2423 || 1.2837 || 22103 | 77897 || 10 || 40 51 . 2728 . 7272 5942 - . ()546 || 2415 || 2840 2121 7879 || 9 || 36 52 . 2751 7249 5936 || . ()594 || 2408 || 2843 2139 7861 || 8 || 32 53 2774 7226 .5930 | .. 06:12 .2400 .2S46 2157 7S42 || 7 || 28 54 . 2796 || . 7204 | 5924 . ()690 || 2393 || 2849 21 76 7824 || 6 || 24 55 .62819 || 37181 || 1.5919 || 80738 || 1.2386 || 1.2852 || 22194 || 77806 || 5 | 20 56 . 2S41 . 7158 .5913 || . ()786 || 2378 || 2S55 2212 778S | 4 || 16 57 2864 , 7136 .5907 | .. 0S34 || 2371 || 2858 2230 7769 || 3 || 12 58 . 2887 || . 7113 .5901 . ()SS2 | .2364 || 2861 22:49 7751 | 2 S 59 2909 | . 700) .5896 || . 0930 || 2356 || 2864 2267 7733 || 1 4 60 . 2932 | . 7(68 .5890 . (1978 || 2349 || 2867 f. 2285 7715 || 0 || 24 M | Cosine. Vrs, Sin. Secante. I Cotang. Tangent. Cosec'nt i Vrs. Cos Sine. M M.S. 128 Natuural. 519 3 i å- - NATURAL LINES. 2h 399 Natural Trigonometrical Functions. 140° 9h M.S. M I Sine. Wrs. Cos. iCosec'nte Tang. Cotang. Secante.jVrs. Sin | Cosine. I M M.S. 36 || 0 | .62032 .37068 1,5890 .80978; 1.2349 | 1.2867 .22285 |.77715 || 60 24 4 1. 2955 | . T()45 .5884 |. 1026 .2342 . .2871 |. 2304 | . T696 || 59 56 8 2 : . 297'ſ 7(323 .5879 |. 1075 .2334 .2874 |. 2322 | . T678 5S 52 I2 3 . 3000 '7000 .5873 |. 1123 .2327 .2877 |. 2340 | . T660 57 || 48 16 4 | . 3022 6977 .5867 | . 1171 .2320 | .2880 . . 2359 7641 || 56 || 44 20 5 .63045 .36955 | 1.5862 |.S12.19 | 1.2312 1.2SS3 .22377 || 77623 55 40 24 6 || - 3067 | . 6932 .5856 | . 1268 .2305 .2886 |. 2395 7605 || 54 || 36 28 7 | . 3090 | . 6910 .5850 | . 1316 .2297 || 2SS) |. 2414 T586 || 53 || 32 32 8 . . 31.13 . 6SS7 .5$45 1364 .2290 . .2892 |. 2432 7568 || 52 || 28 36 || 9 | . 3135 | . 6865 .5839 |. 1413 . .2283 | .2895 |. 2450 |. 7549 || 51 || 24 40 || 10 | .6315S .36S42 | 1.5833 .81461 | 1.2276 | 1.2898 || .22469 .77531 || 50 | 20 44 || 11 . 3180 68.20 .5S28 1509 | .226S .2901 |. 2487 7513 || 49 || 16 48 || 12 | . 3203 | . 6797 .5S22 1558 .2261 .2904 . . 2505 | . T494 || 48 || 12 52 || 13 | . 3225 | . 6774 .5S16 1606 . .2254 .2907 . 2524 | . T476 || 47 8 56 14 | . 3248 6752 .5811 1655 . .2247 .2910 |. 2542 . '7458 || 46 4 37 || 15 | .632.70 | .36729 | 1.5S0.5 ! .81703 | 1.2239 1.2913 .22561 .77439 45 || 23 4 || 16 | . 3293 6707 .5799 . 1752 .2232 .2916 |. 2579 , 7421 || 44 || 56 8 || 17 | . 33.15 | . 66S4 .5794 | . 1800 .2225 | .2919 |. 2597 . 7402 || 43 || 52 12 || 18 , 333S 666.2 .57SS 1849 | .2218 | .2922 |. 2616 | . T3S4 || 42 || 48 16 || 19 . 3300 6639 ,5783 1898 || .2210 .2926 2634 | . T365 || 41 || 44 20 20 ! .63383 . .366.17 | 1.5777 |.S1946 1.2203 | 1.2929 | .22653 |.77347 | 40 || 40 24 21 . 3405 | . 6594 .57'71 1995 | .2196. .2932 2671 '7329 || 39 || 36 28 22 . 34.28 . 6572 5766 2043 .2189 .2935 2690 | . T310 || 38 || 32 32 || 23 | . 3450 | . 6549 57.60 . 2092 | .218.1 293S 270S . Iº92 3 28 36 || 24 . 34.73 | . 6527 .5755 | . 2141 .2174 .2941 27.27 7273 || 36 24 40 || 25 .63495 .36504 || 1.5749 .82190 | 1.2167 | 1.2944 .22745 # 77.255 || 35 | 20 44 || 26 | . 3518 | . 6482 .5743 |. 2238 | .2160 .2947 2763 'IX36 || 34 16 48 27 | . 3540 | . 6459 .5738 || - 22S7 | .2152 .2950 2782 721S 33 || 12 52 || 28 . .3563 | . 6437 .5732 |. 2336 .2145 | .2 . 2800 | . TIQ9 || 32 8 56 || 29 | . 3585 6415 .5727 . 2385 .2138 .2956 . . 2S19 7181 || 31 4 38 30 .63608 .36392 | 1.5721 |.S2434 || 1.2131 | 1.2960 | .22837 .77162 || 30 || 22 4 || 31 . 363() 637() .5716 . 24S2 .2124 | .2963 2856 7144 || 29 || 56 8 32 3653 | . 6347 .5710 | . 2531 .2117 | .2966 2874 . . 'II25 28 || 52 12 33 || - 36.75 6:325 .5705 | . 2580 .2109 .2969 |. 2893 | . TI()7 27 || 48 16 34 3697 6302 .5699 2629 .2102 .2972 2912 ||. T088 26 || 44 20 i 35 | .63720 .36280 | 1.5694 82678 || 1.2095 | 1.2975 .22930 .770.70 || 25 | 40 24 || 36 3742 | . 6258 .50SS 2727 . .2088 .2978 |. 2949 |. 7051 || 24 36 28 || 37 | . 3765 . . 6235 5683 |. 2776 .2081. .2981 |. 2967 7033 || 23 || 32 32 || 38 . 3787 6213 5677 | . 2825 .2074 | .2985 2986 . 7014 || 22 || 28 36 || 39 . 38.10 6190 .5672 | . 2S74 .2066 | .2988 3004 6996 || 21 24 40 | 40 | .63S32 .3616S | 1.5666 .82923 | 1.2059 | 1.2991 .23023 .76977 20 || 20 44 || 41 || - 3854 |. 6146 .5661 | . 29721 .2052 | .2994 3041 | . 6958 || 19 || 16 48 || 42 | . 3877 6123 .5655 3022 .2045 .2997 3060 6940 18 l 12 52 || 43 : , 3899 6101 .5650 | . 8071 . .2038 .3000 3079 | . 6921 || 17 8 56 44 3921 6078 .5644 |. 3120 . .2031 | .3003 3097 | . 6903 || 16 4. 39 || 45 .63944 .36056 | 1.5639 .83.169 | 1.2024 | 1.3006 31.16 || 76884 || 15 21 4 46 | . 3966 | . 6034 .5633 . . .3218 .2016 | .3010 3.134 | . 6865 || 14 || 56 8 || 4-7 || - 3989 | . 6011 .5628 |. 3267 .2009 | .3013 |. 3153 | . 6847 || 13 || 52 12 || 48 . 4011 5989 5622 | . 3317 | .2002 | .3016 3172 . . 6828 || 12 || 48 16 || 49 | . 4033 5967 .5617 | . 3366 || 1995 || 13019 , 3190 6810 || 11 || 44 20 50 | .64056 | .35944 || 1.5611 . .83415 || 1.1988 | 1.3022 | .23209 |.76791 10 | 40 24 || 51 4078 5922 .5606 | . 3465 i ,1981 .3025 3227 . . 6772 || 9 || 36 28 52 | . 4100 5900 .5600 | . 3514 .1974 .3029 . . 3246 | . 6754 || 8 || 32 32 53 4123 5877 5595 |. 3563 .1967 | .3032 |. 3265 | . 6735 || 7 || 28 36 || 54 | . 4145 | . 5855 .5590 | . 3613 .1960 | .3035 i. 3283 , 6716 || 6 || 24 40 || 55 || ".64167 | .35833 1.5584 || 83662 | 1.1953 1,3038 . .23302 | .76698 || 5 || 20 44 || 56 . . 4189 5810 .5579 3712 | .1946 3.04.1 ! .. 3321 6679 4 || || 6 48 57 4212 57SS .5573 3761 .1939 .3044 |. 3339 6660 || 3 || 12 52 58 4234 5766 ,5568 |. 3811 .1932 .3048 |. 3358 | . 6642 || 2 8 56 59 . 4256 57 43 .5563 3860 .1924 .3051 i. 3377 | , 6623 || 1 4 40 # 60 - 4279 5721 ,5557 . 3910 1917 | .3054 j. 3305 | 6604 || 0 || 20 Mſ. S. M I Cosìne. Wrs. Sin." Secante, Cotang. Tangent. Cosec'ntiVrs. Così Sine. Aſ i M.S. 8h ||1299 Natural. 50° 3h NATURAL LINES. 40° Natural Trigonometrical Functions. 1399 M | Sine. Wrs, Cos.;Cosec'nte, Tang. | Cotang. Secante. Wrs. Sin| Cosinc. M 0 | .64279 .35721 | 1,5557 |.83910 | 1.1917 | 1.3054 .23395 76604 || 60 1 | . 4301 |. 5699 ,5552 |. 3959 . .1910 | .3057 i. 3414 6586 59 2 . . 4323 5677 .5546 4009 .1903 .3060 |. 3433 6567 58 3 | . 4345 5654 | .5541 4059 .1896 .3064 |. 3452 6548 || 57 4 | . 4368 |. 5632 | .5536 |. 4108 .1889 .3067 l. 34.70 |. 6530 56 5 | .64390 | .35610 | 1.5530 | .84158 || 1.1882 I.3070 |.23489 |.76511 || 55 6 | . 4412 |. 5588 .5525 | , 4208 .1875 | .3073 |. 3508 6492 54 7 | . 4435 | . 5565 .5520 |. 4257 .1868 .3076 |. 3527 | . 6473 || 53 8 . . 4457 |. 5543 .5514 |. 4307 | .1861 || 3080 |. 3545 |. 6455 52 9 . 4479 5521 .5509 | . 4357 | .1854 | .3083 3564 6436 || 51 10 .64501 | .35499 1,5503 |.84407 | 1.1847 | 1.3086 | .23583 |.76417 | 50 11 | . 4523 5476 .5498 . 4457 .1840 | .3089 3602 6398 || 49 12 | . 4546 5.454 .5493 | . 4506 | .1833 .3092 3620 | . 638() || 48 13 | . 4568 |. 5432 .5487 | . 4556 | .1826 .3096 3639 6361 47 14 . 4590 54:10 ,5482 4606 . .1819 | .3099 3658 6342 46 15 .64612 | .353SS 1,5477 | .84656 | 1.1812 | 1.3102 .23677 .76323 45 16 | . 4635 5365 .54.71 4706 .1805 | .3105 3695 6304 || 44 17 4657 | . 5343 .5466 |. 4756} .1798 || .3109 3714 6286 || 43 18 4679 |. 5321 .5461 4806 .1791 || 3112 |. 3733 6267 || 42 I9 4701 , 5299 .5456 . 4856 || IFS5 || 3115 3752 6248 || 41 20 | .64723 .35277 | 1.5450 |.84906 | 1.1778 | 1.3118 .23771 || 76229 || 40 21 . 4745 5254 . .5445 4956 . .1771 3121 3790 t;210 || 39 22 || - 4768 |. 5232 .5440 5006 | .1764 | .3125 3S08 619.1 ! 38 23 | . 4790 | . 5210 .5434 5056 | .1757 .3128 3S27 6173 || 37 24 4S12 | . 518.8 .5429 5107 ..I'750 .3131. 3846 6154 36 25 .64834 .35166 1,5424 .85157 | 1.1743 | 1.3134 .23865 .76135 | 35 26 4856 5144 .5419 5207 .1736 .313S 3884 6116 || 34 27 4878 |. 5121 .5413 5257 | .1729 3141 3903 6097 || 33 28 4900 . 5099 .5408 5307 .1722 3144 3922 6078 || 32 29 | . 4923 5077 .5403 5358 .1715 .3148 394() 6059 || 31 30 .64945 .35055 1,5398 |.85408 || 1,1708 || 1.3151 .23959 . .7604.1 || 30 31 4967 . 5033 .5392 5458 . .1702 .3154 3978 |. 6022 20 32 4989 5011 .5387 5509 | .1695 .3157 3997 6003 || 28 33 5011 4989 .5382 5559 .1688 .3161 4016 59S4 27 34 5033 . 4967 .5377 5609 | .1681 | .3164 40.35 | . 5965 || 26 35 | .65055 .34945 | 1.5371 .85660 | 1.1674 || 1,3167 H.24054 .75946 || 25 36 | . 5077 | . 4922 | .5366 5710 .1667 .3170 4073 5927 || 24 37 5099 4900 .5361 5761 | .1660 | .3174 4092 5908 || 23 38 5121 | . 4878 || .5356 5811 | .1653 | .3177 4111 58S9 || 22 39 5144 . 4856 .5351 5862 | .1647 | .3180 4130 5870 21 40 .65166 | .34834 || 1.5345 .85912 | 1.1640 | 1.3184 | .24149 .75851 20 41 5188 |. 4812 .534() 5963 | .1633 . .3187 4168 5832 || 19 42 | . 5210 | . 4790 | .5335 | . 6013 I626 .3190 4186 5813 || 18 43 5232 476S .5330 | . 6064 1619 | .3193 4205 5794 17 44 5254 |. 4746 .5325 6115 .1612 .3197 4224 5775 | 16 45 .65276 | .34724 1.5319 .86L65 | 1.] 605 | 1.3200 | .24243 || 75756 || 15 46 529S 4702 .5314 6216 || .1599 .3203 4262 5737 || 14 47 5320 | . 4680 5309 6267 .1592 .3207 |. 42S1 5718 13 48 5342 . . 4658 . .5304 6318 1585 .3210 |. 4300 5699 || 12 49 5364 |. 4636 | .5299 6368 .1578 .3213 |. 4319 5680 11 50 | .65386 .34614 | 1.5294 .864.19 | 1.1571 | 1.3217 | .24338 .75661 || 10 51 5408 |. 4592 .5289 6470 .1565 | .3220 |. 4357 5642 || 9 52 5430 |. 4570 .52S3 |. 6521 | .1558 .3223 4376 §623 || 8 53 5452 | . 4548 .5278 |. 6572 | .1551 .3227 . . 4396 5604 || 7 54 . 5474 | . 4526 .5273 | . 6623 || 1 .3230 l. 4415 55S5 || 6 b5 .05496 | .34504 || 1.5268 .86674 || 1.1537 1,3233 . .24434 .75566 5 56 | . 5518 , 4482 .5263 | . 6725 | .1531 | .323'ſ I. 4453 5547 || 4 57 . 5540 |. 4460 | .5258 |. 6775 | .1524 | .3240 4472 552S 3 58 . , 5562 |. 4438 .5253 | . §826 .1517 | .3243 |. 4491 | . 5509 || 2 59 . . 5584 |. 4416 .5248 | . 6878 .1510 | .3247- |. 4510 |. 5490 || 1 60 5606 4394 .5242 | . 6929 | .1504 .325() |. 4529 54.71 || 0 M I Cosine. 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Cosec'nte | Tang. | Cotang. Secante. Wrs. Sim | Cosine. I M M. S. 48 || 0 | .66913 | .33087 | 1.4945 .900.40 | 1.1106 | 1.3456 .25685 | .74314 || 60 | 12 4 1 , 6935 , 3065 .494() 0.093 .1100 .346() , 57.05 . 4205 || 59 56 8 2 | . 6956 |. 3044 .4935 0.146 .1093 3463 . 5724 . . 4275 58 || 52 12 3 | . 6978 |. 3022 .4930 | . 0198 .1086 3467 . 5744 4256 || 57 || 48 16 4 | . 6999 3000 .4925 0251 | .1080 .3470 . 5763 | . 4236 || 56 || 44 20 5 .67021 | .32979 || 1.4921 |.90304 || 1.1074 | 1.3474 .25783 .74217 || 55 | 40 24 6 | . TO43 . 2957 .4916 | . 0357 | .1067 | .3477 |. 5802 . . 4197 || 54 || 36 28 7 | . '7064 | . 29.3 .4911 |. 04:10 | .1061 .3481 |. 5822 | . 4178 53 || 32 32 8 . . 7086 |. 2914 .4906 |. 0463 .1054 | .3485 |. 5841 4.158 || 52 28 36 9 | . TI()7 2893 .4901 | . 0515 . .1048 .348S |. 5861 4,139 || 51 24 40 || 10 | .67] 29 .32871 || 1.4897 .90568 || 1.104.1 | 1.3492 .258SO .74119 50 | 20 44 11 7150 | . 2849 .4892 |. 0621 .1035 | .3495 |. 5900 4100 49 16 48 || 12 7172 2828 .4887 - 0674 .1028 .3499 i. 5919 4080 || 48 || 12 52 || 13 . 7]94 2SOG .4882 ()727 .1022 | .3502 |. 5939 |. 4061 i47 8 56 14 7215 2785 .487'ſ ()780 . .1015 .3506 5959 4041 || 46 4 49 || 15 .67237 .32763 | 1.4873 . .90834 || 1.1009 | 1.3509 .25978 || 74022 || 45 || 11 4 16 7258 2742 .486S 08S7 .1003 .3513 ||. 5998 4002 || 44 || 56 8 || 17 7280 . 2720 .4863 0.940 .0996 .3517 6017 39S3 || 43 52 12 || 18 7301 2699 .4858 . 0993 .0990 .3520 6037 3963 || 42 || 48 16 || 19 7323 2677 .4S54 1046 .0983 | .3524 6056 3943 || 41 || 44 20 | 20 | .67344 | .32656 | 1.4849 .91099 || 1,0977 | 1.3527 | .26076 | .73924 || 40 || 40 24 21 | . T366 2634 .4844 1153 .0971 .3531 6096 3904 || 39 || 36 28 22 | . T3S7 2613 .4839 1206 .0964 3534 61.15 3SS5 || 38 || 32 32 23 . 74().9 2501 .4835 1259 .0958 .353 6135 3865 : 37 || 28 36 || 24 '7430 2570 .4830 1312|| .0951 .3542 615.4 3S45 || 36 || 24 40 || 25 .67452 | .3254S 1.4825 | .91366 | 1.0945 | 1.3545 | .26174 .73826 || 35 | 20 44 26 '7473 2527 .4821 | . I419 0939 .3549 6.194 3806 || 34 || 16 48 27 | . I495 25()5 .4S16 | . 1473 .0932 . .3552 6213 37S7 || 33 12 52 28 '7516 2484 .481.1 1526 .0926 | .3556 6233 3767 || 32 8 , 56 || 29 . 7537 2462 .4806 15S0 | .0919 | .3560 6253 37.47 || 31 4 50 i 30 | .67559 .32441 | 1.4802 | .91633 1.0913 | 1.3563 .26272 | .78728 30 10 4 || 31 75S0 | . 2419 .4797 | . 16S7 | .0907 .3567 6292 37US 29 56 S 32 7602 | . 239S .4792 | . 1740 .0900 .3571 6311 3688 || 28 || 52 12 || 33 7623 2377 .4788 1794 .0894 | .3574 6331 3669 || 27 || 48 16 || 34 | . TG45 |. 2355 .4783 | . 1847 . .0888 . .3578 (3351 3649 26 || 44 20 || 35 | .67066 .32334 || 1.4778 .91901 | 1.0881 | 1,3581 |.26371 .73629 || 25 | 40 24 || 36 | . T688 . 2312 .4774 | . 1955 . .0875 .3585 6390 3610 || 24 || 36 28 || 37 | . TTO9 |. 2291 .4769 2008 || .0868 .3589 6 || 10 3590 || 23 || 32 32 38 | . T730 2269 .4764 2062 0862 .3592 . 6430 35.70 || 22 || 28 36 || 39 '7752 2248 .4760 2116 .0856 .3596 |. 6449 3551 21 || 24 40 | 40 | .67773 .32227 1,4755 .92170 | 1.0849 | 1.3600 l.20469 73531 || 20 | 20 44 || 41 | . TT94 | . 2205 .4750 |. 2223 | .0843 | .3603 |. 6489 3511 || 19 16 48 || 42 | . T816 | . 218.4 .4746 | . 22.77 .0837 .360, . 650S 3491 18 || 12 52 || 43 . 7837 | . 2163 .4741 . 2331 .0830 ! .3611 . . 6528 3472 17 8 56 44 | . T859 2141 .4736 . 2385 | .08.24 | .3614 |. 6548 3452 16 4 51 || 45 | .67880 .32120 | 1.4732 .92439 || 1,0818 | 1.361S .26568 || 73432 || 15 9 4 || 46 | . T901 | . 2098 .4727 | . 2493 ,0812 | .3622 |. 6587 3412 || 14 || 56 8 || 47 | . T923 . 2077 .4723 |. 2547 .0805 | .3625 6607 3393 || 13 || 52 12 || 48 7944 | . 2056 .4718 |. 2601 | .0799 || 3629 6627 3373 || 12 || 48 16 || 49 7965 | . 2034 .4713 | . 2655 | .0793 | .3633 6647 3353 || 11 || 44 20 || 50 | .67987 .32013 | 1.4709 .927.09 || 1,0786 1.3636 | .26666 || 73333 || 10 40 24 || 51 | . 8008 1992 .4704 |. 2763 .0780 | .3640 6686 3314 || 9 || 36 28 52 | . 8029 . . 1970 .4699 ||. 2S17 | .0774 | .3644 6706 3294 || 8 || 32 32 || 53 | . 8051 | . 1949 .4695 |. 2S71 .0767 .3647 6726 3274 || 7 || 28 36 || 54 8072 . 1928 .4690 . 2926 .0761 | .3651 6746 3254 || 6 || 24 40 || 55 | .68093 .31907 || 1,4686 | .92980 | 1.0755 | 1.3655 | .26765 . .7323 || || 5 || 20 44 || 56 8115 . I385 .4681 |. 3034 . .0749 | .3658 6785 3215 || 4 || 16 48 || 57 | . 8136 | . 1864 4676 |. 30SS .0742 | .3662 |. 6805 |. 31.95 || 3 | 12 52 58 8157 | . 1843 4672 |. 3143 | .0736 .3666 i. 6825 | . 3175 2 S b6 || 59 . , 8178 . 1821 4667 | . 3197 || 0730 .3060 | . 6845 3155 || 1 4 52 60 | . 8200 | . 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T 183 | . 2817 || 44 || 56 8 || 17 | . 8561 |. 1439 || 4586 |. 4180 | .0618 .3737 l. 7203 |. 2797 || 43 52 12 | 18 | . 8582 |. 1418 | .4581 |. 4235 | .0612 | .3740 l. 7223 |. 2777 || 42 || 48 16 19 . 8603 | . 1397 || 4577 |. 4290 . .0605 | .3744 i. 7243 |. 2757 || 41 || 44 20 20 .68624 .31376 | 1.4572 |.94345 | 1.0599 || 1.3748 .27263 | 72737 || 40 | 40 24 || 21 | . 8645 |. 1355 .456S | . 4400 | .0593 .3752 i. 72S3 |. 2717 | 39 36 28 22 . 8666 . 1333 .4563 |. 4455 .0587 .3756 |. 7302 |. 2697 || 38 32 32 || 23 | . 8688 |. 1312 | .4559 |. 4510 | .0581 .3759 |. 7322 | . 2677 || 37 28 36 24 | . 8709 | . 1291 .4554 |. 4565 | .0575 | .3763 i. 7342 . . 2657 || 36 || 24 40 25 .6873() .31270 | 1.4550 .94620 | 1.0568 1.3767 | .27362 .72637 || 35 | 20 44 26 | . 8751 | . 1249 .4545 |. 4675 .0562 | .3771 |. 7382 | . 26.17 | 34 || 16 48 27 | . 8772 . 1228 .454.1 |. 4731 .0556 .3774 |. 7402 |. 2597 || 33 || 12 52 28 | . 8793 |. 1207 || 4536 |. 4786 .055() .3778 |. 7422 | . 2577 ||32 || 8 56 29 |. SS14 | . 1186 .4532 |. 4S41 .0544 .3782 |. 7442 |. 2557 31 || 4 54 30 | .68835 | .31164 || 1.4527 .94.896 1.053S | 1.3786 .27462 | .72537 || 30 || 6 4 || 31 . 8856 |. 1143 .4523 . 4952 . .0532 .3790 |. 7482 |. 2517 || 29 56 8 || 32 | . 8878 . 1122 | .4518 |. 5007 || 0525 .3794 i. 7503 | . 2497 || 28 52 12 || 33 | . 8899 |. 1101 | .4514 |. 5062 .0519 .3797 |. 7523 . 24.77 || 27 48 16 || 34 i. 8920 |. 1080 .4510 | . 5118 .0513 | .38U1 i. 7543 . 2457 || 26 || 44 20 || 35 | .68941 .31059 || 1.4505 | .95173 | 1.0507 | 1.3805 .27563 | .72437 25 40 24 || 36 | . 8962 . IO38 .4501 | . 5229 | .0501 | .3809 |. 7583 . 2417 | 24 || 36 28 || 37 | . 8983 |. 1017 | .4496 |. 52S4 .0495 .3813 i. 7603 | . 2397 || 23 || 32 32 || 38 . . 9004 | . 0.996 .4492 |. 5340 | .0489 .3816 |. 7623 . 2377 22 28 36 || 39 . 9025 | . (975 .4487 | . 5395 .04S3 | .3820 |. 7643 . 2357 || 21 24 40 | 40 .69046 .30954 || 1.4483 .95451 | 1.0476 1,3824 .27663 | .72337 20 | 20 44 || 41 | . 9067 . 0933 | .4479 |. 5506 || “.0470 .3828 |. 7683 |. 2317 | 19 16 48 I 42 | . 9088 |. 0912 .4474 |. 5562 .0464 .3S32 i. 7703 . 2297 | 18 || 12 b2 43 | . 9109 | . 0891 .4470 | . 5618 .0458 | .3836 |. 7723 . 2277 || 17 | 8 56 || 44 . 9130 | . 0.870 .4465 |. 5673 .0452 .3839 |. 7743 | . 2256 | 16 || 4 55 || 45 .69151 .30849 | 1.4461 | .95729 | 1.()446 | 1,3843 | .27764 | .72236 15 5 4 || 46 | . 9172 . . 082S .4457 | . 5785 | .0440 | .3S47 i. 7784 . 2216 || 14 56 8 || 4-7 | . 9193 ) . 0807 .4452 |. 5841 | .043 p .3851 |. 7804 |. 2196 || 13 52 12 || 48 . 9214 | . 07S6 .4448 -|. 5896 .0428 .3855 |. 7824 |. 2176 || 12 || 48 16 || 49 . 9235 | . 0765 . .4443 |. 5952 . .0122 | .3859 i. 7844 |. 2156 || 11 || 44 20 || 50 | .69256 .30744 1.4439 .96008 || 1.0416 | 1.3S63 .27864 .72136 || 10 | 40 24 || 51 . 9277 | . 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IS99 1885 3670 8096 3 99 || 69 || #16I 980S g063' || 6530 gz99 * | I69p" || 3 Igſ) 1856 " | I # 09 || #86 Il 990.83" .. 3068 I gç80'I 69g.96 g69:5"I tºg03 99769 || 0 *S*PW JW "ouſsoo ligS's IAI" ºut:00S ‘5utºgoO || “5ueſ, oku,oosool-soo's. A “outs | W: u6 o98.I "suogoum, I Tuoi.1%auouro31.1L Turnºunſ off, £93 'SSINIT IVºIſhiv N. 25; MECHANICS.—STATICS. - M E C H A N I (; S. . Mechanics is that branch of Natural Philosophy which treats of the action of force, motion, and power. Mechanics is divided into four parts, namely, Statics the science of forces in equilibrium. Dynamics, the science of forces in motion, it produces power or effect. Hydrostatics, the science of fluids in equilibrium Hydrodynamics the science of fluids in motion, its causes, power or effects. Statics, Levere Momenturme * Lever is an inflexible bar, supported in one point called the Fulcrum, or, centre of motion. The length of a lever is measured from the fulcrum to where the force or resistance acts, (when the force acts at right angles to the lever) or, the length of a lever is measured from the fulcrum at right angles to the direc. tion of the force. W= Weigl and l = lever for W e F = gº and L = lever for %} See Fig. 139. Momentum is the product of force or weight, multiplied by the length of the lever it acts upon. The products Wl and FL are called Statics Momentwms; when these ino- mentums are equal there will be no motion, and the weight W will balance the force F. When one momentum is greater than the other, there will be a motion, and the velocity of that motion is measured by the difference of the In 10 mentulus. Levers are of three distinct kinds, with reference to the relative positions of the Force F. Weight W, and Fulcrum C. 1st. Fulcrum C, is between the force F, and the weight W. 2d. Weight W, is between the fulcrum C. and the force F. 3d. Force I'', is between the fulcrum C, and the weight W. Example 1. Figure 139. The weight W = 68 pounds, the lever Z = 3-86 feet. and L = 10 feet 6 inches. Required the force F=? W7 68×3-86 Jormula 1. F=+=-ija- = 25 pounds nearly. a = distance between the force F and the weight W. The formula 3, 4, 7, 8, 11, 12, are for finding the fulcrum C, when the force F, weight W, and the distance a, are given. Eacample 2. Fig.140. The force F= 360 pounds, W = 1870, and a = 8 feet, 4 inches. Required the position of the fulcrum cf __F'a 360X8.333 2999-988. to: Formula. 7. l = WLF - 1570-360 - Ti5IOT - 1986 feet. L = 8.333-1-19-86 = 28-193 feet, the answer. Example 3. Fig.144. The weight of the lever is Q = 18 pounds. The centre of gravity is a = 2.25 feet from the fulcrum. W= 299 pounds, l = 5-5 feet, and I.- 11°95. Required the force F = ? in pounds. Wl — Qz 299X5'5–18×225 134. Jº'- E–– 11-95 =134:25 pounds. Inclimed Plannee Example 4. Fig.163. A load W = 3466 pounds, is to be drawn up an inclined plane, l- 638 feet long, and h = 86 feet high. - What force is required to keep the load on the inclined plane? h W 86x3466 F=T = ~633 = 4672 pounds. MEUBANICS.–STATICS. 255 Example 4. Fig. 167. A Cylinder of cast iron, weighing W- 5245 pounds, is to be rolled up aſ inclined plane; the angles v = 18° 20' and v' = 8° 10' What force is required to keep the cylinder on the plane? F= W: sin.(v4-v) =5245Xsin.(18° 20' +8° 10') = 2340 pounds. Eacample 5. Fig. 168. An iron ball which weighs 398 pounds, is tied to an in- clined plane with a rope; the angle of the rope and the inclined plane is v' = 16° 40', and v = 14° 30'. What force is acting on the rope? _ Wsin.v 398Xsin.14° 30' F= *...* = **śiº = 104 pounds. Eacample 6. Fig. 153. What force F is required to raise a weight W = 8469 pounds, by a double moveable pulley? F= # W- #X8469 = 2117-25 pounds. Example 7. Fig.156. How much weight can a force F= 269 pounds lift by three compound moveable pulleys? W= 2*F = 28×269 = 2152 pounds, the answer. Screwe Example 8. Fig. 172. What force is required to lift a weight W- 16785 pounds, by tº screw, with a pitch P= 0-125 feet, the lever being r = 5 feet. 4 inches 7 __WP 16785X0.125 so. Jº'- Tºrr T 2×3-14535333 T 62.62 pounds, the answer. Including friction the force F will be P = W(P+f d m) 2 ºr r " Find the friction f oth page 267. d diameter of the screw in feet. Wedge. Example 9. Fig. 169. The head of the wedge a = 3 inches, and length l_z= 16% inches; the resistance to be separated is R = 4846 pounds. Required the force F = ? (Friction omitted.) _ 4846×3 16-5 Uncluding friction the force F will be, l46 F-Rſ 4 +f(2+ £)] in which the friction f is to be found on page 267. Catenarias Example 10. An iron chain 256 feet long, weighing 1560 pounds, is to be sus- ponded between two points in the same horizontal line, but 196 feet apart. How deep will the chain hang, unler the line of suspension, and with what Jorce will the chain act at the points of suspension? Figure and Formula 161.we have given, W= }×1560 = 780 pounds, l = #x256 = 128 feet, and a = }X196 = 98 feet. li h = 0-6525)./1282 – 98* = 53.73 feet, the required depth under the horizontal Il6, ū 2X53-73 cot.w = &: = 1.096, or v = 44° 44', and 2v = 89° 28′. The required force will be, F= = 881 pounds. 7SOXsin.440 44" Tin Gº" = 5:49 pounds. 256 LEVER AND STATIG Monentum, - H-. > W 7 FA, * {Z * F. F = L 2 W--F, 1, 2, £ WT 13 -: *, + = "º. 8.4 T WHF, T WTF? 2 * 9 140. F : W = l : L, FL = Wł, W 7 FI, F = — W = - L 9 7 * 5, 6, W a º F'a ++ ºr i = Wºr 7, 8, 141. F: W = l ; L, FL = W l, W 7 FI, = -E-, W-ir, 9, 10, W 0. _Fa T = FHF, l = F * if: 11,12. 142. af-Half-Ha'f'' = br-EU''--ly'r”. If the sum of the momentums that act to move the body in one direction are equal to the sum of the momentums that act opposite, the acting forces will be in equilibrium; c being the centre or fulcrum. 143. To find the fulcrum c when three forces act on one lever R. a = Q(a—b — a Y--P(a — ar), * = Qa4-Pa – Q b ~ TRIQ+P 144. Q = weight of the lever. . a = distance from the centre of gravity of the lever to the fulcrum. Bulance the lever over a sharp edge, and the centre of gravity is found. a WZ –Qº w. FL-Qa l -: —E-, W-----. LEVER AND STATIC MOMENTUM. 257 S. 145. Fºwl rººf, Fr Wºr, TWr W1. ==#, R = }, w-º', ; – J. Tr 146 W 7, r F R R' = ++, W= ** 71 = number of revolutions of the wheels, m : mº = '': R, v : v' = rſ' : RR', v = velocity of W, v' = velocity of F. iT48. W r +'," W = F R R'R'' =-E-F#7, " - –Frºſſ-, ’’ = 7'7" : R R", v : v' = r r"r" : &c. = radii of the pinions. R R'R'' &c. = radii of the wheels. Let P and Q represent the magnitudes and direc- tions of two forces which act to move the body B. By completing the parallelogram, there will be ob- tained a diagonal force F, whose magnitude and di- rection is equal to the sum, P and Q. F is called the resultant of P and Q. 149. If three or more forces act in different directions to move a body B, find the resultant of any two of them, and consider it as a single force. Between this and the next force find a second resultant. thus: P. Q, and Ie are magnitudes and directions of the forces. P+Q = r, r-HR = F= P + Q+R, or F. is the magnitude and direction of the three forces, P, Q, and R. 150. A force Q acting (alone) on the body B, can move it to a in a unit of time, another force P is able to move it to b in the same time; now if the two forces act at the same time, they will move the body to c. c is the resultant of a and b. PULLEYS. 151. PULLEYs.—A single fired Pulley. F: W = R : R, or F = W, * = v. 152. A single moveable Pulley. F: W = R : 2R, or F = % W, If the force Fheing applied at a and act upwards, the result will be the same. v' = 2x. 153. A double moveable Pulley. F: W = R : 4R, or F = }W, w’ = 4v. A double moveable Pulley. F: W = R : 4R, or F = 4 W. F - ". w: v' = 1 : 2w. 2w 155. Quadruple moveable Pulley. F = W. F. : W = R : 8R. Let w = any number of rnoveable pulleys, then F-7 2. w : v' = 1 : 2w. 156. Compound Pulleys. 11 = number of moveable Pulleys. W F– gº, W–2°F, p : to" - 1 : 2". FUNicular AND CATENARIAN. 259 157. An oblique fired Pulley. F: W = sec.z : 2, * - F-Wseez. SéC.Z. W = 158. W: F = sin.(a +v): sin.a., _ W sin.a. f = W sin, v sin.(TTV)' " Tsin.(ºvy W: f = sin.(a +v): sin.v. 159. = * sim.a. sin.[a-Fv)' W sin.w f sin.(u-Hz) 160. W = P+Q+R+S, F=F", f=f', - Winº, f_Wint. T sin.(w-Hv)’ sin.(a +v) 161. l= }(2a+ Vaº F9F). t = length, and W- weight of half the chain, fº, 2h W. Sin. V • F - :- -- - t º ºg COt.2) , F sin. 2 V’ W'tan.” 62 162. W. Sin.v F sin.2v ' f = W tan.v, 2h cot. w := —e & | W = weight of halfthe number of balls. 260 I NºLINED PLANE. 168. F_ wº = W sin.v, W– £' -- ", A sin.v 20 = y = W COS.v. 164. F = W sin.(v4-v'), -- F Tsin.(v4-v')' W = W cos.(v4-v'). 165, g _2~ F = Wsinº, COS, 25 _ F cos.v' sin.” ” w = W (cos.v-Hsin.v. tanv'). 166. º solve an Inclined Plane by diagrams. F = magnitude and direction of the force, which is obtained by completing _y the parallelogram. X- By calculation see Formula, Fig. 163. 167. W = weight of the body, and direc- |tion of the force of gravity; to be drawn 'at right-angles to the base b, and F par- allel to F. By calculation see Formula, Fig. 164. sº g 168. w = the force with which the body presses against the plane, to be drawin at right-angles to the plane l ; then the parallelogram is completed. By calculation see Formula, Fig. 165. WEDGE AND SCREWS 261 169. Wedge. F 7 ** , R – "º. (3 ſ' = F = force acquired to drive the wedge. 170. Let the line I'represent the magnitude and di- rection of a force acting to move the body B on the | line CD; then the line a represents a part of F which presses the body B against CD, and the line b represents the magnitude of the force which actually moves the body B. b = V F* – dº, b = F cos.v. 171. F. : W = h : b = sin.v : cos. v = tan.v. F = º = W tan.v. F = F. —-mºsé W= Fº == F = F cot.v. /. tan.v 172. Force by a Screw. P = Pitch of the screw, r = radius on which the force Facts. F: W = P : 27, r. WP F27. W = —ºtº- F=-2; --- P e 173. Force by Compound Screws. P = Pitch of the large screw, p = Pitch of the endless screw. R = radius of spur-wheel for the endless SCI'êW. W; F = 472 R r : P p. W. P. r W = F4° ºr Ry 472 R r" --PF- On the spur-wheel is a cylinder by which the weight W’ is wound up, the formula will be, r' = radius of the cylinder, and RA F. : W = p r": 2n R r. F = W. gr. w_127 ºr 27, R r p r f := Ajºs Sº tº 3. G ſ º º ~ # *: º º º * § *ºss _F 262 T)YNAMICS. Dynamics. Quantity is that which can be increased and diminished by homogeneous parts. Quantity is of two kinds,-geometrical and physical. Element is that which cannot be dissolved into two or more different quantities. Function is the product of two or more different elements. Jorce, motion, and time are simple physical elements; and Power, space, and work are functions or products of those elements. Force is a mutual tendency of bodies to attract or repel each other. Its physical constitution is not yet known. We only know its action on matter, which is recognized as pressure and measured by weight. Force is the first element of power and work. It is here denoted by the letter P'in pounds avoirdupois. Motion is a continuous change of position, recognized as velocity. It is the second element of power and work, and here denoted by Vin ft. per second. Time implies a continuous action recognized as duration. It is the third element of work, and here denoted by Tin Seconds. Power is a function of the two first elements, force F and velocity V, or the product obtained by multiplying together the force F and velocity V, denoted P = F W. Power P is expressed in footpounds, and called dynamic effect, of which there are 550 in a horse-power; or, if the velocity is measured in feet per minute, there will be 33,000 footpounds in a horse-power. Space is a function of the second and third elements, velocity V and time T. denoted by S = V T, which means that the Space S is the product of the velocity V and time T, expressed in linear feet. ork is a function of the three elements force F. velocity V, and time T. denoted by K = F W T, which means that the work K is the product obtained by multiplying together the three elements F. V., and T. Work may also be de- noted by K = F S, where it appears as if the work was independent of time; but the time is included in the space S = V T. Work is also denoted by K = PT, or the product of the power P and the time T, where it appears, as if work was independent of force and velocity, which latter are included in the power P. Either of the three cases expresses the work K in units of one pound lifted one foot, called footpounds. For large quantities of work, this unit is too small. The work done by one man in a day may amount to some two millions of footpounds. It is therefore proposed to adopt an additional unit for work, namely, the work a horse can perform in the time of one hour, which is equivalent to that of eleven men working one hour, or of one man working eleven hours, to be called a Workman day, and denoted by the letter k. This unit is to express great amounts of work, such as the building of a house, bridge, canal, ship, large fly-wheels, and to express the magnitude of steam-boiler and gunpowder explosions; also the capacity of heavy ordnance, to be noted in workmamdays. Qme horse-power = 550 effects = 11 men's power. One man's power = 50 effects = 0.0909 horse-power. Example 1–What force F- ? is required to generate a power of H = 54 horses, with a velocity W = 15 feet per second? Find the formula for force, which contains the given quantities H and V, which is, 550 EI 550 × 54 JF = —F-- º = 1980 pounds, the answer. Example 2-The pull of a belt over a belt-pulley is F = 350 pounds, the ra- dius of the pulley is r = 1.5 feet, making m = 52 revolutions per minute. Re- quired the horse-power H = ? transmitted by the belt. See formulas for circular motion. _F r n 350 × 1.5 × 52 H==555- - TE250- = 5:2 horses. Example 3–How many workmandays k = ? are required to raise a load of F= 72968000 pounds, a height of S = 25 feet. F. S. 72968000 × 25 k = 1930000 T 1980000 = 911'212 workmandays, the answer. IXYNAMICAL FORMULAs. 263 4 Dynaamical Formulas for Uniform Motion. Space in Feet passed through in the Time T. & = V T. F=-“.. V v=_5. T S T = −. y P = F W. P H = −. 550 A – F. V. T. _ F. W. T. TT930000 y = ** r *. 60 P = *** * * 60 F 5250 H. 7" 7", n = revolutions per minute. N = total no. of revolutions. s = **'. s = 380 "A. s = K. AF F . F Force or Pressure in Pownds. f _ 550 H. F – Fº Ye F_1980000% y S S Velocity in Feet per Second. F F F. T Time of Duration in Seconds. T = S F. • -: _F S_ . T = _K e An 550 FI F. W. Power in Effects. P = **i. P = 550 H. P = #. Z! Borse-power. H = * * __* 8 = * 550 550 Ti 550T Work in Footpounds. A = P 7. A = F S. Jº — 550 H. T. Workmandays. - F. S. _ P 7 k = F. T 1980000 “TT930000 “T 3600 Circular Motion, n = * Z. = * Z. S-2 ºr N. 2 TT r 2 T 7, F2T r *. F* * R = F2Trr N. 550X60 5250 T 60 N. _5250 H - A tº 7. F ºr F 2 ºr " Tr = 3•14159265. r = radius of the circle in feet. º d | 264 OBSERVED RESULTs of PowLR. () BSERVED RESULTS OF POWER. Work. Description of Works. hrs. per Force. Veloc'y day. F V A man can raise a weight by a single fixed pulley, º * º º tº & e tº 6 50 0.8 A man working a crank, . - e tº 8 20 2.5 A man on a tread-wheel (horizontal), e 8 144 0.5 A man in a tread-wheel (axis 24° from ver- tical), • e º e e º e e 8 30 2.3 A man draws or pushes in a horizontal direction, . * º e e © º º 8 30 2 A man pulls up or down, . º © te 8 12 3.7 A man can bear on his back, . e º 7 95 2.5 A horse in a horse-mill, walking moderately, 8 106 3 A horse in a horse-mill. running fast, º 5 72 9 An ox in a horse-mill, walking moderately, 8 154 2 A mule Kg {{ {{ 8 71 3 An aSS $6 £6 {{ 8 33 2.65 On bad Foot-roads, like those in Peru. A man can bear, . º e © ſº º 10 50 3.5 Llama of Peru can bear, . º o e 10 100 3.5 Donkey can bear, . ſº º º e o JO 200 3.5 Mule can bear, . © e e º & 10 400 5 Flour Mills, For every 100 pounds of fine flour ground per hour, require, . . One pair of nuill-stones of 4 feet diameter, making 120 revolutions per minute, can grind 5 bushels of wheat to fine flour per hour, . One pair of mill-stones of 4 feet diameter, making 120 revolutions per minute, can grind 5 bushels of rye to coarse flour per hour, . Saw Mills, alternative. For every 20 Square feet sawed per hour, in dry oak, there re- quires, . e e - º -> e º e e e o e iry pine, 30 square feet per hour, g e e e e e Circular Saw. A saw 2.5 feet in diameter, and making 270 revolutions per minute, will saw 40 square feet in oak per hour, with . tº o In dry spruce, 70 square feet per hour, . e º © e Threshing Machime. Velocity of the feed-rollers at the circumference, 0.55 feet per second. Diameter of threshing-cylinder 3.5 feet and 4; feet long, making 300 revolutions per minute, can thresh from 30 to 40 | bushels of oats, and from 25 to 35 bushels of wheat, per hour, º one man with a flail can thresh half a bushei per hour (wheat), Rolling Mills. Bar iron-mills. Two pair of rough rollers, two pair of finishing rollers, six puddle furnaces, two welding furnaces, making 10 tons of bar iron per 24 hours, rollers making 70 revolutions per minute, Te ulre, o o º © Q ſº o º º e © e © & Hºm requires about five horses per square foot of plates rolled. Hargest size plate rollers should not make over 30 revolu- tions per minute. P 40 50 72 69 60 44.4 237.5 318 648 308 213 87.4 550 550 550 550 2200 70 29000 Effects. Horses. FI 0.072 0.090 0.130 0.125 0.109 ().080 offin 0.165 0.558 0.308 0.160 1.000 4.36 2.91 1.000 1.000 1.000 1.000 4,000 0.127 80 I)REDGING MACHINEs. 265 T) R. E. D G IN G M A C H IN E S. Letters denote. Formulas. T= tons of materials excavated h and raised per hour. FH-T (Fù +k) - - - 1 h = hight in feet in which the ma- 700 terials are raised above the bottom of the excavated 700 H. channel. T = —— • - - 2 0-1 for hard clay with gravel. h—HT00 k 0.07 for hard pure clay. 0-05 for common clay or sand. 550 H 0-04 for soft clay or loose sand. F= 0-03 for very loose materials. 7) horse power required for ex- cavating and raising the ma- 550 Tk terials. F = F= force in pounds required to 19 feed the Dredge ahead. v = velocity of the buckets in feet H , h, per Second. k = ~ * = * i - H e tº 5 Ea'ample 1. What power is required to excavate T–160 tons of hard pure clay per hour, and raise it up h-25 feet above the bottom of the channel? For hard clay k=0.07. & 25 H = 160 (700 +0.07) = 16:9, or 17 horses. Ea'ample 2. What force F=? is required to feed the Dredge ahead for the above example when the buckets move v-1 foot per second. F= * L E A T H E R B E L T S. Detters denote. b = breadth of leather belt in inches, H = horse power transmitted by the belt. w = velocity of the º in feet per second. d = diameter in inches n = revolutions per intute} of the smallest belt pulley. F= force in pounds transmitted by the belt. a = number of degrees occupied § the belt on the small pulley. S = safe working strength in pounds per inch of width of belt, which, for oak-tanned leather # inch thick, cemented and riveted joint, can be taken at 100 pounds, and less in proportion for weaker belts. 5- 80,000,000 II, - 1 | H = d.” F. = 92.95 pounds. * - - →-, - - - - - 4 d n a S 126,500' b = |*4, - - - - - 2 | H = ****ś-, - - - - - 5 w a S - 30,000,000 F=º004, - - - - - 3 | H = **ś, - - - 6 d n. 130,000 Example. A leather belt is to transmit H = 75 horse power over a pulley d = 36 inches in diameter, making n = 80 revolutions per minute, angle of contact a = 170°, and the safe working strength S = 100, pounds per inch of width. Itequired the width of the belt. Formula 1. Width b ==*** =46 inches, nearly. 36 × 80 × 170 × 100 - 266 FRICTION. F RI (; T I 0 N. THE resistance occasioned by Friction is independent of the velocity of mor tion; but the re-effect of friction is proportional to the velocity. Friction is in- dependent of the extent of surface in contact when the pressure remains the same, but proportional to the pressure. This law was established from experi- ments by Arthur Morin in the years 1831-32 and 1833, from which a summary is contained in the accompanying Table. Jetters denote. a = Fibres of the woods are parallel to themselves, and to the direction of motion. b = Fibres at right-angles to fibres. c = Fibres vertical on the fibres which are parallel to the motion. d = Fibres parallel to themselves, but at right-angles to the motion, length by length. e = Fibres vertical, end to end. Example. A vessel of 800 tons is to be hauled up an inclined plane, which inclines 9° 40' from the horizon; the plane is of oak, and greased with tallow. What power is required to haul her up? The coefficient for oak on oak with continued motion is f = 0.097, say 01, then, 800Xsin.9° 40′ = 800X0-16791 = 124:328 tons, the force required if there were no friction, and 800×cos.90 40"XO-1 = 800X0-9858×0.1 = 78.864 tons, the force required for the friction only, and 134'328 78.864 213-192 tons, the force required to haul her up. f lºst lost by friction in axle and bearings is expressed simply by the OTIIllll:8) º Tz d Wºn f_ Wä n f 12-60 T 230 2 in which W- the weight of pressure in the bearing, d = diameter on which the friction acts in inches, n = number of revolutions per minute, and f = co- efficient of friction from the Table. In common machinery kept in good order the coefficient of friction can be assumed to f = 0.065, then _ Wän H = Wd n_ T T353) , T 1941500 Eacample. The pressure on a steam-piston is 20000 pounds, and makes n = 40 double strokes per minute. Required the friction in the shaft of d = 8 inchest 20000×8×40 1941500 Friction in Guides. P = JH = =33 horses, the loss by friction. W= pressure on the steam piston in pounds. S = stroke of piston in feet. ! = length of connecting rod in feet, H = horse power of the friction. - W. S. 7, 350000.75x sº Ezqmple, . The pressure on a steam piston being W = 30,000 po tº S = 4 feet, length of connecting rod l = 7 feet, j making 50 §: Iminute. Required the horse power of the friction H = ? 30000X4X50 º *-āś- 113 horses. FRICTION. 267 TARLE OF FRICTION IFOR PLANE SURFACES IN CONTACT. . Rind of Materials in contact. *:::£cated zºº.ºw. Oak on Oak, - - - - 0. 0. 0.478 0.625 44 £6 * * * * , tallow 0.097 0-160 & 66 º & gº º 99 lard 0-067 © tº º ºs & 66 sº º ºs tº b O 0-324 0-540 6& {& e ‘º e - 2, unctuous 0-143 0-314. &G “ - - - - » i tallow 0.083 0-254 64 £6 se tº ſº - 2, Water, 0-25 tº º - © 6& 66 º tº sº - d O 0°336 tº ºn tº $ & &g e sº º º 0 O 0-192 0.271 66 {& º tº cº º (3 O º º e & 0°43 Cast-iron on Oak, * * Q: O 0.400 0.570 <& 6& tº º » 808p 0.214 e - e G &g &g tº tº - 2 | tallow 0.078 0-108 Wrought-iron on Oak, - - 59 O 0.252 © tº º e (£ 6& gº º » | tallow 0.078 tº e º 'º Wrought iron, together, - - 0. O 0-13 0-13'ſ 6& 6& e - a unctuous 0.177 © e º 'º {{ 66 º º 25 tallow 0-0S2 - © tº e &ć £g º - » olive oil 0-070 0-115 Wrought on cast-iron, - - Q, O 0-194 0-194 {{ 46 tº º » unctuous 0-18 0.118 &&. & 4 tº º 3, tallow 0-103 0-10 6& 4& tº sº. 2, Olive oil 0.066 0-100 Cast-iron on cast-iron, - - Q Water 0.314 0'314 &ć &G tº - 22 | SOtup 0-197 * - e. e. 66 &&. e tº » | tallow 0-100 0-100 {& &C. e - » olive oil 0.064 tº º º Wrought-iron on brass, - - Q, O 0-172 e e º & {{ & 4 º º 33 unctuous 0° 160 e e º 'º 6& &&. º º 33 tallow 0.103 G. e. e. e. &ć & º º » lard 0.075 * @ e {& £& tº - » | Olive oil, 0.078 e e º 'º Cast-iron on brass, * - Q. O 0-147 © tº º 66 6& º º ºr » unctuous 0-132 º e º º 6& &G --> tº º 22 | tallow 0-103 • e º º 6& & 4 - * > º 53 lard 0.075 • * ~ * &g - ge - G - » olive oil 0.078 e a • e Brass on brass, º e ºs Q, O 0-201 e - © tº {{ “ - - - - 25 unctuous 0-134 & © tº e &6 * - - - - » olive oil 0-053 • * * > Steel on cast-iron, - - - 33 O 0-202 tº e º ºs 4. &é º º - 35 tallow 0-105 • * ~ e &ć &é - tº - 35 lard 0 081 tº e º º 6& 66 - º a olive oil 0.079 e FRICTION OF AXLES IN MOTION. Oil, Tallow, or Hog's Lard. Dry or slightly Supplied in the The grease Designation of surface in greasy, or wet. ordinary continually contact. 7).0.717&r. running. Brass on Brass, - - - e tº gº tº º 0.079 e e - 4 - “ on cast-iron, - • * * * * 0.072 0.049 Iron on Brass, - gº º 0-251 0.075 0-054 “ on cast-iron, - - * * * > e 0.075 0-054 Cast-iron on cast-iron, º 0-137 0.075 0-054 &ć on Brass, - tº 0-194 0.075 *654 Iron on lignum-vitae, e 0.188 0.125 ! . . . . . Cast-iron on “ tº 0.185 0-100 0-092 Lignum-vitae on cast-iron, . . . . . 0-116 0-170 268 STRENGTH OF MATERIALS, S T R E N G T H 0 F M A T E R I A L S. TABLE I, shows the weight a column can bear with safety; when the weight presses through the length of the column. The tabular number is the weight in pounds or tons per square inch on the transverse section of a column vº a length less than 12 times its smallest thickness. Table Is RESISTANCE FOR COMPRESSION. 174. Kind of Materials. Pownds. Tons. Oak, of good quality, - - - - 432 0-1885 Wr Oak, common, * º - º º 2SO 0°125 Spruce, red (Sapin rouge), - - - 540 0-241 & & C white, (Sapin blanc), - - 140 0-6256 Iron, Wrought, - * * - -> tº 14400 6-4 Iron, cast, - tº ºp 4- uº º 28750 12-85 Basalt, - - - - - - - 2S75 1.285 Granite, hard, e e º 'º º 1000 0'446 6& common, - - - - - 575 0.256 h Marble, hard, - - - - - - 1435 0-640 G& common, - gº - tº de 431 0.192 Sandstone, hard, - - - - - 1295 0.577 & C loose, - tºº tº- - º 5-6 || 0-0025 Brick, good quality, is e º º 175 0.078 “ common, - º - e º 58 0.0259 Lime-stone, of hardest kind, - - - 720 0.321 &G common, - tº º sº 4; §, Plaster-Paris, - cº- * > • - tº- 6 •03 Mortar, good quality, and 18 months old, 58 0-0259 $ Do. common, - ū - e tº 36 0.016 = When the length or height of the column is more than 12 times its smallest thickness, divide the tabular weight by the corresponding number in this Table. 30 36 42 48 54 60 2-8 4 5 6 8 12 Example. A building which is to weigh 2000 tons is to be supported by piles of Sapin rouge Spruce 18 feet in length, and 12 inches diameter. How many piles are required to support the building? Length}(thickness 12 | 18 24 Divide by 1.2 | 1.6 2 *xr.sºn = 17 tons, the weight which each pile can bear, and * = 118 piles. Professor Hodgkinson’s Formulae for Crushing Strength of Cast Iron Pillars. The ends of the pillars should be perfectly flat and square, and the load to bear even on the whole surface. T=crushing weight in tons. D=outside and d inside diameters in inches. l-length or height of pillar in feet. T=46.65 (? 3.58–d 3.45 ) ! 1" -—— * * 24 Table showing the Weight in tons which Cast irom Pillars or Tubes can bear with Safety, Diameters in Imches. For Tubes subtract the weight due to the bore, L 1 2 3 4 5 6 7 s 9 10 1 11 12 13 14 15 16 17 18 20 24 e- tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons, tons tons, 1 22 |218 1156|3256||7272/14...!244. |395..]6..... 88... 124. 169.226..|296.1379..[478.59....]73... [1...... 2..... ; 6-8 (82.5 |355 1000|2238|4315||7500||1315|85.1.2.1438.323-397.|910.|113-14ſ. 13.22.32....[63. 3 ||3-5 141-5 178 |500 1123|2166||3744:61019322}136.162.262.1350. 457.|586.739.1920. 113.165.318. 4 2-1 125'5 ii) 308 |689 |1328|2313|3741|5716}8354il] 77|16.1..]2148|280.359..]453.564. |693. 101..] 195., 5 1-4 17.4 75 211 |471 |908 |1583|2560|3912|5716|8056||110..jl470|1919;246.310.jë86.j474..[693.133. 6 |1-0 |12:8 55 155 346 |666 1161|1878|2869|4192|5909|8082|1078|1407|1804|2277|283..]347. 508..]980. # 0-81 |9-8 42 |119 |266 513 S94 ||1444|2207|3226|4547 6219.8296 ||1083||1388|1752|217.. 267. 391. 754. § 0.65 |7.8 36 |95 |212 |408 |7|12 |1151|1760|25713624|4956|6611 8633||1106il 396 173. 213. 311. 601. § 0.53 |6-4 28 |78 |174 |335 |583 942 |1440.2104|2966|4057|5412706619059|1142|142.174.255.491. 10 |0-44 5-3 |23 |65 |145 |280 487 788 1200||1760|2479) 3391 |4524|5907|7574|9554|118. I46..]213. 411. 11 |0-38 4-6 |20 |55 123 |238 |414 |670 1024|1496 |2108|2884|3847|5024|6441|8125;101.124..]181.j949. 12 0.32 3.9 |17 |48 iſ 06 |205 |357 |578 |883 |1292|1818|2487|3318|4333|5555 |7008|8717|107.156.301. 13 028 13:4 |15 |42 |93 179 |312 507 |771 |1126|I587|217 1|2896||3782|4848]6116|7608|9346||136.263. 14 025 3-0 |13 ||37 |82 158 |275 445 |679 |993 |1400|1914|25533333|4274|5392|6707|8240|120.232. 15 0-22 |2-7 11:5 (33 73 140 i250 is 96 j604 |883 |1244; 170222712965|38014795|5965||7328)102.206. 16 |0:20 |2-4 10 |28 65 |126 |218 |354 |541 (792 |III5|1526|2035|2657|3406|4297.5345.6551}9590|184. 17 |0-18 |2-3 |9.4 |26 |58 113 |198 || 320 489 |714 ||1006| 1376|1836|2397 |3073|3876|4712|5923|8656|166. 13 |0-16 |2-0 |8-5 |24 |53 |103 174 290 |443 |648 |913 | 1248|1665|2175|2788|3517|4375|5375||7854|15I. 19 |0:14 |18 ||7-8 |22 |49 93 |163 |265 |404 |591 |832 |1139||1519||1984|2543|3208|3991|4903|7165||138. 36 |0-13 |1-6 |7-1 |20 |45 |86 150 |243 ||371 |541 |763 1044|1392|1818|2331|2940|3658||4497|6066|126. These values are about half of that by Prof. Hodgkinson's formula. The points after the numbers mean ciphers. : § 270 STRENGTH OF MATERIALS. Table IIe COHESIVE STRENGTH PER SQ.. INCH OF CROSS-SECTION. ſ: g Just tear asunder. With safety. Eind of Materials. Pownds. | Tons. || Pownds., Tons 17 Cast Steel, - - - 13425b 59-93 || 33600 || 14-98 5. Blistered Steel, tº º 133152 || 59°43 || 33300 || 14-86 Steel, Shear, - - - 128632 || 56.97 || 32160 || 14-24 Iron, Swedish bar, - - 65000 || 29°2 16260 7-3 “ Russian, * * * > 594.70 || 26'ſ 14900 6-7 “ English, ge ſº 56000 || 25-0 14000 6°29 “ common, over 2 in. Sq., 36000 | 16:00 || 9000 4-0 “ sheet, parallel rolling, 40000 || 17-85 || 10000 4:46 “ at right angles to roll, 34400 15:35 || 8600 3-84 Cast iron, good quality, - 45000 || 20:05 || 11250 5:00 6& inferior, - - 18000 || 8'03 4500 2°0 Copper, cast, - - - 32500 14:37 8130 3-6 “ rolled, º e 61200 || 27-2 15300 6-8 Tin, cast, - - - - 5000 || 2:23 || 12500 0-56 Lead, cast, - - - 880 || 0-356 || 220 0-09 “ rolled, - - - 3320 | 1-48 || 830 0-37 Platinum, wire, - - 53000 || 23-6 I3250 5-9 Brass, common, - - 45000 || 20:05 || 11250 5-0 Wood. Ash, * Gº tº º 16000 || 7-14 || 4000 1-87 Beach, - - - - 11500 || 5-13 || 2875 1.28 Box, - - - - - 20000 || 8-93 || 5000 2°23 º Cedar, - tº as tº II400 5-09 2850 1-27 Mahogany, - wº ſº 21000 || 9°38 5250 2:34 {{ Spanish, tº 12000 5-36 3000 1°34 Oak, American White, - 11500 5:13 || 2875 1-28 “ English 6t. ſº 10000 4:46 || 2500 1-11 “ seasoned, - - - 13600 || 6-07 || 3400 1-52 Pine, pitch, tº tº 12000 || 5°35 3000 T-34 “ Norway, - - gº 13000 || 5'8 8250 1°45 Walnut, - - - - 7800 || 3-48 || 1950 0-87 Whalebone, - - - 7600 || 3:40 1900 0-85 Hemp ropes, good, - - 6400 || 2.86 || 2130 0-95 Manilla, ropes, & e tº . 3200 I-43 1100 0.49 Wire ropes, - - 4. 38000 || 17 12600 5-36 Iron chain, - sº gºe 65000 29 21600 9-38 “ with cross pieces, - 90000 | 40 30000 | 13-4 To Find the Cohesive Strengthe RULE.—Multiply the cross-section of the materials in square inches by the tabular number in Table II., and the product is the cohesive strength. Example. An iron-bar has a cross-section of 2.27 sq. in. How many tens are required to tear it asunder, and how many pounds can it bear with safety? English iron 2,272.25 = 56-75 tons, which will tear it asunder, arid it will bear 2:27.2:14000 = 31780 pounds. with safety Inches and 16ths, whi. per fathom. Chain. 220 246 278 302 332 365 399 435 , 472 553 638 729 825 1072 1288 1559 1854 Diam. : •º, 0 : IR er cent. s acturers, Philadelphia. : in. II • CHAIRs, HEMP AND WIRE ROPEs. 271 Price per fathºnºurs. l Chaim. Herap. Wire, ||Chain. Hemp, Wire. Stysin. Pounds Pounds'Pounds s cts.] § cts.] § cts.]] Cwt. 0.23 || 0-08 || 0-06 || 0-15 || 0-06 || 0-08 || 2-6 0-93 || 0:47 || 0-24 || 0-25 || 0-12 || 0-15 9 2-11 || 1-06 || 0-54 || 0°36 || 0-17 || 0-22 20 3.75 | 1.89 I-10 || 0-48 || 0-25 || 0-32 35 5-86 2-94 | 1.83 || 0-60 || 0°33 || 0°43 55 8’45 || 4-52 2.56 || 0-96 || 0:42 || 0-54 80 11-5 6.09 || 3:42 || 1:25 || 0-48 || 0-62 || 109 15-0 7.55 || 4:39 || 1:44 || 0-60 0-78 || 138 18-8 9°56 5-48 || 1-60 0-76 0-90 || 160 23-0 |11.8 7-00 || 1-86 0-95 || 1:20 || 218 27-7 ||14-3 8-38 || 2:16 || 1:14 || 1:50 || 187 33-0 |I'7-1 9,90 || 2:43 | 1.37 1-80 || 254 38°5 |19.9 |II-9 2.70 | 1.60 2-10 || 293 44-7 (23-1 13-6 3.06 | 1.85 2-28 || 335 51-1 ||26-3 |16-0 3-70 2-10 || 2:45 || 397 58-0 |30°2 |18-6 4’33 || 2:42 2-73 iſ 440 65-6 |34°1 |21-3 4-68 2-73 || 3-10 || 492 73-7 ||38.2 24-2 5'58 || 3:06 || 3:50 || 545 82-1 42-6 |27-4 5-86 || 3:40 || 3-91 || 604 91-0 |47-1 |30-7 6'42 3°77 || 4-35 || 668 100 |52-0 |35. 7.08 || 4:16 || 4.89 || 730 110 |57-1 |38-7 7-75 || 4-57 5'35 || 798 120 |63-4 42.6 8°42 5-07 || 5-86 || 869 131 |67-9 |46-7 9:15 || 5'44 || 6′35 || 944 154. 79-8 |56.4 ||10-07 || 6’ 38 || 7-63 ||1105 178 92.6 66-0 ||12-38 || 7-40 || 8.83 ||1275 205 ||106 |76-5 ||14-15 8-48 ||10-00 ||1457 232 || 21 88-0 || 16.00 9.70 || 11:50 || 1650 293 |153 |112 ||20-75 10-25 ||14-60 ||2] 41 363 |189 ||140 25° 15-10 |18-00 ||2575 4.38 229 172 30-25 | 18-30 |21-80 ||3117 522 1272 |205 ||36:00 21-80 |25-90 ||3708 Hemp. Wire, Circºm. Circm. 0-10 || 0:4 1.6 0-8 2-1 || 0-12 2°12 || I-1 3°7 | I-6 4-2 || 1:10 4-15 || 1:14 5.8 2.2 6-3 || 2.6 6°14 || 2: 11 7-9 || 2:15 8’4 || 3-3 8°15 || 3-8 9-10 || 3-12 10°5 4-1 4-6 11-11 || 4-11 12-6 || 5 in. 13-1 || 5-5 13- 12 || 5-10 14-7 || 6 in 15°2 || 6-5 15-15 || 6-10 16'8 6:15 17-14 7:10 19°4. 8.4 20-10 8'14 22- 9.8 24-12 10-12 27-8 12 in. 30°4 ||13-4 33. 14'8 The prices of the chains are taken from that in England Price of hemp ropes from Weaver, Fitler & Co., Rope manu- The prices of Wire ropes are deduced from the price list of John A. Roebling, Patent Wire Rope Manufacturer, Trenton, N. J. The Safety proof is here taken one half of the ultimate strength which may be trusted on for new ropes, but when much ºr use only one nuarter or less should be trusted upon for safety. and added 50 272 STRENGTH of MATERIALs, LATERAL STRENGTH OF MATERIALS. The formulas for lateral strength are here reduced to the simplest pos- sible form, and are in consequence subject to conditions which must be particularly attended to. In calculating the strength of beams of ir- regular sections as shown by the figures 210 to 217 on page 173, it is neces- sary to maintain the proportions marked on the figures and the calcu- lation will be correct. For the sections 206 to 209 any proportion will answer in the formulas. The weight of the beam itself has not here been taken into consideration, for which allowance must be made if considerable. Letters denote. ! = length of beam in feet. See figures. h = height, b = breadth or thickness in inches of the beam, where the Strain is acting. k = coefficient for the different materials and sections of beams, to be found in the tables. a = modulus of elasticity of materials. See Table. f= elastic deflection in inches. = weight in pounds which the beam can bear with safety, being about one quarter of the ultimate strain at which the beam would break. Ea’ample 1. Fig. 200. A rectangular beam of oak fastened in a wall projects out l-6 feet 4 inches, h =8 inches, and b =5 inches. Required what weight it can bear on the end W-7 W= 30.* = 1509 pounds, with perfect safety. Eacample 2. Fig. 201. A beam of section fig. 211, with thickness b-1'25 inches, height h-22-5 inches, supported at the two ends in a length 1–25 feet. Required what weight. W=3 it can bear in the nºiddle. For cast iron coefficient k=260. _4×260×125×22.5° 25 - Eacample 3. Required the elastic reflection for the same beam and con- dition as in the foregoing example? See Table, modulus of elasticity a;=2285 for cast iron. See page 276. - 26325×25. 16X2285×125X22.5% Eacample 4. Fig. 204. A wrought iron girder of section fig. 217, consist- ing of four angle irons of a =3:5X0:5X2X4=14 square inches, the plate being 0.5:1.35 =0-37 inches thick, and h-18 inches deep by 1-22 feet. Re- §: how much weight evenly distributed the girder can bear with gaſety 3 - W 26325 lbs.-11-8 tons nearly. = 0-80 inches, nearly. w—ºxºgº =73309 lbs.-32°75 tons. If plates being riveted to the angle iron at top and bottom, add that area to a. * Ea'ample 5. Fig. 222. The crank R=35 feet, force F=3860 lbs., length of the shaft l =61 feet, diameter D-=5:25 inches. Required the twisting in degrees. The shaft being of wrought iron for which ac-4110. 3’5X 6 Degrees _425X3860X 5X64 = 11.769. 4110X5’25° º STRENGTri ANāy ELASTICITY OF MATERIALS. 273 Etrº. N 200. º k; 5 hº * , I- º • Tºº # , ºº) wº, he eT+D T = −) “ - J. Tº - -++-Ti ------, -----, f I. # T---- ~f r=#. elastic set in inches. i 201. 4 : b hº W. P. l *= amºa. *—— — , , W= 4 m n I | { tº j W P. J–13. Tº ... elastic set in inches. - -I-202. II. 8 & b hº W. P. wYEE| W--→ = . T & 5-T- tº = ºn / X* H- f =º ... elastic set in inches. —T- | — —- 32 & b hº 3– 203. - H-T- 2 k \, h^ W. *E=º HE FEEE I W= l 3. 34, 6) T-HT! r ; i i tº ====E. T. ſ. > 5 W lº tº & & =H f = 2.7 F : elastic set in inches. § 204. * RAf 8 k, b hº E | W--tºº-, L P-L-T-1–t— h L–– +++ (. * 5 W is & tº , ºe Tº J-52. . . elastic set in inches. l —º- E |- i. 205. | —l Wr [I. I6 & b hº ====}==== W-***— T THE:EEE-I-T- FE 'h. 5 W is e ſº q-ºº: l- f TÉ4 x , ; elastic set in inches. | 18 DIFFERENT FORMS OF BEAMS.. 206, Coefficient k. Cast iron, 1 50 Wro’t iron, 120 Wood, * 212. Coefficient k. Cast iron, 236 Wro’t iron, 189 207. Coefficient k. Cast iron, 150 Wro't iron, 120 Coefficient k. B Coefficient k. Cast iron, 88 Wro’t iron, 70 Wood, 30 Cast iron, 250 Wro't iron, 200 b h”= Sº. 208. 214. Coefficient k. Coefficient k. Cast iron, 88 Wro’t iron, 70 Wood, 18 Cast iron, "700 Wro't iron, 560 by h”—D". 209. 215. Coefficient k. * Coefficient k. Wro’t iron, 700 Wood, 18 Cast iron, 900 b hº-D"—d”. 210. 216. Cast iron tube, Cast iron, 260 Wro’t iron, 208 \ k= 800. b h"—a h. 211. 217. k=800, Coefficient k. b h"—a h, ae-area of all the four angle irons in Square inches. STRENGTH OF WIATERIAI.S. 275 218. A beam fixed in one end and loaded at the other, should have the form of a Parabola, in which l = abscissa and h = ordinate. y=depth, ac=length from W. 2: l 3) = h 219. __k b hº_k b hº Tl cos. v T b/ 220. } b h w_*** Divide the length into 24 equal parts, \5*| place 14 in the middle and 5 at each end. 221. To cut out the stoutest rectangular beam from a log. 1st case, divide the diameter in 3 equal parts, and draw lines at right-angles as represented. 2d, divide the diameterin 49qual parts. 1, b=1.414 b, non-elastic. 2, h=1-73 b, elastic beams. 222. 3 3 D=4 FR S0 H y V ºf 3: 72 * * * * 425 F. R. l Twisting in degrees = - E-- 223. --- D: d=3/R : Wr, 3 / H. D==80A / –, & 78 - & R & º 2233000 A l ) Twisting in degrees = Tz n wº 276 STRENGTH OF MATERIALS. Absolute and Ultimate Strength of Materials. Kind of Materials. Wrought iron, . ę Cast iron, º Cast steel, soft, . e Cast steel, hardened, Blistered steel, soft, . Brass, . e o Copper, . 0. e Zinc, . e e Tin, . º º º Lead, . o e Ash, . e © e Hickory, e e Chestnut, sweet, e Oak, white, . & Oak, English, . e Canadian oak, • Pine, white, . e Yellow pine, º Teak, e º e Coefficient k. Safety. Inter. Pr. Cir. º & 120 162 240 º g 150 200 300 © e 385 619 170 e e 1050 1400 2100 º g 175 233 350 * e 58 75 113 o g 53 71 106 e º T5 20 30 © e 17 23 34 e º 4 6 9 º & 45 56 S5 e º 67 90 135 & º 42 56 85 tº º 50 66 Iſ)0 º & 25 33 50 © º 37 49 73 e º 34 45 67 e e 38 50 75 - g 51 68 I02 Ultimate. 488 600 1540 42ſ)0 700 226 212 150 205 Flasticity. 3. 4110 2285 4300 6000 4200 1280 2160 2360 100 221 300 24S 283 '268 316 The absolute safety weight is here taken one-quarter of the ultimate breaking weight, but when the weight is acting at short intervals one-third might be relied upon, or in pressing circumstances one-half, when the materials in the beams are known to be of good quality; but the latter never to be exceeded. Properties of some South American Woods, Taken from the borders of the rivers Perene and Madre de Dios, and ea peri- Peruvian Names of the Woods. Color. Chonta (Palm), . . . [Black, . . . Balsamo, . . . Shacaranda, . Jebe (ind rubber tr Amarillo, . . . Caoba, . . . . . . [Light brown, Huachapeli, . . Oak, . • e Nogal, . Dark brown, Jebe (best Ind-rubb ery M. Barigon, . . ee)* Brown, . . Brown stripes, Light yellow, Yellow, e White, . . . White, . . Specific | Wt. per gravity. cub. ſoot. lbs. 1.564 || 96.75 1.207 75.25 0.991 61.75 0.797 49.65 0.734 45.75 0.613 38.20 0.566 35.25 0.551 34.35 0.527 32.85 0.282 17.58 mented upon by the author of this Pocket-Book. Hard- | Ultimate Elas. ness. strength. ticity. H k 3. 28 450 64() 22 422 || 492 I8 343 322 15 351 305 13 334 300 11 I28 e - e 10 134 180 10 131 158 9 162 262 6 62 92 * There are different kinds of trees which give India-rubber, but of different quantity and quality. The woods were perfectly dry. Tour experiments on each were made. The hardness, II, is compared with that of substances on page 333. The coefficient, k, is the ultimate lateral strength of the woods. a = modulus of elasticity determined near the ultimate strength. __Wl 2––ºf 4b h” T 16.fb hº Meaning of letters is the same as that on page 272. Fig. 201, p. 273. PILE TORIVING. 277 PILE DR1 wing. Letters denote. Ms= weight of the ram in pounds. S = fall of the ram in inches. m = weight of the pile in pounds. s = space in inches which the pile sinks by the blow. r = resistance of the ground iu pounds. a = section area in sq. in. of the pile, sharpened to a point not more than 45°. k = coefficient for the hardness of the ground. h = depth to which the pile is driven. W= weight in pounds which a driven pile can bear with safety after the last blow when the pile sunk s inches. W = velocity in feet per second by which the ram strikes the pile. Ram and pilehead considered non-elastic and perfectly hard. W = 8 WS º - I 1j = 8 M VS cº º 4 M s_ _*S 2 +m. Tr(M-Em)3 , M*S 5 Ts (M-Em)? _ M*S 3 +m 6s (M-Him)” r= ak Wh - - - - 6 Ea’ample 1. A wooden pile 18 feet long by 12 inches square, driven h=12 feet into common natural ground imbedded with tenacious clay for which may be assumed the coefficient k=50. Required how much the pile will set S=4 into the ground at a blow with a ram of M=3500 lbs. falling S=42 inches. The weight of the wooden pile will be about m=18×40=720 lbs. Area of the pile a=144 square inches. Resistance r-144X50/12 =23840 lbs. The resistance sought from this formula 6, cannot be depended upon for calculating the weight the pile can bear with safety. 3500°X42 23840 (3500+720)” Suppose the set to be 8-4:23 inches at the last blow, required what weight the pile can bear with safety : 3500°X42 ' Tex4:23 (3500+720)” This can be depended on with safety, if calculated from the actual set of the pile at the last blow. For ordinary pile driving a heavy ram and short fall is the most effec- tive, but in some cases when the ground itself is elastic, or when º; piles in pure sand it is found more advantageous to use a high fali o the Tam. The set s = =4°23 inches. = 3984 lbs. Approximate Coefficients. 7: In coral formations, . e e * tº . 120 In hard clay with gravel, . º e . 100 In hard pure clay, . º º e * . 70 In common clay or sand, . e e e 50 In soft clay or loose sand, º e e . 40 In very loose materials, - - & e 90 278 . . 4. WROUGHT IRON BEAMS. T. A. T E R A L S T R E N G T H. For wrought iron beams, letters denote. W= weight in tons with safety, uniformly distributed on a beam rest- ing on two Supports. f S = compressive strain in tons per square inch of top 0:5 (a+.) a = section area in square inches of top and bottom flanges of the beam. Top and bottom flanges to be alike. a'- section area in square inches of web or stem. h = height of beam in inches. = length in feet between supports. J = deflection in inches of the beam in the centre, when the weight is uniformly distributed. º = weight in pound per square foot of flooring to be supported by the beams, which in ordinary cases is estimated to P=140 lbs. B = distance in feet between the beams. w = weight of the whole beam in pounds. 3 h. a/ H7 J3 = - - - - 1 - -— # (a++) f 46 hº(3 a-Ha') ' 5 Sh a/ I? *E – - º 2 - - - tº- - tº a 63 # (a++) 7-465, h. -- (a+.) 2200 W w = 3'384 l (a+a'.) tº- 4 --HT. e- º º 8 Formula 6 gives the safety deflection of a wrought iron beam, which should never be exceeded. Ea’ample. A flooring of l-32 feet by 60 feet long to be constructed to Support P=175 pounds per square foot. Required what kind of beams and how many are necessary? and what will be the cost of them? In the table Will be found the nearest star to 32 feet span is a 12 inch beam bearing PV-871 tons, when the distance between the beams in the flooring will be, Formula 7. B = 2240x871 = 3-5 feet. 175X32 Number of beams = # — = 16 about. Add one foot to each beam for the supports at the ends, and the cost will be, 33X16X1'90=1003-70 dollars. The following Table contains sections of iron rolled by the Phoenix Iron Company. Office 410 Walnut Street, Philadelphia, Rules, The price per foot multiplied by 5280 gives the price per mile. The weight in pounds per foot multiplied by 2:36 gives the woight in tons per mile. - The price per foot multiplied by 2240 and divided by the weight in pounds per foot gives the price per ton. STRENGTH or DIFFERENT SECTIONs of WROUGHT IRON BEAMS. 279 Strength of different Sections of Wrought Iron Beams Made by the Phaenia Iron Company, for Sustaining with Safety a Load tinformijºjistributed. Compound Girders. Solid Rolled Beams. Dis. 800 667 550 490 296 S08 168 84 48 tº W-i-W-#|W–F||7–5–7–7-|W-i-W-Fºr-Fºr-i- ºp h=18 i.}}=15 i.]h=12 i !!h=15 i.lh=12 i.|h=9 i. |h =9 i |h = 7 i.jh = 6 i feet. tons, ſtons. tons, to D8. to Il S. tons. tons, tons. tons. 10 || 30-00 | 66-67 55.33 || 49-00 |29.60 || 30-80 | 16-80 | 8:40 || 4-80 12 || 66-66 55-56 44.44 || 40-83 24-66 25-69 14:00 || 7-00 || 4:00 14 || 57.14 |47.61 | 38-09 || 35.00 21-14 22:05 | 12:00 || 6-00 || 3:43* 16 || 50-00 || 41.67 || 33-33 || 30-63 | 18.50 | 19-25 || 10-50 || 5-25 || 3:00 18 || 44.44 || 37-04 || 28.52 || 27-22 | 16.44 || 17-11 || 9-33 || 4-66 2-66 #9 40-00 || 33-33 ||26-66 || 24.50 || 14-80 || 15-40 | 8-40 || 4:20 || 2:40 36-36 || 30-30 |24.24 ||22-27 | 13:45 || 14-00 || 7-63 3.81* 2:18 24 || 33-33 27-77 22-22 || 20-42 | 12:33 12.85 || 7-00 || 3:50 || 2:00 26 || 30-77 |25-64 |20-05 || 18.85 Il-38 11-sº | 6’46 || 3:23 | 1.84 38 || 28.57 || 23-80 | 19-05 || 17.50 | 10:57 | 11:00 6:00+| 3:00 | 1.71 80 ||26-66 22-22 || 17-77 || 16-33 9.86 || 10:26 5-60 2-80 | 1.60 32 || 25-00 |20-83 16-66 || 15-31 || 9-25 | 9-62 || 5-25 || 2.62 || 1:50 34 i23-53 | 19.60 | 15-65 || 14-41 8.71%| 9.06%| 4.94 || 2:47 || 1:40 86 || 22-22 | 18-52 || 14-26 || 13.61 | 8-22 | S-55 || 4-66 || 2:33 | 1.33 38 || 21-05 || 17-37 || 14-00 || 12.90%| 7-80 | 8-11 || 4:42 2-21 | I-26 40 || 20-00 | 16-66 || 13-33 || 12-25 || 7-40 || 7.70 || 4-20 2-10 || 1-20 42 || 19-05 || 15-87 | 12.70%|| 11-67 || 7-05 || 7-34 4-00 || 2-00 || 1:14 44 || 18-18 15-15 | 12:12 || 11-13 || 6-72 || 7-00 || 3-S1 || I-91 | 1.09 46 || 17:37 || 14:48# 11:44 || 10-66 | 6’43 6.70 || 3-65 | 1.83 | 1.04 48 || 16.66 || 13 88 11-11 || 10-21 || 6-16 || 6-42 3-50 | 1.75 1-()() 50 || 16.00% 13-33 10-66 || 9-80 || 592 6-16 || 3 36 I 6S •96 Pe?" || 78 º; 71 lbs. 59.57bs l'Aºstos 40 lbs. 507bs. 293 lbs' 20 lbs. 13375s, Foot || $4.68 $4.26 $3:57 || $2.40 $1.90 $2.40 $1.50, $1.00 $0.66 The above Table gives the weight in tons, sustained by the several kinds of beams, uniformly distributed over them as in a floor. weights given are what may be used in practice, being only 9 tons per square inch of that part of the metal subjected to a crushing force. Under these weights the beams are within the limits of perfect elas- The ticity, and the deflections are therefore in direct proportion to the load. If it be intended to apply the entire weight at the centre, the figures in the Table must be divided by two ; if at any other point, the weight at the centre is to the weight at any other point, as the square of half the beam is to the rectangle of the two parts from where the weight is applied. The prices are subject to changes of the market and agree- ment. * When the span of the flooring is given, the star in the Table gives an approximatibn to what beam ought to be employed ; for instance, =38 feet spam should have beams of h-15 inches high, able to bear =-12-9 tons uniformly distributed. 280 T)IMENSIONS, WEIGHT, AND PRIce of ROLLED IRON. Angle Iron. ! Variety of Forms. |Price Per Foot. | t Blumensions. Weight, Price. Section, Area. Wit.p.ft. Wt, p. Mile, |Per Ft. re, Sq, In, lbs, Tons $ cts, Inches. lbs, cents, Figu *(14+13) | 1.77 | 6’13 || || || ||74 ||25 | 117.86 || 0:56 3. ) gxi &S # (1%+1}) || 2:32 || 6-74 2 -ſh- 5-9 | 20 92.7 0-45 *(14+13) | 2:09 || 6-07 3 I 7-1 || 24 113.14 || 0-54 *(14+13)|| 3:49 || 101 || {} | ( 2+2 ) || 3-17 | S-60 4 º' 1-95 6-6 31-45 || 0-16 # (2+2 ) 4:59 13-3 ||5 A-'541 | 183 | 8548 || 0:45 *(***)|** | *4 || Be- |444 15 70+1 | 0.37 is(24+2}) 6.84 19-9 7 *-* || 4-22 || 14-3 || 67.75 || 0-35 7:00 || 23. t * # (3+3) | 9:32 27.1 s sº-d 23.6 111°57 || 0-58 ; (3}+3%) | 840 24.1 |9 -6-º- 532 18 Chair. 0-72 #(334.8%) | 12:2 34.9 ||10 |- 965 | 32.6 | Channel. 1.16 7- ( 4+4 * : * I's ( + ) 11-2 32.5 ill- 5'41 | 18-3 Channel. 0-65 | # (848)|718 207 | 5 4+4 15.5 & # ( ) 45-0 12 2-66 || 9 Purlin. 0-35 Shipl"rames. lp Iºrames : , 2.66 9 T iron. 0°32 +1}++’s 23 2.5 || 7-95 1-ºf- 0.65 || 2:2 || Window-| 12 #2++3 || 4:36 | 13 8 & & –4– 0.50 1-7 Sashes. | 12 § 2%ti's 83 6.68 || 21.2 |19 • * • & * S } 3}++’s * | 8.85 28.1 ||16 -- 0-89 || 3 ash bar. 12 7 - 3–- 3 - * e Tö 3+; 5+ 11-0 || 35-0 ||1, Bºs 2:07 || 7 Shoe. 0-25 ****" | 164 || 510 ||is -k- |386 225 Girder lost A This is the beam for which the formulas and table are set 20 h up. Top and bottom are alike. A. This compound Girder is 2 made to order of any size, for 2] H about 6 cents per pound. # 20 intermsdiate bises, sº-sº re-sº-º-º-º- === SPHEREs. 2S1 spheres, Balls—Surfaces, Capacity and Weight of. Diameter. Inches. 1 in. 1.125 1.25 1.375 1.5 i 5 : :: 55à 5 i . i. i * * I ; Surface. Sq. inches.’ 3.1416 3.9760 4.9087 5.9395 7.0686 8.2957 9.6211 11.044 12.566 14.186 15.904 17.720 19.63.5 21.647 23.758 25.967 28.274 30.6S0 33.183 35.785 38.4S4 41.282 44.179 47.173 50.265 56.745 63.617 70.882 78.540 86.590 95.033 103.87 113.10 132.73 153.94 I 76.71 201.06 226.98 254.47 283.53 3.14.16 380.13 452.39 530.92 615.72 706. S4 804.24 853.96 I017.8 1134.1 1256.6 Capacity. Cub. Inches. 0.5236 0.745.5 1.0226 1.3611 1.7671 2.24.67 2.8061 3.4514 4.1888 5.0243 5.9640 7.0143 8.1812 9.4708 10.889 12.442 14.137 15.979 17.974 20.129 22.449 24.941 27.612 30.466 33.510 40.194 47.713 56.115 65.450 75.766 87.114 99.541 II.3.10 143.79 I 79.59 220.89 26S.08 321.55 381.70 44S.93 523.60 696.91 904.7S 1150.3 1436.7 1767.1 2144.6 2572.4 3053.6 3591.3 4.188.8 Cast iron. l’ounds. 0.1365 0.1943 0.2673 0.3550 0.4607 0.5861 0.7325 0.8000 I.09.20 1.3124 1.5592 1.8334 2.1328 2.4725 2.84.00 3.2512 3.6855 4.1721 4.6835 5.2612 Lead. Pounds. 0.2] 47 0.3062 0.4200 0.5579 0.724.8 0.9227 1.1528 1.4156 1.7180 2.0631 2.4482 2.8811 3.3554 3.8S92 4.4623 5.1056 5.7982 6.556S 7.3623 S.2521 9.2073 10.231 11.323 J 2.500 13.744 16.482 19.569 23.035 26.843 31.089 35.729 40.856 46.385 58.976 73.659 90.598 109.95 131.38 156.55 184.12 214.75 285. S3 371.09 71.80 589.27 T24.78 S79.61 1055.0 1252.4 1472.9 1718.0 Water. Pounds, 0.0188 0.0264 0.0368 ().0490 0.0636 0.0809 0.1050 0.1242 0.1508 0.1809 0.214.7 0.2525 0.2945 0.3410 0.3920 0.4479 0.5089 0.5752 0.6471 0.7246 0.8081 0.8979 0.994.1 1.0968 I.2064 1.4470 2S2 WEIGHT of Roll.ED IRoN, PER Foot. *º-ºº-º-º-º-º: Bido in Weight in , Side in Weight in | Diameter Weight in Diameter | Weight in inches pounds. inches. pounds in inches. pounds. in inches. _pounds. Tº 0:013 || 3; 44,418 || T's 0-010 || 33 34-886 # 0-53 3% 47-534 # 0-041 || 3: 37-332 †: 0-118 33 50-756 || # 0-119 || 33 || 39-864 3. 0-211 || 4 54,084 || # 0.165 || 4 || 42:464 § 0-475 || 4 || 57.517 # 0.373 || 4 || 45°174 # 0-845. 4+ 61-055 % 0.663 44 47-952 # 1-320 4; 64-700 # 1.043 43 50-815 # 1901 || 4} | 68°448 # 1.493 || 4 || 53.760 # 2.588 || 4g | 72.305 || 4 2-032 4; 56-788 1. 3-380 43 || 76.264 || 1 2-654 || 4 || 59-900 l; 4:278 || 4g | 80-333 || 1: 3'360 || 4 || 63-094 1#. 5°280 || 5 84.480 || 1+ 4°172 5 66-752 l; 6-390 5% 88-784 1; 5-019 5% 69-731 1% 7-604 5} 93-168 1% 5°972 53. 73-172 13 8-926 5} 97-657 1; 7:010 53 76-700 1; 10-325 5% 102-24 1; 8. 128 5} S0-304 l; 11.883 53 106-95 1; 9-333 5; 84-001 2 13:520 5: 111.75 2 10-616 5% 87-776 2} 15-263 5; 116-67 2} II '988 5% 91-634 2} 17-112 6 121-66 2} 13° 440 6 95° 552 23 19-066 6+ 132'04 2# 14-97.5 6# 103-70 2% 21:120 6% 142°82 2} I6-688 6% II2-16 23 23°292 6# 154°01 2# 18-293 6: I 20-96 23 || 25-56 7 165.63 || 23 20.076 || 7 I30-05 2g 27.939 || 7 || || 19014 || 23 21-944 || 7 || 149-33 3 30°416 || 8 2I 6'34 || 3 23-888 || 8 169-85. 3} | 33-010 | 8% 244-22 || 3 || 25-926 8} | 191-81 3+ || 35-704 || 9 273-79 || 3+ 28-040 9 215-04 33 || 38-503 || 10 337'92 || 33 || 30-240 | 10 266-29 3} 41-408 12 486-66 || 3} 32.512 12 382-21 Rule for Finding the Weight of Pipes. The diameter of the pipe in inches, measured from inside to outside, mul- tiplied by the coefficient for the metal, will be the weight in pounds per linear foot. Coefficients. Lead, . & e . 0.1005 Brass, rolled, e . 0.0985 Copper, . © . 0.0989 Iron, rolled, . . 0.0876 Brass, cast, . e . 0.0882 Cast iron, . ſº , 0.0811 Cast steel, ſº . ().0891 Tim, rolled, & . 0.0821 Clay, burnt, . . 0.0214 Zinc, rolled, tº . 0.0808 WEIGHT PER Foot, IN Pounds, of CAST-IRON cyLINDERS AND PIPEs. 283 Diam. 0 00000 2.5132 9-8989 22-205 39'544 61° 584 88-825 120-90 157.91 199-86 246'73 298.55 355-29 416'98 483-73 528° 15 631-64 712-79 799-30 S90-70 986-95 1088-1 1194-2 1305-2 1421°5 1492-1 1667-9 1798.7 1934.4 2075-1 2220.6 2371-1 2526-6 2687-0 2852°3 3022-5 31 97-5 3377-S 3562-9 3752.2 3947.7 41.47°5 4352°3 4562'2 4776-7 4996'3 5220.9 5450-2 5684-6 % •03804. 3.1227 11°145 24°093 41-984 64'807 92°564 I25-26 162-88 205'44 252-94 305-38 362.72 425' 02 492°24. 564°44 641°54. 723-59 810-56 902-48 99.9°30 IT 04:2 1207-8 1319-4, 1436-0 1557-5 I6S3-9 I815°5 1951-7 2093-0 2239-2 2390-3 25.45-7 2707.4 2873-3 3044°2 3219.4 3400°4 3586. I 3776-8 3972.5 41.73-0 4378.4 458S-6 4803-7 5024-0 5249-2 5473-1 5714-1 % 15418 3'9047 12°491 26,059 44' 566 68.005 96-380 129-69 168-15 211. II 259-23 312°28 370-23 433-15 501-02 573.81 651-53 734-19 821-79 914-29 1011-6 1114-6 1221.5 1333.7 1451-0 1572-1 1700-1 1832.2 1969. I 2III:0 2257.8 2409.6 2566-2 2727-8 2894.4 3065.9 3242-0 3.423.3 3609-6 3S01-0 3987-0 4 198-4 4.404-1 4615-2 4831. I 5051-9 5277-8 550S-4 5744-1 A solid cast-iron cylinder 42 Subtract inside cylinder 40 Weight of pipe 134 in. thick will be % 4-6620 13-947 28-104 47-227 71-282 100-27 134°20 173-06 265°59 319-24 377-83 44.1°39 509'84 583-76 661-5S 744-86 838-17 926-23 1024-3 1127.3 1235-2 1348-1 1466-1 1588-7 1716-4 1849-0 1986-5 2129.1 22.76.5 2428-9 2586-2 2748-4 2927.3 3087-7 3264-3 3446-6 3633-5 3835-1 4022-1 4.223.8 4,430-2 4641-9 4S58°4. 5079.9 5306-3 5537-6 5773-9 #3 in. diameter weighs 44$2.6 pºul % in. diameter wei 34675 | |216.86 . % 61669 5'5512 15°419 30-225 49°963 74°537 104-24 138-79 178.29 222.68 272-03 326-28 390-50 449°64 518:77 592.78 671-73 755-80 844-45 938-20 1036-9 1140-5 1249-1 1362.6 14SI:0 1604-4 1732-7 1865.9 2004. I 2147-2 2295-2 2448-3 2606-1 2769-0 2936-8 3I 09:5 32S6-9 34.69-5 3657-0 3849-7 4046-9 4249-2 4456-6 4668-6 4.885-7 5] 07:S 5335-0 5566'S 5S03-7 % 96352 6'5476 I6'999 32°420 52.778 78.06S 108-29 143°45 183'55 228.57 278'54 333°40 393.26 458'04 527-72 602-36 681-94 766-44 855-86 950-27 10.49°5 1153-8 1263-0 1376-9 1496-1 1620-2 1749-1 1882-9 2021-7 2165-4 23.14.1 2467-9 2626.3 2789-7 2958-1 3.131-5 3309.5 3492-7 36S0-9 3873-8 4071-6 4275-0 4482.6 4695-6 4913-3 5.136-1 5363-6 5596-1 5833°5 ghn 3972.5 510-1 % 1.3876 7-5414 IS-658 34.695 55-629 81-577 112-42 148-19 188-91 234°56 285-13 340-64 401-08 466°46 536'80 612-04 692-24 777-3S 867 -42 962-42 I () tº 2-3 11 67-2 1277-0 1291.7 1511-4 1635-S 1765-5 1900-0 2039.4 21S3-8 2333-1 2461-3 2646-4 28] 0-4 2979-5 3143.7 3332-2 3516.0 3704-8 3S 98-3 409-1-1 4300-7 4509: () 4722-7 49.41 1 5 164° 1 5393.5 56.25-6 5863-7 ${ {ſ % |Diam. 1.8975|| 6 8-7012| 1 20-392| 2 37.03S} 3 58-637 4 84-84S 5 116-62 6 153-02| 7 194-34| 8 240-50|| 9 291.81 IO 34.7-92. II 408.69; 12 475.00. 13 545:94; 14 621-71| 15 702-61 16 788-35. 17 879-04 || 8 974.64; 19 1075-0; 20 IISO-7 21 1291-1 22 1406-4. 23 1526-7 24 1651-9| 25 1782-1| 26 1917-2. 27 2057-2| 28 2202-2| 29 23.52-0|| 30 2506.9 31 2666-7| 32 2S31-3; 33 3001-0|| 34. 3175-5| 35 335-4-3| 36 3539-2 37 372S-6 38 3922-8| 39 4122-3, 40 4326°5| 41 4540-5] 42 47.49°7| 43 4968-7| 44 519.2°4| 45 5421-4. 46 5655-1; 47 5S93°S 48 lds per foot. Weight of Flat Rolled Iron pcr Foot, Look first for the thickness, and follow that line until the column where the breadth is on the top ; is the number of pounds per foot. 14 1; 11:00 13 9.610,10:30 | 1% º, sºlº 13 6'970 7:55.81% 8-713 I} sººn 1, 4,752|5-2275-703 tºº 1 |3-8024-224|4-646.5-069 5.492.59148.8% 3 |2% 3°326,3-696 |4-065|4°435|4-805.5°178 5-544 § 321ſº 2-850;3:168|3°484|3°802 4.ii), 4354752 § 1.584|1846.4.1122-375]2-640|2-904|3.1683.432|3-6963-960 % 1.0% 1.265|1-477|1:690|1-901 2112 2.325||2-535|2-746|2-957|3-168 * B |0-6340-792|0-950|1-108||1-267|1'425|1-584|1742|1-9002-059|2-2182-376 # 0.317 º 0-633|0.738||0-845|0'950|1-0561-161|1'266|I-372|1-479||1-584 | 1% 0-1620-237 03160.8% 0-4740-553|0-633|0-712|0-792|0-870|0-949||1-029||1-109|1-188 l # |0.081 0.1080-1880.211 0.264.03180.369,04220475 0.528,0-580 0.633,06860.739,0792 | 2 1267 11.83 10-98 10:14 9°294 8.448 7-604 6-758 5°914 5-069 4°224 3-379 2-534 1.689 1.267 0.845 23 2} 18.06 2, 16-1617-05 12:36:15-21:16:05 1846'1426,1505 12-57,13:31|14-04 11-67;12:36,13-04 10-77|11-40;12-04 9°874|10-45||11-03 8-977|9-505|10-03 8-0798-554|9-028 7-181|7-604|8-025 6-2836-653|7-022 sºsºsºs 6-019 4.488|4-752|5"916 3.591|3.8024°013 2.693|2-851|3-009 1-795|1-900|2-006 1°346||11425 0.9500 898 1-504.1-584 1-00 1,056 2% 20-06 T 9-01 17.95 16-89 15-84 14-78 13-73 12-67 11-62 10-56 9°504 8.448. 7:392, 6:336' 5-280 4°224 3-168 2-112 ź 285 WEIGHT OF FLAT Rol.I.En IRON PER Foot. •------- [ 989. Z I08-8 690.9) †69. 1, 83-01 19-ŻI 03.gI † 1,-1,I 43.0% I8-ŻŻ, g8.gº, 06- 1,3 Źý.08 98-38 6ý.98 I0.88 Ģ9.68 69. Ziff Ź8.ŤŤ 9]|-}}} 9 6%ſ,% 8ř9.8 Sg8.Ť },8%-ºſ, 9 I || -6 į I. ZI }g.ýI 00-1, I 8ř.6I 98. IŻ, 63.5% 34.9% 9.I.6% 89. [8 I().f8 8ý. 38 98.88 60. IŤ Ź8.8† g6.gif įg 838-3 g8ſ, 8 1ř9.Ť 0!!6~9 †6%-6 Ț9. II †6.8I 93.9 I 69-8I 16.0% 83.8% 99.9% 88- ), Z 0%.08 99-38 g8-79 1.I-18 657.68 ZS-IŤ †I.ţŤ Ķg SIZ-Z 438.9 9$ff.† 899.9 I 18.8 60. II Ț8-8I Ź9. ĢI † 1,- /, I 96.6ſ 8I-ZZ 68.ŤŽ I9-9% 88-83 90. [8 93.88 8ț¢. g8 04-18 Z6-68 8.T.ZŤ {g ŽI] •Z 1900-Z 891.8|600.8 #ZZ.jſ|$][0..jº 988.9||610.9 6ýř,8||930.8 99.0'[|80. () I 19. Z.Iļf0, ZI 8). Þ{|f0.jſ I 06-9T|90-9 I I0.6I|90-8I ZI-I?|1,0.03% 83-84||10-3% g8.gº, |80-†Z 9Ť.), Z|80.9% 1,9.6%|60,8% 89. I8|60.08 64,88||0|[-38 06. g8[II-Ť8 Ź0,88||3|[-98 8.I. Off|ZI-88 g | $ſ; T06. I I98. Z 308.8 801-9 †09. 1, 109-6 IŤ.II T 8-8 I IZ. ÇI II. AI I0-6 [ [6.0% I8.3% [1,-74 [9,9% I9.8% IŤ.08 [8.38 ŹŹ.ý8 ZI.98 ## g64. I 869. Z I69.8 988.9 I8I-1, $46.8 14-01 99. ZI 98.jŽI 9 I-9{ 96- 1, ſ[ † 1.6I † 9. IZ 88.8% 81.9% Ź6-9% 34.8% Z9-08 Zg-Z8 II.Ť8 ## ()69-I 999. Z 088.8 690.9 69 !, -9 Çțý.8 į I. 0Ț 88. II Z9.8I I3.9I 06.9I 69-8I 83-0% 16-IZ 99.9% 99.9% į70, 13 Z 1.8% IŤ.08 0[,%8 † 189. I ÇgŤ. Z † 23.8 0I6.jſ 1, ; 9.9 88 [.8 ŹŹ8°6 9ý. II 60-8I † 2...fº'I 1,8-9][ 00,8I † 9.6I † 8.08 § 8 †89. Í 948-Z S9 I, 8 Ź9 )...? 988.9 IZ 6. || 909.6 60. Il 19.ZI 13.ŽI †8. gȚ Zț¢. 1.I [0.6I 69.0% 8Ț. Zº 94.8% 98.9% $j>.9Z [9.8% 69.6% #8 *---- [89. I 1,6%. Z 390.8 †6g.jſ gŽI.9 499. / 88 I. 6 Z || .0 L 0%-ZI 64-81 [8.9 I † 8.91 1,8-8I 06.6I ††. IZ 1,6-ZZ 09. řZ 81.9% 99- 1,3 †8.8% $g 61, †. I 8 IZ-Z 496-3 9$f.jſ ÞI6.g 868, 1, I 18.8 98,0 [ 88.] I [8,8][ $1.jŽI 93.9 I † /, /, I Z3.6] 01-03 8.I, ZZ Ç9.8% 8I, 9% I9.93 60.8% #8 9ZŤ.[ 89.I.Z. [98. Z 1.1%.jſ 801-9 6ZI. 1. †gg.8 086.6 Off. II 88.ZI ÇZ.ŤI 89.9 || 0I. AI 89.8I 96.61 88. IZ [8,3% £Z.7% 99.9% 8I. 13 § 8 £18-I 690. Z 9ț¢). Z 6LI-Ť Ź6Ť.g Ç98.9 18Z-8 0Ț9.6 86.0I 98. 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[IįŞI ĶI ȘI ĶI $1 șZ {Z §Z Weight of Flat Rolled Iron per Foot, RULE.—Look first for the thickness, and follow the line until the column where the breadth is on the top, is the num- ber of pounds per foot. 3} * |34-33 3 |31-9432-95 2; 29.42|30°49|31-57 23, 26-72|27-8829.0430-20 *::::::: 26-61|27-72128-83 23°23'24-29|25°34′26.40.27-46 2; 2}|22:18 3# 37-07 35-64 34-21 32-78 31-36 29-93 28-51 3% 39-92 38-44 36.96 35-48 34.00 32°53 31-04 29'57 3g 42-87 41-34 39-81 38-28 36-75 35-21 33-69 32-15 30-62 33 45-93 44-35 42-77 41-19 39-60 38-02 36-43 34-85 33-26 31.68 33 49-09 47°46 45'83 44-19 42°51 40-92 39.28 37-64 36-01 34-37 4}. 4 |57:45 52°38' 55-65 50-69,53-86 49:00 52-06 47-31|50-27 45°6248-47 43-93446-68 42-24|44-88 40°55|43'09 38-86|41-29 37-1739-50 35-4837-70 32-73 33-79|35-91 4}. 64'63 60-83 58-93 57.03 55-13 53°23 51-24 49°25 47.43 45°62 43-72 41°82 39-92 38-02 4: 72°23 68-22 64-21 62.20 60°20 58-19 56-18 54-17 52-17 50-16 48°16 46-15 44°14 42-13 | 40-13 5 80-26 76°04 71-81 67-59 65-97 63.36 61.25 59-14 57.03 54-92 52.80 50-69 48-58 46-47 44-35 42°24 | 5} 88-71 84.27 79-84 75-40 70-97 68-75 66-53 64-31 62-10 59-88 57-66 55°44 53.23 51*01 48-79 46-57 44-35 5% 97.58 92-93 88-29 83-64 78-99 74°35 72°02 102.0 97.16 92.30 87°44 sº 77-73 75-30 69-70|72.87 67:37170-44 65-05/68-01 62-73|65-58 60°41; 63-15 58-08|60-72 55-76|58-30 53'43|55-87 51*11|53°44. 48-79||51.01 | | | 46-47,48°58 6 5% 116.6 106'9' 111-5 106-4 101'4 96-41 91-24 86-17 81-11 78-58 76°04 78:51 70-97 68°43 65.89 63-36 60-84. 58-31 55-77 53-23 50-69 Nº. $g Jº WEIGHT OF MATERIALs. Weight Per Square Foot in Pounds. #. Cast Iron | "... sheet copper. sheet Lead. sheet zine 1's 2-346 2-517 2.890 3-694 2.320 * 4-693 5' 035 5'781 7-382 4*642 * 7.039 7 552 8-672 11-074 6-961 + 9°386 10-070 II* 562 14-765 9-275 †, 11.733 12:588 14'453 18°456 11.61 # 14° 079 15-106 17-344 22-148 13.93 1% I6'426 I7-623 20-234 25.839 16-23 # I8-773 20 141 23-125 29'530 18-55 †: 21° 119 22°659 26-016 33-222 20-87 # 23°466 25-176 28-906 36'913 23-19 +4 25°812 27-694 31-797 40-604 25-53 # 28-159 30-211 34-688 44.296 27.85 # 30-505 32-729 37-578 47-987 30-17 § 32°852 35-247 40°469 51-678 32.47 }} 35-199 37-764 43°359 55-370 34'81 I 37-545 40°282 46.250 59-061 37-13 l; 42°238 45°317 52-031 66.444 41-78 1+ 46-931 50°352 57.813 73-826 46'42 l; 51-625 55-387 63'594 63-594 51-04 1} 56-317 60-422 69-375 88-592 55'48 15 61:01.1 65°458 '75-156 95.975 60-35 1; 65-704 70-493 - || 80-938 103-358 65.00 13. 70-397 75-528 86-719 110-740 69-61 2 75-090 S0-563 92:500 118-128 74.25 Weight of Copper Rods or Bolts per Foot, Diameter. Weight. I Diameter. weight. Diameter. Weight. Diameter Weight. Inches. Pounds. Inches Pounds. Inches. Pounds. inches- Pounds. # 0-1892 —l 3-0270 l; 10-642 3# 34°487 is 0.2956 || 1 , || 3:4170 2 12-108 || 3} 37-081 # 0.4256 1; 3-8912 2# 13-66S 3} 39.737 I's 0.5794 || 1 , || 4-2688 || 23 15.325 || 3: 42.568 # 0-7567 || 13 4.7298 2} | 17-075 33 45°455 ſº 0.0578 Tº 5.2140 2} | 18-916 || 4 48-433 § 1-1824 || 13 5'7228 || 25 | 20-856 || 44 53-550 }} | 1.4307 || 1 , || 6-2547 || 23 22,891 || 4} 61.321 # 1.7027 * | 6-8109 2g 25.019 4; 68-312 # | 19982 | 1.1% | 7-3898 || 3 || 27.243 || 5 76-130 # 2.3176 § 7-9931 3# 29.559 5% 91°550 # | 2:6605 || 13 9-2702 || 33 31.972 | 6 * 288 AMERICAN WIRE GAUGE. Gauge | Size Rolled Plates. IDrawn Wire, num. inches. Weight per square foot. Weight per 1000 feet. No. In. Iron. Steel. Copper|| Brass. Iron. Steel. Copper|Brass. Lbs. Lbs. Lbs Lbs. Lbs. I,bs. Hibs. Lbs. 0000 .4600 18.75 | 18.97 21.36 20.84 566.3 571.7 646.8 634.1 000 | .4096 16.70 | 16.90 | 19.01 | 18.56 449.1 || 453.3 || 512.9 || 502.9 00 .3648 14.87 | 15.05 | 16.93 | 16.52 356.1 || 359.5 | 406.8 || 398.8 0 | .3249 13.24 13.40 | 15.08 || 14.72 282.4 || 285.1 || 322,5 : 316.3 .2893 11.79 || 11.93 13.43 || 13.11 224.5 226.1 || 255.8 || 250.8 .2576 10.50 10.63 || 11.96 || 11.67 177.6 || 179.3 202.9 || 198.9 .2294 9.354 || 9,464 10.65 10.39 140.8 || 142.2 | 160.8 || 157.8 .2043 8.330 | 8,428 9.486 || 9,255 111.7 || 112.7 | 127.5 | 125.1 .1819 7,418 || 7.505 || 8,448 || 8.242 88.59 | 89.43 || 101.2 || 99.20 2 3 4. 5 G | .1620 6,606 || 6.6S3 || 7.523 || 7.340 70.26 || 70.92 || 80.25 78.67 '3 .1443 5.882 || 5.952 | 6,699 || 6.536 55.7 L | 56.24 || 63.64 || 62.38 8 . .12S5 5.238 5.300 5.966 5.821 44.18 || 44.60 | 50.46 || 49.48 9 .1144. 4,665 || 4.720 5.313 || 5.184 35.04 || 35,37 | 40.02 || 39.24 0 .1019 4.154 4.203 || 4.731 || 4.016 28.26 28.05 || 31.73 || 31.11 II .0907 3.700 || 3.743 || 4,213 || 4.110 22.03 || 22.24 || 25.16 || 24.6S Ilº .080S 3.294 || 3.333 3.752 || 3.661 17.47 17.64 || 19.95 || 19.57 13 | .07.20 2.934 || 2.968 || 3,341 || 3.26%) 13.85 || 13.99 15.82 | 15.52 l:4 | .0641 2.613 2.643 2.978 || 2,903 10.99 11.09 || 12.55 | 12.31 15 .0571 2,327. 2,354 2.650 | 2.585 8.717 | 8.S99 || 9.953 || 9,761 | 16 || 050S 2.072 2.096 || 2.359 || 2.302 6.913 | 6.978 || 7,896 || 7,741 lſº | .0452 1.845 | 1.867 2.101 || 2.050 5.481 5.532 6.261 6.137 18 .0403 1.643 | 1.662 : 1.872 | 1.826 4.347 4.387 || 4.965 4.867 19 .0359 1.463 | 1.480 | 1.666 | 1.626 3.447 || 3,479 || 3.937 3.861 20 .0320 1.303 | 1.318 : 1.4S4 | 1.448 2,735 | 2.761 || 3.125 || 3.064 21 .02S5 1.160 | 1.174 | 1.321 | 1.289 2.168 2,188 || 2.476 2.428 22 | .0253 1.033 | 1,045 | 1,176 | 1.14S 1.720 | 1.73G | 1.964 | 1.926 23 .0226 .9.203 | .9310 | 1.04S | 1,023 1.363 | 1.376 | 1.557 | 1.527 24 .020.1 ,8195 .8291 | .933 .9105 1.081 | 1.091 | 1.235 | 1.211 25 .0179 .7298 } .7383 . .S311 | .8109 .8575 | 8656 .9795 .9603 26 .0159 .6499 || .6575 .74ſ)1 | .7221 .6801 | .6864 .7768 . .7616 27 | .0142 .5787 | .5835 | .6391 . .6430 .5393 | .5444 | .6160 | .6039 28 .0126 .5154 | .5214 | .5S60 .5726 .4277 | .4317 | .4885 | .478) 29 .0113 .4580 | .4643 .5227 | .5099 .3391 . .3422 .3873 . .3797 30 .01.00 .4087 | .4135 | .4654 | .4541 .3699 || .2714 | .3072 | .3012 31 | .0089 .3640 | .3683 | .4145 .4044 .2134 .2153 | .2437 .2389 32 .0080 .3241 .3270 | .3691 .3601 .1691 .1707 .1932 .1894 33 | .0071 .2887 | .2920 | .3287 .3207 .1341 .1354 .1532 | .1502 3+ . .0063 .257 0 | .2600 | .2927 | .2856 ..1063 | .1973 | .1216 .1192 35 | .0036 .2289 | .2316 | .2606 | .2543 .0845 .0853 | .0965 .0947 36 .0050 .2039 || .2062 .2322 | .2265 .0669 || 0675 .0764 .0750 37 | .0045 .1816 .1837 .2067 | .2017 .053 .0536 | .0606 | .0594 38 .0040 .1617 | .1636 .1841 .1795 .0118 .0424 .0480 . .0471 39 .0035 .1440 .1456 | .lbl.0 .1600 .0334 || 0337 L.03S1 | .0374 4-0 | .0031 .1282 | .1297 .1460 .1424 .0268 .0267 | .0302 | .0297 Spec. grav. 7.828 7.92 | 8.917 S.T.() '7.85 7.93 S.96 8.78 The American Wire Gauge is introduced and manufactured by J. R. Brown & Sharpe, of Providence, R.I., and is to be had in the principal hardware stores in the country. It is adopted by most manufacturers of plates and wire, and is now con- sidered the American Standard Gauge. BIRMINGHAM GAUGE. 289 Birmingham Gauge for Wire, Sheet Iron and Steel. Weight per Square Foot in Pounds. tº "ºnlºº) sº I dº: she lead.";" No. 0000 0.454 18.267 18.259 20.566 26.75 7 : 16 OOO 0.425 17.053 17.280 19.252 25.06 27:64 () () 0.380 15.247 15.451 17.214. 22.42 3 : 8 O 0.340 13.7 14.0 15.6 20.06 11 : 32 I 0.300 12.1 12.4 13.8 17.72 5 : 16 2 0.284 || 11.4 11.7 13.0 16.75 9 : 32 3 0.259 10.4 10.6 11.9 15.26 1:4 4 0.238 9,60 9.S0 11.0 14.02 7:32 5 0.220 8.85 9.02 10.1 12.98 7 : 32 6 0.203 8.17 8.33 9.32 11.98 7 : 32 7 0.180 7.24 7.38 8.25 10.63 3 : 16 8 0.165 6.65 6.78 7.59 9.73 3 : 16 9 0.148 5.96 6.08 6.80 8.72 5 : 32 IO 0.134 5.40 5.51 6.16 7.90 5 : 32 11 0.120 4.83 4.93 5.51 7.08 I : 8 12 0.109 4.40 4.50 5.02 6.42 1 : 8 13 0.095 3.83 3.91 4.37 5.60 3 : 32 14 0.083 3.34 3.41 3.81 4.90 3 : 32 15 0.072 2.90 2.96 3.31 4.25 1 : 16 16 0.065 2.62 2.67 3.00 3.83 1 : 16 17 0.058 2.34 2.39 2.67 3.42 1 : 16 18 0.049 1.97 2.01 2.25 2.90 1 : 16 I9 0.042 1.69 1.72 1.93 2.48 3 : 64 2O 0.035 1.41 1.42 1.61 2.04 3 : 64 21 0.032 1.29 1.31 1.47 1.S9 3 : 64 22 - 0.028 1.13 1.15 1.29 1.65 I : 32 23 0.025 1.00 1.02 1.14 1.47 I. : 32 24 0.022 0.885 0.903 1.01 1.30 1 : 32 25 0.020 0.805 0.820 0.918 1.18 1 : 32 26 0.018 0.724. 0.73S 0.826 1.06 I : 64 27 0.016 0.644 0.657 0.735 0.945 1:64 28 0.014. 0.563 0.574 0.642 0.826 29 0.013 0.523 0.533 0.597 0.767 30 0.012 0.483 0.493 0.551 0.708 31 0.010 0.402 0.410 0.480 0.600 32 0.009 0.362 0.370 0.420 0.532 33 0.008 0.322 0.328 0.370 0.472 34 0.007 0.282 0.288 0.323 0.413 35 0.005 0.230 0.235 0.262 0.309 36 0.004 0.170 0.173 0.194 0.236 Birmingham Gauge for Silver and Gold. Thick. Thick. Thick. Thick. Thick. ick. No. . No. i..] No. i...] No. ..] No. ...] No. º 1 | .004 7 .015 || 13 .036 || 19 .064 || 25 .095 || 31 .133 2 .005 8 .016 || 14 | .04.1 || 20 .067 || 26 .103 ||32 | .143 3 .008 9 |.019 || 15 .047 || 21 .072 || 27 | .113 ||33 | .145 4 | .010 || 10 | .024 || 16 | .051 || 22 .074 || 28 .120 || 34 .148 5 | .013 || 1 || | .029 || 17 .057 || 23 .077 || 29 | .120 || 35 | .158 6 | .013 || 12 .034 || 18 .061 || 24 .0S2 || 30 .126 || 36 .167 I9 290 PAPER, TIN AND GLASS. P A PER. 1 ream = 20 quires = 480 sheets. 1 quire = 24 sheets. Drawing Paper. Cap, te e * 13 × 16 inches. | Columbier, . . 34 × 23 inches. 20 t & 15 {{ 33 & G 26 4% Denny, gº * • 2 Atlas, & •º te Medium, . . 22 “ 17 “ Theorem, . . 34 “ 28 “ Royal, . . . 24 “ 19 “ Double Elephant, . 40 “ 26 “ Super Royal, e 27 “ 19 “ Antiquarian, . e 52 “ 31 “ Imperial, . . . 30 “ 21 “ JEmperor, . de . 40 “ 60 “ Elephant, . . 28 “ 22 “ Uncle Sam, . . 48 “ 120 “ Continuous Colossal Drawing Paper, No. A and No. B, 56 inches wide, and of any required length. No A of this paper is excellent for mechanical drawings. Price, from 40 to 50 cents per yard. Tracing Paper. Double Crown, . . . . 80 by 20 -) Glazed or Crystal. Double Double Crown, . º ge 40 “ 30 Yellow or Blue Wove. Double Double Double Crown, . . 60 “40 “ Finest French Vegetable Thracing Paper. Grand Raisin (or Royal), 24 in. by 18. Grand Aigle, 40 in. by 27. Mounted Tracing Paper. This paper is mounted on cloth, and is still transparent; it will take ink and water-colors. It is 38 inches wide, and of any required length. Vellum Writing Cloth. Adapted for every description of tracing; it is transparent, durable and strong. It is 18 to 38 inches wide, and of any required length. Weight and Marks of English Tim-plates. Plates Length Weight Plates | Length Weight Brand. per and per Brand. per and per Box. | Breadth. | Box. e I - No. D. I,bs. No. In. Lbs. 1 C. • 225 | 133X10 112 1 XX. . . 225 133X10 I61 2 C. . tº 225 13+ “ 9; 105 || 1 XXX. . 225 13: “IO 182 3 C. o 225 | 12# “ 9} 98 I XXXX. e 225 13; “10 20.3 H. C. . º 225 | 133 “10 119 || 1 XXXXX. 225 | 13; “10 224 H. X. . 225 133 “10 157 1 XXXXXX. 225 13; “10 245 1 X. . tº 225 13+ “10 140 DC. & g 100 | 16; “123; 98 2 X. © 225 | 13+ “ 93 133 DX. . sº 100 | 16# “12} 126 3,X. . 225 | 12: “ 9} 126 I) XX. . e 100 | 16; “12; 147 Leaded IC 112 || 20 “ 14 112 || DXXX. . 100 | 16# “12} 168 “ IX. 112 || 20 “14 140 || DXXXX. . 100 | 16; “12; 189 ICW. . 225 | 133 “10 112 SDC. tº 200 | 15 “11 168 IXW. e 225 | 133 “10 140 || SDX. . . 200 | 15 “IT 1SS CSDW. 200 15 “11 I68 SDXX. . 200 15 “ 11 209 CIIW. sº 100 | 163 “123 105 SDXXX. e 200 || 15 “ 11 230 XIIW. . 100 | 163 “123 126 SDXXXX. 200 || 15 “ 11 251 TT. . te 450 | 134 “10 112 SDXXXXX. 200 15 “II 272 XT.T. . 450 | 13; “10 126 || SDXXXXXX. 200 15 “I1 293 When the plates are 14 by 20 inches, there are 112 in a box. Thickness and Weight of Window Glass. . Number of the glass or weight in ounces per square foot. 12 13 15 16 17 19 21 24 26 32 .059 | .063 .071 .077 .083 .091 | .100 | .111 | .125 | .154 Thickness in decimals of an inch. 36 42 .167 | .200 ColºrFICIENT FOR CAPACITY AND WEIGHT. 29] mºvº-r----> Coefficient for Capacity and Weights © Names of Substances. F2. 22. Cubic inches, - - 5 903-7| 0:523 Cubic feet, - - - 1. ".. º 44 || 0:523] ..... 3 Gallons, - - - - º º º 34; 3.91 || “..226 Water, fresh, - - g e º º 32.7 || 0-019 Water, salt, - - - ge e,f 4 × º • * e 33-6 || 0-02 Oil, - - - - - - || 57.5 || 0:4 0-033 45-1 || 0-313 0.02630 || 0-017 Cast-iron, - - - 450 3.12 || 0:26 353 2:45 0.204 || 235 || 0-136 Wrought-iron, - - || 487 3.37 0.281 || 382 2.65 0.221 || 255 || 0-147 Steel, - - - - - || 490 3'4 0.283 385 2.67 0.222 257 || 0-149 Brass, - - - - - || 532 3.68 || 0.307 || 417 2-9 0-241 278 0-161 Tin, - - - - - || 456 3-16 || 0-263 || 358 2.48 0.207 || 239 || 0-138 Lead, - - - - - || 710 4-92 || 0:41 557 3-87 0-322 || 371 0.215 Zinc, - - - - - || 440 3-05 || 0°254 345 2-4 0-2 230 0-133 Copper, - - - - || 556 3-85 0-321 436 3-03 0-252 291 0-168 Mercury, . - - - || 850 5-9 0.491 666 || 4-63 0-385 445 || 0-257 Stone, common, - . 156 1-08 || 0-09 122 0.85 0-071 82 0-047 Clay, - " - - - || 135 0.936 || 0-078 106 0-735 | 0-061||70 || 0-04 Earth, compact, - || 127 0.88 || 0-0733 99 0-692 || 0-05S 66 || 0-038 Earth, loose, - - 95 || 0-66 || 0.055 74 0-517 0.043 50 0.029 Oak, dry, - - - 58 || 0-4 0.033 0.316 0.026 30 || 0-017 Pine, - - - - - 30 0.208 || 0-017 24 0-163 || 0-014 || 16 || 0-009 Mahogany, - - - 66 || 0:457 0.038 ± 52 0-36 0.03 ||34 || 0:02 Coal, stone, - - - 54 || 0-375 0.031 || 42 0.294 || 0-024 || 28-2 || 0-016 Charcoal, - - - - 27.5 ! 0-19 || 0-016 || 21 0-15 0-012 : 14-4 i O-008 To Find the Weight and Capacity by this Table RULE. The product of the dimensions in feet or in inches, as noted in the columns, multiplied by the tabular coefficient, is the capacity of the solid, or Weight in pounds avoirdupois. Example 1. A cistern is 6 feet long, 27 inches wide, and 20 inches deep. How many gallons of liquid can be contained in it? 6×27×20×0.052 = 16848 gallons. Example 2. A cast-iron cylinder is 4.5 feet long, and 7-5 inches diameter Required the weight of it? 4.5×7-52)×2:45 = 620 pounds. SELECTION 0F WATER COLOURS. Blue. Iteal Ultramarine. Red. Rose Madder. . French Blue. “ Light Red. & Indigo. r * Cobalt Blue. Brºwn. Hºnºr. Green. Olive Green. Black. India Ink. Yellow. Cadmium. 46 Blue Black. &ć Gamboge. 66 Ivory Black. £6 Ochre. * Lamp Black. JR'ed. Carmine. White. Chinese White. 4& Crimson Lake. 292 PROPORTION OF BOLTS AND NUTS. Proportion of Bolts and Nuts. Number of Threads per Inn. Diameter. sº 3 inch. 4% 5} 5 71's 2} 10 2} 4; 4} 4; 6; 2} 9 2} | 3} || 4 || 4 || 5}} || 2 || # 9 2} | 3% 3% 3% 5; 13 || 1: 8 2 || 3 || 3% 3} | 4} | 1; # 7 1% 2} | 3} || 3 || 4%; 1} | {} 7 1} | 2% | 3 || 23 3% | 13 || # 6 1; 2% 2% 2; 3}} | 1+ # 9 6 1; 2} 2; 2% 3% 1; 1. 8 5 # 1; 2 2} 2} 31%; 11's 1} 7 4 : 1+ 1% 2} || 2 || 2 || 1 || 14 || 7 || 3% #1} | 1 || 1 || 1 || 2 | | || 1 || 6 || 8 # 1 || 1 || 14 || 1 || 2: | | || 13 || 5 || 23 5 § | 1.4%; 1} | 1% 2} # 2 4% 2} # 11, 1} | 1.1%; 1% # 2% 4 2 # | 1 1} | 1.4% 1+} | } 3 3% *; # | 1 1 11's # || 3; 3} # # # # 1} | # 4 3 t’s # # # 11's # 4; 2% # +'s § # T's Hºr 5 2% # *; § * +} Hºs 5% 2} # # 1 # # 1 +} | } || 6 || 2% The above proportions of bolts and nuts are established by Sir Joseph Whitworth. Weight in Pounds of Nut and Bolt-Head. Diameter of Bolt in Inches. 1 1 Head and Nut. Zſ # 3. # # # 1 1} 1% 1} 2 2% 3 Hexagon, . . .017|.057 .128} .267 .43 .73 || 1.1 2.14 3.77|| 5.62| 8.75] 17.2| 28.8 Square, . . . .021|.070|.164! .3211 .553] .882| 1.31| 2.56; 4,42| 7.00) 10.5|21. 36.4 f SCRFW-TITREADS. sº 293 Proportion of Screw-Threads, Nuts and Boltheads. Diam. Of Threads | Diamet. Width Outside Inside Diagonal height Screw. per inch. of core. of flat. diamet. diamet. "*ē " of h’d. •5 ſi6 | 18 •240 •0070 '3ſ 8 || 16 •294 || 0078 •7 ſ.16 || 14 *344 || -0089 •] ſ? 13 •400 || 0096 •9 ſI6 | 12 •454 || 0104 •5 ſs 11 •507 || 01.13 •3 ſ.4 10 •620 -0.125 -7ſ 3 9 •731 || 0140 1. 8 •837 || -0156 1:1f8 7 •940 || “0180 1-1 ſa, 7 1.065 •01SO 1.3 ſ 8 6 1-160 •0210 1:1ſ2 6 1.284 || 0210 I.5ſ 3 || 51ſ2 1:389 || 0227 1.3 ſq 5. 1°490 •0250 1:7f8 5. I-615 '0250 2. 41ſ2 1-712 || 0280 2:1f4 || 4:1ſ2 | 1.962 || 0280 21ſ2 4. 2-175 •0310 2:3 ſ.4 4. 2-425 || 0310 3. 31ſ2 2.628 0357 3-1 ſq 31ſ2 || 2:878 || 0357 31ſ2 | 3:1 ſq | 3:100 || -0384 33 ſ.4 || 3: 3.317 || -0.410 4. 3. 3-566 •0410 4.1 ſq 27 ſs 3798 || 0435 41ſ2 || 2:3f4 || 4:027 | 0460 4:3 ſq 2.5 ſ 8 || 4-255 -0.480 5° 21ſ2 4,480 || 0500 51ſ 4 || 2:1ſ2 || 4730 || 0500 51ſ2 23 ſs 4,953 0526 5:3 ſ4 || 2:3 ſ 3 5.203 || 0526 6. 2:1 ſq 5:423 || 0555 || 10-9 ſi6 9-1ſ 8 || 12:9 ſlo 4.9 ſ 16 Mr. Whitworth makes the angle of the threads 55°, with round top and bot- tom; whilst William Sellers makes the angle of the thread 60°, with flat top and bottom, and of the following proportions, which are recommended by a special committee appointed by the Franklin Institute of Philadelphia. For full infor- mation see Journal of the Institute, May, 1864, and Jan., 1865. Notation of letters. All dimensions in inches. D = outside diameter of screw. * = inside diameter of hexagon, or d = diameter of root of thread, or of side of square nut or bolthead. hole in the nut. s = diagonal of square nut or bolthead. p = pitch of screw. h = height of rough or unfinished bolt- t = number of threads per inch. head. f = flat top and bottom. The height of finished nut or bolt- o = outside diameter of hexagon nut head is made equal to the diameter D or bolthead. of the screw.} WI6 DTIO – 2.909 1 e 16-61 t p s = 1.414 . 1.299 . 3 D . 1 & £ d = D t º, 2 + = O = 1°155 2. f= } * Whitworth makes the height of the nut about half the hexagon diameter v. 2)4 e GEARſNG. G EA RING. Letters denote. P = pitch, –the distances between the centres of two teeth in the pitch circle. 1) = diameter C = circumference . M = number of teeth & the Wheel. JW = number of revolutions d = diameter c = circumference 4 tº 1n = number of teeth : of the pinion. m = number of revolutions --if - 1 C = P M - 5 Pitch Circum. P ºr D © C = az D - 6 == M ge 4 M--2 - 3 –PM - 7 P T. No. of teeth Diameter M-4. . . D--- . . - B * TTz sº D. : d = C ; c = M ; m = n : N Example 1. A wheel of D = 40 inches in diameter, is to have M = 75 teeth. Required the pitch P = ? g º 4X Formula 2. Pitch P= slºw Eacample 2. The pitch of teeth in a wheel, is to be P = 1-71 inches, and having M = 48 teeth. Required the diameter D = ? of the wheel, º l'ſ 1X48 Formula, 7. Diam. D Tà-IT = 1.66 inches nearly. =26:14 in. of the pitch circle -v- &EARng. . 295 Construction of Teeth for Wheelse Draw the radius I? r, and pitch circle PP. Through r draw the line o o' at an angle of 75° to the radius R r. _ 180 wheel, v = -t:-. - - - - I Half the angle be- M * tween two teeth in the 180 inion. V =—" - º * - 2 p j 272, D: d = sin. V: sin. v. wheel, D = d sin. ". . . . 3 e 3???, ?) Diameter of the | * * D sin. v pinion, d = ± - - - 4 Pitch (chord) of teeth { wheel, P = D sin. v. - - - 5 in the pitch circle pinion, P= d sin. V. - - - 6 Approximate pitch in the wheel P= 0.028 D. - - - 7 wheel, M = *}” - cº- • 8 Number of teeth about • * d Mſ pinion, m =-B- º º , 9 Thickness of tooth, a = 0:46 Pº - - - - 10 Bottom clearance, b = 0°4 P. º º gº º - 11 Depth to pitch line, c = 0-3 P - - - - - 12 P(m+6) 2 (m-II)” Distance r o', e = 0-11 P V n - -> º º - 14 13 Distance ro, d = * If a wheel of more than 80 teeth is to gear a pinion of less than 20 teeth, and the wheel and pinion are of the same kind of materials; take the thickness wheel, a = P (0.42 + #) - - 15 of the tooth in the | º - © 7 pinion, a = 0:5 P(l T 350 J. - 16 A rack is to be considered as a wheel of 200 teeth. 296 GEARING. sºmºsº Example with Plate I. Example. A wheel of D = 48 inches diameter is to gear a pinion about 8 revolutions to 1. Required a complete construction of the geariag? Approximate pitch P= 0.028×48=134 in. - - - ? 3.14×48 Number of teeth - M = -TăI- T 112. • 8 in the pinion, m = * = 14 wº tº 9 º º v - – --1-36, sin–0.028. l * * * (pinion y- º =12051'.'sin–0-2224. 2 Diameter of pinion d = *...* = 6'043 in. - 4 Draw the pitch circle for the wheel and pinion so that they tangent one an other at r on a straight kine between the centres of the circles. Pitch in the gearing P= 48×0.028=1344 in. * 5 Take this chordial pitch in a pair of compasses, and set it off in the pitch circles. Thickness of wheel a = 1°344 (0.42 + # )-0.59%in. 15 tooth | tº e 14 plnion a = 0.5×1844(1-#)-0.645in. 16 Set off the thickness of tooth in the corresponding pitch circles. Bottom clearance b = 0°4X1'344 = 0-5376 in. - 1] Depth to pitch line c = 0.3 \l'344=0'4032 in. . - 12 d = 1:344(112+6) Distances r o and Z(IIZEII ) r o' in the wheel . ...------ © e = 0.11x1'344 V112 = 0-7126 in. 14 Set off these distances on the line o o' from r,-d beyond and e within the pitch circle for the wheel; then o is the centre and o m radius for the flank ºn. o' the centre and o' m radius for the face n. Draw circles through o and o' con- centric with the pitch circle of the wheel. _ 1:344(14+ 6) • A C : tº Distances r o and º TZ(IHEIT) - 4'48 in. 13 ' in the pini * * * * *** 'e = 0.11x1.344 VII = 0.356 in. 14 Proceed with the pinion similar as the wheel On the plate is a scale of inches and decimals, which will be ton- venient for the above measurements. = 0-7851 in. - 13 /*/ø//, /. % . Ž % %; % % % % : º % 2... § § § lº &%% &/2% % % % º 2 g %% % % / ſ.l.it/x/// . ill- i § º * &tº s -j- ºs- }* : sº + }sº ---T -- * eº- |-- sº ** l |- ss as “s • * * * > . t ^. ; STRENGTH OF TEETH. 297 On the Strength of Teeth in Cast-iron Wheels. P=pitch, a = thickness, and h = face of teeth in inches. feet per second in the pitch circle. S=Strain in pounds, H = horsepower, and V-velocity in Pitch IP, Thickness a , Face he P=–3– CI, E —º- g # =–3– & 460 h. 1000 h. 1000 a. P_1% H. a 0.58 H. h–* H. h, V h V P V Strain S. Horsepower H. Velocity V. h, a V 0.55 H. - H = tº = —. S = 1000 a. h. 0.55 V h a S = 460 Ph. h P. V. 1.2 H. FI- —. - ſº 1.2 y }, P The face h is generally made = 2.5 P= 5.435 a. a = 0.46 P, and P=2.1739 a. To Find the Diameter of Axles and Shaft. Letters denote, d = diameter in inches, in the bearing; and the length of the bearing 1.5d. |W- weight in pounds, acting in the bearing. d= ??of cast iron. Common f d = Y Wof cast iron. Water- 18 Machinery | 24 wheels. * in good sms-mº d = ºo: wrought irop. Order. d = ſºror wrought iron. Example 1. A water-wheel weighs 58,680 pounds, and is supported in two bear- ings. Required the diameter of the wheel axles? Tho weight acting in each bearing will be 58680: 2 = 29340 pounds, and se $º e /29.340 e g diameter d = ++ = 8.15 inches of wrought iron. 21 © Example 2. Fig. 226, page 307. Required the diameter of the axle in the wheel, when the weights P + Q = 4864 pounds? If the wheel is supported in two bearings W-4864 : 2 = 2432 pounds. V2.432 diameter d = gs-- 1.76 inches of wrought iron. 298 STRENGTH OF MATERfALS. *== Fºcample 3. The pressure on the steam piston, in a walking beam engine is 25000 pounds. Required the diameter of the beam journals? diameter d = viºſ = 5:64 inches the centre one. d = vº; 00 = 4 inches at the ends. º this example it is supposed that the beam is worked by a fork connecting i’OCl. & FI: - 5, */H. 4 70, D = inches wrought iron. It = radius of crank in feet. F = force from the steam piston, lbs. D = D: d = W T : V7 p=4&M/{ Il H = horse-power transmitted. n = number revolutions per minute. When an axle or shaft not only serves as a fulcrum, but effect is transmitted by the act of twisting it, the diameter is to be caluulated as follow. Example 4. The pressure on the piston in a steam engine is F = 45,600 pounds, applied direct on a crank of Ið = 3 feet radius. Required the diameter of the shaft and crank pin f 3/ TET2 AWNZº Diameter of the shaft D = zºº = 12.9 inches. v 45600 .28 Example 5. A steam engine of 368 horses is to make 32 revolutions per minute. Required the diameter of the main shaft? 3 *-* - ſº 8 tº Diameter D = 5 * = 11% inches. Ezample 6. A cogwheel of R = 6.5 feet radius is to gear with a pinion of r = 1.25 feet radius, and to transmit an effect of 231 horses with 42 revolutions per minute. Required the diameter of the wheel and pinion shafts? The force Fis acting uniformly at the periphery. * * * , */[23] . . * Diameter of wheel shaft D = 4:35 —t:- = 7-66 inches Diameter of the crank pin d = = 7-63 inches. 42 D : d = {/TE : y Tr" 3 y-Tº- Diameter of pinion shaft d = 7-66 N/ #– = 4:41 inches. Y". Roofs of WooD AND IRON. 299 R00FS OF WOOD AND IRON. illustrated for iron roofs. from 30 to 80 feet. Span in feet. - -*. The Figs. 1 and 2 illustrate the common form of wooden roofs, as constructed over spans of from 30 to 80 feet. When the span exceeds 60 feet, a proportionate number of struts and tie-rods must be inserted, as shown by the dotted lines, or as Table of Timber Dimensions, in Inches, for Roofs over Spams • { Name of timbers, 30 || 35 | 40 || 4–5 || 50 55 60 70 80 King's rod, I 1. I lº, 1+ 1$ 1% 1; 2 Bolts, . . # # # # | g I Tic-beams, a |5×6.6×7|6X8|7X8|8×9|8). 129 x 1110 × 11|10×12 Truss rafter, b |5 × 5|5× 6|6×7|7X 78x8|SX 9|9 × 9 9 × 10|10X11 Collar-beams | c. 5 × 5|5 X 6|6 X 7 || 7 X 7 |8 X88) ( 99X 9| 9 × 10110X11 Com. rafter, d 2x 5|2×5|2×6|2X62 X62× 7|2X 7|2}× 8| 3 × 9 Purlins, . e 5× 6|5X 6|5X 7|6X 7|6× 86 X 8|6X 9| 6X 9| 6X 9 Struts, . . . f 3 × 4]3 × 5|3}× 64× 7|4X8||5|X 8.5 × 9 6 × 9 6 × h .1; I} 13 9 Load on Roofs in Pounds per Square Foot, exclusive of Framing. Pounds. Pounds. Lead covering, . . . 8 Tiles, . e tº . 9 to 16 Zinc covering, . e g 2 Doarding, + thick, . 3 Corrugated Iron, . . . 3.5 Boarding, 1% thick, . 6 Slates, . . - e. 1() Pressure of wind, . 40 of 10 pounds per square foot per foot of depth of the snow. º In high latitudes the roofs may be covered with Snow, which makes a pressure 800 STRAIN ON ROOFS. Ž ' £ A& Jigure 3 is most in use. L'- length of principal rafter. JR = rise of roof above centre of tie- rod, which is generally one-fifth of the span. S = half the span, or one-half the length of tie-rod between Sup- port on the walls. IV = number of bays or divisions in the whole span, which is 8 in the diagram. s, s' and s” = compression on the cor- responding Struts. Compression on Tension on Tie- Principal Rafter. rod. I, W. WS =— . T= −. O. 2R 2R O =; C– “. C W T 4. cº-o-º: N' N & 3C 2T // = O – …. t” – T'--. C N N STRAIN ON R 00 F S. The above figures illustrate four different kinds of pointed iron roofs, of which FINE LINES IN TENSION AND THICE LINES IN COMPRESSION. Notation of Letters. W= load, uniformly distributed on the rafter, including weight of framing. C = compression at the end of prin- cipal rafter. T = tension at the end of tie-rod. c, c' and c” = compressions between the corresponding divisions of the rafter. t and tº = tensions between the cor- responding divisions of the tie- rod. The formulas will answer for any units of weight and measure. Tension on King and Compression on Queen Rods. Struts. R_*W. s=#". N JN Q–15% s'=º. N N ? Jy 2.66 W ^ = −. s’’= −. Q N IN The principals to be placed not more than 7 feet apart, BRIDGES. - 30] BRIDGES. . | 2 3 S S T S | | l | 3 S S S S ºf \ A^\ /? T. Tº TTTLA- The Warren Girder. FINE LINES IN TENSION AND THICK LINES IN CoMPRESSION. The Warren girder consists of fifteen equilateral triangles formed by the trusses and ties, which make eight divisions or bays in the span. The depth of the girder is 0.10825 of the span. The load uniformly distributed on each girder, multiplied by the tabular number, will be the strain on the corresponding part. Parts in Compression. Parts in Tension. Top-beam. Ties. S Sl S2 S3 t #1 #2 #3 0.5 0.875 | 1.125 | 1.25 0.8 0.6 . 0.4 0.2 T TI T12 Tº S s! s? 83 Bottom Tie-rod. - Trusses. Parts in Tension. Parts in Compression. Weight of one pair of Warren’s Girders in tons, Jor a single track of railway on the top or on the bottom (approximate). On Span of the girder in feet. º the 50 | 60 70 80 || 90 100 110 120 || 130 140 || 150 | 160 -a *- : - . *- : *- Top, 11 || 15 18 || 23 27 || 32 38 || 44 || 51 || 58 66 75 Bottom, 15 19 || 24 29 35 | 41 || 48 56 | 64 | 72 80 | 89 Table of Dimensions in inches of Rolled Iron, Jor roofs on spans from 30 to 80 feet (figures 3 to 6, page 300). Name of Span in feet. ll'Oll. 30 40 50 60 70 SO Rafter T-iron, L. 24x2}X#| 3}X3X} || 4×3}×{} | 5X4}x} | 5}×5×3 6×6X; Struts T-iron, S|2}X2#X; 3X2}X# 3X3Xà || 4X4X# 4}X4}X#| 5X4}X; King bolt, K| 1 I+ 1} 1#. I# li Queen bolt, { Q # # 1. 1; 1#. 13 Q' § # # l l; 14 - l tº 7 Tie-rod, { T 1; I} l; 1; 1: 1#. tf 1 * l; 1+ 1$ 1; Iš Weight, lbs., 1500 3000 4800 7000 9550 12400 The last line shows the approximate weight in pounds of each principal when the rise of the roof is 0.2 of the Span. 302 SUSPENSION BRIDGES. SUSPENSION BRIDGES. Notation of Letters. W = total load on the bridge. EI = versed sine, or height of abutments T = tension on the chain in the centre. above centre of chain. t = tension at any point of distance, D and d = co-ordinates for any point of D, from the centre. the chain. t’ = tension at abutments. v = angle of suspension at abutments. S = span. (The angle of the counter-chain L = length of chain between the ought to be equal to that of sus- piers. pension.) The formulas will answer for any system of weights and measures. - #. - L=3(S+ VO25SETJHF) 2 ! l 2 I'= 3 W cot.w. #/= y cotwV (; ) +1 I/2. NT t = TN(#) + 1. IV 2" sin.” 4 H \? T. W.S. W(#)+1 8EI S WS & H=#. v=#V(#)+i. 8 H W \ S. S JH = 13? generally. 8H tº I/AIHNXT 4 HT W_*#W(#)+ cotan.v = Ts' STEAM HAMMER. 303 BOLLMAN’S AMERICAN TRUSS BRIDGES. = total load uniformly º *I- lengths of tension and coun- the bridge. ter-tension rods. w = load on each point of suspension. | H = depth of truss, which is usually S = span. one-seventh of the span. I) and d = distances from abutments A | N = number of points of suspension. and B to point of suspension. T'and t = tensions on the rods R and r A and a = cross areas of the tension respectively. and counter-tension rods in C = compression on the top at centre. Square inches. These formulas will answer for any system of weights and measures. w D R. wd r S W 1) = -- - -– e t = —- C = −. S H. S H. 8 H ºhen T and t are tons, A and a = square inches, D, d, S, H., R and r = feet, €Il W. D. R. a War . W. 50 N.S H. W. 54 NSIH. 5.N.S H 5.N.S H. T dr T DR S T E A M H A M M E R. A heavy steam hammer with short fall produces a better forging than a light hammer with a high fall, although the dynamical momentum may be the same in both cases. This is accounted for by the inertia of the ingot forged. The effect of blows of a heavy hammer and short fall will penetrate through the metal, and nearly with the same effect on the anvil side, while a light hammer and high fall will affect the metal on or near the surface of the blow, because most of the momentum is in the latter case discharged in the inertia of the ingot forged. In forging a large shaft, it is generally piled up with iron bars sometimes rolled into a segment form to suit the pile. When placed under the hammer in a welding heat, very light and gentle blows are first given, then the momentum of a light hammer may be discharged in the bars nearest to the blow, while a heavy hammer will squeeze the whole mass together throughout, and a sound welding will be produced. The additional expense of a heavy hammer is fully compensated by the waste of labor and materials under a small one. I have often seen, in broken shafts, the bars in the centre as clear and unwelded as when first piled, which is a sure indi- cation that the shaft has been forged by a too light hammer. In crank-shafts for propeller engines, forged under a light hammer, when brought to the machine- shop the best part of the metal is worked away by planing and turning, and the poorest left for the engine; but if forged under a heavy hammer, the difference in quality of metal will not be so great. Cases of this kind are well known in the United States mavy. Weight of Steam Hammers. The weight of a steam hammer in pounds should be at least eighty times the Square of the diameter of the shaft in inches. 304 GRAVITATION. G R A W IT A T I 0 N. Gravity or Gravitation is a mutual faculty which all bodies in nature possess, to attract one another; or Gravity is the force by which all bodies tend to approach each other. A large body attracting a comparatively very small one, and their distance apart being inconsiderable, the force of gravity in the small body will be very sensible compared with that in the large one; such is the case with the body, our earth, attracting Small bodies on or near her sur- face. Gravitation is not periodical, it acts continually ever and ever. A body placed unsupported at a distance from the earth, the force of gravity is instantly oper- ating to draw it down, and then we say, “the body fell down " If it were possi- ble to withdraw the attraction between the body and the earth, it would not fall down, but remain unsupported in the space where it was placed;—giving the body a motion upwards it would continue that, and never come back to the earth again. Law of Gravity. The force of Gravity is proportional to the mass of the attracting bodies, and in- verse as the square of their distance apart. This law was discovered by Sir Isaac Newton. It is this law that supports the condition of the whole universe, and enables us to calculate the distances, mo- tions and masses, &c., of the heavenly bodies. The unit or measure of force of gravity is assumed to be the velocity a falling body has attained at the end of the first second it falls; this unit is commonly denoted by the letter g; its value at the level of the sea in New York is g = 32.166 feet per second, in vacuum. The space fallen through in the first second is #g = 16:083 feet. This value augments with the latitude, and abates with the elevation above the level of the sea. l = latitude, h = height in feet above the level of the sea, and r = radius of t]ue earth in feet, at the given latitude l. = 20887510(1--000164 cos.21), g = 32.16954(1-000284 cos2)(1– *) Letters denote. S = the space in feet, which the falling body passes through in the time T. a = the space in feet, which the body falls in the Tth second. V= velocity in feet per second, of the falling body at the end of the time T. T = time in seconds the body is falling. st V S T R& The accompanying Diagram is a good il- 221 lustration of the acceleration of a falling body. The body is supposed to fall from a 1 2 1 1// to b, every small triangle represents the space 1608 feet which the body falls in the first second; when the body has reached 3 4 4. 2" the line 3" seconds, it will be found that it has passed 9 triangles, and 9X16:08 = 14472 feet the space which a body will fall in 3” 5 6 9 3// seconds. The number of triangles between \ each line is the space w which the body has fallen in that second. Between 3/ and 4// 7 8 16 47 are 7 triangles and 7X16:08 = 112:56 feet, the space fallen through in the fourth sec- t ond. Under the line 3' will be found 6 tri- 9 10 25 5// angles, which represents the velocity V the 0 body has cetained at the end of the third second or 6×16:08 = 96.48 *: ºil b For every successive second the body wi 11 12 36 6// gain two triangles or 2×16:08 = 32-J 6 feet per second. GRAVITATION. 305 Formulas for Accelerated Motion. v=o T-*#– Vºſs-s02/s. • . . . 1. s=94: = Y4 – Yº = -º, . . . . 2. 2 2 2 g 64.33 T - Y - 28 = 2s – V8, e º e º . 3. 9. y 9. 4.01 tº = g(T – #), T-3 tº . . . . 4. Example 1. A body is dropped at a height of 98 feet. What velocity will it have when it reaches the ground, and what time will it take to fall dewu ? Formula 1. W = 8,02)/S = 8.02/98 - 79.39 feet per second. Forºwula 3. 77 = WS -: V98 2.46 seconds. 4.01 4.01 Example 2. A body was dropped at the opening of a hole in a rock, and reached the bottom after 3.5 seconds. How deep was the hole? JFormula 2. S = ag: - alsº = 196.98 feet. Retarded Motion. A body thrown up vertically will obtain inversely the same motion as when it falls down, because it is the same force that acts upon it, and causes retarded mottoº when it ascends, and accelerated motion when it descends. W = the velocity at which the body starts to ascend. v = velocity at the end of the time t. Z = time in seconds in which the body will ascend. t = any timize less than T. s = height in feet to whicle the body will ascend. s = the space it ascends in the time f. Formulas for Retarded Moțiorh, v= y-at-#–4, º e º © © 5. y- #2 g/ (2 = Wrt –9 & = t v 4-9 ºf 3 2 v -- 2 3 e e e © e 6. º º e e º 7. - _ 8 Q & W = u + 9 t = & + 2 2 3. & *-***=}-\º-#, . . . 8. 9. 9. gº 9. Formulas for T and S are the same as for accelerated motion. Facample 3. A ball starts to ascend with a velocity of 135 feet per second. At what velocity will it strike an object 60 feet above? Find the time t by the Formula 8. – 135 — 135° 2 × 60 32.16 T V35.IGs Tº.15 = until it strikes, and from Formula 5 we have w = 135 – 32.16 × 0.41 = 121.83 feet per second. & 0.41 Seconds, 20 306 w GRAVITATION. Eacample 4. A ball thrown up vertically from a cannon, occupied 20 seconds, until it arrived at the same place it started from. IIow high up was tho ball, and at what velocity did it start? One-half of 20 = 10 seconds. Formula 2. 2 s-ºr-1608 feet high. V=32.16x10=321.6 feet per second. If a cannon-ball be shot from A, in the direction AB, at an angle BAC to the horizon, there are two forces acting on the ball at the same time, namely — the force of gunpowder, which would propel the ball uniformly in tho direction AB, and the force of gravity, which only acts to draw the ball down at au accelerated motion; these two different (uniform and accelerated) motions will cause the ball to move in a curved line (Parabola) AgC. Fig. 225. W = velocity of the ball at A. W = weight of the ball in pounds. S = the greatest height of ball over the horizontal kine A. C. t = time from A to C, vià a. p = pounds of powder in the charge. = the distance from A to C, called horizontal range. TV V2 º TV = 2800W., p=#, b–87.06 sina cosz ºft. W 78.40000 W Eacample 5. The cannon being loaded sufficiently to give the ball a velocity of 900 feet per second, the angle 2 = 45°. Required, the distance b = ? and the tinue t = ? - 9002 × sin.45° x coS.45° 32.16 It will be observed that the distance b will be longest when the angle & is 45°, because the product. of sine and cosine is greatest for that angle. Sin, 45° X cos. 45° = 9.5. Example 6. What time will it take for a ball to roll 38 feet on an inclined plane, angle a = 12°20', and what velocity has it at 38 feet from the starting- point? Fig. 222. * T= N —#–– 2 s 8 =3.33 seconds. g Sln.” 32.16 × sin.12° 20' V=g Tsina = 32.16x3.33x sin.12°20′ =22.8 feet per second. Resistance of Air to the FTight of Projectiles. A = area of resistance of the projectile in Square inches. q = angle of resistance of the projectile, which for flat surfaces sin.9% = 1, for sphere sin.” (p = 0.5. For a pointed projectile of parabolic form, and when the ordinate is double the abscissa sin.” (p = 0.25. W = velocity of the projectile in feet per second. It = resistance to the projectile in pounds. R=A V* sin.”g, 57000 Let T denote the time of flight in seconds, and W = Weight in ponnds of the rojectile. p } = distance in feet which the projectile is retarded by resistance of air in the time T. 32.166 RT′′ 16 FT" 2 W. W b = =1259 feet, the distance from A to C. ! I)= FoEcg of GRAVITY. . 307 222. *. y ~: 3. º sin.” == M2g S sin.ar, &. W *** AS S - g T's - y 2 JC 2 sin.a 2 g sin.a.” L =#|T_ _V_ _ , / 2S g sin.” g sin.” Q 9NSg/ 223. A body will fall from o the distances a, b, c b and d, in equal times. /2d T - ^ – e. # 224. Cº. A body will fall from a to b via c in the short- < P ð, est time, if the curve is a Cycloid. *- Tw S=4d, the length of the Cycloid. * a v - Tø –P- - 77 - = 77 e * C 29. 2ſt g 2V2sin.a cos.a. 3 V sin.a. s — V'sin.” y -- T2g sº © T. F. V*M, 2M T 2g F F /2g SF V M vº g F g F * W. M. _2S M Tº Tg. Tº 808 TABLE Fort FALLING BODIE8. ſ | Velo. Space fall- | Time in velocity |Space fall- Time in I velocity |Space fall. Time in city 1.2577 || 0-281 50 3S-820 || 1:555 I-2858 || 0.283 51 40-388 || 1-588 1-3143 || 0-287 52 || 41-987 | 1.6 19 1.3430 || 0-290 53 43-618 1.650 1.3720 0.293 54 45-279 || 1-680 1°4041 || 0-296 55 46.972 | 1.711 1-4310 || 0-300 56 48-695 1-742 1°4610 || 0-302 57 50-450 1774 1.4913 || 0-306 58 52°236 1-805 1-5219 || 0-309 59 55.058 1.835 1.5528 || 0-312 60 || 55-900 i”868 •24844 || 0°1246 •26102 || 0-1278 •27391 || 0-1309 •28571 || 0-1339 •30062 || 0-1371 "31444 || 0-1403 •32857 || 0-1433 •34301 || 0-1465 •35776 || 0-1496 •37.282 0-1526 •38820 || 0-1559 at the on through seconds, * en through | seconds. º en through seconds, V S T V S T V S T 0-1 || 00015 || 0-0031 5-1 '40388 || 0-158 11 || 1:8789 || 0-342 0.2 || 00062 || 0.0062 5.2 '41987 || 0-162 12 || 2-0652 | 0.373 0-3 || 00139 || 0-0093 5-3 || “43618 || 0-165 13 2-6242 0-405 0°4 || -002.48 || 0-012.4 5-4 "45279 || 0-168 14 3-0435 | 0°436 0°5 || -00388 || 0-0155 5.5 46972 0-171 15 3-4938 0°46'ſ 0-6 || -00559 || 0-0187 5-6 || 48695 || 0-174 16 || 3-9751 || 0°498 0-7 || -00761 || 0-0218 5-7 || “50450 || 0-177 17 || 4'4S'76 || 0-530 0°8 || -00994 || 0-0230 5.8 || "52236 || 0-181 18 5-0310 || 0-560 0-9 .01257 || 0:0280 5-9 °55057 || 0-184 19 || 5-6056 || 0-591 1. •01552 || 0-0311 6- *55900 0-187 20 6-2112 || 0-622 1°1 | "01879 || 0-0342 6-1 || '57779 || 0-190 21 6-8478 0-654 I-2 •02065 || 0-0373 6-2 | "59689 || 0-193 22 7:51.55 0-685 1°3 || 02624 || 0-0.404 6-3 || 61630 || 0-196 23 8-2143 || 0-7 16 1°4 -03043 || 0.0436 6.4 63602 || 0-199 24 8-94.41 || 0-747 I-5 -03493 || 0-0467 6.5 | *65606 || 0-202 25 9-7049 || 0-778 1-6 || -03975 || 0-05 6.6 '67639 || 0-205 26 || 10-497 || 0-810 l'7 || 04487 || 0-052 6:7 | 697.05 || 0-209 27 11-320 || 0-840 1-8 || 05031 0.556 6.8 || 71801 || 0-212 28 12.174 || 0-872 1-9 || -05605 || 0-0591 6-9 || 73928 || 0:215 29 || 13-059 || 0:903 2- •06211 || 0-0623 7. •76087 || 0-218 I. 30 13-975 || 0-933 2-1 || -06847 || 0-0654 7.1 | 78276 0-221 31 || 14-922 || 0-965 2-2 || -07515 || 0-0685 7.2 '80497 || 0.224 32 || 15-900 || 0-996 2-3 || 08214 || 0-0717 7-3 || 82748 || 0-227 33 | 16.910 | 1.025 2-4 || 08944 || 0-0747 7.4 || 85031 || 0:231 34 | 18-789 || 1:058 2.5 || 09705 || 0-0780 7.5 -87.344 || 0-234 35 | 19.022 | 1.091 2-6 || 10497 || 0-0810 7-6 || “.89689 || 0-237 36 20-124 || 1:120 27 | 11320 || 0-0841 7-7 || '92065 || 0-240 37 || 21,258 || 1:151 2-8 ; 12174 || 0-0872 7-8 '94472 || 0:243 38 22°422 || 1:184 2-9 '13059 || 0-0903 7.9 96910 || 0-246 39 23-618 1-213 3' | 13975 || 0-0934 8. •993.79 || 0-250 40 24.844 || 1:243 3-1 || 14922 || 0-0966 8-1 || 1:0187 || 0-253 41 26-102 || 1-276 3-2 15900 || 0-0997 ,8-2 || 1-0441 || 0-256 42 27.391 | 1.308 3-3 | "I6910 || 0-1025 8-3 || 1:0697 || 0-259 43 28.57 I-338 3°4 || 18788 || 0-1059 8.4 || 1:0956 || 0-262 44 30.062 || 1:370 3’5 19022 0.1092 8 5 || 1:1218 || 0:265 45 31-444 || 1:400 3-6 || 2012.4 || 0-1121 8.6 || 1:1484 || 0-26S 46 32-857 || 1:431 3-7 || -2.1257 0-115.2 8.7 | 1.1753 0-271 47 34-301 || 1:463 3-8 22422 || 0-1185 8-8 1-2015 0.274 48 || 35-776 || 1:495 39 •23618 0°1214 8-9 || 1:2299 || 0-27S 49 37.282 | 1.525 k” 9- 4." 9. 4” 9. 4. 9. 4° 9 4' 9 4. 9. 4. 9. 4. 9. 4. 9. 5 l DYNAMICS OF MASS AND WEIGHT IN MOVING BODIES. 309 DYNAMICS OF MASS AND WEIGHT IN MOWING BODIES. Let a constant force, F, be applied to a body, W, free to move, then the body will start and continue with an accelerated velocity until the force ceases to act, when it will continue in the same direction with a uniform velocity equal to that of the final action of the force. Any force, however small, is able to set in motion any body free to move, or to bring to rest any moving body, however large. The direction of the force F Inust pass through the centre of gravity of W, otherwise the body will be set in rotation. A force applied obliquely from space to the surface of the earth would change the axis of rotation, which is actually the case with meteors falling on the earth. No force is required to maintain a uniform motion of a body free to move; but force is required to bring a body from rest into a uniform motion. If force is applied to maintain a body not free to move in uniform motion, such force is expended in overcoming the friction and resistance of the medium in which the body moves. A steamboat or a railway train in motion is thus suspended between the action of two opposite forces—namely, the driving force on the one side, and the friction and resistance on the other. When the opposite forces are equal, the motion will be uniform; and any change of velocity is due to a disparity between these opposing forces. Mass means the real quantity of matter in a body. It is proportionate to weight when compared in one locality. The mass of a body is a constant quantity, whilst the weight varies with the force of attraction or gravity. A body weighed on a spring balance at the equator will weigh less than if weighed at the poles, because the radius of the earth is greater, and consequently the force of gravity is less at the equator. - A body weighing one hundred pounds at the surface of the earth would weigh only twenty-five pounds at a height of 3956 miles (radius of the earth) above the surface, whilst the mass of the body remains constant. The weight of a body is inverse as the square of its distance from the centre of the earth when weighed above the surface; but if weighed below the surface, the weight will be only - W r R W= weight at the surface; w = weight below the surface; R = radius of the earth, and r = radius where the body is weighed. A body weighed at a height h above the surface of tho earth would weigh W F2 QU E ... 2 (R+ h)? and the force of gravity, or acceleratrix g, at the height h, is _ 32.166R" 9 (ET). Therefore the mass M of a body is QU) JM – W. which is a constant quantity. 9 Below the surface of the earth the acceleratrix g is 32.166r R 310 DYNAMICAL TERMS. Attraction of the Sun and Moom on the Earth, Assuming the mean radius of the earth as a unit for the distances to the Sun and to the moon, and the attraction in pounds per 100,000 pounds of material in the earth, then the data will be as follows: Location of Attraction in the centre or ends of Attraction of the Il Distances to the the diameter the earth pointing to the sum º: of arth pozntvm.g Sun. Moon. Sun. Moon. In the centre of the earth, or where the *} 11.3273 || 0.0769971|| 24000 60 or moon is seen to set or rise, . e e On the surface of the earth nearest to the |11.3283 0.0794.466|| 23999 || 59 Sun or moon, i.e. * * e * On the surface of the earth farthest from |113264 0.0744936|| 24001 61 the sun or moon, * e © g ge Greatest difference, . . . 0.00190 || 0.0049530 2 2 It is these differences of attraction which, in co-operation with similar differ- ences of centrifugal forces, cause the tidal waves of the ocean. Although the Sun's attraction on the carth is one hundred and ninety-five times that of the noon, its difference of attraction in the diameter of the earth is only one-half that of the moon, for which reason the moon causes the highest tidal waves. A weight of 100,000 pounds weighed on a spring balance at sunrise would weigh only 100,000 — 11.328 = 99,988.67 pounds where the sun is in the zenith; and if the same weight, is weighed at midnight in a latitude opposite the sun, it would weigh 100,011.328 pounds on the same spring balance. But when substances are weighed by weights, there will be no difference where they are weighed. Dynamic Momentum in a body free to move is of two kinds—namely, Momentum of time F T = M V momentum of motion. A force multiplied by its time of action on a body free to move is the momentum of time, which is equal to the mass of the body multiplied by its uniform velocity after the force has ceased to act, the momentumn of motion. Vis-Viva, or living force, is a term intended to express the quantum of work concentrated in a moving mass, and is usually denoted by M P2, which is twice the true amount of work; but as there is no living force in a dead body, the term is improper and confusing. See Journal of the Franklin Institute for 1864 and 1865. Morment of Inertia is another confused term not always properly under- stood. In substance it means work imparted to or given out by a revolving body. It is denoted by M R2, in which M = the mass, and R = radius of gyration. Inertia in a body free to move is equal to the force applied to change its motion. A force multiplied by the lever it acts upon is called static momentum, which is analogous to the force of inertia multiplied by the radius of gyration in a revolv- ing body, which seems to have a better claim to be called moment of inertia. A List of Confused Dynamical Terms, which ought to be abolished in our school-books. Quantity of motion. Rate of work (which is power). Total quantity of work. Quantity of moving force in a body (in- Actual total annount of work. ertia or work). Virtual velocity. Vis-viva and principle of vis-viva. IIeat a mode of motion. Moment of inertia (if you please). See Nystrom's FLEMENTs of MeciſãNICs, which establishes strict precision in the meaning of dynamical terms, and abolishes the ideal vocabulary heretofore used in text-books on Mechanics. The subject of confusion in dynamical terms has been thoroughly discussed, and may now be considered exhausted. DYNAMICAL TERMs. 311 Proper Dynamit al. Terms. All the terms necessary in statics and dynamics are as follows: JElements. I'unctions. Force = F. Power, P = F V. Velocity = W. Space, S = VT. Time = T. Work, W = F VT. Static momentum = Fl. Dynamic momentum, FT= M V. These terms include all cases in dynamics, whether by mechanical force, gravity, uniform accelerated, retarded, straight or curved linear motion. In the static momentum Fl, the lever lamay mean radius of gyration in revolv- ing bodies. In irregular motion, V means the mean velocity in the line T. Force is any action which can be expressed simply by weight, such as pressure, attractions, repulsion, gravity, inertia, eacertion, cohesion, electricity, magnetism, strain, stress, strength, thrust, burden, load, squeeze, pull, push, resistance, compresswom, etc. Power is any action of force and velocity, such as dynamic momentum of inertia, impetus, dynamºc effect, traction, propulsion, impulsion, labor, etc. Work is any action which includes the three simple elements force, velocity and time, such as energy, vis-vivie, labor, throw, cast, hawl, drag, draw, occupation, eacercise, toiling, lift, raise, heave, cultivate, to till, etc. The term effect is used in three senses—namely, effect of force, effect of power and effect of work. Much inconvenience has been experienced for the want of distinction between elements and functions in dynamics, and sufficient room cannot be allowed for a full explanation of the subject in this Pocket Book. In the Elements of Mechan- ics before mentioned the subject is fully elucidated. The Work concentrated in a moving body is equal to the work expended in bestowing its motion, and is equal to the work required in bringing the body to rest, which is derived from the primitive formula F. W. T.; but for accelerated motion V means the mean velocity in the time T, which is just one-half of the final velocity F, when the acceleratrix G or g is constant. The following table of formulas will show what a variety of problems are con- nected with a force acting on a body free to move. When a body is left free to the action of gravity in falling or rising, the accel- eratrix G = g, and the force F= W. Ea:ample 1. What force F= ? is required to give a body W = 1689 pounds a velocity of V = 36 feet per second in a time T = 5.6 seconds? Find in the formulas under constant force the one which contains the given quantities W, W and T, which is the second formula. F. W. W. 1689 × 36 T g T 32.166 × 5.6 Earample 2. A projectile of JP = 150 pounds is fired horizontally from a rifled gun of S = 11 ſect in length, in which it receives a velocity of P = 950 feet per second. Required, the mean force F = 2 of the powder acting on the projectile, when the friction in the rifle is 230 pounds. JV V2 150 × 950? - - = 191302-1-230 – 191532 d 29 S 2 × 32.166 × 11 302-H23 91532 pounds, the force required. 2 =337.55 pounds, the answer. 312 MOWING BODIES. Example 3.-The moving parts in a propelker steam-engine, such as the steam- piston, piston-rods, cross-heads, connecting-rod, &c. &c., weigh W = 8456 §. Stroke of piston = 4 feet, making m = 52 revolutions per minute. What force Fis required for each stroke, to set in motion and bring to rest the moving mass 7 The velocity of the moving mass at half stroke will be (formula , page 263) _2 if r n 2 × 3.1416 X 2 X 52 y 60 60 = 10-79 feet per second. The time for each half stroke will be 60 T = 4 × 52 = 0-28846 seconds, Then the required mean force of the momentum will be yyy 8456 × 10-79 F g T 32-166 × 0.2846 9966.8 pounds. For high grade of expansion of steam, this force acts beneficially to the move- ment of the engine. Example 4.—The mean force of gunpowder in a rifled gun is known to be 231400 pounds, on a projectile W = 180 lbs. The friction of the projectile through the gun is estimated to 264 pounds, leaving F = 231400 — 264 = 231136 pounds. The length of the gun is S = 12 feet, elevated to an angle 2 = 6° 30'. Required the velocity W = ? of the projectile when it leaves the gun. | F . \ 231136 *- -- - SłIN.3, -> - in. O f V = 29 s (; / Vºxºxeſ 1SO sin.6 w) = 995-64 feet per second, the answer. Ixample 5.-What velocity W = ? can a steam-engine of H = 56 horses im- part to a body W = 9 tons in a time T'- 30 seconds? P= 56 × 550 = 19800 effects, and W- 9 × 2240 = 20160 lbs. 2g PT |2 × 32-166 × 19800×30 F- - ^ 20160 - V = 43’538 feet per second. Example 6.-A body War 3685 lbs. is moving with a velocity V = 56 feet per second. What time T = ? is required to bring that body to rest, with a force F= 128 pounds? Wy 3685 × 56 Rſ). T = 7F - 32-165 × 125 T 50-121 pounds, the answer. Example 7–What power P = ? is required to drive a centrifugal gun to throw out balls of W- 50 lbs. every T = 8 seconds, with a velocity V = 785 feet per second (friction omitted)? W 72 50 × 7852 P = 377 - 2×32-166 × 5 = 59867 effects, divided by 550 = 108.85 horses, the power required. Example 8.—A sledge of W = 20 lbs. strikes a spike into a log S = 0.08 foot, with a velocity of W = 25 feet per second. Required the forco F = ? with which the spike was driven into the log, omitting the weight of the spike. Jy y2 20 × 252 Example 9.-A body starts to ascend vertically with a velocity of 860 feet per second. What will be its velocity at the end of T = 5 seconds? V= G T = 32.166 × 5 = 160830 feet per second, and 860 – 160-83 = 699-17 feet per second, the answer. I)YNAMICAL FORMULAS. 313 Dynamical Formulas for Accelerated or Retarded Motion. Constant Force in Pounds acting on a Body free to move. *-** = == = 2 PW 2 K K. * *E* *- º * g T T G T2 T S Final Velocity in the Time T, or Uniform Velocity of a Moving Body. —a m_9_* T 2 S. |29 S F_ _P T | 2 g P T |2g K W= G, T- W==N #==1/2 GS-EF=AH;--AF# Time-in Seconds in which the Force acts on the Body free to move. T=% = lºſ- 2 WS 2 S 2 F. S. _ K 2 S W_ W.K. To ºf \ g F. T a - V K - P - ºr-Wºr, Constant Acceleration of the Force Fin Feet per Second. *=s 2 S Y Y” 9 PZ º' Yê 9 K 2 K. - W - T2 - T - 2 S - WS - TET - WS - TFTA Space in Feet in which the Force acts on the Body free to move. * — Y " — 7° 9 FT% P T g PT4 – 9 K — K = −2− = -2 = 2-3 - -2TF - TF - THFF = GF = -F- Weight in Pounds of the Moving Body. W = 9 º' – 94'4”-39’8–g F T g PTº 9 Fº T-2 (, K_g Tº K sma- ------' ass= -- =s= - - - m. 3 S- - -2TP---Fā---53 Mean Power in Effects during the Time T, or in the Space S. P_ F & 9 F#4' 2 WS” Ty Prº 2 K. J" S 9 Fº T' 2 WS” W. Wº 2 K T K V K = F * * - T - 2 W - a T3 - 377 - T - 2 s - 3 - a 7 Work in Footpounds concentrated in a Moving Body. W 7.2 F W T G Ty FT. F. G T3 g F2 T2 2 S. P * K-Fs====================Pſ. The Body moving in an Inclined Direction of an Angle x. Applied Constant Force in Pownds. ++ inz)= w(#4 inz)= w (z++ in.) == — -- * == — == tº. := -j- e e F W (;% + SIIl-2, w(# Sill.30 Jy 2 g S T SRD1.33 Final Velocity in Feet per Seconds when the Force F ceases to act. y–97 (## sinº) - asºn: := Q | 29 s (; FF sinz). Time of Action in Seconds. Acceleration, ——º-- # 6–9 (## sin.) T g(FF Wsin.a.) T \| g (FFTWsin.…) = g \;= sna). Space in Feet. Work done by F. _ 9 Tº (F -e ss ) - p- * _ Fg TA (F - ... ) s=== F. T S111.3, j . r=ws(## sing)– 2 (## SIIl,36 || - Use the upper sign when the direction of motion rises above the horizon, and the lower sign when the direction of motion dips under the horizon. 314 FLY-WWIIEELS, Force and Work in Revolving Bodies. Cenatre of Gyration. Fly-Wheels. Centre of gyration is a point in revolving bodies in which, if all the revolving matter were there collected, it would obtain equal angular velocity from, and sustain equal resistance to, the force that gives it the rotary motion. The centre of gyratvon in different forms of bodies will be found by the for- mulas on pages 204 and 205. JF = constant force in pounds, acting to rotate the body as in figs. 249 and 250, or the mean force on a steam-piston. & * = radius in feet upon which the force Facts. For a steam-engine the mean radius will be r = 0.63661 X the radius of the crank, or 0.3183 S, when S = stroke of the steam-piston in feet. W = weight in pounds of a fly-wheel, or other rotating body. a: = radius of centre gyration in feet. * T = time in seconds in which the force Fis applied from the first start, or the timo in which the velocity is accelerated. M = number of revolutions in the time T. m = number of revolutions per minute. R = work concentrated in the revolving body. f = irregularity in a fraction of the mean revolutions n. For a double-acting single-cylinder engine, the fly-wheel in its regular course of running has an irregular velocity through each revolution. Its smallest velocity is when the crank is at an angle of 40° from the beginning of the stroke, and its greatest velocity when at 40° from the end of the stroke. The larger the fly-wheel is for a given velocity, the more regular will the machinery run without limit. But the fly-wheel may be made so small that its accumu- lated work cannot carry the machinery around, which will be the case when the irregularity f = 1. In ordinary practice make irregularity f = 0-1 to 0-01. Example 1.—What force F = ? is required to give a body W = 3600 pounds an angular velocity n = 76 revolutions per minute in a time T = 24 seconds, the radius of gyration being a = 12 feet, and the force Facting on a radius 7" = 3 foot ? TV a.2 m 3600 × 122 × 76 T 307-1 T 7- 307-1 X 24 × 3 1779.5 pounds, the answer. Example 2.—Required the weight W = ? of a fly-wheel for an engine of D = 36 inches diameter of cylinder double acting, with steam-pressure p = 50 lbs. per sq. in. S = 6 feet the stroke of piston. Area of steam-piston 1017-8 sq. in., and the force F = 1017-8 × 50 = 50890 pounds. Radius of gyration a: = 10 feet, and m = 48 revolutions per minute. Assume f = 0.05. 2542 I'S 2542 × 50890 × 6 x2 x2 f T 4S2 × 102 × 0.05 Should the steam be used expansively, the fly-wheel ought to be so much heavier, as the initial pressure is greater than the mean pressure. The radius of gyration in a ſly-wheel, including the arms, can in practice be assumed to be the inner radius of the ring. Example 3–What time from the start of engine is required to give the fly- wheel in Ec. 2 a velocity of n = 48 turns per minute? » – 0.3183 S= 1-9098 ft. T W 22 ºz. 67376.2 × 102 × 48 10-85 d 35FTTFF = 50F.T. EOS,05. Tººs = ** seconds. Ea-ample 4.—Let the steam-engine in the preceding examples be applied to a rolling-mill, geared two to one of the rollers. An iron plate is rolled through with N = 8 revolutions of the engines, after which the revolutions were ſound to be reduced to nu = 36 per minute. Bequired the work done in rolling the plate; and what time is required for the engine to regain the m = 48 revolutions? Work done by engine, JC = 2 F S N = 2 × 6737.62 × 6 × 8 = 6468.1152 footps. Work done by fly-wheel, R = W a.2 (n2 — mi2) -: 737.62 × 102 (482 – 362) 5866'5 5866-5 to which add 6468115.2 = 7625786.2 footpounds, work consumed in rolling plate. The time requircd for the engine to make up the n = 48 revolutions will be Waº (n — na) 6737.62 × 102 (48 –-36) 307-17 Fºr TT 307-17 × 50890 × 1-9098 W = = 67376.2 pounds, the weight required. = 1157671 footpounds, T = = 2.71 seconds. *=== Circular MOTION. 315 Formulas for Accelerated Circular Motiome Fºrce F, in pounds, acting on the Lever or Radius r, to rotate the Body. F W 22 ºn W 22 N_ _60 K. RC ~ 307-1777. Tº 2.5GT37. Tº Triº, T ~ 2 TFTW' Final Revolutions per Minute in the Time T. a 120 N 307:17 F4'r 60 K |5866.5 K. — —#–– W 22 - Tºr T. F. T Wa:2 Total Number of Revolutions in the Time T. Time of Acceleration, in Seconds, from the Start of Change of Motion. z_ _W ºn ||Wºy 60 K . . . YYK T 307-17 Fºr T V 2.56 Fºr T ºr 7 m FT T2 Fºr " Radius of Gyration, in Feet, of the Revolving Body. 307-17 Fºr T Nº. - | 2.56 F ºr T2 R. 77 327-78 K. 2 = *-*-* := == * V WN 4 N }/WNFr n y Wm T Fr Weight, in Pounds, of the Revolving Body. w_307-17 T Fr 2.564°Fr_38663 K_ _K Tº * ... T = T. WFT ~ TºârâT ~ 2.15.1 : Nā’ Work in Footpounds, concentrated in a Revolving Body. Tºy 32 m2 2.454 Wa:2 V2 ºr r n F. T K=-Hsiaº-- T2 ==== 2 + r N F Fly-Wheels for Steam-Engines. Fly-Wheel for a Single Acting Steam-Engine for Uniform Work. *===m, w_ºsº S., - 78° ſº. 2 – 78° ſº. F-5sºs T 2,222 - a Nw -- a VW - T. WF' Fly-Wheel for a Double Acting Steam-Engine for Uniform Work. w_*FS., 5042 ſº. 4 – 5942 ſº. F-342 º'S Tº ºf “T 2. Aſ W7F * T n VW = .577 Jºly-Wheel for a Double Acting Two-Cylinder Engine for Uniform Work. Jr. 117%;" S. n_34% |; _ 34-23 |; 1172 Fs T nº ºf “T # 7-iº = NW7 °- 77, 316 CENTRE OF GYRATION'. -- |239. # 2 A line or bar. H– a = 0.5775l, < |l & JC, # a = 0.28877. 240. A circumference round its diameter, A circle-plane round its centre, A cylinder round its aris. a = 0.7071 r. 241. A circle-plane round its diameter. a = 0°57. 242. A Sphere round its diameter. r = 0-8165r, a = 0-6324r. Convex surface, Solid, 243. Parallel piped. 4/*--59 E.T.' *; ***ra i 2} = 37 - #244. Cylinder. 41 +3r: 12 > l°-L3ra TI2. T º * = CENTRE OF GYRATION. 317 245. Cone. ‘L = 2}FT3 º - 20 ° * = 12.h4+3R8 20- 246. Conic Frustum. Q? = ‘h / R*-tSR r--R ra Vº *-i-R r-i-ra )+ 3 R. — r" 30(r — rº ) 247. Cylinder and Sphere. w = Maºrº, z = V a”+£r”. 248. Wedge and Ring. a = 0.204 VIZFTEIS, - R24-ra •-Vº 249. Fly Wheel. _, /Rºſſ --Vº, F G : W g = w”: s”. 250. Fly Wheel with Arms. R2+7° 4r24-5." a"(W.Hw) = W 2–4 w "I2 2 * - , /ö/(Riº+w(4r45°). 12(W--w) >k 3.18 CENTRIFUGAL FORCE. {} E N T R I F U G A L. F 0 R C E. Central Forces are of two kinds, centrifugal and centrepetal. Centrifugal Force is the tendency which a revolving body has to depart from its centre of motion. Centripetal Force is that by which a revolving body is attracted or at- tached to its centre of motion. The Centrifugal and Centripetal forces are opposites to each other, and when equal the body revolves in a circle; but when they differ the body will revolve in other curved lines, as the Ellipse, the Parabola, &c., according to the nature of the difference in the forces. If the centrifugal force is o while the other is acting, the body will move straight to the centre of motion; and if the cenitripe- tal force is o while the other is acting, the body will depart from the circle in a straight line, tangent to the circle in the point where the centripetal force ceased to act. The central forces are distinct from the force that has set the body in motion. If the centrifugal force be made use of to produce an effect, such effect will be at the expense of the one producing the rotary motion. letters denote. F = Centrifugal force in pounds. M = the Mass or weight of the revolving body in pounds. v = Velocity of the revolving body, in feet per second. Ič = Radič of the circle in which the body revolves, in feet. m = number of revolutions per minute. Example 1. Required the centrifugal force of a body weighing 63 pounds, and making 163 revolutions per minute, in a circle of 4 feet, 4 inches radius 2 Aſ E m2 : 63×4:33X1632 2933 2933 Eacample 2. A Railroad train runs 43 miles per hour on a curved track of 115 feet radii. What should be the obliquity of the track? Miles? 432 or a = 13° 10', the obliquity of the track. Example 3. A governor having its arms l = 1 foot, 6inches, how many revol- utions muſt it make per minute to form an angle a, a 309? F= $3 = yºs = 47°5 revolutions per minute. 227. º a - F ~...Q. oo 34.1 M R ºn? #" M* E M v. tº tº . l g R 32-TöR y 4M R 77° n° M R nº x & F-º-º-º-, - 2, M–Pg 8...º. tº º 3, º) &xºn M v × 2933F R-#-F# - - - , 2533TF FR g * -\/-MH-, * = V*#, 5, CENTRIFUGAL FORCE GOVERNORS. 319 w 228. ^ Centrifugal force of a ring. } M na VR Tra - -4150- 229. Centrifugal force of a grinding stone, circle-plane, cylinder, rotating round ltS Centre. M R m2 F– #. 230. Centrifugal force of a cylinder rotating round the diameter of its base. – M172 . 5866 231. Centrifugal force of a ball, Jſ m? R. I'-ºº: 232. Governor. ºw ºn sº sº sº sº. 4 º' -- ºr ººs--" ""Živ - 60 º -º- 54:16 54-16 * ~ 2. A T v h M 7 cos ºr " h * T ~...~" ‘T ºcos.; T cosº.' } cos, -º-;, r = Vlº - ?". 32ſ) YPENDULUM. P E N D U L U M. ' Simple Pendurum is a material point under the action of gravitation, snd suspended at a fixed point by a line of no weight. Compound Pendulum is a suspended rod and body of sensible mag- nitude, fixed as the simple pendulum. Centre of Oscillation is a point in which iſ all the matter in the com- pound pendulum were there collected, it would make a simple pendulum oscil- late at the same times. Angle of Oscillation is the space a pendulum describes when in mo- tion. The velocity of an oscillating body through the vertical position, is equal to the velocity a body would obtain by falling vertically the distance versed sine of half the angle of oscillation. Letters denote. * = length of the simple pendulum, or the distance between the centre of sus- pension, and centre of oscillation in inches. t == time in seconds for n oscillations. * = number of single oscillations in the time t. Example 1. Required the length of a pendulum that will vibrate seconds? here m = 1, and t = 1". & = 39.109 ; = 39°109 inches, the length of a pendulum for seconds. na Example 2. Require the length of a pendulum that will make 180 vibrations per minute 7 here t = 60" and r = 180. _ 39.109ta 39.109×602 ! 772. 1802 = 4:346 inches. Example 3. IIow many vibrations will a pendulum of 25 inches length make in 8 seconds? _ 6254% 6254×8 T W T y25T Example 4. A pendulum is 137.67 inches long and makes 8 vibrations in 15 seconds. Required the unit or accelleratrix g = ? 0-8225l nº 0-8225X137.67%.8% g=–g— = 152 = 10 vibrations. = 32°209, Example 5. A compound pendulum of two iron balls P and Q, having the centre of suspension between themselves: see Fig. 238. P = 38 pounds, Q = 12 pounds, a = 25 inches, and b = 18 inches. How long is the simple pendulum, and how many vibrations will the pendulum make in 10 seconds? a P-b Q 25X38 - 18X12 * = -Fra- = 38-H 12 = 14.68 inches. _ as P+U2 Q _25°X38+18°X12 _ sw.es : * = ±（q) = *āšîă, - 37-68 inches, the length of the single pendulum. •254ff. •254X1 12, F- 6.254. &== º = 10-193 vibrations in 10 seconds. }/l V37-68 If a compound pendulum is hung up at its centre of oscillation, the former centre of suspension will be the centre of oscillation, and the pendulum will Oscillate the same time. PENDULUM AND CENTRE OF OSCILLATION: 321 233. 236. -º- Sumple Pendulum. ||x_12–w 1772 m2 !!! 9 = Tâ72-, || || – 12g tº - 89.1% & T 12, | | | &= zanº T -a-, K7 .2 | \ * (Z/ !, ,-ºr | l, t = nºv l | | | 6.25° ..., |o = centre of suspen- I \ ſº S207&e | | | 6.254t *([C] 21. f 72 = T-3 l = a-- - #--- ºw-º vl “' 5a' O 234. 237. "T" |A = centre of gray- ity. 2 ** B = centre of gyra- t = * P+5° Q. tion. a P--5 Q .4 * C = centre of oscil- B}–9 lation. P and Q expressed w—#C. in ds bi a : 5 = 5 : l pounds, or cubic • ‘’ contents. b = Val-1-1432a, V----- l = #a. 235. 238. Compound Pendu- lum. — rad; & a P — 5 Q r dº of cylin- * --Prº-. , - 16443r. ! - a Płº Q 12a | *(PIQ)” l == 4a, r*. ~ 3 +42. -e—t— At the Equator, lat. 0° 0' 0" “ Washington, lat. 38°53'23" “ New York, lat. 40°42' 40” “ London, lat. 51° 31' 66 lat. 45° - - * Stockholm, lat. 59° 21' 30" . . W = 39°127 – 0.09982 coS.2 Iength of a Pendulum vibrating seconds at the level of the sea, in various places. lat. for seconds. 39.0152 inches. 39.0958 “ « » 39-1017 cc tº 39-1393 * ge 39-1270 * 39-1845 ° 21 $22 Cor. LISION OF Bop IES IN MoTION: collision OF BODIES IN MOTION. When bodies in motion come in collision with each other, the sum of their concentrated momentum will be the same after the collision as before, but their velocities and sometimes their directions will differ. on the accompanying page the bodies are supposed to move in the jºight line, and the formula illustrates the consequences after COlliSIOIl, Letters denote. M and m = weight of the bodies in pounds. V and v = their respective velocities in feet per second. V” and v' = respective velocties of the bodies after impact. K and k = coefficient of elasticity, which for perfectly hard bodies k=0 and for perfect elastic bodies k=1, therefore the elastic coefficient will always be between 0 and 1. . When the bodies are perfectly hard their velocities after impact will be common. MV ºnly M. K = —— For m. k = –—. For y M ( VLW), or m, 77, (v–W') Ea'ample 1. Fig 191. The non-elastic body weighs M=25 pounds, and moves at a velocity V-12 feet per second; me=16 pounds, and v=9. Re- quired the bodies’ common velocities, v'-3 after impact. y_M VH-mv_25X12+16×9 IM+m. 25–H16 Ewample 2. Fig. 195. The perfect elastic body M=84 pounds, V-18, jº, and v=27. Required the velocity V-1 after impact with the Oqy m. = 10-83 feet per second. 18 (84–48) —2X48×27 V= = — 23°64. 84–H48 the negative sign denotes that the body will return after the collision with a velocity of 23.63 feet per second. Eacample 3. Fig. 196. The Fº elastic body M-38 pounds and W-79 feet per second, will strike the body in rest m=24 pounds; what will be the velocity v'=4 of the body m, its elasticity being k'-0-6. y_79×38 (1+0.6) 38-H24 When a moving body strikes a stationary elastic plane, its course of departure from the plane will be equal to its course of incident. = 70.6 feet per second. S-g Ab ºr tºº.º.º.º.º. ^ sº AB so that it will depart to the given point b ; *~ 2^ #2 required its course of incident from a A ~gº £-B Draw ba, at right angles through AB, make * * £ 'S l cd=bc join a and d ; then ad is the course of ins `-d Gident, and eb, the course of departure, and the body will strike in e. IMPACT OF BODIEs. DYNAMIcs. 3Cy 191. 4 The bodies move in the same direction. vſ. (M-H-m) = MV--mv, v'— MV+mu º: M+m Ö 192. g The bodies move in opposite directions. g: &a v' (M-H-m) = MV-mv, ×3 Cºo H v'— MV—ºnv g * M+m 73'- M+m tº : 195. ~ The bodies move in opposite directions. tº- Y(M-Km)—vm(1+K) 3. º V7= 3. ſº M+m § 1– MV(1+k)—v (m—kM) 2)" -: M-H-m. 196. º * V Only one body in motion. É. V(M – Km) 9 V7= —º-PI-, M+m VM(1+k) 2)/ = - M-H-m 4. 324 CENTRE or GRAVITY. C E N T R E 0 F P E R C U S S I 0 N. Centre of Percussion is a point in which the momentums of a moving body are concentrated. Centre of Percussion is the same as centre of oscillation, and to be calculated by the same formulas. Take an iron bar in one hand, and strike heavily over a sharp edge, if the centre of percussion of the bar strikes over the edge, the whole momentum will there be discharged, but if it strikes at a distance from the centre of percussion a part of the momentum will be discharged in the hand, and a shock felt. It is sometimes of great importance to properly place the centre of percussion. If it is dislocated, the moving body not only fails to properly transmit its effèct, but the lost momentum acts to wear out the machinery. -> Q- C E N T R E OF G R A W IT Y. Centre of Gravity is a point around which the momentums of all matters (under the action of the force of gravity) in a body, or system of bodies, are equally divided. A body or system of bodies suspended at its centre of C2/ gravity, will be in equilibrium in all positions. A body or system of bodies, suspended in a point out of its centre of gravity, will hang with its centre of gravity ver- tical under the point of suspension. \ A body or system of bodies suspended in a point out of its centre of gravity, and having two different positions, the two vertical lines through the point of suspension will meet in the centre of gravity; thus if a plane be hung 251 () up in two different positions, the vertical lines a, b, and c, d, will meet in the centre of gravity o. 2 = distance to the centre of gravity as noted in the figures. a." Example1. The radius of a circle being 3 feet, how far is ~~~ its centre of gravity from the centre of the half circle 7 O z = 0.6367X3 = I-91 feet. Example 2. How far from the bottom of a cylindric shell, open at one end, is its centre of gravity ? The cylinder is 4 feet long, radius r = 0.8 feet. J. 4 = — = — = 0.625 f * - FT5, 68EX4 eet. Example 3. Fig. 264. An irregular figure weighing P = 138 pounds, is sus- pended between a fulcrum and a weight, l = 5.6 feet, W = 57 pounds. Re- quired the distance to the centre of gravity z =? 57X5'6 o. § = 188 = 2.31 feet. CENTRE OF GRAVITY, 33.5 Quadrangle.—a and b parallel. h h /b — a 2–3 —; (+). 253. Triangle. 2-4 ~ 3 - 254. e- Half a circle plane or Elliptic plane. z = 0-424r. b - 255. 256. Circle Segment. a = area. 3. z – 4– T 12a" Irº s Q. - a = h-i-z — r. - - 257. Parabola. ſº - 2h * 2 = --e ‘ ^ 5 For haira Prabola -º % % ** 3, t'a I’8, ola w = go. 326 GENTRE of GRAVITY. 258. Half Sphere. Convex surface . . . z = }r. Solid tº º º tº sº tº © 2 == #r. 259. Spherical Sector. - h Solid, 2-3 (r–3). |260. Spherical Segment. Convex surface - #. º h 2.94. }% Solid z = 5. [...] 261. Cone. Convex surface z = #. Solid 2 = #. 262. Comic Frusthum. C _ h_h IR-r | on sur. 2–; 6 : REr’ º -- h TR-4-r(2R+3r) | Solid z = i. Rºr(RIT) ' 263. Twº Pyramidic Frustum. A and a = area of the two bases. solid 2-4 *#) 4 H. A+a+VA a * * * *-*-- ~~~~ ... CENTRE OF &#Avrº Y. 327 264. Irregular Figure. P : W = J : 23, 2 = W. J . | - F. 265. To find the Centre of Gravity of two bodies, P and Q. – Q4. _ _P a * - Fºy, "--Éiº 266. To find the Centre of Gravity of a sys- tem of bodies. * - #4, 2-84 i P+R’ TPIRIQ' 267. Half a circumference of a Circle or Ellipse. z = 0.6367 r. 268. Circle arc or Elliptic arc. _c r_c(cº-táh’) * = y = -s. F-. 269. Cylindric Surface with a bottom in one end. h * = rº" $28 SPECIFIC GRAVITY. S P E CI FI C G R A W IT Y. Specific Gravity is the comparative density of substances. The unit for measuring the specific gravity is assumed to be the density of rain water, or distilled water. * d One cubic foot of distilled water weighs 1000 ounces, or 62.5 pounds avoir- uponS. To Find the Weight of a Body. RULE 1. Multiply the contents of the body in cubic feet by 62.5, and the product by its specific gravity, will be the weight of the body in pounds avoirdupois. RULE 2. Multiply the contents of the body in cubic inches by 0.03616, and the product by its specific gravity, will be the weight of the body in ponnds avoirdupois. RULE 3. Divide the specific gravity by 0.016 and the quotient is the weight of a cubic foot. Example 1. A bottle full of mercury is 3 inches, inside diameter, and 6inches high. How much mercury is there in the bottle in pounds? One cubic inch of mercury weighs 0.491 pounds, and by the formula for Fig. 119 we have the weight = 0:491X0-785×32X6 = 20-85 pounds. Example 2. Required the weight of a cone of cast iron, diameter at the base d = 1:33 feet, height h = 4 feet? One cubic foot of cast iron weighs 450.5 pounds, and by formula for Fig. 117 we have the weight = 450:5X0:2616X1:332X4 = 834 pounds. Example 3. The section area of the lower hole in a steam boat is 245 square feet; how much space must be taken in the length of the hole for 131 tons of anthracite coal? Anthracite coal are 42-3 cubic feet per ton. length = *gg = 22.6 feet, the space required. Weight and Bulk of Substancess Cubic . Cubi º º: joot | feet OO! eet JNames of Substances. in per Names of Substances. in per pownds. tom. pounds. ton. Cast iron, e sº 450-5 497 Sand, tº º tº 94-5 23-7 Wrought iron, tº 486.6 || 4:60 || Granite, - - - 165 13.5 Steel, e - ſº 489.8 4.57 Earth, loose, - - 78-6 28°5 Copper, - - - 555° 4:03 || Water, salt, (sea) - 64.3 || 34.8 Lead, - - - 707-7 3-16 “ fresh - - 62.5 35.9 Brass, - - - 537-7 || 4-16 || Ice, - - - - 58:08 || 38-56 Tin, - - - - 456 4.91 |Gold, tº º º 1013 || 2:21 Pine, white - - 29:56 || 75-6 Silver, - - - 551 4'07 “ yellow, - - 83-81 | 66-2 || Coal, Anthracite - 53 42-3 Mahogany, - - 66.4 33-8 “ Bituminous - 50 44-8 Marble, common, - 165 13-6 * Cumberland - 53 42-3 Mill-stone, - - 130 17.2 * Charcoal º 18-2 123 Oak, live - - - 70 32.0 Coke, Midlothian - 32.70 | 68-5 “’ white, - - 45.2 || 49.5 ** Cumberland - 31-57 || 70.9 Clay, - - - 101.3 22.1 “ Natural Virginia || 46-64 || 483 Cotton Bales, - - Conventional rate of Brick, - - - 100 22.4 Stone coal, 28 bushels| e 105 21-3 l (5 pecks) = 1 ton, - 43-56 Plaster Paris, - SPECIFIC GRAVITY. 329 To Find the Specific Gravitye W= weight of a body in the air. = weight of the body (heavier than water) immersed in water. S = specific gravity of the body. Then, & $ºm o - W. º tº e gº tº W – w : W = 1 : S. S= IPH, 1, Example 4. . Required the specific gravity of a piece of iron-ore weighing 6845 pounds in the air, and 4.935 pounds in water, S = ? 6°345 & * S = 6:345 - 4935 T 4-5 the specific gravity. When the body is lighter than water, annex to it a heavier body that is able to sink the lighter one. S = specific gravity of the heavier annexed body. s = specific gravity of the lighter body. W = weight of the two bodies in air. w = weight of the two bodies in water. W = weight of the heavier body in air. w = weight of the lighter body in air. º W–w-g Example 5. To a piece of wood, which weighs v = 14 pounds in the air, is annexed a piece of cast-iron V = 28 pounds; the two bodies together weigh w = 117 pounds in water. Required the specific gravity of the wood? W= V-Hv = 28--14 = 42 pounds. S = 7-2 specific gravity of cast-iron. Formula 2. S = —*— = 0-529, the specific 28 42 — 11°7 —- 7 7.2 gravity of the wood, (Poplar White Spanish.) A simple way to obtain the specific gravity of woods, is to form it to a parallel rod, and place it vertically in water, then when in equilibrium, the immersed end is to the whole rod as the Specific gravity is to 1. Eacample 6. A cylinder of wood is 6 feet, 3 inches long, when immersed verti- cally in water it will sink 8 feet, 9 inches by its own weight. Required its spe- cific gravity. _ 3-75 3.75:625 = S : 1, S = --> = 0-600. tº 25 To discover the Adulteration in Metals. or to find the proportions of two Ingredients in a Compound. wº v= *=*=”, - - - - - - 3, 1–3 Evample 7. A metal compounded of silver and gold weighs W- 6 pounds in the air, and in water w = 5-636 pounds. Require the proportions of silver and gold 7 S = 19-36 specific gravity of gold. s = 10-51 specific gravity of silver. 6 — 10.51(6–5.636) 10-51 T 1936 and 1.245 pounds of silver. weight V = = 4.755 pounds of gold. 1 330 SPECIFIC GRAVITY. Specift Weight Specift |Weight * Specific per Specific per Names of Substances. gravity. cubic Names of Substances. gravity, cubic ºnch. tnch. Metalse - Platinum, rolled - - 22:669 || 798 || Alabaster, white - - || 2:730 || 0987 6& wire, - - || 2H-042 761 6& yellow - - 2’699 •0974 46 hammered, ' 20:337 | 73 Coral, red - - - - - || 2:700 '0974 66 purified, 19:50 | 706 || Granite, Susquehanna | 2-704 •0976 & crude, grains; 15-602 || 565 6 Quincy - - 2.652 “0958 Gold, hammered - - ) 19:361 :700 66 Patapsco - - || 2:640 "0.954 “ pure cast - - - || 19°258 -697 66 Scotch - - - || 2:025 || 0948 “ 22 carats fine - 17:486 || 733 Marble, white Italian 2-708 •0978 h. 6. ojoo 0.393 ii.395 JEacample 6. Iſow much water will flow over a 8. 0.666 || 0.390 | 1.7464 weir of b = 5 feet, h = 0.5 feet, in one minute? 9. 0.750 || 0.385 2.0331 Q = k b t = 1.1295 × 5 × 60 = 338.35 cubic feet. 12. 1.000 || 0.376 || 3.1350 On the Velocity of Water in Rivers. IVotation of Letters. F = fall of the river in feet per mile. R = hydraulic radius in feet, or the area of the cross-section of the river in square feet, divided by the wet perimeter in feet. W = mean velocity of the water in inches per second. M = mean velocity in miles per hour. - 72 = 10.9 1/F R. I'= —. y 1/PIR, 118.8 R M = 0.619 VFR, F= * - ve 2 T 3.83 R' The mean velocity of the water throughout the whole section of the river is to the velocity at the surface in the middle of the river as 84: 100, or as 100: 120. Evample 1. The cross-section of a river is measured to be 560 square feet, and the wet perimeter 196 feet; the fall of the river is 5 feet per mile. Required, the hydraulic radius and the mean velocity of the water in miles per hour? 560 Hydraulic radius R = Tø6 = 2.86 feet. Mean velocity M = 0.619 1/5 X 2.86 = 2.34 miles per hour. Eacample 2. The velocity of the surface in the middle of a river is 36 inches per second; hydraulic radius R = 2 feet. Required, the mean velocity and the fall of the river per mile 7 Mean Velocity V = 36 × 0.84 = 30.24 inches per second. 2 Fall F=# = 8.8487 feet per mile. 118.8 × 2 HYDRAULICS. 341 Obstruction ira Rivers. JE = rise of water in feet caused by obstruction. * A = sectional area in square feet of river unobstructed, and a = that when ob- structed. W = velocity in feet per second of the water without obstruction. R = (; + 0.05 )((+)- 1) Resistance to a Planne Facing a Current of Water or Moving in Still Water. A = area of the plane in square feet. I? = resistance in pounds. W = velocity in feet per second. R = A. V’, in fresh water. R = 1.032 A. V*, in salt or sea water. When the plane is set at an angle of less than 90° to the direction of motion, the resistance will be, when p = angle of the plane, R = A (V sin.6)”, in fresh water. R = 1.032 A. (V sin.6)”, in salt water. Theoretical Velocity of Water, due to Head of Fall. See table for falling bodies, page 308, in which the column S represents the head of fall in feet. To find the Number of Gallons of Water G which can be raised per Hour from a Well of Depth D, By a Suitable Double-action Force-and-lift-pump. D may also denote the height to which water may be raised in water-works. 18000 D º - 36000 A donkey working a gin, G = ~~~~. D 126000 D 190000 — , L) One man working a crank, G = A horse working a gin, G = Per steam horse-power, G = or 0.8 of the natural effect. Example 1. How many gallons of water can be raised per hour from a well 150 feet deep by a horse working a gin? 126000 G = - 150 Example 2. How many gallons of water can be raised per hour to a height of D = 150 feet by a steam-engine of 120 actual horse-power? 190000 × 120 -*. 150 = 840 gallons, the answer. = 152,000 gallons, the answer. G 842 HYDRAULICS. MOTION OF WATER IN PIPES. Letters denote— Q = cubic feet of water passed through the pipe per minute. D = inside diameter of the pipe in feet. L = length of the pipe in feet, increased by 50 diameters. JH = head of fall in feet. W = velocity of the water in the pipe in feet per minute. ... , THD5. 1 5 IQWL THIDT =356 Wł, D =::\% V-8000 Wł. Q L zºv FI L Example 1. A water-pipe of D = 1.75 feet in diameter, L = 36,000 + 50 × 1.75 = 36087.5 feet long, head pressure H = 390 feet. Required, how much water it can discharge per minute? 300XI.75, = 235 Alº = 992.26. Q 356 36087.5 992.26 Example 2. At a distance of 27960 feet from a water-work is required Q = 564 cubic feet of water per minute, head pressure being H = 256 feet. Required, the diameter of the pipe? L = 27960 + 50 = 28010 feet. p_ _1 & 564"> 28010 22.329 NT 256 Jºcample 3. A water-pipe of D = 0.75 feet in diameter, L = 8650 + 50 = 8700 feet, has a head pressure of H = 128 feet. Required the velocity v = ? of the discharge. TV = 3000 *:::::: = 315.13 feet per minute. = 1.4436 feet. Consumption of Water in Cubic Feet per head of Population, Including all Uses, as for Manufactories, Fires, etc., in 24 hours. January, 2.58 April, 2.73 July, 4.58 October, 4.46 February, 2.40 May, 3.37 August, 4.75 November, 4.12 March, 2.64 June, 3.50 Sept., 4.61 December, 3.61 On the Flow of Water in Bends of Pipes. Notation of Letters. L = the whole length of pipe in feet, straight and curved or bent, increased With 50 D. It = radius in feet of the bend of the centre-line of the pipe. q = angle of deflection or bend of pipe in degrees. Should the pipe have several bends, add all the angles to b. Sin. © to be used only up to 90°, and disappears in the formulae for greater angles. D = inside diameter in feet of the pipe; W = velocity of the water in feet per uminute; and H = head of fall in feet; Q = cubic fect of water dis- clarged per minute. Q = 2356 W#(º) (l + #) p + 90 JR + 10 v=w Wºº)(; ; ;) The formulae will answer for a pipe of the form of a screw-spiral. THE HYDRAULIC RAM. 343 THE HYDRAULIC RAM. This hydraulic motor appears to be too little known in many parts of the world. The author of this book has been in the interior of many countries where Water is raised in a very rude and laborious way, and where the hydraulic ram would be of great utility. The useful effect of the ram, like that of water-wheels and tur- bines, depends much upon its construction. In ordinary cases it returns about 50 per cent. of the natural effect. That is, the quantity of water (q) multiplied by the height (h) of the delivery above the ram will be about 50 per cent. of the quantity of water (Q) working the rami, multiplied by the head of fall (F), in the same unit of time. qh = 0.59 F. q=º. Q and q can be expressed in any unit of volume or weight. F and h can be expressed in any unit of length. But let us assume Q and q to be cubic feet per minute, Jº and J. = fall and height in feet, L = length in feet and D = diameter in inches of the supply-pipe S, l = length and d = diameter of the delivery-pipe d. _2 gh Q==#. 5 EO2 (LT5]) 5 ºf 2-77-TET Then D = Vºdºrº , and d = Nº. Description of the Hydraulic Ram. Reference to the figure above. The water working the ram is supplied through the pipe S, and escapes through an opening at o, until it has gained a velocity sufficient to raise the valve of ball JB, which suddenly stops the current, and causes an excessive pressure in the ram R, which opens the valve or ball C; the water is forced into the vessel and air- chamber A, and finally through the delivery-pipe d to its destination. When equilibrium of pressure is restored between S and R, the ball B falls, and the operation is repeated. The ram can make as much as 200 strokes per minute, depending upon its size. The length of the supply-pipe S should not be less than five times the height of the fall F, because it is the dynamic momentum (see page 310) in the pipe columns of water which works the ram. But the pipe may be made 10 times, or more, the height of the fall. |- 544 HYDRODYNAMICS. H Y D R 0 D Y N A M I C S. Water Power. The natural effect concentrated in a fall of water is equal to the weight of the quantity of water passed through per second, multiplied by the vertical Space it falls. Fig. 297. Let Q be the quantity of water which passes through the orifice a in the time t = 1/second, in cubic feet of 62.5 pounds each. h = the vertical space the water falls; then the value or natural effect of the fall is at the orifice a. P– 62.59 h, effects of power. But, Q = 5.06 a hy/h; Then We have P=315.5 a hy/h. This will be in horse power. H-- 0.573 a hy/h, H= 0.11349 h, 3 ITT2 h - –– |H. h - –4– e 1.07 WTº º 0.1134 Q Ea:ample 1. In a creek passes 18 cubic feet of water per second. IIow high must that creek be dammed up to produce an effect of ten horses? 1() h = 0.II34 × Is = 4.9 feet, the answer. Comparison of Columns of Water in Feet. Mercury in inches, and pressure in pownds, per square inch. Pounds Water. |Merc'ry.||Water. Merc'ry. Pounds. ||Merc’ry | Water. Pounds. Pr. sq.in. Feet. Inches. Feet. Inches. Pr. sq. in. Inches. Feet. Pr. sq. in. l 2.311 2,046 1 0.8853 || 0.4327 1.1295 || 0.4887 2 4,622 4.092 2 1.7706 0.8654 2 2.2590 0.9775 3 6.933 6.138 3 2.6560 1.2981 3 3.3885 1,4662 4 9.244 S.184 4 3.5413 1,7308 4 4,5181 1.9550 5 11.555 10,230 5 4.4266 2.1635 5 5.6476 2.4437 6 13.866 12.2276 6 5.3120 2,5962 6 6.7771 2.9325 7 16.177 14.322 7 6.1973 3.0289 7 7.9066 3.4212 8 18.4SS 16,368 8 7,0826 3.4616 8 9,0361 3.9100 9 20.800 | 18.414 9 7.9680 3.8942 9 10.165 4.3987 10 23.111 || 20,462 T0 8.8533 4,3273 1() 11.295 4.8875 11 25.422 || 22.508 11 9.7386 || 4.7600 11 12,424 5,3762 12 27.733 || 24,554 12 10.624 5.1927 12 13.554 5.8650 13 30.044 26.600 13 11.509 5,6255 I 3 14,683 6.3537 14 32.355 28,646 14 12,394 6.0582 14 15.813 6 S425 15 34.666 30,692 15 13.280 6.4909 15 16.942 7.3312 T6 36.977 || 32.738 16 14,165 6.9236 16 18.072 7.8200 17 39.288 || 3.4 TS4 17 15.050 '7.3563 17 19.201 8.30S7 18 41,599 || 36.830 18 15,936 7.7890 18 20.331 8.7975 19 43.910 || 38,876 19 16,821 8.2217 19 21.460 9.2862 20 46.221 | 40,922 20 17.706 8,6544 20 22,590 9.7750 21 48,532 || 42 968 21 18.591 9,0871 21 23.719 || 10.264 22 50.843 45.014 22 19.477 9.5198 22 24,849 || 10,752 23 53.154 47.060 23 20.362 9,9525 23 25,978 || 11.241 24 55.465 || 49.106 24 21.247 | 10,385 24 27.108 || 11.7300 25 57.776 || 51,152 25 22.133 10.818 25 28.237 12.219 26 60.087 || 53.198 26 23.018 || 11.251 26 29,367 12,707 27 62.398 || 55 244 27 23.903 || 11.683 27 30,496 || 13,196 28 64'709 || 57.290 28 24.789 12,116 28 31.626 13.685 29 67.020 | 59,336 20 25.674 | 12.549 29 32.755 14.174 30 69.331 61.386 30 26,560 | 12.981 30 33.885 14.662 WATER-WEIEELS, 345 W A T E R - W H E E L S. Water-wheels are of two essential kinds, namely, Vertical and Horizontal. The Vertical are subdivided into * Undershot-wheels, Breast-wheels, and High-breast and Low-breast TMJ/100/S, The Horizontal are with Floats, Screw-wheels, Turbine, Reaction-wheels, dºc. Waterwheels do not transmit in full the natural effect concentrated in a fall of water; under most favourable circumstances 80 per cent. has been utilized, but under poor arrangements only 20 per cent. Imay be expected. Example 1. Fig. 302. The vertical section of the immersed floats of an under- shot-wheel in a mid-stream is a = 27 Square feet, velocity of the stream V– 8-6, and v = 4 feet per second. Itequired the horse-power of the wheel H == 27X4 200 ii U = —- — ?)? = •6 – 412 = º e FT ;(Y v) (8:6 — 4) 11'4 horses Example 2. Fig. 307. On a breast-wheel is acting Q = 88 cubic feet of water per second, the head h = 8 feet, velocity of the wheel at the centre of the buckets v = 5 feet per second; the water strikes the buckets at an angle u = 8° and velocity W = 7 feet per second. Required the horse-power of the wheel, S8 5 ET = #( +jºcos.8°– 9) = €5 horses. Example 3. Required the effect of Poncelet's wheel, the head h = 4 feet, and the orifice a = 5 square feet, the velocity of the wheel at the centre of pressure of the floats is v = 678 feet per second 7 V= 6.91 VT = 13.82 feet per second. Q = 6.5×5×VI = 65 cubic feet per second. 65×6-78 H = 197 (13.82 – 6-78) = 15.8 horses. Example 4. Fig. 309. A saw-mill wheel is to be built under a fall of h = 18 feet, and to make n = 110 revolutions per minute. Required the proper diam- eter of the wheel. - _ 100 T 110 at the centre of pressure of the buckets. * Velocity V = Sy18 = 33-94 feet per second. W18 = 3.857 feet, Velocity ^) –8wºup = 22-2 feet per second. The fall discharged 30 cubic feet of water per second. Required the horse- power of the wheel. H = ? 30×22.2 “93 — 22:2) = & 200 (33-9 22:2) = 39 horses H = How many square feet of dry Pine can it saw per hour? See page 264. 30X39 = 1170square feet. The saw is meant to be applied direct on the wheel shaft. It YT)R.AULFCº. 302. Undershot wheel in a mid-stream. 3}, t) º H = ... (Y-9) y When V = 2w about, the effect will be, FI = fº a = area of float. 303. Undershot-Wheel. Qj H-ºv-º, H.- ... (V-2) ſh; - *-s º When V = 2w, about, H = (Z # 304. Poncelet's Wheel. FI = #(V- v), when h > 5 feet, H = ;V-0 when h 3 5 feet, Q = 8m a Vh, W = 6.91 Vh. — 305. Breast-Wheel with Parabolic drain. _ Q 7) | H = | +3*(V-1)} Q = 6.5a VY. HYDRAUL103. 347 306. Low-breast Wheel. H = +&HA ++(V cosu-w )] Q = kb. W = Q. See table for weirs. Q. 307. Breast Wheel. * _ Q 25 *s H = TI-4 [**-ā- V cosu •)] 308. Over-shot Wheel. Q Q) H = rºl ** 31.5 ( V cosu — ºl 35 D + 100. Proper velocity about n = D revolutions per minute. 30ſ). Saw-Mill Wheel. Q v V — H-j- (V-1) Proper diameter of the Wheel, D = lº.J. in feet, n = revolutions per min. 348 Turt BINE WIIEELS. TU R B IN E S. Letters denote. Q = cubic fee? of water passed through the turbine per second. h = height of fall in feet. D = diameter in inches of circle of effort in the turbine. a = area in sq. in. of the conduit passage into the turbine wheel. b = depth in inches of turbine buckets. c = depth in inches of leading buckets. r = breadth of turbine buckets in inches. m = number of buckets in the turbine wheel. m/= number of leading buckets. * = number of revolutions of turbine per minute. S and s = height of conduit and discharge in inches. t = thickness of steel plate buckets in 16ths of an inch. #= i. hºpower Of the turbine. = length in fee * - d = diameter in inches } of conduit pipe. d’= diameter in inches of the discharge pipe. W= Hydraulic pressure on the turbine wheel bearing on the end of the shaft. T 11 a -398, - - - 9 m = 5 VD, - - 17 }/h. (7. A -- D= tº cº- m/= 4.5/D, - - 18 0.436 r" 2 a =*** - - 10 5 -- E. ' 0-625 D n=%", gº tº ºn 3 b=== º 19 I) a = 0°436 Dr, - 11 _20 k Q — ano’” •), a * Tº = a D’ 4 a = m/rs, e - - 12 a 978 D 20 y my’ (1, a'= m r s, - - - 13 = –- 5 y =- lº - tº 0°436D a'-0-98 a, - a 14 s = 0-86 S, 21 46 k Q 2 y --— =-Bº; " - 6 ai/h d = D+r-i-Vl, 22 &="... • c e 15 º d'= D–H2 r, - c. 23 r=# to: - = 7 +2 r, Q-ºº. - - 16 w_P* 24 t=% - - - - 8 20 k ’ =-a-, - 10 H = 0-1134 Q h natural effect of the fall, gº ºn - 25 30 Q° H = 25T’ actual horse power, hyſ. 66 per cent of the natural. 27 0. The coefficient k can vary from 800 to 1200 without seriously aftecting the per centage of the ultilized power, but it is best between 900 and 1000. This is a great advantage of the turbine over water wheels, that under the same head of fall it can run at different velocities and słiii utilizing the maximum effect. Whatever coefficient k adopted it must be kept the same throughout the construction of the turbine. - TURBINE WHEELS, 349 Jonval's Turbine has so many advantages above other hydraulic mo- tors that it is considered sufficient to describe the construction of that one only, but the principal formulas will answer for any kind of turbines. On the accompanying plate is a drawing of a Jonval Turbine such as the Author of this Pocket Book has built in Russia. The buckets are not supported by concentric rings, but are fastened only on one side, which is considered more simple and convenient for replacing new buckets. For falls over 30 feet it may be better to make it with concentric rings. When a turbine is to be constructed we have on the one side given the natural effect of the fall, and on the other side the actual work to be done, which latter should not exceed 66 per cent. of the former. Between these two points the turbine is to be so proportioned as to utilize the greatest possible effect with smallest expense of Machinery. Jonval’s turbine in good condition generally utilizes 60 to 80 per cent. Suppose a fall of h =25 feet, discharging Q=12 cubic feet of water per second, the natural effect will be, H= 0:1134)(12X 25 = 34 horses, of which 34X-66=224 horses to be counted upon as the actual effect of the turbine. Turbine shaft to make n=200 revolutions per minute with the assumed coefficient k=960. From these dates we will obtain all the principal dimensions of the turbine, namely, 960 y 25 g 48 º - -– = tº tº r = —- = 4'6 in. - - - 5 D 200 24 inches, 1 0.436x24 0-625X24 & 20 960 b= ———— = 3 IIſle • * * * 19 24×200 - •=*** = 4 inches. - - 20 m = 5/24=24.5 say 25. - - 17 }/ 22 m’ = 4.5/24=22 buckets. - 18 t–º-25, 16ths. - - - - 8 In calculating the breadth r from formula 5, it must come inside of formula T, if not the diameter D must be altered. Now proceed with the construction as shown at the bottom of the plate which represents a section of the buckets through the circle of effort o the turbine. The drawing of the turbine is 3 of an inch to the foot, and the construc- tion of the buckets 3 inches to the foot. Draw the base line AB, set off the angle of the leading buckets=10°. The distance between the leading buckets will in this case be 24X3:14:22= 3'43 inches, set off this from S towards A, draw the straight part of the second bucket parallel to the ſirst one, draw from S the line d d at right angle to the buckets, and e will be the centre for the curved part. From the centre of S draw the line o to the end of the second buckets, divide this line into eight equal parts take five of them as radus and draw from the end of the second bucket a circlearc of about 50°, which will be the propelling part of the turbine wheel bucket. Distance between the wheel buckets will be 24X3:14:25=3-02 inches, set off this from A towards S, draw the second propelling arc. Set off from A the depth of the wheel buckets b-3 inches, set off 2 b to s, which will be the length of the first wheel bucket. Set off from s to w the distance betwoen the buckets 3-02 inches. Make S=0-86 S. Draw from w a curved line in the form of a parabola that will leave the space s and tangent the propelling circlearc somewhere about al., Care must be taken that the discharging area a' of all the wheel buckets will be about 2 per cent. less than the conduit area a of all the leading buckets. The surface of the buckets should be made as smooth as possible, or even polished. For very high falls the Hydraulic pressure WBecomes very considerable 350 * TURBINES. and may necessitate another arrangement, namely, to lay the shaft horizontally and place on it two turbines, so that the leading buckets are either between or outside of the wheels; but then comes another disadvantage, namely, that the number of revolutions will be greatly increased and may be required to gear it down 10 to 20 times to the proper speed of the main shaft. To avoid this as much as possible, take k = 800, and make r = #. g One great advantage with Jonval’s turbine is that it can be placed almost any- where between the high and low levels to suit the location, though it should not be more than 20 feet above the lower level; then, in order to utilize the whole fall, Care must be taken to make the discharge-pipe perfectly air-tight. It is not neces- Sary to make the discharge straight down from the turbine: it can be carried hori- Zontally or inclined, as may suit the location. The author has built turbines Similar to that represented on the accompanying plate, at General Maltzof's es- tablishment, Kaluga, Russia. Approximate or Proportionate Price of Turbines, as fitted and delvered at the foundry, without shaftings or gearings, is— 4001/H, $=++, 9/E in which H = horse power of the turbine and F the height of fall in feet. Example, Required the price of a turbine, H = 100 horses, to work under a fall of F- 25 fect. * s_400VION_400×10 H---gº-- 1375 dollars. ſ/25 & -IPrice List of Turkpines in Dollars. Horse IIead of ſall in feet, F. power. 5 IO I 5 26 30 40 50 75 I00 I50 H. $ $ $ $ $ $ $ $ $ $ 1. 234 || 186 163 148 130 II.7 II.0 95 86 76 2 330 263 231 209 183 167 154 || 134 122 107 4 467 372 || 326 295 258 235 218 I90 I72 151 6 55.2 455 400 262 || 316 2S8 266 232 211 185 S 660 526 || 462 | . 418 365 332 30S 269 244 213 12 S08 642 || 565 510 || 4:47 405 377 329 300 261 16 935 742 654 590 616 468 434 380 345 302 20 1045 830 730 660 77 524 || 4S5 425 3S5 338 30 1280 1020 894 S10 705 642 695 520 472 414 40 1480 || 1180 | 1035 932 815 740 6S6 600 545 476 50 1650 | 1320 | I].55 || 1045 913 828 768 71 610 532 60 1810 || 1440 | 1264 || 1140 | 1000 90S 840 73 66S 584 80 2090 | 1645 1460 | 1320 || 1155 | 1050 97 848 770 674 100 2340 1860 | 1630 || 1480 | 1295 || 1175 || 1090 948 860 753 125 2620 | 2080 | 1820 | 1650 || 14-40 || I310 || 1220 1060 984 845 150 2860 2280 | 2000 | 1810 || 1580 || 1440 | 1330 1170 1060 924 175 3100 || 2460 || 2150 1950 || 1700 1550 | 1.440 | 1260 1140 || 1000 200 3310 || 2630 || 2300 | 2090 | 1825 1655 | 1540 || 1350 | 1220 | 1066 225 3510 || 2790 || 2440 2215 1935 | 1755 1630 || 1430 | 1300 | 1130 250 3700 2940 2570 || 2335 2035 1850 1720 | 1500 1365 1195 275 3890 || 3090 || 2700 2450 || 2135 | 1945 1800 | 1580 1430 1255 30: 4060 || 3225 | 2820 || 2560 2235 | 2030 | 1890 | 165(y | 1500 1300 350 4380 || 3480 || 3035 2760 2410 2190 | 2035 | 17 S0 | 1610 || 1410 400 4680 || 3720 || 3250 || 2950 25S0 || 2340 || 2175 | 1900 1725 | 1510 450 5080 3950 3450 3130 2740 || 2480 || 2310 | 2015 | 1825 | 1600 500 5240 || 4160 || 3610 || 3300 2890 2620 2535 | 2125 1925 | 1685 A/e/Z Jonval's rugbing. as constructed by John W. Nystrom. THE ATMOSPHERE. 351 T H E A T M 0 S P H E R E. The mean height of the atmosphere is about 302 feet greater at the eqnator than at the poles, which is caused by the difference of the earth's attraction at the two places, and also by centrifugal force. The mean height of the atmosphere in 45° latitude is 60158.6 feet; at the poles, 60007.6 feet, and at the equator, 60309.6 feet. The temperature of the atmosphere is greatest at the surface of the earth, and decreases with the height above the surface. The compression of the air by the upper layers of the atmosphere generates heat in the lower layers, as explained in the article on Air and Heat. The rays of light from the Sun, passing through a densor air near the surface of the earth, also generate more heat by friction, as it were. The temperature of congelation of water being 32°, which is marked by the perpetual snow-line on high mountains, as shown in the accompanying table. Heights of Snow-Lime in Different Latitudes. Latitudes of snow-line on high mountains. 15o | 25o 35o 400 | 450 55o | 650 | 75o 850 14,760 | 12,560 | 10,290 9,000 7,670 5,030 2,230 | 1,016 || 120 Heights of snow-line in feet above the sea. 5o 15,210 New-fallen snow occupies eight times its volume in water. Heat is constantly absorbed from the atmosphere by evaporation of water on the surface of the seas, which heat is carried up and warms the atmosphere above; heat is also absorbed by support of the growth of vegetation on land. It is this opera- tion of consuming and generating heat which causes the winds and difference of weather. As the atmosphere is a material substance, it is subject to the action of the force of gravity, which causes a pressure of 14.75 pounds to the square inch at the level of the sea; or a column of air one square inch base and of the height of the atmo- sphere weighs 14.75 pounds, which balances an equal weight of a column of mer- cury 30 inches high at the temperature of 60° Fahr., or a column of water of 34 feet high. § Columns of Air, Mercury and Water. A is a Wessel full of mercury, in which is placed verti- cally a glass tube about 3 feet high above the surface l; in the glass tube is fitted an air-tight piston a, just one square inch area, which can be moved by the piston-rod c; now the piston stand is at a on the level l, and in contact with the mercury in the tube; raise the piston by the pis- ton-rod and handle c, the mercury in the tube will follow until the height of 30 inches, the piston still continues to move higher in the tube, but the mercury will maintain its position at 30 inches from l. Now it may be supposed that it is some force of the piston that draws the mercury up in the tube; if so, why did it separate at 30 inches? If the column becomes too heavy, it could separate at l, and the 30 inches of mercury follow the piston; as this is not the case, but the weight of the atmosphere pressing on the surface l and forcing the mercury up in the tube until they (the mercury and the atmosphere) come in equilibrium, which occurs at the 30 inches; and the piston only served to remove the atmospheric pressure in the tube; hence we - =l have the weight of a column of atmospheric air with one : E-T E tº c. --> square inch base equal to the weight of a column of mer- cury 30 inches high and one sq. in. base. One cubic inch of mercury at 60° Fahr. weighs 0.941 pounds; this, multi- plied by the height, 30 inches, gives 14.73 pounds, the weight of the column of mercury or atmosphere; this is generally termed “the atmospheric pressure por; square inch.” The specific gravity of mercury at 60°Fahr. is 13.58, and 13.58 × 30 13.58 X30 33.95 feet, the height of a column of water required to balance the atmosphere. º: -- := == - #. --- :- E - --- -> --- # § . # 12 352 WIND, AERODYNAMIC. g WIND, A E R O D Y NAM I C. The motions and effects of gases by the force of gravity are precisely the same as that of liquids. (See Hydraulics.) , The altitude or head of the atmosphere at uniform density will be the altitude of a column of water 33.95 feet, divided by the specific gravity of the air, 0.0012046, or, —º-–28188 feet. 0.0012046 The velocity due at the foot of this head will be — V= 8.02 V28.183 = 13464 feet per second, the velocity at which the air will pass into a vacuum. Velocity of Wind. When air passes into an air of less density, the velocity of its passage is meas- ured by the difference of their density. H and h = density of the air in inches of mercury; t = temperature at the time of passage; and V = velocity of the wind in feet per second. H — h s V = 1346.4 W#" 1 + 0.0020s) . . . 6. l, The force of wind increases as the square of its velocity. a = area exposed at right angles to the wind in square feet; F = force of the wind in pounds; H = horse-power, and v = velocity of the plane a in direction of the wind, + when it moves opposite, and — when it moves with the wind. F= 0.002288a V*, when v = 0, . av(V –– v)” F= 0.002288a(V+ v)”, 8. _*(X=x)", 241400 Ecample. A rail-train running ENE 25 miles per hour exposes a surface of 1000 square feet to a pleasant brisk gale NE by E. Required the resistance to the train in the direction it moves, and the horse-power lost. E N E – N E by N = 3 points = 33°45'; W = 14 feet per second, a brisk gale; w = 25 × 1.467 = 36.6 feet per second, and F= 0.002288 sin. 233°46' X 1000 (14 + cos. 33°45' X 36.6)? = 305.1 pounds. 305.1 × 36.6 & IBI = 20 horses. 550 Table of Velocity and Force of Wind, in Pounds per Square Inach. * Force - I' * º: º: Common Application of * *::: : Common Application of • hour. second #. m force of Wind. hour. second #. the force of wind. ardly percept- 18 26.4 | 1.55 1 | 1.47 || 0.005 { ible. 20 || 29 34|| 1.968 }vº, brisk. 2 2.93 0.020 g 25 || 36.67| 3.075 à | . . . . |} Just perceptible. || 3 || ºil º - 4. 5.87 || 0.079 35 || 51.34 6027 High wind. 5 7.33 0.123 U Gentle pleasant 40 58.68; 7.873 6 8.8 0.177 wind. 45 66.01 || 9.963 * 10.25 0.24! 50 73.35||12.30 Very high. 8 || 11.75 || 0.315 55 80.7 |149 1. # º 60 | SS.02; 17.7 | Storm. 0 14.6 O 492 & * 66 || 95.4 |20.85 #|iº is Pºnt brisk 70 102.5 24.1 Great storm. 14 | 20.5 || 0 964 gºt 10. 75 110 |27.7 IIurri 15 22.00 | 1.107 80 117.36||31.49 UII"I'l C8,110. 16 23.45 1.25 J 100 l 140.66'50. Tornado. WISTD-MILLS. 35% WIND-M ILLS. The Sail-shaft of vertical wind-mills should have an inclination from 120 to 15° with the level when built on low flat ground; on high ground, elevated from 1000 to 1500 feet within a circle of about two miles, the sail-shaft should incline from 39 to 69 with the level Effect of Windº Millse Metters deaote. 4 = projecting area of sails exposed to the wind, in square feet. W = velocity of the wind in feet per second. H = horse-power of the mill. R = extreme * * sº ºf ºt = inlaer }radii of sails in feet. w = V #r. , radius of centre of percussion in feet. m = number of revolutions of sails per minute. v = mean angle of sails to the plane of motion. The angle of the sails should be from 20° to 30° at the inner radius r, at the extreme radius R from 7° to 12°, and the mean angle v = 15° to 17°. H=4 ! n sin.v cos.w ( V — 6 ºn sin.tv y * * * *=ºssmºmºsºm-ºm- 1,540,000 9-5 assume the mean angle v = 16°, we have the horse power. __4 * / 2 ºz \ 2 5,800,000\ 34-5 In order to utilise the maximium effect of wind, it is necessary to load the *:: * number of revolutions of the sails are proportional to the velocity Q3t iſł0 Yllº Gle Proper revolutions will be found by n=-33° ºf l sin. v. Ifo =16°, n = ***, H = 4.7., and A = ***. l 1,135,000 Prs Iºrample 1. A wind-mill is to be built of six horse power in brisk wind, W = 20 feet per second. Required the area of sails A = ? =*999×8 – 851 sq. feet. 4 203 Q. Example 2. Four sails * = 212-75 sq. feet each. 212.75 = 6 feet wide by 35-5 long, dimensions of the sails. . Inner radius r = 5 feet and AE = 5-H35°3 =40-5 feet. Required the radius of centre of percussion l = ? i t = V age = V832°5–28.85 feet. Example 3. The mean angle of sails to be p = 162. Required the proper nº revolutions of the sails per minute in brisk wind of V-20 feet por second 21, = ? Revolutions ºn —º- 8 per minute. Example 4. A wind-mill has an area of A = 750 sq. feet exposed to high wind of W = 50 feet per second, and makes & = 26 revolutions per minute, centre of percussion l = 25. Required the horse power of the mill H = ? and proper number of revolutions per minute n = ? __750X25X26 (50 25X27 \*_ 7-8 horses. 5,800,000 34-5 ſº —iº- 23 rev. per minute. –750X25X* (50-3ºx”)"-826 horses. | * §I= 5,800,000 ( 34°5 ) 354 TELESCOPES. Improvements in Telescopes. $º mººlºº gº Yº =º Biº º A simple or astronomical telescope consists of two essential parts, namely: the object-glass and the eye-glass or eye-piece, which combination gives an inverted image. A terrestrial telescope consists of three essential parts, namely: the object-glass, the eye-piece and the erector. The erector is generally composed of two plano-convex lenses placed between the object-glass and the eye-piece, for the purpose of erecting the inverted image in the field of the telescope. In engineering instruments, cross-hairs are placed in the focus of the object-glass, for the purpose of defining the line of collima- tion. When an observation is made, the telescope must be focused so that the image fall in the cross-hairs, and that the focus of the eye-piece and erector also fall into the cross-hairs, for which pur- pose the most important parts—namely, the object-glass and eye- piece—have both been made movable; which condition cannot constitute a very reliable instrument, and the result is that it fre- quently requires adjustment, by which the correctness and progress of the work is impaired. An important improvement in telescopes has lately been made, in which the object-glass and eye-piece are both permanently fixed, and only the erector is moved for the purpose of focusing the telescope. Another improvement consists in fixing the cross- hairs in the eye-piece, by which a greater leverage is obtained for accuracy in observations. These combinations make a firm and reliable instrument. The improved telescope is manufactured by W. J. YOUNG & SONS, 43 North Seventh St., Philadelphia. THE BAROMETER. 355 T H E BAR 0 M ET E R. The barometer measures the pressure of the atmosphere, as described in the former editions of this Pocket Book. The English have graduated the barometer to indicate weather as follows: JBarometer in inches. Weather. At 28.3 = Stormy. At 28.7 = Much rain. At 29.1 = Rain. At 29.5 = Change of weather. At 29.9 = Fair weather. At 30.3 = Set, fair. At 30.7 = Very dry. The following guides in predicting weather-changes are selected from the “Barometer Manual * of the London Board of Trade: I. If the mercury, standing at thirty inches, rise gradually while the thermometer falls, and dampness becomes less, N.W., N. or N.E. wind; less wind or less snow and rain may be expected. II. If a fall take place with a rising thermometer and increasing damp- mess, wind and rain may be expected from S.E., S. or S.W. A fall in winter with a low thermometer foretells Snow. III. An impending north wind, before which the barometer often rises, may be accompanied with rain, hail or snow, and so forms an apparent exception to the above rules, for the barometer always rises with a north Wind. IV. The barometer being at 29% inches, a rise foretells less wind or a change of it northward, or less wet. But if at 29 inches, a fast first rise precedes strong winds or squalls from N.W., N. or N.E., after which a gradual rise with falling thermometer, a S. or S.W. wind will follow, especially if the rise of the barometer has been sudden. V. A rapid barometric rise indicates unsettled, and a rapid fall stormy, weather with rain or snow; while a steady barometer, with dryness, indicates continued fine weather. VI. The greatest barometric depressions indicate gales from S.E., S. or S.W.; the greatest elevations foretell wind from N.W., N. or N.E., or calm weather. VII. A sudden fall of the barometer, with a westerly wind, is sometimes followed with a violent storm from the N.W., N. or N.E. VIII. If the wind veer to the south during a gale from the E. to S.E., the barom- eter will continue to fall until the wind is near a marked change, when a lull may occur. The gale may afterward be renewed, perhaps suddenly and violently; and if the wind then veer to the N.W., N. or N.E., the barometer will rise and the ther- mometer fall. IX. The maximum height of the barometer occurs during a north-east wind, and [. the minimum during one from the south-west; hence these points may be consid- ered the poles of the wind. The range between these two heights depends on the direction of the wind, which causes, on an average, a change of half an inch; on the moisture of the air, which produces, in extreme cases, a change of half an inch; and on the strength of the wind, which may influence the barometer to the extent of two inches. These causes, separately or conjointly with the temperature, pro- duce either steady or rapid barometric variations, according to their force. $56 HYGROMETRY. HY GROMETRY. On the Humidity and other Properties of Air, deduced from Glaisher's Tables of the Greenwich Observatory. Mason's hygrometer, consisting of wet and dry bulb thermometers, is considered the best for determining humidity in the air and the dew-point. Example. The temperature of the air being 75°, and the wet-bulb thermometer showing 639, or 12° cold; barometer 30 inches. Required, the humidity of the air, the dew-point, weight of vapor per cubic foot, and the weight of a cubic foot of the air in grains troy 7 - Table I., 759 and 120 cold = 55 per cent, of humidity. Table II., “ {{ = 579 temperature of dew-point. Table III., weight of dry air = 516.7 grains per cubic foot. “. . * “ Saturated air = 511.4 “ “ £& Difference = 5.3 × 0.55 = 2.915 grains. Weight of the air 511.4 + 2.9 = 514.3 grains per cubic foot. Table III., 9,31 + 0.55 - = 5.12 grains of vapor per cubic foot. The weight of air of equal temperature and humidity is inverse as the height of the barometer. TABLE I. Humidity of the Air, or Percentage of Full Saturation, At Different Temperatures, indicated by the Dry and Wet Bulbs of the Hygrometer (Glaisher). Temp. of * * the air Difference in Temperature, or Cold on the Wet-bulb Thermometer. 3. - Fahr. | 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 |10|11|12||1314|15|16|17|18|19|20121122|23 30 86||73 |55 || | 35 91 |83}76||70}64|57 |53|48 40 938679 74|68; 63.58||53|50|46 45 93|86|80|74|69|64 |59 |55 51 |48 50 93|87|81 |76||7|1||66||61|57|53|49 |46 55 9.4|S8|S3 '78 |75|69 || 65|60|56|53|50 49 |46|44|41 |39;36 60 9.4|89|84|80|75|| 71 || 67 |63|59|56|53 50 |47|45|42|40|38|35|33 65 95|89|85 |81||75||72|69|| 65||61|58||55|52|49|47 |44|42|40|37 ||35|34 |32|30}28 T() 95||9|1868.278,7471 |67 6461 |58||55 |52|49 |47 45 |42|40||37 ||35||34 || 31|29 75 95 9086 i82; 78!'7471 (68; 6461 |58||55 52}49 |48|47 44; 41.39||37 ||35 |32 |30 80 95.90 (87 83|79|75||72|68|65|62|59|56|53|50|49|48}44|42|40|38|| 36||33 ||31 S5 96|91|8783|79|75||72|68|65|62|59|56|54|51|49|46|44|42|40|38|36||34}32 90 96|91 |87|83|79 |75||72|68| 65|62|59|56|54|51 |49 |46|44|42|40|38|| 36||34 |32 Percentage of Humidity. TABLE II. Temperature of the Dew-point, At Different States of the Hygrometer. º g Difference in Temperature, or Cold on the Wet-bulb Thermometer. cut I Fahr. 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 |10|11 12||13 14|15 16 17|18 |19|2021 22:23 - - - |- - * 30 |25 |21 35 |32 |30}27|25 |22|20|17|15; 13 40 ||37 ||35 33:31|29|27|25 |22'20118 45 |43|{1|3937|3||32|30|28,2524'22 50 |48|46!44;42|40|38|36||34|32.30 28:26 55 |53|52|50|48}46|45|43|4140,38 35|34|33 |31|29|28|26|24 60 |58||56|55|53|51:50.48|46|45,43,4139||38|36||34|3331|29|28|26 § |º]}}|...}}}|...}}|##25949.4|4|4|+}}||38||38|3}}#}}}}}|28 70 |68|6765|64|62|61|59:58:56:55:53:52:50.49;4||46|44|434140||38||37 |35 Tà |73 |12|70|69|6||66|64|63|61:60.58||5|Tiā554|52|31:49|48|46|45|43.42140 80 |78||77||75||7472; 71 |69 |68; 66 65 63 62’60|59:57 |56; 5453 |51|50 4847 45 35 |848#328i/30|75||75||77|76 7517473.73|Titolčj6867;6665,6463.6% 90 89|88,87|86|85|84|S3|82|S1,80,7978,7776||75||74|73 |72|71|70;69|68|67 Temperature of Dew-point. * HYGROMETRY. 357 TABLE III. . Properties of Air, by Glaisher, Greenwich Observatory. Barometer 30 inches, at 60° Fahrenheit. Weight 4 Weight |w- Temp. i. §of ...] We per cub.ft. Temp. *::: i. of j Wt. per cub.ft. º inches of P. º Dry i. * inches of P.º Dry | Sat'd * mercury sat. air, ; *. air. * |mercury sat. air. air. air. Fahr. Inches. || Grains. |Grains. Grains. || Fahr. Inches. || Grains. |Grains. Grains. 10° 0.089 1.11 |590.0 589.4 520 0.400 4.56 || 540.5 537.9 11 0.093 1.15 |588.7 || 588.1 53 0.414 4.71 539.4| 536.7 12 0.096 1.19 |587.5 586.8 54 0.428 4.86 538.3| 535.5 13 0.100 1.24 |586.2 || 585.5 55 0.442 5.02 || 537.3/ 534.4 14 0.104 1.28 |584.9 || 584.2 56 0.458 5.18 536.2} 533.2 15 0.108 1.32 583.7 582.9 57 0.473 || 5.34 || 535.1 532.1 16 0.112 1.37 |582.4 || 581.6 58 0.489 5.51 534.1| 530.9 17 0.116 1.41 |581.1 || 580.3 59 0.506 5.69 || 533.0 529.8 18 0.120 1.47 |579.9 || 579.1 60 0.523 5.87 || 532.0| 528.6 19 0.125 1.52 |578.7 || 577.8 61 0.541 6.06 || 530.9 527.5 20 0.129 1.58 |577.4 || 576.5 62 0.559 6.25 529.9| 526.3 21 0.134 1.63 |576.2 575.3 63 0.578 5.45 528.8; 5.25.2 22 0.139 1.69 |575.0 | 574.0 64 || 0.597 6.65 527.8| 524.0 23 0.144 1.75 |573.7 || 572.7 65 0.617 6.87 || 526.9| 522.9 24 0.150 1.81 |572.5 571.5 66 0.63S 7.08 || 525.8||521.7 25 0.155 1.87 |571.3 570.2 67 0.659 7.30 524.7| 520.6 26 0.161 1.93 |570.1 ſ 569.0 68 0.681 7.53 || 523.7| 519.4 27 | 0.167 2.00 |568.9 || 567.7 || 69 0.704 || 7.76 522.7. 518.3 28 0.173 2.07 |567.7 566.5 70 0.727 8.00 || 521.7 517.2 29 0.179 2.14 |566.5 565.3 71 0.751 8.25 || 520.7| 516.0 30 0.1.86 2.21 |565.3 || 564.I 72 0.776 8.50 || 519.7. 514.9 31 0.192 2.29 |564.2 562.8 73 0.801 8.76 518.7 513.7 32 0.199 || 2.37 |563.0 561.6 74 0.827 9.04 || 517.7| 512.6 33 0.207 2.45 |561.8 566.4 75 0.S54 9.31 || 516.7| 511.4 34 0.214 2.53 |560.7 | 559.2 76 0.882 9.60 || 515.7| 510.3 35 0.222 2.62 |559.5 558.0 77 0.910 9.89 || 514.7| 509.2 36 0.230 2.71 |558.3 || 556.8 78 0.940 10, 19 || 513.8; 508.0 37 0.238 2.80 |557.2 555.6 79 0.970 || 10.50, 512.8 506.9 38 0.246 2.89 |556.0 || 554.4 80 1.001 || 10.S1 || 511.S. 505.7 39 0.255 2.99 |554.9 || 553.2 S1 1.034 || 11.14 || 510.9| 504.6 40 0.264 3.09 |553.8 || 552.0 82 1.067 || 11.47 || 509.9| 503.4 41 0.274 3.19 |552.6 550.8 83 1.101 || 11.82 508.9| 502.3 42 0.283 3.30 |551.5 || 549.6 84 || 1.136 | 12.17 50S.0|| 501.1 43 0.293 3.41 |550.4 || 548.4 S5 I.171 12.53 507.0|500.0 44 || 0.304 3.52 |549.3 547.2 S6 1.209 || 12.91 || 506.1 498.9 45 0.315 3.64 |548.1 ! 546.1 87 1.247 || 13.29 505.1 497.7 46 0.326 3.76 |547.0 | 544.9 SS 1.286 || 13.68 504.2| 496.6 47 0.337 3.88 |546.0 || 543.7 S9 1.326 14.08 || 503.2|495.4 48 0.349 4.01 |544.8 542.5 90 1.368 || 14.50 502.3| 494.3 49 0.361 4.14 |543.7 541.3 91 1.411 || 14.91 501.3| 493.2 50 0.373 4.28 |542.6 || 540.2 92 1.456 | 15.33 500.4|492.0 51 0.386 4.42 |541.5 || 539.0 93 1.502 l 15.76 |499.4| 491.9 358 CLIMATE AND SEASONS. MEAN TEMPERATURE AT DIFFERENT SEA- SONS OF THE YEAR. December, January, February, Summer. March, April, May, Fall. June, July, August, Winter. September, October, November, Spring. June, September, October, December, March, July, Mean Temperature, Fahr. IIemis- º: LOCATIONS. Year. Spring. Sum. Autm. Wint’r. phere. º Algiers, . º { } ſº . 63.0 | 63.0 74.5 || 70.5 || 54.0 N. 310 Berlin, . e de º tº 47.5 464 63.1 47.8 30.6 N. 128 Berne, • * & iº . . 46.0 || 45.8 60.4 || 47.3 || 30.4 N. 1918 Boston, . ſº wº e tº 49 48 66 53 28 N. 71 Buenos Ayres, e © . 62.5 59.4 | 73.0 64.6 || 52.5 S. e e s Cairo, . * e g gº 72.3 || 71.6 | 84.6 || 74.3 58.5 N. & is e Calcutta, . . . e . . . . 78.4 82.6 83.3 80.0 | 67.8 N. e & © Canton, . * tº tº © 69.8 || 69.8 82.0 | 72.9 || 54.8 N. T0 Christiania, g tº © . . 41.7 || 39.2 || 59.5 42.4 25.2 N. 74 Cape of Good Hope, . e 66.4 || 63.5 || 74.1 | 66.9 || 58.6 S. tº a s Constantinople, . . . 56.7 51.8 || 73.4 60.4 | 40.6 || N. 150 Copenhagen, . . . . 46.8 || 43.7 || 63.0 || 48.7 || 31.3 | N. 20 Edinburgh, . . . . . 47.5 45.7 57.9 || 48.0 38.5 | N. 288 Jerusalem, . g & º 62.2 || 60.6 | 72.6 | 66.3 || 49.6 N. 2500 Jamaica (Kingston), º . 79.0 78.3 | 81.3 | S0.0 | 76.3 N. 10 Lima, Peru, . tº e tº 66.2 63.0 | 73.2 69.6 || 59.0 | S. 511 Lisbon, . ſº * e . 61.5 59.9 || 71.1 || 62.6 52.3 N. 236 London, º ſº e * 50.7 49.1 62.8 51.3 || 39.6 N. 50 Madeira (Funchal), . e tº 65.7 63.5 70.0 67,6 61.3 N. tº tº ºn Madrid, . © º * ſº 57.6 57.6 74.1 56.7 42.1 N. 2175 Mexico, City, . te e . 60.5 53.6 63.4 || 65.2 || 60.1 N. 6990 Montreal, te e & e 43.7 || 44.2 |. 69.1 47.1 17.5 N. tº ſº tº Moscow, . Q e tº . 38.5 43.3 || 62.6 || 34.9 13.5 N. 480 Naples, . º e © tº 61.5 || 59.4 || 74.8 62.2 49.6 N. 180 New Orleans, . . . . . 72 73 84 72 5S N. 20 New York, . e º tº 53 50 72 56 33 N. 20 New Zealand, . * tº . 59.6 60.1 | 66.7 58.0 | 53.5 S. tº is e Nice, . tº ſº © º 60.1 || 55.9 || 72.5 63.0 || 48.7 N. e 9 º' Nicolaief (Russia), . e . . 48.7 49.3 71.2 50.0 || 25.9 N. tº º is Paramatta (Australia), © 64.6 | 66.6 | 73.9 || 64.8 54.5 | S. * * * Palermo, . * tº º g 63.0 59.0 7.43 66.2 52.5 N. 180 Pekin, China, e & tº 52.6 || 56.6 || 77.8 54.9 || 29 N. 97 Paris, e g gº e . 51.4 50.5 | 64.6 || 52.2 37.9 N. 210 Philadelphia, * e te b5 52 '76 57 34 N. 30 Quito, Ecuador, e tº . 60.1 || 60.3 60.1 62.5 59.7 S. 9560 Rio Janeiro, . G tº e 73.6 72.5 || 79.0 || 74.5 | 68.5 S. 10 Rome, te ğ tº & . 59.7 57.4 | 73.2 61.7 || 46.6 N. 174 San Francisco, ſº tº e 57.5 58 59 ſ:0 53 N. 150 St. Petersburg, . & e . 38.3 || 35.1 60.3 | 40.5 | 16.7 N. 10 Stockholm, . gº cº tº 42.1 || 38.3 || 61.0 || 43.7 || 25.5 N. 134 Trieste, . e © de . 55.8 || 53.8 71.5 56.7 39.4 N. 2S8 Turin, . © e tº gº 53.1 || 53.1 || 71.6 || 53.8 || 33.4 N. 915 Vienna, . ſº tº {º . 50.7 || 49.1 62.8 51.3 39.6 N. 480 Warsaw, © * e º 45.5 44.6 | 63.5 46.4 27.5 N. 397 Washington, . iº * . I 59 60 79 58 38 N. gº tº º Seasons. SOUTHERN IATITUDE. SEASONS. NORTHERN LATITUDE. August. November. May. January, February. April, RAIN AND MELTED SNOW. 359 Raira and Melted Snow. Fall in Inches at Different Places. - LoCATIONS. Year. Spring. Summ’r. | Fall. Winter Albany, North America, . . . 40.67 9.79 12.3 10.3 .30 Algiers, e º e - e. 37.04 8.34 0.60 10.3 17.8 Baltimore, North America, tº 42.00 11.2 11.1 10.52 9.31 Berlin, Prussia, e º • 23.56 5.66 7.21 5.45 5.24 Bergen, Norway, . . . . . . 87.61 15.7 18.6 29.8 23.5 Bombay, India. º tº º 110. * tº º tº ºf * * ~ & 9 º' Boston, North America, . . . 44.48 10.8 II.8 12.57 9.89 Buffalo, {{ º © 27.35 5.90 8.45 7.48 5.52 Canton, China, . . . . . 69.3 18.8 27.9 19.3 3.3 Charleston, North America, . 4S,29 8.60 IS.7 11.6 9.40 Copenhagen, . e e -> 18.35 2.84 6.86 5.13 3.52 Dover, England, . º e o 38. º e e & © & tº e ºp tº G & Dublin, Ireland, º e - 25. tº tº e * 6 º' © e e - © G Edinburgh, Scotland, . . . . 28. * @ 9 e tº e © tº tº tº dº tº England, . te º e º 33. • * * * * * º e º © º g Glasgow, . dº • * . . 28.9 5.43 7.13 8.95 '7.39 Granada (Colombia), e & 115. tº & & - e ºs * - - tº tº º Liverpool, . * º & e 34.1 6.19 9.78 10.8 7.32 Lima, Peru, . . o 13.5 5.1 0.2 1.2 .0 London, e e e e e 20,69 4.09 6.00 6.15 4.45 Madeira Islands, - • © 30.87 5.11 2.30 6.96 16.5 Manchester, England, o . I 36. 7. 9. II. - Milano, Italy, . º & º 3S. 9.04 9.18 11.7 .05 Mississippi State, . º º * 53.00 10.9 14.2 9.50 18.4 New York, ſº º º º 42.23 11.5 11.3 10.3 .63 New Orleans, & º º . 52.31 13.3 16.I. 10.8 12.6 Ohio, State, gº © e º 39.69 || 10.4 10.9 9.03 .91 Pekin, China, * º e * 26.9 2.67 20.5 3.22 .53 Peru (Interior), Carabaya, º 355. 88. 120. S7. 60. St. Petersburg, . e e * 17.65 2.S9 6.73 5.1.1 2.93 Paris, º cº & º 22.64 5.53 5.92 6.51 4.68 Philadelphia, e º e . . 4S.00 13. 12. 11. 12. Rio Janeiro, Brazil, . e & & G & - - - º e e * * * 10.76 Rome, Italy, ū e e tº 30.S7 7.27 3.4 I0.9 9.3 Stockholm, e e • © 19.67 2.17 7. SL 6.94 2.75 Tiflis, Caucasus, . • • º 19.26 6.25 '7.62 3.51 1.SS Washington, . tº º & 41.20 10.4 I0.5 10.2 11.1 San Francisco. California, . . 83. 22. 1. 15. 45. Volume of Evaporation and Rain-Fall. Inches X 2.323.200 = cubic feet per square mile. Inches X 17,335,019 = gallons per square mile. Inches X 3630 = cubic feet per acre. Length in Miles of the Principal Rivers. EUROPE. Noºn ASIA AND AFRICA. Volga, Russia, . . . 2000 || Missouri, , . . . . 2900 || Yang-tse-kiang, . . 2800 Danube, . . . . 1600 || Mississippi, . . 2800 || Lena, . . . . 2600 Don and Dnieper, . 1000 || Mackenzie's, . . . 2500 || Obe, Hoangho, . 2500 Rhine, . . . . 950 || St. Lawrence, . 2200 || Yencsei, . . . . 2300 Dwina, . . . . . 700 || Rio Grande, . . 1800 || Amor, . . . . . 2200 Petchora, Elbe, Loire, 600 || Colorado, Cal., . 1100 || Cambodia, . . . . 2000 Vistula. Tagus, . . 550 || Alabama, . . . 600 || Indus, Irrawaddy, 1700 Dniester, Guadiana, 500 || Amazon, . . . 3600 || Nile, . . . . . 3000 Rhone, Po, Seine, . 450 || Rio de la Plata, . 2250 || Niger or Joliba, 2600 Mezene, Desna, . 400 || Orinoco, . . 1500 || Senegal, . . . . 1200 Dahl, Bug, . . . . 300 || Araguay, . . . 1100 || Orange, . . . 1000 Thames, . . . . 233 || Magdalena. . . 900 || Gambia, . 700 360 EVAPORATION, Evaporation on the Surface of water in the Opera Air. When the surface of water is freely exposed under the atmosphere, the dry air in contact with it becomes charged with vapor, and consequently becomes lighter (see Table, page 357), rises, and gives place to drier air, to repeat the same opera- tion, by which moisture is constantly carried up into the air from the surface of the Water. The rate of this evaporation depends upon the temperature of the water, the dryness, the temperature and the velocity of the air. Evaporation of Water in Decimals of an Imeh, per 24 Hours, on the surface of fresh-water lakes, rivers and canals, at different temperatures of the water and currents of the air. Water. Velocity of wind in miles per hour on the water. Temp. Calm. 10 20 30 40 50 60 329 0.012 0.014 0.016 0.017 0.019 0,021 0.023 35 | || 0.020 0.023 0.026 0.029 0.032 0.035 0.038 40 0.040 0.046 0.052 0.058 0.064 0.070 0.076 45 0.068 0.078 0.088 (),098 0.109 0.1.19 0.129. 50 0.I00 0.115 0.130. 0.145 0.160 0.175 0.190. 55 || 0.133 0.153 0.173 0.193 O.213 0.233 0.253 60 || 0.177 ().203 0.230 9.256 0.2S3 || 9,310 0.336 65 # 0.225 0.259 0.292 || 0.326 0.360 0.394 0.42'ſ TO 0.278 0.320 0.361 0.404 0.444 0.486 0.527 75 || 0.335 0.385 0.435 0,485 0.535 0.585 0.635 80 0.400 0.460 0.520 0,580 0.640 0.700 0.760 85 || 0.468 0.538 * 0.608 0.679 0.749 () 819 0.8S9. 90 0.540 0.621 0.703 || 9,784 0.865 Ö.946 1.025 95 || 0.620 0.713 0.808 0,900 ().995 1.088 1.180 100 || 0.700 0.805 0.912 1.015 | 1.123 1.225 1.332 The evaporation on the surface of salt water on the ocean is about 0.8 of that in the table. t . The quantity of water evaporated on the surface of all the waters on the earth is equal to the quantity of rain-fall. Area in Square Miles of the largest Imland Lakes. IAKES. Sq. Miles. | LAKES. Sq. Miles. º Tonting, China, de e 1200 Eastern Hemisphere. Wenern, Sweden, . . 2400 *Aral Sea, Tartary, . . 16630 || Wettern, Sweden, . . 1045 Azov Sea, Russia, . . . . $99 || Zaizan, Mongolia, . . 1600 Baikal Sea, Siberia, . wº 13000 - Balkash, Mongolia, . . 5300 || Western Hemisphere. Black Sea, Turkey, . gº 113006) Athabasca, N. America, . . 3200 Caspian Sea, Russia, Q 138000 Erie Lake, N. America, 7000 Constance, Switzerland, . . 456 Great Bear, N. America, . 4000 Dead Sea, Palestine, . 370 Great Slave, N. America, 12000 Dembia, Abyssinia, . © 13000 Great Salt Lake, e * 1880 Enare, Lapland, . * 70 Huron, N. America, . 22800 Geneva, Switzerland, . • r 400 Maracaibo, S. Annerica, . . 6000 Hjelmaren, Sweden, . 900. | Michigan, N. America, 22600 Tchad, Africa, . * te 11600 Nicaragua, Cent. America, 3905 Ladoga, Russia, . tº 6200 Ontario, N. America, . . 4950 Loch Lomond, Scotland, . 27 Otehenantekane, N. Amer., 2500 Lough Jeaugh, Ireland, 80 Superior, N. America, . 30000 Onega, flussia, . º * 3300 || Titicaca, Peru, . * * 5400 Ouroomia, Persia, . . 1000 Winnipeg, N. America, | 7200 DIFFERENCE of LEve:Ls. 361. BAROMETRICAL OBSERVATIONS. For Determining Difference of Levels. Notation of letters for the complete formulae of La Place, in French and English. 7??&Q St.07"6S, - h = height of barometer = h/ Lower station, - T = temp. of barometer = T' > Upper station. t = temp. of the air = tº H = height of barometer at the upper station reduced to the temperature of the barometer at the lower station. When the height is read on a brass scale, the reduction will be in French measures. English measures. H=h/[1+0.0001614 (T-TV)].] H-h/[1+0.00008967 (T-TV)]. Mean radius of the earth = 6,366,200 metres = 20,886,860 feet. Mean height of the atmosphere = 18,336 metres = 60,158.6 feet. L = mean latitude between the two stations. Z = difference of level between the two stations. I’rench measures. 1 t –– #) - —I- -— º © tº º e Z= log.- : X 18336 × ( (1 + 0.00251 X cos. 2L) ×. . H. - 1 + 2 + #) ( 6366200 / e - © Tº e JEnglish measures. — — (1++*)× te e e 2. |2=log #x001586 (1 +90°ºo. 21)× . . 8. __ 529; (l +##). . . . 4. 20886860 The factor (1) gives the difference of level when the observations are made in a temperature of 32°Fahr., or 0° Cent., the freezing-point of water, and in latitude 45°, without the factors of correction (2), (3) and (4). The factor (2) is the correction for temperature of the air above or below the freezing-point. - The factor (3) is the correction for latitude above or below 45°. The factor (4) is the correction for the decrease of the earth's attraction. This correction is included in the following Table I., to suit any level of the stations. There are some other barometrical corrections not included in the above for- mulae, such as for humidity of the air, capularity and boiling of the glass tube, for the hour of the day and season of the year, all of which are so insignificant, uncertain and complicated that I have concluded to omit them. T- 262 DIFFERENCE OF LEVELS. Explanation of the Baronnetrical Tables. The tables have been calculated in Peru, under actual practice, by the author. Table I. is calculated from the factors (1) and (2), which gives the approximate heights above the level of the sea, in English and French measures, for every tenth of an in 3h from 11 to 31 inches. The mean temperature of the air and of the barometer is assumed to be 60° Fahrenheit = 15.555 Centigrade, and in latitude 45°, The barometer is assumed to be 30 inches = 7.62 centimetres at the level of the sea, but when it is observed to be higher or lower, make the corresponding addi- tion or subtraction for difference of levels in the tallyle. Table II. contains the correction for difference of level in feet or metres at dif- ſerent temperatures of the air above or below 60°Fahr. Table III. contains the correction for heights in different latitudes above or below 45°. Tables IV. and W. are logarithmic corrections for temperature and latitude. Table VII. gives the height of a column of air in metres, corresponding to a dif- ference of one milimetre of mercury at different heights of the baroneter. Table VIII. gives the height of a column of air in feet, corresponding to a differ- ence of one-tenth of an inch of mercury at different heights of the barometer. 'Tables X. and XI. contain the correction for the mercurial column at different temperatures of the barometer above or below 60° Fahr. = 15.555 Cent. This correction must be made before the barometrical height is applied to Table I. Table XII. contains tho approximate mean temperature of the air at the level of the sea for every month of the year in different latitudes. This table has been deduced from observations of Mr. Dove, Iſumboldt, Raimondi. and other distin- guished authors. The table agrees very well with the mean temperatures on the Atlantic and Pacific coasts, but will not auswer for the North Sea and the Baltic, where the temperatures are much higher. I have found a great deal of inconve- nience in the interior of South America, for want of a tablo of this kind. When barometrical observations are made far inland, some means must be resorted to for estimating the temperature of the air at the level of the sca in the latitude of observation, in order to make proper corrections for difference of level. Front all the meteorological observations of different authors it appears that the mean temperature of 24 successive hours is near 9 o'clock in the morning, and that the mean temperature of the day from 9 to 5 P.M. is at noon. The variation of temperature throughout the day varies with the latitude, that is, the higher the latitude, the greater is the variation. Example 1. On the 14th of March, 1869, 2h. 15m. P. M., in Oroya, Peru, lati- tude 11° 30', the barometer stood 19.46 inches, the temperature of the air 623, and that of the barometer 60°. Required, the height of Oroya above the level of the sea in feet 7 Tablo I. Barometer 19.4 in., = 12099 ($ feet. Correction 0.0 × diff. 142.6 feet, = 85.5 feet. Approximate height, = 12014.1 feet. Temperature at Oroya, 629 Fahr. Table XII. Latitude 11° 30', 14th of March, 799 Mean temp. of the column of air, 141 = 70.50. Table II. Correct mean temp. 709 10000 feet = 208.2 feet. 2000 feet = 41.6 feet. of the air, 10 feet = 0.2 feet. 10000 foet = 23.6 feet. Table III. Correct for lat. 11° 30', 2000 fect = 4.8 feet. 10 feet = 0.0 feet. Sum of corrections, . gº * e º * > º 278.4 feet. Approximate height, . tº tº º º tº . 12014.1 feet. Height Oſ Oroya, te º * º tº tº gº 12292.5 feet. ANEROID BAROMETER. 303 Example 2. In the city of Paucartambo, Peru, the barometer was observed to stand 21.272 inches, the temperature of the air 70°, and that of the barometer 69°, in latitude 13° 18' south. About three miles from the city, on the mountain Huanacaury, the barometer stood 18.224 inches, the temperature of the air 62°, and that of the barometer 64°. Required, the height of the mountain above the city of Paucartambo 7 Barometer at the lower station, . 21.272 inches. Correction for 699, Table XI., subtract .017 inches. Height of barometer at 60°, . e 21.255 inches. Barometer at the upper station, . 18.224 inches. Correction for 64°, Table XI., subtract . .006 inches. Height of barometer at 60°, . 18.218 inches. Barometer. Heights. 18.218 1384.5.0, upper station. Table I. { 21.255 9565.3, lower station. Logarithms 3.6314.133 = 4279.7 feet, approximate height. Table IV. 0.0053929 = 669 mean temperature. Table W. 0.0009888 = 13° 18' latitude. 3.6377950 = 4343.1 feet, the height required. Aneroid Barometer. The aneroids made by Negretti & Zambra, London, are compensated, and show the height of a column of mercury at the temperature of the freezing-point of water, 32° Fahr., or zero Centigrade. The aneroid is not affected by different temperatures. When the aneroid is used with the accompanying Table I., a correction must be made to convert the column of mercury from 32° to 60° Fahr., namely: Height of a column of mercury as indicated by the aneroid. 18 || 19 20 21 22 ... 25 26 27 | 28 29 30 .050 | 053 l .056 .059 .061 .064 .067 l.070 .073 l .075 l .078 l .081 OS4 Correction in fraction of an inch, always additive. I6 || 17 .046 .048 JExample. Suppose the aneroid to indicate 25.261 inches. Correction from the table, g .070 Height of a column of mercury, 25.331 inches at 600 Fahr. Heights of the Principal Mountains and Volcanoes. North AMERICA. | Feet. EUROPE. Feet. Wolcanoes, Active. Feet. Mount St. Elias, 17,860 || Elbruz, Caucasus, 17,776 || Aconcagua, Chili, 23,100 Mt. Brown, R. M., 16.000 || Mont Blanc, Alps, 15,668 || Gualatieri, Peru, 22,000 Sierra Nevada, Cal., 15,500 || Malhaven, Spain, 11,678 || Cotapaxi, Equador, 19,500 Fremont's Peak, 13,470 || Mt. Maladetta, Py., II.436 || Misti, Peruº, . . 18,136 Long's Peak, R. M., |12,500 || Mt. Caballo, Alps, 10,154 || Popocatapeti, . 17.735 Cibao Mt., Hayti, 8,600 || Mt. Scardus, Tur., |10,000 || Pichincha, Equa., |16,000 Cierra del Cobre, 7,200 || Ural MIts., Russia, 5,397 || Kliutchewaskaja, 15,763 Black Mt., S. C., § ASIA. M. : Fuego, 14,000 Mt. Washington, y in ori nor p * auna Loa, S. Isl., 13,440 tºº, º||...}}|sºli: & s * * *-J - 5 * tº . Peak of Otter, Vt. 4,200 || Hindo Kºo, Cabui,"|20000|ºndrapura, Sun, |12300 Mt. Ararat. Tºur.” |17|3|10|Teneriffe, Can. Isl, 12.182 SOUTH AMERICA. Mt. Íebanon, Syr., iž006 Erebus, Vie. Land, 12.400 Illimani, Bolivia, 8 |24,100 Cartago, Q. Amer., |11,480 Ausangatí, peruº ||22:150 AFRICA. Etna, Sicily, . . [10,874 &. E., |giº ||}}.}ºy., Hºll Heela, Iceland, || 5 liq š. ºi." |giº ||.º.º. |}}}|Soumiere, Guid., 5.108 i., §§j, isº, Talbº Walla, Aby, 12,000|Jurolo, Mexico," | 4205 Cerro de Potosi, 16,150 OCEANICA. Vesuvius, Italy, 3,948 Cerro de Pasco, 13,780 || Mt. Ophir, Sum., |13,842 Organ Mt., Brazil, 7,500 || Mt. Semero, Java, 13,000 * Measured by the author of this Pocket-book. 364 BAROMETRIC AND ATMOSPHERIC HEIGHTS. TABLE I. Barometric and Atmospheric Heights. FRENCII. IöNGLISLI. PRENCH, lèNGLIS iſ. Dif Altitude. I Bar. Bar. Altitude. Dif. Dif Altitude. Bar. Bar. Altitude. Dif * I metres. m.m. l in. feet. * " I metres. m.m. l in. feet. * 8488.09 |279.4 || 11. 27848.5 --, 53.18.11 406.4 || 16. 17448.2 ſº $41,148 |3}}}| ||37.91% § § 5263.39|408,9| |||1313: #: ##|:35, 3:#| }}}}}}}|† ||...}}}.}}|###| |}|}}}}|†† #;|' ;9| 33.9}}|º]}}}}}}}}}|#9| |}}}.}}|ičjº #. §§§ 2.É| #3;{2|3: Häij}}}}}|###| #|####|išč #. §1.9% |}}}}| |}}}}}}|...}}...] §§ {}}}}}|...}} #3;|3}}}|...}|| |}}}}|33 ||#6; ; º #; 66.7 71.98 # § : § 236.2 || 50.50 ; # $|igoj || 33.7 71.43 ... [...'. I ..., |2343||50.29) is jºſº | jij} |1947 #6; 7822.19 |30}}|..., 9%;3|3;[...] § {}}}|...9|}}}}}}|†† #|ſºlº ſº; 13,3:3:... [...;|#####|17||##|ić §§ ſºlºſº |3|13| ||}}}}}}|...[2.4] };}|##| |}}}}}|ići. §§ 131392 |3098 || 3|}}}}}}|...}|...}|#9%|##$| #####| |ići. '''', 7543.02 |312.4] .324747.9 |3:..., ||...}}| 4657.17|439.4|| 3|15279.7 ||. ;|º] 4:3;|##|º] };}; §|14}}.}}|...}} {{#993 ||3:... ii.;|#9%|###| |}}} |. ‘... [7339.18 |320.0; .6|24079.1 |5. ||...}} 4511.75:447.0 .6|14802.6 ||..."; §§§ 3; #|##|º] jiāº; #3 § 20:33 ºil sº;|##|#1.29% Hºli. ... [7140.08 |327.6|_ _ 9|23425.9 |3. Hijº. 4368.80454.6 || 9 ||14333.6 ||. § 7074.76 |330.2 : 13. 23211.6 #. #; 4321.65|457.1 | 18. 14178.9 # e 9 g t 2 * fº •) * ;|††: 3;3|}}}|...}}|†: 3; #. e & * * t g * ; ; ; ; ;|#|:#| ||#### 6693.01 |345.4 .6|21959.1 ...". . . . 4044.25 |472.4 .6||13268.8 |j.” 61.97 º'. 203.3 || 45.35 ..., |148.8 Šiš, ; ; ; ; ; ;|ºil;|##| |}}}}}|iº $1.99 |gº, jol jºij |}}}||#99|3.jãs) of gliºſº ||{ſ} #: gift so [355.6/14 |2tíšíð #! #: §§§iſsºl 19, i2678 i #: jśī; ; ;|ij|{ij|####| |####|ſº § §§§ 3; 3|º #: ; 3:31.3313903 || 3|12333i #: * t º e * S) e g §§§| ||...}|iº ||ºil;| |##|iº §§2 ºf Šºš Šišjº |}}}}|{}}|ºjjī; list;... [13].3 57.73 ſº |}}}}| |iji |}}}}||#3% j|º] }|iº 1 ||41.2 37.49 jº, º #|iggii.3 |}}}}||##$3| 3:53:5. $|ii.33.6 || 95 ; §%g: 373 Al jiບi #. #; §§§ 4| |ii; }: tº e *ls * i tº s ºf §§§[*]; is...};|;|*||##|i; §§§ ºo, ji žišij |}}}||4}}|†ojö| |iojº. ||37.3 #4, #|...] §§§39||##|:#| ||...}|{#2 55.11 |2:...”. [...”. I * |180.3 || 41.60 . . . . ." st "..., |136.5 #; ; ; ;|riº;| #: ##|;|jč| |išišio ||38||4}}}|31.10; }| |ij ||3:0 : §§§7| |iº, #: # # §§| }|†. # § {}}}}}|##| |}}}}|iší|º]}}}}|##| |}}}}|iš6 § 5371.15|403.8| 9||17622.2 #6||...}}|3057.87|530.8 .9|10032.6 ||. The columns Bar. is the beight of the Barometer in inches and milinnetres. The columns Altitude is the corresponding height of level above the sea in feet and metres. The altitude in metres can be read from the barometer in inches; or, the altitude in feet can be read from the barometer in milimetres. BAROMETRIC AND ATMOSPHERIC HEIGHTS. 305 TABLE I. Barometric and Atmospheric Heights. FRENCII. ENGLISH. FRENCH. FNGLISH. & Altitude. I Bar. Bar. Altitude. * 4, Altitude. Bar. Bar. 1 Altitude. g Dif. metres. m.m. l in. feet. Dif. Dif. metres. m. In in. feet. Dif. 3017.50 |533.4 |21. 990ſ).1 º 1210.61 j660.4 26. 3971.9 § 2977.33 |535.9 .I 9768.3 #| §§§ 13sº ; 333i 2937.34 538.4 .2| 9637.1 130.6 3% j.1145.81 665.4 .2|3759.3 105.7 39.65 2897.53 |541.0 .3| 9506.5 i. 3203 1113.59 |668.() .3 || 3653. 105; 35.4ſ 2857.88 |543.5 .4| 937 6.4 i29.4 31.93 1081.50 670.5 .4 3548.3 iº 36.23 2818.44 |ā46.1 .5 92.47.0 1337 3i: 1049,52 673.1 .5| 3443.4 1046 35.11 2779.21 |548.6 .6 9118.3 13:3 31.76 1017.64 675.6 .6; 3338.8 104.2 33.96 2740.10 551.1 .7 | 8990.0 i2.6 31.84 985.888 678.1 .7 3234 6 ió33 3874 27.01.21 |553.7 .8 || 8862.4 1271 31.52 954.251 680.7 .8| 3130.8 ió3.4 §§§3 2662.47 |556.2 .9| 8735.3 išā 3139 922.734 683.2 .9 3027.4 ičº 3537 2623.94 |558.8 22. 8608.9 1253 31 ºn 891.341 6S58 27. 2924.4 102.6 § 23; #3| || 3:339|##||##|:39.979 |G:33| ||28313 ||3: 3300 2547.37 563.8 .2| 8357.7 1247 31.03 828.919 690.8 .2| 2719.6 ióis 37.88 2509.37 566.4 .3| 8233.0 124.3 36.94 197891 693.4 .3 2617.8 ió1.5 37.67 24.71.49 568.9 .4 §§ 1236 30.8; 766 953 #695.9 .4 25163 101.1 37.5% 2433 82 |571.5 .5 T985.1 123.ii.3072 736.140 698.5 .5| 24.15.2 ió0s 3737 2396.30 |574.0 .6 || 7S62.() 1236 30.60 705 416 701.0 .6; 2314.4 ió04 37.16 2358.93 576.5 .7| 7739.4 ižić 30.48 674.815 ITU3.5 .7 2214.0 ióó0 37.06 2321.77 579.1 .8 7617.5 121.6 3641 644, 335 |7(36.1 .8| 2114.0 jji 3; 32%. 33.31.9 Tºº Häjäjś 108.61...9|}}}}} | . 3.35 2247.89 |584.2 || 23. 7375.l 120. || 36.18 583.682 '711.2 28. 1915.0 39% 36.5% 2211.20 586.7 .1 | 7254.7 i2.0 30.05 553,503 (13.7 || 1 | 1816.0 §§§ §§ 2174,61 |589.2. .2| 7134.7 iſº, 2333 523.454 716.2 .2| 1717.4 383 3. 2138.22 591.8 .3| 7015.3 iišš 3º4 493 523 718 S .3 | 1619.2 gig 36.0% 2102.01 |594.3 .4 | 6896.5 iš4 25.75 463.683 '721.3 .4| 1521.3 97.6 § 20:33: 23° 3. ... ii., §§ 33% I239; £ 4:37 | }. § 2.3% ºft| || 3:33 liń||####399 Tºº $4%; j 㺠1994.20 |601.9 .7| 6542.8 iić's 35.45 374.785 '728.9 .7 1229.6 966 3. 1958.60 |604,6 .8 || 6426.0 iič4 2. 32 45.332 '731.5 .8 1133.0 96.3 35.30 1923.14 |607.0 .9| 6309.6 ilă's 2: 33 316.010 '734.0 .9 || 1036.8 §§§ § 1887.34 || 09.3 ||34, ºli;|3.386.18, 736.6|29, 340.9 j §§ 1852.63 |612.1 ..I 6078.3 ii. 25.0% 257.672 #739.1 .1) 845.4 §5? 34.50 T817.61 614.6 .2| 5963.4 11 4.5 2šš3 228.657 '741.6 .2| 750.2 34.9 § 1782.71 ||617.2 .3 584S.9 1140 2šš4 199.731 |744.2 .3% 655.3 34.5 31.56 1747.96 (319 7 .4 5734.9 ii.3.5 3. 17 O.S98 '746.7 .4| 560.7 94.2 § 1713.37 622.3 .5| 5621.4 113t 28.6% 142.186 749.3 .5 466.5 §§§ 3 1.32 1678.90 #624.8 6| 55083 ii.26 33.3 113.566 jºiâl.8 .6 || 372 6 §§§ 34.17 1644.58 627.3 ..'I 5395.T 112.1 ||23.43 85.037 '754.3 .7 279 0 §§ 34.05 1610.41 |629.9 .8 5283.6 iii.; 2S 35 56.600 '756.9 .S.; 185.7 §3) §§ 1576.36 |632.4 .9 || 5171.9 iii.3 2S.25 2S.254 Tâ9.4 .9 92.7 gº 33.78 1542.44 |635 0 iz5. 5060.6 iio's 2šić 0.0000 |762.0 i 30. 0.0000 §2. § 5% ºl '!' Hºli.láš: 2:31:45] 1 .335 | jº 㺠1475.05 |640.0 .2| 4S39.5 ióð Žsoi 56.295 |767.() .2 184.7 šío 33.37 1441.55 (612.6 ,3; 4729.6 io9.5 3. S4,305 '769,6 .3 276.6 gi.6 33.35 I40S.18 |645.1 .4| 4620.1 ioji Žiš 112.225 TT2.1 .4! 368.2 Šiš § 1374.93 |647.7 .5| 451.1.0 ióši 27.7 140.053 |774.7 .5 ! 459.5 gii §oi I341.80 |650.3 .6|| 4402.3 ič 27.6 I 67.820 |'''I'ſ.2 .6 550.6 go's 33.33 1308.79 |652.8 .7| 4294,0 ið. 27.58 195.495 #779.7 .7 641.4 go's §4 1275 96 655.3 .8 || 41S6.3 ió7.1 Žiž 223.07.9 |7S2.3 .S. 731.9 goš 3. 1243.22 |657.9 .9 || 4078.9 ióio 3.33 250.601 |784.8 .9 S22.2 § * 32.61 7.0||27,43|2|3.33 ||37 || 31|| || 12.2 90.0 The difference in the Bar. m.m. column is 2.5 milimetres; therefore, multiply the difference of altitude in metres by the exceeding milimetres and by 0.4; subtract the product from the tabular altitude, and the remainder will be the altitude of level in metres, corresponding to the reading of the barometer in milimetres. CORRECTION ITOR TEMPERATURE. TABLE II.-Correction for Mean Temperature. Temp. IIeight in feet or metres. Temp. Fahr, I Cent|| 1000] 2000 3000 | 4000 || 5000 || 6000 || 7000| S000; 9000 Cent. I’ahr. 61|| 6.1|| 2.08| 4.17 6.25 S.34|| 10.42 12.50|14.59|16.68||18.76. 20.85|| 15.0;59 62|16.6|| 4.17| 8,3112,49|16.61 | 20.82 24.98|29.15|33.32|37.48 14,458 63; 17.2|| 6.25 12.49||18.74|24.98 || 31.23 || 37.48|43.72}49.97 |56.21 13.S. 57 64|17.7|| 8.33 16.67 24.99|33.34 41.65 |49.98||58.3166.6474.97 13.3|56 65||18.3||10.41|20.82; 31.24|41.65 52.06 || 62.48; 72.89|83.30|93.72 12.7 |55 66 18.8||12,49|24.99 i37.48 49.98 || 62.47 74.96 |87.46|99.96||112.4 12.2|54 67|19.4||14.58|29,15|43.43|58.31|| 72.89 || 86.86|102.1|116.6|130.3 11.6 |53 68|20,0||16.66i.33.32|49.6.166.64 83.30 || 99.22#116.6.133,3|148.8 11.1|52 69| 20.5; 18.74|37.49|56.2374.98| 93.71 112.4|131.2|149.9||168.7 10.5151 70|21.1||20.82|41.6462.46|83.28| 104.1 | 124.9|145.7 |166.5|187.4| : 10.0|50 g Tl|21.6||22.91|45.81|68.72|91.63||1145 ||1314|160.3|183.2|206.1|: 9.4|49 : 5 72|22.2||24.99 |49.98 74.77|99.96 124.9 || 149.5/174.9|199.9|224.9 8.8|48 = # 73|22.7|27.07 |54.14|81.21108.3||1353 |162.4|189.5|216.6|243.6 : 8347 5 # 7423.3||29.15|58.31|87.46||116.6|| 145.7 174.9|204.1|233.2|262.4|291. 7.7 |46 #. à Tâ123.8||31.2462.47|93.71 |124.9| 156.2 187.4|218.6|249.9|281.1|31: 7.245 = § 76|24.4||33.32|66.64|99.96||133.3| 166.6 || 199.9233.2266.5|299.9 6.644 & * 77|25.0||35,40|70.80|106.2|141.6|| 177.0 212.4|247.8|283.2|318.6 6.l|43 a £ 78|25.5||37.48|74.98||112.4|149.9| 187.4 |224.9|261.4|299.8|337.3 5.5|42 5 35 79|26.1||39.57 |79.13||118.7 |158.2 197.8 ||237.4277.0|3.16.5|356.1 5.0|41 - 3 S0|26.6||41.65|S3.30|124.9|166.6| 208.2 || 249.9|291.5 333.2|374.8 4.4|40 & * 81|27.2||43.73|87.46||131.2|174.9| 218.6 262.4|306.1|349.8|393.6 3.8|39 g E 82|27.7||45.81|91.63||137.4+183.2 229.0 |274.8|320.7|366,5|412.3 3.3|38 : 5 83|28.3||47.90/95.79|143.3|191.6] 239.5 286.6}.335.3|383.2|431.1 2.7|37 3 § 84|28.8||49 98.99.96] 149.9 |199.9| 249.9 || 299.8|349.8|399.8|449.8 2,2|36 : 3 85|29.4||52.06|104.1156.21208.2| 260.3 || 312.4364,4416.5|468.6 1.6:35 3 * 86|30.0|54.14|108.2|162.4|216.5| 270.7 |324.8|379.0.433.1487.3 1.1|34 ce 3 87|30.5||56.23|112.4|168.6|224.9; 281.1 |337.2|393,6449.8||506.0 0.533 (5 ... 88|31.1||58.31|116.6|174.9|233.2 291.5 |349.8|408.2466.5|524.8 0.0|32 : º: 89|31.6||59.37 || 118.7|178.1 ||237.5| 296.8 || 356.2|415.5 474.9 |534.3 –0,5}31 : * 90|32.2||61.77|123.5/185.3|247.1|30s.8 |370.6|432.4494,1555.9 —1.1|30 - 91 |32.7 (64.56|129.1|193.7 |258.2| 322.8 387.3|451.9}516.5|581.0 —1.6|29 ; 92|33.3||66.64|133.3|199.9|266.5| 333.2 || 399.8|466.5i 533.1|599.7 —2.212S 93|33.8||68.72|137,4206.11274.9| 343.6 412.3|471.0549.7|618.5 —2.7 27 94|34.4||70.80 141.6|212.4|283.2| 354.0 424.8|495.6}566.4637.2 —3.3|26 95|35.0||72.89|145.8|218.7|291.6||364.4 |437.8||510.2583.1|656.0 —3.8|25 96||35.5||74.83|149,6|224.5 299.3| 374.1 || 449.0|523.8598.6|673.5 ––4.4|24 97|36.1||77.05 154.1|231.1308.2| 385.2 462.3|539.3}616.4|693.4 —5.0|23 98|36.6||79.13158.2|2374|3.16.5 395.6 |474.8|553.9,633.0|7|12.2 —5.5|22 99|37.2|S1.22|162.4|243.6|324.9 406.1 |487.3|560.5649.7 |731. —6.1. 21 100|37.7||83.30|166.6|249.9333.2| 416.5 499.8||583.1666.4 749.7 —6.6|20 Fahr, Cent|| 1000| 2000|3000 || 4000|5000 | 6000 || 7000 S000| 9000 Cent. l'ahr. TABLE III.—Correction for Mean Latitude. Mean IIeights in feet or metres. Mean latitude. 1000 | 2000 || 30 5000 || 600 000 S000 10000||latitude. 44 0.09 || 0.18 || 0.27 | 0.36 || 0.45 0.54 || 0.63 0.76 || 0. 0.90 || 46 . 42 0.27 | 0.54 0.81 | 1.08 || 1.35 | 1.62 | 1.89 || 2.16 || 2. 2.7 48 § g; 40 0.44 || 0.88 | 1.32 | 1.76 2.20 || 2.64 || 3.08 || 3.52 3.96 4.4 50 E. 3 38 0.62 | 1.24 | 1.86 2.48 || 3.10 || 3.72 || 4.34 4.96 || 5.58 6.2 52 : = 36 0.79 | 1.58 2.37 3.16 || 3.95 || 4.74 || 5.53 || 6.32 7.11 || 7.9 54 & s 34 0.96 | 1.92 || 2.88 || 3.84 4.80 || 5.76 || 6,72 || 7.68 8.64 || 9.6 56 do .< 30 1.28 2.56 3.84 5.12 6.40 || 7.68 || 8.96 i 10.2 | 11.5 | 12.8 || 60 & § 28 1.43 2.86 4.29 5.72 7.15 || 8.58 10.0 | 11.4 || 12.9 14.3 || 62 ºf * 24 1.71 3.42 5.13 | 6.84 8.55 || 10.26 12.0 | 13.7 | 15.4 17.1 || 66 5 ... 20 1.95 || 3.90 5.85 || 7.80 | 9.75 11.7 | 13.6 ſ 15.6 17.5 | 19.5 || 70 * § 18 2.06 4.12 || 6.18 || 8.24 || 10.3 || 12.3 || 14.4 | 16.5 l8.5 .6 || 72 5 - 14 2.25 || 4.50 | 6.75 9.00 | 11.2 13.5 15.7 | 18.0 20.2 22. 76 : ~ 10 2.39 || 4.78 || 7.17 | 9.56 || 11.9 14.3 | 16.7 | 19.1 21.5 23.9 || 80 .3 * 6 || 2.49 || 4.98 || 747 996 || 12.4 || 14.9 17.4 || 19.8 22.4 24.9 || 84.3 2 2.54 5.08 || 7.62 10.2 12.7 15.2 17.8 || 20.3 22.9 || 25.4 || 88 BOILING WATER. 367 TABLE IV.-Logarithmic Correction for Temperature of the -Atmosphere. Always positive. Temp. Loga- Temp. Loga- Temp. Loga- Temp. Ioga- Cent. , Fahrl rithms. || Cent. Fahrl rithms. || Cent. , Fahr rithms. || Cent. Tahrl rithms. —2.2 28 || 9.97005 || 8.33 || 47 || 0.98808 || 18.8 | 66 || 0.00539 || 29.4 || 85 || 0.02.204 —1.6 29 || 9.97 102 || 8.88 || 48 9.98001 || 19.4 || 67 || 0.00628 || 30.0 86 0.02:230 —1.1 30 9.97198 || 9.44 || 49 9.98993 || 2).0 6S 0.007 17 || 30.5 87 || 0.02376 –0.5 || 3 | | 9.97294 || 10.0 || 50 | 9.99.086 || 20.5 | 69 || 0.0')S06 || 31.1 88 0.02461 0.0 32 || 9,97391 || 10.5 || 51 9.991.78 || 21.1 70 0.00895 || 31.6 | 89 0.02547 + 55 33 || 9.97.487 || 11.1 || 52 9.90270 || 21.6 || 71 || 0.00984 || 32.2 | 90 0.02632 1.11 || 34 9.976S2 || 11.6 53 9.99362 || 22.2 | 72 0.01.072 || 32.7 91 0.02717 1.66 35 9.97678 || 12.2 54 9.994.54 || 22.7 | 73 0.01.16() || 33.3 92 || 0.02802 2.22 || 36 || 9.97773 || 12.7 || 55 9.99.545 || 23.3 74 0.01.248 || 33 8 || 93 || 0.02886 2.77 || 37 || 9.97868 || 13.3 56 || 9,99637 || 23.8 || 75 0.01.336 || 34.4 94 0.02971 3.33 || 38 9.97963 || 13.8 57 9.997:28 || 24.4 76 0.01423 || 35.0 | 95 || 0.03055 3.88 || 39 9.98058 || 14.4 || 58 |9.99819 || 25.0 77 || 0.01511 || 35.5 96 ().03139 4.44 || 40 || 998152 || 15.0 || 59 |9.99909 || 25.5 || 78 || 0.01598 || 36.1 || 97 || 0.03224 5.00 41 9.98247 || 15.5 60 0.00000 || 26.1 79 0.01685 || 36.6 98 0.03307 5.55 || 42 || 9,98341 || 16.1 | 61 || 0.00090 || 26.6 80 || 0.01772 || 37.2 99 || 0.03301 6.11 || 43 9.9S434 || 16.6 || 62 || 0-00180 || 27.2 81 || 0 0 1859 || 37.7 | 100 0.03475 6.66 44 9.9852S || 17.2 63 0.002.70 || 27.7 82 0.01945 || 38.3 || 101 || 0.03558 7.22 || 45 || 9.98622 || 17.7 || 64 || 0.00360 || 28.3 83 || 0.02032 || 38.8 102 || 0.03641 7.77 || 46 || 9.987 15 || 18.3 || 65 || 0.00450 || 28.8 || 84 || 0.02118 || 39.4 i 103 || 0.03724 TABLE V.—Logarithmic Correction for Mean Latitude of 1L i : Log. ||Lat. Observation. Always positive. Log. || Lat. 0.00111 || 15 |0.00096 || 30 0.00110 || 16 |0.00004 || 31 0.00110 || 17 |0.00092 || 32 O.00110 || 18 |0.00089 || 33 0.00109 || 19 |0.00087 || 34 0.00109 || 20 |0.00085 || 35 ().00108 || 21 |0.000S2 || 36 0.00107 || 22 |0.00079 || 37 0.00106 || 23 0.00077 || 3S 0.00105 || 24 |0.00074 || 39 0.00104 || 25 |0.00071 || 40 0.00103 || 26 |0.00068 || 41 0.00101 || 27 |0.00005 || 42 0.00093 || 28 |0.00062 || 43 0.0009S || 29 |0.00059 || 44 Log* 0.00055 0.00052 0.00048 0.00045 0.00041 0.00038 0.00034 ().00030 0.00027 0.00023 0.00019 0.00015 0.00011 0.0000S 0.00004 Lat. Log. 45 ().00000 46 |9.99990 47 |9.99992 4S |9.99988 49 |9.99984 50 |9.99981 51 |9.99977 52 (9.99973 53 (9.99969 54 |9.99966 55 |9.99962 56 9.9995S 57 |9.99955 5S |9.99951 59 |9.9994:S . Log. ||Lat. 60 |9.99944 || 75 61 |9.9994.1 || 76 62 (9.99938 || 77 63 |9.999.35 || 78 64 |9.99932 || 79 65 9.99929 || 80 GG |9.99926 || S1 67 |9.99923 || S2 6S 9,99920 || S3 69 9.999.17 || S4 70 |9.999 15 || 85 'Il 9,999.13 || S6 '72 |9.09010 || S7 73 |9.9990S | SS 74 |9.99.406 || S9 Log. 9.99904 9.99902 9.99900 9.99SuS 9.99$97 9.99$96 9.998.94 9.99 S93 9.998.92 9.998.31 9.99S9 | 9.99S90 9.99.SS9 9.99$S9 9.99SS9 TABLE VI.—Temperature of Boiling water, Corresponding to the Height of the Barometer at 60° Fuhrenheit. French Measures. Diff. 9.61 9.77 Height | Temp, Baronn. Water. M. M. Cent. 434.07 || 85.00 443.68 || 85.55 453.45 | 86.11 463.41 | S6.66 473.55 S7.22 483.86 S7.77 494.33 || S8.33 504 99 || 88.99 515.84 || 89.44 526.92 || 90.00 53S.16 || 90.55 549.62 | 91.ll 561.26 91.66 573.11 || 92.2.2 585.18 92.77 JEnglish Measures. Temp. Height Water | Baronn. | Diff. Falir. Inches. r: 185 17.09() ,378 1S6 17.46S 187 | 17853 |...}} 188 || 1S 245 189 | 18.644 4öö 190 | 19.050 Ai 191 || 19.462 420 192 || 19.SS2 127 French Measures. M. M. Height | Temp. Diff. Baronn. Water. . . Cent. 597.45 || 93.33 609.94 | 93.SS #|&#| ºf išiš |º 95.09 i.; ºff; ºš iš | 6′3.92 | Qºll išš, ºf 36.6% i., §349 || 9.3 ii. 1935. ºſ. ‘.... 717.82 98.33 '732.35 | 9S SS #; 747 13 99.44 i.; 762,]] |1999 iš; III+3 |1993 ** 792.95 || 101.6 English Measures. Temp. Height Water | Baronn. Diff. Fahr. | Inches. 200 || 23,522 192 201 || 24.014 | 202 || 24.514 203 || 25.022 | " 201 |35.51 | }} 205 26,067 || “... 200 || 26,602 207 || 27.146 208 27.699 * 209 28.261 | * 2io || 2: 3:3 ||373 211 || 29.415 212 || 30.007 213 30.60S 214 || 31.319 || 011 30'08 00'09 96'66, * 18°68, II'09 | * $6'63, 00'03 | * 60'09 *saucuſ ‘spuu e & º • ‘ū ISI ºut 3sa A • ‘BIonzaue A ‘uoposaS a3.1aqzqūds * * *XIIojS • ‘pubtloos * ‘eyssnidſ ‘niodſ 6S'63, 01, 6% 00'09 G1'66 00'03 30’0S 66'6% II'09 ‘souduſ • ‘‘KeAIoM ‘putteo I • ‘ĀIt]I ‘pub(uoal{} ‘ ‘ooutºſ ‘puttp:5ugſ • ‘Hieutua (I • ‘eupu O II'09 9I'09 QL'O3 00'09 MI'09 OI’03 9%, ’09 “saulou I • ‘odopſ poor) eduo * * *spuels I KIuut:0 • ‘IIzbig * * *eſſe.14sny ‘sage, S UJeugnoS • ‘sage, S utougio N. “Fr AI “spoo ongunlly • ‘ulou! ION ‘80ſ.IJ.V. ‘947??p.toduº tºpºſ oog of pup ‘oos oſ, ſo 72.27 277 of poompa. ‘sav.trºnoo wo.aſp wº ...toº eulo.rug out, Jo quºia H tragaru-ºx I (HTHIVI, IOI 20I f0L Q0ſ 10T 6t).I ZII #II QIt SII 0&I GZT ()9][ 98.I QFI IQI SGI 99.I L!I SSI | o00I 00I l() { 90 I #OL 90I S0 I III 66 00I ÇOI SOI Q0I 10 I 0E I ZII #II 9II SIL ZZI 13 I #3I OFI Sf I QQI o06 S6 ($6 I()[ ZOI #0I 90I 6DI III $H [. GIL 1.II IZI 9&I 3&I 69. I 9.5I QGI &QL ILL 9SI o98 16 S6 00L IOI 90I Q(){ SOI 0II ŽII #II 9II Q 0&I QöI ISI 19:I GFI ZGI 09 I 01.I ISI IQI 69. I, SQI 6LI o0S o91 Q6 96 S6 66 IOT SOI QOI SOI 60T ZII QII LII ÇI &I QSI ZłT 6f I 19T 99T SLI o01. 91, I --- o99 fg|I 99T # II Z6 96 QG 96 S6 00I 30I. iOU 90T SOI OLL #II SLI QZ, I ISI 1.8L #FL 39T T9T 3LI 0LI §§f 6S 06 36 9:6 Q6 16 S6 IOI 90I GOI 10 L III QII IZI 13. I 99 I SQL SQL 99T 19T o09 o99 o09 o9 f o0f ‘Jagau (O.Iuq ol tº pub Jºb out, Jo or nºted Uuaq q9UIUloat[\}{ ‘so.unm.todiual two.13(lºp ſo .torouto.tvq our uy ſowy wo ſo Intoſ-owo on 6wºpwodso...100 *339,81 up II W Jo Utulatoo u go quisia FI-'IIIA (HTAIWAL g'FI f"II 1, IL O'ZI §"ZI 1, ZI 0’9 U #’QU 6'9I gift Sºl g"gl S’QI 8'91 6'9"I 9".[ g"SI I’6 I l'O3, O' LZ ()"33, 98 QºII 9°II 6’II Ž'ZI 9 &I 6 &I Q'ºf S’9 L g"fL *i; I 3'QI J. GI Z'QI S’9 L g"l'I Z’S [. 0'6'ſ O'OZ 6'0& 6"IZ, 03 Q’II 9 II 6"II I'...I 9'ZI 6 &I 9'g|I L'91 If I 9‘FI Lºg I 9 GI I'9I 1. QI # I.I. I'SI 13 3'II G II S’II I'ZL Q'º, I S’3I Z'QI 9'QI O'FI g"f I I’II # II I, II 6 II #"ZI l. 3i lºgl g"gſ 6'9"I fºL 6"f I #"g I 6 GI Q'QT 3 J.L 6 I L’SI 9 6 I g’03 Q'IQ, 0 l I 8’ II 9°II #"Iz, #z, T3 SI QI 6'0'ſ Ç'II Q"II S’Il 3 ZI : QºI 6'3L 3I 6'0 I 3’ll Q’II f'SI £6 L 3 Z'03. Z"IZ. 6 S"Of ITI # II I, II 3"6 I I'OZ I’IZ, S’()l I’II gº II Q" It 0°3I 9 &I I'3L LºgL G'QI 0"IZ, -* g-H 1'OL 0.II 9 II 9° II 6'03 *- 0 "Jayouoluol out, pub Ipu out) Jo olnºtt.Iodual opulägue O 9'OI 6'0E Ž II g"II | 8–. 9– 9'OI 0S ). 09), Of), 031, 001. 0SQ 099 ()#9 039 009 'sa.in/p.2duo, quotaſlip qu ‘tojoulo.inq on up 2.1jouglyw awo or 6w8pwodso...too ‘sorportſ uri Riv Jo uvulatoo tº Jo qušioh—“ILA GITAVJ ‘x?Inoug|N (INV HIV Jo SNNanoo CORRECTION FOR THE RAROMETER. 369 TABLE X.—Correction for the Mercurial Column in Millimetres at Different Temperatures of Barometer. Tenn. IIeight of barometer in millimetres. | 9*|| 415|440|465. 490. 515 540|565| 590|615| 640i 665|600|715||740; 765 16||0.030.03|0.040.04|0.04|0.04|0.04|0.050.050.05|0.05}0.05|0.06.0.06'0.06 # 17||0.10|0.11|0.110.1210.12|0.130.140.1410.15|0.15} 0.16|0.17 9.179:18 0.18 # 18||0.16|0,180.190.2050.21 |0.22|0.230.240.25:0.26|0.27|0.28;0.29'0:00.30 § 19||0.240.25|0.26; 0.28||0.29 |0.30 Q32,933 0.35 0.36 || 0.38||0.39 |0.40 042 0.43 #20 0.35|0.37 (0.39;0.42;0.44 |0,460.48;0.50|052,054 || 0 56}0.59;0.61;0.630.65 § 21|0.42|0.45|0.42}0.51|0.53 |0.560.59|0.61|0.640.66 iO.690.710.740.76'0.78 3, 22|0.49|0.52|0.55|0.58|960|0.63|0.66|0.69|0.72|0.75 |078||0.81|0.84 9.8.0 §§ # 23||0.55|0.58||0.62}0.650.69 |0.72|0.75|0.79;0.82|0.85 |0.89;0.02|0.950.991.02 : 24||0.62|0.660,700.73|0.77|0.81|0.850.88||0.920.96 |1.001.03||1,071.11||1.15 S 25||0.69 ().730.77 O.S1 ().S5 || 0.89 |0.940.98 1.02|1.06 || 1.10 1.14|1.19 1331.27 § 26,0.75|0.80|0.84|0.89 |0.93 ().98|1.02|1.07 |1.11||1.16|| 1.201.25||1.30|1.34'1.39 § 27||0.82|0.870.920.97|1.02 || 1,08||1.121.17 | 1.22|1.27 | 1.32|1.37|142|| 1.47 1.52 £28||0.89|0.941.00|I.05|1.10|1.16||12||126||1.321.37 |1.421.481.53 1.591.64 § 29||0.98|1.01 |1.07|1.13||1.19 || 1.24|1.30|1.36||1.41 147 | 1.531.591.04 ifoliig g 30||1.02|1.09||1.151.21|12||133|140|146||1531.58||1.94:1.1911.7%|1831.89 31||1.09|1.16||1.221.29|1.85 [142|148|155 (1.61|1.6s 1.74 § 1.94 2.01 § 32||1.16||1.23|1.30|1.36|l.43 1.50|1.56|1.64 1.71 | 1.78 || 1.85 1.92; 1.99;2.06 2.13 § 33||1.231.30|1.38||1.45|1.52 | 1.60|l.07||1.741.821.89 | 1.97 2.04:2.112.19, 2.26 Ž 34||1.29 |1.37|1.441.52; 1.6() 1.68||1.76 1,831.9l 1.99 || 2.07 2.142 22:2.30 2.38 35; 11.36||1.441.52 |L60 | 1.69 1.76; 1.85 | 1.93;2.01 |2.10 2.17|2.25 2.33}2.42 2.50 Temp Cent. TABLE XI.-Correction for the Mercurial Column in Thou- sands of an Inch, at Different Temperatwres of the Barometer above or below 60°. Temp. Height of the barometer in inches. Temp Fahr. || 16 || 17 | 18 19 20 || 21 22 || 23 24 25 26 27 | 28 29 30 ||Fahr. 3 62 || 003|| 003|| 003; 003|003 || 003 004 004 004 |004 || 005 || 005 || 005: 005 |005 || 58 ă 64 || 006| 006 006 007 |007 || 007 || 008 008; 008 |009 |009| 009| 010; 010|010|| 56 g § 66 ||Q09|003|009| Q10|011 |011|Q12|012|Q12013 |Q14|Q14|Q15|Q15|Q16|| 54 g # 68 || 011|012|013|013|014 |015 016 016| 017,018 018 019; 020 020021|| 52 # # 70 || 014] 0.15} 0.16|| 017 (918 ()19 020 ()20 021 ()22 || 023| 024 (.25 026 (326|| 50 ° ă 72 || 017| 018 019 620 021 |022 023| 024 024027 |02S] 029| 030) 031|032|| 48 # # 74 ||Q20|^21|Q23| 024925 |926|02, Q28|Q29|031|03: 9:4|933 036|038||46 É g 76 || 023| 024 026i 027 |02S 030| 031|| 032| 034036 |037| 039| 040 041043|| 44 3 5 78 || 026| 027| 029 030|032|033| 035| 036; 038'040 log2 044) 045 046048 42 : # 80 || 029| 030| 032 034036 037| 039 || 0:41 043,045 046|| 04S} 050} 052|0.54|| 4ſ) = & 82 || 031|| 033| 035 037039 || 041 04:3| 045 048;049 || 051 053| 055 (57|059|| 38 3 3 84 || 034| 036|| 0.39| 041043 |045 (.47| 049 052'054 |056; 0.58|| 060 062|064|| 36 g § 86 ||Q37| 040|Q42 Q44040 1049. 951j953|Qā6|938 900 963|963|06T010|| 34: # 88 || 040|| 043 045, 047 |050 052; 055 057 061|063 |065| 068| 070 072|075|| 32 É 3 90 || 043 (146|| 04S} 051}054 || 056 || 059 || 061 065 067 070 072| 075 077 OSO || 30 5 # 92 || 046 (49) 051|Qā4057 |060 QG3 065|Q69|971 |974|977|QSO 083|086 28 . ... 94 || 049| 052| 054|| 057 060 064; 067 || 069 ()74 ()76 || 079 (182 085| 08S 091 || 26 5 § 96 || 051 i ().55 05S] 060}063 º 07.1) O73) 07S 080 | 084 OS7) 090) 0931097 || 24 g # 98 || 054| 058| 061 (64,067 071 (75|| 077; 0820S5 || 088| 092 095| 098|102|| 22 < 5, 100 li 057 | 061| 064| 068,071 || 075 079 oš3| 0860s) 093 097 || 100| 104 107 || 20 Heights in Feet of the Principal Waterfalls. Gavarny, Pyrenees, 1260 || Gray Mare's Tail, 350 || Rupin, Himalayas, 120 Lauterbrun, Switz., | 912 || Hepste, tº -> 300 || Kakabika, S. Am., 115 Staubbach, Switz., 900 || Nakchikin, Kamch. 300 || Lidford, England, 100 Ruican, Norway, 800 || Terni, Italy, . 270 Genesee, N. York, 100 Seculego, Pyrenees, 795 || Montmorency, Can., 242 Oyapock. S. Amer. 80 Luleå, Sweden, . 600 || Foyers, Scotland, 207 || Rhine Lauffen, Swi. 65 Tequendama,Colum. 540 || Wilberforce, N. A., 160 || Trollhetta, Sweden, 60 Tosa, Piedmont, . 470 || Cetina, Dalmatia, 150 || Parana, Paraguay, 52 Missouri, N. Amer., 400 || Niagara Falls, . 145 || Tivoli, Italy, e 50 Powerscaurt, Irel, 380 Il Tendon, France, 125 || Cataracts of Nile, 40 24 $70 TEMPERATURES. * *- TABLE XII.-Mean Temperature of the Air at the Level of the Sea. Months in the North Latitude. South Latitude. Ther year. 60 | 50 | 40 30 20 || || 0 || | 0 || 10 |20 30 40 e January, . 25 46 62 | 72 78 80 80 80 || 77 || 75 | 72 February, 28 || 48 || 63 || 73 || 78 || 81 i 80 || 80 77 74 || 71 March, º 32 || 50 | 64 || 74 79 82 81 '79 || 76 | 72 || 69 April, , . 38 55 | 67 || 76 81 83 82 79 75 70 64 is May, . . . 48 61 | 72 78 || 83 84 83 || 78 74 | 68 61 3 June, . 58 67 || 75 80 | 84 || 85 84 || 78 || 72 | 66 || 65 5 July, . . . 61 || 69 || 76 || 8 || || 85 | 86 84 77 | 73 || 64 52 g August, . 59 | 68 75 80 | 84 85 83 || 78 73 || 64 || 51 3 September, . 52 64 || 72 78 || 83 | 84 82 78 || 72 62 || 54 ſº October, . . . 44 57 | 68 || 76 81 | 83 : 81 || 79 || 71 || 63 59 November, . 35 | 52 65 || 74 || 80 | 82 SO || 79 73 66 | 66 December, 28 || 48 63 | 73 || 79 || 81 i 80 || 79 || '15 || 7 || || 71 Jamuary, . 1–3.8| 7.7| 16.6 22.2 25.5; 26.6 || 26.6| 26.4|| 25.3| 23.8 22.2 February, –2.2| 8.8| 17.2 22.7| 25.8. 27.2 26.8 26.6|| 25. 23.3| 21.6 March, e 0.0|| 10. 17.7 23.3; 26.1 27.7 || 27.2| 26.1| 24.4, 22.2| 20.5 - April, . +3.3| 12.7|19.4|24.4| 27.2|283 || 27.7|26.1|23.8. 21.1. 17.7| 3 May, . © 8.8|16.1|22.2 |23.5|28.3|28.8 28.3|25.5|23.3|20. 16.1| 3 June, e 14.4| 19.4; 23.8 || 26.6}. 28.8| 29.4 28.8 25.5| 22.2| 18.8] 18.3 § July, . . 16.1|20.5 24.4 27.2| 29.4|30. 28.8|25. 22.7| 17.7 11.1 : August, . 15. 20. 23.8 26.6|28.8| 29.4| 28.3|25.5|22.7|17.7|19.5 s September, . 11.1 || 17.7 22.2 25.5 28.3| 28.8 27.7 || 25.5 22.2| 16.6|| 12.2 CŞ October, . 6.6|| 13.8) 20. 24.4; 27.2| 28.3 ; 27.2} 25.8. 21.6] 17.2; 15. November, . 1.6|| 11.1| 18.3 || 23.3| 26.6| 27.7 26.8. 26. 22.7| 18.8| 18.8 December, —2.2| 8.8; 17.2 22.7| 26.1 | 27.2 || 26.6}. 26.1 23.8. 21.6} 21.6 Heights of Natural and Artificial works. HEIGHTS ABOVE LEVEL OF THE SEA. | Feet. HEIGHTS ABOVE THE GROUND. | Feet. Green in a balloon, 1837, . . 27,000 || Tower of Babel, said to have been | 680 Gay-Lussac, Paris, 1804, o 22,900 || Pyramid Cheops, Egypt, . º 520 Highest flight of condor, . (21,000 || Tower of Baalbec, Syria; . 500 Humboldt in the Andes, e 19,500 || St. Peter's Cathedral, Rome, . 500 Growth of vegetation, . e I'7,000 || Spire of Strasbourg, . “e 486 The author in the Andes,” ... [15,120 || Cathedral, Antwerp, ë © 76 Lake Manasarooa, Thibet, . 14,500 || St. Stephen's spire, Vienna, 465 Pine and birch grow, e . 14,000 || Highest chimney, Glasgow, . 455 Highest habitation of people,’s 14,000 || Spire of Salisbury, e - 450 Potosi silver mine, Bolivia, . 13.350 || Cathedral, Milan, . * º 438 Lake Titicaca, Peru,º . tº 13,000 || St. Mary, Lübeck, . © º 404 La Paz, Bolivia,” e º . 12.400 || Cathedral, Florence, . e e 384 Poplar grows at tº º 12,000 || St. Paul, London, . o e 366 City of Cuzco, Peru,” e . 11,500 || Hotel des Invalides, Paris, . 344 Oak grows at . * e e 11,000 || Cathedral, New York, . * 325 City Riobamba, Andes, . ... 10,800 || Dome of Capitol, Washington, 287 Quito, Equador, . e & 9,560 || Trinity Church, New York, 286 City St. Bernard, Switzerland, 8,600 || Notre Dame, Paris, . o & 220 City Santa Fe de Bogota, o 8,350 || Column City of London, e 202 Wild monkeys found atº . . 8,000 || Porcelain, China, . . . 200 City of Mexico, . . º e 6,990 || Leaning Tower of Pisa, e 188 St. Gothard, Alps, . g º 6,900 || Alexander Column, St. Petersb'g,| 175 Lake Lucon, France, . e 6,220 || July Column, Paris, . • 157 Palm and bananas grow at . . 2,500 || Column Napoleon, Paris, . . . 13S * Measured by the author of this Pocket-book. L. JoBNSON & Co.'s PROPORTIONS OF TYPE. 371 Diam. Pearl. Agate. Nonp, Min. Brevier Bourg. L.Prim. S. Pica. Pica, English Gr, Prim. ; ; ; ; ; 1 1 1 1 1 || 2 2 2 2 ; ; ; ; ; ; ; ; 2 2 3 I § 5 4 § 3 3 2 } . . ; ; 4 4 3 3 8 7 ; 6 5 5 4 4 4 3 ; : ; ; ; ; ; ; , ; ; 4 ; ; ; ; ; ; ; 6 5 4 ; ; ; 11 .9 8 7 6 #–E–3—3–10–3— — — 7– 9– 6T 5 ; : ; # II }} 10 g 8 7 i i ; ; ; ; ; ; ; ; ; 6 9 14 ; ; ; ; ; ; ; ; ; ; ; ; 7 4 2 ; I6 15 14 12 10 9 * § ; # 17 16 14 I0 i–3– * : * is 17–1 5–14–1 2 I 1– 9 i-º-º-º-º-º-º-º: 15 i : ; ; ; 20 19 i. 4 I2 I () ; ; ; ; ; ; 2) is 16 1 13 * 6 3 22 24 22 21 17 15 II ; ; ; ; 25 23 22 19 18 16 14 ; ; ; 32 35 24 23 20 I2 ; ; ; ; ; ; ; ; 19 17 15 ; ; ; ; ; ; ; 22 39 is 16 13 3–5–4–3–3–3–3, 5–31–3–2– ; ; ; ; ; ; ; 24 22 14 ; ; ; ; ; ; ; 25 23 20 18 i i ; ; ; ; ; ; ; 21 is 15 † : ; ; ; ; ; ; 25 22 20 16 ; : " 45 37 34 32 2 26 23 ; ; ; ; ; ; ; ; ; ; ; 21 17 9 i-º- i-º-º-º-º-º-º-º: 5–22– l 8 # ; ; ; ; # 35 ; 29 23 ; ; ; ; # 33 37 : 30 26 I9 i ; ; ; ; ; ; ; ; 2. 34 6 45 41 ; ; ; ; ; ; ; ; ; ; 25 20 § 3 ; ; ; ; ; ; : 26 21 8 ; 7 49 37 30 27 4. 8 60 45 43 . * 22 § – –gl—50—46—†—38–35–31–. -*. i iſ ºf Tºi º żº $ 3.23 3 : 3 & # 48 46 40 37 29 4- - * * **- B72 LIGHT AND COLORS, LIGHT AND C 0 L () R. S. Light is the sensation transmitted to the eye, and produces the sense of seeing. Light is a component part of heat, and a compound imponderable substance whose ingredients depend upon the composition of the burning substance; or, burning substances can be analyzed by decomposition of its light in a spectrum. Decomposition of Light in the Spectrum. Colors. Maximum ray. Combination of Colors. Violet. Chemical. Primary. Secondary. Tertiary. Yndigo. Blue. lº * Riu. Electrical. Yellow. Green. 3. * .* Teel], Green. Elue. Purple Yellow. Light. Red. ple. Orange. Yellow, • Brown. Red. Heat, Red. Orange. All the colors of the spectrum mixed together make white, which is proved by the decomposition of white light, which makes the seven colors. The velocity of light in planetary space is 192500 miles per second. The velocity of light through transparent bodies is not known, but probably varies inverse as the square root of the specific gravity of the transparent substance. Light passes from the sun to the earth, 95000000 miles, in eight minutes, at which rate of velocity light cam pass around the earth in one-eighth of a second. The intensity of light is inverse as the square of the distance from the luminous body. The standard unit for measuring the intensity of light is assumed to be that produced by a sperm candle, “short 6,” burning 120 grains per hour, A spermatic candle 0.85 in diameter burns about 1 inch per hour. MOTION OF GAS IN PIPES. Letters denote, Q = cubic feet of gas passed through the gas-pipe per hour. L = length in feet, D = diameter in inches, of the pipe. H = head of water in inches which presses the gas through the pipe. s = specific gravity of the gas, air being 1. n = number of candles required for giving the same light as Q cubic feet of gas per hour. Q = 780D? |Hip Q= V/n D=-|- ; SL Q. WSL. n= gº 14.35 W. H. Example. At a distance of L = 6450 feet from the gas-work is required a - 940 cubic feet of gas per hour. Head of water being H-1 inch, specific gravity s = 0.5. Required, the diameter of the pipe D = ? _ _1_*|0.5×6450×940° 14.35 1 Each light in a room consumes about 4 cubic feet of gas per hour, and ordinary street-lights 5 cubic feet. = 5.396 inches. SOUND. - 873 S 0 UN D. Velocity of Sound through Air. v = velocity in feet per second. t = temperature of the air, Fahr. scale. D = distance in feet the sound travels in the time T. v= 1089.42/1+ 0.00208(t–32). Velocity of sound in water is about 4 times that in air, and 8 times that through solids. Intensity of sound is inversely as the square of the distance. D= 1089.42T1/1+0.00208(?–32), T=P Q) Facample. A ship at sea was seen to fire a cannon, and 6.5 seconds afterward the report was heard; the temperature in the air was 60°. Required, the distance to the ship. D=1089.42 × 6.5/I-F.00208(60°–32)=7300 feet, or 1.38 miles. Audible at a distance of Descriptions of Sound. Feet. Miles. A powerful human voice in the open air, no wind, 460 0.087 Report of a musket, * e e e te g 16,000 3.02 Drum, * e te º º tº & e & 10,500 2 Music, strong brass band, . . te tº e 15,840 3 Cannonading, very strong, º * g tº { } 575,000 90 In a barely observable breeze a strong human voice with the wind can be heard. . * e & e 15,840 3 Ringing Bells—Weight, Dimension and Key-note. Place, and when Weight. Diameter. Sound-bow. Key- || Wibrat. Cast. Jy. L). S. k. DOte. 72, pounds. inches. inches. S : D. In Ote. vib. Moscow, 1736, . . . . . 432,200 272 23 0.084 D 18.21 “ St. Ivan's, 1817, 127,350 185 14.750 0.080 G; 25.1 Nishmi-Novgorod, Russia, 69,664 || 151.25 12.125 0.080 B 30.5S Olmutz, Bohemia, . . 40,320 | 121 9.125 0.075 D; 39.4 Vienna, Austria, . . . . 40,200 118 9,500 0.080 D; 38.6 Westminster, Eng., 1856, 35,625 113.5 9.375 0.083 E 40.3 Erfurt, 1487, . . . . 30,800 | 103.5 7.75 0.075 E" 42 Paris, 1680, . . . . 28,672 | 103 7.5 ().073 }} 40 Montreal, 1847, . . . . 28,560 103 7.8 0.082 F 42 York, Eng., 1845, . . 24,080 | 100 8 0.0SU F: 46.4 New York, City Hall, . 22,300 || 108 8.3 0.07'ſ I. 41 St. Peter's, Rome, . . 18,600 97.25 7.5 0.077 F; 46 Oxford, Great Tom, 1680, 17,024 S4 6,125 0.073 G#. 50 Cologne, 1447, . . . 16,016 95 7.2 O.076 G. 49 Brussels, Belgium, . . . 15,848 81 5.75 0.071 G; 51 Lincoln, Eng., 1834, . 12,096 82.5 6 0 073 G; 51 St. Paul’s, Eng., 1716, . 11,500 81 6,08 0.075 A 54 Exeter, 1675, . . . . 10,08 76 5 0.066 G; 50 Old Lincoln, 1610, . . . 9,856 '75,5 5.93 0.078 b 60.4 Westminster, 1857, . . 8,960 72 5.75 0.079 lS 60.4 } t 374 MUSIC.—ACOUSTICS. Music-Acoustics. Let a string L, fig. 312, be stretched over two edges, a and b, and made to vibrate a musical note; place a stall c in the middle of string L, and the two parts will vibrate an octave above that of the whole string. To Divide a String into Musical Tones. L = length of the whole string in any unit of measure. l = the part of L which vibrates any other tone above that of the whole string. p = position of any note above the fundamental note, counted in half notes. If the whole string sound the note C, thon p = 5 for F. If L sound A, then p = 7 for E, and so on. Then the length l will be— log. l = log. L — 0.30103 #. for the geometrical scale of vibrations. The length L, multiplied by the decimal in the table, will be the correspond- ing length l, by which the stalls on a guitar can be divided. The Toning of Musical Instruments. A musical instrument toned by the harmonical scale of vibration cannot produce good music except in the one key in which it is toned, because that scale does not arrange the notes in equal distances apart; but when toned by the scale of geo- metrical progression, it will make equally good music in all keys, because that scale divides the notes equally. In toning a piano and striking the quint C and G in perfect harmony to the ear, G will be a little too high for the geometrical scale, which ought to be cor- rected in the toning. In the terce C and A, A will be too low for the geometrical scale. The case is the same in toning a guitar or violin. See the following table, where the harmonical tone is lower and where higher than the tone in the geometrical scale. All musical instruments should be toned in the geometrical scale. The column m” is the standard number of vibrations per second, established at Stuttgart, Germany. Table of Comparison of the Geometrical and Harmonical Scales of Vibrationn in Music. GEOMETRICAL SCALE. Difference. IIARMONICAL SCALE. No. p. Note. W. % m and n/ m’ l Note. I m// O C L = 1 64.000 0.000 64.00 L = 1 C 264 I C# 0.946.30 67.790 C; 2 D 0.89087 71.840 || – 0.160 '72. 0.8887 D 270 3 D; ().840.91 76.11() D; 4 E 0.79369 80.636 || + 0.636 80. 0.8000 E 330 5 F 0.74917 85,430 || + 0.097 S5.33 0.75 F 352 6 F: 0.70712 90.510 F; 7 G 0.66745 95.890 || – 0.11 96. 0.6666 G 396 8 G# 0.62997 || 101.59 G; 9 A 0.59462 | 107.63 + 1.03 106.66 0.60.142 A 440 10 A # 0.56125 | II.4.03 A: 11 B 0.52973 || 120.82 + 0.82 120. 0.5333 B 495 12 C 0.50000 || 128.00 0.000 128. 0.5000 C 528 Approximate Weight of Bells in Pounds. Diameter. 0 I 2 3 4. 5 6 7 8 9 inches. | 000 || 0.025 0.2 0.68 1.6 || 3.12 5.4 8.6 | 12.8 | 18.2 10 25 || 33.3 || 43.2 || 55.0 | 68.6 | 84.4 || 102 || 122.8 146 || 171.5 20 200 || 231 266 304 || 346 391 || 440 492 || 549 610 30 675 745 820 | 900 9S2 1072 | 1166 1244 || 1372 1483 40 1600 || 1723 1852 1990 || 2129 2278 2433 2595 || 2765 2941 50 3125 3316 3500 || 3722 || 3936 || 4160 4390 || 4630 || 4880 || 5134 60. 5400 || 5675 5958 6250 || 6553 | 6865 7187 7520 7866 | 8212 70 8575 | 8948 || 9331 || 9723 |10|130 10547 |10974 |11413 |11864 12323 80 12800 |13286 13784 ||14294 || 1482 I5353 |15901 |16462 |17037 |17624 90 |18225 |18838 19467 |20109 |20764. 21434 |22118.22817 |23529 |24257 100 |25000 |25757 |26530 1273.18 |28121 |28940 |29775 |30626 |31493 ||32335 375 MUSIC, 312. 6 d. a b-：- -º-dºc., dºc. f g-ø--ſº -ſ iſ” n––“_º_º_ſ_ſ_ſ_º_º_º_º_º_º_2.- * * * ± + + E ---- -*----|----+---------•––º–ſ，--i–––1–––1–––1–––D- ÞÆ--EËEĘIII~~HTATTFÖTT！”L-I-L-L--------~r Š–!---„LLE LOEI， LIL-ITTTTTTTTTTT | IVSTĀV--|----+• --º---+------|------------~：=?=~= --61- v ### : ;---5-7-T-T-T-T-T-T-E-a-a-z-z-z-z, § 1 Chrommatic Soale in Treble Clef. ©AL–1––––- 1––1––1–––1！114， ––––––1––3|-----|----|-----|------|----|--------–ſſo--º---- H-ſ-））+$TI-I-ĒĒĒ±Fºj-№ſſº-ſº-#ø-----†== |-ĢŠjL.*#E-#~~ ----±r—#---- –4–－----，--4：— —T|× ########35 "6 7 8 9 10 11 į 1 2 3 4 5 82 1 -：- -º# e- -09-9-_ſ2_^•----#●● Q––# º---º--#-2-#· · * * ?* # # # # # # # …, e − № º---T-ſt--;I--TI-TE---T+L− | | __/A|--|-----|–––1–––1––––1–––1，~~~~ EGPA---!----+---+--→– ©/6 78 9 TT107īIT21 23 4 56 7 đo, đo, d. Chromatio Scale im Bagg Clef, 62€. �- ſº «№b|Çſ,9.fº ed.e b� |-sæsbF •& b ſº-ſo-�h a• I----|----|-L–––1––––1–––––1L—| 1 （NoIL-T- ZT-z-IV-2|I/I،[Ll_III_（_DOE_LOET，LITTI !！！!-----ȘILITI-TIIT-TIEſº”TÆLL （)ai–––––1––1–––1––––1 | TLZ----|-Š-----|----|----!！ Ț|-----|-----|------------º---º-bºſ---G-D-（−）- LLC−•r•- r-TÅr--]-!|-v_yyw(）VOEL 482l12 34 ő6 7 89 10 11 1 1 . 284 576 BELIS. RING IN G B ELLS. Lciters denote. D = diameter of the beh at the mouth, in inches. d = diameter of the bell at the crown, in inches. h = heighth of the bell from the mouth to the crown in inches. S = thickness of sound bow in inches. W = weight of the bell in pounds avoirdupois. n = number of vibrations per second, corresponding with the key note of the bell, and to be found in the accompanying table I. k = from 0.07 to 0-08, or a coefficient expressing the relative thickness of the Round bow to the diameter of the bell. In peals of bells, the sound bow is generałky S = 0-0SD for the triple, and S = 0:07D for the tenor; the iuterme- diate bells in the peal having the intermediate proportions of sound bow. Example 1. Required the weight of a bell D = 62 inches in diameter, and S = 4\in. thickness of the sound bow, W = ? Formula 1. W= 0.25X622X4+5 = 4324.5 pounds. Example 2. A bell of 2,500 pounds is to be constructed with a sharp note, taking the sound bow k = 0.075. Required the diameter of the bell D = ? 37 TNZºº Fºrmula 10. p = V*.*= 510s, inches l Example 3. It is required to construct a bell with the key note p: in the first octave above zero, n = 152'22. To be of light weight with a full good note, for which latter case take k = 0.07. Required the diameter of the bell, D = ? _ 58000XO-07 Formula 11. D = *...* Example 4. Required the key note of a bell with D = 36-5 in. diameter and S = 2.75 in., n = ? = 26-665 inches. Formula 4. * = 58000X # = 11.9-7 vibrations. In the table the nearest number 120-82, in the first oetave below zero, answers to the key note B, which will be the note of the bell. Frample 5. A bell of 6860 pounds is to be constructed with the key note O’ in the first oetave below nero n = 64, see table I. Required the diameter of the bell D = ? 860 Formula 9. D = 21-947 Vº = 70-6175 inches Example 6. Required the thickness of sound bow for the bell in the pre- ceding example? D = 70.6175 inches and n = 64. S = ? 64×70.6175? – 3-know s Formtºla, 12. S = T55000T = 5-5027 inches. Example 7. Required the weight of a bell D = 48 inches diameter at the mouth, d = 25 inches at the crown, and h = 34 inches height from the mouth to the crown, S = 3.5 in., W = ? Formula I'ſ. W=48×25x3-5 (0.5–0.002816×25) +0.00375×34×25mx35–2126'226 pounds. BELlº. 377 Formanulas for Ringing Bellso W’ = 0.25 DaS - - - 1 --w .S. 2, D2 == &=ºm: tº º ºs 7 ----- e. * * * * 12 w.e. Pº D = 2 S 58000 ** annº - - - - 2 * 232000 S J. W. * sº *= & 8 > = n e º sº e W = 0.25D3% - - - - ap = 2,088 7), S. 1)2 13 4 / Tº * n = 58000; . º º ſº. ‘p-aeº/4 gº g|S= k D • - - - - 14 33 * = S. W. 3. 4 Jy k -- e. tº E tº sº ºn 15 * = 282000; - - - - 5|p = + . . 10 ” 4 W. n = 58000; - - - - op-58000; - - - - 11%–º - - - - - 18 W= D d S (0.5–0.00028.16d)+0-00375 h de S - • - 17 Table I. Vibrations per Second = n. Key IBass. 6 Descant. ſº note. 3rd Oct. 2nd Oct. 1st Oct. § 1st Oct. 2nd Oct. 3rd Oct. C 16-000 32°000 64'000 128-00 256-00 512:00 C# 16-947 33-385 67-790 135°58 271-00 || 542-32 D 17-960 35'920 II*840 143-68 287-36 574-72 D# 19°027 38'055 || 76° 110 || 152-22 || 304-44 60S-88 E. 20-159 40°318 80-636 161-27 322-54 645-09 F 21.357 42-715 85-430 170-86 341-72 683°44 F# 22-627 45°255 90-510 181-02 362-04 724'08 G 23.972 47-9.45 95-890 191-78 383-56 767-12 G# 25-398 50-797 101*59 203-19 || 406-37 812.75 A. 26'908 53.817 107.63 215-27 430-53 861-07 A# 2S-508 57-017 114-03 228-07 456-13 912-27 B 30°204 60°409 120-82 241-63 483-27 966-5-4 O 32,000 64'000 128-00 256-00 512-00 1024.0 Table V. Abscissa Ordinate Thickness of Metal. 20 gy S = 1 S = 0-07 D S = 0-75D 1 S = 0-087) 1 0°4142 l 0-700 0-750 0-800 14 0-686 0-800 0-500 0-600 0.640 2 0-867 0-653 0°459 0°490 0-522 24 0-974 0°54'ſ 0-3S2 0°410 0°437 3 I-025 0-474 0-331 0-355 0-379 3| 1*030 0-423 0-295 0.317 0-338 4 1-000 0-380 0-266 0-2S5 0304 4} O'955 0-351 0.245 0-263 0°281 5 0-S75 0-327 0.228 0-245 0-261 5% 0-775 0-301 0-211 0-226 0 241 6 0-665 0-291 0-203 0-218 O-233 64 0-5.30 0.286 0-200 0-214 ſ]-223 | 7 0-390 0.279 0-195 0-209 0-223 7| 0-235 0.272 0-190 0.204 0-2 iſ 8 0.075 0.267 0-186 0-200' 0-213 8-74 0-78 0.333 0.233 0.250 0-266 J ſ 378 BELLs. To Construct a Bell. When a bell is to be constructed, we generally nave the weight or rey note glven by contract, the diameter and sound bow are calºuiated º the preceding formulas and examples, and then ready to proceed wit the construction. See fig. 1. The diameter of the bell at the mouth, is divided into 10 equal parts, called strokes, which then is the scale and measurement for the con- struction. Make a decimal scale, as shown on plate VII. Shrinkage to be allowed for 3 sixteenths of an inch per foot. The section of a bell is generally laid out on a piece of board repre- sented by the dotted lines a, b, c, d, which then is cut out and used for turning up the mould for the bell. The board should be about 11 strokes long, and 2-5 strokes wide. Through the centre of the board draw the line p, q, parallel to b, c, bisect the line p, q, and set four (4) strokes from the bisecting point towards each end, divide the strokes into halves, and number them as shown on the accompanying drawing. Through each division draw lines at right angles to p, q, set off the corresponding ordi- mates y expressed in strokes, Table II. and join them by a curve-line, which then will be the centre of thickness of metal in the bell. At the end of the first ordinate, as a centre, draw a circle with a diameter equal to the desired thickness of the sound bow, which should be from 0.7 to 0-8 strokes. At every succeeding ordinate draw a circle with the diameter noted in Table II; for instance, if the thickness of the sound bow is 4} inches, then the thickness of metal or diameter of the circle at the third ordinate will be 4.5X0:474=2-133 inches; but if the sound bow is 0.7, 0-75 or 0.8 strokes, the thickness of metal at the third ordinate will be 0.331, 0:355, or 0.379 strokes. When all the thicknesses are thus drawn, draw the two lines tangenting the circles on each side of the centre line of the metal. From 0 to 1 make a moulding of 0-1 stroke thick over the dotted line as shown by fig. 2. Prolong the 63 ordinate, and set off 1.79 strokes to e, which then is the centre for the curve on the top, draw the arc through the centre of the small circle at the 8th ordinate; joine, 8, set off from e, 0.46 strokes to the centre for the inside curve at the top. Thickness of metal of the top should be 0.3 the sound bow at 8, and 0.333 at r. Draw the ordinate at 8:74, set off-0-78 to r, join r and the abscissa ; º: prolong the line through r, then finish the drawing as shown on e plate. When the board is cut out and ready for turning the mould, it must be carefully set, so that the outside diameter of the crown will be half the diameter of the mouth of the bell. This form of Bells gives the greatest possible gravity of tone with the least possible quantity of metal. Bells can be made almost in any form without seriously affecting the quality of tone, but the thickness of metal should always be in proportion as the square of the diameter taken at the centre of the metal as in fig. 3. Proportions of a Peal of Eight Bells, Bells. | Keynote. | 70, | k | S. in. D. in. W. lbs. Clapper. Tenor, D 71-84 || 0-070 3.95 56°5 3156 63 lbs. 2nd, E 80-64 || 0-071 || 3-62 51* I 2366 48°6 3rd, F# | #| #| 3: 46-1 1765 || 37.2 4th, G 95-89 || 0-073 || 3-22 44°2 1575 34° 1 5th, A. 107.63 || 0-075 || 3:08 40°5 1262 28-1 6th, B 120-82 || 0-077 2.85 37-0 976 22°4 7th, C # 135°58 || 0-079 2-67 33°8 763 18-2 Triple, J D 143*68 || 0-080 || 2:58 32-3 673 16-8 Clapper. The weight of the clapper should be from one fortieth to one fiftieth the weight of the Bell, the smaller bells take the largest clappers. . Metal. Thirty of Tin to one hundred of Copper, is a good pro- portion. lſ - ſ &I (G º # ºr * Fi - - - -> • • • • • • • • • • • • • • • • ** - * =：= * ... • * -- * → „ *---- … *§ • |-- > L • ^<> ·:Ç - *• • •• × \șNew•§. ~a- ... • *�••«Nº ... • •*, SÅ ... • **NĂ … •.”· - „ • *• •^ _ • ” ’ ’.~; ••，•· … *wtº ... • • ••• … • *--* * _ - r●wł ... ~~！… • **A ：)-*«>. - **FTv• *● ….！!！-* -•• ... - *ųą ų• ! ~~��?ºſ-… • ×',\,� ºfW - ~~~~）;！•-<■*�•$1' ----wos ºſº -~~ ~ ·**<\> --~~~~ {！* • !！！w. •�::!*. … * * 'ſ， W§, !!<>^ •• *{• » .●§:: :，:)）;{ SĂ;|… º！', ſº，！: ~~~~ · · · · · ·, <.*'* * * * *---- •№.3 &&)\! ！!*¿. '№ſºſ*}} · ！ 0 .. ' •--><3; ; • " - $$ ſlºj|- ！º）. \， y, Nitric acid, . © º Fahr. 26069 Cent. - - - . .. *_ Expansion of BoDrºs BY HEAT. *—.------------- ~~~~ EXPANSION OF BODIES BY HEAT. All bodies in nature expand when heated, and contract when cooled. Solids vary but little by the difference in temperature; liquids vary more, but gases are extremely susceptible to the impression of heat and cold. There is a very singular fact connected With the expansion and contraction of Substances at and near the temperature of fusion, which may be.illustrated in the accompanying figure. - Let A B represent the absciss-axis of temperature, C. D the ordinate axis of ex- pansion or contraction, and the origin O the temperature of fusion, O A the tem- perature of the solid, and O B that of the liquid. Let a solid of volume and temperature at a be heated, it will expand until it - reaches a maximum volume at b, after which it contracts toward the temperature of fusion O. The temperature still increasing, the liquid will continue to contract until it reaches a minimum volume or maximum density at d, after which it will expand toward e. The lines a b 0 and 0 d e are parabolas, of which the absciss-axis A B, passes through the focuses f. The formula for the parabola is y = a,", in which the exponent n depends upon the nature of the substance operated upon, and also whether it is linear or volume expansion, a representing the temperature and y the volume. Ice melts at 32°Fahr., and the water reaches its maximum density at d = 390 (as now accepted, but d is nearer 40°). Ice reaches its maximum volume at b = 249. Ice and water are of equal density at the temperatures 16°, 32° and 489. Ice generally floats in water, because the difference in temperature is less than 329; but if ice of less than 16° is put into water of more than 48°, it will sink. The same phenomenon takes place with other substances; for instance, solid cast iron put into molten cast iron will float, but if the fluid cast iron is at a white heat, like that in a pneumatic furnace (Bessemer), the Solid iron Will sink. The following formulas are deduced from experiments which have not extended through the temperatures of fusion, except that for Water, page 392. Notation of Letters. L = linear expansion of solids and liquids, per degree Falir., between any temperatures. . ! = linear expansion per degree between 32° and 212°, as contained in the accompanying table. D and d = absolute temperature in degrees Fahr. 4. m = exponent of expansion, which varies inversely with the rate of expansion of bodies. 15.6 Mercury. ' 14.1 Platinum. 2.77 Copper. 2.6 Exponent n e Iron. 1.04 2.5 for Water. Glass. Linear expansion per degree from 329 to TO will be = — T/L). ** 105,000 V Linear expansion per degree between any temperature is l ºn /- ºn ,- L===, (WD–V d 10580000 The linear expansion per degree multiplied by 2 will be the surface expansion. | The linear expansion per degree multiplied by 3 will be the volume expansion, DILATATION or ExPANSION OF SUBSTANCES, 3Rf, Dilatation or Expansion of Substances, Per Degree of Fahrenheit Scale. Tºº- Solids. Linear, l. Surface, a Volume, v. 320 to 2120 0.00000478 ().00000956 0.00001434 212 “ 392 | X-Glass, e e g e 0.00000546 || 0-00001093 || 0.00001639 392 º 572 | }... --~~~~...~... ....... … . #; 0.00001320 || 0.00001980 32 “ 212 & .00000656 || 0.00001312 0.00001968 § a #3 |} Wrought iron. . . jº, ºff jº, 32 “ 212 |Soft, good iron, T. ". ſº 0.00000680 || 0.00001360 || 0.00002040 32 “ 212 Cast iron, . . . e * 0.00000618 || 0.00001.236 0.00001854 32 “ 212 |Cast steel, . . . . . . . 0.00000600 0.00001200 0.00001800 32 “ 212 |Hardened steel, . . . 0.00000689 || 0.00001378 || 0.00002067 32 “ 212 }co er 0.00000955 0.00001910 0.00002865 32 “ 572 | }.9°PP* * * * * 0.00001092 || 0.00002184 0.00003276 32 “ 212 |Lead, tº e º º 0.0000158() 0.00003160 || 0.00004740 32 “ 212 |Gold, pure, . . . Q 0.00000815 0.00001630 || 0.00002445 32 “ 212 Gold, hammered, . ge 0.00000830 ; 0.00001660 || 0.00002490 33 “212 |Silver, pure, * e e 0.00001060 0.00002120 0.00003180 32 “ 212 |Silver, hammered, . dº 0.00001116 0.00002232 || 0.00003348 32 “ 212 |Brass, common cast, . ë 0.00001043 || 0.00002086 || 0.00003129 32 “212 Brass, wire or sheet, tº 0.00001075 0.00002150 0.00003225 32 “ 212 } Platinum Uire 0.00000491 || 0.00000982 0.00001473 32 “ 572 || || Pºº" " ' | 0.00000520 | 00000040 0.00001560 32 “ 212 |Palladium, © tº & | 0.00000555 || 0.00001110 ! 0.00001665 32 “212 |Rorian cement, . © & 0.00000797 || 0.00001594 0.00002391 32 “ 212, Platinum, hammered, . . 0.00000530 ().00001060 0.00001590 32 “ 212 |Zinc, pure or cast, . º 0.00001633 || 0.00003266 || 0.00004899 32 “ 212 |Zinc, hammered, . 0.00001722 : 0.00003444 0.00005166 32 “ 212 |Tin, cast, . . . e tº 0.00001207 || 0.00002414 0.00003621 32 “ 212 Tin, hammered, o e 0.00001500 0.00003000 || 0.00004500 32 ° 212 Fire brick, . tº º º 0.00000275 0.00000470 0.00000705 32 “ 212 Good red brick, e tº 0.00000305 || 0.00000610 || 0.00000915 32 “ 212 |Marble, • . . . 0.00000613 || 0.00001226 || 0.00001839 32 “ 212 |Granite, . & & & 0.00000438 0.00000$76 0.00001314 32 “ 212, Bismuth, . & & e 0.00000773 0.00001546 0.00002319 . 32 “ 212 Antinuony, © gº & 0.00000602 || 0.00001204 || 0.00001806 32 “ 212 |Palladium, . ge º & 0.00000555 || 0-0000IIIQ || 0-00001665 32 “ 212 || 0.00003333 0.00006666 0.00010000 212 “ 392 | X-Mercury, . . . 0.00003416 0.00006833 0.00010250 392 “ 572 . 0.00003500 || 0.00007000 0.00010500 32 “ 212 0.00008306 || 0.00017612 || 0.000264.20 212 “ 392 Water, . . tº º 0.00017066 0.00034.133 0.0005102() 392 “ 572 : 0.00018904 || 0.00037808 || 0.00056713 32 “ 212 Salt, dissolved, tº tº 0.00009250 0.0001S500 ().000277SO 32 “ 212 Sulphuric acid, . . . G.00011111 || 0.00022222 || 0.00033333 32 “ 212 Turpentine and ether, . (3.00012966 || 0.00025933 0.00038900 32 “ 212 |Oil, common, † g 0.00014814 || 0.00029629 || 0.00044444 32 “ 212 || Alcohol and Nitric Acid, 0.00015151 0.00030302 || 0.00055555 32 “ 212 |All permanent gases, . 0.00(369416 || 0.00138832 || 0.0020S250 Force of Temperature. It is the force of temperature which expands the bodies, and not the quantity of heat. See pages 379 and 392. Temperature is convertible into force. Let P denote the force of pressure in pounds per square inch, and T' temperature Fahr. * 6 *...*), and T=202sºp-105.1. This force, multiplied by the space of expansion, is the work done by the heat. Then, P= ( 25 - ---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - – . . . . . --- 886 PROPERTIES OF HEAT. Conducting Power of Different Substances for Heat and - Electricity. Metals. Quartz sand, . . . . 35.56 Liquids. Silver, fine, . 100 ||Limestone, . . . . . 19.8 ||Water, . . . . . 1.000 Gold, “ . . . 98 ||Lime, . . . . . . 24.00 Mercury, ... • - - 2.80 Gold, .991, . 84 ||Quartz crystals, . . . 80.0 |Proof spirit, . . . . 0.847 Copper, ham’d, 85 ||Slate, . . . . . . . 10.00 Alcohol, pure, . . ().931 Copper, cast, 81 ||Keen's cement, . . 1.901 ||Nitric acid, . . . | 1.5 Mercury, . . 68 ||Plaster and sand, . 1.870||Sulphur. acid, . . 1.7 Aluminium, . 66 ||Plaster Paris, . . . . 2.026|Sulphur. ether, . . . 2.1 Zinc, hammered 64 º cement, . . . 2.080|Turpentine, . . . 3.1 Zinc, cast Ver- ||Asphalt, . . . . . . 4.52 Gases. tical, , , ... 63 ||Chalk, . . . . . 5.853||Air, . . . . . . 0.9855 *...* hori- 60 Woods, Radiating Power. Le. cast. * 20 Fir, cross grain, . . . 1.10 ||Water, . . . . . 100 &amium. . . § ||ir, with the fibre, . 3.10 ||Lampblack, . . . 190 Wrought iron, 43 Pine, ..., , ; , , . . . 3.99 ||Paper, writing, . . . 98 Tin .*.*. . ||Qak, with the fibre, 3:30 ||ºosin, ... . . . 96 steel." 36 ||Elm, {{ “. . 3.2 ||Sealing-wax, . . . 95 platinum.'. ' 40 Ash, {{ ‘. . 3.1 ||Glass, common, . 90 ðast iron... ." 36 #. º º g ; º ink, . . . . ; . & 9 * * Sbon º g Ce, - - - - - - Alº, cast 21 Hºad, . . . 0.112||Red lead, . . . . . . 80 Anº cast Cross: with fibre=1: 3, Graphite, . . . . . 75 h orizontai, 19 ; . . . . . . 4.10 *. tempered, . ; German silver 10 ack oak, . . . . 3.2 || erºury. . . . . Bismuth 3. 6 Chestnut, . . . . 3.0 ||Lead, polished, . . 19 3 * * Spanish mahogany, 2.8 ||Iron, polished, . . 15 Stone dé Crystals. Walnut, . . . . 3.3 ||Tin and silver, . . . 12 . Marble, . . . 12.21 Hare’ Fur. 946 º: and É. . . 12 Glass, . . . 9.65||Hare's fur, . . . . |0.0 beflecting Powers. Common brick, 8.422|Eider down, . . . . 0.0668||Brass, . . . . . 100 Fire-brick, . 6.05||Beaver's fur, . . . 0.0675||Silver, . . . . . 90 Fire-clay, . . 6.61||Raw silk, . . . . |0.0692||Tinfoil, . . . . . 85 Porcelain, . 7.55||Wool, sheep, . . . 0.0778||Tin, . . . . . 8() Wood-ashes, . 0.8359|Cotton, . . . . . |0.0834||Steel, . . . . . 70 Coal, anthracite|| 19.25||Lint, . . . . . . |0.0846||Lead, . . . . . 60 Coal, bitum., . | 16.84|Sewing-silk, . . . 0.0955||Glass, . . . . . 10 Coal, charred, 0.738||Flannel, . . . . |0.395 ||Glass, oiled or waxed, 5 Coke, . . . . 19.80|Horse-hair, . . . . [0.304 ||Lampblack, . . . () Miscellaneous Temperatures. Fahr. - Fahr. In the Bessemer furnace, . | 40009||1 alcohol, 1 water freezes, . |— 79 Puddling-furnace, tº tº 3500 || Mean temp. of the poles, . — 13 Cupola, . g g . 3000 || Temp. outside atmosphere, . – 58 Heat of common fire, . . 1100 || Greatest natural cold, . . . — 56 Red heat in daylight, e . 1070 || Winous fermentation, . |— 65 Iron red in dark, . . . 752 || Acetous fermentation begins, — 78 Mean temp. of the earth, g 50 || Acetification ends, . . . . – 88 66 & “ torrid zone, 75 || Phosphorus burns, gº º — 43 “ {& “ temp. “ 50 || Greatest artificial cold produced, -166 st st “ polar region, 20 At 50°, Miztures of - . Temp, of ignition, tº ſº 636 || Nitrate of ammonia, g } º Highest temp. of wind, . tº 117 || Water, . . e g te Temp. of the human blood, 98 || Sulphate of soda, . Q 8 60 A comfortable room, • e 70 || Muriatic acid, . • - 5 Mean temp. of ocean, . e 62 || Dilate sulphuric acid, ... 5 23 Snow, tº º . . 4 AIR AND HEAT. - 387 346s- - s: R3.<\;….º. ** * : * > .oO 2.6.3 G. * - - sº tº a tº |*::= ******ON AIR AND HEAT. esses Dry air expands or contracts uniformly 0.0020825 its volume per degree Fahr. in difference of temperature, or 0.0037485-perúčgree Centigrade under constant | pressure. Assuming the expansion per degree Fahr. as unit, the primitive volume | will be- R&vkº.g. --------- = 4-732. Reawolvul P Y. .9 C & & 2.7 & - * : - - - — = 48 ce 2;...&iled tºw-o,366s 0.0020825 * W and v = volumes of dry air of temperatures t and T. Fahr., and pressure p and | P above vacuum. The volumes and pressures in the following formulas may be expressed in any units of measures. * Volume and Temperature Variable under Constant Pressure. T—t - 480 (V-1) v=o ( 1) and (T- t) = −4. 1. * (*in Fl), (T– t) v ' Example 1. v = 18 cubic feet of air, of t = 36°, is to be heated to T-849 under constant pressure. Required, the volume V: V= 18 (*** +1) = 19.8 cubic feet, the answer. volume and Pressure variable under Constant Temperature. (T'—t) = 0, the pressurcs will be- | P 7. º #=+, and # =#, 2. y p P=p-, and Q) = V+, 3. Pacample 2. V+= 150 cubic feet of air, of pressure p = 1475 pounds to the square inch, is to be compressed to 50 cubic feet. The heat generated in the compression to radiate through the vessel until the temperature of the compressed air is equal to that in V. Required, the pressure P : - Formula 3 P-p+=1475× *: = 44.25 lbs. - Q) When the Temperature, Volume and Pressure are all variable, we have— {{# ) (; ) = p —ſ -— —– 1 d (T-t) = 480 ſt–º — 1 e P=p +(−1), and (T-0 p V / 4 It must be distinctly understood in all these formulas that the volume v belongs to the pressure P and temperature T, and the volume V to p and t. The prim- itive quantities are v, P. T. It may happen in the Formula 4 that v × V, t > T', and p) > P. Example 3. V = 1000 cubic inches of air of p = 14.75 and t = 590, is to be reduced to v = 320 cubic inches, and the temperature increased to T-369°. Required, the pressure P per square inch Le 1000 / 369–59 * Formula. 4. Pi— 14.75 #( sºmºss-s-s-s-s-sºus smºs 1) = 75.8 lbs. & - 320 480 - Eacample 4. ... v = 290, P= 88.5 to be increased to V = 838, and p = 18.4. Re- quired, the difference of temperature (T-t) * 88.5 × 290 Formula. 4. ( T-t ) = 480 — 1 = 320°. 'ormula 4. (T-t) (*** ) 320 $88 - - . . AIR AND Hear. On the Compression and Expansion of a definite weight of air enclosed in a vessel. In this treatment no heat must be lost or gained by radiation from the sides of the vessel in which the air is enclosed. Let D and d represent the degrees of absolute temperatures of volumes v and V of the air to be experimented upon. | The absolute zero is 461° below Fahr. Zero, and 274° Cent, below the freezing- point of water. D = 461 + T., d = 461 + t, and D – d = T–t, Fahr. scale. - Volume and Temperature. . . y (#)" Q) ( d y- - – E - || - d - - — te I-UT) , an V : \D 5 | º D\2.45 . d \2.45 Expansion V-v (#) , Compression v = (#) t \/ 6. - 2.45 ITTF - 2.45 ITAT |Compression D =iº 5 Lxpansion d-D + 7. Q) - - Example 5. To what fraction must air of t = 65° be compressed, in order to fire | tinder at a temperature of T-550°, d = 461 + 65 = 526°, D = 550 + 461 = 101.19% 2.45 IFormula 5. + - (# ) = 0.20, the answer. Example 6. How much must air of T-80° be expanded to reduce the temper- |ature to t = 32°, or freezing-point of water? - 2.45 Formula 5. *= (#) = 1.3308 times, the answer. v \493 Example 7. v =360 cubic inches of air of temperature T– 380°, or D =841°, is to be expanded until the temperature becomes t = 80° or d = 5419. Required, the volume V, corresponding to that temperature? 821 \2:45 Formula, 6. V = 360 (#) = 1025.9 cubic feet. Example 8. W= 20 cubic feet of air of t = 32°, or d = 493, is to be compressed. | to v = 12 cubic feet. Required, the temperature T'of compression ? 2.45 ; ) Formula. 7. D=493 N; =60729°, or T– 146.29°. Pressure and Temperature. 3-42 3.42 + =(?)" and #-(?)" s. p d P d • ID 3.42 • Q \3.42 Compression P=p (#) * Expansion p = P ( #) - *-*. 3.42 p - 3.42 TD' - Compression D= d N# , Expansion d-D w/Hº- 10. p P Example 9. A volume of air of pressure p = 15 pounds to the square inch, and of temperature t = 62°, is to be compressed until the temperature becomes T=120°. Required, the pressure P per square inch at T=120°: d=461–1–62=523, and D=461-I-120 = 581. 3.42 . Formula 9. P=15 (#) = 21.49 lbs. pr. sq. inch. ---.... . . . . . . . . . . . . 889 –––.---------…. Ars and Heat........ Jºrample 10. . A volume of air of pressure P=45 pounds to the square inch, and of temperature 4–250°, or D=711°, is to be expanded to a pressure of | p=25 pounds. Required, the temperature t of the expanded air? - - - ... 3.42ſ25. * . . . . Formula 10. d = 711 N # =59372, and t=598.72–461=137.72°, the temperature required. TE’ressure and Volume. Nº _*Ip. and Ji- Vá 11 P. Al Q) V P 'V V A P * 14|P e - 14 p. - Expansion V-E a N —, Compression v = E. : : 12. p . V \14 º Q) \l-4 Compression P= p(#) , Expansion p = F(#) . 13. Example 11. A volume v = 50 cubic inches, and of pressure P= 80 pounds per Square inch, is to be expanded until the pressure becomes p = 15 pounds. Required, the expanded volume V: . 1.4; ºf I'ormula 12. W= wV #. = 165 cubic inches. Example 12. What will be the pressure of a volume of air expanded 1.3308 times? - le+ Formula 13. p == (Hºw) =0.5324 of the primitive pressure. 1.3308 - Volume and Weight of Dry Air At different Temperatures, under a constant Atmospheric Pressure of 29.92 inches . in the Barometer, the Volume at 32° Fahr. being the unit. Temp. : Wt. per || Temp. Wt, per || Temp. . . . Wt. per. Volume. Cub. ft. Volume. Cub. ft. . Wolume. Cub. ft. Fahr. Pounds. || Fahr. | Pounds. || Fahr. - Pounds. Oo .935 | .0864 1629 1.265 .0368 550° 2,056 .0384 12 . .960 ,0842 172. 1.425 .0628 . 600 2.150 .0376 22 .9S0 .0824 182 I.306 .0618 650 2.260 . . .0357 32 I.000 .0807 192 I.326 .0609 '700 2.362 .033S 42 1,020 ,0791 202 1.347 .0600 800 2,566 .0315 52 1.041 .0776 212 1,367 .0591 900 2,770 .0292 62 1,061 .0761 230 1.404 .0575 1000 || 2.974 | .026S 72 1.082 .0747 250 1.444 ,0559 1100 3.177 .0254 82 1.102 .0733 275 1.495 .0540 1200. 3.3S1 .0239 92 1,122 .07.20 300 1.546 .0522 1500 3.993 .0202 102 1.143 .0707 325 1.597 .0506 1800 4.605 .01.75 I12 1.163 .0694 350 1.648 .0490 2000. 5.012 .0161 122 1.184 .0682 375 1.689 .047'ſ 2200 5,420 .014.9 132 1.204 .0671 400 I.750 .0461 2500 6.032 .01.33 # 142 | 1.224 || 0659 450 | 1.852 | .0436 2800 || 6,644 | .01.21 : 15 1.245 || 0649 || 500 1.954 ,0413 3000 || 7,051 .0114 For weight and Volume of air at Low Temperature, see Hygrometry, page 357. 390 ; - SPECIFIC HEAT. SPECIFIC HEAT OR GAL0 RIC. Different bodies require different quantities of heat for equal difference in tem- perature. This difference is called specific heat. The specific heat of bodies varies nearly inverse as the specific gravity. The specific heat in all bodies increases slightly with the elevation of temperature. One pound of water elevated from 32° to 212° requires 180.9 units of heat instead of 180. The specific heat increases nearly in the same ratio for all solid and liquid bodies. The specific heat of water from 32° to T'9 will be— & T - 32)1.67 + 1167713 The specific heat of water between any temperatures T and t will be— *-º-º-º: I-67 — sºmsº 1.67 s=1+4. 32) (t — 32) 2. 1167713 The following table gives the specific heat of different substances between the temperature 32° and 212°, compared with water as unit. When the specific heat of a body is required between high temperatures, it is necessary to calculate first the specific heat of water between such temperatures, which multiplied by the number in the table will give the required specific heat of the body. Specific Heat or Caloric of Substances. Water, . . . . 1,000 || Lead, . . . . 0.030 || Sweet oil, . . . . 0.310 Ice of water, . 0.513 || Steel, . . . . . 0.118 || Oil of turpentine, 0.472 Cast iron, . . . 0.140 || Diamond, . . . 0.147 || Gases of constant Wrought iron, . 0.110 || Arsenic, . . . . . 0.081 wolwºme and wºnder | –, Cobalt, . . . . . 0.150 || Iodine, . . . . . 0.054 || almospheric pres- º Nickel, . . . . 0.103 || Sulphur, . . . . 0.200 || sure. ,2-57s Zinc, . . . . . . ; 0.093 || Glass-crystals, . . 0.193 || Oxygen, . . . . 0.230 Tin, . . . . . . 0.047 || Glass, common, 0.177 || Hydrogen, . . . 3.30 Antimony, . . . 0.051 || Woods, average, . 0.500 || Nitrogen, . . . . 0.275 Bismuth, . . . . . 0.030 || Brick, common, 0.200 || Carbonic acid, . . . 0.221 Tellurium, . . 0.091 || Firebrick, . . . . 0.220 || Carbonic oxide, . 0.288 Gold, . . . . . |0.029 || Coal, . . . . . . 0.261 || Olefiant gas, . . . 0.421 Silver, . . . . . 0.057 || Beeswax, . . . . . . 0.450 || Nitro-oxide, . . . 0.237 Platinum, . . . 0.034 || Alcohol, S. g. 0.81, 0.700 || Gas of oils, . . . 0.421 Brass, . . . . 0.094 || Sulphuric acid, . 0.335 || Sulp. hydrogen, . 0.242 Mercury, . . . . . 0.030 || Nitric acid, . . . 0.661 || Steam of atm. pr., 0.475 Let two different substances of known weight or volume and temperature be mixed together; the temperature of the mixture will dissolve the relative quantity of caloric in the ingredients. Mixture of the same Substances, W= weight or volume of a substance of temperature. T. w = weight or volume of a similar substance of temperature t. t = temperature of the mixture W+ w. We shall have— Copper, . . . . Q.994 Lime, burned, . 0.217 || Atmospheric air, . 9:250 | WT t t’(W-i-w) = WT-H wi, 3. t'=++---#: y 5. w(t’ — t) w(t/— t) / .* W=#-F#, T===44. 6. º Raw 8PECIFIC HEAT. 391 Eacample 1. Let W-4.62 cubic feet of water at T=150° be mixed with w = 5.43 cubic feet at t=46. Required, the temperature of the mixture t'= y_4.62×150°-H 543X46° 4.62 + 5.43 Ea-ample 2. How much water of T-1079 must be mixed with w = 27.3 gallon of t = 58°, the mixture of the water to be 75°t 27.3(75–58) W= −A-4. 107 – 7 = 97.6°, the answer. = 14.5 gallons. Mixture of IDifferent Substances. W and w expressed by weights only. S and s = specific caloric as given in the accompanying table. We shall have— WST-Hwa t —#/) = t’— e ^ =-1–––. e WS(T-t') =ws( t), 7. t WS-H-w8 9 t’(WS-H-ws)—w st w8(t’—t) T– . 8. = — 10. WS W S(T-t)’ Example 3. To what temperature must W- 20 pounds of cast iron be heated to raise w = 131 pounds of water of t = 54° to a temperature t'= 64°? T'- ? From the table we have s = 1, and S= 0.14. _64(20 × 0.14 + 131) — 131 × 1 × 54 *E* 20 × 0.14 the required temperature, supposing no vapor escapes from the water. If any chemical action takes place in the mixture, these formulas will not an- swer, because part of the sensible caloric may become latent, or latent caloric may be set free. JEcample 4. The temperature of 5 pounds of copper is to be elevated from 60° to 80°. How many calorics will be required 7 See table for copper 0.094 (80–60) = 1.88 calorics, the answer. T = 532°, -sº Specific Heat of Gases. When heat is applied to a constant volume of gas enclosed in a vessel, the specific heat of that gas increases as the square root of the pressure generated by the heat. _0.9585 When the volume, pressure and temperature are all variable, the specific heat of air will be— S iº 11. - WP º 12. g — tº #(#+1). Q) (#. + For any permanent gas of s = specific heat under atmospheric pressure, the specific heat under any other pressure and volume will be— 3.834s - VP /T-f *( #+1). 13. & 392 DYNAMICS AND UNITS OF HEAT AND WORK. The weight of 320 cubic reet of air at 590 is (see table) 0.076 × 320 = 24.32 pounds. The calorics consumed in the operation will be— Formula 13. h = 1.08 × 24.32 (369–59)= 1050.625 units. The specific heat of any other gas, under different volumes, pressures and tem- peratures, is equal to that of air multiplied by the specific heat in the table and divided by 0.25. The number of calorics h required to elevate the temperature of W pounds of gas from i9 to Tº, will be- h = S W(T—t). . . . . 13. When the pressure is constant, and the volume is increased by heat, and S= Specific heat of the primitive volume of air, then the calories will be- - —all-º- (*- h = S W(T 0+...( •), . . 14. in which the last term expresses the calorics expended in expanding the volume under the pressure P. Example 6. One cubie foot of air of t = 32° is enclosed in a cylinder of one Square foot area of piston, and under atmospheric pressure P= 14.75 pounds to the Square inch. Let heat be applied to the air until 7–51.1°, when the volume will be about double, the piston being well balanced to move with the constant pres- sure. Required, the number of calorics imparted to the air, and the heat ex- pended in moving the piston with the pressure 14.75 pounds. Formula 14. h = 0.25 × 0.0807 × (511–32) + #; # — 1 \ = 9.65 + 2.75 = 12.4 calorics, of which 2.75 was expended in moving the piston one foot. DYNAMICS AND UNITS OF HEAT AND WORK. The ordinary English unit of heat is that required to elevate the temperature of one pound of distilled water one degree Fahr. from 39° to 40° (Fahr. ib.), and called one calor:c. The German unit of heat is that required to elevate the temperature of one pound (German pfund) of water one degree Centigrade, from 4° to 5° (Cent. Íb.) or (Cent. pfd.) The French unit of heat (called calorie) is that required to elevate the tempera- §: Of * (2.2047 fos.) of water one degree Centigrade from 4° to 5° Cent, kilo.). A combination of French and English units of heat is sometimes expressed by Fahr. scale and French weight (Fahr. kilo.). Heat is dynamic work, or the product of the three simple elements, force, velocity and time, in which the temperature of the heat represents force, and the cubic contents of the units of heat represent the product of time and velocity, which is space. The English unit of dynamic work is one pound raised one foot, called footpound (Ft. ib.). One caloric = 772 ft. fos. The French unit of dynamic work is one kilogramme raised one metre, called kilomet. One horse-power will consume or generate 2564 calorics per hour. Comparison of Different Umits of Heat and Work. English Calorics. French Calorie. Prussian. Dynamic Work. Fahr. Ib. Cent, fº. N'abr, kilo. 1 Cent. kilo. Cent, pfd. Ft. Ib. Kilomet. I 0.5555 0.4536 0.2520 0.5769 772 T06.51 1.8 T. 0.8165 0.4536 1.0385 I389.6 I91.71 2.2047 1.2248 I 0.555 1.2719 1702 348.066 3.968 2.2047 1.8 I. 2.2894 3063.6 626.52 1,733 0.9630 0.7862 0.4368 l 1368.3 273.66 .0012953 .0007196 | .0005876 . .0003264 || 0007473 i 0.13825 ,0093896 | .0005205 | .0004250 | .0002361 | .0005405 7.233 i cruns AND GUNPOWDER. --- 393 Performamce, Weight and Dimensions of Heavy Ordnance. Diam. Length | Weight in Pounds of DESCRIPTION. Of Of Bore. Gun. Gun. |Proj’tile Powd. Welocity Bore. Inchest Ft. in. | Pounds. | Pounds. Pounds | Ft. per sec American, rifle, . . 44 7' 4" 3,089 | . . . . . . . . . . . Rifle. {{ “ . . 6 9' 6// 7,970 | . . . . . . . . . . . . . Rifle. Eng., wrought iron, 8 9' 10" || 14,560 I80 30 1324 |Smooth. American, cast iron, 9 : 10/ 9,084 . . . . . . . . . . $3 English, {{ 10 11/ 40,320 400 60 1298 $6 American, £6 11 || 11’ 6” 16,511 . . . . . . . . . . . &ć Russian, ££ 11 11’ 6” 55,800 496 82.5 1362 (g English, {{ 12 || 12' 55,800 600 67 1180 66 American, “ 13 13’ 3” | 16,511 . . . . . . . . . . . &6 tº {6 15 15’ 6” 42,100 | . . . . . . . . . . £6 {& {{ 20 20' 3" | 115,000 936 120 | 1131 {{ {{ mortar, . 13 2’ 10” 17,198 | . . . . . . . . . . {{ Rus. brass, Moscow, 30 || 25/ 80,000 || 3000 * & J C & E. {{ Effect of Gunpowder. The dynamic work of different kinds of gunpowder, utilized in heavy ordnance, varies between 150,000 and 200,000 foot-pounds per pound of powder. Let K de- note the dynamic work in a charge of powder, W = weight in pounds of the ball or projectile, V = velocity of the projectile in feet per second, then JR 64 K WV2 W= 8vº, W= +, and JK = * > W V2 64 The length of the gun for these formulas should be at least 12 times the bore. Force of Gunpowder. The force of gunpowder depends much upon its quickness of burning and resist- ance to its expansion. Gunpowder enclosed in a strong vessel, and burned in its primitive volume, may reach a pressure of 100 tons to the square inch ; but when the gas of powder is subjected to an excessive pressure, it gets cracked, as it were, and loses the property of expansion due to a permanent gas less strained. This is a very important fact in the use of heavy ordnance, where the gun may be double strained with a loss of effect in the projectile. - Quick powder may strain a gun over 30 tons to the square inch, whilst slower powder will strain it only 15 tons, and give a greater velocity to the projectile. It appears that the charge ought to be so arranged in a gun that a slow powder is first ignited, and then a quicker and quicker until the quickest at last, by which the gun need not be strained to more than 15 tons to the square inch, with full benefit of the expansion property of the gas, greater velocity of the projectile and less risk in bursting the gun. The work done by the gas of powder in a gun should be treated under the same laws as that of steam in a steam cylinder. This special subject is too extensive for proper treatment in this Pocket-book. Composition of Gunpowder. The composition of gunpowder varies in all proportions between the limits of 70 and 78 parts of saltpetre (nitrate of potash, RO, N05), 13 and 15 parts of charcoal, 9 and 20 parts of sulphur in 100 parts of the powder. Chinese powder, 62 saltpetre, 23 charcoal, 15 sulphur. Thé different proportions depend much upon the purpose for which the powder is used, and also upon the ideas and experience of the manufacturers and users of the powder. The quickest powder requires the highest proportions of saltpetre. Size of Gunpowder-grains. USE OF POWDER. SIZE OF SIEVE. SIZE IN INCHES. COMPOSITION Sporting, . . . No. 1 to 2. 0.03 to 0.06 77.13.10 Mortar, tº te No. 2 to 3. 0.06 to 0.1 76.13.11 Cannon, . wº tº No. 4 to 5. 0.25 to 0.35 75.13.12 Mann.nnoth, . g No. 6 to 7. 0.6 to 0.9 74.14.12 Fine gunpowder is also moulded into lumps to fit the chamber of the gun. The Russians mould fine powder into hexagon blocks for heavy ordnance. B94 PROPERTIES OF WATER AND STEAM. PROPERTIES OF WATER AND STEAM, * In Relation to Heat. The following six pages of tables for water and steam have been calculated by the author whilst stationed in the Bureau of Steam Engineering of the United States Navy Department, under the direction of Chief Engineer Isherwood. The tables have been improved for this Pocket-book. Properties of Water. Column h’ contains the calorics required to raise each cubic foot of distilled water from 32° to temperature T, under the pressure P. Column h contains the calorics required to raise each pound of water from 329 to T'9. This column is calculated from the formula deduced from Regnault's experiments, namely: 9 — º, *. 2 * — ſy/- º, A-Tº-32°4′º", ("-ºr-º-º: 1. or h/ = T-t/ in which the last term is a parabola of exponent n = 2.67, and parameter p' 1167713. log. p = 6,0673350. h’ = calorics required per pound of water of temperature t', and raised to T'9. Column c contains the fractional cubic feet per pound of water of temperature T. Column w contains the weight in pounds per cubic foot of water of temperature T. Water of the maximum density at 39° weighs 62.388055 pounds per cubic foot. Column v contains the volume of water of temperature T, that at 399 being unit. This column is calculated from the Formula 2, deduced from Kopp's experi- mentS. gºsºsºme 2 v = 1 + (T – 39) ( . . 2. 2000000 [0.23 + 0.0007 (T-39)] Columm t contains the temperature of the steam and water, Celsius’ scale. Columns i and p give the steam-pressure indicated on the safety-valve or mercury-gauge. -- means pressure above the atmosphere. — means vacuum under the atmosphere. Properties of Steam. Column P contains the total steam-pressure in pounds per square inch, in- cluding the pressure of the atmosphere. Column I is the same pressure in inches of mercury, The specific gravity of mercury at 32°Fahr. is 13.5959, compared with water of maximum density at 39°. One cubic inch of mercury weighs 0.49086 pounds, of which a column of 29.9218 inches is a mean balance of the atmosphere, or 14.68757 ibs. per sq. in. Column T' contains the temperature of the steam on Fahr.'s scale, deduced from Regnault's experiments. Column V contains the volume of steam of the corresponding temperature T, compared with that of water of maximum density at 39° Fahr. This column is calculated from the formula of Fairbairn and Tate, namely: 49513 = 25.62 — . tº © ge 3. y + I + 0.72 Column W contains the weight per cubic foot in fractions of a pound; and Column C the cubic feet per pound of saturated steam under the pressure P and temperature T. Column H contains the calorics per pound of steam from 32° to temperature T and pressure P, calculated from the formula— EI = 1081.91 -- 0.305 T. . wº o . 4. Column H/ contains the calorics per cubic foot of steam from 32° to tempera- ture T. - PROPERTIES OF WATER AND STEAM. 395 The columns H and H/ give the calorics required to heat the water from 32° to the boiling-point and evaporate the same to steam under the pressure P and of temperature T. Column L contains the latent units of heat per pound in stcam of temperature T and pressure P. The latent heat expresses the work done in the evaporation, or the difference between the calorics per pound in the steam and in the water of the same temperature. Column LZ contains the latent heat per cubic foot of steam. Latent heat L = H – h, the calorics required to evaporate each pound of water from the boiling-point into steam. The maximum work K, which can be realized per caloric in steam without expansion, is— 144 P(V – 1) R = —--—-. º tº * > 5. * FIZ V Example 1. Required, the maximum work K = ? that can be realized per caloric in steam of P = 50 lbs. per sq. in.” V = 508.29 and H/ = 14.3.3. 144 × 50 (508.29–1) = 50.14 footpounds, 143.3 × 508.29 or, 50.14 : 772 = 0.0649 of the natural effect. The maximum work which can be realized per caloric in steam with expansion will be- * 144 P(V – 1)(2.3 log. ; + 1) R = HZ V y . . 6. in which S = stroke of piston, and l = part of the stroke with full steam. N K. The natural effect of a steam-engine in horse-power is - T550’ of which from 50 to 75 per cent. is realized in ordinary practice. N = number of calorics passed through the engine in the time r in seconds. JExample 2. Let the steam in Example 1 be expanded S: l = 3 times. We have log. 3 = 0.47712, and 2,09737 X 50.14 = 105.16 footpounds per caloric. Sup- pose each stroke of the piston to use 4 cubic feet of steam expanded 3 times, and making 90 strokes per minute. Then ºxº = 439.8 calorics per second, and the power will be 439.8 × 105.16 = 84 horses. 1 X 550 This is the effect of steam when raised from water of 32°, but when the feed- water is of higher temperature, calculate the calorics from the Formula I, h’, and add the latent heat per pound of the steam; the sum will be the calorics required in generating the steam. The late Professor Alexander, of Baltimore, gave a very simple and clear formula for temperature and pressure of steam, which may be as reliable as the experiments of Regnault, from which it differs very little, namely: T 990 \6 6.-- 180 1695 The inches of mercury IX 0.48875 = pressure in pounds per square inch- T = temperature of the steam, Fahr. The pressure in atmospheres A will be— T 561.91\6 6,-- - - – —H – d T = 317.13 1/A — 105.13. A. (aſſ + #) , an 1% 3 396 PROPERTIES OF WATER FROM FREEZING TO BOILING POINT, Temp. Wolume Units of heat. IPounds Culic ft. Fahr. | 1 at 39° pr. 1b. pr. cub. ft.) pr. cub. ft. pr. 1b. To Q) h h/ Q1) C 32 | 1.000109 || 0-000000000; 0.00000 || 62.387 0.01603046 33 1-000077 1-000000867 62.383 62-383 0.01602994 34 || 1:000055 2:000000545| 124-77 62.384 0.01602956 35 | 1:000035 || 3:00001609 || 187°16 62-385871 0.01602927 36 | 1.000020 || 4:00003468 249-55 62-386791 || 0-01602904 37 || 1:000009 || 5:00006294 || 311-99 62-387493 || 0-01602886 38 || 1.000002 || 6-00010241 || 374-33 62-387.930 || 0.0160287.4 39 || 1:000000 || 7-00015455 43672 62-388055 0.01602871 40 || 1:000002 || 8.00022076 || 499-12 62-387.930 || 0-0160287.4 41 1-000009 || 9-00030234 || 561-51 62-387493 || 0-01602886 42 1-000019 || 10-00040056 623-89 62-386869 || 0.01602902 43 || 1:000034 || 11.00051663 | 686-28 62-385933 0.01602926 44 1-000053 | 12.00065175 || 748.66 62.584.748 || 0-01602956 45 1-000077 | 13,00080704 811-03 62-383251 || 0-01602994 46 || 1-000104 || 14.00098362 873°40 62-3S1567 || 0-016.03038 47 || 1-000136 | 15:00.1326 935-70 62-3795.71 || 0-01603088 48 || 1.000171 | 16-0014050 997.77 62.377388 || 0.01603.146 49 1.000211 17:0016518 1060-0 62-374893 || 0-01603210 50 | 1.000254 | 18-0019242 1122-8 62-372212 || 0-01603278 51 | 1,000302 19.0022230 1185-1 62-3692.19 0.01603355 52 | 1.000353 20.0025,493 1248-0 62-366039 || 0.01603437 53 | 1.000408 || 21.0029241 1310-1 62-362611 0.01603525 54 1-000468 22:0032880 1372-3 62.358871 0.01603621 55 | 1.000531 || 23-0037024 1434-3 62-354944 0.01603723 56 1-000597 || 24-00414.79 1496'4 62.350831 || 0-01603828 57 | 1.000668 25-0046256 155S-6 62-346.407 || 0-01603942 58 1-000740 26'005.1362 1620.9 62-341921 || 0-01604057 59 | 1.000S19 27.0056808 1683-2 62-337000 || 0-01604.184 60 | 1.000901 || 28.0062600 1745-5 62-331893 || 0-01604.316 61 | 1.000986 || 29-0068749 1807-8 62,326620 || 0-01604451 62 | 1-001075 || 30.0075.263 1870-1 62.321059 0.01604594 63 | 1.001167 || 31°0082149 1932°4 62-315333 || 0-01604741 64 1-001:262 || 32-0089416 1994.4 62-309420 || 0-01604894 65 || 1:00.1362 || 33-0097073 20566 62-303,198 || 0-01605054 66 || 1-001464 || 34°010513 2118.7 62.296852 0.01605218 67 || 1:00.1570 35.011359 2180-8 62.290259 || 0.01605.388 68 1-001680 || 36.012246 2242-9 62.2834.18 || 0-01605564 69 || 1-001793 37-013175 2305-0 62°276293 || 0-01605748 70 1-001909 || 38'014148 2367-1 62-269183 || 0-01605921 71 || 1:002028 39-015164 2429-2 62-261788 0.01606122 72 | 1-002151 | 40-016224 2491°2 62.2541-16 || 0-01606318 73 || 1:00.2277 || 41:017330 2553-2 62°246.320 || 0-01606521 74 1-002.406 || 42-018482 2615.2 62°23S309 || 0-016067.28 75 | 1.002539 43.019680 2677-1 62°230052 0.01606941 76 || 1:00.2675 44,020926 2739-2 62-221612 || 0.01607158 77 | 1-002814 || 45°022220 2801-0 62.212987 || 0-01607382 78 1.002956 || 46-023563 2862-8 62.204.179 || 0.01607610 79 || 1:003101 |47.024956 2924'6 62-195187 || 0-0160784.1 80 | I-0032.49 || 48-026398 2985-4 62-186012 || 0-01608078 81 | 1.003400 || 49.027893 3048.2 62-176654 0.01608321 82 | 1.003554 || 50-029.438 31.11-0 62-167113 || 0-01608567 83 || 1-003711 || 51-031039 3172-8 62-157388 0.0160SS20 84 1-003872 || 52-032688 3234'4 62.147.420 || 0-0160907'ſ 85 || 1:004035 | 53-034394 3296.2 62-137330 0.01609338 86 || 1:004.199 || 54-036154 3358-2 62-127182 0.01609601 87 1-0043.70 || 55-037969 34.18.7 62-116605 || 0-01609875 88 || 12004542 56-03984.1 3480:4 62-105969 || 0-016.10151. 89 1-004717 57.04.1769 3542-1 62-095152 || 0-01610.432 90 || 1:004894 | 58.043754 3603-8 62-084.214 || 0-016.10715 Temp. Celsius. t 0.000 PROPERTIES OF WATER FROM FREEZING TO BoILINg PoſNT. 397 Temp. Volume |Units of heat. Pounds Cubic ft. Temp. Fahr. 1 at 390 pr. 1b. pr. cub. ft.) pr. cub. ft.) pr. 1b. Celsius. To ºy h A/ QU C t 91 || 1:005094 59.045797 3665-0 || 62-071860 || 0-016.11036 32-777 92 || 1:00:5258 60-047899 3726.6 62°0617:34 || 0-01611298 33-333 93 || 1:005444 61*050061 3788-2 62-050252 0.01611597 33-888 94 | 1.005633 62-052282 3849-8 || 62-038591 || 0-016I1900 34°444 95 || 1:005825 63.054564 39.11.2 62-0267.49 || 0-01612208 35'000 96 || 1:006019 64-056907 3972.6 62-014787 || 0-016125:9 35-555 97 1°006216 65'059312 4033-9 || 62-002646 || 0-01612834 36°111 98 || 1:006415 66.06.1780 4095°2 61°990386 || 0-01613153 36-666 99 || 1:006618 67°064311 4156°5 61°977885 || 0-01613478 37-222 100 1-006822 68-066906 4217-7 61°965322 || 0-01613S06 7-777 101 || 1:007030 69-069565 4278-9 61,952528 || 0:01.614140 38-333 102 || 1:007240 70'072290 4340-1 61-93961.2 : 0.01614.475 3S-888 103 || 1:007553 71°075080 4.401-3 61-920370 0.01614977 39:444 104 || 1.007668 72:07.7937 4462.5 61-913303 0.0161516.1 40’000 105 || 1:007905 73-080861 4523-0 || 61.8987.45 || 0-01615541 40°555 106 || 1:008106 74.083852 4585-0 61°886403 || 0-016.15863 41*111 107 || 1:008328 75-086912 4645'9 61-87.2778 || 0-01616220 41-666 10S 1°00'S554 76°090044 4706-8 61-858913 || 0-016.165S1 42-222 109 || 1:008781 77.093239 4767-7 61-844994 || 0-01616946 42-777 110 || 1:009032 78-096509 4828-6 || 61.829609 || 0-01617348 43.333 111 || 1-009244 79,0998.46 4889-5 61°816622 || 0-01617677 43-888 112 || 1-0094.79 80-103255 4950'4 61°802231 || 0-01.618064 44'444 113 1-0097.18 81-10674 5011°3 61-787.602 || 0-01618.447 45°000 114 | 1.009956 82-11029 5072°2 61-773042 || 0-01618829 45°555 115 1-010197 83-11392 5133-0 61-75S305 || 0-01619216 46-111 116 || 1:0104.42 | 84*11762 || 5193.7 || 61-743331 0.01619608 || 46-666 117 || 1:010688 85-12140 5254-3 61-72S302 || 0-01620003 47-222 118 1.010938 86-12525 5314.9 61°713037 0.01620403 47,777 II9 1-01.1189 87-12918 5375-5 61-697719 || 0-01620806 48°333 120 | 1:01.1442 88.13318 5436-1 || 61-682286 || 0-016.21211 48°888 121 || 1:01.1698 || 89-13726 || 5496.6 || 61-666678 || 0-01621621 49-444 122 || 1:01.1956 90° 14141 5557-1 61-650956 || 0-01622034 50-000 123 1-0.12216 91.14565 5617-6 61-635123 || 0-01622.451 50°555 124 || 1:012.478 92-14996 5678-1 61.6.19170 || 0-01622871 51*111 125 | 1:012743 93-15435 5738-6 61-603047 || 0-01623.296 51-666 126 1-013010 94-15882 5798-9 61-586S10 || 0-01623724 52.222 127 1*013278 95-16338 5859-2 61.5S0516 || 0-01624153 52-777 128 || 1:013550 96-16801 5919.5 61-553998 || 0-01624590 53°333 129 1*013823 97.17272 5979-7 61°5374.23 || 0-01625027 53°S88 130 || 1-014098 98.17752 6040:0 61-520735 | 0-01625468 54'444 131 || 1-014358 99-18239 6100-2 61-504966 || 0-01625.884 55.000 135 | 1-015505 || 103.20274 6340-3 61-435497 || 0-0162772.4 57-222 140 1-016962 || 108-23009 6639-6 61°347282 || 0-01630.064 60-000 145 || 1:018468 113°25965 6937.9 61.256765 || 0-01632473 62-777 150 | I*020021 118°29.147 7215-1 61-163500 || 0-01634961 65°555 155 1-021619 || 123-32562 753.1-2 : 61-067829 || 0-01637 523 | 68.333 160 | 1.023262 | 128-36217 || 78.26-2 || 60-969776 || 0-01640156 || 71*111 165 || 1:02.4947 | 133,401.19 8098-1 60°S695.42 || 0-01642857 73-888 170 J-026672 138-44273 8412-8 60-767270 || 0-01645623 76.666 I75 || 1:02843S | 1.43-48687 8704-2 || 60-662047 0.0164S477 79°444 180 1-030242 148-53666 8994-9 || 60-556699 || 0-01651345 82-222 185 i 1-032083 || 153-58316 928.1°9 || 60'448679 || 0-01654296 85.000 190 1-033960 15S-63545 9571-6 || 60-33S944 || 0-01657.305 || 87-777 195 || 1:035873 || 163-69057 9858-5 60-227513 || 0-01660370 90°555 200 | 1.037819 || 16S-74858 || 10318 60-1145S1 || 0-01663489 || 93,333 205 || 1,0397.98 || 173-80956 10428 60-000168 || 0-01666.662 96'111 210 | 1.04] 809 || 178-87355 10712 59-88.4350 0.01679885 98’SS8 212 || 1:04.2622 | 180-90000 18824 59.837654 || 0-01681160 | 100'000 g; + | 19'I6+ | 91%I | 6810"I II8'19 || 08LIO' 66"Z9% 80%g I ( 89 &6% +f + | #9'68 + | g LºffL | g810"I 8pS’19 || 6′ZLL()” 18°I92 || 65|LGI 8t"I6% £f + | IQ 18 + fggfl. 81.10"I 618'13 | SZ1.IO" IL'09% I609. 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IOf0'I 816'69 || 699 [0° 69'51.I 8680T | 1.1'90% 9 – | [[I’9 - || Ig'g1, Q890"I Z10°09 || 999.10° 19'01, I | 98IOI | 06’I0& † — | 6f I'8— || 09'81, 6990"I 69 L’O9 || 999L0° Zg'99I g166 09 16ſ g — 8L'OI — I j9‘Il 6990"I ZSZ'09 || 69.910° 18"I9'I 9916 0%’96. I 9 – ZZ'ZI — 6+'69 99.90°I 3.18'09 || 999 [0° 16°09'E | 1896 98’SSI 1 - || 93'fl. — I 60' 19 6090"I fLQ'09 || Zgglo' Zg"IGI 90Z6 96%8T 8 – 6% 9L – I gif"f 9 88&O'I 199°09 || Mig 10° 95'gi-I ZISS 16'91.I 6 – 38°SI — I ſh'I9 1930'I Z91'09 | ##910' 69'S9I IZīS 93'01, I Oſ - || 19.0% – || 0()’89 IfZO’I 036'09 | 899 IO’ 68'03T | f 161, IQ'Z9I II — If'ZZ — I 06'gg OIZ0"I TOI"I9 || 139TO’ 89'IZI | 1881, 18, 99T ZI — , fiºſº. — I # 1'85 fºLIO'I 118'19 || 089 IO’ I6'60'I £899 19. If I 2I– | 8p'92, — 18"If 09:IO'I 989"I9 || 52910' 698"f6 Iggg TZ’9&I #I - ZQ'82, — I 98'09 I100"I 8f 8"I9 || LI9'10" Ogi’69 103? 99"IOI 3. 7 Q. Q?, 9 7| // J, ‘uſ ‘bs | Aſmoſom 'orgos of 3 48 || || 'quo 'qi Igd punod | "J º '9180S 'Id 'sql | Soubuſ | SmISI60 | E="40A | id 'sql "j ‘qū0 10d J0 ‘JúðI populoxo 'Souſºv ū0 Oum(0A qqā19N XIII.g. '100'ſ Jo Spuſ, ‘duigi *ssaid "oppur I *...toº AA. . **IGILVAA TO SQLIL?IGIdOºſcL 868 PROPERTIES OF WATER. Temp. Fahr, Scale. 293.66 2.94.73 295.78 296.82 297.84 298.85 299.85 301.81 302.77 303.72 304.69 305.60 306.52 303.35 309.22 310.11 Units of heat, 0 r per Qub, ft. pound. Itſ h 15265 26 ſ.10 15321 265.20 15377 266.27 15432 || 267.34 15485 268.39 15536 269.42 15588 270,45 15639 || 271.46 1569() || 272.46 15739 273.44 15789 74.42 15839 275.40 15888 276.35 15936 277.30 15983 || 278.22 16029 || 279.14 16075 280.07 16120 || 280.98 16165 281.87 16209 || 282.78 16254 283.66 16298 || 284.54 16342 285.41 16384 || 286.27 16426 287.12 16467 287.96 16507 || 288.80 16547 || 289.62 16587 290.44 16637 || 291.26 16677 292.06 16717 || 2:02.85 16756 293.65 16795 294.43 16834 || 295.2.1 16872 295.96 16910 || 296.75 16947 || 297.51 16984 29S.26 17020 299.01 17056 299.75 17092 || 300.50 17127 || 301.23 I'7162 301.95 1719.7 302.67 17231 || 303.38 17265 304.10 17299 || 304.80 17333 305.50 7366 || 306.19 17399 || 306.8S 17465 308.34 17497 || 308.91 17529 || 309.60 17688 || 312.87 17840 || 316.04 17993 319.12 18136 322.13 18278 325.06 18413 327.91 18549 330.75 Water. Bulk | Weight cub, ft. lbs. pr. per lb. cub.ft. C Q0 ,01731 57.786 ,01732 57.769 ,01733 57.742 ,01734 57.714 .01735 | 57.687 .01735 57.660 .01736 57.633 .01737 || 57.606 .01737 7.58() .01738 || 57.554 .01739 7.529 .01739 57.504 .01740 || 57.480 0.1741 || 57.456 .01741 57.432 0.1742 || 57.410 0.1743 57.388 0.1743 57.364 0.1744 57.344 01745 || 57.322 0.1745 || 57.300 0.1746 57.278 0.1746 || 57.254. 0.1747 || 57.232 0.1748 57.210 .01748 || 57.188 0.1749 57.166 0.1750 57.144 0.1751 || 57.122 .01752 57.101 .01752 57.080 .01753 57.059 0.1753 57.03S .01754 57.017 .01755 56.996 .01756 || 56.975 .01756 || 56.954 0.1757 || 56.933 .01757 56.912 .01758 56.891 0.1759 56.871 .01759 || 56.862 .01760 56,844 .01761 || 56.826 .01761 || 56.808 .01762 56.790 ()1763 56.772 .01763 || 56.754 .01764 56.735 .01765 56.716 .01765 56.699 .01767 56.664 .01768 56.647 .01769 56.629 .01772 56.549 O1775 || 56.469 .01778 56.3S9 .01781 || 56.309 .01784 56.220 .()1786 56.146 .01788 56.073 Wolume Wat.=1 at 399 7) 1.0794 1.0799 1.0804 1.0809 1.0814 T.08.20 1.0825 1.0830 1.0835 1.0840 1.0844 1.0849 1.0854 1.0859 1.0863 1.0867 1.0871 1.0875 1.0880 1.0884 1.()SSS 1.0892 1.0897 1.0901 1.0905 1.0909 1.0913 1.0918 1.0921 1.09.20 I.0929 1.0935 1.0937 Temp. Celsius Scale. t 145.37 145.96 146.54 147.12 147.69 148.25 148.80 149.34 149.89 150.43 150.95 151.48 152.00 152.51 153.01 153.51 154.01 154.50 154.99 155.48 155.95 .156.42 156.90 157.36 157.82 15S.28 I58.73 159.17 139.02 399 Indic. press. Atmos. excluded inches lbs. pr. mercury | 84. Im. Ž 19 + 93.71 + 46 + 95.75 | + 47 + 97.78 || + 48 + 39.82 + 49 + 101.8 + 50. + 103.9 || + 51 + 105.9 + 52 + 108.0 + 53 + 110.0 + 54 + 112.0 + 55 + 114.1 + 56 + 116.1 | + 57 + 118.1 | + 58 + 120.2 + 59 + 122.2 + 60 + 124.3 + 61 + 126.3 + 62 + 128.3 | + 63 + 130.4 + 64 + 132.4 + 65 + 134.4 + 66 + 136.5 + 67 + 138.5 + 68 + 140.5 + 69 + 142.6 + 70 -H 144.6 + 71 + 146.7 + 72 + 148.7 | + 73 + 150.7 -H 74 + 152.8 + 75 + 154.S | + 76 + 156.8 + 77 + 158.9 + 78 + 160.9 + 79 + 163.0 | + 80 + 165.0 + 81 + 167.0 + 82 + 169.1 + 83 + 171.1 + 84 + 173.1 | + 85 + 175.2 + 86 + 177.2 + 87 + 179.2 + 88 -H 181.3 + 89 + 183.3 + 90 + 185.4 + 91 + 187.4 || + 92 + 189.4 + 93 + 191.5 + 94 + 193.5 + 95 + 195.5 + 96 + 199.6 + 9S + 201.6 | + 99 + 203.7 + 100 + 213.9 || + 105 + 224.1 | + 110 + 234.2 + 115 + 244.4 + 120 + 254.6 + 125 + 264.8 + 130 + 275.0 | + 135 PROPERTIES OF STEAM, 300 Total pressure lbs. pr. inches Sºl, llì. mêT. P I I 2.037 2 || 4,074 3 6.11.1 4 8.149 5 10.1S 6 12.22 7 14.26 8 16.29 9 18.33 10 20.37 11 22.41 12 24.44 13 26,48 14 28.52 14.7 || 29.92 15 30.55 16 32.59 17 34.63 1S 36.67 19 38.71 20 40.74 21 42.78 22 44.82 23 46.85 24 48.89 25 50.93 26 52.97 27 55.00 28 57.04 29 59.08 30 61.11 31 63.15 32 65.19 33 67.23 34 ($9.26 35 71.30 36 73,34 37 75.38 38 77.41 39 79.45 40 81.49 41 S3.52 42 85.56 43 87.60 44 89.64 45 91.67 46 93.71 47 95.75 48 97.78 49 99.82 50 101.86 51 |103.90 52 |105,93 || > 53 107.97 54 |110.01 55 |112.04 56 |114.08 57 116.12 58 118.16 59 |120.19 60 |122.23 Temp, Fahr, Scale. T 101.36 126.21 141.67 153.27 162.51 170.25 176.97 182.96 188.36 193.20 197.60 201.90 205.77 209,55 212.00 5 1213.04 216.33 219.45 222.40 292.58 Wolumo Wat, =1 at 39° p 17983 10353 72S3.S 5608.4 4565.6 |. 3851.0 |. 3330.8 2935.1 l. 2624.0 2373.0 |. 2166.3 |. Steam. Weight lbs. pr. Gub, ft. Hy .00347 .00602 .00856 .01112 1993.0 .031: 1845.7 1718.9 1641.5 1608.6 1511.7 1426.2 |. 1349.8 1281.1 1219.7 . 1163.8 1112.9 1066.3 . 1023.6 984.23 . 947.86 914.14 |. 882.80 | .07 853.60 . 826.32 I.07 S00.79 |.07 766.83 . 754.31 l. 733.09 |. 713.08 i. 694.17 l. 676.27 ||. 659.31 |. 643.21 . 627.91 |. 613.34 |. 599.46 j. 586.23 . 573.58 . . 561.50 . 549.94 . 53S.S7 |. 528.25 . 518,07 ||.1. 508.29 . .l. 498.89 |. 489.85 . 481.15 472.77 |. 464.69 |. 456.90 449.38 44.2.12 l. 4'25.10 ! . 428.32 Bulk Qub. ft. pr. lb, C 2S8.21 165.94 116.75 89.895 73.180 61,742 53.388 47.046 42,059 38.037 34.723 31.945 29,584 27.551 26.311 25.784 24 230 22,859 21,636 20,539 19.550 18.654 17.838 17.092 16.407 15,776 15.193 14.652 14.150 13.682 13.245 12.835 12.451 12,090 11.750 11.429 11.127 10,840 10.568 10,310 10.064 9.8310 9.6086 9,3963 9.1938 9.0002 8.8149 8.6374 8.4673 8.3040 8.1472 7.9966 7.8517 7.7122 7,5779 7.4468 7.3236 7.2030 7.0866 6.97.41 6,8654 Units of heat, from 32° to T Latent pr. Total pr; pound, Aſ 1112.8 1120.4 1125.1 1128.7 1131.5 1133.8 1135.9 II37.7 1139.4 1140.8 L142.2 1143.5 1144.7 1145.8 1146.6 1146.9 1147.9 1148.8 11:49.7 1150.6 1151.4 1152.2 1153.0 1153.7 1164.7 1165.2 1166.0 1166.4 1166.8 1167.2 1167.6 1167.9 116S.4 116S.'ſ 1169.0 1169.4 1163.8 1170.1 1170.5 1170.8 1171.2 Gub. ft, Pſ 3.8614 6.7449 9.6308 12.551 15.456 18.156 20.846 24.176 27.083 29.980 32.895 35,791 38.691 41.581 43.571 44.476 47.328 50.248 53.138 56.011 58.894 61.758 64.637 67.503 70,367 73.410 76.074 78.913 81,772 84.604 87.444 90.166 93.121 95.861 98.782 101.48 104.38 107.19 109.98 112.79 115.59 118.39 121.17 123.95 126.74 129.51 132.29 135,07 137.83 140.69 143.30 146.08 148.85 151.63 154.48 157.02 159.74 162.45 165.15 167.84 170.58 pound. I, 1043.4 1026.0 1015.2 1007.1 908.18 cub, ft. I/ 3.6337 6.1165 8.6901 11.199 13.714 16.113 18.194 20.937 23.352 25.728 28.099 30.450 32.789 35.435 36.706 37.421 39.690 42,012 44,393 46.698 48.655 51.924 53.282 55.529 57.743 59.942 116.51 118.50 120.49 122.47 124.43 126,40 I2S,38 130.33 132,28 0 3 1 5 PROPERTIES OF STEAM. 401 Steams Press Total pressure | Tomp, Wolume IWeight | Bulk Units of heat, from 329 to T ob. at. lbs. pr. inches] Fahr. wat-i |lbs, pr. Cub. ft. Total pr. atent pr. lbs. pr. sq. in. mer. Scale. at 39° Gub.ft. pr. 1b. pound. flub, ft. pound. cub. SH. llì. P I T' V Wy O. H | HZ || I, L/ p 61 |124.27 1293.66 421.75 .14792 | 6.7601 || 1171.5 173.27 907.40 || 134.22 + 46 62 |126.30 |294.73 || 415.40 |.15018 6.6583 || 1171.8 175.96 |906.63 136.16|| + 47 63 |128.34 |295.78 |409.25 |.15244 6,5597 || 1172.1 || 178.65||905.87 | 138,09 || + 48 64 130.38 296.82 |403.29 |.15469 6.4642||1172.5 | 181.34|905.13 140.01 |-|- 49 65 |132.42 |297.84 397.51.15694 | 6.3715||1172.8 184.03 |904.39 || 141.93| + 50 66 |134.45|298.85 || 391.90 |.15919 || 6.2817 | 1173.1 | 186.72 |903.66 || 143.85 |+ 51 67 |136.49 |299.85 |386.47 |.16130 6.1994|1173.4 | 189.40 |902.94 || 145.64|+ 52 68 |138.53 |300.84 381.18 |.16366 6.1099 || 1173.7 192.07 |902.23 147.66 |-|- 53 69 |140.36 |301.81 376.06 |.16590 || 6.0277 || 1174.0 | 194.74 |901.53 || 149.56 || + 54 70 ||142.60 |302.77 371.07 .16812 5.9478 1174.3 197.42 900 84 151.45 || -- 55 71 ||144.64 |303.72 || 366.24].17035 | 5.8702 || 1174.6 200.08 900.15 153.34|+ 56 72 146.68 |304.69 || 361.53 .17256 5.7948 || 1174.9 202.74 | 899.46 155.21 |-|- 57 73 #148.72 |305.60 356.95 i.17478 || 5.7214 || 1175.1 205.40 | 898.79 || 157.09 || + 58 74 150.75 |306.52 352.49 |.17690 5.6500 1175.4 208.04 |898.13 | 158.88 || -- 59 75, 152.79 |307.42 |348.15 ||.T/919 5.5805 || 1175.8 210.67 $97.57 | 160.83 || + 60 76 |154.83 ||308.32 343.93||.18139 || 5.5129|1176.0 213.30 | 896.83 162.67 || + 61 77 |156.86 |309.22 || 339.81 |.18359 || 5.4468 1176.2 215.93|896.18 164.56 || + 62 78 158.90 || 310.11 || 335.80 .18578 || 5.3825 || 1176.5 218.56 | 895.54 || 166.37 || + 63 79 |160.94 || 310.99 || 331.89 .18797 || 5.3190 || 1176.8 221.19 894.92 | 168.22 + 64 80 |162.98 || 311.86 || 328.08 .19015 5.2588 || 1177.0 || 223.82 | 894.27 170.04 || + 65 81 165.01 || 312.72 324.37 .19233 5.1992 || 1177.3 226.44 | 893.65 171.87 + 66 82 |167.05 || 313.57 |320.74|.19451 5.1410 || 1177.6 229.06 | 893.03 || 173.70 || + 67 83 |169.09 314.42 || 317.201.19668 5.0843 || 1177.9 || 231.68 892.51 175.52|+ 68 84 |171.12 || 315.25 || 313.74 .19885 5.0289 1178.1 234.28 891.82 177.33 || + 69 85 173.16 316.08 310.36|.20101 || 4.9748 || 1178.3 236.89 | 891.22 179.14 || + 70 86 |175.20 |316.90 || 307.07 i.20317 || 4.92.19 1178.6 239.50 | 890.63 180.95 || + 71 87 |177.24 |317.71 || 303.85 / .20532 4.87.03 ſ 1178.8 ſ 242.10 | 890.04 || 182.75 || + 72 88 |179.27 ||318.51 |300.70 | .20747 || 4.8198||1179.1 244.69 |889.46 18453 || + 73 89 |181.31 |319.31 297.62 .20962 4.7704 || 1179.3 247.29 |888.88 || 186.33 || + 74 90 |183.35 |320.10 |294.61 |.21185 || 4.7222 1179.6 || 249.88 |888.31 | 188.12|+ 75 91 |185.38 |320.88 291.66|-21390 || 4.6750 || 1179.8 252.45 || SS7.74 189.88 || + 76 92 187.42 |321.66 288.78 |.21603 || 4.6288 1180.0 255.02 || 887.19 191.66 || + 77 93 189.46 |322.42 285.96 || .21816 || 4.5836||1180.3 257.5S | 886.63 193.43 || + 78 94 |191.50 |323.18 283.21|.22029 4,5394 || 1180.5 260.14 886.08 || 195.19 || + 79 95 |193.53 ||323.94 |280.501.22241 || 4.4961 || 1180.7 262.69 1885.53 196.94 | + 80 96 |195.57 |324.67 || 277.86 | .22463 4.4537 || 1180.9 || 265.23 SS5.00 || 198.71 || + 81 97 |197.61 |325.43 275.27 | .22672 4.4106 || 1181.2 267.77|| 884.45 200 49 |-|- 82 98 |199.65 || 3:6.17 | 272.73 .22875 4,3715 || 1181.4 270.30 S83.91 || 202.18 || + 83 99 |201.68 |326.90 270.24 .23085 4.33.16||1181.6 273.10 || 8S3.38 203.92 || + 84 100 |203.72 |327.63 267.80 .23296 || 4.2926 1181.9 275.52 882.85 205.67 || + 85 101 (205.76 |328.35 | 265,41 .23505 || 4-2543 1182.1 || 277.85 SS2.33 207.39|| + 86 102 |207.79 |329.07 263,07 ||.23715 || 4.2167 1182.3 280.38 || 8S1.81 | 209.12|+ 87 103 ||209.83 ||329.78 260.77|.23924 4.1799 || 1182.5 || 282.90 || 881.29 210 84 || + 8S 104 |211.87 ||330.48 || 258.52 i.24132 4.1438 || 1182.7 285.42 880,78 || 212.55 || + 89 105 |213.91 || 331.18 256-31 ||.24340 4.1083 || 1182.9 || 287.93|SS0.27 214.26 || + 90 106 |215.94 || 331.87 251.14 .24548 || 4.0736 || 1183.2 290.45|879.77 215.96 || + 91 107 |217.98 || 332.56 252.01 . .24750 4.0394 || 1183.4 || 292.94 || 879.27 217.66 + 92 108 ||220.02 || 333.24 249.92 | .24963 4.005S 1183.6 295.41 S79.79 219.36 + 93 109 |222.05 || 333.92 || 247.87 | .25169 || 3.9731 || LIS3.8 297.91 || 878.2S 221.05 + 94 I10 |224.10 || 334 59 245.86 .25375 || 3.9406 || 1183.9 |300.44 || 877.80 222.74 || + 95 111 |226.13 ||335.26 243.8S .25581 | 3.9091 || 1184.2 302.93|877.31 224.42 |-|- 96 113 |230.20 |336.58 || 240.03 |.25991 || 3.84.74 || 1184.6 307.90 || 876.25 227.74 || + 9S 114 |232.24 |337.23 || 238.15 .26204 || 3.S100 || 1184.8 || 310.36 |875.88 229.51 | + 99 115 |234.28 || 337.89 || 236.31|.26400 || 3.7878 1185.0 || 312.S6 875.40 231.10 || + 100 120 |244.4 |341.0 227.56 .27421 || 3.6475 1185.9 || 325.20 873.09 || 239.41 || + 105 125 |254.6 || $44.1 || 219.50 | .28422 3.5184 1186.9 337.39|870.85 247.51 || + 110 130 264.8 |347.1 || 212.07 .294.19 3.3991 || 1187.8 349.44 S68.6S 255.55 |-|- 115 125 1275.0 |350.0 | 205.18 .30406 || 3.2880 || 1188.7 361.42%. 866.56 263.48 || + 120 140 |285.2 |352.8 || 198.78 .31385 3.1862 1189.5 373.34 S64.49 271.32 |-|- 125 145 |295.4 |355.6 || 192.83 |.32354 || 3.0908 || 1190.4 3S5.20 $62.48 278.97 || + 130 150 la(\5.6 |358.4 187.261.33315 3.0001 || 1191.2 | 396.86 l860.45 286.66" + 135 102 * ICxpANSION OF STEAM. E X P A N S I O N OF ST. E. A. M. In order to save steam, or more correctly to employ its effect to a higher degree, the admittance of steam to the cylinder is shut off when the piston has moved a part of the stroke; from the cut-off point the steam acts ex- pansively with a decreased pressure on the piston, as represented by the accompanying figure. Let the steam be cut off at # of the stroke, and 4. ; dº Aa represent the total pressure, º 20 pounds A =T per square inch which will continue to the point == E where the admittance of steam is shut off at E= one-third the stroke S. The steam Aa eIF2, is now Ecº-E acting expansively on the piston, and the pres- sure decreases as the volume increases, when the piston has attained/CC or two-thirds of S, the == pressure C'c=10 pounds, only half the pressure = S Aa-20 because the volume Aa eB is only half of =x-Ectº-c Aa co, and so on until the piston has attained B b G|----- \;=ic the pressure Bºb-34 X20=666 pg.pnds. Cº ºf:= The mean pressure, or the effectual pressure, E throughout the stroke, will be about 13:33 pounds per square inch, or 66 per cent, but the quantity B - b of steam used is only 33 per cent., hence 33 per *1–1* cent. is gained by using the steam expansively. ! = part of the stroke S in feet, at which the steam is cut off. P= pressure per square inch under full admittance of steam. F= mean pressure per square inch throughout the stroke S. .j = mean pressure per square inch during the expansion, which in double expansion cylinder engines will be the average pressure per square inch on the large piston A. p = end pressure per square inch after expansion. S = stroke of the cylinder Piston in feet. Pl FS — Pl Pl E— 2. e & F - = - - e. P=#|280og s—log 0+1] f====, p=s The following Tables are calculated from these formulas. Eacample 1. Required from the Table I. the mean pressure F for P=32 lbs. at five-eights expansion. Ada (**** {-ººom the table i. Mean pressure of 32 lbs. F-23735 the answer. Ea'ample 2. Required from Table II, the mean pressure f, per square inch during the expansion, or on the large piston A in double cylinder engines, when the initial pressure P=75 lbs. and under two-thirds ex- pansion ? f=40-75 Table II. Eacample 3. Required the mean pressure f=? for an initial pressure P=43 lbs. under; ºpºlº f= 18.48 Or P = 40 lbs. = 18- * P = 30 or 3 lbs. f* #} Table II, P = 43 lbs. f= 1986 the answer. The effect gained or fuel saved by expansion and high steam is calculated from the following formulae, in which it is supposed as a unit the work of an engine with P=30 pounds per square inch, or an indicated pressure of 15 lbs, without expansion. c = per cent on 100, of effect gained or fuel saved. For expansion c = 100 (1–. ). For high steam c = 100 a *. The following Table III, is calculated from these formulae, in which the first line from 30 contains the economy per cent. from expansion alone, and the column o contains the economy per cent. from high steam above P=30 lbs. The balance of the table contains the jointed economy of expansion and high steam. Required the jointed economy of P=90 Åbs, under # expansion'ſ 50.5 per cent. the answer. ºxPANSION TABſ. E. I. MEAN PRESSURE F. Press. 1 11 0-9637 1-9275 2-8912 3.8550 4°8185 5-7822 6-7469 7-7106 S-6733 9-6370 10-601 | + | # 0-9333 1°8666 2.7999 3.7333 4-6666 5-5999 6'5334 7-5666 S-3999 9-3333 10-266 11°565 12°528 13°492 14°456 15°420 16-383 17-347 18-311 19-275 20-238 21-201 22°166 23-030 24°093 25-057 26-985 28-912 33.731 38°550 43-368 48-187 53-005 57-822 62-640 67-460 72-278 77-096 81-91.4 86-730 91°548 96-370 101-18 105-99 110-80 120°46 134-92 144°56 192.75 240-93 289°12 11-199 12-133 13.066 13-930 I4-933 15-866 16-799 17.733 18°666 19°599 20-532 21-466 22-399 23-333 24-266 26-233 27-999 32-666 37.333 42-000 46-666 51-333 55-999 60-666 65-333 69-999 75-666 80-333 83-999 8S-666 93-333 97-999 101-66 106-33 115-66 130-66 139-33 186-66 233-33 279-99 Grade of Expansions # 0-918.7 1-8375 2-7562 3.6750 4°5935 5-5122 6°4319 7.3396 8-2683 9-1870 10-106 10.925 11-943 12.862 13-781 14-700 15-618 16-537 17-448 I8-375 19-293 20-211 2I-131 22.050 22°968 23-887 25-714 27-562 32-156 36-750 41-341 45.937 50-530 55-122 59-715 64-300 6S-893 73.500 78-093 82°680 87-273 91-870 96'463 101-05 105-64 114°S3 128-62 137-81 1S3-75 229-68 275-62 # 0-84.65 1-6930 2-53.95 3-3860 4°23'25 5-0790 5°9255 6-7720 7-6185 8-4656 9-3115 10-158 I1 004 11.851 12-698 13’544 14-390 15-237 16'803 16-930 17-776 18-623 19-469 20-316 21-162 22:009 23-702 25°395 29' 627 33-860 38-092 42-325 46'557 50-790 55-022 59.255 G3-4S7 67-720 71-952 76-180 S0-412 84-650 88-882 93-120 9 352 105-81 118-51 126-47 169-30 211-62 253-95 # 0-74.17 1°4835 2-2252 2.9670 3-7085 4-4502 5-2419 5°9346 6-6753 7-4170 8-1597 8-9014 9-6421 10-384 11-126 11-268 12-609 13-531 14.093 14.835 15-576 16-318 17-060 17-802 18-573 19-285 20-769 22-252 25-961 29-670 33-378 37-067 40-775 44°520 48-228 52°419 56-127 59-340 63-048 66-750 70-458 74-170 77-878 81-586 85-294 102-S3 103-84 III-26 148°35 185*43 222°52 0-6991 1°3982 2-0873 2.7964 3°4955 4-1946 4-8937 5-5928 6°2919 6-9912 7-7001 8-3892 9-0.783 9.7874 10-486 IL-185 11.884 12°483 13-183 13-902 14-501 15-300 15-989 16-698 17:477 18-046 19-495 20-873 24°368 27-964 31-459 34°955 3S-450 41°946 45°441 48-937 52-432 55-928 59°423 62-919 66°414 69-910 73°405 76-900 80-395 87-387 97.874 104°S6 139-02 17477 208-73 0-5965 1-1930 1-7895 2-3860 2-9825 3.5790 4°1755 4-7720 5°3685 5-9650 6°5615 7-1580 7-7545 8'5310 8-9475 9°5440 10-140 10.737 11-333 11:930 12:526 13-123 13.720 14-316 14-912 15-509 16.702 17-895 20-877 23.860 26-842 29°825 32-S07 35-790 38-772 41-755 44.737 47-720 50-702 53-680 56-662 59-650 62-632 66"614 69°596 74-562 85-310 89.470 119-30 149-12 178-95 0.3848 0-7697 1*1546 1.5395 1°9240 2-2008 2°6946 3-0784 3°4632 3-84.80 4°2338 4°6186 5.0034 5-3882 5-7730 6-1582 6-5426 6-92.74 7-3122 7-6970 8-0818 8-4667 8-8516 9-4365 9-6210 9-8978 10-775 II*546 13°470 15-395 I7-319 19-243 21-167 23-U90 24-924 26-694 2S-626 30-790 32.714 34°638 36-554 38.480 40-404 42-328 44'252 48-101 53-882 57-730 76-970 96-210 115-46 404 ExPANsroN TABLE II, FoR Dotſple CYLINDER ExPANsion ENGINEs. Mean Pressure f during the Expansion. Pres, #"| # || # 30 || 28-549 24 23-50 20-79 17.60 | 16-31 || 13-86 8-9097 35 || 33-308 || 28 27°41 24-25 20-54 || 19°02 16-17 | 10-394 40 || 38.066 || 32 || 31.83 27-72 || 23-47 | 21-73 | 18-48 || 11-879 45 || 42.824 || 36 35°25 31°18 || 26-40 || 24°46 20-79 13-364 50 || 47-582 | 40 39°16 || 34-65 || 29-33 || 27-16 || 23°10 || 14-849 55 || 52°340 || 44 43-08 || 38-11 32-24 || 30-17 || 25° 41 || 16-334 60 || 57-098 || 48 47-00 || 41-58 || 35-20 | 32-62 || 27.72 || 17-819 65 || 61-853 52 50-91 || 45-04 || 3S-14 || 35-33 || 30-03 || 19°303 70 || 66-616 || 56 54'83 || 48-51 || 41-07 || 38-04 || 32-34 20-788 '75 || 71°371 60 58-75 || 51.90 || 44-00 | 40-75 || 34-65 22-263 80 || 76-128 64 62-66 || 55°44 || 46-94 || 43°47 || 36'96 || 23°758 85 || 80-885 | 68 66-18 58.90 || 49-87 46-19 || 39-27 || 25°243 90 || 86°448 72 70°50 | 62.37 || 52-80 || 48-93 || 41.58 26-729 95 || 90-391 || 76 74'41 || 65-73 || 55-73 || 51-62 || 43-89 || 28-213 100 || 95°160 80 78-33 || 69-30 58-66 54°33 46-20 29.699 I05 || 99-910 | 84 82°24 || 72-76 || 61-57 || 57-33 || 48-51 || 31-183 110 || 104°68 || 88 86°16 || 76-23 64-48 || 60°35 | 50-82 || 32°669 115 || 109-40 || 92 90-08 || 79-69 67°44 || 62.79 || 53-13 34-153 125 || 118-95 || 100 97.91 || 97.02 || 73-34 || 67-95 || 57-75 37-122 140 || 133°23 112 109-6 || 97.02 || 82°14 || 76-08 || 64.68 || 41'576 150 || 142-74 120 117.5 | 103.9 || 88.00 | 81:50 69;30 |44.548 200 || 190°32 | 160 156-6 || 138.6 || 117-3 || 108.6 92:40 59:398 250 || 2:37-07 || 200 195-7 || 173.2 || 146-6 || 135.8 || 115-5 74-247 3öö| 288-16 || 240 235-0 || 207.9 176-0 | 163.1 138-6 | 89-097 Table III. Economy of Expansion and high Steam. Fuel saved or effect gained per cent. ºil o || 3 || 3 || 3 || 3 || 3 | 8 | * | * 30 0 | 12 29-5 32 41 49°3 52 58 67.5 35 || 1:6 || 13-6 || 31 33-6 || 42°6 51 53-6 59-6 || 69°1 40 || 2:5 14.5 || 32 34°5 || 43’5 || 51.8 || 54-5 60-5 || 70 45 || 3-4 15°4 33 35-4 44°4 || 52*7 55-4 61-4 71 50 || 4.3 | 16-3 || 33-8 36-3 || 45°3 || 53°6 || 56°3 62-3 || 71-8 55 || 5-2 || 17-2 34-7 37-2 || 46-2 54°5 57-2 63-2 72-7 60 || 6 18 35-7 38 47- 55-3 || 58 64 73-5 65 || 6-7 | 18-7 || 36.2 38-7 47.7 || 56 58-7 64-7 || 74°2 70 || 7-3 || 19.3 36.8 39-3 || 48-3 || 56-6 || 59-3 65-3 || 74-8 75|| 7-8 | 19.8 37.3 39°8 || 48-8 57-1 || 59'8 65.8 75°3 80 || 8°5 20°5 38 40°5 || 49°5 57-8 || 60°5 66.5 || 76 85 || 9 2I 38°5 41 50 58-3 || 61 67 76.5 90|| 9-5 21.5 || 39 41-5 || 50-5 || 58-8 || 61-5 67.5 77 95||10 22 39°5 42 51 59-3 || 62 68 77.5 100||10-4 22°4 | 40 42.4 51.4 || 59-7 | 62-4 68°4 78 105 || 10-7 22-7 || 40-2 42-7 || 51.7 60° 62-7 68-7 || 78-2 115||11 23 40°5 43 52 60-3 || 63 69 78-5 125 || 11.7 23-7 || 41.2 43-7 || 52-7 || 61 63-7 69-7 || 79.2 150 || 14 26 43°5 46 55 63-3 | 66 72 81-5 200||16 28 45°5 48 57 65-3 || 68 74 83-5 250 || 17-7 || 29-7 || 46-2 49-7 || 58-7 || 67 69-7 75-7 || 85-2 300 ||19 31 48-5 51 60 68-3 || 71 77 86.5 CONSUMPTION OF FUEL. 40; Table IV. -- - -º- ºr ----- *--- T - - - - ---------- g Consumption of Coal in pounds per luorse power per hour, Grade of Expansion. *:::: o || 3 | | | 3 || || 3 || 3 | | | ****- - - Tibs, IITibs, Tibs, Ibs, TTIbs, lbs, lbs, lba, ibs, Tibe, T 30|| 5-6 || 4-93 3-95 3-81 || 3-30 || 2-84 2-69 || 2:35 1.82 35|| 5-5 || 484 || 3-86 || 3 72 || 3:21 || 2-74 || 2:60 2-26 |1-73 40|| 5-46|| 4-79 || 3-81 || 3-67 || 3-16 || 2.70 || 2:55 2.21 |1.68 45||5-41| 4-73 || 3-75 || 3-62 || 3-11 || 2.65 2.50 | 2-16 |1-62 50|| 5-36|| 4-68 || 3-71 3-57 || 3-06 || 2-60 || 2:45 || 2-11 |1-58 55||5-31; 4.63 3-66 || 3:51 || 3:01 || 2:55 || 2:40 2-06 |1-53 60|| 5-26 4'59 || 3:60 || 3-47 || 2.97 || 2.50 || 2:35 || 2:02 I-49 65|| 5-20 || 4-55 || 3:57 3-43 || 2-93 2-46 || 2:31 | 1.98 || 1:45 70||5-19| 4-52 || 3-54 || 3-40 || 2.90 || 2:43 2-28 || 1.94 |1°41 '75|| 5-16 || 4.49 3-5 1 || 3:37 || 2:87 2-40 || 2-25 || 1-91 |1-39 80|| 5-12] 4:45 || 3:47 || 3:33 || 2-83 || 2-36 || 2:21 | 1.88 |I-35 85||5-09| 4-42 || 3-44 || 3-30 || 2-80 || 2:33 || 2:18 | 1.85 |1:32 90|5-0|7| 4-39 || 3-41 3-28 || 2-77 2-31 || 2-16 il-82 1:29 95||5-04| 4:37 || 3:39 || 3-25 || 2.74 2-28 || 2:13 I-79 T-26 100|5-0|| 4:34 || 3:36 || 3:23 || 2:72 2°26 2 10 | 1.77 |I-23 105||5-00 || 4-32 || 3-35 | 3-21 || 2-70 2-24 2-09 | 1.75 |I'22 115|| 4-98; 4.3L 3-33 || 3-19 || 2-69 2-22 || 2:07 | 1.73 |1-20 125|| 4.94| 4-27 || 3-29 || 3-15 || 2.65 || 2:19 || 2-03 | 1.70 |1-17 150|| 4-81. 4-14 || 3-16 || 3:02 || 2:52 || 2:05 | 1.90 || 1:57 1-04 200|| 4-70 4-03 || 3-05 || 2.91 || 2:41 | 1.94 | 1.79 || 1:46 |0-92 250|| 4-60|| 3-93 || 3:01 || 2.81 || 2-31 | 1.85 | 1.70 | 1.36 |0-83 300|| 4.54|| 3-87 2.89 2.75 || 2:24 1-78 || 1-62 | 1.29 |0-75 Table V. Consumption of Coal in tons per 100 horses in 24 hours. * , T: I I # a lbs. tons tons, tons, tons, tons, tons, tons, tons, tons, 30|| 6-00 || 5-29 || 4-23 || 4-09 || 3:54 || 3-04 || 2-88 2.52 |1-95 35||5-90) 5-19 || 4-13 3-99 || 3°44 2.94 || 2-79 2.42 1.86 40||5-85| 5-13 || 4-08 || 3-93 || 3:39 2-90 2-73 ſ 237 |I-80 45|| 5-80 || 5-07 || 4.02 || 3-88 || 3-34 2.84 2.68 2-31 |1-73 50||5-75|| 5 01 || 3-97 || 3-83 || 3-28 || 2-79 2-63 || 2-26 I-69 55||5-70) 4-96 || 3-92 3-77 || 3-22 || 2-73 2-57 2-21 I-64 60||5-64) 4-92 || 3-87 || 3-72 || 3-18 2-68 2-52 2-17 | 1.60 65||5-58) 4-88 || 3-82 || 3:68 || 3-14 || 2-63 2-48 || 2:12 |1-55 70|| 5-56, 4.84 || 3-79 || 3-64 || 3-11 || 2:60 2.44 || 2:08 |1-51 '75|| 5-53| 4-81 3-76 || 3-61 || 3-07 || 2:57 || 2:41 || 2:05 || 1:49 80|5-49| 4-77 || 3-72 3•57 || 3:03 || 2:53 2.37 || 2:01 |I-44 85||5-46|| 4-74 || 3:69 || 3-54 || 3:00 || 2:50 || 2:33 | 1.98 || 1:41 90|| 5-43 4.70 || 3-66 || 3:51 || 3-97 || 2:47 2-31 | i-95 1-38 95|| 5-40 || 4-68 || 3-63 3-48 || 2.94 || 2:44 2-28 || 1'92 || 1:35 100|5-37| 4-65 || 3-60 || 3-46 || 2-93 || 2:42 || 2:26 I-90 1-32 105|| 5-36|| 4.63 || 3:59 || 3:44 || 2:89 2-40 || 2:24 I-88 1.31 115|| 5.34 4-61 || 3:57 || 3:42 || 2-88 2-38 2°22 || 1:85 | 1:29 125"| 5-30 || 4-58 3•53 || 3-38 || 2:84 || 2:34 2-18 1-82 1-25 150 5-16 4-44 || 3.39 || 3-34 || 2:S1 || 2:30 2-04 || I-68 I-11 200|5-04| 4:32 3-27 || 3-12 || 2:59 || 2:19 || 1-92 || 1:56 |0-99 250) 4-93 4-21 || 3-22 || 3:01 || 2:47 || 2:09 | 1.82 I 46 0-89 3öö|4.37 415 || 3:10 295 || 2:30 || 2:01 | 1.74 138 0-83 406 FORCE AND AIR PUMPS. Force or Feed Pumps, Letters denote, d= diameter] of the force-pump, single acting. s= stroke I) = * of the steam-cylinder piston, in inches, double acting, V= volume of steam given in the table at the given pressure. The stroke of the steam-piston is only that under which steam is fully admitted to the cylinder. S. D2 S. d F: 2D -—y F #–. º c e 4 5. V VS 7 º’ 3. Slip-water included in the formulas. Example. Required, the diameter of a force-pump having the same stroke as the cylinder piston s = 38 inches, diameter of cylinder D = 30 inches. The steam is cut off at # the stroke, and the steam pressure + 50 pounds per square inch. Here Y = 437, and S= 19 inches, because steam is cut off at # the stroke. d = 2 × 30 —º- = 2.03 inches. r 437 × 38 To find the Quantity of Comdemnsing Water. = condensing water of temp. t in cubic feet. 1.4Q(990 + T– t’) § = steam of temperature T in cubic feet. := —H·m−, . 6. t’= temperature in the condenser. (t’—t) Dimensions of the Air-Pump. d = diameter l of the air-pump, Roan TT, s = Stroke J single acting. d=23D, $999-E *-*). . . 7. D = diameter of the steam cylinder, V s(t’—t) 2 S= stroke double acting. Assume t'= 100°, and t = 50°, we shall have—- Single acting air-pumps. Double acting air-pumps. a-oxºſ, 8. a-wapVºſſ, 10. VS V’s S(940 –H 940 S = 0.106 D2 sº* 9. [ s = 0.053 D2 sºn, 11. Example. A single acting air-pump is to be constructed for an engine D = 38 inches, S = 45 inches stroke of the cylinder; the stroke of the air-pump can be 32 inches, and tho exhaust steam is 2619. Required, the diameter of the air- pump? V-767. d=0.826×38, 42949+*) = 18.25 inches. g 767 × 32 Bºy Slip-water included. T and V must be taken for the exhaust steam, as the steam may have worked expansively; the area of the foot valve must be calcu- lated from the following formulas. Foot Valve in the Air-Pump. To enable an air-pump to work well and with the greatest advantage, it is neces- sary to pay particular attention to the following formulas. The force by which the water is driven from the condenser through the foot valve into the air-pump is limited by the pressure in the condenser; this pressure is the vacuum sub- tracted from 14.7 pounds; it is noted in the third column, where the temperature in the condenser is opposite, in the first column. I’very pound of this pressure per square inch balances a column of water 27 inches high, which is the head that presses the water from the condenser. AIR-PUMP. * - 407 ºv- - Foot-Valves in Air-Pumps. 3 = area of the air-pump piston. a Dºsn (20+2). a = area of the foot-valve, or bucket-valve. 23000 m V y p JB = diameter of the air-pump-piston. 7m - 0°6 to 0°8 b = diameter of the foot-valve, when round. 5* = stroke of air-pump piston, in feet. 3} = pressure in the condenser at the temperature T. n = number of strokes of the air-pump piston per minute. 3 S m º TB VES n & = Ex-F, 12, U = −, 15 100 V3} 10 */39 2 _100a Vää s_100ü” M3) $ = Tº a 13, T Taij, ’ 16, n = 100: M3, 14, n = 100 "Yº, 17. It Øſ sis £32 S Example. The area 3 of an air-pump-piston is 2:35 square feet, stroke of . piston S = 3-6 feet, to make n = 40 strokes per minute, and the pressure to be 39 = 32 pounds. Required the area of the foot-valve. 2-35X3- F: #– 1-85 square feet. To Find the Velocity and Quantity of the Injection Water through the Adjustage into the Condem sere Letters denote. v = velocity in feet per second. h = head of the press water; + when above, and — below the adjustage. F= vacuum, noted — or negative in the last column, but is positive in the formulas. q = quantity of water discharged in cubic feet, per second. a = area of all the holes in the adjustage in square feet. # = * of the injection pipe, in feet. n = double strokes of cylinder-piston, or revolutions per minute. A, D, and S, dimensions of the steam cylinder, in feet. = temperature, and v = volume coefficient of the exhaust steam. – –– = 5a J2E-FM. . 21 •-5-ºry, 18, 7-5, v.2°E. 5 I, on • -8./3FE 19, d = 0.35 N/º, 22, n S D (940+T) ... n. S D (940+T) q = —557– 20, | 0. 275VV2FF}, ' 23, º p 408 STEALſ, Example; Required the diameter of an injection pipe L = 10 feet long, which shall supply q = 1:3 cubic feet of water per second into a vacuum of 12 pounds per square inch, the head of press water h = 2 feet? - \}* 5 Tox1-8 -*= *H = 8.I.L. i d = 0-35 2×12+2 T 0-3055 feet = 8. # inches. Area of Steam Passages, a = area of the steam pipe, sq. in. * A = area of the cylinder piston, sq. in. d = diameter of the pipe, in inches. I) = diameter, S = stroke of cylinder, in inches. _4 Sz, _ DWSn * = 35000' d=*:::::, . . tº 24, 25. Example. Required the diameter of a steam-pipe for a cylinder I) = 40 inches. Stroke of piston S = 48 inches, and m = 38 revolutions per minute? d = º = 9.2 inches, nearly. Steam Ports to the Cylindere A S m a = #iº, - - - - 26, Safety Valvee Three-fourths of the fire grate in square feet is a good proportion for the safety valve in square inches. Notation of Letters corresponds with Figure 3, Plate V. a = area of safety valve in square inches. P= pressure per square inch in the boiler W= weight on the safety valve lever in pounds. Q = weight of the safety valve and levers -----~ \ ------> = lever for W W --- x - -- Q ; € = “ a P X-in inches. Tº cººla " 32 = 6% Q & Balance the lever over a sharp edge, and the cené of gravity Q is found; measure the distance 2 from the fulcrum G. a P e-Wu Q , 27, W-4 fºr 94, 29, l y a Pe — Q a P – WłłQ *, 28, – " ? -9%, 30, à, € W Example. Area of tho safety valve a = 9 square inches, e = 4} inches, W = 50 pounds, weight of the lever and safety valve Q = 15 pounds, and a = l? inches. Required at what distances l, l and l” will the weight W indicate pres: sures of P = 30, 12' = 40, and P’ = 50 pounds? _9X30X4-5-15X17 - 50 from the fulcrum C the weight W will indicate P= 30 pounds, lſ – 37.9 inches, when P’ = 40 pounds. l// = 45.8 “ * Hºº = 50 66 and thus the lever can be graduated. 7 = 19.2 inches, ExPANSION OF CAST IRON. 409 Linear Expansion or Contraction in Inches of Cast Iron, Lengths in Feet. Length. Difference in Temperature.—Fahrenheit. 1000 || 1500 | 2009 || 2500 30 Oo | 4000 || 5 000 600 o 8000 Feet. Inch. Inch. Inch. Inch. | Inch. Inch. I Inch. | Inch. Inch. I 0.0072 0.01.10 0.0150 0.0192 || 0.0237 0.0336 0.0444 || 0.0561 0.0787 2 0.0144 0.022() 0.0300 || 0.0384 || 0.0474 || 0.0632 || 0.0885 0.1123 0.1574 3 0.0216 || 0.0330 || 0.0450 0.0576 0.0711 || 0.1008 || 0.1332 0.1684 0.2361 4 0.0288 || 0.0440 || 0.0600 || 0.0768 j 0.0948 || 0.1344 || 0.1776 || 0.2246 || 0.3148 5 0.0360 0.0550 0.0750 || 0.0960 0.1185 0.1680 ().2220 || 0.2805 || 0.3935 6 0.0432 0.0660 0.0900 || 0.1152 || 0.1422 || 0.2016 || 0.2664 0.3368 || 0.4722 7 0.0504 || 0.0770 0.1050 || 0.1344 0.1659 || 0.2552 || 0 3108 || 0.3929 || 0.3509 8 0.0576 0.0880 0.1200 || 0.1536 0.1896 || 0.2688 || 0.3552 0.4496 || 0.6396 9 0.0648 || 0.0990 || 0,1350 || 0.1728 || 0.2133 || 0.3024 || 0.3996 || 0.5052 0.7083 I0 ().0720 0.1102 || 0.1502 || 0.1926 0.2376 || 0.3360 0.4440 0.5616 || 0.7872 11 0.0792 || 0,1214 || 0.1652 0.2125 || 0.2615 || 0.3696 || 0.4884 || 0.6177 || 0.8659 12 0.0864 0.1316 0.1802 || 0.2318 0.2853 ().4032 0.5328 0.6739 || 0 9446 13 0.0936 || 0.1417 | 0.1952 || 0.2510 || 0.3090 0.4368 || 0.5772 0.7300 | 1.0233 14 0.1008 0.1519 0.2102 || 0.2703 || 0.3328 0.4704 || 0.6216 0.7862 | 1.1020 15 (). 1080 0.1620 || 0.2253 || 0.2895 || 0.3565 0.5040 ().6660 0.8423 | 1.1808 16 0.1152 || 0.1722 || 0.2403 || 0.3088 || 0.3803 0.5376 || 0.7104 || 0.8985 1,2595 17 0.1224 0.1823 0.2553 || 0.3280 || 0.4040 0.5712 0.7548 || 0.9546 | 1.3382 I8 ().T.296 || 0.1925 0.2703 || 0.3472 0.4278 0.6048 || 0.7992 | 1.0.108 || 1.4169 19 0.1368 || 0.2026 0.2853 || 0 3665 0.4515 || 0.6384 || 0.8436 | 1.0669 || 1 4956 20 0.1440 0.2203 0.3005 || 0.3852 0.4752 0.6720 0.8880 | 1.1232 | 1.5744 21 0.1512 0.2305 || 0.3155 || 0.4045 0.4995 || 0.7056 || 0.9324 | 1.1793 | 1.6531 22 0.1584 || 0.2407 || 0.3275 0.4238 0.5228 ().7392 || 0.9768 1.2394 | 1.7318 23 0.1656 || 0.2508 || 0.3425 || 0.3430 0.5465 0.7728 1.0212 | 1.2915 | 1.8105 24 0.1728 || 0.2610 || 0 3575 ().3623 || 0 5703 || 0.8064 | 1.0656 | 1.3477 | 1.8892 25 0.1800 || 0.2711 || 0.3725 || 0.3815 0.5940 || 0.S400 | 1.1100 | 1.4038 | 1.9679 26 0.1872 || 0.2813 || 0.3876 || 0.4008 || 0.6179 || 0.8736 | 1.1544 | 1.4600 || 2.0467 27 0.1944 || 0.2914 0.4026 || 0.4200 0.6415 || 0.9072 1.1988 15161 || 2 1254 28 0.2016 || 0.3016 || 0.4176 || 0.4393 || 0.6553 ().9408 || 1.2432 | 1.5723 2.2041 29 0.2088 || 0.3117 0.4326 || 0.4585 0.6890 0.9744 || 1.2876 | 1.62S4 2.2829 30 0.2160 || 0.3304 || 0.4507 || 0.5778 || 0.712S | 1.00S0 | 1.3320 | 1.6848 2.3616 31 0.2232 0.3405 0.4657 0.5970 0.7365 | 1.0416 || 1,3764 | 1.7409 || 2.4403 32 0.2304 || 0.3507 | (),4807 || 0.6163 0.7603 | 1.0752 | 1.4208 1.7971 2.5190 33 0.2376 0.3608 || 0.4957 0.6355 0.7841 I.1088 || 1.4652 | 1.8533 2,5977 34 0.2448 || 0.3710 || 0.5107 || 0.6548 || 0.8078 | 1.1424 | 1.5096 | 1.9094 2.6764 35 0.2520 | 0.3811 0.5258 || 0.6740 || 0.8316 1.1760 | 1.5540 | 1.9656 2.7552 36 0.2592 || 0.3913 0.5408 || 0.6933 0.8553 | 1.2096 || 1.5984 2.0.217 | 2.83.39 37 0.2664 0.4014 || 0,5558 0.7125 || 0.8791 | 1.2432 | 1.6428 2.0779 || 2.91.26 38 0.2736 || 0.4116 || 0.5708 || 0.7298 || 0.9028 | 1.2768 || 1.6S72 || 2.1340 2.99.13 39 0.2S08 || 0.4217 | 0.5858 || 0.7490 || 0.9266 | 1.3104 || 1.7316 || 2.1902 || 3.0701 40 0.2880 || 0.4406 || 0.6009 || 0.7704 || 0.9504 || 1.3440 | 1.7760 2.2464 || 3.1488 45 0.3240 || 0.4957 | ().6760 || 0.8667 | 1.0692 I.5.120 | 1.99$0 2.5272 3.5424 50 0.3000 || 0.5508 || 0.7512 0.9630 | 1.18S0 | 1.6800 2.2200 2.8080 3.9360 55 0.8960 | ().605.9 || 0.8263 | 1.0593 | 1.3068 || 1.8480 || 2.4420 || 3.0888 4.3296 60 0.4230 0.6610 0.9014 | 1.1556 | 1.4256 2.0160 2.6640 3.3696 || 4.7132 65 0.4680 || 0.6665 || 0.9765 | 1.2519 | 1.5444 2.1840 2.8860 | 3.6540 5.1068 70 0.5040 || 0.7711 | 1.0517 | 1.34S2 | 1.6632 2.3520 ! 3.1080 3.9312 5.5104 75 0.5400 0.8262 | 1.1268 || 1.4445 1.7820 | 2.5200 || 3.3300 || 4.2120 | 5.9040 80 0.5760 0.SS13 | 1.2019 || 1.54(38 | 1.9008 || 2.6880 || 3.5520 4.4948 || 6.2976 85 0.6120 0.9364 | 1.2770 || 1.6371 || 2.0196 || 2.7560 3.7740 4.7756 6.6912 90 0.64SO || 0.9914 | 1.3521 | 1.7334 2.13S4 3.0240 || 3.9960 5,0544 7.0848 95 ().6840 | 1.0465 | 1.4272 | 1.8297 2.2572 || 3, 1920 || 4.2180 || 5 3352 || 7.4784 100 0.7200 | 1.1016 | 1.5024 | 1.9260 2.3760 | 3.3600 || 4.4400 || 5.6160 7.8720 0.00000600 || 612 626 642 600, 700 740 780 S20 Expansion per Degree.—Fahrenheit. Multiply by 1.1 for wrought iron, 1.5 for copper, 1.6 for brass and 2.6 for zinc. $10 HORSE-POWER. Table of Pressure and Temperature of Steam, calculated by the Alexander formula, with a slight modification to accommodate the volumes of the component gases, oxygen and hydrogen. Pressure Temp. Pressure Temp. Pressure Temp. Pressure Temp. per sq. in...] Fahr. H|per sq. in. Fahr. per sq. in. Fahr. per Sq. in. Fahr. P. T. P. T. P. T. IP, T. 100 328.2 700 503.0 4500 714.1 30,000 1019 150 358.6 800 508.4 5000 728.3 35,000 1ſ)48 200 381.5 900 520.6 6000 754.0 40,000 1074 250 400.0 1000 531.8 7000 776.5 45,000 1098 300 415.6 1500 576.4 8000 796.4 50,000 1119 350 429.2 2000 610.0 9000 814.3 60,000 1157 400 44.1.1 2500 637.2 10,000 830.6 70,000 1190 450 452.2 3000 651.3 15,000 896.2 80,000 1219 500 462.1 3500 680.1 20,000 915.4 90,000 1245 600 479.6 4000 697.8 25,000 987.7 100,000 1285 In the above table it is supposed that the steam is superheated from an indefi- nite volume of water. The formulas on pages 394 and 395 are not reliable above a pressure of 500 pounds to the square inch. The formula of Messrs. Fairbairn and Tate, for the volume of steam, is incon- Sistent with the physical laws involved. The steam in Fairbairn and Tate's exper- iments' has evidently been moist with globes of water, which made the steam- volume too small and the formula wrong. For low pressure and temperatures of aqueous vapor, see Hygrometry, page 357. , HORSE-POWER IN STEAM-ENGINES. Horse-power in machinery is assumed to be about the effect a horse is able to produce, and has been estimated and established by Mr. Watt to be 33,000 lbs. raised one foot per minute for one horse, which will be the same as 550 lbs. raised one foot per second. Mr. Watt adopted a standard steam-pressure of 7 lbs. per square inch, cStablished a simple rule for the nominal horse-power of engines, which is, “ The square of the diameter of the cylinder in inches multiplied by the cube root of the stroke in feet, and divided by the constant number 47, is the nominal horse-power. This rule agreed very near to the actual performance of engines in those days, but as the improvements advanced we found that the steam-piston can move with a greater velocity, and the steam-pressure gradually increased—that our day’s engines greatly exceed the above rule. Nominal Horse-Power. Assume a standard steam-pressure of 30 lbs. per square inch expanded two- thirds, the velocity of the steam-piston to be 200 WS feet and revolutions per minute m = 100, We will arrive at a formula of nominal horse-power. f/Sº for condensing engines, which will agree very near with the actual Dºg/S 10 performance of our present condensing engines. The following tables are calcu- lated from this formula. For high-pressure engines I will assume the steam-pressure to be 80 lbs. persquare inch, expanded one-half, which will give the nominal horse-power— 2 3/TKT H–DºS 4 The horse-power in the accompanying table, divided by 0.4, gives the nominal power of high-pressure engines. The diameters D are contained in the first col- umn in inches, and the stroke S in feet and inches on the top line. Indicated Horse-Power Is that imparted by the steam on the cylinder-piston, without friction and working the pumps. JH = , high-pressure engines. HoRSE POWER. 411 A C T U A. L H O R S E PO W E R. One actual horse power is 33000 lbs. raised one foot in one minute. This applied to steam engines will be the mean steam pressure on cylinder piston in pounds, multiplied by the velocity of piston in feet per minute, divided by 33,000, is the horse power imparted by the steam. From this we snaii’deduct 25 per cent. in condensing engines, and 13.1 per cent. in high pressure engines, for working friction and pumps, the balance to be termed the actual horse power. Eacample 1. Fig. and formulae 318. Area of steam cylinder A=1809 square inches, stroke of piston S=4 feet, indicated pressure of steam 30 lbs. to which add the atmospheric pressure 15 lbs. or P=45 lbs. expanded #, the mean pressure will be F=31.459 lbs. (see Expansion Table I.), vacuum v= 12 lbs. the engine making n=45 revolutions or double stroke per ; Required the actual horse power, H=? W=-31'459-H12–147= *759 lbs. 4 H= 1809×4×28:759×45. = 425-6 horses. 22000 In this example the actual horse power is 11-6 per cent. more than the nominal power from the table. Ea'ample 2. Fig. 318. A high pressure engine of cylinder piston A=314 square inches, stroke S=3 feet, steam pressure 80 lbs, per square inch, to which addišibs, fºg; ibs. expanded #, the engine making n==56 revo- lutions per minute. Required the actual horse power? From the ex- pansion table we have the mean pressure F=80.412 Ibs., from which sub- tract the atmospheric pressure 14-7 lbs. W=65-712 lbs. H_314X3X65-12x56 19000 = 180°8 horses. Annular Expansion Double Cylinder, Fig. 319. These kind of engines are now sometimes made in Europe with a view to economise fuel, and to extend the expansion of steam. . The outer cylinder A, A, is annular, similar to that made by Mouslay, but in this case it is employed only for expansion, the inner cylinder a is used for high pressure only. It is so arranged by steam valves that the high steam is acting the whole stroke on the small piston a, after which it is conducted to the annular cylinder where it acts expansively on the large piston A, A. The two pistons being connected by rods to one common crosshead as shown by Fig. 319. This arrangement has been successfully carried out º Mr. Jägerfelt in Nyköping, Sweden. The inner cylinder can be con- sidered an ordinary high pressure engine where the utilized steam is set free into the air at each stroke; but in this case the exhaust steam ac- complishes a second engagement in the annular cylinder, which according to the grade of expansion may greatly exceed the original effect im- parted in the small cylinder during the first engagement. Eacample 3. Fig. 319. Area of the high pressure cylinder piston a=254.4 square inches, the annular expansive piston A==763.2 square inches, stroke of pistons S=3 feet, the high steam pressure P=60 lbs. vacuum v-12 lbs., making n=65 revolutions per minute. Required the actual horse power of the engine H=? The grade of expansion will be 1–. = }, for which the mean pressure on the annular piston will be f=32-62 lbs. See Expansion Table II. The effective pressure on the two pistons will be V=763-2 (32.62+12–14-7) +2544 (60–32-62) = 29800 lbs. _29800×3×65 22000 Eacample 4. Now we will reject the annular expansion cylinder, and take the effect of the steam without expansion, when the effectual pres- sure will be 60—14-7=45-3 lbs. and the actual power, H 354.4×3×453X65 * 19000 = 264 horses. = 118 horses. 41% HoRSE POWER. If we onsider the last result as unit we shall have 264–118=146 horses or nearly 124 per cent. gained by the expansion, omiting the loss of steam in the steam passages. In the first case about 11 per cent. was gained by vacuum, but that ad- vantage is rather in favour of the utility of expansion, because the high steam cannot so well be introduced into the condenser. The economy will be in the same proportion when the same grade of expansion is used in one cylinder. I do not mean to maintain that this high per centage of economy is al- ways fully realized in practice, as I am well aware of cases where expan- sion is of little use, caused by misconception and carelessness in its em- ployment. There are many circumstances about an engine which are in favour of expansion, for instance, the steam passages between the main valve and cylinder, and the clearance between the piston and cylinder heads, contains a great deal of steam which is a total loss, but when ex- pansion is used, that steam expands into the cylinder, and is consequently utilized. The expanded exhaust require a smaller air pump than would be necessary for high steam introduced in the condenser. Half Trumk Expansion Engines. Fig. 320. This kind of engines has been introduced by Mr. Carlsund, and are ex- tensively used in Sweden, they are well suited for Gunboats where the machinery is required to be below the waterine. The high Steam is em- ployed throughout the stroke in the annular space around the trunk, after which it is conducted to act expansively on the large piston A Fig. 320. Eacample 5. Fig. 320. Area of the annular piston a-562 square inches, and A=2248 square inches, stroke of piston S=4 feet, steam pressure P=90 lbs., making m-68 revolutions per minute. Required the actual horse power? 562 Grade of expansion = *~22.5 = %, From the Expansion Table II. we have f-41'58 lbs. mean pressure on A. sº pressure will be V=2248 (41'58–14-7) +562 (90—41-58) = 639 lbs., high Dressure 3 high p H-ºº- 627'3 horses. Pouble Cylinder Expansion Engines, Fig. 321. This kind of engines are now made in England and are said to be ve economical. The small cylinder is used for high pressure, from which the steam is conveyed to expand in the large cylinder. In the figure it is arranged so that the pistons follow one another in one direction, when the steam must be conveyed from the top of the small cylinder to the bottom of the large one, and vice-versa; but it is sometimes arranged so that the pistons move in opposite direction, when the steam is conveyed direct at the same ends from the small cylinder to the large one, which has the advantage of making the steam passages shorter, but is more complicated in concentrating the motion. Ea'ample 6. High pressure cylinder, { a = 962. square inches. S = 5 feet. Expansion cylinder, { $= ; guar e inches. Steam pressure in the small cylinder P=40 lbs., vacuum v- 12 lbs., making m-21 revolutions per minute. Required the actual horse power, H=? 2 Grade of expansion _1__962X5 z- 3848×10 From the Expansion Table II. we have f-11.879 lbs., mean pressure on A. w = 3848×10 (11.8794-12–14:7) +962X5 (40–11.879) = 366767 lbs. of mo- mentum. 366767X21 C− = 350 horses. 22000 SeS arºº-w- ºr " ----, e. 1HORSE Powert OF ENGINES.” - ~~ 13 A. 318. One double acting Cylinder. A S W ºn 4. H==;º- cond, engs. power. _A S W n, high pr. en- TTT9000" gines. W=F4-v–147 for cond. engines. Hº- W=F–147 for high pressure engines. | V S n e Actual F=-25000, cond. engines. 319. Annular expansion double Cylinder. F-ºtºs (log. 4.—log.a)+1]. F'A-P a V=A(f-Hv-147) -* +-a(P–f). T A–a ’ horse V S n £20206ſ. I Jºſe ". i or ſº © 19000' high pr. engs 320. Halftrunk expansion Cylinder. =# [2:3(log.A–log.a)+1]. Fa—Pa V=A(f{-v–14-7) f====. +a(P–f). TV S m º Actual H= A4000' cond. engines. horse V S ºn power. +: 7 high & ſº II. 3S000 high pr. engs 321. Double Cylinder expansion. P a s - e .A.S.—loa. 1 A S [2:30.og. AS–log.as)+1] F A–P a w-AS(f-Hv–14-7) J–-a-i- Hasóp-f). 20 72 g Pi— , cond. engines. F'- Actual 22000 horse 207 72 power. 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[6 |7.18 |L.98 |1-08 |I-81 ||g.g. 9-31 |7.69 |6.99 |0.69 |9-19||5& 9-06 |7.18 |0.58 |8-08 |8.91, I-71 |8. Il (8.69 |1-99 |1-89 |g-09 |0-19 |6-Zg|8& 8.38 |0.08 |8.91, 9.81, 8.69 |8.79 |9.99 ||f.89 |0. I9 |8-89 |7.99 |I-69 |F-87|3.3 f'g! |8.31 |0.01 |0-19 |9.89 |l-I9 |8-69 ||8.19 |9-gg |I-89 |g-09 |9-17 |[.77||13 f.89 |0.99 |g.89 |1-09 |1.1 g |0.99 |3.79 ||P.Zg |#.0g 3.8p |8.gif|I-87 |0.0%|03 A-I9 |9.69 |g. 1g S.p.g. |I.Zg |9-09 |0.6p ||8. Lip g.gf; |g.8F |8. If |6.88 |[.93||61|| j.gg |g-gg #. Ig|3.6% |8.9% W.giº |0.7% ||g.&#|8-0p |0.69 |I-13 6-78 |p.Zg|8I j.6f 1-17 |6.gif|6.87 |!. If |g.07 ||3-63 ||6.13 p.93 |8.78 |I.88 |I. I? |6.83||.T 8.8% |Z.Z; 9.0p |6.83 |0.13 |8.99 |1-73 ||g.89 |&-Z8 |8-09 |3.6% 7.1% |9.93||9|I 9.88 |I-13 1.g3 |[..f3 |7.73 |g. Ig |g.09 ||g.6% |8.8& |I-13 |8.92 |6. P& |g|3.3||9|I 9.88 W.Zg |I. Ig 1.62 |8.88, p. 18, 9.93 || 1.g3 |1.f3 |9.8% |p.g3 |I-I& |9.61||f I 6.8% |6.18, 8.93 |9.gz |f.f3 |1.8& |6-ZZ ||[.3% |8.IZ |3.0% |8.6L 6-8 I |6-9T]|3|I 9.7% |8.8% (6.3Z |6. Iz |S.0& 3.0& |g.6I ||6.8I I-8I #.1. I |g.9I (9.9 I W.W.I./3T 1.03 |0.08, 3.6 [|3.8L |f. 1 I |6.9L |ff.9L ||8.g|I|Z-g|I|9. PI |6.3L 0.8 L |I.ZIIII I./ I |9.9L |6.9I Z.g|I|y.j I |0.j I |9.8L ||[.8 I |9.3, I |0.3I |f. II |8-0|I|0I i 6-8I j.8I |6-ZI |8.8, I | 1. II 3. Il |0. II ||9-0I |3.0I '91-6 ||Z.6 |61-8 |I-8 lló 0-II |9-0T |I.0I |& 1.6 |83.6 |96.8 |89.8 ||69.8 |90.8 |IA. 1 #38.1 |06-9 |f|.9 ||8 88.8 |0 I-8 |81-1 ff. 1 || 10.1 (98-9 (99.9 ||37.9 |! I.9 |06.9 |I9.g. 13-9 |6.7 ; 9I-9 |V6-g|Il-g|1}.g|6.I.g|70.g. |88.j, |z1-# 89.7 |88.7 |&I.'ſ 88.8 |9.8 ||9 H | H | H | H | H | H | H | H | H | H | H | H | H uſ 19 ſo ºwl w '9 & 2 aga,9 allºcal & G Tºo ºne iſ at q "1999 up s u01.sidſ repuy IKo go exioiºs lung -- ‘sºn Ibn QI BMISNGCINOO Jo HgIAAodgsuoh Ivnixſon #If 3> NOMINAL, HORSEPOWER OF CONDENSING ENGINEs. 415 D || 6’ 7, Stroke of Cylinder Piston S in feet. 8/ 9/ in. H 30|| 163 32|| 186 34|| 210 36|| 235 38|| 262 40|| 290 42|| 320 44|| 352 46|| 384 48|| 418 50|| 554 52|| 491 54|| 529 56|| 570 58|| 611 60|| 654 62|| 698 64|| 744 66|| 791 68|| 840 70|| 890 72|| 942 74|| 995|1048 76||1050|1105 '78||1105||1165 80||1162|1225 84||1282|1350 88||1407|1423 92||1538|1619 96||1674|1763 100||1817|1913 104||1964|1969 108||21192231 112||2279|2399 116|24.45|2574 120 2616|2754 124||27.93|2941 128|12977|3|133 132,3166/3333 H 172 196 221 248 276 306 336 3.71 405 441 478 518 558 600 644 689 736 784 834 885 938 99.4 136||3360|3538 140||3561|3749 144||3767|3966 148||3980/4190 152|4198|4402 156||44.214655 H 180 204 231 259 289 320 352 387 423 461 500 541 583 637 673 720 769 819 871 925 980 1037 1095 I 155 1219 1280 1411 1549 1693 1843 2000 2163 2333 2509 2691 2880 3075 3.277 3485 3699 3920 4147 438] 4621 4867 #; 4768||5020 51285399 174 # 6198 180|5887 5249 5645 | 6055 6480 JHI 187 213 240 269 299 333 365 402 440 479 520 H 194 220 249 273 311 344 380 417 460 496 538 562 606 652 700 749 800 852 906 960 1019 1078 1139 1201. 1265 1331 1467 1610 1761 I917 2080 2250 2426 2609 2799 2995 3.198 3408 3624 3847 4077 43.13 4556 4805 5062 5458 5870 6297 6539 582 628 (375 724 775 828 882 938 10' | 117 || 12' 13\ H 200 227 257 288 32] 355 392 430 470 512 555 601 648 697 648 800 855 911 968 996 1023 1055|1089 1116||1153 1179|1218 1244|1284 1310||1353 1378|1423 I520 1569 1668|1722||1773 1823|1882 I985|2049|2010 21.54|2224||2290 2349|2405||2477 2512|2594||2671 27022790 2871 2898|2992||308.1 3312|3202 ||3297 3.101|3419||3521 3529|3643||3752 3753|3875||3990 3984|41.13||4235 4222|4359||4488 4466|4611||4748 4718|4871||5016 49765138||5291 5242|5412||5573 5653|5836||50l 0 6079|6277||6463 6521 |6733||6933 6979|7205||7419 H 206 252 264 296 330 366 404 453 484 527 572 619 667 718 770 824 879 938 997.|1024 1059|1087 II22|1152 1187|1218 I254|1287 1322|1358 1393|1430 1465||1504 1615||1658 1820 1938; 1990 21 66 2351 2542 2742 29.49 31 63 3385 3614 3852 4096 43.48 4608 4875 5179 543] 5721 6170 6635 7117 76.17 H 211 241 272 306 341 377 416 461 497 541 588 635 685 737 791 846 903 963 --- 14/ 15/ | 16/ 18/ IH 217 246 276 312 348 385 425 466 51() 555 602 651 700 755 810 867 925 987,1010 1049.1074|1999 1114|1140,1165 1171|1208||1234 1249|1278 1319||1350 1392 1466|1500 1542|1578 1700||1741 1866|1909 3243 3949 4190 4457 4716 4997 5279 5568 2039|2086 2272 2466 26.66 2873 3092 3315 3550 3790 4038 4295 4557 4832 5111 5399 5696 6000 6469 $802,6958 7296,7464 7809.7989 ZU.5% 222I 2400 2608 2S06 3023 34.71 3706 5865 6324 Hl 222 253 285 319 356 395 435 477 522 569 617 667 719 775 830 888 948 1024 1306 1380 1424. 1455 1533 1612 1778 1951 2133 2322 2520 2725 2939 316] 3391 362S 3874 4128 4390 4654 4939 5225 5519 5821 6132 66] 3 7112 7629 8164 H 226 258 291 326 363 403 444 494 533 580 630 681 734 790 847 907 968|1007 1073 1141 1211 1283 1358 1434 1512 1594 1676 1848 a'a a ºn Ara 236 268 303 339 378 419 462 507 554 603 655 708 764 811 880 943 HTTHT 1043 1101 1182 1254 1329 1406 1485 1567 1649 1737 1914 Z U A J 2258 2414 2620 2833 3056 32S6 3525 3772 4028 4292 4565 4846 5135 5432 5739 6053 6376 6876 70947.659 7932,8216 8488;8793 2100 2297 2474 2714. 2935 31.65 3404 3651 3908 4172 4446 4728 5019 5319 568I. 5944 6270 6604 7122 20\ 244 278 313 351 391 434 478 525 587 625 677 733 790 850 912 977 416 APPROXIMATE HORSE-POWER. Approximate Horse-Power of small high-pressure engines. H = 0.1D2 j/ S. Steam pressure not less than 80 pownds to the square inch. Pºn. Strol:e S of piston in inches. Inches 3 || 4 5 6 7 - 8 9 10 | 12 || 14 | 15 | 16 | 18 2 .577 |.634 .684 ,727 .765 . .800 | .832 ||.862 |.915 .964 . .985 | 1.00 | 1.05 2} .900 .990 | 1.07 | 1.13 | 1.20 | 1.25 | 1.30 || 1.34 | 1.42 | 1.50 | 1.54 | 1.57 | 1.63, 3 1.30 | 1.43 | 1.54 | 1.64 | 1.72 | 1.80 | 1.87 || 1.94 2.06 || 2.17 2.22 2.27 | 2.36 3# 1.77 | 1.94 | 2.10 || 2.22 || 2.34 || 2.45 2.55 || 2.64 || 2.80 2.95 || 3.00 3.09 || 3.21 4 2.31 i 2.54 i 2.74 2.90 3.06 ig.20 i 3.33 || 3.44 i 3.66 j 3.85 3.94 || 4.05 || 4.19 4} || 2.92 || 3.21 3.47 3.68 3.87 || 4.05 || 4.42 || 4.36 4.64 4.88 5.00 5.10 || 5.30 5 3.60 | 3.96 || 4.27 4.54 || 4.78 || 5.00 5.20 || 5.38 5.72 | 6.02 || 6.16 || 6.30 6.55 6 5.19 || 5.70 || 6.15 6,53 || 6.89 || 7.20 || 7.55 || 7.82 8.31 || 8.75 8.95 9.15 9,50 7 7.08 || 7.78 || 8.40 | 8.92 || 9.40 9.80 || 10.2 || 10.6 || 11.2 | 11.8 | 12.1 | 12.3 12.9 8 9.25 || 10.1 | 11.0 | 11.6 | 12.2 12.8 13.3 iſ 13.8 14.6 || 15.4 15.7 | 16.1 | 16.8 9 11.7 | 12.9 || 13.9 || 14.7 || 15.5 | 16.2 | 16.8 || 17.4 18.5 | 19.5 | 20.0 20.4 || 21.2 10 14.4 || 15.9 || 17.1 | 18.2 | 19.1 | 20.0 | 20.8 || 21.5 22.9 || 24.1 || 24.6 25.2 26.2 II 17.5 | 19.2 | 20.8 || 22.0 23.2 || 24.2 || 25.2 || 26.1 || 27.7 || 29.2 | 29.9 || 30.5 || 31.6 12 20.8 22.9 || 24.7 26.2 27.6 || 28.8 || 30.0 || 31.0 || 33.0 || 34.8 || 35.5 36.3 || 37.8 The horse-power of small engines, as counted by the English, is only 0.4 of that in this table for the same size cylinders. To Approximate the Size of Steam-Engines. Eacample 1. It is required to build a river steamer of displacement T = 1000 tons to run M = 16 nautical miles per hour. Required, the size of the cylinder for an ordinary overbeam engine 7 From the table of steamship performance will be found the required actual power H = 1798 horses. From the table of Nominal horse-power select the approximate size of cylinder, which may be D = 88 inches, diameter of cylinder by S = 14 feet stroke, which answers to H = 1866 horses nominal. In this case the nominal horse-power can be considered the same as the actual. Example 2. A propeller steamer is to run M = 10 nautical miles per hour, with a displacement T' = 3400 tons. Required, the size of the cylinders ? From table of steamship performance EI = 992 horses, to be divided into two cylinders of 496 each. Select from table of Nominal horse-powor D = 60 inches diameter of cylinders and S = 2' 19' stroke of piston, which answers to H = 504, or 504 × 2 = 1008 horses of the two cylinders. After these approximations are made, make a careful calculation from the original formulas. Example 3. Suppose the propeller for the steamer in the preceding Example 2 makes n = 60 revolutions per minute. Required, the diameter of the propeller- shaft? See Table, page 418, for wrought-iron shafts, for 1000 horses and 60 revolu- tions, the shaft should be 12.8 inches. Example 4. A steamer of T = 2500 tons is to run M =9 nautical miles per hour with an indicated steam-pressure of 20 lbs., or P = 35 lbs. per square inch, expanded #. Required, the consumption of fuel in tons per 24 hours? Table of steamship performance H = 585 horses. Table W., page 400, consumption of fuel, 3.44 tons. The required consumption will be 5.85 X 3.44 = 20.124 tons per 24 hours' steaming. FRESH WATER ConDENSER. 417 ! & FRESH WATER COND ENSER. ; Fig. 1. Fig. 1, is a longitudinal, and Fig. 2, a transverse section of a fresh water con- densor with horizontal tubes. A, air-pump. a, fresh water. 2, exhaust-pipe. 6, hot well. T, tubes. c, injection pipe. d, strainer. The tubes are of copper one inch outside diameter, thickness of metal, No. 22 or 24 wire guage, weighs 5+ ounces per foot. The space occupied by the tubes should be about cubical, that is, the sides of the tube-plate should be about the length of the tubes. Between the injection and the tubes is a horizontai strainer, to spread the cold water uniformly over the tubes. Steam inside the tubes. Letters Denote. A = condensing area of all the tubes in square inches. * = length of tubes, height and breadth of tube-plate in inches. JN = number of tubes in the condensor. I} = diameter of steam cylinder. S = stroke of piston, in inches. n = number of º of engine per minute. T == temperature of exhaust steam ;I jñjeje.” see steam table, page 248. 4-4.4(90+2'), t = 0.85894, N= 0.5128.1°, 4 – 161 v. Example. A fresh water condensor is to be constructed for an engine of D = G2 in. S = 76 in. making n = 34 revolutions per minute; T =230°, T=1225. Required the condensing area of tubes, A = ? _622×76XV8ſ O W —42CNS ºf T55x1225 ( 940-1-230 )=295,709 square inches. Required the length of tubes, and sides of tube plate, l = ? ! = 0.853;295.7 69 = 57 inches nearly. Required the number of tubes in the condenser, N = ? JW = 0.5128×572= 1666 tubes. se t 57 no -- 57 Number of tubes in the top row IF=I5 - 38, side row =#5-- 44 tubes. The tubes to be placed zigzag, as shown in Fig. 2. Should the location fºr the condensor not permit the cubical form, we have, Length tubes l = A • Breadth of tube plate b = rºm Height h *I-6Iſ, D2 S rº, * tº I64 W 2 gallons per minute, or about 75 per cent of the feed water. Temperature in the hot well about 1100 to 115°. 35 of fresh water 120° to 130°. For fresh water condensers the capacity of the air pump should be about 10 per cent larger than by the rules on page 250, - – A. 1.6lb/, Fresh water produced G = 27 418 PIAMETERs of WroughT-IRON SHAFTs. Diameters in Inches of Wrought-Iron Shafts. Horse powo- II. *-ºse 1 2 4. 3. 4 5 6 7 8 9 I () 1000 1200 1500 2000 2500 3000 3500 4000 4500 b000 LO in. 2.32 2.92 3.40 3.68 3.97 4.22 4.44 4.64 4.82 5.00 5.34 5.72 6.30 || 5 6.79 7.21 7.59 |6. 7.94 8.25 8.55 9.08 9.57 10.0 10.4 10.S 11.4 12.0 12.6 13.1 13.6 14.6 15.5 16.3 17. I 17.8 18.4 19.0 19.6 20.7 21.5 23.3 24.7 26.6 29.3 31.5 33.5 35.2 36.8 38.4 39.6 Number of royolutions por minute of 15 : 20 In. .93| 5.39 1.84 2.32 2,66 2.92 3.15 3.35 3.52 3.69 3.S3 3.98 4.22 4.54 5.00 5.72 t 6.03 6.30 6.55 6.79 7.21 7.59 5||7-94 8.25 S. 55 9.09 9.57 10.0 10.4 10.8 11.6 12.3 .3||13.0 13.6 14.1 14.6 15.1 15.5 16.4 17.1 18.5 19.6 21.1 23.5 25.0 26.6 28.0 29.3 30.4 31.5 25 ln. !.71 2.16 2.47 2.72 2.92 3.11 3.27|, 3.42]: 3.56|3.: 3.69 3.92 4.22 4.64 5.00 5.32 5.60 5.85 6.09 6.30 6.7() 7.05 7.37 7.67 7.94 8.44 S.S.9 9.29 9.67 10.0 10.8 11.5 12.0 12.6 13.1 13.6 14.0 14.4 15.2 15.9 17.1 I9.2 19.6 21.5 23.3 24.8 26.0 27.2 28.3 29.3 30 In. 1.61 2.03 2.33 2.55 2.75 2.92 35 In. 1.53 1.92 2.21 2.43 2.61 2.78 2.92 3.06 5 ! 3.18 3.29 40 ***g* f : i 3|5.20 5.38 5,72 6,03 6.30 6.55 6.78 7.21 3|7.59 7.94 S.26 8.55 .63|9.22 11.2 11.7 9.80 10.3 10.8 11.2 11.6 12.0 | 2.3 13.0 13.6 3|14.6 15.6 16.7 18.4 19.8 21,1 22.2 23.3 24.1 25.0 45 In. 1.41 1.72 2.03 2. 2 9 : : ; : 16.1 17.7 19.1 20.3 21.4 22.3 23.3 24.1 2 wrought-iron shafts. *sºg •&* 7.94 i. ; 9.58 : ! 11.1 11.5 12.1 12.6 13.6 14.5 15.5 17.1 18.4 19.6 20.7 21.6 22.4 23.3 55 In. 1.31 1.66 1.90 2.09 G0 In. 1.28 1.61 1.71 1.S() 1.88 2.06 2.12 2.27 2.40 2.75 2.92 Aiº : : 4.28 4.55 .: : 3 :4 19.9 º : 96.' 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I f$“I II, "I gg “I Qg"I 80' I • uſ QZ | ‘8).JöUS UIO.1]-4115mo.IA Jo olnului Iod SuognIOAO.I. Jo 19quin N 00I 0.00% 00gif 000? 0.098 ()008 0.09% 0.00% 009 I ()0%I 000 I ‘...toAod 08.10II “sºyeus uro. II-qušrao. IAA Jo saviouſ up sixaqaukuyoſ 6 If? ‘SLIVIIS NO3(I-LH9ſ).0%IAA 10 S&IGILGIWVIGI 420 SLºC VALVDS. S L I D E W A L W E S. The slide valve motion is one of the most important features in causing a steam engine to work well, and to employ the effect of steam economically. The author of this book being well acquainted with disarrangements on this point, has here endeavoured to give a good working-drawing of the proper pro- portions and arrangements of slide-valve motion. (See Plate IV.) Maian Valvee It will be best to assume a certain size cylinder, and at the same time give the proportions for any size. D = 34 inches, diameter of the cylinder. S = 18 inches stroke of piston.* = 56 double strokes per minute. We have the area of the steamports m, from Formula 26, page 252. . . º _34*X0-785X18X56 &\, = 30600T = 30 square inches, nearly. D+S 34+18 & 7), - –25 ––a– = 2 inches, the width of the steamport; if the quotient gives a fraction take the nearest quarter or eighth. 8, 772, = 15 inches, breadth of steamport. *. . r = } m about = 1 inch, the exhaust port o = 2m — ºr = 34 inches, and f = 0 + 2r = 5% inches, h = f – #r = 5; inches, k = 14m = 3 inches, and t = h--2k = 113 inches, e = m = 2 inches. * The stroke and diameter is here rather out of proportion, but we wifi maintain them in the cakeulations as they suit the drawing, which is purposely made to show the slide valves on a large Scale. The rules will however suit any propor- tions of diameter and stroke. To Find the Stroke of the Eccentrice s = stroke of the eccentric in inches. s = i – f – #1 = 5% inches. The lap L = }( — f--2m) = # inches. The lead of the valve, or opening of the steamport when the crank pin stands on the centre should be about == º == * = # inches, nearly. Having finished the main valve and ascertained the stroke of the eccentric, it is now required to find the position of the centre b, (Plate V.) of the eccentric, to the crank-pin. Suppose the crank pin of the engine stands atºa on the centre nearest to the cylinder, and the eccentric rods are attached direct to the valve rods; draw the line da, at right-angle to the centre-line aa" of the engine, them 3. I-2. the angle, sin.W-*tº– gº = 0.409, or W- 240 10'. See Plates IV. and W. To Find the position of the Craml;s Pima at the maomen at the Main Valve operas. _ _Sl 18X025 9 - scos, W- 5.5×öð123 from the centre line. = 0-9 inches, nearly, a S//e ///ia3. tº t º // §, º \ 4 - QEZ N AºA Şişº º ſº ; !!! | º (i. º - Hººtº -º-º: ~ n |G: % lii) à º l * | | sK<<|-SKK|W º S->º -º J.ſ.līs/roſſ, | * - - - - - Aºaze / 2" . 2?2&_radic-gū ºf L-z’ - - - - .” Ærſaw ºf 6/?ca ./I.'ºz, ºr: SLIDE WALVER. - 421 I- | To Find the position of the Cranic at the moment the - Exhaust operas. S. 1 18 1 2 = - sin. W--(f-h) = ~ 0.409 — —(5.5–5. = ** t #(º #(ſ 9) #( ;6% **) 3-27 inches from the centre line. | To Find the position of the Crank Pin when the Main Valve cuts off the Steame . o 7 Ar=** = ** = 5·72, inches. sm-ºss-m-- * * sms sº-sº-sº ºme * S 5°5 . To Find at what pårt of the Stroke the Main Valve Cuts. g off the Steam, º ovz\ * Will cut off at = 1 — º: = 1 — (#) = 0-899 of the stroke. The greater the lap is, the sooner will the main-valve cut off, but if the lap is increased the stroke of the eccentric must also be equally increased. It does not work well to cut off much by the main-valve, especially when the engine works fast; for very slow motion it may answer to cut off at # the stroke. - It will be noticed that the centre of the eccentric is always ahead of the crank pin with an angle 90°,+w. Hence when the engine is to be reversed, the centre b must have the same position on the opposite side of the centre-line, or the eccentric must be moved forwards an angle of 90° — 2w. Cut-off Valvee The width of the cut off ports should be about d = &m = 1; inch, and their - a 30 sº - = − = 12 breadth 2d 2X1+ 12 inches, when two ports are used. Proportions of the Valve. a – b = c – d. a+d = b +c, and a = 2d, and the stroke of the cut-off valve cccentric s = 2b, we shall have a = 24, b = 23, c = 13, c = 13, and s == 4} inches. e Let us assume the steam to be cut off at # = l of the stroke S, the position of the crank-pin a' will then be sin.w = 21 = 0.666, or w = 700 30'; at the same time the position of the centre c’ of the cut off eccentric will.be l. sin.2 = d-Hc T- 1}+1} == 0.612, or z = 379 50', - & 4}. and V = w — z = 70° 30' — 37° 50' = 32°40', the position of the centre c when the crank-pin a is on the centre. This Table will show the positions of the centre a and c, at different cut offs. Letters correspond with Figure 1, Plate VI: TCut off º stroke of F. at l. t) S1El...?) 6CC873. S. 2 ,13 * 10. J. 22d 10° 0.377 25 370 50/ 600 0°5880 0-250 # 320 40" | 0-539 2b 37o 50° 700 307 || 0-6914 0-333 # 31O 55/ Q:527 C+0. 43. 35° 750 30/ 07332 0-375 # 42° 35' | 0-675 b-i-c 47° 25' | 902 . . . . 0.8350 || 0:500 # 460 307 || 0-71.93 a-i-Ö – 6 58° 1040 30/ | 0-910 0-625 # 500 307 || 0-7933 a--5 – c || 5So 30' | 109° 30' ' 0-9S5 O'666 It will now be observed that the effectual pressure F in this Table is less than in the Table on page 239, owing to the valve not cutting off the steam instantly, but gradually, so that the density of the steam in the cylinder is slready diminished at the cut off point. The valve will cut off quicker the less the angle 2 is. * - See Figure 2, Plate WłII. The actual pressure will not form a sharp corner at | e, or follow the line e,e,e, as would be due when cut off at ; the stroke, but the line ff'ff will be the true diagram. Including the steam in the ports and steamchest, the density at the end of the stroke will correspond nearly with the Table. 422 STEAM BOILERs. ST E M M B 0 I L E R S. The accompanying proportions are averages of a great number of good marine boilers. Letters denote. D = diameter of the steam-cylinder in inches. S = stroke of piston under which steam is fully admitted, in inches. * = number of double strokes, or revolutions per minute. to = pounds of water evaporated per pound of coal, per hour. V= volum coefficient from the steam table, EI = fire grate in Square feet, for each cylinder, and with natural draft. To Find the Area of Fire Grate, I)” S m. *6620 WE E = P.S. n_40%" H. . . . . 1, 2. 4'66 wV Da S + Example 1. A steam engine of D = 54 inches diameter of the cylinder, and stroke of piston 96 inches, cut off at #, S-48 inches; is to make 22 revolutions per minute. Anthracite coal to be used, that evaporates w = 7 pounds of water per pound of coal, and to carry 27 pounds of steam per square inch, W = 649. Required the area of fire grate E = ? in square feet. E = 54*X48X22 4'66X7X649 Eacample 2. A steamboiler of El = 128 square feet, is to be used for an engine of D = 36 inches diameter, and 64 inches stroke, --cut off the steam at $ then S = 42-66 inches. Steam pressure to be kept at 25 pounds per square inch V = 679. w = 6-5. Required for how many revolutions per minutes can the steam be kept at 25 pounds? a 4-66-6.5×679×128 -: 36°X42'66 Horse Power of the Fire Grate. = 145-34 square feet. = 4.7-6 revolutions. H = horse power of the fire grate. P = pressure in the boiler in pounds per square inch, excluding the atmosphere. p = Wacuum in the condensor in pounds per square inch. - D + Ea--,-,-Hº, H==Yººt *P) 3 a. Vw (P+ 0.8 p) QC # the stroke. a = 27700. saves 55 c. *.ſit":E;"ºlº steam at § º . 2 = 45500. . 26 of fuel. # 53 35 32 = 49100. 35 20 Steam admitted throughout the stroke a = 61700, , 0 per cent. Erample 3. Steamboilers are to be constructed for an engine of 650 horses, the steam to be cut off at # the stroke, and P = 36 pounds per square inch, V= 544, w = 7.5 pounds of water evaporated per pound of coal. Required the fire grate in the boilers EI = ? in Square feet. 650 X. 38400 Ea-Hā:########III -188 square feet. —— * STEAM BoII.ERs. 423 x-x * Arº-. --> Example 4. Required, the horse-power of a fire grate E1 = 112 square feet, to carry 18 pounds steam, and cut off at 3% the stroke? W.2-8T0, w = 7 pounds. H_112X18X810X7 45500 Consumption of Coal. C= coal consumed in pounds per hour. o_3D: Sn. 14H2 5, 6 w y ' T V w(P+ 0.8p) 3 * * Example 5. A steam-engine of D = 42 inches diameter, and 48 inches stroke, cut off the steam at 3% S= 16 inches, is to make n = 65 revolutions per minute with a pressure of 34 pounds per square inch, V-564, and w = 6 pounds. Required, the consumption of coal in pounds per hour C= ? o 3×4% Xl6X65 sºmºs 6 X 564 Example 6. A pair of steam-engines of H = 260 horses are to be worked with P = 28 pounds per square inch, cut off at 3% the stroke, V-635, the coal to evap- orate w = 6.5 pounds of water per pound of coal. Required, the cousumption of coal in pounds per hour G =? __ 14 × 260 × 31400 630 × 6.5(28 + 0.8 × 10) It will be observed in the Formulas 4 and 6 that the higher steam used, the less fuel and fire-grate is required for the same power—the proportion of fuel will be nearly as the square root of the steam pressure, and still more fuel is saved by cutting off the steam at an early part of the stroke. = 251.2 horses. = 1625 pounds per hour. =775 pounds per hour. Heating Surface O Compared with Grate. In common stationary boilers, . . O = 20E. Returning flue boilers, . . . . O = 254–1. Tubular boilers (marine), . ... O = 30H. With vertical tubes (Martin), . . . O = 35E. Cross-area of Flues (Calorimeter). In the common single returning flue boilers, the cross-section area of the first row should be, . . . . . . .18E. Returning row, flues or tubes, . . . 0.13 E. Cross-section area of chimney at the top A = 0.16EI. Height of Chimney. O2 H2 _g = ---- h=IH -% *=ir, c=2HVA-F2 O = 1.45A1/h. A = TL = —. H= 1.45AH/h, 1.451/h' 2}/h -- 2 Example. Area of fire-grate E = 140 sq. ft., to consume C=2100 pounds of ccal per hour. Required, the lieight h of the chimney? 4 × 140% = 56.3 feet, the answer. h = 2100% £24 • STEAM-BOILERS. Standard Horse-Power of Stearm-Boilers. The power of a steam-boiler ought to be graded by the dimensions of the areas of the fire grate and heating surface, like that of a steam-engine is graded by the diameter and stroke of the steam-piston, without taking into consideration the evaporative power of the fuel, expansion of the steam, etc., which are independent of the size of the boiler, as well as that of the engine. Let =| denote the area of the fire grate. O = the area of the heating surface in square feet. P= pressure of steam in pounds per square inch above vacuum. Then tho standard nominal horse-power II of a steam boiler can be expressed by— Tº 2.7F ºr 2 2 2 \ . H-V* e-jº o-º; P-(*#)| 10 O VP EyP {E}O Evample. Suppose E = 100, O-3000 and P=75, Then, BI = w lºvić = 510, the standard nominal horse-power. Ordinary Performance of Steam-Boilers. Natural draft consumes about 12 to 15 pounds of coal per square foot of grate per hour, and generates about 4 to 5 horse-power per square foot of grate. The heating surface should be about 4 to 5 square feet per horse-power, and evaporate 4 to 5 pounds, or 92.5 to 115.5 cubic inches, of Sea water pe. sour, at the above-mentioned rate of combustion. Good coal evaporates about 6 to 8 pounds of water per pound of coal. Each horse-power requires the consumption of about 3 to 4 pounds of coal per hour. To find the Ultimate Bursting Strength of a steam-boiler—shell, tube or flue. IVotation of Letters. W= ultimate tensile strength in pounds per square inch of the boiler iron. = thickness of the plate iron in decimals of an inch. D = diameter of the boiler in inches. P= bursting pressure (internal) in pounds per square inch. := #. for single-riveted, and P= ly. for double-riveted boilers. Example 1. The diameter of a single-riveted boiler being D = 72 inches, thick- mess of plates t = 0.375 inches, and the ultimate strength of the iron W = 45000 pounds to the square inch. Required, the bursting pressure of the boiler? P= 45000 × 0.375 72 A double-riveted boiler of the same dimensions will burst with 1.3 × 234.4 = 3'34.72 pounds to the square inch. The Thickness of Boiler-Plates required for Bursting Pressure. =234.4 pounds to the square inch. t–PP, for single-riveted, and FV t--tº for double-riveted boilers. 1.3 W g | STEAM-BOILERS. * 42; HT Ultimate Strength of Tubes and Flues for Ecternal Pressure to Collapse. ſotation of Letters. D = diameter of tube or flue in inches. L = length of the tube or flue in feet. t = thickness of iron in decimals of an inch. P= external collapsing pressure in pounds per square inch. p_200,000t’, and f = V. PD 1/L. D VI, 447.2 Evample 1. A flue of D = 15 inches diameter, and L = 12 feet Iong, thickness of iron t = 0.25. Required, the collapsing pressure ? 5 2 p_200000x025?–941 pounds to the square inch. 15 × 1/12 Eacample 2. D = 9, L = 10 and t = 0.2. p-200000x004–2s2 pounds. ** 9 V10 Example 3. D = 6, L = 6, and t = 0.2. Required the pressure P2 P= 200000x004 = 504 pounds. 6× 1/6 Staying Steam-Boilers. d = diameter of good iron stay-bolts in inches. d–D V P § 74 I} = distance apart in inches in salt water on flat surfaces. wº o gº º {º tº L) == 74d. 1/ P 2 P=pressure of steam in pounds per square inch. p-ºº: - The following table is given by Mr. Fairbairn, as exhibiting the strongest form and best proportions of rivet joints, as deduced from experiments and actual practice: & Thickness Diameter of Length of rivet Distance from Quantity of lap in of plate. rivet. from head. centre to cent. single riveted. double riveted. in. 16ths. in. Ratio. in. Ratio. in. Ratio. in. Ratio. in. Ratio. 0.19 = 0.38 2 : 0.88° 4.5 1.25 6 1.25 6 2.10 10 0.25 = ().5() 2 1.13 4.5 | 1.50 6 1,50 6 2.50 10 0.31 = 5 0.63 2 | 1.38 || 4.5 | 1.63 5 1.SS 6 3.15 I() 0.38 = 6 0.75 2 | 1.63 4.5 | 1.75 5 2.00 5.5 3.33 9.2 ().50 = 8 || 0.81 | 1.5 2.25 || 4.5 2.00 4 2.25 4.5 3.75 T.5 0.63 = 10 || 0.94 | 1.5 2.75 4.5 2.50 4 2.75 4.5 4.58 7.5 0.75 = 12 | 1.13 | 1.5 ! 3.25 || 4.5 ! 3.00 4 3.25 4.5 5.42 7.5 426 FUEL AND TIMBER. WOOD FOR COMBUSTION. A cord of wood is 8 feet wide by 4 feet high and 4 feet deep, or the wood is 4 feet long. The cord contains 8 × 4 × 4 = 128 cubic feet, of which only 74 cubic feet is solid wood and 54 cubic feet of space. Two cords of wood evaporate about the same quantity of water as one ton of anthracite coal. The best pine wood evaporates 5 pounds of water per pound of wood consumed in a steam-boiler furnace. One cord of wood can be consumed per hour on 60 square feet of grate. Weight in Pounds per Cord of Different Woods. Woods, Seasoned. lbs. Woods, Seasoned. lbs. Woods, Seasoned. lbs. Shell-bark IIickory. 4469|| Iſard Maple . . . .2878||Cedar . . . . . [1910 White Oak . . . . .3821|| Beech . . . . . [2875|| Yellow Pine . . . [1904 Red-heart Hickory. 3705|| Hazel . . . . . 2870; White Pine . . . .1868 Southern Pine . . .3375|| Virginia Pine . . 2689|| Spruce . . . . . [1685 Red Oak . . . . .3254||New Jersey Pine . .2137 || Hemlock . . [1240 Wood requires 32 per cent. more fire grate than mineral coal, for equal genera- tion of steam. The furnace should be 60 per cent. of cubical space more for wood than for coal, or about 4.5 cubic feet per square foot of grate. Properties of Fuel. : ‘s § 3 . # 3 * tº-t g-i $4 s e ## | ###| ||s|###| ##| #5 Kind of Fuel, § 3 ; ; : : §§ 3 ; ; ; ; g | * * # * * | ##3 ||38|# E = | ##| # 3 Bituminous coal * © 11600 7 to 9 80 265 50 44 Anthracite Coal . & & 13340 8 to 10 92 282 54 40 Coke . * º º e T 2420 8 to 10 86 245 31 72 Coke, nat. Virginia. . wº II600 8 to 9 80 260 48 48 Coke, Cumberland . º 11600 8 to 10 80 250 32 70 Charcoal . tº © tº 13920 5 to 6 96 265 24 10.4 Dry wood . † g g 6380 4 to 5 44 | 1.47 20 100 Wood with 20 per ct. water 4930 4. 34 115 25 100 Turf, dry . & º © 7395 6 51 I 65 28 8() Turf, 20 per ct. water tº 5S00 5 40 132 30 | . T 5 Oil, Wax, Tallow & te III65 14 77 200 59 37 Alcohol (from market) . . 84.10 9.56 58 154 52 42 Chemically, one pound of carbon burnt to carbonic acid requires the oxygen of 153 cubic feet of atmospheric air. * Timber, Green and Seasoned. Timber. Green. Seasoned. American Pine . . . 44.75 30.69 e tº * º * ſº (). * & * ; . #. º Comparative weight per cu- Cedar tº & © e 33.00 38.25 bic foot in pounds of green jish oak . . . . .I.; 43.50 | | , and seasoned timber. Riga. Fir . tº º 48.75 35.50 Board Measure. Multiply together the three dimensions, width and thickness in inches and the length of the lumber in feet; divide the product by 12, and the quotient will be | the board measure. *== Combustion and Effect of Fuel. 427 Combustion is the rapid chemical combination of substances with oxygen. Carbon C and hydrogen H, are the substances most generally employed for generating heat. Carbon is fully consumed when combined with oxygen O, to form carbonic acid gas CO2, and partly consumed when in the form of carbonic oxide gas CO or smoke. h = units of heat generated of one pound of fuel. The heat necessary to raise one pound of water one degree Fah, is one unit of heat, w=pounds of water at 212° evaporated per pound of fuel. A =volume in cubic feet and a-weight in pounds of atmospheric air required for the perfect combustion of one pound of fuel. C, O, and H are in the four first formulas, fractions jn one pound of the compound fuel. Perfect Combustion. Imperfect Combustion. - _ 9. - 560–210 A=149| C+3(H #)], . 1 (CO)=. - -, - - - 5 * O 12 a=12|[C+3(H–- )], - - 2 330– 44C h=14500[C+428(H-4)], 3 12 . " h h=3960(CO.)+1650(CO), - *966 h=8002:50–6820C, - - 8 When oxygen is supplied to carbon in a proportion between CO and CO2, both the gases will be formed separately in the proportion of the formulas 5 and 6, when the heat generated will be as formulas 7 and 8, in which C, O, CO and CO2 are expressed in pounds, for instance: 0–20 lbs. of oxygen united with C-12 lbs. of carbon will form 56X12–2 |X20 (CO)= 7 2D =15(G+428(H-4)] (CO)= 12 =21 lbs. carbonic oxide. 33X20–44X12 * (CO)= 12 =ll lbs. carbonic acid, and will generate h-8002:5X20–6820X12=78210 units of heat. One unit of heat—772 foot pounds, if generated per second will be H= 1.4 horses, of which we in our days practice utilizes about one-twentieth. The following table will show how important it is to fully consume the Combustibles to acid. One pound of carbon consumed to oxide will gen- erate only 1:72 horses, instead of 5-66 when consumed to acid. Properties of Combustion, per Hour. C | CO | CO, O Cl, A h 70 H ! lbs. lbs. lbs. lbs. lbs. cub. ft. heat. Ibs. horses. # 1 3-666; 2. 666|| 12 149 || 14500 15 5 - 660 jø （1 2-666 1-333| 6 74°50 4400|| 4-55 1-720 0.433 . H 0.566|| 2:550 31-65 | 1650; 5.633 1-200 0-272 I 0.727 3-275|| 40-56; 3960; 4.100 1-545 1-750 1-375|| || || 3-500 43.33 5440 5.633 2-125 0.445 0-392|| 0.222| || 12-38|| 1210 1-250, 0-472 •0358 0246|| 0231 - 0808 II 97.3 0-100 0-03S 0-584 0.244 0-170 0-800; 9.920. 966|| || 0-37S 1-550 0-651 0-470 2-120 26-30 2558; 2.645 | 428 PADIATION OF HEAT FROM STEAM-PFPES. RADIATION OF HEAT FROM STEAM-PIPES, Boilers or Steam Cylinders. Notation of Letters. D = outside diameter of steam-pipe, without casing, and limited to not more than 12 inches. T = temperature of the steam, Fahr. scale. t = temperature of the external air. h = calorics radiated per square foot per hour, on uncovered pipe. A = outside area in square feet of steam-pipe. II = horse-power lost by radiation of heat. Wind. Exp. m. n = exponent of the wind, which varies with the cur- Calm. 1.20 rent of air or draft about the steam-pipe, as in Gentle. 1.22 the following table: - Brisk. 1.24 Storm. 1.26 The loss of heat will then be per hour— h = 0.001122 (450 + (12–D)*] (T-t)". . . . . 1. A h 2564 One horse-power consumes or generates 2564 calorics per hour. By logarithms the Formula 1 is reduced to— log. h = log. k + m log. (T'- t). . º e e • 3. The log. k is contained in the second column of the accompanying table for different diameters of pipes. For any uncovered plane or cylindrical surface above 12 inches in diameter the radiation in units of hoat per square foot per hour will be— h = 0.505 (T – t)*. . . . . . . 4. The effect of thickness of metal is inappreciable for practical purposes. Example 1. The California S. N. Co.'s steamer Julia has a steam-pipe 40 feet long by D = 9 inches in diameter, and two branch-pipes 12 feet long by D = 4 inches each, all uncovered. Pressure of steam, 100 fos. T = 337°. Temperature of the external air t = 70°. Required, the loss of heat and power by radiation? In calm wind n = 1.2. See table for 77. h = 0.001122(450 + (12–9)* (3370–709).2 = 420.34, the units of heat lost per Square foot per hour. Area of pipe, A = 0.75 X 3.14 × 40 = 94.24 Square feet. Horse-power lost H = Power lost, H = º = 15.5 horses. The branch-pipes lose 4.6 horses. The total loss of power 20.1 horses. The same pipes covered with 2-inch-thick felt would gain 20.1 × 0.93 = 18.7 horse-power. Example 2. In the factory of Bellavista, Peru, are 150 feet of uncovered steam- pipes Đ = 3 inches in diameter. Steam-pressure, 45 lbs., T = 292° Fahr. Temperature of external air t = 68, and wind gentle, n = 1.22. Required, the horse-power and fuel lost? Formula 3. log. h = 0.77408 — 1 + 1.22 log. (292–68) = 2.32805, or 173 units of heat lost per hour. Area of steam-pipes, A = 0,785 X 150 = 117.8 square feet. Formula 2. FI = *...* = 8 horse-power nearly, which is lost by radiation. RADIATION OF HEAT FROM STEAM-PIPES. 429 The same pipes covered with one-inch felt will gain 8 × 0.89 (see table) = 7.12 horse-power. The steam-engine in Bellavista works without expansion, and con- sumes about 10 ibs. of coal per horse-power per hour = 7.12 × 10 = 71.2 pounds, and for 8 hours' working = 569.6 lbs. of coal lost per day. The radiation of heat from steam-pipes causes a condensation of steam to water, and the weight in pounds of water so condensed is equal to the units of heat radiated, divided by the latent heat of the steam in the pipe. The Formula 1 will also answer for calculating the quantity of heat radiated from steam or water- pipes for heating rooms. Percentage of Heat or Power Gained by covering steam-pipes with felt and canvas outside. Diam Logarithm Thickness of felt covering in inches, I) k. } | } | } | # | 1 || 13 || 2 || 3 || 4 || 6 1 0.80663–1 65 || 76 || 81 86 92 || 94 || 96 || 98 99 || 100 2 0.79561–1 63 74 80 85 90 93 95 97 98 99 3 0.77408–1 61 72 79 84 89 92 95 96 98 99 4. 0.76096–1 59 71 77 83 88 92 94. 96 97 99 5 0.74809–1 57 69 76 82 87 91 94 96 97 99 6 0.73670–1 || 54 || 67 || 74 81 | 86 91 || 9 || || 95 || 97 90 7 0.72668–1 52 66 73 81 85 90 93 95 97 99 S 0.7183.S.–1 || 50 | 64 || 71 || 80 | 85 90 | 93 95 . 97 || 99 9 0.71179–1 47 62 70 79 85 89 93 95 97 99 T0 0.70705—I 45 || 61 69 '78 | 84 || 89 || 92 95 || 96 || 90 11 0.70417–1 || 42 || 59 || 67 || 78 83 || 88 92 || 94 96 || 9S I2 0.70321–1 || 40 || 58 | 66 || 77 83 8S 92 || 94 | 96 || 98 Lap-welded American Charcoal Iron Boiler Tubes. PASCAL IRON WORKS TASKER IRON WORKs, Philadelphia. New Castle, Del. Diameter of Heating surface Thickness Length of tube | Area of cross- | Weight the tube. per foot of length. of metal per square foot. section. - per foot of Outside | Inside. Outside | Inside. g ()utside. Inside. Outside Inside. length. Inches. |Inches. | Sq. ft. Sq. ft. Wg. inches. I'eet. Feet, Sq. in. Sq. in. Pounds. 1 0.856 0.2618 0.2241 | 15 0.072 3.S19 4.460 | 0,785 0.575 0.708 1.25 | 1.106 || 0.3272 0.2895 15 0.072 3.056 3.455 | 1.227 || 0.960 09 1.50 | 1.334 || 0.3926|| 0.3492 || 14 0.083 2.547 2.S63 | 1.767 | 1.396 1.250 1.75 | 1.56() 0.45S0 0.40S4 || 13 0.095 2.1S3 2.448 2.405 | 1.911 1.665 2 1.804 || 0.5236|| 0.4723 13 0 098 1.909 2.11S | 3.142 2.556 1.9S1 2.25 2,054 || 0.5890|| 0.5377 || 13 0.09S I.69S 1,850 | 3.976 3.31.4 2.238 2.50 2.283 || 0.6545 || 0.5977 i2. O.109 1,528 1,673 || 4.909 || 4.09.4 2.755 2.75 2.533 || 0.7200 0.6631 | 12 0.109 1.390 1.508 5.940 5.039 3.045 3 2,783 || 0.7853| () 7285 | 12 0.109 1.273 1.373 7.069 6,083 3.333 3.25 3.012 ().850S} 0.7885 || || 0.119 1,175 | 1.26S S.296 || 7.125 || 3.95S 3.50 | 3.262 0.91 63 0.8430 || 1 || 0,119 1.091 1.171 9.621 S.357 4.272 3.75 || 3.512 0.9S17 || 0.9.194 || 1 || 0.119 | 1.018 | 1.0SS 11.045 9.687 4.590 4 3,741 | 1.0472. 0.9794 || 10 0.130 0.955 1.023 12.566 |10.992 5.320 4.50 4.241 | 1.1781| 1.1105 || 10 0.130 0.849 0 901 15.904 14.126 6.010 5 4,720 | 1.3680) 1.2357 |9.5 0.14() 0.764 0.S09 || 9,635 | 17,497 || 7.226 6 5,699 || 1.5708 || 1.4920 9 0.151 0.637 0.670 2S.274 |25.509 9.346 7 6.657 | 1.S326|| 1.7428 ||7.5 0.172 0.545 0.574 |38.484 |34.805 || 12.435 8 7.636 2,0944|| 1.9991 || 7 0.182 || 0.478 0.500 50.265 |45,795 15.109 9 8.615 2.3562; 2.2553 |6.5 ().I.93 0.424 0.444 |63,617 ).5S.291 | 18,002 10 9,573 2.5347| 2,5022 |5.5 0.214 0.3S2 O 399 || 78,540 |71.975 22.19 The length of tube and thickness of metal can be varied to suit orders. The heating surface of a boiler tube is that exposed to the fire. Safe ends of thicker metal welded on the ends of tubes as may be required. .* 430 BLOWING OFF. INCRUSTATION. BLOWING OFF. SALT WATER. INGRUSTATION. Sea water contains about 0.03 its weight of salt. When salt water boils, fresh water evaporates and the salt remains in the boiler; consequently the proportion of Salt increases as the water evaporates, until it has reached 0.36 weight to the water; the Salt will then commence to saturate in the boiler, and the water in so- lutiou will hold 0.36 weight of salt to 1 of water. To provent deposit in the boiler, it is necessary to keep the salt below this pro- portion, which is overcome by withdrawing (blowing off) part of the supersaited Water, whilo less salted (feed) water is replaced. It is found in practice that when the proportions are kept 0.12 of salt to 1 weight of water, the deposit will be very slight. To obtain this it will be necessary to blow off— * fº 0.03_ 0.25 parts of the feed water, or, 0.12 if a brine-pump is used, it should be at least 0.25 of the feed-pump. I)2 S 77. W= cubic feet of supersalted water to be blown off per minute. W = } tº D, S, n and V, as before, we shall have— 3000 V Eacample. , D = 30 inches, stroke of piston 36 inches, cut off at half stroke Sa- 18, making 14 revolutions per minute, with a pressure of 30 pounds per square inch, W = 610. IIow much water must be blown off per minute? = 30°X18 × 14 0.124 cubic feet. 3000 × 610 Heat Wasted by BIowing Off. Letters denote, w = water evaporated and W = water blown off t = temperature of the feed water. T= 4 { “ blowing off. BI = heat wasted, per cent. _ _ W(T-1) -- w(990 + T-t) Jºcample. Let the quantity of water blown off be 3% of the feed water, we have W= 1, and w= 2; the boiling-point of the water will then be T– 215.50; let the feed water taken from the hot-well be t = 100°. Required, the quantity of heat lost? _ _1(215.5°-100) 2(990 + 215.5 — 100) This is a very trifling quantity of heat lost. }in cubic feet per unit of time. = 0.052 or 5.2 por cent. . Heat Wasted by Incrustation. The conducting power of iron for heat is about 30 times that of saturated scales, hence a considerable portion of heat is lost when the scales become thick in it boiler. * * t = thickness of the scale in 16ths of an inch. _ t? JH = per cent, of heat wasted. T & Tº Jºvample. The scale in a boiler is 5-sixteenths of an inch thick. IIow much heat is lost by it? 52 32 + 52 which goes out through the chimney. This is merely to show that the heat lost by blowing off is but trifling compared = 0.438, or 44 per cent., nearly, with the heat lost by Saturation of scales, which additionally injures the boiler by softening and fracturing the iron, and final explosions. When boilers are taken good care of by cleaning and blowing off at short inter- vals, the scales need not exceed 1-sixteenth of an inch. L– BELL SIGNALS. 431 Tºroportions of Salt in Water : its boiling-point and weight per cubic foot. Salt Boiling Weight Spe- Salt Boiling Weight | Spe- in 100 tenap. per cific in 100 temp. per cific Weights. Fahr. cub. ft. graw. weights. Fahr. cub. ft. gray. pounds. pounds. 0 2129 59.837 1.00 21 218,304 72.224 1.21 1 212.205 60.431 1 ()1 22 218.690 72.728 1.22 2 21:2.4.22 61,024 1.02 23 219.082 73.395 1.23 3 212,649 61.617 1.03 24 219.483 73.980 1.24 #r. 212.887 62.209 1.04 25 219.887 74.565 1.25 b 213.136 62 801 1.05 26 220.296 75.148 1.26 6 213.394 63.393 1.06 27 220.713 Tà,732 1.27 7 213,664 63.984 1.07 28 221.131 76.316 1.28 8 213.942 64.575 1.08 29 221.558 '76.899 1.29 9 214.229 65.166 1.09 30 221.9S4 77.482 1.30 10 214,526 65.756 1.10 31 222.4.19 '78.064 1.31 11 214.801 66.346 1.11 32 222,857 78.646 1.32 12 215.145 66,935 1.12 33 223.302 79.228 1.33 13 215.446 67.524 1.13 34 223.733 '79,810 I 34 14 215,797 68,113 1.14 35 224.208 80,390 1.35 15 216.132 68.701 1.15 36 224.668 8(J.970 1.36 16 216,477 69.289 1.16 37 225.139 81.550 1.37 17 216.826 |* 69.877 1.17 3S 225.6+1 S2.130 1.38 18 217.1S6 ’70 464 1.18 39 226.057 82.709 1.39 I9 217.550 71 051 T.19 40 226.572 $3.288 1.40 20 217.924 TI 377 I.20 Saturates with 40 parts of salt. Water does not increase in volume by addition of the above proportions of salt. To Command the Engineer how to Manoeuvre the Engine in a Steamboat. Go ahead, . e - º e ſº one stroke. Back, . º tº l -º- tº º ... two strokes. Stop, . • - º e e one stroke. Slowly, . º º J. te o ... two short. Full speed, ſº --- e º three short. Go ahead slowly, - -- tº e ... one long, two short. Back slowly, wº 2 --- º tº two long, two short. Go ahead, full speed, - - . -º- o ... one long, three short. Back fast, . e --- -º- -º- ... two long, three short. Hurry, . . . -- J"- -- ... three short repeated. It is also customary to have two bells in the engine-room—a large bell for the long strokes, and a smaller for the short strokes. 432 BRASS TUBES, STRENGTII OF IRON AND COPPER. Weight, Size, Price and Surface of Copper and Brass Tubes, 10 feet long. ()htside ... Weight, of tube, Pri Whole HOutsid e Weight of tube, Pri Whol Sºlºyºlºr|ºſºvº ºr ſº eter. 8° Brass. Cop. tube face. Cter. * | Brass. Cop. tube. face. Inches. No. Lbs. Tilbs. scts, sq. ft.) Inches. | No || Los Timbs. | Scts.sq. ft. 0.625 | 18 || 3,478 ; 3.6S1 || 2 30 | 1.636 * 14 | 18.84 19.95 || 8 ()0 5.236 0.75 17 4.950 || 5.241 || 2 97 1,963 || 2,125 || 14 || 19.07 || 20.18 || 8 20 5.563 .8125 || 17 || 5,372 5.679 || 3 00 2,127 || 2.25 14 || 21.18 22.42 || 8 90 5.890 0.875 | 17 | 5,775 || 6.114 || 3 40 | 2,290 || 3.375 || 14 22.32 || 23.65 || 9 45 6.217 .9375 | 16 || 6.954 || 7,362 || 3 60 | 2,454 || 2.5 14 || 23.53 || 24,89 || 9 95, 6.544 1. 16 || 7.418 5.854 || 3 92 || 2.618 || 2.625 || 14 24.67 26.12 10 45 6.872 1.125 16 || 8.354 || 8,835 || 4 40 || 2,945 || 2.75 14 || 25.83 || 27.35 | 11 00 7.200 1.25 15 || 10.21 || 10.83 || 4 70 || 3.272 || 3. 13 || 37.00 || 39.17 |13 70 || 7.854 1,375 | 15 11.23 || 11.91 || 4 95 || 3,600 || 3.25 13 | 40.00 42.34 ||14 85 | 8.508 1.5 15 | 12.28 13.00 || 5 20 | 3,927 || 3.5 13 || 43.10 i 45.61 |16 00 || 9,163 1.625 | 15 || 13.30 14.08 || 5 65 || 4.254 || 4, 12 49.39 52.30 |17 00 | 10.47 1.75 14 | 16.5 || 17.45 || 7 00 || 4,581 || 4.5 12 || 55.55 || 58.8 |19 10 | 11.78 1.812 || - 14 17.08 || 18.08 || 7 20 4,745 || 5. 12 || 61.44 65.00 |21 00 || 13.08 1.875 || 14 || 17.72 | 18.75 7 5() || 4,908 || 6. 11 || 81.58 || 86.35 |26 00 | 15,71 1.937 14 | 18.26 || 19.32 || 7 75 5,072 || 8. II 108.8 || 115.0 34 50 20.95 Seamless-Drawn Brass Tubes for Plumbing, In lengths of 10 feet. Screw-couplung on one end of each length. Price per tube. Diameters, inches. # | # # 1 | 1+ | 1% Scts. Scts. Scts. Scts. $ cts. $ cts. Plain tubes, . . ſe 2 50 3 ()() 4 50 6 (10 7 00 8 00 Tinned tubes, . 3 00 3 50 5 00 7 00 8 00 9 00 Price of Taps, Dies and Stocks. Diameters, inches. § # § | l 1#. | 1% Scts. Scts. Scts. Scts, Scts. Scts. Taps, . • * 2 50 2 75 3 00 3 50 4 50 6 00 Solid dies, § e 3 50 3 50 3 50 3 50 3 50 4 00 —i. Stocks, $8, net. Price for Each Extra Coupling. Diameters, inches. # | # | # 1 1} 1; $ cts. Scts. Scts. $ cts. Scts. $ cts. Straight couplings, . 20 25 35 40 45 5() Jºlbows, . e ſº 26 35 48 53 65 SO Tees, . e * g 30 40 55 60 S5 1 00 Cross couplings, 45 60 85 90 1 50 2 20 more than of brass. The prices are only approximate. The price of copper tubes is 12 to 13 per cent. Brass and copper tubes are manufactured at the American Tube Works, Boston, Mass.; Merchant & Co., 507 Market street, Philadelphia, agents. Proportionate Temsile Strength of Rolled Iron and Copper, In pownds per square inch, at different temperatures, Fahr. and Centigrade. Fahr. Cent. Iron. Copper. Fahr. Cent. Iron. 32 0. 55,000 32,800 800 427 51,800 100 37.7 58,200 32,300 90ſ) 483 45,000 200 93.3 62,800 31,000 I000 540 37,000 300 149. 65,750 29,500 1200 650 25,000 400 205. 67,000 27,400 1500 820 16,500 500 260. 66,000 25,300 2000 1090 7,000 600 316. 62,700 23,0 2500 1370 2,500 700 370. 57,800 20,100 3000 1650 Fused. Copper. 17,200 14,000 11,000 7,000 3,000 0.0000 Fused to liquid. NAIIs, RiveTs, IRON, CoPPER, ZINC. 433 Composition Nails, Copper and Iron Rivets. Composition Nails. || Braziers' Copper Rivets. Iron Rivets. In 10 #. º Diameter. Length. ibs. # Thick. Length. Diameter. Length. N o Inches. Inches. Num iſ Inches. Inches. Num. . . Inches. Inches. Num. 3.f4 290 3 fló 1.f3 23S4 O 3f 16 1.f3 3280 .05 260 || 1.f4 1.f3 || 1018 1.f4 1.f3 | 1276 0.06 || 1 inch. 212 || 1.f4 9 f16 983 1.f4 9 f 16 || 1130 0.07 1.1.f3 201 || 5 f 16 9 ſió 573 5 fl6 9 fló 654 ().08 1.lf4 || 199 || 5 ſ 16 5.f3 516 5 f 16 5f 8 589 0.09 || 1 inch. 190 || 3/8 7f8 357 7f8 407 0.10 | 1.lf.8 184 || 3 f$ 15 fló | 334 3. f6 15 f 16 || 380 0.10 | 1.1.f4 168 || 7 f 16 || 1 inch. 210 7 f16 || 1 inch. 239 0.11 1.1 ſ 2 | 110 || 1ſ2 1.3 fló I41 1.f3 1.3 fió 160 10 : 0.11 1.5 ſ.3 101 9 ſld 1.5 fló 99.5 9 fló 1.5 fló 112 11 : 0.12 1.3 ſ4 74 || 5 fs 1.7 fló | 71.9 || 10 || 5 fg 1.7 fló | 81.7 12 0.12 |2 inches 64 || 11 fló | 1.9 ſ 16 || 53.8 || 11 || 11 fló | 1.9 fló | 61.3 13 0.13 2.1.f4 59 || 3.f4 1.3 f{ 41.6 || 12 3f4 1.3.f4 47.3 # O 0 3. 7 f 8 3 f 8 iá | 0 iſ 2 if 2 || 5 || 13 fig 1jišfig | 32's 13 || 13 fig 1.f3 fig | 373 15 0.15 2.3.f4 43 || 7 f$ 2.1.f 16 || 26.3 || 14 7 f$ 2.1 fló | 30. 16 || 0.16 13 inchest 35 || 1 inch. 2.3 f$ 16.7 | 15 || 1 inch. 2.3 f6 19. Length in Inches of Penny Nails. 1 in. |1.25 | 1.5 |1.75 || 2 |2.25 || 2.5 2.75 || 3 || 3.25 3.5 || 4 || 4.25 || 5 || 5.5 | 6 2 d. | 3 d. 4 d. | 5 d. 6 d. 7 d. 8 d. 9 d. 10 | 12 | 16 20 30 | 40 || 50 | 60 Sheet Zinc and Iron. - SHDET ZINC. RUSSIA SHEET IRON. Size 84 in. by 24, 28, 32, 36 and 40 inches. Size 28 × 56 in. = 10.88 sq. feet. Zinc Width of Sheet. Bir. W. j| Russian Weight per Bir. W. gauge. 24- 32 40 gauge. gauge. Sheet. Sq. Ft. gauge. No. Pounds. | Pounds. | Pounds. No. No. Pounds. | Pounds. --- 8 .2 9.68 12.1 28 7 6.25 0.574 29 9 7.20 Il.2 14.0 27 8 7.25 0.666 28 10 8.00 124 15.6 26 9 * 8. 0.735 27 11 8.90 13.8 17.3 25 10 # 9. 0.827 26 12 10.1 15.7 19.7 24; IL 10. 0.91S 25 13 11.1 17.3 21.6 23 12 10.75 0.987 24; 14 12.4 19.3 24.1 22 13 II.75 1.08 24 15 16.2 25.2 31.6 21 14 12.5 1.15 23# 16 17.4 27.1 33.9 20 15 13.5 1.24 223 18 21.9 34.0 42.6 18 16 14.5 1.33 21; To find the Weight of Castings, by the Weight of Pine Patterns. RULE.— 12 for Cast Iron, Multiply the weight # . #. and the product is the of the Pattern by 12.2 for Tin, 2 weight of the Castings. 11.4 for Zinc, Reductions for Round Cores and Core-prints. Iºwle. Multiply the square of , the diameter by the length of the Core in inches, and the product by 0.017, is the weight of the pine core, to be deducted from the weight of the pattern, Shrinking of Castings. Cast Iron, . ; * Brass -º- & Pattern-Makers' Rule * . . .” of an inch longer per should bo for Leatl, . . . ; linear foot. rºws l Tin, . . . Tº Zinc, . . # 434 WEIGHT OF BOILERS AND ENGINEs. To Approximate the Weight of Steam Boilers. The area of fire grate gives a nearer approximation to the weight ef Marine boilers, than the heating surface. * Letters denote. E = total fire grate in square feet. W = weight of the boiler in pounds, including fire bars, doors, smoke pipe, fire tools and appendages, but without water. WT300 É g Ea’ample. Required the weight W-4 of a steam boiler of E =250 square feet, grate surface. W=800X250=200,000 lbs. F. of the water is about three-fourths of W or of the total weight of boilers. Weight of rivets, braces or stays, doors and fire bars, is about one quarter of W or of the total weight of boilers. To Approximate the Weight of Engines. Letters denote JD = *} of cylinder in inches. S = stroke W= weight of engine in pounds, including engine room tools, oil and tallow tanks, wheels, propeller and shafts. * cocfficient k. Trunk and oscillating engines, - - - - - - - 4 Direct action paddle wheel engines, - gº º º gº - 4:25 Horizontal direct action propeller engine, - - - - 4. Geared propeller engines, - - - - - - - - 5 American overhead beam engines, - - - - - - 5. 6 3 Side lever engines, - - * * * = tº º sº gº Horizontal direct action high pressure, - - - - - W = k Dº yS. Eacample. Require the weight. W=% of a pair of Horizontal direct ac- tion propeller engines of D=72, S-36 inches, k=4-5. W= 4.5×72°/36 = 139968 lbs. for one cylinder, multiplied by 2–2,9936 lbs. the weight required. For trunk engines must be taken the largest diameter. Practical Thickness in Decimals of an Inch of Good Plate Iron in Steam-boilers, Single Riveted. P= steam pressure in pounds per square inch above atmosphere, Press. Diameter of Boiler in Inches. P. 10 | 15 20 25|30} 35|4-0|50|60|70|80}90 100||120|150|200 PUNCHING ANI). SHEERING. 435 | Punching Iron Plates. To punch iron plates of from # to 1 inch thick requires 24 tons per square inch of metal cut ; that is, the circumference of the hole multi- plied by the thickness of the plate is the area cut through. Letters denote. d = diameter of the punch or hole. D = diameter of the hole in the die. t = thickness of the iron plate. All dimensións in 16ths of an inch. W= weight or force in pounds required to punch the hole. W= 660 t d. D =d–H0-2 t. Ea'ample 1. An iron plate of t =6 sixteenths of an inch thick, and the hole to be d-12 sixteenths in diameter. Required the force W-4 W=660×6X12=47520 lbs., the answer. Eacample 2. Under the same conditions require the diameter D=? of the die. D=12+0.2X6=13-2 sixteenths. Eacample 3. Required the diameter of piston for a direction action steam punch, for the plate and hole as in example 1, pressure of steam to be 50 lbs. per square inch. Force 47520=AX50 of which area of piston will be A-tº-soo's square inches, which answers to a diameter of 34.8, say 36 inches. Shearing Iron Plates. It requires the same force per section cut, for shearing as for punching, namely, 20 to 24 tons per square inch. If the shears are good, sharp, and well adjusted, 16 tons may be sufficient. When the cutters in the shears are inclined to one another, the area cut, will be the square of the thickness of the plate multiplied by half the cotagent for the angle of the shears. Let v-angle of the shears, W and t same as for punching. W=88 tº cot.ºy, * Eacample 4. What force is required to cut a halfinch plate t—8 sixteenths with a pair of Shears forming an angle of v = 12°. Cot.12°=4-7, W=88×8”x4-7–26470 lbs. Atmospheric Columns. Water=33.95 feet. 2'3 feet for 1 lbs. per square inch. Seawater=33-05 ft. 2:23 $6 & 4 Mercury at 60°=30 inches. 2:05 inches, { % Atm. air=28183 feet. 1912 feet, £ 6 & { { Atmospheric air Required for each. Blacksmith's forge, - - 100 to 200 Charcoal forge, - - - - 400 to 500 * & Finery forge, - - - - 800 to 1000 Cubic feet per minute. Charcoal furnace, - - 1000 to 3000 Anthracite furnace - - 2000 to 5000 Cupola.- In a cupola of 3 feet 4 inches diameter, and 10 feet high, can be melted down 1000 lbs. of cast iron, 200 lbs. of bitumninous coal per hour, with a blowing machine of 4.5 horses making 1700 cubic feet of air per minute into a pressure of 2.25 inches of mercury at which the temperature of the blast will be about 70° Fah. 436 STEAM-BOILER EXPLOSIONS. Steam-boiler Explosions. THE steam-boiler is a reservoir of work. Each unit of heat in the steam and water is equivalent to a work of 772 footpounds. The steam-table gives the units of heat per cubic foot, or per pound, in the steam and water at different temperatures and pressures. Work is the product of the three simple elements force F, velocity V, and time T, or K = F V T. x- when the force of the work will be F = #. When the pressure in any part of a steam-boiler is suddenly removed, the entire work in the steam and water is at the same time started with a velocity proportionate to the removed pres- sure. The steam and water, in the form of a foam, strike the sides of the boiler, by which the work is suddenly arrested. If the time of arresting the work is infinitely small, we see from the above formula that the force of the work will be infinitely great, and thus the boiler explodes. Steam-boiler explosions are caused in various ways, namely: 1st. By long use boilers become corroded and, from neglect, give way in some unexpected place. 2d. The general construction with staying and bracing of steam-boilers is often very carelessly executed and results in explosion. This kind of explosion is often indicated, long before the accident occurs, by leakage of the boiler; when the engineer, not suspecting the approaching danger, limits its remedies generally to efforts to stop the leak. The leakage from bad calking, or pack- ing, is easily distinguished from that of bad or insufficient bracing. In the latter case the fire onght to be hauled out, the steam blown off, and the boiler secured with proper bracing. 3d. Explosion is sometimes caused from low water in the boiler, but more rarely than generally supposed. When the fire-crown and tubes, subjected to a strong heat and not covered with water, the steam does not absorb the heat fast enough to prevent the iron from becoming so hot that it cannot withstand the pressure, but collapses from weakness. 4th. Steam-boilers often burst by strain in uneven expansion or shrinkage, occasioned by the fire being too quickly lighted or extinguished. 5th. It is a very bad practice to make boiler-ends of cast-iron, composed of a flat disc of from two to three inches thick, with a flange of from one to two inches thick, with cast rivet-holes. The first shrinkage in the cooling of such a plate causes a great strain, which is increased by riveting the boiler to it. Any sudden change of temperature, therefore, either in starting or putting out the fire, might crack the plate and thus occasion an explosion. Such accident may be avoided by making the cast-iron ends concave and of even thickness. 6th. In cold weather, when the boilers have been at rest for some time, they may be frozen full of ice: then, when fire is made in them, some parts are sud- denly heated and expand, whilst other parts still remain cold, causing an undue strain, which may also burst the boilers. Such accident can be avoided by a slow and cautious firing. 7th. Sometimes a great many boilers are joined together by solid connections of cast-iron steam-pipes, which expand when heated, whilst the masonry en- closing the boilers contracts. Should such a steam-pipe burst from expansion or shrinkage, explosion will likely follow in all the connected boilers, of which numerous examples have occurred. Such accident may be avoided by making the connection elastic, or free to expand or contract without moving the boilers. Steam-boiler explosion is thus not always caused by the pressure of steam alone, but often by the expansion and contraction of the materials of the boiler. A steam-boiler which is perfectly safe with a working pressure of 200 lbs. may explode with a pressure of 20 lbs. to the square inch. The bursting of a boiler is a preliminary process to explosion. A boiler may burst Without exploding. A boiler full of steam may burst, but never explode. It is the work in the heated water which makes the explosion. The sudden disturbance in the water by the forming of foam generates a high heat, and consequently a corresponding high steam-pressure is formed over the original pressure, and thus causes a disastrous explosion. It is evident from the results of explosions that a much ligher pressure had been acting than the normal Working pressure. IJESTRUCTIVE WORK OF STEAM-EOILER EXPLOSION. 437 Destructive Work of Steam-boiler Explosion. When a steam-boiler explosion takes place, the enclosed water is resolved into one volume of boiling hot water, and one volume of steam, as follows:— IVotation of letters prior to caplosion. W/= weight in pounds of the water under full steam pressure in the boiler. w/- pounds of water evaporated in the explosion. h = units of heat per pound in the water W’. FI = units of heat per pound in the steam of pressure P. EI/= units of heat per cubic foot in the steam P P = pressure of steam in pounds per sq. in. W = Volume coefficient of steam. W H (h – 180-0) I / — Then 70’ = 824.8 * The destructive work K will be in footpounds. Wy/ H. P 2 K=####0–1809)(7–1)(28 log P-16348298). The number which expresses the destructive work of an ordinary steam- boiler explosion in footpounds is so large as to be incomeeivable to the mind; for which it is proposed to express explosions loy a larger unit, as workmandays, of 1980000 footpounds each, or 2564.75 units of heat. (See page 262.) The workmandays of explosion will be W. H. P(H – 1809)(7–1)(23 log, P-16848298) 3 - 11341440 HIV V e Example.—In the explosion at CORNELIUS & BAKER’S, Philadelphia, April 25, 1864, the boilers contained about W = 14750 lbs. of water; the steam- pressure P = 80 lbs., H = 1177.05, H/ = 223.82, W = 328:08, and h = 282-78. Required the workmandays of the explosion. 14750 × 1177.05×80(28278–1809)(328:08–1)(2.3 × log.80–16848298) 11341440 × 223-S2 × 328-08 - 149-63 workmandays; or 150 men in one day could do the work of the explosion. The dostructiveness of an explosion is thus proportioned to the quantity of water and units of heat in the boiler. The steam in a boiler, prior to the ex- plosion, does very little or no damage in the explosion. The work concentrated in a given volume of steam expanded into the atmosphere is R = C. H. (0.1518 log.P.- 0:1771987). 4 C = cubic feet of steam in the boiler. le k = Precaution against Fire on Steamboats. Each steamer should have three buckets for every 100 tons measurement, plus 10 buckets. That is, a steamer of 800 tons should have 8 × 3 + 10 = 34 buckets. Also one axe for every 5 buckets. U. S. Steam-boilers Inaspector’s Rule for Strength of Boilers. Multiply one-sixth (V6) of the lowest tensit strength found stamped on any plate in the cylindrical shell, by the thickness expressed in parts of an inch of the thinnest plate in the same cylindrical shell, and divide the product by the radius or half the diameter of the shell expressed in inches, and the quotient will be the steam pressure in pounds per square inch allowable in single riveted boilers, to which add twenty per centum for double riveting. S = breaking.strain in pounds per square inch stamped on the plato. = thickness of the plate in fraction of an inch. D = diameter of the boiler in inches. P = steam pressure in pounds per square inch. = 8* D = ** t – 3 D.P S – 3 DP ºmºmºmºmº l 3 D 3 P S t #38 SUPERFIEATING. S U P E R H E A T E D STEAM. The Author's experience in superheated steam has been sufficient to convince him of its great importance. It appears that in order to utilize the maximum effect of steam or at least to attain the maximum quality of expansion, it is not necessary to overheat it after a pure steam is formed, that is, when all the small particles and bubbles of water are evaporated. Water which accompanies the steam in such a form has the same temperature as that due to the surrounding steam pressure, prevents it to vaporise; but when it passes through the superheating apparatus the temperature is greatly increased while the pressure re- mains the same because it being in connection with the steamroom in the boiler allows the water to vaporise and a pure steam may be formed. If steam with particles of water is admitted into the cylinder part of the stroke and then allowed to expand, it is generally found that the end pressure does not come up to that due by theory, from which it has been pronounced that the expansive quality of steam ãoes not follow that of a perfect gas, and that steam has condensed during the stroke ; but if we knew the cubic containt of all the particles of water and subtracted that from the cubic containt of the steam it might be found that its expansive quality is not so far from that of a perfect gas. It appears also that the expansive quality is diminished by overheating pure steam. The small particles of water contain a great deal more caloric per volume than the surrounding steam, consequently when admitted into the condenser a good vacuum cannot be formed so well as with pure steam. It is therefore of greatinportance to pay particular attention to the superheating of steam, otherwise economy by expansion will not be realized to the extent herein given by formulas and tables. It is also of great importance for expansion that the piston and steam valves are per- fectly tight. SU P E R Eſ E A T IN G A P P A R A T U. S. The accompanying figure represents a superheating apparatus such as the Au- thor has built it in Russia, and is found to answer exceedingly well. The figure is a section of the forend of an ordinary tubular boiler with steamdrum and up- take. The chimney is made a great deal wider in the steam drum and contracted to the usual size at e, of 0-16 times the area of the firegrate; if a strong fan blast is applied it may be better to contract it to 0-114-H. In the inside of the chimney are placed a number of copper tubes a, a, b, b, with flanges screwed to the side; the area of these tubes should be about four times that of the steam pipe c. In the steamdrum is riveted steamtight a conical plate d, d, so that the stoam can- not pass to the top without passing the superheating pipes. This superheating apparatus is in successful operation in three first class passenger steamers on the River Volga in Russia, each of 500 actual horses, and one in a steamer of 100 actual horses on the Black Sea. The steamdrum can be placed around the chimney separately from the boiler and the Rteam led either above or below ~~~ the plate d, d, by pipes from the steam- room, as may suit the circumstances. This superheating apparatus may also be well suited for locomotives. Furnuco GIFFARD INJECTOR. 439 THE GIFFARD INJECTOR, Sellers’ Patent. The following table has been furnished by William Sellers & Co., Philadelphia, manufacturers of this injec- tor. It gives the quantity of water injected per hour in cubic feet. The first column No. is the size or diameter of the throat in French millime- ters. The last column is the size in 16ths of an inch. Capacity and Size of Giffard's Injector. Size Pressure of steam in pounds per Square inch above atmosphere. Size No. 10 || 20 30 || 4-0 || 50 60 || 70 80 90 100 | 110 120 130 150 16ths 2 || 8.3 9| 9.7 10.4 II.L. 11.8; 12.5 13.2| 13.9 || 14.6; 15.3| 16.0; 16.7 | 18.1; 1.26 3 |19.3 21.0 22.8 24.6; 26.3| 28.1| 29.9| 31.6|| 33.4| 35.2; 37.0 38.7 40.5| 44.1: 1.89 4 36.6|| 39.6| 42.7 45.9| 49.0|| 52.1 55.3| 58.4| 61.6| 64.7 | 67.8 71.0| 74.1 S0.4; 2.5 5 §7.6| 62.5| 67.4|| 72.3| 77.2| 82.2| 87.1| 92.0| 96.9| 102 107 || 112| 116; 126|| 3.6 6 S3.5| 90.6 97.7 105; 112| 119 126|| 133 140|| 147 155 162| 169; IS3; 3.78 7 || 114 124; 133| 1.43 153 162] 172| 182; 192| 201| 211| 221 231: 250|| 4.41 8 || 149 | 162| 174 IS7 200| 213. 226] 239 251 | 264. 277 200 303| 32S 5.04 9 || 189| 205 221 237 || 254| 270] 286 302. 318|| 334 351 367 || 3S3 415 5.67 10 || 234 254| 274| 294; 313| 333| 353. 373| 393 413| 433 453| 473| 513; 6.30 12 || 337 366|| 395 || 423| 452; 481| 510 || 539| 567 596 625 654|| 682| 740; 7.56 14 || 451 491 || 531|| 571 611 651 691. 731|| 771 S11 || 851 | 891 931|1011: S.S0 16 600|| 65|1| 703; 784 805 || 857 908 959|1010; 1062 1113||1164|1215||131S 10.l 18 760| S25 | 800; 955|1020 1085 |1149|1214;1279 1344. 1409 1474 1539|1669 11.3 20 | 939|1019 [1099|1179||1259 |1339||1420 |1500|1580) 1660|1740; 1820, 1900 |2061 12.6 Method of Working the Injector. FIRST.-See that the steam-plug is closed down, and waste-valve stem is raised. SECON D.—Admit steam from boiler to Injector, which should cause the water to flow from the waste pipe. THIRD.—Turn up the steam-plug until the waste valve can be closed without causing the Injector to cease working. FourtTII.—Turn up the steam-plug to increase the delivery, and down to decrease it. When this Injector has to lift its supply water, the steam valve between the Injector and boiler must be opened very slowly, until the water flows out of the waste pipe. N. B.-A failure to work will always he indicated by an escape of steam and water from the waste check attached to check Valve in water-supply pipe. 440 BLOWING MACHINES. B L O W IN G M A C EIT N E S. & © Letters deſtote. Q- º, ‘. inches.) of blowing cylinder double acting. y ! = part of the stroke S under which the air compresses from the atmospheric density to that in the reservoir. F= mean resistance in pounds per square inch of the air on the cylinder piston. P = pressure in pounds per square inch of the blast in the reservoir. C = cubic feet of air of atmospheric density, delivered from the blow- ing cylinder to the reservoir per minute. H = actual horse power required to work the blowing engine, includ- ing 13 per cent. for friction. d = diameter of blast pipe in inches. n == number of revolutions or double stroke per minute. A = area of supply valve to the blowing cylinder in square inches, at each end of cylinder. p = vacuum in pounds per square inch, on the supply side of the cylinder piston, which should not exceed 0-1 lbs. W = velocity of the blast through the tuyeres in feet per second. v = velocity of the air through the supply valve A, in feet per second, which should not exceed 100 feet. a = area of the orifice or tuyeres in square inches. h = height of mercury in inches, in the gauge on the blast reservoir. L = length of the blast pipe in feet from the receiver to the tuyeres. k = volume coefficient, see Table. t = temperature Fah. of the blast caused by compression or heating. Eacample 1. Formulae 8. For an Anthracite blast furnace is required 4000 cubic feet of air per minute, under a pressure of 6 inches mercury. Required the horse power necessary for the blowing machine The ef- fectual resistance F=2'365 lbs. see Table. Assume the vacuum to be 19=0°09 lbs. 4000 (2.365+0-09) 198 Example 2. Formula 10. Suppose the blast cylinder to be D=144 inches diameter with S=15 feet stroke. Required the number of double strokes per minute n=? We have H = = 49°6 horses. _96X4000 T 1442×15 Eacample 3. Formatlac 9. Under the above conditions, require the area of the supply valves A=4 when the velocity v=105 feet, per second. 2X 15X 12-3 A 144*X15X = 911 square inches. 40X105 = 12'3 the answer. Capacity of B1ast Reservoir should not be less than the following proportions, but more is better. For one single acting cylinder, 20 For one double acting cylinder, 10}-times the capacity of one cylind’r. Two double act. cyl. cranks at 90° 5 One double acting cylinder, same as two single acting. The more cylinders the less capacity required for blast reservoir. V=246/E (I-F07002087), t=32–H 493 (k—1), P= 14-7 (k—1), t = 33:55 P-- 32, _t–32 k = P T 33.55? ~ 14.7, BLOWING MACHINES. 44 I. Formulas for Blowing Machimes. Sh C=1:83ah(30+h),6, a_v^C+10 E 11 For TT1 7 3 *g, j P=0.49%, - - 2 19000 v=350 vp - 12 Q D° S 71. c_P.. ", - - 3'H_* †P), s y 13 96 198 40 A 1 De S n Dº Sº nº C= *H, - - 4 A= - y - 9 P-55aaaaaaa-Tº F+p 40w 180000000 A. h = ***, . 5|n="' - - 10 * ***, . - 15 26 D” S | 30 Table for Blast and Blowing Machines. volume and temperat, TGuage in inches.T. Teressure ibs, sqinch. T stroke. Twotocity. k | t *Dater. h P F l y I-002 330 I 0-073 0-036 || 0-032 || 0-0024 72 1-005 34°5 2 0-147 0-079 || 0-063 || 0-0049 I 02 1.007 35'5 3 0-220 0-108 || 0-095 || 0-0073 I25 1.010 37 4 0°294 0-144 || 0-128 || 0-0097 144 1-0 12 38 5 0.36S 0-180 || 0-159 || 0-0121 I 61 1-015 39°5 6 0-441 0-216 || 0°] 91 || 0-01.45 I 76 1-020 42 8 0.588 0-288 || 0-253 || 0-0192 20.4 1-025 44'5 10 0.736 0-360 0.309 || 0-0239 228 1.030 47 12 0-884 0.432 0-379 || 0-0287 249 1*035 49' 5 l4 I •030 0. 503 0-437 || 0-0334 269 1.043 53-5 17 1:250 0.612 || 0-531 || 0-0.400 297 1-051 57-5 20 I-470 0.719 || 0.623 || 0-0467 322 I-062 63 24 1-766 0.863 0-745 || 0-0556 352 I-074 69 28 2-060 1-008 || 0-865 || 0-0643 3S1 1-082 73 31 2-281 1-116 0-955 || 0-0706 401 1-09] 77.3 34 2-501 1-223 1-043 || 0-0769 420 1-100 82 37 2.720 I-332 || 1-130 || 0-0833 438 1-109 86.5 46 3.000 1-4.70 | 1.205 || 0-0908 460 1-116 90 47-5 3•500 I-715 1-431 || 0-1045 496 1.132 98 54".3 4'000 1.961 | 1.636 || 0-11.78 530 1-165 || II.4°5 67-7 || 5,000 2.450 2-010 || 0-1431 593 I-200 || 132 8] -4 || 6-000 2-941 2-365 || 0-1667 650 1-265 | 164°5 108.5 | 8-000 3.925 3-0SS || 0-2105 751 1-400 232 J 63 I2-00 5-9 (10 || 4-389 || 0-2S59 918 ! I •500 2S2 203.7 15.00 7-375 6.875 || 0-3333 1077 1-625 || 344°5 254-6 || 18.75 9-217 | 8-831 || 0-3846 1393 1-750 407 305-5 22:49 I 1-06 || 10-67 || 0-42S5 I 590 1-875 || 469-5 356-4 26-24 13-90 11-64 || 0:4666 1760 { 2°000 || 532 407-4 30.00 14-75 12-50 || 0-5000 1955 442 FANS OR VENTILATOR. tºº- *-*- — F A N O R W E N T I ILAT O R. Fans constructed as the accompanying figure have been ſound by the Author who has made several of them, to be the most effective. The vanes are each one quarter of an arithmetical spiral with a pitch twice the diameter of the fan, that is, each vane should be constructed in an angle of 90° from centre to tip. Length of fan to be from # to ; the *A diameter. Inlet to be half the diameter of the fan. Number of vanes to be not more than six, and not less than four. Six vanes work gofter and better, but they give no better effect than four. NO2 The housing should be an arithmetical spiral with sufficient clearing for the fan at a, and leaving a space at b about # of the diameter. Fans of this construction make no noise. Letters denote. d = diameter }=;”) of fan in inches. #= tºº, } of blast pipe, to be as straight as possible. a = area in sq. in. tuyeres or outlet. C= cubic feet of air delivered per minute. h = inches of mercury. v = velocity in feet per second through a. k = volume coefficient, see Table, page 441. n = revolutions of fan per minute. H= actual horse power required to drive the fan. Formulas for Fams. *-**= "d l T 1 | a_n Vd.* d l T 50000000 25 a-Fāī 28-86 25a-Hål' d l h m n_2,000 H, • - 7 FI = 24000' º º 2 d l h 7) a k 24000 HI T 2.6 ° tº º tº º • 8. h=+...+, - - - - 3 C= 94 a k}/h. 9 v=244WD - - - - 4 A = C /T, 10 5 94 E\ h A = a VL - us * - Eacample 1. A fan of d-S6 inches diameter, l =12 inches, making n=-725 revolutions per minute, area of tuyere being a =25 square inches. Re- quired the density of the blast in inches of mercury h-? 86×725° 36×12 T 50000000\/ 25×25+86x12 Ea'angie 2. Under the same conditions require the cubic containt of air delivered per minute, C=? k=1:01 the nearest in the Table. Formulae 9. C=94X25X1-01)/0.242 = 1167.7 cubic feet. Required the horse power H=? Formulae 1. h. = 0.242 inch CS. •242 O Formula: 2. II = ºxº = 3'16 horses. IRON FuRNAcES. 44 B.T., A S T O R. I R O N E U R N A C E S. It is almost impossible to set up the many variable circumstances con- nected with the performance of Blast Furnaces, into a table form. The datas herein given are deduced to an average from the performances of a great many furnaces both in America and Europe. The accompanying Tables are so arranged that the numbers in Table I., multiplied by the numbers in Table II., gives the corresponding charge Iron ore, lime stone, coal, and the produce of pig iron in pounds per 24 hours, with the consumption of air in cubic feet per minute. Table iſ contains the effective capacity of blast furnaces in cubic yards. Eacample. It is required to construct a blowing machine for an Anthra- cite blast furnace of 12 feet diameter of boshes, height of stack 45 feet, to be worked with hot blast. Required the produce of pig iron per 24 hours cubic feet of air per minute and actual horse power of the blowing engine? º of pig iron 155 Table I.x123 Table II.-19065 lbs. or 8-5 tons per 24 hours. & Consumption of air 20X123=2460 cubic feet per minute. Suppose the blast to be blown into the furnace at a pressure of P = 2.94 lbs., vacuum in the supply side in cylinder to be p =0:07 lbs. we shall have the required actual power. 2460 (2.26+0.07) Forumla 8, p. 287. H 30-2 horses. 198 Table I. Iron or Blast Furnaces. Tºhe unit being the capacity Charge and produce per 24hours. Air of the Furnace in Iron Pig #mé | Coal per cubic yards. Ore. Iron. | Stone. * | minute. ibs, T Tibs. - lbs. lbs. cub. feet." Cold blast, 535 215 196 400 24 Soft charcoal {{..., 700 292 256 350 19 Cold blast 670 270 2.45 400 24 & I 3. f { Hard charcoa. {\. blast, 875 365 320 | 350 I9 Goke Cold blast, 268 I ()S 98 515 26 Warm blast, 350 146 12S 397 20 e * Gold blast 252 101 92 515 24 Pituminous warm biºt, ...? | | | | 1% . 19 •,• 2: Gold blast, 287 115 105 515 26 *** { warm blast, 3% 15, ij || || || 3. Table I 1. Capacity and Dimensions of Iron Furnaces. Diameter o Height of stack in feet. Bºsbes in ft. 25 30 35 40 45 50 55 60 8 40 44 47 51 54 58 62 65 9 50 55 60 64 69 73 7S 83 1() 62 68 74 79 75 91 96 102 11 75 82 89 96 103 110 117 123 12 90 98 106 114 123 130 139 147 13 105 115 125 134 144 153 I 63 172 14 121 133 145 I 55 167 178 189 200 15 140 I 53 166 I7S 19| 204 217 230 ió I 60 17.4 189 203 217 232 247 261 17 280 197 213, 229 245 262 279 295 l 18 202 220 239 257 275 293 312 330 sº ------sº-º-º- — £44 PARABOLIC CONSTRUCTION OF SIIIPS. ON THE PARABOLIC CONSTRUCTION OF SHIPS. The water-lines, frames, areas of frames and water-lines, and the displacement of a vessel, are all parabolas of different orders and power, with the vertex in the greatest cross-section or dead flat frame. The accompanying tables contain ordinates for different parabolas, with the cor- responding areas, displacement, centre of gravity, meta-centre, tangent for the angles of resistance and inflection. The lengths from ſº to the stem and sterm, and the draft from the water-line to the keel, are each divided into eight equal parts, at which the ordinates are calculated. The first column in the tables contains the exponent n for the parabolas. The higher the exponent is, the fuller will the vessel be. The power q of the parabolas makes hollow lines when greater than 1, and the higher the power is, the hollower will the lines be. The power of the water-lines should never be less than 1. The top-line contains the number of the ordinates, which are counted from the stem or stern to ſº, or from the keel to load-water line. The half breadth of the beam is the unit for the ordinates of the load-water line, and for the frame Ø. It requires some experience in selecting the proper exponents and powers, but let us suppose, for example, the exponent n = 2 and the power q = 1, for the for- ward load-water line of a vessel of length l = 100 feet from X) to stem, and the beam B = 30 feet. Then multiply 15 feet by the ordinates in the table, and the products will be the corresponding ordinates for the water-line, the whole area of which will be (see column a), a = 100 × 30 × 0.6666 = 1999.99, or 2000 square feet. The centre of gravity of that area will be (see column e)— * = 100 × 0.3750 = 37.5 feet from Q. Now let us select the exponent m = 5 and tho power q = 1 for the frame (X), and let d = 12 feet depth from the water-line to keel. Then multiply 15 by the ordinates of that exponent and power, and the products will be the corresponding ordinates for the frame (X). The area of the frame (X) will be (see column a), Q = 30 × 12 × 0.8333 = 299.988, or 300 square feet. The depth of the centre of gravity of ØØ will be (see column e)— e = 12 × 0.4286 = 5.14 feet under the load-water line. Let us select the exponent n = 2.25, and power q = 1 for the forward displace- ment. Then multiply (X) = 300 square feet by the ordinates for that exponent and power, and the products will be the corresponding areas at each frame. The forward displacement will then be (see column a)— D = 300 × 100 × 0.6923 = 20769 cubic feet. The distance from (X) to the centre of gravity of the forward displacement will be (see column e), e = 100× 0.3823 = 3$23 feet. The operation is precisely the same for the after-body of the vessel, only that fuller exponents are generally selected. After the lines are laid out as described, the divisions of the frames are made as required in the building of the vessel. For simplicity in our illustration, let us suppose that the exponents and powers are respectively the same for the after-body of our vessel, and 100 feet from (X) to stern ; then the displacement will be D = 20769X 2 = 4153S cubic feet, the whole length L = 100 × 2 = 200 feet. The height of the meta-centrum will be— o, LL bºm__200x15"x0.3021 JD 4135S The coefficient m = 0.3021 is taken for the exponent and power of the load-water lime in column 7m. The tangent for the mean angle of resistance to the vessel in motion in water will be— & t 300× 1.447 2 l/d 2 × 100 × 12 tang. 10° 15', the mean angle of resistance. = 4.9305 feet. tang. v = = 0.18087 = PARABOLIC CoNSTRUCTION OF SHIPs. 445 The coefficient t = 1.447 is taken for the exponent and power of the displace- Iment is column t The whole area of the load-water line is a = 2000 × 2 = 4000 square feet The depth of the centre of gravity of the whole displacement will be— d 12 -- == = 5.28 feet, under the wa- 2( 2––P ) 2 (2– 41358 ) ’ tier line. a d 4000 × 12 To Calculate the Ordinates for the Frames, IIaving given the half area of the frame, the ordinate of the load-water line and the depth d, multiply the ordinate by the depth, and divide the given area by the product, which will give a decimal fraction. Pind this fraction in the columns a, or area in the tables for frames, multiply the ordinates of that area by the given ordinate for the load-water line, and the products will be the corresponding ordinates for the frame, Example. The given area = 64.5 square feet, the ordinates of the water-line = 6 feet, and the depth = 18 feet. Then #; = 0.599. Find this area in table q = 1 for frames, and it will be found to correspond nearly with the ordinates of exponent n = 1.5, which will give a full line: but if it is desired that the frame should be a reverse curve, then select the area of higher power—for instance, In table q = 1.25, the area 0.599 corresponds nearly with the ordinates of the expo- nent m = 1.75, in which case the frame will reverse at 18 × 0.77 = 13.86 feet from the water-line. To lay down the Ordinates of all the Water-lines and frames on the drawing direct from the tables without calculation. Construct a scale or diagram like that on Plate VII., divided into 100 equal parts each way. Set off half the beam b of the same scale as the drawing, from B toward C, say at a, join a with A, then the line a A, measured from the base A B, is the scale for all the ordinates in the load-water line and greatest cross-section ſº. Let us se- lect the exponent n = 2.5 and power q = 1.5 for the forward load-water line, which will be slightly hollow and reverse at 0.782 × V from ſº. Measure off the ordi- mates from the corresponding number on the base A is to the line a 4, and set them down on the drawing, Set off the ordinates from B toward a, and join them with A ; then each line forms a scale for the ordinates of the corresponding frames. The fourth ordinate \.7469 will be the line g Il, and the incline line 4 is the scale for the fourth frame. Turn the scale round, and set off the half beam b from D toward A; join it with C, and proceed in the same way for the after-body of the ship, only select higher exponent for the aft load-water line. Scales or diagrams like that on Plato VII., but of larger size, have been printed, and can be had in Philadelphia. To find the Exponents and Powers from the lines of a vessel constructed without regard to the parabolic method, Divide the lengths from [X] to the stem and stern, and the depth from the load- water line to the keel, each into eight equal parts, as before stated. Suppose lalf the beam of the vessel is b = 20 feet. Measure the 2nd and 4th ordinates in the load-water line, say 6.68 feet and 15.31 feet, respectively. Divide those ordinates by the half beam 20 feet, which will give 0.334 and 0.765. Find these ordinates in the tables, which will be found to correspond with exponent n = 3 and power q = 2. The area of the water-line will be, supposing the length to be 130 feet from Q to stern, 130 × 40 × 0.6428 = 3342.56 square feet. All the other exponents and powers are found in the same way. For the dis- placement, divide the ordinate areas of the frames by [X' and find the correspond- ing ordinates in the tables, and thus all the properties of the displacement a e found by the parabolic method. 446 PARABOLIC CONSTRUCTION on SHIPS. Recording Formula. The form of any vessel may be recorded by one general formula, as follows— #} l (# ) l/ W.m.q. D.m.g.) \@ n. q. D.m.g. WV.n.d. The first part, # } l, represents the form of the after-body of the displace- ment. W.n.g. = exponent and power of the load-water line, D.m.g. = exponent of the displacement, and l = length from Q to stern-post. The middle part, ..B.d ). represents the length, beam and depth of the dis- placement, (Q. m.g. the exponent and power of the frame Ø. The last part, l' { #. represents the fore-body of the displacement. The recording formula is not intended for arithmetical operation. Recording Formula of the Ironclad “Dictator.” W 5.5 × #} 96 (.. × 42º) 144ſ W2.75 ×1.5 D4.75 X 2.25 & 2.75 × 0.5 D 3 × 2 From these data a similar vessel to the hull of the “Dictator” can be constructed by any one familiar with the parabolic method. Sailing-Yacht. A well-proportioned sailing-yacht may be set up as follows: W 3 × 2 30(*#) W 2.75 × 2 — y OU — mºmºmºsºsºmsºmºmºmºsº D 2 × 2 Q 3 × 4 I) 2 × 2 The Formula for the Steam-Propeller on Plate X. is— r: W 2.5 × 1 } 65.625(*. × 30 × #)-1875 { TV 2 × 1.5 D 2 × 2 £º 3 × 2.5 JD 2 × 2 Angles of the Limes. The angle of entrance in the stem, or delivery in the stern, of any water-line, is— b tan,”) – #: & The angle of the line, at any distance y from Q, is— b 77, n — I ºft\ q - 1 tany-tºº--ſ 1–4 !” ln The mean angle of resistance or delivery of the disp'acement is— * ºf” 2g – 2 n/2n - 2 tan.v = q^b aſ 1–7. !/ dy. lſº |2n The column tº in the tables is calculated from this formula. The distance y from Ø, to the point of inflection, is— º m — I Q) = m q–1 PARABOLIC CONSTRUCTION OF SITIPs. 447. Explanation of Tables. Table I. gives the greatest buoyancy and stability of the displacement with the least proportionate resistance. Tables II. to IX, inclusive, give hollow lines of which those with high power and low exponents are too hollow for the load-water line, but the high exponents with high power will suit for the aft load-water line. Tables X. to XIII. should not be used for water-lines, but for frames. Tables XIV. to XIX. are for frames exclusively. Table XX. for elliptic stern and sheer of vessels. Table XXI. contains the coefficient of displacement, or the fraction it occupies in the parallelopipedon, length, breadth and depth. For light draft and-high speed the coefficient should be selected toward the corner .490, and for freight and light draft toward the corner .797. The first column ºn contains the exponent for ſº, and the tope-line that for the displacement, supposing q = 1 in both cases. Table XXII. The first column C contains the coefficient of the vessel, for which the tabular number, multiplied by the cube root of the displacement in tons, wiłł give the length of the vessel in feet. IExplanationn of the Plates VIII., VIII., IX. and X. Plate VII. is a scale for laying down the parabolic lines directly from the tables without calculation, as before described. Plate VIII. illustrates the sharpness of water-lines and cross-sections. The num- bers on the figures correspond With the exponent n in the first column of the tables. Plate IX. Fig. 1 illustrates the sharpness of full lines from Table I. Fig. 2 are cross-sections from the same Table I. Fig. 3 is a scale of displacement and draft of water, referred to Table [., in which the exponent n (or ratio r) is taken for the areas of the water-lines. This scale also corresponds with the Formulas 6 to 9, for finding the displacement P and draft 6. Fig. 4 is a scale for laying out the frames. Fig. 5 is the spring or rise of beams, ſor which the ordinates should always be taken from the exponent 2, Table 1. Plate X. illustrates a propeller steamer constructed on the parabolic method. º Steamship Performance. For similar proportioned vessels the resistance is a function of ſº M2, and the horse power a function of ØØ M 8. The displacement of a vessel is a function of the cube of any linear dimension of the same, and the greatest immersed section ºf a function of the square of any linear dimension of the displacement; consequently the y T is a function of any linear dimension, and (¥ T)2 = º T3 is a function of (X) ; therefore, the resistance is a function of M 2% T2, and the horse power a function of M 8% T2. The tables of steamship performance are calculated by this last formula, in which it is assumed that the displacement of the vessel is about one-half of the parallelo- pipedon, and the length about eight times the beam. *|22SH Mºtº T— N(#).3 = ~ | & JHI & M=\; 228 Jſ 3 On the Form Waves, or the Wave-line. v = velocity of the wave in feet per second. l= length of wave in feet from top to top. h = total height in feet of the wave from top to bottom. Ab. Ordinates. || Ab. Ordinates. .00.06(7 9 -59755 .03S06 IO .69134 .0S.426 11 .77778 .14644 12 .S5355 .22221 13 .91573 .30SU6 14 .96.194 .40245 15 .990:39 .50000 16 1.0000 v= 1.82/t, l = 3.14h, h = 0.3183. The accompanying table shows the ordinates for the wave line, the abscissa or base of the line from the lowest part to the highest, or half the length of the wave being divided into 16 equal parts. The ordinates are expressed in parts of the height h. 448 PARABOLIC CONSTRUCTION OF SIIIPS, The Vessel. A = area of the immersed hull. a = area to load Water-line. a = area of water-line at tho draft 6. B = breadth of beam. b = half breadth of beam B. d = draft of water, omitting depth of keel. 5 = any draft less than d. 88 = difference of draft of water, which should not be taken more than 0.25 feet for the largest, and only 0.1 for vessels of 100 tons. b = differential. L = length of vessel in load-water line. l = length from ſº to stern-post. l' = length from ſº to stem. (X) = area of greatest immersed section. D = displacement, cubic feet. D = displacement forward of Q. Q = displacement aft of Ø. D = displacement at the draft 8. T = displacement in tons. b A= differential of displacement. Centres of Gravity. S = distance of centre of gravity of dis- placement from the stern-post. s = distance of centre of gravity of G from [X]. s' = distance of centre of gravity of D from [X]. = centre of gravity in the tables. e' = depth of centre of gravity of dis- placement under the load-water line. Parabolas. m and q = exponent and power. B = any ordinate at distance y from Ø. r = cR poncut of the displacement from load-water line to keel, supposing g = 1. feet from [X] described, to be s = 48 feet aft of Formula 5. A =[a-2(X) +d L)] Explanation of the Formulas, with Examples. The Whole method of the Parabolic Construction of Ships is based upon the Formula 1... The formula looks very simple, but when developed into its combina- tions with the form of a ship, it becomes a very complicated affair, which requires a separate treatise on ship-building for proper explanation. Pºmple fºr Formula 5. The length of a vessel is L = 250 feet, of which l = 100 º to stern post, and W = 150 feet from ſº to storm. The centres of gravity of the ſore and after body of the displacement have been found, as before - £º, and 8' = 68 feet fore of Ø. Required, the Centre of gravity of the whole displacement from the stern-post 2 & 100 (100–68)+150 (100+48) Notation of Letters for the Formulas. All dimensions are in feet, Square feet or cubic feet. Stability of the Vessel. n = meta-centre for load-water line in the tables. m/ = height of meta-centre above Centre of gravity of displacement. P= the real stability of a slip. Q = momentum of stability in foot-tons. g = height between centres of gravity of vessel and displacement when in equilibrium. o = horizontal distance between the two centres of gravity when out of equilibrium. o' = horizontal distance between centre of gravity of displacement when in and out of equilibrium. 2 = careen-angle of the vessel. C = coefficient of displacement, Tablo YXI. Performance. Eſ== horse-power required to move the vessel. M= speed in knots per hour. R = resistance to the vessel in pounds. t = tangent in the tables. w = mean angle of resistance to the ves- sel moving in water. W = mean angle of delivery of the aft displacement. k = coefficient of friction in pounds per square foot of the immersed sur- face of the Yessel. Surface. Coefficient, k |For polished metallic surface, . Ordinary copper sheeting, . . . 0.005 Wood, smooth-planed, . . . . 0.007 Rough castings of iron or brass, . 0.009 Fouled with barnacles and sea- 0.003 Weeds, . . . . . . . . . 0.015 Fouled with barnacles and oyster- Shells, . . . . . . . . . . 0.020 | P_. WBI, d 250 = 116 feet. PARABOLIC ConSTRUCTION of SHIPs. $4% Formaulas for the Parabolic Construction of Ships, Slabtlity and Performance. n Displ ~~~ + 1, 24 smallſ Stability : 6== b ( –% ) ". ºr tº l dif. of Draft. zlºty **** of 53 Centres of Gravity. 6– º: 6 m’-iſm l/+m l). 10 r = —— . . tº tº 2 & 7" º a d-ly _ T |Q= Tsin.2(m/+g).11 d?' _bt jól © ^ = 2r-1’ º sº-ºvº. tº e 7|o-sin.2(m/+g). 12 d 3 *- / — an /e: / = -* r O’ = m/sin.2. . . 13 € 2(? #) © **-aº. * ~ * ad P–*-*. . . 14 – e’ f rTNT 77', s-º-ºººº. 54-ax. . . . 9 L d Q= To. . . . 15 Resistance to and Performance of Vessels in Motion. R=2.858 M* [x(0.9/ºroivº V) +4 °. . . 16 WL 3X) : R. Mſ 2.2450EI tan.v = −. . . . 17|H=+ . . . 19|R = +: . . nº–7, 224.50 9|R-## 21 any-º. . . 1su-ºº. . 20 Fielion-4*... 22. & d WL. Example for Formula 8. A vessel of p = 150000 cubic feet of displacement, a = 10000 square feet area of load-water line at d = 20 feet draft. Required, the exponent r? and how much be can the vessel be loaded per inch of difference of draft at 6 = 15 feet draft 7 15000 Formula 2. fº = =3, the exponent. 10000 × 20 — 150000 3 TTE Formula 8. h B- = wº#-728 cubic feet, which in salt water will be 728 : 35 = 20.8 tons per inch of draft at 8 = 15 feet. Exampie for Formula 10. A vessel of D = 150000 cubic feet, half beam b = 20 feet, W = 170 feet, n = 3.5, q = 2, for which meta-centre in the table is m = 0.350, and l = 180, fº = 2:5, q = 2, for which m = 0.277. Required, the height of the meta-centre above the cent. gr. of the displacement? 3 Formula, 10. m/ = ſº (0.350× 170+0.277 × 180) = 5.832 feet. 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I |&9%g 0013 || 88.19 || 1966"|LIQ6' |S| GS’ Iñ81 |Z099 || Ofg?' |z103' g"Z () 198 |835"I |f|QIg" | 0903" | 9999 || OI66"|S996 |S648' Iggſ, 0609 || Zg LF" |I96I. g19'Z 81,78 |(}89"I ()f03" | 9396 || $309" | #SS6"|Lºft;" |##9S' I giff!'}90Sg" | 6968 |ZgSI" || gz’, G&S 98.3"I II67, Z2.98" | 6630 || ()g86 Ifºgó ()158" | ZzZL | L999" | 1918" | If LL || 93I'z, g3 LS' ISSZ"I 181.17, j Qzgg" | g fºg" || g()S6 gzzó" Hºlzs' 6169 |iggg" | 89.98"|199 [' ‘Z “soul I AOLIoH *gz"I = 0 T3AAOc{ *II @HQiu Ji, s (313.8 [0909 |Ozit'ſ 3156 || 000 Iſ OOO'IQ00I $666. If 666 |0066 (6188, '91 # |g|LZ’9 |SQ19 |gº Igzé || 000 I 000 IG666 1666 |tg66 |8896 |9861. ‘z, : |ggz"g |gSgg" || $Sgf"| 0606 || 000"I 6666 S666 ()666' 6066 18:f6 6991. "Of ; Z1%’ſ fggg" || 009:5"| 8SSS" i. 6666' | 6666 |9606 || IQ66"|19||6 || 6668"|f 999' "S 3 |01.3 |000g. gigſ. Ilg8 || 6666 | 1666 3166 iſsºlvoró |0338|&Igg. ‘9 * |gll & |f ºf 1987;"| 98.98" | 6666 0666 |9&66 1896 |9F06 || 1391 |ILSã' *g # |iggz, Olg; 10&#| 1818 || 8666' 0866 |61S6 |6egó |tó18, 10931; licy gºj, § |g|SZ'2, 1913 f : 991?' | 0008" || 1666' | 1966 |Z086 | g 196" flipS 98.89'83If" ‘f 3 |99 L’ô j993f." J 08Iſ' | #681 || 9666 G#66 liff.6 99.6 |f 838 0009' 6963' || gl’8 5: (#0% iffTſ". I60f 1.1.1.1" | g666 Zz66"|1196 || 91L6 |010S 9fgg' |gglg’ g"g à |z76'I |010F| LIOf' | 1591 ||8866 | 6886"|1896 |6f6s 6281 |#109 |Izggſ || gzig § {008" | | 1983" | (100%" | 009 || || 0866"| fiſs6"|9|f|6 || 091S [$991. ISA9' 0633” *g g |S&L'I |911g. flóg|6|f| || glé6"| #186 |f|OV6 |1308|IIłl 1299|g818 || glSº 3 |Z89'L |8893" | Słó9 £831 || 1966 6116 |9386 glg8 fig7.1 || 09F9' |{103' || glº, g |z29. I |&I93. 6168. līgº i 1966. 1816.8836, 81:8.880", 100gg|Eggſ ||gg.93 g ºg I K0% $$$g ºf It. Igúð... 1896 |ºtº. 3838|$169. 6.19.18883. g"Z # |009. I |fs& 1988 || 1:01 || 8366 88.96:19:06 |z108 |$319 |09ñf 9113 ||913 & § liff'I |$833' £88 $369 || 1066 8896 |668S | 868, 1799 g91;"|gógz' || gz'Z § |zó8.1 |Z918. 8813. Q089 || 61.86 tº 6.19918, 101.1.1.189 |f|lgû, olfa ||ggiº *_{338 L I403' |0318|9909 ||ff{6 |918.6. f638' 00.gllfö09 |g|18Flif{3. ‘Z "uoſº | ? wa || 2 ||a ºu || Z. ſº I Tº 2, I || ºt —oaguT! •se'ſ "thoſwl'.13 'Ol ‘poidſ ‘Sº buyp,[O dxCI *saw. H II thºſ *I = 0 (9AAodſ *I 3IQºI, ‘SqLIIS IO NOILoſiº.LSNOO OIIOdVºIV.I 09:5 PARABOLIC CONSTRUCTION OF SIIIPS. 451 Table IV, Power q = 1.75. Hollow lines. | Exp. Ordinates. Prod. C. gr. Meta. Res. Inflec- 70, I 4. 6 7 ||aº D| e i m t | tion. 2. .0789).2353 .4203.6044 || 7670} .8932.97.28 || .5592 .3252 .2412; 1.228|.6345 2.125 ||.0865 .2544 |,4476].6340 |.7926 .9098.97.90 || .5761 .3308|.2538|| 1.26() .6600 2.25 || 0944.2733 .4739|.6617 | .8154 .9240.9838 || .5919 .3358 .2665; 1.294 .6822 2.325 ||.1022.2921 |.4994].6874 |.8359] .9359.9875 || .6065 .3400 . .2786| 1.329 .7018 2.5 .1104.3108 .5240,7115 .8559] .9460.9904 || .6200 .3446 .2900; 1.363 .7200 2.625 || .1184.3218 .5475 |.7337 .8705 .9545).9925 || .6320 |.34S'ſ .3000; 1,400 .7364 2.75 || .1269|.3476 .5702).7545 .8851] .961.7 .9943 iſ .6431 .3529 | .3105; 1.438; 7514 2.875 || .1350 |.3656 .5919 |.7738 . .8989 .9677 .9956 || 6542 .3568 .3200 | 1.475 | .7650 3. .1437 ſ.3833 .6128|.7916 . .9102} .9728}.9966 || .6t,46 .3604 || .3295] 1.511 .7777 3.25 ||.1609|.4179 |.6517.8234 |.9289| .9807 |.998() || .6833 j.367() .3464 1.591 .79S7 3.5 .1825|.4513 |.6871.850.5 ! .9442} .9864.9988 || .7000 .3736 | .3618| 1.591 .8175 3.75 ||.T.959}.4833 i.7193).8736 |.9562} .99()4}.9993 || .7158 |.3782 .375S 1.673; S310 4. 2135.5139|.7484.8932|.9057| .9932.9996 || 7293 ||3834 |.3883| 1.850).s.p.30 4.5 .2483).5710 ! .7985).9241 .9789} .9966.9998 || .7521 |.3922 . .4108' 2.010| 8613 5. .2S401.6225 |.8391 |.9460 .9870; .9983 .9999 || .7732 .4006 . .4’300; 2.171}.S754 6. .3526 .7696 .8980 .9728 .9951 | .9996 || 1.000 || .805() .4129 .4610; 2.500) .89S9 8. .4787 .8314 .9596 ſ.9932 : .9993; .9999 il.00() iſ .84.75 .4296 .5041 3.I.75 .9258 10. .5861|.9()09 | .9841.9983 | .9998 1.000; 1.000 || .87:34 .4420 # 5313; 3.878 .9431 12. .6746).9454 .9938 .9996 .9999 || 1.000 | 1.000 || .893S .4503 || 5.528; 4.550 .9533 16. .80261.98.25 | .99901.9999 || 1.000 | 1.000 | 1.000 .9185 .4615 j .5821) 6.050 .9643 Table V. Power q = 2. Hollow lines. 2. .0549].1914 .3713.j.5625 .7385} .8789}.9630 || .5333 .3125 .2273 1.219) .5773 2.125 || .061()|.2092 .3990/.5940 i.7667 .8976i.9761 : .5504 .3160 .2415 | 1.242} .6038 3.25 || 0673] .2271 .4260 |.6237 .7920 | .9136.9815 ; i .5664 .3189 .254.1 | 1.273 .6292 2.375 || .0738|.2450 | .4523|.6516 8l47 | .9271 .98.57 || .5813 | .3278 .2662 | 1.301 | .6517 2.5 ,0806|,2630 | .4777|.6777 8371 .9385.98.9() iſ .5952 . .3333 .2770; 1.355] .6723 2.625 || ,0873|.2810 : .5024].7020 i .S534 .9 |S| |..9915 . . .6()S3 .3388 .2S75 | 1.666 .6904 2.75 ||.09.44).29SS .5262}.724S .8698 .9563 .993.4 #} .6205 .3418 .2979| 1.402| 7066 2.875 ||.T014.3166 i.5492}.7460 |.8343| .9632!.993.4 #| .6320 .3460 | .3074 | 1.436| .7222 3. .1090|.3342 .5713|.7656 .8980 | .969) : .9961 || .642S .3500 .3164 1.472] .736S 3.25 || .I240|.368) : .6130|.S(\OS .9192|| .9779 .9977 || -6627 .3571 .3255 | 1.545 .7988 3.5 .1394.4(28 .6512.8310 ! .9365 | .984.4.9986 || .6805 |.3636 .3500 1.620) .S175 3.75 || 1555 .4356 .6862].8569 .950.1 ! .9890 9992 || .6966 .3695 .363S 1,697 .8319 4. .1712|.4673 .71S1 |.8789 . .9608 .9922; .9995 || 7111 |.375() .3765 1.773 .8013 4.5 .2045/.5270 .7733|.9137 j .9759 .996.1 .9998 |} .7363 1.3S46 .4000: 1.929) .S315 5. .2373}.5804 .8184.9385 .9852| .99.80; .9999 || .7576 .3929 .4.196 2.088} .84SS 6. .3038 .6729 SS43 .9690 .99.45| .9995 9999 || .7912 .4062 .4517 2.407 .8767 8. .4308|.8098 .95.40|.9922 .9992 | .9999 || 1.000 .8366 .425() .4962; 3.060; .90SS 10. .5431,8905 .9819 |.99.80 .9999 || 1.000 1 000 || S6 (3 .4375 | .5249 || 3.643; 927 9 12. .6377 .9379 .9929 .9995 . .9999 || 1.000 1.000 : ..SS61 .4464 .5463} 4.390 .941S 16. .7778|.9S00 .99$9|.9999 i 1.000 | 1.000; 1.000 .91.26 .4629 .5763; 5.7 I5 | .9356 Table VI. Power q = 2.25. Hollow lines. 2. .0382}.1557 328 lj.5235).7 111 | .864S}.9652 .5100 2815 .2155 | 1.226) .5328 2.125 || 0430|,1721 .3557 .5566 .7416 .SS56}.97.31 || .5288 .2861 | .2300 | 1.248 .5657 2.25 || 0481 i.18S7 .3S29 i.5SS() |.7693 .92.43 .9792 .5450 292S .2425 | 1.272 59:23 2.375 ||.0533.2055 .4095; .6176 .7942} .91831.98.40 || 5600 .2998 .2515 | 1.298 .6164 2.5 .05SS|.2226 .4356.6455 .8187 9311 .9876 it .574.2 .3046 .2657 1.325 .6363 2.625 ||.0644,2398 |.4610|.6716 j.8367 | .94.13.9904 || .5878 .3115 .2762. 1.354 .6547 2.75 || 0703 .2570 .4857 .6962 || 85.47 .9510 .9926 . . .6005 | .315S .2S63 | 1.385 .67 22 2.S75 || .0762,2742; .5096].7191 .870S .95S7 .99.43 || 6124 .3207 .2360 | 1.413 .6885 3. .0826|.2914 | .5327 |.7405 . .8967 .9652: .9956 || .6235 .3250 | .3053 | 1.453 .7035 3.25 || .09551.3257 .57 661.7789 . .9095 .9752.j.997.4 .6443 |.3333 .322S 1.524 .7294 3.5 .1090; .3595 .6172}.S120 | .92SS} .9S25 .9985 .6631 | .3415 .33SS| 1.596 .7490 3.75 ||.1229.3926,6547 |.8403 : .94.40] .9876}.9991 .6799 || 3480 . .3585, 1.663| .7658 4. .1373|.4249 .6S90 ,S648 .9560 | .9912|.9004 || .695() .3543 .3665 1.725 .7800 4.5 .16731.4865 .7488 .9035 .973) | .9056.9998 || 7213 .3659 .39:20) 1.SS3 .8048 5. 1982|.5436 .79S1 .9311 .9834 .997S .9999 || 7435 | .3753 .4105 2.038 .S258 6. 2618|.6434 || S624.9652 .9938 .9095 |.9999 iſ .7794 .3907 | .4437 || 2.352| S622 8. 3S18.7SS7 .9484.9912 |.99.91 .9999; 1,000 || .827.1 |,412S .4895 2.98S} .8950 10. 5032|,8777 | .9797|.997S .9999 || 1.000||1.000 || .8565 |.42; 0 | .5192 3.623| .9144 12. 6029 .9304 .992() .9995 .9999; 1,000 1,000 || .8794 |.4373 .5413 4.270 | .931.5 16. 7538|.9776|.9988}.9999 || 1.000 | 1.000||1.000 || .9072 |.4524 | .5774 5.500|.9485 452 PARABOLIC CONSTRUCTION OF SIIIPS. Talble VII. Power q = 2.5. Hollow Lines. Exp Ordinates. Prod. C. gr. Metal Res. |Inflec- 9? I 4. 6 7_i|alſº D_e 7)?, t tion. 2. .0266!.1266 .2899 i.4S71 . .6846|.8510 |.9614 || .4913 |.2912 .2053| 1.248} .5000 2,125 || 0303.1415 | .3171 |.5215 H.7174 .8737 .97.02 || .5092 |.2954 .2182| 1.266] .5300 2.25 ||.0343|.1568 .3442.5543 |.74.72i.8932 .9769 || .5260 |.3003 | .2310|| 1.287|.5568 2.375 ||.0384.1724 . .3709!.5854.7741).9007 1.8922 || .5414 .3084 | .2480 | 1.309| .5807 2.5 .0429|.1884 | .30721.6149 .8007|,9237 |.9862 || .5558 |.3134 || .2545| 1.333 .6008 2.625 ||.0474).2046 |,4230|.6426 |.8203}.9356 .9894 || .5695 |,3198 || .2655; 1.360 .6227 2.75 ||.0521.2210 ! .4482.6688 |.840ſ)}.9497 .991S [] .5825 |.3235 . .2762| 1.380 . .6424 2.875 || .0572; .2375 .4728|.6933 .8576.954.2 .993.7 || 5940 .3286 | .2861 | 1.420 .6605 3. 0626.2541 || 4967|.7162 .8742: .9614 |.9051 .6068 .3325 | .2954 1.452 | .6767 3.25 || .0735|.2875 . .5424.7576 . .9000],9725 |.9974 || .6282 |.3405 .3138|| 1.520 | .7044 3.5 .0852.3209 .5850.7935 | .921.2|.9806 |.9983 || .6476 .34S0 .3296 | 1.590 .7254 3.75 || 0974|.3339 |.6246|,8244 |.9380 .9863 |.9990 .6650 |,3540 | .3441| 1.653| ,7432 4. .1102}.3864 |.6610|,8510 i.9513].9903 |.9994 || .6806 |.3605 | .3577| 1.710| ,7582 4.5 .1371 .4501 || 7252| S913 .9700 .9951 .9998 || .7077 .3715 3820 1.861 || 7846 5. .1656 .5080 || 7784|.9237 .98161.9976 |.9999 || 7311 |.3806 | .4027 | 2.014 | .807.0 6. .2255 |.6126 .8576|.9614 .9931 .9994 | 1,000 || .7692 .3955 .4365 2.312 | 84.72 S. .349] |.7682 .9428|.9903 || .9990|.9999 || 1.000 || .8188 .4166 || 48.35| 2.935 | .8827 10. .4662|.8651 .9774|.9976 .9998 || 1.000 | 1.000 || 8496 |.4303 .5145 || 3.554 .9029 12. .5699 |.92.29 .991] |.9995 .9999 || 1.000 | 1.000 || .97.32 |.4402 | .5368| 4.180 .9223 16. 7305 (.9751 t .998.7 .9999 f 1,000 | 1.000 II.000 iſ .9026 i.4558 ,5738 5,365 94.19 Table VIII. Power q = 3. Hollow Lines. 2. .0129|.0837 .2263|.4219 .6347 |.8240 |.9539 || .4571 |.2739 .1894| 1.330] .4472 2.125 ||.0151.9567 .252()|,4579 |.6713|.8505 |.9643 || .4758 |.2787 .2030 | 1.341|,4771 2.25 || 0175|.1082 | .2781.4927 || 7049 |,8732 |.9724 i ! .4933 |.2S62 |.2151 | 1.357| .5037 2.375 || .0200.1213 .3042|,5260 . .7355,8926 .978.7 || .5097 |.2927 | .2274| 1.379 .52S6 2.5 .0229 |.1349 .3302}.5579 .765)|.9092 .9S35 || .5252 .297.0 [ .2387 | 1.400 .5508 2.625 ||.0258|.1490 | .3561|.5882 .7S85|.9232 |.9S73 || ,5397 .3042 .2498| 1.424 .5723 2.75 || 0290|.1634 .3817|.6171 .8112|.9352 .9902 || .5534 |.3090 .2600|| 1450; .5926 2.875 ||.0323|.1782 | .4070}.6443 |.83.16|.9453 |.992 || || 5663 |.3135 | .2702| 1.476 ,6117 3. .0359|.1932 |,4318|.6699 || 851ſ)}.9538 |.994] .5785 |.3180 | .2708 || 1.504; ,6290 3.25 ||.0436|.224l .4799 |.7166 i.8812.96.75 .9965 || 6011 .3267 .2973| 1.563) .6591 3.5 .0520.2556 |.5255).7576 j.9062| 9767 j.9079 || ,6213 j.3350 . .3136|| 1.625 | .6832 3.75 ||.06ll|.2875 |.5684.7932 . .926.1 |.9835 .9988 || .6397 |.3420 .329() 1.705|.7039 4. .0709|.3194 .6085}.8240 |.94.10|.9883 .9993 || .6564 |.34S2 .3427 | 1.755 | 7227 4.5 .0921|,3826 .6800|.8734 . .9641|.994.1 |.99%)7 iſ .6S55 |.3603 | .3682) 1.890} .7514 5. .1156|,4437 || 7403|.9092 .97.79.9971 |.9999 || 7 102 |.370() .389.3 2.025 TT60 6. .1674.5555 . .83.16|| 95.39|.99 || 7 | .9993 1.000 || .7514 .3860 .4244| 2.312| 8155 8. .2828|.7287 .9318|.9883 . .9988.9999 || 1.000 || .8042 |.4000 || 4734| 2.SSS|.8635 10. .4002}.8404.97.30|.9971 . .9999; 1.000 | 1.000 || 8378 .4238 .5060) 3.476] .8845 12, .5093.9083 .9894.9993 . .9999 I.000 |1.000 || S622 4345 .5293 4.065 .9082 16. .6868}.9702 .9084.9999 || 1.000||1.600 1,000 || .8940 |.4490 | .5062| 5.260} .9310 Table IX. Power q = 4. Hollow Linnes. 2. .0302|.0366 . .1379 |,3164 .5454.7725 .9390 4063.24.6l T685 Tºp55,5778 2.125 ||.0372|.0438 |.1592|.3529 |.5878|.8058 |.9527 || 4257 .2534 IS10| 1.461 || 4107 2.25 || 0453].(516 .1815|.3891 |.6273|.8346 .9634 || .4440 |.2607 .1930, 1469| 4400 2.375 || 0544.0600 . .2045/.4246 |.6639|.8594 |.97.17 || .4612 .2572 .2047 | 1.478] .4648 2.5 .0649.0692 |.2282|,4593 .70')7|.8807 |.97S1 || .4774 |.2735 .2160 | 1. 190|| 4865 2.625 || .0762i.0789 . .2524.4599 |.7284.8090 .9831 || 492S .2796 .2270| 1.505| 5078 2.75 || 0892|,0893 .2769].5253 || 7565.9145 |,9869 || 5073 .2852 |.2374] 1.522. 5282 2.875 ||.0130 1024 .3016|.5564 || 7820.9275 .9899 iſ .5210 |.2906 .2475|| 1.556 .5477 3. .01.19.1117 | .3264|.5862 |.80651.9389 |.9922 || .5340 |.2955 .2570| 1.569| 5665 3.25 .0154].1361 .3757|.6413 |.8449.9564 |.9954 || .5581 |.3046 .2747| 1.618| .5985 3.5 .0194|.1622 |.4241|,6906 | .8770,0691 (.9972 || 5799 i.3131 | .290S 1.670|,6261 3.75 || 0241.1897 || 4709|.7342 . .9027|.9781 |.9983 || 5997 .3211 .3056 | 1.725] .6488 4. .0293|.2184 .5157|.7725 | .9232|,9845 .9990 || .6178 .3288 . .3198| 1.787| 6687 4.5 .0416.2778 .5981.8349 |.9524.9922 |.9996 || .6494 |.3415 | .345S 1,906] .7000 5. .0563}.3384 .6697.8807 |.9707.996.1 .9999 || .6764 |.3529 || 3607 2.036|.7275 6.5 .0923.4566 .7821 |.9390 .98S9.9990 | 1.000 || .7196 .3711 | .4057 2.309|.7754 8. .1856].6558 . .9101|.9845 |.9984.9999 |1.000 || 7788 |.3966 .4575| 2.845] .8304 10. .2949|.7931 |.9641}.9961 .9995).9999 || 1.000 || .8174 |.4138 j.4924| 3.403| S637 12. 4067 .8796 | .9859.9990 | .9999||1.000 |1.000 || .8445 .4260 . .5174| 3.987 .8859 16. .6050|.9605 |.9978|.9999 || 1.000:1.000 |1.000 || .8803 |.4424 i.5507| 5.087| 9142 PARABOLIC CoNSTRUCTION OF SHIPs. 453 Table x. Power q = 0.25. Full Limes. Exp. Ordinates. Prod. C. gr. - ”, l l 3 4- _6_ _ 7 Area. *—l— 0.25 ||.4257 .5053 .5770.6316 ||.6829 |.7357 .7979 |} .6126 .4164 0.375 || .4697.5655 |.6196|.6916 |.7448.7979 |.8578 || .6524 .4210 0.5 .5041.6046 |.6765 .7357 .7890}.8408 .8967 || .7011 .4249 0.625 ||.5315|.6224 .7103}.7700 |.82281.8725 .9235 |} .7317 i.427.7 0.75 || .5556.66:37 . .7383|.7979 |.8495|.8967 .94.22 |} .7540 .4304 0.875 || 5757|.6868 ||.7620.8202 ||.8712}.9155 .9567 || 7721 .4328 l. .5946|.707I .7825|.8409 |.8891}.9306 |.9672 || 7885 |.4351 1.125 || .6111.7251 .8005|.8557 .8039 |.9420 .9750 || .8019 .4323 1.25 ||.6262|.7413 | .8164.8725 | .9172|.95.25 |.9809 || .8128 |.4393 1.375 ||.6398.7560 .83024.8854 .92764.9606 .9853 || .8224 .4412 1.5 .6527 .7694 ||.84291.8967 |.9368].9672 .9887 || 8313 .4428 1.75 || 6756 |.7930 ||.8653.9156 | .9517 .9771 .9933 . . .8463 |.4455 2. .6955|.8133 |.8835".9306 .9628 .9840 .9961 iſ .8586 .4478 2.25 || .7137|.830S .8988.9427 .94.13|.9887 .9977 iſ .8678 |.4499 2.5 .7299; .8462 .91.18|.95.25 . .9780 .992L .9986 || .87.60 .4519 2.75 ||.7446,8599 || .9229|.9606 . .9827.9944. .9992 || 8836 .4537 3. ,7580|.8720 |.9292 |.9672 .9866|.996.1 .9995 || .8902 .4554 3.5 .7817|.8925 | .9478.9771 | .9918|.9980 .9998 || .9023 .45S3 4. .8020 .9093 ||.9594.9840 .9950 .9990 |.9999 || .91.21 |.4606 5. .83541.9345 .9752|.992L i .9981 1.9997 .9999 || | .9203 |.4634. Table XI. Power q = 0.375. Full Limes. 0.5 .3579/.4702 | .5564.6310 1.6693}.7709 |.8491 || .003] .4003 0.625 || .3S74|.4478 .5986}.6757 |.7463 |.8150 .8875 || .6410 |.4090 0.75 || .4141.5407 |.6343|.7128 ||.7S30}.8491 .8840 || .6722 |.4150 0.875 || 4368 .5692 .6652.74.41 |.S132|.8761 | .935S .6993 .4182 1. .4585 .5946 . .6922}.771 t S3S4.8977 |.9511 || .7203 .4215 1.125 ||.6014|.6147 .7162].7945 |.S593}.9142 .9626 || 7388 .4240 1.25 || 4955|.6383 . .7377 | 8150 | .87S1 9290 .9714 .7537 .4262 1.375 || 5118.6573 . .7564|.8331 |.S934|.9414 9781 || .7673 .4281 1.5 .5273}.6749 .7738|.S491 .906S .9511 .9832 || .7797 .429S 1.75 || .5541 |,7062 | .9049 |.9761 .92S4|.9659 |.990() || S()00 .4330 2. .5S02|.7334 .8305|.S977 .94.48].9761 |.9941 iſ .8172 .4355 2.25 || 6030|.7573 . .8521 .9153 .9572; .9832 .9965 || 83.12 |.43S1 2.5 .6236 .7785 .8707 .9297 .9672|.98S2 |.997.9 || .8434 |,4405 | 2.75 ||.6425.7973 |.8866|.9414 | .9742;.9916 .9992 || .8534 |.4427 3. .6599 |.8142 |.8957 .9511 | .9800ſ.9941 .9995 || .8620 | .4448 3.25 || 59341.7794 |.SS48.9460 .9791 |.9944 .9994 || S693 .445S 3.5 || 6611|.8432 | .9227 .9659 i.9878.9971 .9997 || .8759 .4486 4. .7183}.8671 ||.9398.9761 .9925).9985 .9999 || SS69 |,4523 4.5 .7423.886S |.9529).9832 | .9954}.9993 | 1,000 || .8960 | .4553 5. .7636|.9034 ||.9633.9881 .9972.9996 |1,000 || .9033 .45S2 Table XII. Power q = 0.5. Full Lines. 1. .3535|.5000 | .6124.7071 i.7906].8660 |.9354 iſ .6666 [.3966 1.125 ||.3734.5258 .6408|.7358 .817)}.8873 |.9506 || .6S60 | .4023 1.25 || 3921,5496 ||.6665|.7613 |.8406}.9073 |.9621 || 7048 || 4050 1.375 || 4093),5716 |.6892|.7839 .8605.9227 .9709 || 7218 .4076 1.5 .4260].5920 |.7104.8040 .8777 .9354 .9776 || 73S2 | 4002 1.625 ||,4414|.6111 | .7308|.8221 |.S927 .9496 |,9827 || .7518 .4125 1.75 || .4565}.6289 |.74SS}.S383 . .9057 |.954S 1.9868 || 7646 .4148 1,875 ||.4703 |.6457 |.7653|.S528 .9171.9621 .9898 || 7755 .4170 | 2. ,484.1 |.6614 |.7806|.8660 .92.70).9682 |.99.21 || .7854 |.4.192 2.25 ||.5094,6903 .8079|..SSS7 |.9434|.9776 .9953 || .8013 .4234 2.5 .5327 .7161 .8314|.9()73 .9565}.9842 .997:2 || S147 || 4270 2.75 || .5544|.7394 |.8517 | .9227 . .9657 |.9889 |.99S4 || .8262 4302 3. :57.45}.7603 ||.8634|.9354 9735}.99.21 .9990 || S376 |,4331 3.25 ||.5934,7794 | .8848|.9460 | .9791 |.9944 .9994 Ti .S.465 |,4357 3.5 .61.10: T966 || $983|.9526 .9837|.9961 |.9996 || S554 .4383 3.75 ||.6276|.8124 .9102}.9621 .9373}.997:2 .9998 || .8630 |.4407 4. .6433|826S .9205|.96S2 |.9900.9980 .9999 || .8704 .4430 4.5 ,6721.8520 .9377|.97.77|| 99.39|.9999 |.9999 || SS19 |.4474 5. .6979].87.33 .9511|.9S42|.9963|.9995 |1.000 || .8910 .4512 6. ".7424.9067 | .9697 .992.1 |.9986|.9999 || 1.000 || .9054 |.4562 454 PARABOLIC CONSTRUCTION OF SHIPS. Table XIII. Power q = 0.75. Full Lines, Exp. Ordinates. C. gr. 72 I 2 3 4 || 5 || 6 || 7 || Area. | 6 || Infi. 1.5 .2781 .4555 . .5988 .7209 .822.3 |.9t)47 .9666 || .6594 |.38.19 1,625 || .2932 |.4777 |.6247 |.7284 .8434 .9201 |.9742 || .6788 |.3853 1.75 || 3070 / .4988 || 64.79 |.7675 |.8619 |.9329 |.9802 || .6884 .389.3 1,875 || .3225 .5188 || 6695 |.7876 |.8782 |.94.37 .9848 || .7028 .3929 a 2. .3367 |.5379 .6897 |.8059 .8925 | .9527 .9883 || 7175 |.3963 : 2,125 || 3504 |,5562 .7085 |.8226 .9051 .9603 |.9909 || 7282 |.3993 : 3 & 2.25 || .3636 |.5735 | 7261 |.8378 |.9l 63 .9667 |.9930 || 7400 |.4020 33 2.375 ||.3762 |.5902 |.7426 |.8516 |.9261 .9720 |.9946 || .7504 |.4050 3 2.5 .38SS .6060 .7580 .8446 |.9355 | .9765 .9958 || .7604. .4071 §§ 2.75 || .412S |.6358 |,7860 |.8S63 |.9490 .9834 .9975 .77.70 || 4128 #3 3 .4355 .6630 || 802.3 |.9047 .9605 | .9882 .9985 .7910 |.4168 || 3 3 3.25 || 4570 |.6880.8323 .9201 .9689|.9917 |.9991 || .8032 |.4207 || 3 gº 3.5 .4776 |.7110 ! .8514 |.9329 .9757 |.994.1 |.9995 .8142 .4238 £3 3.75 || 4972 |...}22|.8683 -94.37 |.9G75||9958 -9997 || 8:44 |.4268 g 4 .5159 |.7518 |.8S32 |.9527 | .9851 .9971 .9998 .8341 .4292 gº 4.5 .5510 |.7865 .908] | .9667 .9909 . .9985 .9999 || -85.25 .4333 5 .5830 |.S161 .9276 |.9765 .9944 .999:3 |.9999 || -8652 .4372 6 ,717.8 |.S633 . .9120 .9882 |.9979 | .9998 |1.000 || .8835 .445) 8 .7292 |.9239 .9S25 |.9971 |.9997 .9999 || 1.000 90.36 .4562 10 .7954 .9575 .9932 .9993 .9998 | .9999 || 1.000 || .919.4 |,4623 Table XIV, Power q =1. Frames. .125 ||.0165 |.0353.0517 |.0830 |.1154|.1591 .2289 || .llll .2647 g; .25 ||.0320 |.0694.1109 |.1591 |.2174 .2989 |.4054 | .2000 |.2777 ; ; .375 ||.0487 .1023; 1616 .2289 .3077 .4074 i.5415 || .2727 .2849 || 3 .5 .0646 |.1340; .2094 |.2929 |.3876 .5000'i.6464 || .3333 |3000 | * .625 ||.0798 |.1646 |.2545 |.3516 .4583 i.5795 .7274 || .38 £6 |.3095 i ; .75 || .0953 .1941 || .2971 |.4054 .5208 ||.6464 |.7S98 || .4286 |.3182 3 .875 ||.1099 |.2225 .3372 .4547 576 ll .7027 .8379 || 4666 .3261 $2 1. .125%) .2500 .3750 .5000 |.6250 .7500 |.8750 iſ .5000 .3333 strai't 1.125 ||.1394 | .2764 .4106 |.5415 |.6683 .7873 |,9036 l; .5294 |.3400 1.25 ||.1537 .3020; .4443 (.5795 || 7065 |.8232 .9257 || 5555 .3461 g; 1.375 ||.T676 .3267 ||.4750 |.6144 |.7404 .8513 |.9427 || .5789 .3518 || > 1.5 .1815 .3505 5059 |.6464 |.7704 | .8750 |.955S || .6000 |.3571 || 3 1.625 ||.1948 |.3734|.5341|.6758 || 7968 | 8949 .9657 || 6190 |3621 || 8 1.75 2084 |.3955 .5607 |.7()27 |.S203 || 9,116 |.9737 6363 .3666 a 1.875 ||.2211 .4169 |.5857 |.7274 |.8410|.9257 |.97.97 || 6522 .3710 || 3 Table XV. Power q = 1.25. Frames. .125 ||.0059 |.0137 l.0279 |.0446 |,0671 .1005 || 1583 || .0765 .2120 l is .255 ||.0140 |.0329 .0640 |.1005 |.1485 .2155 H.3235 1500 |.2421 || 3: . .375 ||.0229 .0578 |.1024 |.1582 .2292 .3242 |,4645 ,2222 i.2555 33 .5 .0326 |.0808 |.1417 |.2155 .3059 |.4.201 i.5796 || .2755 i.2687 #3 .625 ||.0424 |.1048 |.1808 |.2707 |.3771 .5054 |.6717 || 3254 .2800 s: .75 ||.0530 |.1288 . .2193 .3235 .4424 |.5797 |,7445 || .3691 |,2895 §'s .875 ||.0633 .1528 .2569 |.3734 .5019 .6434 .8095 || 4080 .2991 g a .1 .0743 |.1768 .2934 |.4204 |.5557 |.6980 |.S.463 || .4432 .3073 º 1.125 || 0852 .2004 |.3287|,4645 |.6033 .7417 |.8810 || 4755 |3145 || 3332 1.125 ||.0963 |.2239 .3627 .5056 |.6478|.7841 j.9080 || .5040 j.3212|,5135 1.375 ||.1072 .2470 .3943 .5440 |.6868 || 8178 |.9289 || .5299 |3273 ||6175 1.5 .1185 |.2697 |.4254 |.5796 |.7217 | .8463 .945.1 .5523 .3333 j.6$86 1.625 ||.1294 |.2919 . .4566 |.6127 |.7529 .870.8 .957.4 || 5726 .3386 || 7382 1.75 ||.1408 |.3137 4S52 |.6434 |.7807 |.S908 |.9673 || 591.0 .3435 jºinio 1.875 ||.1516 |.3350 .5124 |.6717 |:8054 90S0 |.9747 .6084 .34S3 .7966 PARABOLIC CONSTRUCTION OF SHIPs. 455 Table XVI. Power q = 1.5. Frames. Exp. Ordinates. C. gr. _* || 1 || 3 || 3 || 4 5 |_6_|_7. Area. e | Infl. .125 || .0021 |.0058 . .01.36 |.0239 |.0391 |.0635 |.1095 || 0650 .1915 5 .25 || 0059 .0166 |.0369 |.0635 .1014 |.1585 .2581 .1252 .2125 i = 3 .375 ||.0.107 |.0327 | .0818 |.1094 i.1707 |.2581 .3985 -1810 .2290 33 .5 .0164 |.0489 .0958 .1585|.2413 .3532 .5197 || .2313 .242S] = 3 .625 ||.0225 |.0668|.1284 .2085 .3102 |4412 .6203 || .2792 |.2552 | g = .75 ||.0294 |.0855; .1619 |.2581 .3758 .5197 .7019 || .3238 .2658 || 3 2. .875 ||.0364 |.1050 |.1958 |.3067 .4372 |.5890 |.7670 || .3638 .2763 || 5 * 1. .0442 |.1250 | .2296 .3535 | .4940 |.6489 |.8185 || .4000 |.285) | * 1.125 |}.0521 |.1453] .2632 .39851.5454.6986 |.8590 || 4328 |.2937 l.206S 1.25 ||.0603 |.1660; .2961 |.4412 |.5939 .7469 |.8906 || .4626 .3015 .3333 ". 1.375 ||.0686 .1867 || 3274 |4816|.6371 || 7855 1.9153 || 4892 |3091 || 4444 $" 1.5 .0773 .2075 . .3586 .5197 |.6761 .8185 .9344 || 5131 |.3160 |.5429| 1.625 ||.0860 .2282 .3903 |.5555 . .7II3 .8465 .9491 .5348 |.3224 .6115 1.75 || .0951 |.2488 .4198 |.5890 .7429 .8704 |.9608 || .5545 / .32S5 .6534 1.875 || 1040 ! .2692 .4483 | .6203 | .7713 | 8906 |.9698 || .5727 .3342 . .6838 Table XVII. Power q = 1.75. Frames. .125 || 0008 |.0025 |.0067 .0128}.0228 |.0401 .0758 || .0500 |.1623 5 .25 ||.0025 |.0084 |.02.13 |.0401 |.0693 ).1166 .2060 || .1000 |.1876 2 3 .375 ||.0050 |.0185 |.0412 |.0767 |.1272 .2060 |.4403 || .1482 .2058 || 33 .5 .00S3 |.0296 |.0648 |.1166 1904 .3039 |4660 || .1955 .2200 | #3 .625 || 0120 |.0425 | .0912 [.1605 |.2553 ||3850 |.5729 || .2418 .2337 | g = .75 || .0164 |.0767 ||.T195 .2060 | .3193 .4660 |.6617 || .2860 .2456 || 5 s .875 ||.0210 |.0721 |.I492 .2518 .3809 j.5393 |.73.38 || 3272 |.2563 || 3 F 1. .0263 .0884 |.1797 |.2973 | .4393 |.6044 .7916 || .3636 |.2663 | * 1.125 ||.0318 .1054 |.2107 .3418 i.4929 |.6581 |.8375 | .3973 |.2757 |.1619 1.25 ||.0377 .1231 .2418 .3850 |.5445 |.7115 j.S736 || .4276 |.2848 . .2888 1.375 ||.0439 |.1412 .2718 .4264 |.5910 |.7545 .9019 || .4556 |.2937 .3S45 1.5 .0505 |.1596 .3022 |.4660 h.6334 .7916 .924) || .4796 |.3)08 .4575 1.625 ||.0571 i.1784 .3337 |.5037 | .6721 |.8234 .9409 || 5015 .3072 .519.3 1.75 0643 |.1973 .3633 5393 | .707) -8505 .9545 || .5222 |.3137 .5676 1.875 ||.07 l? '.1718 l.3922 .5729 .7386 .8736 |.9648 || .5414 | .320) .6052. Talyle XVIII. Power q = 2. Frames. .125 ||.0003 |.00111.0033 |.0069 [.0133 1.0253 |.0524 || .0222 |.1470 5 .25 0.011 |.0043 |.0113 |.0253|.0473 .0858 .1643 || .0666 |.1606 || 2: 3 .375 ||.0024 |.0105 |.0261 |.0523 .9471 .1643 .2932 || .116S |.1842 g: .5 0.042 |.0 l'79 |.0489 .0858 ||.T.503 .2497 |.4179 .1666 .2000 || 5 § .625 ||.0064 |.0271 |.0648 |.1236 l.2100 |.3359 .5323 .2136 .2145 | g = .75 0.091 |.0377 i.0652 .1643 | .2712 |.4.179 |.6237 .2571 .2273 || 5 s .875 || .0121 |.0495 ||.T1:37 .206S .3319 .4938 |.7021 .2969 .2395 || 5 * 1. .0156 |.0625 |. 1406 .2500 .3906 |.5625 .7656 .3333 || 2500 || 9 1.125 ||.0194 |.0764 .1686 .2932 j .4456 |.6199 |.S165 3665 .2600 .1291 1.25 || 0236 .0912 .1974 .3359 i.4992 .6777 |.8569 3968 .2691 .2350 1.375 ||.0281 .1067 .2256 .3775 |.54S2 .724S .8887 42.45 |.27SO | .3214 1.5 .0330 |.1128 .2547 .4179 |.5934 .7656 |.9136 4500 |.2S60 | .3968 1.625 i.0380 .1394 | .2853 .4567 .6350 .800S .9327 4734 .2930 .4587 1.75 ||.0434 .1565 .3144 |.4938 . .6729 |.8310 j.9481 J949 |.3000 .5092 1.S75 ||.0489 |.1738, 3431 .5291 .7073 .8569 |.9593 5149 .3062 .5473 Table XIX. Power q = 2.25. Frames. .375 ||.0011 |.0059 [.0166 |.0362 .07.05 .1311 .2515 || 1022 |.1237 5 s .5 0021 .0108 ||.0297 |.0631 |. 1213 |.2099 .3747 || .1441 |.1546 | E3 .625 ||.0060 |20172}.0460 |.0952].1728 |.2931 .4SS6 || .1863 |.1629 S3 .75 || .0051 |.0251 .0481 |.1342 . .2304 .3747 |.5881 iſ .2270 .1855 # = .875 ||.0069 |.0340].0866 |.169S .289\ |.4521 |.6717 || .2675 .1990 | #'s - .0095 .0442}.1100 .2102 j .3473 |.5235 .7405 || 3078 .2110 || CŞ * 1.125 || 0119 |.05541.1350 .2515 .4027 .5839 |.7961 || .3412 | .2216|.1078| * 1.25 || 01:48 |.0676 |.1612 .2931 .4577 |.6455 i.S405 || .3717 | .233l 2015 1.375 || 0180 |.0807 ||.1830 3343; .5085 .6962 |.S757 || 4000 .2422 .2833 1.5 .0215 [.0945 .2147 .3747 || 5560 |.7405 |.9033 || 4251 | .2512 | .35}5 1.625 ||.0252 |,IO90 .2439 .4141 .6000 |.7789 |.9246 || .4489 .2590 t .4032 -456 PARABOLIC CONSTRUCTION OF SHIPs. Table XX.-For Elliptic Stern of Vessels. IE X- po- Ordinates of Ellipses of Different Order. *| Index ment bl × - 3. 72. % I. 2 3 4. 5 6 7 # 2, .3398|.4840|.6616.7808|.8660.9204|.9682|.9922||.7854| 1.603 | | 2.25 |.4108.5490|.7147.8274.90041.9495.9801j.9958||.8154| 1.544 | 2.5 .4670.6042.7657,8627.9252.9546.9873.9978||.8382| 1.495 | 2.75 |.5174|.6514.8029.8901].9434.9749|.9932.9989||.8564| 1.453 | 3. . .5604|.6911.833}}.9019.9565.9821j.9948.9994.8709| 1.418 | 3.25 |.5991.7252.8578.9275.9664.9871.9973.9996.8833 1.388 3.5 |&333.7548.8782.9406.9740.9907.9978.9998||.8935. 1.362 4. |,6096.8021,9003.9595.9840.9950.9995.9999||.9075| 1.318 3 ſ 30° |0149].0582|1311||2374.3740.5449||7531 Sheer º | 45° |.0157|.0539|.1221 i; .5227|.7 #} of 60° |.0160,0474.1086|.1972|.3190|.4794.6946 vessels. Table XXI.--To Approximate Size and Shape of Vessels. - - & || Exponent for displacement n. q=2. d º/ || 2 2.5 || 3 || 3.5 , 4. 5 6 8 : 10 3 ſ 2...] .356 |.397 | .429 |.453 |.474 .500 |.528 .558 .577 ; : { 2.5||.381 |.425 | .459 |.486 |.508 .541 |.566 |.597 .620 : ". U 3 ||.400 | .447 |.482 |.510 | .533 .563 |.594 |.627 | .650 F # ( 3.5||.414 .462 .500 |.528 .552 |.589 .616 .650 .673 ‘s 3 & 4 || 427 |.476 .514 |.544 .569 .606 |.635 | 668 .693. s: # U 5 ||.444 |.496 | .535 | .567 .592 |.631 .660 |.696 || 722 ; : ſ 6 ||.458 |.509 |.550 .583 : 610 | .649 .679 |.717 | .750 P #| 8 || .474 .529 .571 |.605 |.632 .673 .704 || 743 .770 3 U10 ||.490 .547 |.590 .625 .654 |.696 .720 .759 .797 Speed and Freight and Freight and Purpose. passengers. passengers. slow speed. Table XXII.-Length of Vessels=Tabular Number X}^T Proportion of draft and length of vessels. 'Coefficient C. - 8 12 18 26 36 48 64. 82 I02 # (.356 ||14.9 | 19.0|23.7 |28.9 |34.0|38.9 |43.6 |47.2|48.0 ## 425 || 13.9 17.7 22.1 26.9 317 | 36.2 |40.6 |43.9 |44.5 . . . .482 || 13.3| 17.0 |21.2|25.8 30.3 34.7 ||38.9 |42.1 |42.7 ## ſ.5% ||129 | 16.5 20.5 |25.0 29.4|| 33.7 ||37.7 40.8 |41.3 ## 569 || 12.5 16.0 20.0 24.4 28.7 30.8 || 36.8 39.8 40.3 5 * (.631 || 12.1 | 15.5 | 19.4|| 23.5 27.6 || 31.6 || 35.4 | 88.3 ||38.8 *: { .679 || 11.8 15.1 | 18.8 22.9 26.9 30.8 34.6 37.4 38.0 §§ { .723 || 11.6 14.8 | 18.5 22.5 26.5 30.3 34.0 || 36.8 37.2 Hå U$797 || 11.2 14.3 17.9 || 21.8 25.6 29.3 || 32.9 35.6 || 36.0 Vessels for Ordinary River steamers, Condition. deep water. navigation. light draft. Play?e V// Plate Wii A/o//ow Waterlines 72/72 /V -EH Q 1 2 3 4 5 6. 7 6 5 4 3 2 E & I . Óross-sections & Za &/e / º | W • 10 J. | º - E. E = (ross-se & 72 / /e ZV º = /'laſe /X. „º???!!! 22， <！-- ~- - - - - - - - -=== · ſae· @76， Plate I 'Q %?/ ºd/t/， /º/ //）????? //ºzo) î， Z '^/ ///ø/22//273，2%）， To CoNSTRUCT A DISPLACEMENT SCALE. 457 TO CONSTRUCT A DISPLACEMENT SCALE. D = displacement of the vessel in cubic feet. 8 = displacement in cubic feet per inch of difference of draft. a = area of load water line in square feet. d = draft of water in feet. Q = area of any water line at draft y and displacement a. = exponent of the displacement scale. a d. D ==- n = D - _ g” – lº. * =#, D, ! = VF Cſ 'ſ D 3/ 7. —1 Py". ô = + Oſ, d” 12 d” , Example. The area of the load water line of a vessel is a = 6400 square feet; draft of Water d = 17 feet, and the load draft displacement D = 80,500 cubic feet. Required the draft exponent n = ? and at what draft y the displacement is a = 45,000 cubic feet? n=%22&11. =1.85, gy = 1 80500 Construct a scale as shown by the accompanying figure, and draw the ordi- nates a ; the draft d being divided into eight equal parts. Assuming the displacement as unit, the ordinates a are found in the following || table, Opposite the given exponent m. After the exponent is known, the displacement can be expressed in toms, and the load draft displacement multiplied by the tabular number gives the displace- ment aſ at the corresponding draft y. RULE. Multiply the load draft displacement, expressed either in tons or cubic | feet, by the tabular number for the given exponeut and water line, and the pro- duct is the corresponding displacement. 7 1-35/ Tº i’45000-1105 feet, the draft required. 80500 Displacement Scale. 72 = a d Ordinate Waterlines. Dead rise I) I 2 3 4. 5 6 7 º 1.00 .1250 | .2500 .3750 .5000 | .6250 | .7500 .8750 | Flat lyottom. 1.05 .1127 | .2333 .3571 .4830 .6105 .7393 | .8692 1.10 .1015 | .2176 | .3300 .4665 j .5963 .72S7 | .S634 1.15 .0915 | .2031 | .3237 .4506 .5824 .71S3 .S577 of as 1.20 .0825 | .1895 .3082 .4353 | .5689 .7080 .8512 † : 1.25 ,0743 | .1768 . .2935 | .4205 .5557 .6980 .8463 ; :- 1.30 .0669 .1649 | .2794 | .4061 542S .6880 | .8407 #3 1.35 .0604 || 1539 .2660 | .3923 ,5302 | .6782 .8351 3.3 1,40 .0544 .1436 | .2533 .3789 .51.79 | .66S5 | .8295 § 2 1.45 .0490 | .1340 | .2303 .3660 | .5047 | .6589 .8240 J. 5 1.50 .0447 .1250 | .2297 .3535 | .4941 | .6495 .S185 5.2 I.55 039S | 1166 .218.7 .3415 || 4826 .6402 | .8130 ; : 1.60 0.359 | .1()S8 .2082 .3299 .4714 | .6311 || 8076 .#3 1.65 0323 .1015 .1982 .31S6 .4605 .6221 j .8023 §: 1.70 0291 .09.47 .1887 | .307S | .4498 | .6132 .7969 -- ~~ 1.75 0257 .0SS4 .1797. .2985 .4393 || 6044 .7916 3 = 1,80 .0233 .08.24 .1711 | .2S72 | .4291 | .595S .7SG4 E+ º- I.S5 .0213 .0769 .1629 .2774 .4129 .5873 . .7811 1.90 .0192 | .0718 .1551 | .2680 | .4094 | .5789 . .77.59 1.95 .0173 .0670 .1477 .2588 .4008 .5706 || 7708 || Highest 2.00 .0156 | .0625 | .1406 | .2500 | .3906 | .5625 | .7656 | dead rise. 458 APPROXIMATE LENGTIIs of VESSELs. i f T i 120 122 132. 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Iodi dipuſsuuuags up 1944 odosio]H ‘IONVNaorºad dihswVäIS 095 STEAMSHIP PERFORMANCE. 461 Horsepower im Steamship Performance. º: I Nautical miles or knots per hour. tons. 11 | 12 || 13 || 14 | 15 16 17 18 19 20 T IH | H TH TH H IH H H H H 1 || 5'85 7'59| 9'63| 12:0 14'8 17-9 21-6 25-6 30:1 35.1 2 || 9:28 12.0 , 153 19 1 23°5 28°4 34 2 40 6 7-8 54-7 3 || 12:2 15.8 || 20 0 || 25-0 30-8 38-3 44-8 53-3 62.6 73. 4 || 14-8 192 24.4 3(1-3 37-4 43-1 54'3 64'5 75-9 S8-4 5 17.2 22.2 283 || 35.2 43'4 52'4 630 74-9 88-0 97.8 | 6 || 19.4 25:1 31.9 39 7 49-0 59-1 71-1 84-5 99.2 116 '7 || 21-4 || 27-8 || 35-3 || 44-0 54'0 65-5 79:0 93.7 110 128 8 || 23°4 || 30°4 || 38-6 || 48:1 59.3 68-7 86-2 102 T21 140 9 25-3 || 32.9 || 41.8 52.1 64-0 77.5 93-2 110 30 152 I0 27 2 || 35-3 || 44 8 || 55'8 6S-8 83-2 100 119 J40 163 11 : 29-0 || 37.6 47-8 || 59-7 73-5 89-0 107 127 150 174 12 || 30-7 || 39.9 50 6 || 63-2 77-7 94'4 113 134 158 184. 13 || 32°4 42-0 || 533 66-6 82-0 99.6 120 142 167 194 14 || 34 0 || 44'2 56-0 || 70-0 86-0 105 126 149 176 203 15 35-6 || 46'3 5S'ſ 73-5 90-0 109 131 156 183 213 16 || 37°2 || 48-3 61-3 || 76.5 94-0 114 137 163 192 223 17 | 387 50.2 63.8 || 79:6 98-0 120 143 170 200 233 I8 || 40-2 52'2 | 66-2 S2’ſ 102 124 T48 176 207 242 19 || 41-7 || 54'ſ) 6S-7 || 85°8 106 128 154 182 215 250 20 || 43.2 56-0 || 71.0 | 88-9 111 132 159 189 222 258 25 50-0 || 65-0 || S2-5 T03 127 154 184 194 258 205 30 56.5 | 73°4 || 93-2 117 143 173 20S 248 291 339 35 | 62-6 || 81-3 || 103 130 159 192 230 274 322 377 4-0 || 68°4 || 8S-8 113 141 T'73 209 252 300 350 410 45 || 74.0 | 962 | 122 152 188 228 273 324 382 445 50 || 79.4 || 103 || 131 164: 201 242 293 346 410 476 55 || S4-6 || 110 || 140 174 215 200 312 370 437 509 60 || 90-0 || 117 | 1.49 1S5 226 285 330 393 464 538 65 || 94.7 | 123 156 I95 240 292 349 414 4SS 568 70 || 99-6 || 130 | 164 206 252 306 367 437 512 599 75 || 104 || 135 | 171 214 264 320 383 455 36 624 80 || 109 || 141 | 180 224 276 333 400 467 561 653 S5 113 | 1.47 187 234 287 348 4.17 496 584 680 90 118 153 | 194 243 29S 362 433 516 607 707 95 || 122 | 158 201 25]. 309 376 448 533 629 732 100 || 126 164 207 259 318 3S7 464 551 64S 756 II 0 || 135 | 175 222 277 34() 414 495 58S 693 807 125 || 146 190 241 3(10 370 450 539 640 '753 STS 150 | 165 215 273 342 42() 494 6619 '724 S52 Q92 I'75 || 1 S3 || 238 || 302 378 464 564 675 S()2 946 | 1.10() 20 0 || 200 260 || 330 412 506 615 737 S'ſ 5 1027 | 1201 225 || 217 | 281 || 358 4:47 54S 666 S00 947 11] S | 1300 250 || 232 || 301 || 3S4 478 5SS 714. S55 1016 || 1200 J400 275 248 || 322 409 510 627 762 912 1087 12S6 1490 300 || 262 340 || 432 540 662 S06 966 || 1146 1347 1573 325 77 360 457 570 '700 S52 T010 || 1213 || 1428 1665 350 || 290 378 480 60() 73 896 || 1073 || 1276 | 1500 || 1750 375 || 305 || 395 502 62 770 936 1122 || 1332 1570 | 1830 400 317 || 412 || 522 654 S03 976 || 1170 1402 1632 1907 450 || 343 || 446 567 70S 87() 106t) 1265 1500 | }'770 2005 500 || 368 || 478 || 607 759 932 1131 135S | 1611 || 1896 2213 550 393 || 510 || 64S S10 995 || 121 () 1450 | 1720 | 2025 || 2362 600 || 415 || 540 | 684 856 || 1036 1280 | 1532 1820 21:40 || 2500 650 || 440 57 0 || 724 905 || 1111 || 1350 | 1618 1923 || 2265 2636 '700 || 460 599 || 759 93S 1166 1417 1700 2016 || 2373 2770 750 483 || 627 797 995 || 1220 14S5 || 17SO | 2113 2490 || 2900 800 503 || 654 830 || 1038 1274 1548 1857 2206 || 2593 3026 850 525 | 680 | S66 1080 1330 | 1620 1935 || 2300 2710 || 3152 900 545 || 70s 898 || 1123 1380 1675 2009 || 2385 2803 3274 950 || 565 | 734 || 933 1170 I430 174() 2080 247S 2920 i 3400 462 STEAMSITIP PERFORMANCE. Horsepower in Steamship Performance. Displace- ment in tons. T 1000 1100 Nautical miles or knots per hour. 1 2 3 4 5 6 7 8 H H H HI HI IH H IH 0°438 3•50 | 11-8 28-0 || 54-9 94.6 150 225 0.456 || 3-75 | 12.5 30-0 || 5S-4 100 | 160 239 0°500 || 4-00 || 13'4 || 32-0 || 62-0 107 170 254 0.515 4-12 || 14-0 || 33-0 || 65.3 112 || 179 267 0°548 || 4:38 || 14:9 35-0 | 68-7 119 189 281 0.562 || 4:50 | 15.5 36-0 || 71-9 124 || 197 205 0.578 || 4-62 | 16.2 37-() 75-0 130 || 200 307 0°594 || 4-75 16:9 || 38-0 || 7S-1 135 | 215 320 0.625 5:00 17.5 40’0 81-2 140 224 332 0-634 || 5-25 | 18-1 || 42-0 | 84-2 145 || 231 345 ()'700 5-60 | 18-8 44-0 || 87.0 150 || 239 356 0 719 || 5'75 19-4 46 () 90.0 155 247 369 0.735 5'88 20-0 || 47-0 || 92-7 160 255 3S() 0.765 || 6-12 || 20-6 || 49-0 || 95-6 165 262 391 0-788 || 6-28 || 21-1 50 2 | 98-4 170 270 402 0-805 || 6’44 || 21.8 51-5 101 17 277 414 O'S28 662 22-4 53.0 104 179 285 424 0'851 | 6′S1 || 23-0 54'5 106 184 292 436 0-872 || 6-98 || 23.5 55-8 109 1SS 299 446 0.876 || 7-12 24.0 57.1 III 192 || 306 457 0-909 || 7-35 | 24-6 || 58-8 114 197 313 467 0.931 7:45 || 25-1 59-8 IIT 201 || 320 478 0.952 || 7-62 25-6 G1-0 119 205 327 48S 0.972 || 7-78 26 1 62.2 121 209 || 334 498 0.992 || 7-94 || 26-8 63-5 124 214 340 508 1-01 S’10 27.2 64-8 127 218 || 347 518 1:03 S-25 || 27-8 (36-() 129 222 || 354 528 1*05 8-39 2S-2 67.1 131 226 || 360 538 1’OS 8-60 27-8 68-5 133 230 || 367 548 1'09 8-7() 28-9 69.6 135 234 || 373 558 1:11 S-85 29.9 70 8 138 23S | 3S0 567 1'13 9°01 || 30-4 || 71.1 140 242 S 6 577 1:14 9°14 || 30-9 73-1 l42 246 392 586 1-16 9-30 || 31.4 || 74 4 T45 250 398 595 1:18 9-42 31.9 75-5 T47 254 404 604 1-19 9°56 32°4 || 76-5 150 258 || 410 613 1-22 9-72 32-8 77.7 I 52 261 416 622 1-23 9'S6 || 33-4 78-9 154 26G | 422 631 1-25 | 10-0 33-9 80-0 T56 27() 428 640 1°28 10:1 34°4 S1:1 T5S 274 434 649 1-30 || || 0-3 34 S 82-7 160 277 || 44() 658 1°32 || 10:6 35-6 85-0 165 283 || 455 70 1:36 || 10.9 36.4 || 87.5 171 290 || 469 700 1-40 || 11.2 37-5 90.0 176 298 || 483 '721 1'42 | 11-4 38.0 92.8 181 303 || 497 742 1'47 || 11.9 40-2 95°2 188 322 || 512 7.62 1°52 | 12.2 41-2 97.8 191 330 526 7S2 1'56 | 12:5 42-4 100 196 339 540 802 1:60 | 12-9 43.2 103 202 346 || 554 S22 1 64 13-1 44'4 105 205 355 566 842 1'6S | 13.5 45.5 108 21() 364 579 861 1:72 13 S 46.5 II () 215 372 599 879 1°75 || 14-0 47-4 112 220 379 603 899 1'78 14.2 48-4 115 224 387 615 918 1.81 || 14.5 49-4 116 229 395 628 929 1-84 || 14.9 50-0 119 233 403 640 955 1’S8 || 15.2 51-1 122 238 411 653 973 l'92 || 15 4 52.2 124 242 418 668 991 1-95 || 15-6 53-2 126 246 426 683 || 1008 2:05 || 16.4 55-1 131 255 441 714 || 1044 1488 Of£9L | LS68L f()6II 0800I Iggs ZS89 | SOLG | Slfiſ | 6&g3 || 0Z13 || 0000I gll.GI g0gg| | #6f II | QSQ6 S908 gi'99 g9%g fºr 10#3 9&SZ 0096 £6f 9 L f{}{.8 L | 68&II ZIG6 936, 9Ig9 ºf £g | LFäf 9599 61SZ O gºé IL&QL | #309:I fSOIL ()fgé gS11, S0ft) || 3&g | Ollf 9SZg 8863 || 0006 ZZ6f I 91.1% I gzSOL 29I6 1991. 9839 &Ig g60f £ZZg | #81% Og 28 #89; L | SãC&L | Z990L | #S6S SSp1. f$19 £30g | Iloſ, 1918 9897, 009S gf£FL || 083&I 1970L 90SS Ofg, Zī09 £36; l863 S603 || S85& 09:8 990FL 39.0. 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Tsin, w T'd Example. The weight W-15 tons, the centre of gravity of which is placed at r= 12 feet from centre on deck and h = 8 feet above the water, which careens the vessel to an angle v = 29. The displacement T=4288.8 tons. Required, the mo- mentum of stability Q, and depth centre of gravity d ? Q = 15(12 X cos. 2°-- 8X sin. 2°) = 183.08 foot-tons, and d = —lº – 1.223 feet, the depth of the centre 4288.8 × 0.0349 of gravity of the vessel, under meta-centre. Momenturm of Wind on Sails Careening a Vessel in Sailing. Let F denote the force of wind in tons, acting at right angle to the vessel on the centre of gravity of all the sails, a l feet above the centre of gravity of the displacement. Then the momentum of the wind will be— Q = Fla- T'd sin. v. Eacample. The centre of gravity of all the sails being l = 35 feet above the centre of gravity of the displacement of a vessel of T= 4288.8 tons. The force of wind on all the sails F-7 tons. The depth of the centre of gravity of the vessel, under meta- centre d = 1.223 feet, as found by oxperiments. Re- quired, the momentum Q of the wind, and to what -- angle the vessel will be careened : Q = 7 × 35 = 245 foot-tons, and, sin, v= –9----—”— = 0.04671 = sin. 2°40'40//, the careen angle required. TONNAGE MEASUREMENT. 465 Tonnage of Vessels.—Old U. S. Measurement. T = tonnage of vessel. L = length of the vessel in feet, from the fore part of the stem to the after part of the stern-post, measured on the upper deck. B = greatest beam in feet, measured above the main-walls. d = depth of the vessel in feet. For double-decked vessels, half the beam B is taken as the depth d. For single-decked vessels, the depth is taken from the underside of deck plank to the ceiling of the hold. Example. L = 186 feet, B = 30 and d = 15, for a double-decked vessel. Re- quired, the tonnage 7 Z’= # (L–0.6B) = *::: (180–0.6 × 30) = 795.77 tons. Custom-House New Tommage Law, May 6, 1864. An Act to regulate the admeasurement of tonnage of ships and vessels of the U. S. Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That every ship or vessel built within the United States, or that may be owned by a citizen or citizens thereof, oa or after the first day of January, eighteen hundred and sixty-five, shall be measured and registered in the manner hereinafter provided; also every ship or vessel that is now owned by a citizen or citizens of the United States, and shall be remeasured and reregis- tered upon her arrival after said day at a port of entry in the United States, and prior to her departure therefrom, in the same manner as hereinafter described: Provided, That any ship or vessel built within the United States after the passage of this Act may be measured and registered in the manner herein provided. Sec. 2. And be it further enacted, That the register of every vessel shall express her length and breadth, together with her depth, and the height under third or spar deck, which shall be ascertained in the following manner: The tonnage deck, in vessels having three or more decks to the hull, shall be the second deck from below; in all other cases the upper deck of the hull is to be the tonnage deck. The length from the forepart of the outer planking, on the side of the stem, to the after part of the main sternpost of screw steamers, and to the after part of the rud- der-post of all other vessels measured on the top of the tonnage deck, shall be ac- counted the vessel's length. The breadth of the broadest part on the outside of the vessel shall be accounted the vessel's breadth of beam. A measure from the under side of tonnage deck plank, amidships, to the ceiling of the hold (average thick- ness) shall be accounted the depth of hold. If the vessel has a third deck, then the height from the top of the tonnage-deck plank to the under side of the upper- deck plank shall be accounted as the height under the spar deck. All measure- memts to be taken in feet and fractions of feet; and all fractions of feet shall be expressed in decimals. SEC. 3. And be it further enacted, That the register tonnage of a vessel shall be her entire internal cubic capacity in tons of one hundred cubic feet each, to be ascertained as follows: Measure the length of the vessel in a straight line along the upper side of the tonnage deck, from the inside of the inner plank (average thickness) at the side of the stem to the inside of the plank on the stern timbers (average thickness), deducting from this length what is due to the rake of the bow in the thickness of the deck, and what is due to the rake of the stern timber in the thickness of the deck, and also what is due to the rake of the stern timber in one- -third of the round of the beam; divide the length so taken into the number of equal parts required by the following table according to the class in such table to which the vessel belongs: Table of Classes. Class I.—Wessels of which the tonnage length according to the above measure- ment is fifty feet or under, into six equal parts. Class 2–Vessels of which the tonnage length according to the above measurement is above fifty feet, and not exceeding one hundred feet long, into eight equal parts. Class 3.−Wessels of which the tonnage length according to the above measure- ment is above one hundred feet long, and not exceeding one hundred and fifty feet long, into ten equal parts. Class 4.—Wessels of which the tonnage length according to the above measure- ment is above one hundred and fifty feet, and not exceeding two hundred feet long, into twelve equal parts. Class 5.—Wessels of which the tonnage. length according to the above measure- ment is above two hundred feet, and not exceeding two hundred and fifty feet long, into fourteen equal parts. 30 466 TONNAGE MEASUREMENT. Class 6.-Wessels of which the tonnage length according to the above measure- ment is above two hundred and fifty feet long, into sixtecn equal parts. Then, the hold being sufficiently cleared to admit of the required depths and breadths being properly taken, find the transverse area of such vessel at each point of division of the length as follows: * Measure the depth at each point of division from a point at a distance of one- third of the round of the beam below such deck, or, in case of a break below a line stretched in continuation thereof, to the upper side of the floor timber, the inside of the limber strake, after deducting the average thickness of the ceiling, which is between the bilge planks and limber strake; then, if the depth at the midship division of the length do not exceed sixteen feet, divide each depth into four equal parts: then measure the inside horizontal breadth, at each of the three points of division, and also at the upper and lower points of the depth, extending each meas. urement to the average thickness of that part of the ceiling which is between the points of measurement; number these breadths from above (numbering the upper breadth one, and so on down to the Iowest breadth); multiply the second and fourth by four, and the third by two; add these products together, and to the sun, add the first breadth and the last, or fifth ; multiply the quantity thus obtained by one-third of the common interval between the breadths, and the product shall be deemed the transverse area; but if the midship depth exceed sixteen feet, divide each depth into six equal parts, instead of four, and measure as before directed, the horizontal breadths at the five points of division, also at the upper and lower points of the depth; number them from above as before; multiply the second, fourth and sixth by four, and the third and fifth by two; add these products together, and to the sum add the first breadth and the last, or seventh; multiply the quantities thus obtained by one-third of the eommon inteſr]val between the breadths, and the product shall be deemed the transverse area. Having thus ascertained the transverse area at each point of division of the ves- sel, as required above, proceed to ascertain the register tonnage of the vessel in the following manner: Number the areas successively one, two, three, etc., number one being at the extreme limit of the length at the bow, and the last number at the extreme limit of the length at the stern; then whether the length be divided, according to table, into six or sixteen parts, as in classes one and six, or any intermediate number, as in classes two, three, four and five, multiply the second and every even-numbered area by four, and the third and every odd-numbered area (except the first and last) by two; add these products together, and to the sum add the first and last if they yield anything; multiply the quantities thus obtained by one-third of the common interval between the areas, and the product will be the cubical contents of the space under the tonnage deck; divide this product by one hundred, and the quotient, being the tonnage under the tonnage deck, shaki be deemed to be the register tonnage of the vessel, subject to the additions hereinafter mentioned. If there be a break, a poop, or any other permanent closed-in space on the upper decks, on the spar deck available for cargo or stores, or for the berthing or accom- modation of passengers or crew, the tonnage of such space shall be ascertained as follows: Measure the internal mean length of such space in feet, apd divide it into an even number of equal parts, of which the distance asunder shall be most nearly equal to those into which the length of the tonnage deck has been divided; meas- ure at the middle of its height the inside breadths—namely, one at each end and at each of the points of division, numbering them successively, one, two, three, etc.; then to the sum of the end breadths, add four times the sum of the even-numbered breadths and twice the sum of the odd-numbered breadths, except the first and last, and multiply the whole sum by one-third of the common interval between the breadths; the product will give the mean horizontal area of such space; then measure the mean height between the plank of the decks, and multiply by it the mean horizontal area; divide the product by one hundred, and the quotient shall be deemed to be the tonnage of such space, and shall be added to the tonnage under the tonnage decks, ascertained as aforesaid. If a vessel has a third deck, or spar deck, the tonnage of the space between it and the tonnage deck shall be ascertained as follows: Measure in feet tho inside length of the space, at the middle of its height, from the plank at the side of the stem to the plank on the timbers at the stern, and divide the length into the same number of equal parts into which the length of the tonnage deck is divided; measure (also at the middle of its height) the in- * Chapman's rule, p. 114. TONNAGE MEASUREMENT. 467 side breadth of the space at each of the points of division, also the breadth of the stem and the breadth at the stern ; number them successively one, two, three and so forth, commencing at the stem; multiply the second and all other even-num- bered breadths by four, and the third and all the other odd-numbered breadths (except the first and last) by two; to the sum of these products add the first and last breadths, multiply the whole sum by one-third of the common interval be- tween the breadths, and the result will give, in superficial feet, the mean horizon- tal area of such space; measure the mean height between the plank of the two decks, and multiply by it the mean horizontal area ; and the product will be the cubical contents of the space; divide this product by one hundred, and the quo- tient shall be deemed to be the tonnage of such space, and shall be added to the other tonnage of the vessel, ascertained as aforesaid. And if the vessel has more than three decks, the tonnage of each space between decks, above the tonnage deck, shall be severally ascertained in manner above described and shall be added to the tonnage of the vessel, ascertained as aforesaid. In ascertaining the tonnage of open vessels the upper edge of the upper strake is to form the boundary line of measurement, and the depth shall be taken from an athwartship line, extending from edge of said strake at each division of the length. The register of a vessel shall express the number of decks, the tonnage under the tonnage deck, that of the between decks, above the tonnage deck; also that of the poop or other enclosed spaces above the deck, each separately. In every registered United States ship or vessel the number denoting the total registered tonnage shall be deeply carved or otherwise permanently marked on her main beam, and shall be so continued; and if at any time it cease to be so continued, such vessel shall no longer be recognized as a registered United States vessel. SEC. 4. And be it further enacted, That the charge for the measurement of ton- nage and certifying the same shall not exceed the sum of one dollar and fifty cents for each transverse section under the tonnage deck; and the sum of three dollars for measuring each between decks above the tonnage deck; and the sum of one dollar and fifty cents for each poop, or closed-in space available for cargo or stores, or for the berthing or accommodation of passengers, or officers and crew, above the upper or spar deck. SEC. 5. And be it further enacted, That the provisions of this act shall not be deemed to apply to any vessel not required by law to be registered, or enrolled, or licensed, and all acts and parts of acts inconsistent with the provisions of this act are hereby repealed. - Emglish Tonnage Measurement. Divide the length of the upper deck between the after part of the stem and the fore part of the stern-post into 6 equal parts, and note the foremost, middle and aftermost points of division. Measure the depths at these three points in feet and tenths of a foot, also the depths from the under side of the upper deck to the ceil- ing at the limber strake; or in case of a break in the upper deck, from a line stretched in continuation of the deck. For the breadths, divide each depth into 5 equal parts, and measure the inside breadths at the following points, viz.: at .2 and .8 from the upper deck of the foremost and aftermost depths, and at .4 and .8 from the upper deck of the amidship depth. Take the length, at half the amidship depth, from the after part of the stem to the fore part of the stern-post. Then, to twice the amidship depth, add the foremost and aftermost depths for the sum of the depths; and add together the foremost upper and lower breadths, 3 times the upper breadth with the lower breadth at the midship, and the upper and twice the lower breadth at the after division for the sum of the breadths. Multiply together the sum of the depths, the sum of the breadths, and the length, and divide the product by 3500, which will give the number of tons, or register. If the vessel has a poop or half deck, or a break in the upper deck, measure the inside mean length, breadth and height of such part thereof as may be included within the bulkhead; multiply these three measurements together, and divide the product by 92.4. The quotient will be the number of tons to be added to the re- sult as above ascertained. For Open Wessels.-The depths are to be taken from the upper edge of the upper Strake. For Steam. Vessels.-The tºnnage due to the engine-room is deducted from the total tonnage computed by the above rule. To determine this, measure the inside length of the engine-room from the fore- mast to the aftermost bulkhead; then multiply this length by the midship depth of the vessel, and the product by the inside amidship breadth at .4 of the depth from the decle, and divide the final product by 92.4. 468 CENTRIPETAL PROPELLER. C ENTRIPETAL PROPELLER. THE Centripetal Propeller has, since the year 1851, fought its way through the usual obstructions to success, and is now approved and adopted by the most advanced engineers in Europe and America. From the course of progress, it ap- pears that the form of propeller now in use has not been reached through scientific investigations, but through the usual and expensive course of trials and errors, by which it has gradually approached the form represented on Plate XI., and accord- ing to the present rate of progress that shape will no doubt be reached and finally adopted within a few years more. - e The propellers constructed by John Roach for the Pacific Mail Steamship Com- pany are upon the centripetal principle, a full description of which is given in a work entitled “Education and Shipbuilding,” published in the year 1866, by H. C. Baird, Philadelphia. The helicoidal or propelling surface in the common propeller is formed by a straight generatrix at right angle to the axis; whilst in the centripetal propeller that surface is formed by a spiral generatrix constructed in an angle w, Folmula 7. In practice this angle can be assumed to be, w = 309 for the fore-edge, and w'- 45° for the after-edge of the propeller. The difference between the angles w and w' makes the pitch expanding from the centre to the periphery. IIaving given the spirals a and e, the spirals b, c and d are obtained by dividing the angles into four equal parts, as will be understood by the illustration. A straight generatrix inclined to the axis will give the same helicoidal surface as that of the curved generatrix at right angles to the axis; but the inclination of the straight generatrix must be according to Formula 8. The dotted lines fgh i represent a centripetal propeller with straight inclined generatrix. Propellers constructed either as the dotted or drawn lines, or between the two cases, will produce the same propelling effect in the water. When the propeller is constructed between the two cases represented on the drawing, the blades will appear curved in both views. The length I, of the propeller should be from 0.2D to 0-25 D, and the pitch from 1.5D to 2D. For very sharp vessels constructed for speed, and when the draft of water is over one-half the beam, the pitch may be made 2.5D. One quarter of the pitch is set off on the centre line from 0 to 8, and the helix constructed in the ordinary way. The outer edgo of the blades should not follow the true helix, but be made slightly concave, as shown in the drawing, which makes the pitch expanding in the direction of the axis. Tºwn pitch of the propeller should be calculated by Formula 3, making r = 0-7 R. Prºgle 1. The diameter of a propeller is 10 feet 6 inches, and the angle at the periphery. Required the pitch P= in feet? P= cot. 580X3-14 X 10.5 = 20.6 feet. Example 2. The propeller on Plate XI. is of dimensions D = 15 feet, L = 5 feet, W= 57° 30', the slip is 38 per cent. or S= 0:38. What power is required to drive it 40 revolutions per minute, H =? 153X403 T 480000 Example 3. A propeller of diameter D = 12 feet, angle W- 649, and length L = 3 feet 6 inches, is to be driven by a steam engine of 450 horses, the slip S= 0.28. How many revolutions will it make per minute, n = ? — R * * * (xoskº 30/ +on) = 509 horses, nearly. 78 3 450 g n = −75– = 61 revolutions 12 (3.5×0.28×cos.649+011) per minute. ºn- /77,772 X/. ar, Prºp é//e/ ('eſ). trip 7//// % ! | ! | _ - - ??- r ~ ~ ~ ~ ►. – – + * 2. ^ \ T - >7 Áº ●~ Ti » J ---- — — — º — ~ > �£ ！= • • • • * * * * • • • • • • •=> • <- .-- ~~~~ • • • → • • •=== → → → → → → → → → → → • *-- （- ~ ~ ~ = = = = = =” --> ~ ~ ~ ~ ~~~~）; !|- Formulas for Propellers. 469 Pitch. Angles. Areas. P=r Dot. W 1|cº. W=+, 5|a-#" - - - - - 9 TD P 360 L 360 L L) m P – - 2 v = −. - 6 A=#(L X) - - 10 v ' P ' 3’5 + 2 º P= —#" " -, 3 w_P^*. - 7 a=#(P+ ºpºp). 11 Vº–L” 102°4 275"TV 2 2.5D2 P=*** - 4|Cº-º, slo–––º-, - - - 12 8. 180 D VT2 D?-- P2 Horsepower and Revolutions. D2 nº 78 3 H FH = . W. on) 13 | n = + - ||—. 1 iºſ' Scos. W--0-11 |, 13 n LScos. W--0-11’ Horsepower of Friction. R L km n° ( h = 311-7 RAE –– 26-42 R2 P2 P) - - - 15 59,400,000 P \ + + D = diameter, R = radius, L = length, and P= pitch of the propeller in feet. W = angle of the blades to the centre line. v = projecting angle of each blade. | w = centripetal angle for the curved generatrix. q = angle of inclination of the straight generatrix. a = projecting area of all the blades. A = helicoidal surface of the propelling side of all the blades. a = helicoidal suſrface of one whole convolution. O = acting area at right angles to the axis. All areas in square feet. a = length of any helix at radius r, and m = number of blades. Aſ= length of external helix of the blade. m = number of revolutions per minute. FI = horsepower required to drive the propeller. h = horsepower required for friction in the water. k = friction coefficient. See page 448. The pitch of the propeller is equal to the tabular number opposite the given angle W, multiplied by the diameter. W | Pitch. W Pitch. || W Pitch. Il W Pitch. H| W Pitch. I W | Pitch. 30 5-45 || 40 || 3-74 || 50 || 2-63 || 60 | 1.81 || 70 1-14 || 80 || 0-55 31 || 5°23 || 41 || 3-62 || 51 || 2:54 || 61 | 1.74 || 71 || 1 -I I II 81 0-50 32 5-03 || 42 3•50 || 52 2-45 || 62 | 1.67 || 72 1-02 || 82 || 0-44 33 4-85 || 43 3-27 || 53 || 2:37 || 63 1.60 || 73 || 0-96 || 83 || 0-37 34 || 4-66 || 44 || 3:20 || 54 2-28 || 64 || 1:53 || 74 || 0-90 || 84 || 0-33 35 || 4-50 || 45 3•14 || 55 || 2:20 || 65 | 1.46 || 75 0-84 || 85 || 0-27 36 4°33 || 46 || 3-09 || 56 2-12 || 66 1-40 || 76 0-78 || S6 0-22 37 || 4-17 || 47 2-93 || 57 || 2:04 || 67 || 1:33 || 77 || 0-72 || S7 0-16 38 || 4-02 || 48 2-83 || 58 I-96 || 68 || 1:27 || 78 || 0-67 || 88 0-11 39 || 3-88 || 49 || 2-73 || 59 1-89 || 69 || 1:20 || 79 || 0-61 || 89 || 0-06 470 - - CHEMISTRY. Sixty-five Simple Elements, with their Symbols, Equivalents and Specific Gravity. Sym- |Equiv- §. |º-|sp. gr. ELEMENTS. bols. |alent ELEMENTS. bols. ulent. Sp. gr. Aluminium, . . . Al. 13.70 || 2.50 || Nickel, . . . . . Ni. 29.50 | 8.S0 Antimony, . . Sb. 129.00 6.70 || Niobium, . . . Nb. 48.80 | . . Arsenicum, . . . AS. 75.00 5.80 || Nitrogen, . . . . N. 14.00 0.971 Barium, . . . Ba. 68.50 || 4.70 || Norium, . . . No. * . e. * * Bismuth, . . . . . Bi. 210.30 || 9.80 || Osmium, . . . Os. 99.40 | 10.00 Boron, . . . . Bo. 10.90 2.00 || Oxygen, . . O. 8.00 1.10ST Bronnine, . . . . Br. 80.00 3.187 || Palladium, . . . . Pd. 53.20 | . . Cadmium, . . . Cd. 56.00 | 8.60 || Phosphorus, . . . P. 31.00 | 1.83 Caesium, . . . . . Cs. |123.40 | . . || Platinum, . . . . Pt. 98.60 | 21.5 Calcium, . . . Ca. 20.00 | 1.57 || Potassium, . . K. 39.00 || 0.855 Carbon, . . . . . C. 6.00 3.52 || Rhodium, . . . . Ito. 53.20 | 11.00 Cerium, . . . . Ce. 46.00 . . . . It ubidium, . . Rb. 85.36 | . . Chlorine, . . . . . Cl. 35 50 2.44 || Ruthenium, . . . Ru. 52.11 8.60 Chromium, . . Cr. 26.30 | 6.8 || Selenium, . . . Se. 39.70 || 4.8 Cobalt, . . . . . Co. 29.50 8.9 || Silicon, . . . . . Si. 14.00 | . . Copper, . . . . . Cu. 31.70 || 8.9 || Silver (Argentum). Ag. 108.00 10.5 Didymiumi, . . . . D. 48.00 . . . Sodium (Natr.), a. | Na. 23.00 0.972 Erbium, . . . E. * . . || Strontium, . . . Sr. 43.80 || 2.54 Fluorine, . . . . . F. 19.00 | 1.31 || Sulphur, . . . S. 16.00 2. Glucinum, . . . Gl. 4.70 |0.0692 || Tantalum, . . . Ta. 68.80 | . . Gold (Aurum), . . Au. |196.44 || 19.34 || Tellurium, . . . . Te. 64.5 6.6 Hydrogen, . . H. 1.00 |0.0692 || Terbium), . . . | Tb. * * , , º is Indium, . . . In. 74 | . . || Thallium, . . . Th. * * © Iodine, . . . . I. 127.00 || 4.94 || Thorinum, . . . . Th. 59.50 | . . Iridium, . . . . . Ir. 98.60 | 18.68 || Tim (Stannum), Sn. 59.0() || 7 Iron (Ferrum), . | Fe. 28.00 || 7.80 || Titanium, . . . . Ti. 25.00 || 5. Lanthanum, . . . Ila. 46.00 | . . . ] Tungsten (Wolf.), W. 92.00 17. Lead (Plumbum), Pb. 103.60 | 11.44 || Uranium, . . . U. 60.00 10 Lithium, . . . . . I. 7.00 0.593 || Vamadium, . . W. 68.5 |, . Magnesium, . . Mg. 12.16 | 1.70 || Yttrium, . . . . Y. tº tº ; : | Manganese, . . . Mn. 27.48 8.00 || Zinc, . . . . Zn. 32.60 7.00 Mercury, . . . Hg. 100.00 || 13.59 || Zirconium, . . . Zr. 22.40 | . . Molybdenum, . . . M. 48.00 | 8.60 Proportions of Compounds. Carbon. Hydrogen. Oxygen. Nitrogen. NAMES. C. H O N Olive oil, by weight, is g º & 772 133 95 Spermaceti oil, by weight, . . º 7SO IIS j 02 Castor oil, {{ * Y ſº tº 740 103 157 Linseed oil, {{ tº gº * 76() 113 127 Alcohol, {{ tº º ſº 527 129 344 Sugar, {{ * te e 432 68 500 Atmospheric air, “ * - tº tº º * @ 230 770 Atmospheric air, by volume, . e e G tº º 210 790 Water, fresh, by weight, º e e * e 1 8 Water, fresh, by volume, . g gº e is a 2 1 India rubber, loy weight, . & * 853 147 Blood, by weight. 66.6 iron, e 665 53 II0.4 104 Gunpowder, 75 mitre, 13 charcoal, 12 sulphur. Binary Compounds, with their Formulas and Equivalents, CHEMISTRY. 471 j -Ty - Name of Compound. Water, Binoxide of Hydro., Protoxide of Nitro., Binoxide of Nitrog., Hyponitrous Acid, Nitrous Acid, Nitric Acid, Ammonia, Cyanogen, Sulphurous Acid, Sulphuric Acid, Carbonic Oxide, Carbonic Acid, Light Carb'ett'd Hy. Olefiant Gas, Bisulphide of Carb., Boracic Acid, Chlorous Acid, Chloric Acid, Hydrochloric Acid, Quadrochloride Nit., Hodic Acid, Hydriodic Acid, Teriodide of Nit'gn, Hydrofluoric Acid, Phosphorous Acid, Phosphoric Acid, Phosphoret'd Hydr., Selenious Acid, Selenic Acid, Seleniuret'd IIydro., Sulphuret'd Hydro., Protoxide of Iron, Peroxide of Iron, Dinoxide of Lead, Protoxide of Lead, Quadrotrisoxi’e Le’d §orm ula. HO HO2 NO NO2 N 03 NO4 N 05 NHs N C2 SO2 SO3 CO CO2 EI2 C H2 C2 CS2 BO3 C| 04 Cl O5 H Cl N Cl4 I05 H I N I: II Fl P2 Os P2 O5 H3 P2 Se O2 Se 0s H Se HS Fe O Fe2 Os Pb O . | Ph2 O Pb3 04 I 7-2 26-4 32-1 40-1 14-1 22- : j 3 3 6 7 36-5 156-2 166-5 127-5 393-7 19-7 55-4 71.4 34°4 56-0 6 (-0 41 - 0 17-I 36°0 80-0 | Il-7 215.4 343-I | Equiv.: Name of Compound. | 9-0||Binoxide of Lead, 17.0||Dinoxide of Copper, 22.2|Protoxide of Copper, 30°2||Chloride of Copper, 38-2|Oxide of Zinc, 46-2|Sesquioxide of Ant., 54°2|Antimonious Acid, Antimonic Acid, Protoxide of Tin, Binoxide of Tin, Bisulphide of Tin, Chloride of Tin, Bichloride of Tin, Oxide of Bismuth, Chloride of Bismuth Protoxide Mangan., Sesquioxide Manga. Red Ox. Manganese Binoxide Manganese Protoxide of Cobalt, Peroxide of Cobalt, Protoxide of Nickel, Peroxide of Nickel, Arsenious Acid, Arsenic Acid, Arseniuretted Hydr. Sesquisulphide Arse. Protoxide Mercury, Peroxide of Mercury Bisulphide of Merc., Chloride of Mercury Bichloride of Merc., Oxide of Silver, Chloride of Silver, Teroxide of Gold, Terchloride of Gold, Bichloride Platin’m, Formula. Pb O2 Cu2 O Cu O Cu Cl Zn O Shz Os Sb2 04 Sb2 Os Sn O Sn O2 Sn S2 Sn Cl Sn Cl2 Bi O Bi Cl Mn O Mn2 02 Mn3 04 Mn O2 CO O Co2 0s Ni O Ni2 0s As2 Os As2 O5 Hs As2 As2 Ss Hg O. Hg O2 Hg S2 Hg. Cl Hg Cls Ag O Ag Cl Au O3 Au Cls Pt Cl2 | Equiv. 34°0 76-0 110-0 42°6 37-5 83-0 37-5 83°0 99-4 115-4 78°4 123-7 208-0 216-0 232.2 235-5 271-8 116-3 143-8 220-6 303-1 iggs To Transform Chemical Formulas into a Mathematical Expression. Rule. Multiply together the equivalent, (equiv.) and the exponent, (exp.) of each substance and the product is the proportion in the com- §§ by weight. Divide this by its specific gravity gives the proportions y bulk or volume. Eacample 1. Required its proportioned parts by weight in 1000? cquiv, exp. Carbon C1, =6'12×4=24-48 Hydrogen Hö, - 1X6= 6 Oxygen Oz, = 8X2=16 } 1000:46′48–21-5. proportions. 527 X21.5 & 129 U 31. 1000 The chemical formulae for common alcohol is C; } by weight. He O2. 472 BINARY CoMPOUNDs. Bimary Compounds of Salts and Solids. Chemical Terms. Chemical Formula. Remarks and Popular Names. Biborate of soda, . . . NaO, 2BO3+10HO. : Borax, used as a flux. Nitrate of potassa, . . . . . ISO, NO5. . . . Saltpetre, prismatic crystals. Nitrate of soda, . . . Na0, NOF. . | Soda Saltpetre, cubic crystals. Carbonate of lime, . . . . . CaO, CO2. . . . Common limestone, marble. Sulphate of lime, . . CaO, SO3 ‘īāHo. Alabaster, gypsum, plaster Paris. Oxide of calcium, . . . . . . CaO. . . . . Quick or caustic lime. Chloride of sodium, . . . ClNa. . . . Common salt. Chloride of ammonium, . IIAN, Cl. . . . . Sal ammoniac. Sulphate of soda, . . . NaO, SO3 + 10HO. Glauber salt. Oxide of sodium, . . . . NaO. . . . Soda. Oxide of Natrium. Oxide of barium, . . . . . . Ba0. . . I}aryta, a gray powder. Sulphate of baryta, . . Ba0, SOs. . . . . Heavy spar, mineral used in paints. Sulphate of alumina KO, SO3 + Al2O3, Alum, used for dyeing, calico and potash, . . . 3S03 + 24HO. printing, preserving skins, etc. Bimary Compounds of Liquids. Nitric acid, . . . . . . NO5. . . . Aqua fortis. Nitrolium, . . . . . . . C6H62(NOA)06. . . Nitroglycerine, explosive. Aqua regia, . . . . NO5, -- 2.É. Nitro-muriatic acid. Muriatic acid, . . . . . . . HCl. . . . . Hydrochloric acid. Water, . . . . . . . . . . . HO. . . . . Protoxide of hydrogen. Alcohol, . . . . . . . . . C4H8O2. . . Binary Compounds of Gases. . . . N2O. . . . . Atmospheric air. Nitrous oxide, . . . . . NO. . . . Laughing gas. Nitric oxide, . . . . . . . . NO2. . . . . Extinguishes fire. Carbonic acid, . . . . . C02. . . . Perfectly-consumed coal. Carbonic oxide, . . . . . . . CO. . . . . Partly-consumed coal. Carburetted hydrogen, . . C2H4. . . . Marsh gas, fire-damp. Olefiant gas, . . . . . . . C4H4. . . . Illuminating gas. Cyanogen, . . . . tº º K.02. . . . Produces blue color. Vegetable Acids and Salts. Tartaric acid, . . . . . C4H606 . . . Sour vegetables. Acetic acid, . . . . . . . C2H4O2 . . | Pungent, agreeable odor. Citric acid, . . . . . . . C6H 807 . . . Juice of lemons. Oxalic acid, . . . . . . . C2H204 . . | Powerful poison. Mecomic acid, . . . . . C7H4O7. . . . Exists in opium. Morphia, . . . . . . . C17H19NO3 . Active substance of opium. Opeanic acid, . . . . . . Cao Hio,05 . . . Melts at 284°Fahr. Quinic acid, . . . . . . . Čº. . . . From bark of trees. Quinine, . . . . . • C20H24N202 For chills and fever. Gallic acid, . . . . . . . C7H605 . . . Used for black ink. Lactic acid, . . . . . . CAIH603 . . . From milk sugar, Saccharic acid, . . . . . . CoH 1008 . . . From cane sugar. Saccharose, . . . . . . C12H22O11 . Cane sugar. Gum-arabic. Alcohol, . . . . . . . . . C4H10O2 . . . Intoxicating. Stearic acid, . . . . . ClaRI3602 . Solid fat. Candles. Strychnine, . . . . . . c.fijiº, . I Strong poison. Equal proportions of different atoms may be formed into different orders and make different substances, as cane sugar and gum-arabic; also, strychnine and quinine, which have the same formula. NITRo-GLYcERINE. 473 NITRO-GIYCERINE, CH2(NO00. Nitro-glycerine is an oily liquid of the above composition, which is highly ex- plosive under peculiar circumstances, but can be set fire to and burned like alcohol without explosion. It explodes by concussion or pressure of about 2000 pounds to the square inch, or by the corresponding temperature of about 630°Fahr. suddenly applied. The explosion of nitro-glycerine is imstamataneous, like that of electricity pass- ing between two points, decomposes a small portion of the air and explodes the nitrogon by concussion, which makes the electric spark. Thunder and lightning are explosions of a kind of nitro-glycerine formed by electricity in the air. Small portions of nitro-glycerine, say half an ounce each, placed (any number) within a few feet of one another, if one of them is exploded, all the rest will explode simul-instantaneously. Therefore, when a charge is to be exploded, care must be taken that no more of it is in the neighborhood. It may appear strange that nitro-glycerine can be so dangerous to handle, when it requires the enormous pressure of 2000 pounds to the Square inch to explode it; but the fluid may be squeezed between surfaces of only one 10,000th part of one square inch, when the pressure need be only 3 ounces to explode it. The charge of nitro-glycerine in blasting is exploded by a percussion cap placed on the end of a fuse and dipped into the liquid. The fuse explodes the fulminant in the cap, the concussion of which explodes the charge. On account of the action of nitro-glycerine being instantaneous, no tamping is required in the blast-hole, ex- cept water or loose sand, but even that is not necessary. This explosive is there- fore entirely unfit for use in firearms, which would be blown to pieces without dis- charge through the muzzle. - The many and very serious accidents which have happened by unexpected ex- plosions of nitro-glycerine have caused it to be forbidden transportation on railroads and steamboats, for which a new form of the explosive has been invented, which consists in mixing sawdust and some other solid substances with nitro-gly- cerine, to the form of a moist brown powder, of Ilearly the same specific gravity as that of water. Dynamite. This powder is called dynamite, and is now manufactured by the nitro-glycerine inventor, Alfred Nobel, in IIamburg, and also by the Giant Powder Company, in San Francisco, California. The strength and instantaneous action of dynamite are precisely the same as those of nitro-glycerine, but it is much safer to handle, it is said—more so than common gunpowder. The dynamite powder is made up into cartridges of different sizes to suit the blast-hole, and is exploded by percussion caps like nitro-glycerine, and requires no tamping. It has been employed with great success in blasting im- mense masses of rock in the Andes, Peru. The price of dynamite is higher than that of gunpowder per weight, but its ex- ecution per price is much greater. The blast-holes for dynamite need be only one- half the size of those for gunpowder, with equal execution. . * The instantaneous action of dynamite malies it far superior to gunpowder in blasting, but it is unfit for use in firearms. Any number of cartridges of dynamite placed in a deep blast-hole with tamp- ings of sand between, if one of them is exploded, all the rest will explode simulta- neously. Small cartridges are made for the percussion cap, and called primers, by which the principal charge is exploded. Should a charge fail to explode, put in a new fuse and primer. Blasting under Water. For this purpose the cartridges should be made of strong oiled paper and per-H fectly water-tight, to save the dynamite from moisture. The cartridges should also be ballasted, so as to sink easy in water, which can be done by placing a lead ball in the bottom and pack the dynamite on the top, after which the cartridge is her- metically sealed with some varnish insoluble in water. The cartridges (any num- ber) are guided into the blast-hole through a tube, and finally the primer with the fuse, by which the whole charge is exploded. - Dynamite is insoluble in water, but will not explode if moist with water. It freezes to a snowy mass at 40° Fahr., but its explosive quality is not impaired rºsezeby. At 2129 the nitrogen evaporates and spoils the powder. 474 CEMENT, CoNCRETE AND MoRTAR. CEMENT, CONCRETE AND MORTAR. Roman Cement. Parker's analysis. One part of common clay to 2% parts of chalk, set very quick. Concrete. Eight parts of pebble or pieces of brick about the size of an egg, to 4 parts of scrap river-sand, and 1 part of good lime, mixed with water and grouted in, makes a good concrete, - - - Lime Mortar. One part of river-sand to two parts of powdered lime, mixed with fresh water. - - - Hydraulie Mortar. One part of pounded brick powder to two parts of pow- dered line mixed with fresh water. This mortar must be laid very thick between the bricks, and the latter well soaked in water before laid. - - No. 1. Hydraulic Concrete, by Treussart. 30 parts of hydraulic lime, measured in bulk before slacked. “ sand. t 20 “ gravel. - 40 “ broken stone, a hard limestone. . This concrete diminishes one-fifth in volume after manipulation. The mortar is made first, and then mixed with gravel and stone. No. 2. Another Concrete, by Treussart. - 33 volumes hydraulic lime unslacked. 45 “ Puzzolano (Pozzulano). 22 “ sand. - - 60 “ broken stone and gravel. - Asphalte Composition for street pavement, by Colonel Emy. 2# pints (wine measure) of pure mineral pitch. 11 lbs. of Gaugeac bitumen. . 17 pints of powdered stone-dust, wood-ashes or minion. Cements for Cast Iron. f Two ounces sal-ammoniac, one ounce sulphur and sixteen ounces of borings or filings of cast iron, to be mixed well in a mortar and kept dry. When required for use, take one part of this powder to twenty parts of clear iron borings or fil- ings, mixed thoroughly in a mortar, make the mixture into a stiff paste with a little water and then it is ready for use. A little fine grindstone sand improves the Cement. - - Or, one ounce of sal-ammoniac to one hundred weight of iron borings. No heat allowed to it. •.. - - ‘. . The cubic contents of the joint in inches, divided by 5, is the weight of dry bor- ings in pounds Avoir. required to make cement to fill the joint nearly. - Cement for Stonne and Brick Work. Two parts ashes, three of clay and one of sand, mixed with oil, will resist || weather equal to marble. w - i; - Brown Mortar. One part Thomaston lime, two of sand and a small quantity of hair. Hydraulic Mortar. Three parts of lime, four Puzzolano, one smithy ashes, two of sand and four parts of rolled stone or shingle. º Crushing Weight in Pounds per Square Inch on Portland cement, mixed with different proportions of sand, and at different age of the mixture in months. Age in Parts of Sand to one of cement. - months. O | 1 | 2 |_ 3 | 4. | 5 || 6 3 || 380) 2490 1900 1500 1200 950 | 7so 6 5280 355 () 2750 . 2190 +800 1500 1200 9 5980 445() 3350 2700 22SO 1800 1440 12 6160 5150 3850 3010 245() 2050 | 1600 About 3% of this weight should be depended upon in practice. Some iron filings in a very weak solution of sal-ammoniac, mixed with Portland cement, increases its strength to double or more. - BRICKs. 475 Dimensions. w Common brick, 8 × 4 × 2% inches = 85 cubic inches. Front brick, 8+ X4; X 2; “ 92.8 “ 46 When laid in a wall with cement, it occupies a space of Coumon brick, 8+ X4% X 2% inches == 102 cubic inches. Front brick, 8; X44 × 24 “ = 111 “ {& Weight and Bulk of Bricks. Number of bricks. - by itself. in wall with cement. Tons. Pounds. Cub. ft. C. brick. F. brick. C. brick.__F. brick. 1 224() 22.4 448 416.6 381 347 0.04464 100 I 20 T8.6 17 15; 2.23 5000 50.00 1000 920 850 77 2.4 .. 5376 53.76 1075 1000 914 834 2.62 5872 58.72 113() 1100 1000 913 2.88 6451 64,51 1240 1200 1100 1000 One perch of stone is 24.75 cubic feet. Acids for Soldiering or Timming. TIN. One part of muriatic acid, with as much zinc as it will dissolve, then add | two parts of water and some sal-ammouiac. - BRASS and COPPER. One pound of muriatic acid, four ounces of zinc and five ounces of sal-ammoniac. ZINC. One pound of muriatic acid, two ounces of sal-ammoniac with all the zinc it will dissolve, then add three pints of water. IRON. One pound of muriatic acid, six ounces sperm tallow and four ounces of sal-ammoniac. - GOLD and SILVER. One pound muriatic acid, eight ounces sperm tallow and eight ounces of Sal-ammoniac. Silvering Metals. - Ten parts of nitrate of silver, ten parts common salt, thirty parts cream of tar- tar. Moisten the powder with water when ready to apply. - Glues. Rice glue. Rice flour mixed in cold water and boiled in china or clay pot; stir it well during the boiling. This makes an excellent white glue. Houseblose glue. Dissolve the houseblose in strong alcohol, and apply it hot on the articles to be glued. This makes a very strong glue which is not soluble in Water or moisture. Barrel Measure. Bushel Measure. A barrel of flour weighs 196 pounds. . The following are sold by weight per . A barrel of pork, 200 pounds. bushel : : A barrel of rice, 600 pounds. Wheat, beans and clover-seed, 60 A barrel of powder, 25 pounds. pounds to the bushel. A firkin of butter, 50 pounds. Corn, rye and flax-seed, 56 pounds. A tub of butter, 84 pounds. Buckwheat, 52 pounds. Barley, 48 pounds. - Oats, § pounds. 14 pounds, . & ge 1 Stone. Bran, 20 pounds. 28 pounds, & º . 1 quarter. . Timothy-seed, 45 pounds. 4 quarters, . e e 1 cwt. Coarse salt, 85 pounds. Acre. A square of 208.75 feet each way is one acre. A circle of 235.5 feet in diameter is one acre. 476 ELECTRo-Dynamics. E L E G T R 0-D Y NAMICS. It was intended to devote this page to the elucidation of Electro- Dynamics in its simple form, but the subject is so confused and complicated in our text-books that it would be necessary to reform the whole, and abolish a great number of obscure and curious terms which are established by the highest authorities, which the author does not feel disposed to undertake, at least so long as e- chanical dynamics remains in its present confusion. - The science of Electro-Dynamics can be treated fully within the fundamental formulas and terms which are explained in the begin- ning of page 311. These ideas of dynamics have now been before the public since year 1864, but have not yet been officially con- firmed or acknowledged by any scientific authority. A few attempts have been made to upset them, but without success; the parties |only exposed their weakness on the subject. A list of confused dynamical terms is given on page 310. A similar list for Elec- tricity and Electro-Dynamics would be about four times as long. The greatest confusion consists in that each term has a different definition, whilst in substance it is the same function differently conceived, and thus differently named. An author of independ- ent reasoning finds a strange function in his formulas, which he christens with a highly scientific name; a second author finds the same, but to him strange, function, and gives it another curious name; a third author reads the two books, and copiet the two terms with their respective definitions as two distinct functions, and so the confusion has grown to a mystery. ELECTRICITY. 477 Electro-Chemical Ord I C I T Y. E Simple Substances er of Orde P LECTRO-positive * r of Comducting P gººm. ºw E 2 2.5 3 Metals, b for Electricity g Power odiu o: O A. 8 º #. 3 & # weiß est conductors. . . . $, $ ithium. }={ g § ;: c # P juTInt charcoal Gly º of S : Barium. * | #### É. º £3333 - • * . OD e S Strontium. # 33 §§ 2 P centrated acids. *5 S. s.3 Calcium. É #853 E. É. charcoal # E3 3 jº. #|###|: + 33 3 & Aluminium. à | * g 3 #3; M ine solutions. g #3 # Uranium. 5 || 333 3 : 3 ... OTeS. *::: 3 # o Manganese. É 3 3 = 33 3 S nimal fluids. 3 #75 Cº. --> Zinc. E ... a. 3 5.9 §. * : 5 * : - • *s s Iron. ; ‘35.2 9 : 5 g §§ water. ° 4-, : 3 3 Nickel. * | # 333i: |I in Water. #33;á Cobalt. 5. 3.5 E = 3 ce above 139 F 3 - 5 § 3 Cadmium #| = #####| || $º. ahr. ####3 º Go 3 go º-, 3 $2 “... • : că Lea * ; : 5 & Living y & ; : C © Hºl. § ## 3 #5 3. #; ...tº. # 3.5 33 :-- *: 50%. 25 3.5 St als. ~! -aºs ... 9 Bismuth #| ######|sº = #g g sº & lº q2 * ar 8t - £3 - -> ilver. É. 3 = 3.5 g. 5 || V efied air. tº o, 3 £ 3 #5 Mercur £ | g . g º a 3 apor of alcoh F3 c.45 B: ‘H y. -r: *4 * wa Moi ol. 4- 3 3 ; : Palladium 3 || 3 ºf £893 oist earth and 3 & B 3 35.3 Plati o . . . ; ; ; ; ; Powd nd stones. | * * : 3 * : Gold c- Ö º dº.: "º go Flower Of $– a : Tº j - # 3 # = 3, 5:5 sulphur 3 = 2 3 #,75 Hyd § 5 T - 2 +S Dry metalli * - jää 3.3 : ydrogen. 3 à g : #5 || Oil ic oxides. g5 5- 23 2 Silicon B § 5 : *.3 11S, the heavi 3 . ; 2 &# Titaniu § £ # - * 3 best viest the à Bºg É. #"> 8, Ill till?. bºnde # 9 sº. 3 A º 3: 2 E- 25 - Tellurium lºa o:3'ſ 3.3 : #. 2.9 ~ rº- O - e H-5 * 3– $—s : T3. In 4. r: 2-4 C ~ CD Antimony. 5 | #### 55 sparent crystals : ; ; ; ; ; Carbon g | = §: ... 3 & ice below 136 Fair." : š #3 ~ ; Boron. § | *** 33 || Phosphorus. T. 3 : ; ; ; ; Tungsten. Eb 3 g : 9 Lime. E-i 5 3. #S 2. Molybdenum à | ###. 3. chalk. & ##S 5 Wanadium. s 3 gé 5 s gº.º. 3 3 + 53 Chromium. £ 5 #2 § # º - * = #3 Arsenicum. à | * : * >3 § icious stones. # 35 #g Flºorus # | g =5 #3 #. marble. gº:3 = º - & # ; 5.3 orcelai - ** – odine. # 3 is E £ Bak lain. º, Fºg's a Bromine. : § 9 §§ : Fº wood. … 3 # 9 §. #|#### |#: "" ##### uorine. # ºf 5 3 àtiller. e $33: 353 Nitrogen. # | ###3 § Fººt. gºi; a 3 - ** * - * * * * Selenium. = | # 3 E: : ry paper. E = gé Sulphur. à |#####| ||#.” 3.33 = 3 Oxygen. e --> 3. .g.: 3 Hair: --> cº †: = c ELECTRO-NEGA § 3 ; ; ; Wool. à l'Éjº Ord Artwr. ##### Dyed silk. Å 3 ## 3. or of Compounds, ##### Bleached silk. 2 #. #5 ELECTR , 3.5.3 ° à Raw silk 5 * @ Tºº O-POSITIVE § 3 ; 3. - * * >" ºf it: Fur. º #33 = Diamond. ####3 - S = C §º. ă ă ă ă Yºu; gº. 5 Sog F oollen cloth. tº 5 ° # 5 vitrifications 3.353 > eatſherS. 2 : * : * #lass. e 5 #### 3 Wood. 5 §§ 33 Jet. 5.3 3 T.G 3 wº-4 >, º > -º © GD Kł) Paper. spº 8.5. Wax. § # 2 # s: . - +º * * Silk. ####3 Sulphur. § £5 33° Lac. 33: 3 3 Resius. § 5 = 3 $5 rººf Rough glass s: # sº Amber. § 3 ; # 5 Sul & § 3 ; 3 = Shel ‘Siº 5 ° E 3 3. phur. ššā ā à. lac. Sºº 803 Kl) O ºs C E Tº l'IUU ºl- s.sºr sº to #. &# 3 # # . the worst #. 55 § O-NEGATIVE. : § 3 ; ctor of all. $'s 3 g = i. > S : S. § - 3.5 ± 3 ##### 478 ASSAYING. FIRE-ASSAY OF SILVER AND GOLD ORES. From actual practice by the author in California and South America. Assay Composition. Gold or silver ores, 400 grains. Litharge (oxide of lead), 500 “ Carbonate of soda, 240 Dorax, 110 “ Charcoal, 20 “ º Total, 1270 “ All the ingredients to be well powdered and mixed before placed in the cruci- ble. Should the ore contain much sulphur, stick a 3-inch nail in the assay. The more galena in the ore, the less litharge is required. Smelt the assay, cupel the lead and weigh the remaining button of precious metal. Should the button be pure silver, multiply the weight in grains by 100, and the product is the value of silver in dollars per ton of ore; if pure gold, multiply by 1500, and the product is the value in dollars per ton of ore. When the button contains both gold and silver, the latter metal must be dis- solved in nitric acid, for which the alloy must contain at least 3 silver to 1 of gold, otherwise the acid will not dissolve it. In case the alloyed button does not con- tain sufficient silver, it is necessary to add what is required, and melt it into one button by blowpipe and charcoal. Hammer the button to a thin leaf and boil it in nitric acid; when all the silver is dissolved, the pure gold remains solid. Wash the gold in clean water, dry and weigh it. Suppose the alloyed button to weigh 2.156 grains, and its color being between that of gold and silver, so as to suspect too little of the latter metal; them add, Say, 2 grains of pure silver, and dissolve the button, weigh the remaining gold, which, for example, may be 1.162 grains. Then 2.156–1.162 = 1.994 grains of silver in the assay. Silver, 1.994 + 100 = 199.40 dollars per ton. Gold, 1.162-H 1500 = 1743 $6 $6 Value of the ore, − 1942.40 “ &G About one per cent. of the precious metal is lost in the cupelling. This rule is sufficiently correct for practical purposes. North American Standard. Gold, 387 ounces, 8000 dollars. Pure Silver, 99 ounces, 128 dollars. Peruvian Standard. Pure Gold, 1 ounce, 24.29 pesos = 19.43 soles. Silver, 1 libra, 25.66 pesos = 20.53 soles. One peso = 4 francs; one sole = 5 francs. Assay Table M.–North and South American Measures. The table will answer for any system of assaying weights. Percentage Value of Metal per ton || Value of Metal per quin- Sil of metal in of Ore. tal of Ore. liver per the ore. Gold. Silver. Gold. Silver. Cajon. T'er ct, Dollars. Dollars. Soles. Soles, Marcs. 0.1 602.924 39.709 31.090 2.053 12 0.2 1205.85 79.418 62.179 4.106 24 ().3 ISOS.77 II9.127 93.269 6.158 36 0.4 24.11.69 158.836 124.359 8.211 48 0.5 3()14.62 198.545 155.448 10.264 60 0.6 3617.54 238.254 186.538 12.317 72 0.7 4220.45 277.963 217.627 14.370 84 0.8 4823.39 317.672 248.717 16.422 96 0.9 54′26.31. 357.381 279 80'ſ 18.473 108 i per cent. 6029.24 397.090 3.10.896 20.528 120 North American. South American. SILVER AND GOLD. 479 Suppose the assay to be 112 grammes, and the cupelled button weighs 0.657 of a gramme of silver, then 0.657 × 100: 112 = 0.586 per cent. 0.5 = 10.264 .08 = 1.642 - See Table §: }: Soles per quintal. 0.586= 11.918 0.5 = 198.545 D 0.08 = 31.767 ollars per ton See Table { boºg- 2.33% of ore. 0.586 = 232.694 Table II. For Gold and Silver. Weights. Value Bulk. Avoirdupois. Troy. in dollars. Gold. Silver. Tons. | Pounds. Ounces. Grains. Gold, I Silver. | Cub. ft. [Cub. in. Cub. ft. [Cub.in. T 2000 |29166.614 millions. 602924| 39709 |1.6643 |2875.913.060 |52S7.48 0.0005 1. 14,5833 7000 301.46|18.854 — — 1.43795 | — — 2.64284 — — . 0.06857 I 480 20.6718|1.2929 || — — 0.09859 — — |0.18129 — — . 0.0002 |0.00283 I 0.04306|0.00269| — — |0.00020 — — . 0.00038 — — . 0.00332 |0.04837 || 23.2202 1 1.000 | — — |0.0524 — — . — — — — . 0.05304 |0.77346|| 371.264 1.0000 1. — — | — — . — — 0.1401 0.60085| 1201.7 |17524.8| 8411900 || 362267| — — I 1728 |1.000() 1728 — — |0.695 [3 |10.1416 4867.99 |209 645) — — |0.00058|| 1 |0.0005S]1.00000 0.32679|653.577 |9531.34 4575043 — — 12976.4|1.0000 || 1728 I 1728 — — 10.378.227 5.51581| 2647.59 — — 17.5095 |0.0005S 1.0000 |0.00058! I Table III. Gold, Silver and Platinum. Weight in grains per square inch of sheet, thickness by Birmingham gauge for those metals, and ºn inches. Bir. G. Thick. Gold. Silver. Platin. Bir. G. Thick. Gold. | Silver.|Platin. No. inches. grains. grains. grains. No. inches. grains. grains. grains. 1 0.004 20.68 || 11.52 || 25.50 19 0.063 || 339.5 184.7 397.5 2 0.005 26.93 || 14.40 || 31.26 20 0.969 371.8 201.9 435.0 3 0.0()6 32.19 || 17.28 38.00 21 0.075 404.0 220.0 471.8 4. 0.008 || 42.80 || 23.52 50.43 22 0.081 || 436.1 || 237.2 509.2 5 0.010 || 53.85 | 29.28 58.14 23 0.087 468.8 || 255.0 || 548.6 6 0.012 64.46 || 35.04 || 75.45 24 0.093 500.0 | 272.6 586.0 7 0.014 '75.42 40.80 | 87.88 25 0.099 || 533.3 290.0 625.0 8 0.016 || S6.58 || 46.56 || 100.1 26 0.105 566.6 || 307,8 | 663.0 9 0.018 97.07 || 52.00 113.2 27 0.111 || 596.1 325.5 697.3 10 0.022 || 118.9 64.32 || 138.5 28 0.117 630.0 342 6 T35.0 11 0.025 134.6 | 72.96 157.7 29 0.124 673.3 363.5 783.1 12 ().029 156.3 | 84.96 1892.3 3() 0.130 701.5 3S0.3 817.0 13 0.033 178.2 | 96.4S | 207.4 31 0.136 || 730.0 || 398.4 || S55.0 14 0.038 204.6 111.3 || 2:39.5 - 32 0.142 || 769.5 || 416.3 | S92.0 15 0.043 231.5 | 125.8 270 5 33 0.148 || 798.5 433.3 | 932.0 16 0.048 258.8 || 140.8 302.6 34 0.152 837.6 451.6 || 970.0 17 0.053 285.6 155.3 || 323.8 35 0.160 865.7 470.5 I007 18 0.058 312.6 l 170.0 || 365.3 36 0.166 | 894.0 l 486.0 | 104'ſ California. Rule for Silver and Gold. It is an established custom in California to allow one per cent. for base metal in all gold and silver bars from the mines. The fineness is always stamped in parts of 1000; that is, if a gold bar is stamped 900 fine, it is understood to contain— 900 parts of pure gold, 90 parts of pure silver, 10 parts of base metal, in 1000 parts of the bar. 480 GOLD AND SILVER. To Find the Value of Gold and Silver Bars. Example 1. Required, the value of the pure gold in a bar weighing 989 ounces and stamped 797 fine? 790 fine = 16.33.07 From table 7 fine = .14.47 } dollars. Required value of the bar, 989 × 16.47.54 = 16294.17 dollars. Example 2. A gold bar weighing 366 ounces has been assayed and stamped to 860 fine. Required, its total value? Metals. Bul. Fine. Ounces, per Ounce. Value. Gold, 366 × 860 = 314.76 × 20.67.18 = $6506.65.57. Silver, 366 × 130 = 47.58 × 1.27.29 = 61.51.61. Base metal, 366 × 10 = 3.66 no value. Total amount 1000 = 366 Answer, $6568.17.18. The last two figures in the columns of Table IV. are decimals of a cent. The fineness of gold is also expressed in carats, 24 for pure gold; that is, a piece of gold 18 carats fine is 18X 1000:24 = 750 fine. Table IV.—Value of Gold and Silver, per ounce Troy, of Different Finerness. Finº | Gold silver. Ilº: Gold, silver lºººl gold. silver in 1000. º * in 1000. tº * || in 1000. * & $ cts. $ cts. $ cts. $ cts. $ cts. $ cts. 1 () 2.07 || 0 00.13 290 5 99.48 || 0 37.49 650 | 13 43.67 || 0 84.04 2 || 0 4 13 || 0 (0.26 300 || 6 20.16 || 0 38.79 660 || 13 64.34|| 0 85,33 3 || 0 6.20 || 0 00.39 310 6 40.83 || 0 40.08 670 || 13 85.01 || 0 86.63 4 || 0 8.27 | 0 ()().52 320 | 6 61.50 || 0 41.37 680 14 05.68|| 0 87.62 5 : 0 10.33 || 0 00.65 330 6 82.17 || 0 42,67 690 14 26.36|| 0 89.21 6 || 0 12.40 || 0 00.77 340 || 7 O2.84 || 0 43.96 700 || 14 47.()3| 0 90.51 7 || 0 14.47 || 0 (00.90 350 7 23.51 O 45 25 T10 || 14 67.70 || 0 91.80 8 || 0 16.54 || 0 01.03 360 || 7 44.19 || 0 46.55 720 || 14 88.37} 0 93.09 9 () 18.60 || 0 01.16 37 () 7 64.86 || 0 47.84 730 | 15 09.04] O. 94.51 1() 0 20.67 || 0 01.29 380 T 85.53 || 0 49.13 740 15 29.72| 0 95.68 20 || 0 41.34 || 0 02.59 390 8 06.20 || 0 50.42 750 15 50.39|| 0 96.97 30 || 0 62.02 || 0 (3.88 400 8 26.87 || 0 51.72 760 | 15 71.06| 0 98.26 40 || 0 82.69 || 0 05.17 410 # 8 47.55 || 0 53.01 '770 15 91.73 || 0 99.56 50 | 1 03.36 || 0 (6.46 420 8 68.22 () 54.30 780 16 12.40 | 1 00.85 60 | 1 24.03 || 0 (7.76 430 8 88.89 || 0 55.60 790 | 16 33.07 1 02.14 70 || 1 44.70 || 0 09.05 440 || 9 (99.56 () 56.89 800 | 16 53.75|| 1 0,3,4,& 80 1 65 37 || 0 10.34 450 9 30.23 () 58.18 810 | 16 74.42 1 04.73 90 || 1 86.05 || 0 11.64 46() 9 50.90 || 0 59.47 820 16 95.09 || 1 06.02 100 || 2 06'72 || 0 12.93 70 9 71.58 || 0 60.77 830 ; 17 15.76] 1 07.31 110 || 2 27.39 0 14.22 480 || 9 92.25 || 0 62,06 840 || IT 36.43 1 08.61 I2O || 2 48.06 () 15.52 490 || 10 12.92 || 0 63.35 850 17 57.11 || 1 09.90 130 2 68.73 || 0 16.81 500 10 33,59 || 0 64.65 860 17 77.78] 1 11.19 140 || 2 89.41 || 9 18.10 510 || 1() 54.26 || 0 65.94 870 || 17 98.45|| 1 12.48 I50 || 3 10,08 || 0 19.39 520 |10 74. 94 || 0 67.23 880 18 19.12| L 13.78 160 | 3 30.75 () 20.69 530 10 95.61 O 68-53 890 18 39,79| 1 15,07 170 || 3 52.42 0 21.98 640 |11 16.28 () 69-82 900 | 18 60.46|| 1 1636 180 3 72 ().9 || 0 23.27. 550 |11 36.95 || 0 71.11 910 18 81.14|| 1 17.66 190 3 92.76 () 24.57 560 [1] 57.62 || 0 72.14 920 | 19 O1.81| 1 18.95 200 || 4 13.44 || 0 25.86 570 11 78.29 () 73.69 930 19 22.48] 1 20.24 210 || 4 34.11 || 0 27.15 580 || 1 98.97 || 0 74,99 940 | 19 43.15} 1 21.54 220 || 4 54.78 || 0 28.44 590 12 19.64 || 0 76.28 950 19 63.82] 1 22.83 230 4 75.45 || 0 29 74 600 12 40.31 || 0 77.58 960 | 19 84.50 1 24.12 240 || 4 96.12 || 0 31.03 610 12 60.98 || 0 78.87 970 20 05.1'ſ 1 25,41 250 5 16.80 ,032.32 620 12 81.65 || 0 80.16 980 | 20 25.84 1 26,71 260 5 37.47 || 0 33.62 630 |13 02.33 || 0 81.45 990 20 46.51] 1 28.0ſ) 27 5 58.14 || 0 34.91 640 |I3 23.00 || 0 82.75 || 1000 | 20 67.18] 1 29.29 280 5 78.81 || 0 36 20 CIIEMISTRY. 481 To Refine Silver. Dissolve the impure silver in nitric acid, add chloride of sodium (salt) sufficient to precipitate all the silver in form of chloride; then all the impurities will remain in Solution. Filter, wash and dry the chloride of silver. Fuse in a crucible two weights of carbonate of potash, add gradually one weight of chloride of silver, raise the heat, and the pure silver will melt and collect on the bottom. Tests for Metals in Solution with Acids. The reagents are placed in the liquid, which precipitates the metal in solution. REAGENTS. PRECIPITATES. SOLUTIONS. Sulphate of iron, Gold, as brown powder, Gold in Oxalic acid, Gold in large flakes, a Cit : Potash or soda, Gold, yellow, qua-regia. Potash or soda, Silver, dark olive, Plate of copper Metallic silver, $1 trans ' Muriatic acià," White crudy silver, i. i. Common salt, White crudy silver, ILITIC 31C1 Tincture of nutgall, Brown silver, Potash or soda, Blue cobalt s for opiussiate of potash, Green º' ... Carbonate of potash, Red “ *.*.*.*.*.*.* tº Pure water White bismuth * * Gallic acid,' éreenish yellow, jº Potash or soda, White bismuth, IC a CI Sulphate of soda & suijauric acid," White lead, º: ind Infusion of nutgall, lc àCICl. Plate of iron or zinc, Metallic copper, Potash, Green copper, Copper in Ammonia, Azure-blue copper, nitric acid. Infusion of nutgall, Brown copper, Pure water, White antimony, Antimony in 4 muriatic Plate of iron, Black antimony, acid, 1 nitric acid. Plate of copper, Fºllº ºurs, Mercury in “ iron ark powder * * * * * * - Gallic acid," Örange yellow, Imuriatic or nitric acid. Infusion of nutgall, Black iron, Iron in Ferro-prussiate of potash, Blue iron, # a +i o ºx of Ammonia, ' ' park-red iron, muriatic acid. Acid Test for Strength and Quality of Iron and Steel. This is a subject well worthy of attention by workers in iron and steel. The sample to be tested is filed smooth, or polished on all sides, and placed in di- lute nitric or sulphuric acid for 12 to 24 hours; then wash the sample and dry it. The action of the acid has revealed the structure of the sample, from which its quality can be decided with great precision. The best steel presents a frosty appearance; ordinary steel, honeycombed. Iron presents a fibrous structure in the direction in which it has been worked; the best irou shows the finest fibres. Should the iron be uneven, or made from a pile of dif- ferent kinds of iron, all are exposed by the action of the acid. Hammered blooms show slag and iron ; gray cast iron shows crystals of graphic carbon; other cast irons show different figures, all with marked characteristics. 31 482 CHEMISTRY. - Ironn Pyrites, Sulphurets. There are two kinds of iron pyrites—namely, protosulphuret and bisulphurel, of which the latter is generally richest in gold. All iron pyrites are slightly mag- netic, but the gold seems to destroy the magnetism. The protosulphurot acts sen- sibly on the magnetic needle, whilst the bisulphuret does not, and may therefore be distinguished for gold. The presence of arseniuret of iron in sulphurets indicates richness in gold. Roasting of Sulphurets. When sulphurets contain magnesia, lime or arsenic, sufficient salt should be added to chlorize those substituces, which then evaporate and go out through the chimney. The amount of those impurities should be ascertained beforehand. The salt should be well mixed with the ore before put into the furnace. 'ien pounds of Salt contain six pounds of chlorine and four pounds of sodium. y JPounds required. Ten pounds of Chlorine. I Salt. Those impurities are very ( Magnesia, . . . 3.58 6 injurious to chlorination of - Calcium, © e g 5.78 9.65 the gold in the yat. Arsenic, . e e 10.64 17.6 Chlorination of Gold in Roasted Sulphurets. Free gold is attacked and dissolved by chlorine gas, and forms two chlorides, namely Am, 844 parts of gold. Au. 648.5 parts of gold. Cl. 156 “ chlorine. Clz. 351.5 “ clalorime. Au. Cl. 1000 protochloride of gold. Au. Clz. 1000 terchloride of gold. Gold-bearing sulphurets are roasted for the purpose of obtaining the gold free for the action of chlorine gas. The combination is very slow, and requires the gold to be very fine for the prompt formation of chloride. In some ores, the gold is too coarse for chlorination, when it must be extracted by amalgamation. Composition for Making Chlorine Gas. For each ton of roasted ore in the vat are required 14 pounds of salt, 10 pounds of peroxide of manganese and 5 quarts of sulphuric acid. The composition should be constantly stirred in the gasometer, and kept to a uniform temperature of about 180° Fahr. The chlorine gas thus formed is led into the vat containing the ore. | On account of chlorine gas being much heavier than air, the gasometer ought to be placed at a considerable height above the vat, to facilitate the chlorination of the gold. In California they place the gasometer below the vat, which is de- cidedly wrong. Chloride of gold is soluble in water, and can be washed out from the vat simply by pouring water on the top of the ore and running it into another vessel, where the gold is precipitated with sulphate of iron. Chloride of silver is not soluble in water, and remains in the ore in the wat. There is always some silver in gold sulphurets. Quartz Mills. IEach stamp, weighing about 800 pounds, lifted one foot 60 times per minute, can crush oue ton of quartz per 24 hours with a dynamic effect of two horse-power, This is the average performance. The custon-unill in Grass Valley, California, crushes quartz for about four dollars per ton. The stamps are generally divided into sets of four or five, working in one mor- tar, and called a battery. The shoes and dies in the battery are made of chilled Cast iron. Most of the gold is collected by amalgamation in the battery. The pulp from the battery contains much gold, which is often allowed to run away, but generally the sulphuret in the pulp is concentrated and roasted for culorination; the rest of the pulp is ground in pans and the gold amalgamated. Amalgams. GOLD. One weight of mercury amalgamates with two weights of gold. SILVER. 10 silver to 19 mercury. 7 {{ {{ 20 wº TIN. 1 tin to 3 mercury, for looking-glasses. 1 tin, 1 lead, 2 bismuth, 10 increury, for glass-globes. 1 tim, i zinc, 3 mercury, for rubbers in electric machines. OPTICS. 483 0 PTICS, Optics is that branch of philosophy which treats of the property and motion of light. Mirrorse Frample 1. Fig. 307. Before a cowcave mirror of r = 6 feet radius, is placed an object 0 = 1, at d = 175 feet from the vertex. Required the size of the image I = ? º O r IX6 2- -: -: 2-4 image I = ### =E=#### Example 2. Fig. 308. Before a concave mirror of r == 5:25 feet radius, is placcd an object O = 1, at D = 4.5 feet from the vertex. Required the size of the in- verted image I = ? O r _ 1X5’25 1°4 Example 3. Fig. 309. Before a convex mirror of r = 1-8 feet radius, is placed an object O = 1, at D = 3-15 feet from the vertex. Required the size of tho image I = ?, and the distance in the mirror d = ? 3' 15XI-8 wº IXI-8 º image I*—tº # =0-222 distance d == tºt:- *** 2X3T5-HT3 2X3-15–H1'8 Example 4. Fig. 310. A parabolic mirror is h = 1-31 feet high, and d = 2:15 feet in diameter. Rcquired the focal distance f = ? from the vertex. & d” 2-153X12 sº focal distance f = -º- = titº ºtº = 2.646 inches. jº 16 h 16X1-31 =0.699 ft. Optical Lensese Ecample 5. Fig. 316. A double convez lens, of crown glass, having its radii R = r = 6 inches. Required its principal focal distance f = ? For crown glass the index of refraction is m = 1°52. See table. 6 4- = —”— = 5·768 inch f-gº-o-ººsingles Microscope, A/CCLC?'S 46710te. = magnifying power of a lens. j9 = limit of distinct vision. a = limit of distinct sight, which for long-sighted eyes is about 10 or 12 inches, and near-sighted 6 to 8 inches. For common eyes take & = 10 inches. T = limit distance of the object from the optical centre at distinct vision. Aºzample 6. Fig. 322. Required the magnifying power of a single microscope with principal focal distance, f = 4:3 inches? Mag, power p = -7---is- =3'325 times. 484 OPTICS. "307 Spherical Concave Mirror. 2-’ 1. r = radius, and f = } r, focal distance of the 2’ O º mirror. S. d || 0 * 1 = -2t. D=—ºr >>, — — . . .” ---53. The image disappears when d =f- # r. 308 Spherical Concave Mirror. O r I) ºr - . d = —#––. I 2 D—r 2 D—r When the object is beyond the focal distance the image will be inverted. 309 Spherical Convez Mirror. O r D r I=3 I)+?" d = 2 I)+r 310 Parabolic Concave Mirror. d? d” r=-iſſ k=#7 811 Hyperbolic Concave Mirror. Heat, Light, or Sound emanating from the foci of a hyperbola will be reflected diverged, from the concave surface. 312 Eliptic Concave Mirror. Emanating rays from either of the two foci in an elipse, will be refracted by the convex surface to the other foci. UPTIOs. 485 Astronomical Telescopes and Opera Glassess Example 7. Fig. 325. The principal focal distance f = 0.65 inches of the ' ocular or eye-lens. F = 58 inclues the principal focal distance of the objcctive- lens. Required the magnifying power of the telescope I = ? - O F IX58 e e image I = –7– = g- = 8923 times the object. The telescope is to be used at the limited distance D = 1380 feet and D = oo. Required the proper lengths l = ? and micrometrical motion of the ocular or eye-lens 7 when the limit of distinct sight a = 10in. F = 58: 12 = 4.833 feet, f = 0-65 : 12 = 0.05416 feet. _ _1380×4833 - 10×0.05416 4'89035 T T380–4,833 "TOFOF05416 = 0.05386 When D = 1380 feet, the length l = 4:94421 feet. When D = co, l = 4.8333+0.05386 = 4.887.19 a ſjö5702 , Micromefrical motion of eye lens 3 0-68424 inches, 11 is nearly. Table of Refractive Indicese Substances. Inºx. Substances. y*. * 2-97 Quartz- º * - 1.54 Cromate of Lead - 2:50 Muriatic Acid - - 1.40 Realgar -, - - 2-55 Water - º - º 1.3 Diamond - º º 2-45 Ice * gº * - 1.30 Glass, flint tº - 1.57 Hydrogen - - - 1.000138 Glass, crown - wº 1-52 Oxygen tº º - I,000272 Oil of Cassia - - I-63 Atmospheric air - - 1.000294 Oil of Olives º 1 47 314 Prism. An emergent rays of light a a' falling upon a transparent medium A (say a glass prism) will be transmitted through in the direction a b, and de- livered in the direction bb', parallel to a a' a”. W = angle of incident, v = angle of refraction. Indix of refraction m = ** –y * S27?, ?) ... , 315 Given the direction of the emergent rays a a', : angles e and r.—to find the angles w and ac,-or the direction of the rays b b'. **, cos. u-m cos. (180–2–r). ! — C cos. 2 = a = 180 –(e–-r-ţ-w). When e = w, the angle 2 is smallest. An eye in Ü' will see the candle in the direc- tions by Ö ly". OPTICR. 316 Double Convea: Lens. R ºr the principal = + . Une princip f (m-1)(R+r) focal distance. -: 7" == f Ž(m-I)' when R = r 0 – optical centre of the lens. 317 Plano Convex, Lens. - - + 7° º f 7m-l The optical centre is in the convex surface. * Conver concave Lens (Meniscus.) R + = + V - "Ti-ºf- Draw the radii R' and r’ parallel to one another.—Draw m 0, then o is the optical centre. - 319 Double Concave Lens. R ºr V -- (m=jtºry 320 Plano Concave Lens. 7" º f=-ji=T. The optical centre is in the concave surface. 32] - - Concavo-convex. Lens. = - R r e VTT (m=i) (ELF) Draw R and r" parallel to one another. Draw n o, then o is the optical centre. OPTICS. 487 322 Single Microscope. ||r. A– – 0.f n – d.f. I: O= f°: f_d. I = - : g- f:f. - y J–d y p f-d' – a t.f n – o f * - a.f jº = * */ - ºf . 3} F- J. W. " B ał?' 323 g|. When the object 0 is beyond the focal *||distance the image I will be inverted. I: O = f; D–f, 1-##, a-##. 324 Diminishing Power of a Double Comcave Lens. _ Of _f(O—I) – D D ſtori), d #7. |D f d 2% - O F _ D I: O = F : f, 1--, -, a Pſ. _ D F * a f ( 3- - F + ºf ) * ~ 0–ºr, it a + f" if D = o, l = F a +f * + for astronomical telescope, — for opera-glasses. 326 Opera Glass. D > Formulas are the same as for Astronomical Telescope. 488 - GEOGRAPHY. GEOGRAPHY. The Earth on which we live is a round-ball or sphere, with a mean diameter of |7914 statute miles. The whole surface of the earth is 196,800,000 square miles, of which only one-fourth or nearly 50,000,000 square miles is land, and about 150,000,000 square miles water. Table of Area and Population of the Whole Earth. Divisions of the Earth. * Population. *::::::::A; | America, . . . . . 14,491,000 90,000,000 6. Europe, . g * * g 3,760,000 301,000,000 80. Asia, . {º © * , * * 16,313,000 731,000.000 '45. Africa, . © * wº ge 10,936,000 200,000,000 20. Oceanica, . © tº & e 4,500,000 28,000,000 6.2 Total, . . . . [ 50,000,000 | 1,350,000,000 27. About *6th of the whole population are born every year, and nearly an equal number die in the same time; making about one born and one dead per second. The annual increase of population per 1000 is about 6 in Europe and 19 in | America. Europe loses and America gains by emigration. The annual increase of population in the whole world is about 6 per 1000. The Earth is not a perfect sphere, it is flatted at the Poles. The following are her true dimensions in statute miles of 5280 feet. - .” - ~ Dimensions of the Earth. -: - . . . .” 7898.8809 miles at the Poles. º: * : * Diameter, . . (7911.92 miles mean, or in 45° lat. : : $. > 7.924,911 miles at the Equator. . … . . . . Difference, . . 26.0302 miles Poles and Equator. :: *s * ... … Flatted, . . . 13.015 miles at each Pole. . . . ." * , º, 24802.486 miles round the Poles. & sº ºf Circumference, 33.1% miles mean, ºr in 45° lat. , S; i.e. Qºş . . 24884.22 miles round the Equator. 3 - 3 To Find the Radius of the Earth in Any Given Latitude. I = 3955.96(1 + 0.00164 cos.21), statute miles. Distribution of Population in the world. Br. Amer., Montreal, . Quebec, . . Toronto, . . St. John, N.B. Halifax, N. S. Ottawa, Ont., U.S. Amer., N.York & Brk. Philadelphia, Chicago, . . St. Louis, .. Baltimore, . 13oston, . . Cincinnati, . S’n Francisco, Washington, Buffalo, . . Newark, . . Louisville, . Cleveland, Pittsburg, . Jersey City, Detroit, Milwaukee, . Providence, Albany, . . Rochester, . Alleghany, . Richmond, New Haven, Charleston, . Troy; . . Syracuse, . Indianapolis, Worcester, . Iowell, . . Memphis, . Cambridge, Hartford, . Scranton, . Reading, . . Kansas City, Mobile, . . Portland; . Wilmington, Toledo, . . Columbus, . Dayton, . . Lawrence, . Utica, . . Savannah, . Nashville, . Alaska, . . Sitka, . . RACES IN U. S. White, Colored, Chinese, Indians, Sweden, Stockholm, Norway, . Christiana, 5,800,000 15,000 38.550,000 1,338,600 675,000 299,000 311,000 268,000 251,000 216,000 150,000 120,000 118,000 105,000 101,000 93,000 87,000 82,500. 80,000 71,500 69,000 69,500 62,500 53,200 51,000 | 51,000 31,000 31,600 31,300 30,500 29,000 28,800 28,300 26,000 * 33,570,000 | 4,890,000 64,000 25,500 4,106,000 136,000 | 1,650,000 500 GEOGRAPHY. Denmark, 2,700,000 Copenhagen, 180,000 |Russia, . . . 70,000,000 St. Petersb’rg, 680,000 Moscow, . 400,000 Engiana, 22:33,000 London, . . .] 3,883,000 "Scotland, 3,359,000 Bdinburgh,. 180,000 Glasgow, . 468,000 Ireland, . 6,000,000 Dublin, . . 322,000 France, Rp., 36.595,000 Paris, . . 1,830,000 Germ. Emp 41,000,000 Berlin, . . 702,000 Austria, 20,400,000 Vienna, . . 607,500 Hungary 15,509,000 Pesth, . . 202,000 Holland, . 3,688,500 Amsterdam, 281,800 Bavaria, . 4,825,000 Munich, I71,000 Switzerl’d, 2,670,000 Berne, . . 29, Belgium, . 5,022,000 Brussels, 287,000 Spain, . . 16,642,000 Madrid, . . 317,000 Italy, . . 26,000,000 Rome, . . 240,000 Greece, . . 1,458,000 Athens, . . 42,000 Turkey, E., 15,487,000 Const’tinople, 1,075,000. Turkey, A., 16,463,000 Smyrna, . 150,000 Arabia, . . 8,000,000 Mecca, . . 60,000 Persia, . . 9,000,000 Tabreez, . 110,000 Afgh’mist’m 4,000,000 Candabar, . 100,000 Beloochs'm 1,500 000 Kelat, . . 15,000 Turkistan, 6,500,000 Bokhara, . 100,000 India, . . [172,000,000 Bombay, . 817,000 Calcutta, . . 616,000. Chima, . . 446,500,000 Peking, . . . 1,800,000. Canton, . . 1,000,000 FIong-Kong, 40,000 Japan, . 35,000,000 Yeddo, . . . 2,000.000 Miaco, . . 500,000. Barbary, . 2,800,000 Tunis, . . . 130,000 Egypt, . . 5,195,000 Cairo, . . 313.400 Jerusalem, . 25,000 | Mexico, . . 9,200,000 Mexico City, 200,000 Puebla, . . Guanajuato, Cuba, Havana, . St.Jago Cuba, Porto Rico C. America . Whites, Indians, Negroes, Mixed, | Guatemala. Guatemala, A St Salvador St. Salvador, A Nicar a 2 Managua, . Honduras, Comayagua, Costa Rica, San Jose, S. America, Wild Indians, Whites, Negroes, Mixed, U.S. Colom. Bogotá, . . Panama, . . Venezuela, Caraccas, . Equador, Quito, . Guiana, . Georgetown, Brazil, . Rio Janeiro, Bahia, . . Slaves, Peru, . . Lima, . . . Callao, . . Bolivia, La Paz, . . Chili, . . Santiago, . Valparaiso, . Argentime, Buenos Ayres Paraguay, Asuncion, . Uruguay, Montevideo, Patagonia, Antonio, . . Australia, Melbourne, . Wellington, Jamaica, Kingston, . Hayti, . . P’t-au-Prince Sandwich I Honolulm, . W. Endies, 600,000 20,000 400,000 10,000 350,000 31,369,000 3,500,000 10,000,000 600,000 14,569,000 3,000,000 'iz8000 1,200,000 ? 1,900,000 193,700 3,300 444,000 36,500 4,000,000 *: Measured by the author. 49- LATITUDE AND LONGITUDE. Latitude and Longitude of Places (from Greenwich.) America Latitude. Longitude. Latitude. Longitude. Atl. Coast. [D. M. S. D. M. S. France. D. M. S. D. M. S. Quebec . . . 46.49. N. 71.16. W. Paris, Obs. . . 48.50.13 .N. 0.09.21 E. IIalifax . . . 44.38. “ |63.35. “ ||Cherbourg . . 49.38. “ | 1.37. “ |Chicago. . . 42.00. “ |87.35. “ ||Marseilles. . 43.18. 5.22. “ Boston . . . . 42.21. “ 71.04. “ || Calais . . . . 50.58. * | 1.51. “ New York . . 40.42. “ |74.00.42 “ || Brussels. . . 50.51. “ 4.22. “ Philadelphia . 39.57. “ |75.10. “ ||Antwerp . . 51.13. “ 4.24. “ Cincinnati . . . 39.06. “ 84.30. “ Italy. St. Louis . . . 38.36. “ 89.36. “ ||Turin . . . . 45.04.06 “. 7.42. “ Washington . |38.53. “ TT.00.18 “ ||Florence . . 43.46. “ II.16. “ Charleston . . 32.42. “ |79.54. “ ||Leghorn . . 43.32. “ |10.18. “ New Orleans . |29.57.30 “ |90.00. “ ||Rome . . . 41.54. “ 12.27. “ Georgetown, Br. 32.22.12 “ |46.37.06 “ |Malta . . . |35.54. “ |14.30. “ Nassau . . . 25.05.12 “ 77.21.12 “ ||Naples . . . 40.50. “ 14.16. “ Port-au-Prince 19.46.24 “ 72.11.12 “ l! Palermo . . . 38.08. “ (13.22. “ Porto Rico. 18.29. “ |66.07.06 “ ||Venice . . . . 45.26. “ 12.21. “ Kingston, Jam. |17.58. “ |76.46. “ Austria. Havana. . . . . 23.09. “ 82.22. “ ||Vienna. . . . . 48.13. “ 16.23. “ Vera Cruz . . . 19.12. “ | 96.09. “ ||Trieste . . . . 45.39. * 13.46. “ Mexico, City . 19.26. “ 99.05. “ ||Pesth. . . . 47.28. “ |19.13. “ Colon, N. G. . . 9.22. “ 79.55. Germany. - Para, , 1.28. S. 48.29. “ ill&erlin . . 52.31. “ 13.24. “ Rio Janeiro 22.56. “ 43.09. “ ||Hamburg . . . 53.33. “ 9.56. “ Buenos Ayres |34.36. * 58.22. “|Cologne 50.56. “ 6.58. “ Cape Horn . . [55.59. “ 67.16. “ || Amsterdam 52.22. “ 4.51. “ Pae. Coast. Bremen . . . . 53.05. “ 8.49. “ Valparaiso . . .33.02. “ 71.41. “ ||Berne . . . 46.57. * | 7.25. “ Callao . . 12.04: * | 79.13. {{ Turkey. - Limaš . . . . 12.02.34 “ 79.06. “ i. Constantinople 41.01. “ 28.59. “ Cuzco's . . . |13.31.45 “ 74.15.50 “ ||Ragusa. . . 42,38. “ | 18.07. “ Payta. . . . 5.05. “ S1.10. “ ||Salonica. . . 40.39. “ |22.57. “ Guayaquil . . . 2.13. “ 79.53. “ l'Athens . . . . 37.58. ** 23.44. “ Panama. . . . . 8.57. N.] 79.31. “ ||Smyrna. . . . |38.26. “ |27.07. “ Acapulco . . 16.55. “|9948. “ ||Cairo. . . . . 30.03. “ |31.18. “ San Francisco 37.47. “ | 122.21. “ ||Jerusalem, Pal. 31.48. “ |37.20. “ Alaska. . . . 58. T58 Russia. Behring's Strait 67° 170 St. Petersburg |59.56. “ 30.19. “ China, Ind. Moscow . . . 55.46. “ |35.33. “ Pelzing . . . 39.54. “ 116.28. E. Nish Novgorod 56.20. “ 43.43. “ Canton . . . . 23.07. “ | 1.13.14. “ || Cazan . . . . . 55.48. “ |4S.50. “ Hongkong . . . 22.15. * | 114.12 “|Archangel . 64.32. * |40.14. “ Honolulu . 21.19. “ | 157.52 “ |Jecatherinburg 56.50. “ 60.21. “ Jeddo . . . . 35.40 “ 139.43 “ ||Astracam . . . 46.21. * |47.46. “ Owyhee, S. Isl. 20.23. “ 155.54. W. Odessa. . . . 46.27. “ |30.42. “ Calcutta . 22.34. * | 88.20. I. Warsaw . . . 52.13.05 “ 21.02.9 “ Batavia . . . . 6.08. “ 106.50. “ Sweden. w Sydney . . . |34.00. S. 151.23. ... “ ||Stockholm . . 59.21. “ |18.04. “ Melbourne . . |37.48.36 “ 144,57.45 “ ||Gothenburg . . 57.42. “ 11.57. “ Wellington . |41.14. “ 174.44. “ ||Wisby, Gotland 57.39. “ 18.17. “ Africa. Christiania. . . 59.55. * |IO 52. “ Cp. of G. Hope. 34.22. “ | 18.30. “ ||Bergen . . . . 60.24. “ 5.20. “ Morocco . 39.34. N. 2.23. “ Ystad . . . [ 55.25. “ 13.50. “ Algiers . 36.47. “ 3.04. * || Haparanda . , 65.49. “ (24.11. “ England. < * Copenhagen . 55.41. “ |12.34. “ London, Tower | 51.31. “ 0.06. W. Spain. Greenwich . 51.28.38 * | 0 0 0 Madrid . . . . 40.25. “ 3.42. W. liverpool . . 53.22. “ 2.52. “. . [IBarcelona. . . . 41.23. * | 2.11. E. Glasgow . . 55.52. * || 4,16. “. . Gibraltar . . |36,06. “ 5.20. W. Edinburgh . . 55.57. “ 3.12. * |Carthagena. . . 37.36. “ | 1.01. “ Bristol . . . 51.27. “ 2.35. “ (Lisbon . . 38.42. “ 9.09. “ Dover . . . . 51.08. “ | 1.19. E. Oporto . . . . 41.11. “ | 8.38. “ Dublin . . . . 53.23. * | 6.20. W.I.Terra, Island |27.47. “ 17.56. “ DIFFERENCE OF LONGITUDE IN TIME. 491 Difference of Longitude in Time Between Places. San London. St. Peters- Canton, Francisco. New York. Greenwich. burg. China. H. M. S. H. M. S. H. M. S. H. M. S. H. M. S. Amsterdam . . . . . 8 29 19 || 5 15 32 || 0 19 32 1 41 44 7 13 24 Antwerp . . . . . . 8 27 17 5 13 30 || 0 17 36 I 43 40 7 15 20 Batavia. . . . . . . 8 42 50 | 11 56 37 || 7 07 20 5 6 4. 0 25 36 Berlin . . . . . . 9 3 22 5 49 35 | 0 53 35 1 7 41 6 39 21 Boston . . . . . . . . 3 25 33 0 11 46 || 4 44 14 6 45 30 II 42 50 Buenos Ayres . . . 4 16 19 1 2 32 || 3 53 28 5 54 44 | 11 20 24 Canton . . . . . . 8 17 17 | 11 31 4 || 7 32 56 5 31 40 () () () Calcutta. . . . . . 9 56 53 10 49 20 || 5 53 20 3 52 4 1 39 36 Cairo . . . . . . . . 10 14 59 || 7 112 || 2 5 12 || 0 356 || 5 27 44 Copenhagen . . . . 9 0 3 5 46 16 || 0 50 16 1 11 0 6 42 40 Constantinople . . . 10 5 43 6 51 56 || 0 55 56 0 5 20 5 37 0 Dublin . . . . . . . 744 25 4 30 38 0 25 22 2 26 38 8 18 1S Florence . . . . . . 8 54 51 5 41 4 || 0 45 4 1 16 12 6 47 52 Gibraltar . . . . . 7 47 56 4 34 32 || 0 21 28 2 22 44 7 54 24 Gothenburg . . . . . 8 57 38 5 43 51 || 0 47 48 1 13 28 6 45 08 Halifax . . . . . . 3 54. 27 0 41 40 || 4 14 20 6 15 36 || 11 47 16 Hamburg. . . . . . 8 49 39 5 35 52 O 39 52 1 21 24 6 53 04 Jecatherinburg . . . 11 48 57 7 25 20 || 4 1 16 2 0 0 || 3 31 40 Jerusalem . . . . . . I0 39 7 7 25 20 || 2 23 20 0 22 4 5 09 36 Uima. . . . . . . . 1 43 2 0 12 24 || 5 08 24 7 9 40 | 11 18 40 London . . . . . . 8 9 50 4 56 3 || 0 0 24 2 1 40 7 33 20 Lisbon . . . . . . .] 7 33 11 4 19 24 || 0 36 36 2 37 52 8 08 32 Madrid . . . . . . 7 55 39 4 41 52 || 0 14 8 2 15 24 7 47 04 Melbourne . . . . . 4 2 24 7 16 11 || 9 39 51 || 11 41 31 2 7 0 Naples . . . . . . 9 6 51 5 53 4 0 57 4 I 04 I2 6 35 52 New Orleans . . . . 2 9 37 1 04: 10 || 6 0 10 8 01 26 || 10 26 54 New York . . . . . 3 13 47 0 0 0 || 4 56 3 6 57 19 11 31 01 Paris . . . . . . . 8 19 'ſ 5 5 20 || 0 9 20 1 51 50 7 23 36 Peking . . . . . . 8 4 21 | 12 41 52 || 7 45 52 5 44, 36 0 12 56 | Quebec . . . . . . 3 2 36 0 11 11 || 4 44 49 6 46 5 || 11 19 53 Rome . . . . . . 8 59 35 5 45 48 || 0 49 48 I 11 28 6 43 08 San Francisco . . . . 0 0 () 3 13 47 | S 9 47 || 10 11 3 8 17 17 St. Louis . . . . . 2 8 46 1 5 I | 6 1 1 8 2 17 | 10 26 03 St. Petersburg . . . . . 10 11 3 || 6 57 16 || 2 1 16 0 0 0 5 31 40 Stockholm . . . . . 9 22 11 6 8 24 . 1 12 24 0 58 52 6 20 32 Turin . . . . . . . S 40 38 5 26 51 || 0 30 48 1 30 28 '7 02 08 Washington . . . . . 3 1 46 0 12 5 8 1 7 9 17 | II 19 53 Wellington, N. Z. . . . 5 30 22 8 44 09 || 11 3S 56 || 10 19 48 2 46 52 To Heduce Longitude in Degrees into Time, and vice versa. RULE 1. Divide the number of degrees by 15, and the quotient is the corre- sponding hours. Should the degrees be less than 15, multiply them by 4, and the product will be minutes in time. The minutes of degrees multiplied by 4 will be seconds in time. The seconds of degrees divided by 15 will be seconds in time. RULE 2. The time in hours, minutes and seconds, multiplied by 15, will be the corresponding angle in degrees, minutes and seconds. The trigonometrical table contains the conversion of time and angle. Dacample 1. Required, the difference in time between Philadelphia and Paris? Longitude of Philadelphia, 75° 10' W. {{ “ Paris, . . 2 20 E. Difference in longitude 77° 30' divided by 15 will be 5h 10m, the difference in time. When it is 12 o'clock in Philadelphia, it is 5h 10m o'clock in Paris. Example 2. A vessel sails from New York to Liverpool; after she has been at sea, about one week, her difference in time with New York is found to be 2h 7m 45s. Required, her longitude from New York : 15(2h 7 45) = 31° 58' 15" from New York. - The time is almead in the east, from where the sun rises. The time is behind in the west, toward sunset. e DISTANCES BY SEA. 492 | ſozgſ0879 |OsºgļſzſziQ0871gºzřIIgorígssºffreiſgºgiſgºogtſſoºſ JºgºſQZAZIĻ9řIIļzglī£389°I 1909&t|0，19ogsgael 08g2 |098||ºptºgſ98031irzºtſºbºtſysziļºſziſºzzi|$10，11090）||2001||8||11|2，901||890||36911||9097||0819ſoļAgļegſ 089),Q8160098,088, 100699908 | 1981, [998], |, ſz, [0801, 18889 |0Ť89 |0ý89 ||0099 |0Ť89 ||00Ť9 |8Īř9 |0.38),rodoH pooÐ øł80 | 90091|089řI0,979|0799 |09zg |914） |IIgſ |9I0ff || Zgg |0.348 |8Ť98№ºngOffſg 10ț48 |0țgz |0#0Z |0ç II ſººſdouņuºqsuoO įgoriļsgoºi|sſs|ºrgq |sszy ſzºs ſºirs ſzez ſºssz |8696 pºrº 868， þrjá þigg ſºrgt þ101 ºg Q808||0ř93109ř9009Ť |0zzº9094 |ę0řz 1906 I || Ig I [089I |88řL 088 I |00řI 1003I 1088（IU)[8Iqț¢) Qź08īſgºl0,979|00gſ | 0964, 1988Z ||IĢIZ 1989 I ||№g I |048 I 189II'OIII |090 I 1006 l'uoqsȚI įszřiljººgiļozelogºſ 10008 1986， 1811 ºsóI 1611 ſºggi 318|00, lot!'toođioxyſ gzgŤ1|0，181099 ||008Ť |0Ť38 |}|0ýI|104190）, įſig ||gſſ |8gz |0IŤ kuopuºT Q197||061810I91||Oggſ | 06zg |989 I ||I9ȚI |986 || 69 |931, 1819 fººd |şºrişisei|309||3067 ſºrgs ſøſt ſºrgſ ſig ſººs ſº|ºduæAquy* gºsſiſſoriſºsaegſ ſae ſººrt|ggg |897 logg ſºmquis H 40091ſıgı1509||349 | 1448 |688 ſý89 |68 I || 3 inquoqqoÐ& g6091|ągzg1||818|9189 |9188 1001 |867ua3equado.O"spåisrųoſaſuesomºſ• Ëſ Įºggi ſiglºſſºsºs||189 |1197 || 18 Fusioſtaeºs…en （8 #Ëſ: 9619 I9ģ6911,888||9||099197ºingsløſøā.ºņSøđoH pooÐ go od 80 punoſ * 98691||8981|004||6318 ſºdox aºn•ø9.13øp rođ 09 ºsaſuu (goțqđe）3000 | 907）||90881|029）||suwºſo aºsuſ : quea aqq uosąoeţd queņIoduuſ qsotu oqº, uô0AA494 08160019 ſurohød 80·eºs aeq•••••••ía 0389 ſoos ſouſ 8J ſ UſºŞ - ļºxoqueo- * 493 D18TANCES IN EUROPE. ~ * †99 646 ĢIŤI Iý ||I 9098 88% Z6țZ Å!!8I 1811 I9II 888 88{}}[ ZŤſ, 68), 3/4; †ZA, 388 990I 06|| 1,63 ZIZ 968 #0I}, I6 II1668 939'IĘ88 I 896Īİſő9I AIM,8|ŘII8 0g%|0,06I Þ044|ZIý Z 689 I|1,64 I 668||10|II 81,8I|8}\, gț0I|Igg g6ZI||06}} †g6 |gŤ} Ig6 ſºff †86 |g|Z9 986 |0.49 ††g |Zgº, }9&I/068 8ZI ZIŤ IgI|f09 Iſſ00II |399 ſeg kººſºT OŤ8Z|Z],41|01@I|g18 |gºy ſu03ſequºdo0 SȚII|9,14|gg6T|ggg |Off ſuiſoq:{004Ş Og61|00ZI|ę0ï |3ūmqsioſºa "3S 099.I] QI6 |Á00SOWI 092 | 0880pO 00ZI|0Įdouņu gļsūOO ºedornſ uſ saņ10 uøøAļºq †88I|669||398 |68 ſýIZ || 8Ť9 |6ý8 |8I£ 1689þyrilºr,60), IggI|9|[8][}]|iſ 9 |993 |[$j] | [],II|fºº||g86 |998ºgyſgºſ1,96 986 I ||19|34|380|I||I69 |998 | 9091||66||I|0,18I|[6ZI|8，6I'889 I 6IZZļý8ÝZ|8). ſIſſýII'696 | ††g|I|Zj], [[ZQII] [1,4Iříši，88'ſ, 68!!8!！00Ť|8||18|799468% || 796Z[[618|8IŤZ|889Z|8g|Ig|IgA,Z,z54, 98Ť3|09||3||988IſIIGI,088 I || 0.003||9gSI/660||698 I|2,06|I|00ȚI|g08 9803|0983|09IZþý803|606I || 789 I|99ŤIį60ýI|6,19||0çOI|0001 981, IſºpºgroſII|, |989 | 988 ſý80 ||00Ť |89g. [690] [86], |AUSI GAA 980IſºgiggºZŤ6 |96| |3789 |99Ť |888 ZŤ8 |107 |}Søſuſ, 896gºzī£30160II;896 || IŻg |8Z$ |gŤ || ||Og |outoyſ 99II|[8Ť Iſ&Ig |009 ſýgif | I3Z (S0g (0,1% ſt[0ȚUIn W. 69řiſſzaeſ|I91|619 000 |×19 |209 ſuuuoſa 080||368 1981 |106 808 | 861 ſuſini. ºss pºIII， 300 809 ſouſea 09zI|g191609 |g|, I ſuſog 96ZIîși išg3.Inqub H Z48 ſzțII|d10AQuy gQZ |uoqspI pȚIp'BJN 300 I 1939 609 |s\，t}&{ Įood IQAȚI uopuorņſ søougļsſp ºuȚIIæABIĄ ĮSaigº N ºgøtyrºſ aquaqwgs un sºovxe?,850I Sailing distances between the most important seaports in the United States, in Geographical miles, 60 per Degree. Halifax. Boston. 390 Nantucket light. 107 ||360 Sandy Hook. 193 |284 648 New York.] 18 211 302 | 666 Cape Henlopen. 149 |131 274 |380 650 Philadelphia. 103 || 252 234 || 377 |483 753 Cape Henry|238 135 |276 |258 |403 |509 || 765 Baltimore. 152 | 390 287 |428 |410 555 | 661 917 Washington. 174 158 |396 |293 |434 |416 561 | 667 |923 Norfolk. 178 172 30 268 165 |306 || 288 |433 539 795 Richmond. 90 248 242 100 || 338 235 | 375 358 || 503 609 || 865 Cape Hatteras. 214 || 144 272 266 114 |321 218 340 322 || 426 |624 786 Charleston, 281 |495 |425 || 553 547 || 395 |602 || 499 || 621 | 603 || 707 || 905 || 1065 Savannah. 95 || 376 590 520 | 648 642 |490 697 || 594 || 716 698 || 802 1000 || 1162 Key West. 542 | 622 || 768 || 1348 2288 | 1.406 || 1410 | 1258 1465 1362 1484 || 1466 1570 1768 1930 \ Havana. 85 562 592. | 738 1328 1258 || 1376 || 1380 | 1228 1435 | 1332 1454 1436 1540 1738 1900 New Orleans. 640 640 | 1187 | 1267 | 1.413 | 2003 || 1933 | 2051 2055 1903 || 2110 |2007 2129,211, 2215 2413,2685 yºgrazºº 823 S70 || 1412 || 1492 | 1638 2228 1158 2276 || 2:280 2128 |2335|2232 |2354.2336|2440 |2638 (2800 § 1407; 1370'1200 | 1070 | 1612 | 1692 |1838 2428 2358 |2476 |2480 2328, 2535 24332554 2536 12640 12738 |2635 à Page Missing in Original Volume Page Missing in Original Volume Page Missing in Original Volume Page Missing in Original Volume -> Cements, concretes, Yº Xentirneters and inches, . Centre of gravity, . . * “ “ gyration, . . “ “ oscillation, . “ , “ percussion, . . Centrifugal force, | Chain, surveying, , . Chains for railroads, . €harge of powder, . . Charcoal, . . . . Charge in blast furnaces, hemistry, . e ime of bells, . & º Ahimneys, height of, . Chlorination of gold, . . Chronology of events, . Circle, formulas for, . . $ to Square a, . , sidereal, . . & o , COn Sumption Of, --> $old, artificial, . º º llision of bodies, . e NorS, Spectruma, . º “. . tempering Steel. . º e tº e weight ind built of 338,336 805, 427 __ſ_&#TENTS. –5– PAGE ... 474 | Creation of the world, . tº 38, 364 | Cube and cube root, . º . 324 | Cubic contents, 314 | Cubic inches water, iron, lead, 281 . 320 | Cupola, e ; : Ł). º: PAGE 498 82 60 & º tº - º “kö) 324 Curvature of the earth, 131, 132 e e . 318 Curves for railways, . 116-12] Chains, stren’th, weig't, price, * ; Cut-Off Steam, expansion, 32 Cut-Off valve, . . . . . . . 280 | Cycle of the sun and lunar, Chapman’s rule, area & solid., 1.14 Cycloid, . . º g 3 Characters, signification of, , , 12 | Day and might, length of, 510 06, 393 || Dates, civil, astron m., mar's, 333 Decimal fractions to vulgar, . 443 fecimals of an hour, degree, 470 Deaf and dumb alphabet, hemical formulas, compids, § Declination of the sun, { Degree of the earth’s circle, 423 Departure, . . • . . 4S2 Dew point, . º o o . 499 || Diagram, indicator, Plate V., 4S Diamond, º & © º Diameter Of Wro’t-iron shafts º . . 54 Circumf and area of circles, º Difference in longitude, . 26 Differential calculus, . e 378 Dip of horizon, . . . . . te, 358 Discount or rebate, tº ſº 498 || Displacement of vessels, 444, 457 4; Distances ºn the Almer, coast, #2 U. S. railways, . !coal, from diffent woods, 276 ſ & in the world, . cients of vessels, . 458 4 & in Europe, • heels, © © ( ) . 294 “. Of Objects at Sea, . ūsive strength, . e 270 & & Spherical, . * nS, American and foreign, sº Distance to the sun and finoon • 3S Distillation of coal oil, . 322 Division in algebra, * & . 372 | Divergence of the parallel, 332 Dodecahedron, º & mps, air, water, etc., 344, 351 | Double cylinder expansion, iſºmº, 18 Drain, motion of water in, . Dredging machine, . e 402 420 / 49S 36 208 511 500 32 & Yº 3 * 356 431 Cº) , 418 291 26 131 t 495 , 292 3S3 14 133 55 413 339 265 49S 307 iº"…º. 6 CQNTENTs. *- ~ - PAGE | Elevation of external rail, 119 || Geometry, . tº . ~ ſº | Embankments, º e . 122 * { analytical, . . 14 Engineer's command, º 431 || Geometrical progression, . Engines, steam, of dif. lxinds, 413 & # scale in music, , Epact of the year and month, 496 |Giffard's injector, º • Equation of time, . . . & Girders, compound, iron, . | Equivalents, atolia weights, 470 4 & cast iron, e - Evaporation On Seas, . • 0 || Glass, Window, . . . 290 Events before and after Christ,499 || Glues, . . . . . . . 4T1 . Evolute of a circle, ſe . 47 | Gold metal, º º . . . 332 | \ Expansion of air by heat, 387 Gold and silver, . . 478–480 * . Of Water “ . , 394 | Gold, sil., platin’m, weight of, 479 - “ of bodies “ . 384,409 Golden number, . . . . 498 46 Of Steal)] ** . 402 || Governor, . . . . 319 Explo. nitro-glycer., dynam., 473 || Gravitation, . . . . 804 4 & of steam-boilers, . . 436 Gravity, specific, . . . . .328 Excavation and embank., 122 4 & centre Of, . . 32. Eyes, long and near-sighted, 483 || Gunpowder, properties of, 3 Faces of the moon, . . . . 508 Guns, heavy artillery, ... 3 | Falling bodies, table for, .. 308 Gyration, centre of, . 314, 3T7 Fan, or ventilator, . . 442 || Hammers, Steam, . . 303 Fathom, . . . . . 32 Half-trunk Steam-engine, , , 413 | Feed pumps or force pumps, 406 Hardness of substances, 276, 333 Feet per sec. = miles per hr., 352 | Heat, caloric, . . - Feet and metres, . 38, 40, 36 “ as a mode of motion, 310 Fellowship, e º tº . 17 “ latent, . - e . 38.2 Felt covering for steam-pipes, 428 “ lost by radiation, . 428 Fine, and val. Of sil. and gold, 480 “ specific, • e ſº Fire-engines, . . . 337 “. . units and horse-powe Fixed Stars, . . . . . . 506 Heating surface in boilers, . Flags of nations—plate, . l houses by steam, Flour-mills, . . . . . . 264 Helix of a screw, construct. Flooring, beanns in, . . 279 | Hexahedron, © e © Flow of water in pipes, ". 342 | Hemp ropes, weight and pri $ $ in rivers, .. 340 High water, time of, , . " . Flues for steam-boilers, . 423 Height of the atmosphere, 351, 35 Fly-wheels, . e . . 315 ‘’ measure of by barom., 3 Focal distance of lenses, .. 486 “ of cities, º . 358 Foot-valves in air pumps, 408 “’ of snow-line, º Force defined, & . 262, 311 “ mountains and Volc? Force, quantity of, moving, 310. ** miscellaneous F'Orce pumpS, . . 407 || “. CO - 3 ** Worce of temperature, • Steam-han º CoNTENTs. - - 7 - - PAGE Hyperbola, equation for, . 145 Icosahedron, . e * º Ice, expansion, contraction of, 384 Inches to decimals of a foot, 36 Inches and gentimetres, 38, 364 Igglined plane, . . ; 260 Píndicator diagram, Plate V." 421 Index of refraction, . . 4 Inertia, º º e e e 310 Injection water e o , 408 | Injector, Giffard's, º , 439 | Incrustation in boilers, . 430 | Integral calculus, & 28 | Interest, sim., compound, 16, 24 'Interpolation, . . º & 'Impact of bodies, . o e Irºn Or blast furnaces, * 443 ** acid test for quality, . 418 “, strength of at high tem., 433 Irrigation, vol. of water for, 359 Joints, proportion of riveted, 425 JOInval’s turbine, . . ſº Jupiter’s satellites, . 505, 508 Joule's equivalent of heat, 392 Julian period, e * e ; 6 Lakes, area. Of, . º º - \Land surveying, . . . . 128 , Lap and lead on slide-valves, 420 ... Latent heat, © & 2 Later&T Strength, . 272,278 * Latitude and longitude, . 490 Law Of gravity, . º . 303 • 498 265 486 Leap year, . º º ºg Leather belts, tº G & Lenses, Optical, º e º Letter for printing, . 511 Lever, static momentum, 154, 156 Light and Color, . . . 72 Longitude of places fr. Green.,490 $6 difference in time, 491 Log-line, length Of, . . 32 Lunar cycle, * e º . 49S Magnifying power of lenses, 486 4 & opera lässes & teles’p's, § Mass, explained, . . . . . . 09 Mantissa of logarithms, . 146 Manual labor, . . . . 264 Mariner's compass, . . 130 Mariner's date, . . . . . 498 Maxima and minima, e 30 Mean time, º . . . . 498 Measures, ancient, . . 40 “ foreign, . . . . 40 “. . . and weights, . 32 Mechanics, • - . . 254 Men's power, . º º • 262 Meridian, to find the, . . 509 Natural sines, etc., Liquid measure, . . . 33. Llama of Peru, ... . . 264 Load on roofs, . . . . 299 Locomotive, traction, & 124 Logarithms of numbers, . 148 & 4 trigonometric, 163 & & hyperbolic, . I43 | Pendulum, . o PAGE Meta-centrum of ships, .. 446 || Metals, hardness of, . . . Metre and feet, . . . 38, 40, 364 Metrical System, . . . 37 MicroScope, . . . . . . 486 Mile, statute and nautical, . .32 Miles pr. hour = ft. pr. Second, 352 Miles and kilometers, . . . , 39 | Mills, flour, Saw, rolling, 264 “ wind, . . . . .3 Minerals, hardness of, . 33 Mirrors, convex and concave, 484 || Momentum, Static, . . . 254 dynamic, . 310 $º in bodies, . . 322 Moment of inertia, 310 Money, American & foreign, 34 Moon, elements of, . . 96 Moon's faces, . . . . . . 508 Mortar and cements, . . 474. “ piece of ordnance, 307, 393 Motion of bodies in collision, 322 “ quantity of, mode of, 310 “ of water in pipes, , 342. “, of water in rivers, “ of gas in pipes, Mountains, height of, . Monuments, height of, . . Multiplication in algebra, 13 Music, acoustics, . . . 373, § 2 Nails, penny, length & wei’ht,432 Navigation, traverse, . . . . 128 Night and day, length of, . 508 Nitro-glycerine, dynamite, 473 Nominal horse-power, . . . 410 North and South, to find, .. 509 Notation of numbers, . . 11 Nuts and bolts, size & weight, 392 Nyström's calculator, . " .. 71 || 55.; Octahedron, º e - e. Obstructions in rivers, . . 341 Opera-glasses, . . . 487 Optics, . . . . . . . . 483 Ordnance, heavy, . 393 321 Oscillation of pendulum, . Overshot-wheel, . . . . . 347 Paper, drawing and tracing, 290 T, value of, to 45 decimals, .. 48 Parabola, to construct a, , . 47, 145 Parabolic construc. of ships, 444 Iłll I'l'Or, . . . 4 $6 Vein, . . . . Paradox, hydraulic, . . Parallax, Sun's, . . . Pattern-makers' rule, e & Peal of bells, to construct, . § Penny nails, . . . Perch of stone, . Percussion, centre of, Periphery of circle 4 & 4 & 2.Il e] Performance of CoNTENTs. PAGE Permutation, . . . . . . 1 Piling of balls and shells, 21, 23 File-driving, . . . . 277 Pipes, cast-iron, weight of . , 283 “ of diff’nt metals, wºt of, 282 “ brass and copper, . 43. “ motion of water in, 337, 342 “ and flues, . . . 423 . “ motion of gas in, . 372 ‘‘ steam, radiat. of heat, 429 “, steam, size of, . . 409 Pitch of propellers, . . 467 “ “ screw, . . . 53 * “ spirals, . . 47, 54 * “ screw thread, . .293 “ “ teeth in gearing, . 29. Planetary syst’m, el’ments of, 506 Plane, inclined, • • . 260 “. . sailing, traverse, 127 Platinum, weight of, . . 479 Plumbing, tº e e 432 Points of the compass, . . 130 Polygons, . . o & º 63 Polyhedrons, . . . . 55 Poncelet's water-wheel, 346 POpulation countries & cities, 489 Portland cement, . . 474 j'I’orts, steam, . . . . . . 408 “ high water in, . . 506 Powder, properties of, . . 393 Power, actual horse, . . 411 “ nominal “ . . 410 ‘‘ of Steam-engines, . 413 “ “ locomotives, . 125 “ “a man, horse, 262, 264 “ in moving bodies, . 313 “ definition of, 262, 311 “ for different mills, . . 264 “ “ pumping, . . . . 341 * “ punching, . . 435 “ “ steamboats, . 460 ( & magnifying, . . 486 & & reflecting of heat, .. 386 “ for blowing machines, 440 ‘‘ ‘‘ fans, ventilation, 442 “ “ quartz mills, 482 Pound, avoirdupois, troy, .. 33 Prime, vertical and parallel, 133 Press, hydraulic, . - . 335 Pressure col’m’s wat'r, etc., 344,351 Price of boiler tubes, . . .429 “ copper & brass tubes, 432 ‘ “ couplings for plum'g, 432 * “ gold and silver, . 480 “ “ hemp and wire ropes, 271 * “ rolled iron, diff fºm's, 280 “. . . “ taps, dies, stocks, . “ turbines, • * Wrought-iron gird’rs, 279 for guns, . 06, 393 432 35 66 || Scales in music, PAGE Pumping, water, . . . . $41 Bunching iron plates, . 435 Pyrites, iron, göld-bearing, . 482 Quantity defined, . . sº * of motion, -. . . . ; 482 “ “ work, total, & & “ moving force, Quartz mills, . . . . . . . Iładiation of heat, ..' 386, 428 Radius of the earth, . 309, 488 Roads, bad, in Peru, . . . 264 4 4 traction Oll, º . 126 Rail, elevation of outer, . . 119 Rails, Spring of, . . . . . . 119 “ weight and price of, . ; Railway curves, . . . 116 | Ram, hydraulic, e G . 343 | “ in pile-driving, . . . . .277 Rain, fall of, . . . . 126, 359 Range of a projectile, . 307 Rebate or discount, ... . / 17 | Rectang'r beams from a log, 2.É. Reduction of inches to feet, 36 Reflecting power of heat, ... 386 Refraction, light, atmos., 132, 503. Refractive indexes, . . 4S5 Reg’ng a time-keeper by stars, 507 | Resistance and force Of Wind, 358 in water, . 341 “ in railway curves, 352 | 4 4 - of air to projectiles, 306 Resultant of forces, . . . 257 Retarded motion, . . . 313 Right ascension, Sun's, .. 500 Ringing bells, . . . . 376 Rivers, length of, . . 359 $ $ flow of water in, . 340 “ obstruction in, . 341 Riveted joints, proportion of, 425 Rivets, iron and copper, . . 433 Roasting of Sulphurets, . 482 Roebling's wire ropes, . 271 Roll diron, weight, size, price, 280 Rolling-mills, & © Roman Cement, e . 474 | § { notation, s e 12 Roofs, wood and iron, . . 299 | Ropes, strength, weight, size, 271 Roots, Square and Cube, . . 2 Rule for pattern-makers, . 433 | Russia, sheet iron, . . . 432 Safety-valve, . . . 408 || Sailing distances, . . . . 492 Salt water in boilers, incrus., 430 Sash-bars, iron, for windows, 280 Satellites, Jupiter’s, . . . 508 Saturation in boilers, . . 430 Saw-mills, circ. & alternative, 264 “ , water-wheel, # - . 375 261 53 | 466 | Screw § force by, & $6 elix, . . © e “ propeller, . . . " ; : - - - PAGE , 292 £2 ºr wº Screw threads, . . . . Seasons, . . . . . 35. Secant, natural, . . . 200 Segments of a circle, . . 8 Setting of stars, . . . 507 Shafts, diameter; revolutions, 418 # Sheering iron plates, . . . . 435 Sheet, iron, copper, etc.287,289, 443 Ships, construction of, . . . . 446 Shrinkage of castings, . 433 Sidereal time, clock, year, . 498 “ and solar times, ". 503 Sidings of parallel tracks, . 121 Signs, signification of, . 12 Silver, to refine, . . . . .481 . ...“ , and gold table, value, 480 ‘Silvering metals, . . . . . 475 Simple interest, . . . 16 Simple substances, . . 470, 477 Simpson’s rule, . & . 114 Siºnes, cosines, etc., natural, 209 & 4 i logarithmic, 163 | Slide valves, . . . . . 420 :Slip of propellers, . . 466 §§lope of embankments, 122 :Smelting or freezing-point, 3S3 Snow-line and bulk of snow, 351 Snow, weight of new-fallen, 209 | Solar tin-e to sidereal, . . 503 Solders for bracing, . . . 332 $oldering, acids for, . . 475 † Solidity of révolution, . . Solids, capacity of, * - Sound, velocity of, . . 373 Soundings to low water, . 509 Specific gravity, ... . . . 328 “ heat, caloric, . . 390 Spectrum, colors, e 372 Speed & horse-power of steam .,460 “Spheres, Sur., capac., weight, 281 Spherical distances, . . 138 : “' mirrors, . . . 484 & 4 trigonometry, . 138 , Spiral, construction of, . 47, 54 Spindle, circular, . . . . 62 Spring of rails, . . . 121 Square a circle, to, . º {{ and square roots, . S2 º horse-pow'r of boil., 422 Stability of ships, monentum, 469 Static momentum, . 554, 311 Stars, R. A. and declination, 506 “ setting of, . . . 5 Steam, properties of, . . - $$. Oilers, . • e . 422 Condenser, . . 411 “ engines, . . . 413 6 & expansion of, . . 402 “ hammers, . 303 “ loss by rad. from pipes, 428 {{ pipes and ports, 40S $6 superheated, . . 438 & 4 ship performance, .. 460 400, 410 ** table, . . . º- - CoNTENTs. - * . . . 9 - PAGE Steel, tempering of, . . . 332 Stocks and dies, price of, . . 432 Stone, perch of, . . . 471 Strength of materials, 268-279 * iron, cop., high heat, 433 $ $ boiler flues, . 3. “ Portland cement, . 474 $g of animals, . . 264 String, musical, . . 374 Subtraction in algebra, . 13 Sulphurets, gold-bearing, .. 482 Sun, set and rise, . . . . 510 “ cycle of the, . . . 498 “ distance to, . . . 310 “ parallax of, . . 503 {{ pº Of, . . 496 “. R. A. and declination, 500 Superheating steam, . . 4.38 Surface condenser, 41 1 “ of boiler tubes, 429, 432 * of revolution, . . “. . of solids; . . . . 60 Surveying, tº º chain, . . . .32 Suspension bridges, . . 302 Tangent, etc., natural, . 209 $$. logarithms, . 163 Taps and dies, . . . . ; Teeth for wheel, gearing, . - Telescope, astronomicaſ, 354, 487 Temperature, Inean, climate, 358 | & ſº, color of steel, 332 & 4 force of, . 385 * * * fusion, alloys, 332 “ . miscellaneous, 386 £ 4 correction for, 366 {{ on the ocean, 370 44 boiling, smelt., 383 & £ table, conver'n, 380 $t boil. water bar., 367 Terms, dyn’cal, proper, conf., 310 Tests, chemical, for metals, 481 Tetrahedron, elements of, 55 Thickness of boiler-iron, 434 Thermometer, graduation of, 379 Threads, screw, num. pr, inch, 292 Threshing machine, . 264 Timber, green and Seasoned. 426 Time chronology, e g 49S “ apparent, . . . 504 “ equation of, & . 500 “ sidereal, . e • º “ diff longitude, . . 491 “ to regulate a watch, “. . of high water, . . . . S Tinder, tenniperat’re of firing, 3SS Tin-plates, English, . . . .29ſ) ' Tinning, acids for, . . . e Tonal sys. of weight & mea Toning of musical instru, TOnnage register, Anner Tracing paper, . Traction on roads, . Traveling distanc, 10 CONTENTS, - FAGE | º PAGR, Traverse surveying and table, 90 Water colors, & . . 291 || Triangles, plane, . . . . 13 “ motion of in pipes, 342 | 4 & spherical, . . 139 44 “ “ rivers, 340 | Trigonometry, plane, - . 134 Water-wheels, . . . 346 4 & spherical, . 138 “ injector, . . . . 439 Tropical year, . . . . 498 “ works, . . . . . . .341 || Troy weight, . . . . 33 “ falls, . . . . 3 Truss-bridges, . . . . . . 303 || Wave-line, proper's of waves, 449 Tubes, iron, lap welded, . 429 Weather, prediction of, ... 355 “. . . copper and brass, . 432 edge, . . . . . 255, 261 Turbines, Jonval's, . . . 848 || Weight and measure, • 32 Tuyeres in furnaces, . . 440 $ $ ' ' . & 6. new system, 44 *::::::tiºns ºf . . . . #| ºś y • * 5 d Ul I.] S R Undershot water-wheel, . 346 “ of pipes, º --~~ Units of work and heat, • 392 “ cast-iron, cylin., pip’s,283 | “ of power, dynamics 262 “ fiat rolled iron 284 & 4 workmah-days, ... " .. 262 & & copper bolts, ". . . 287 §§ Stan. Y. and meas., i. . pr º sheet, *-*. 299, # . S. tonnage la W. . ę ſe * &’ and Capaci tº • Valves, lº, . . 407 4 & §. • e y. . 433 $ $ Safety, . . . . 408 ! { steam-hammers, ... 30 . ; : • * ; ... & bulk of º; blast-engines, . te engines and DOlier8, 4. | | Vegetable acids and salts, 472 § { bells, tº & .” #4 Vein of water, contracted, .. 339 “ cub. in. water, etc., 381/. | Ventilator, fan, . . . . 442 6 & heavy Ordnance, . d Vessels, tonnage Of, . . 463 | Weirs, water flowing over, 340 4 * construction of . , 446 Wheels, water, . • *, * 6 Velocity of light and electrºy, 372 | Wind, force by, . . . . 352 4 & of Sound, . . . . 373 ... mills, . . . . . . §§ {{ of water in pipes, 342 || “. velocity of, . te • 35 & & of water in rivers, 340 || Window-sashes, iron, • 280 66 of falling bodies, .. 308 “ glass, . . . . . 290 & 4 of projectiles, i. 393 || Wire-gauges, . 28S & $ virtual, . . . . 310 “ ropes, Roebling’s, . 271 Vis-viva, principles of, . 310 || Wood, for combustion, cord, 426 Vibration of pendulum, . 320 * South American, . 276 & $ in musical tones, 374 || Work, dynamic, 262, 311, 314, 392 Volcanoes, active, height of, 363 || “ actual total amount, 310 Vulgar fractions to decimals, 36 “ total quantity of, . 10 Walking-beam, . . 53, 63 || “ rate of, . . . . . Warren girder, bridges, .. 301 || Yard, feet and inches, * tº Water, properties of, . . . 394 || Years, different kinds of, . 498 “ composition of . 470 Zirac, sheet, weight of, 287, 433 é & fresh, condenser, . 411 - MATHEMATICS. | I N T R O DUCTION. Quantity is that which can be increased or diminished by augments or abatements of homogeneous parts. Quantities are of two essential kinds, Geºmnctrical and Physical. * 1st, Geometrical quantities are those which occupy space; as lines, surfaces, Solids, liquids, gases, &c. - 2nd, Physical quantities are those which exist in the time but occupy no space, they are known by their character and action upon geometrical quantities; as attraction, light, heat, electricity and magnetism, colours, force, power, &c., &c. To obtain the magnitude of a quantity We compare it with a part of the same, this part is imprinted in our mind as a unit, by which the whole is measured and conceived. No quantity can be measured by a quantity of another kind, but any quantity can be compared with any other quantity, and by such com- parison arises what we call calculation or Mathematics. -- F. W. M A T H E M A T I C S. Mathematics is a science by which the comparative value of quantities are investigated; it is divided into : 1st, Arithmetics—that branch of Mathematics, which treats of the nature nd property of ºber; it is subdivided into Addition, Subtraction, Multiplica- $. Division, Jnvolution, Evolution and Logarithms. 2nd, Algebra,_that branch of Mathematics which employs letters to repre- sent quantities, and by that means performs solutions without knowing or noticing/the value of the quantities. The subdivisions of Algebra are the same as in Azrithmetic. 3rd, Geormetryº-that branch of Mathematics which investigates the rela- tjve property of quantities that occupies space; its subdivisions are Lomgimetry, 2Pianºmetry, Stereometry, Trigonometry, and Comic Sections. - 4th, Differential-calculus,— that branch of Mathematics, which ascer- tains the mean effect, produced by group of continued variable causes. 5th, Integral=calculus, -the contrary of Differential, or that branch of Mathematics which investigates the nature of a continued variable cause, that has produced a known effect. A R IT H M ET I C. The art of manaeuvering numbers, and to investigate the relationship of quantities. - Figures—1, 2, 3, 4, 5, 6, 7, 8, 9. Arabic digits, about nine hundred years old. Giphers—0, 0, 0. Sometimes called noughts, it is the beginning of figures and things. - Wºmber is the expression of one or more figures and ciphers. Integer is a whole number or unit. Fraction is a part of a number or unit. When figures are joined together in a number, the relative dignity expressed by each figure, depends upon its position to the others. Thus, i - 3 ad • o: § Ca $– 5 Q g § 'º. ă 3 H 3 d : , c. cº # º > t- ,--AN-º- /*-A-en /*-A-e-N as-A-e Y ~~~ 674,385; 496,345; 695,216 : 505,310: 6 12 - NoTATION. - Y R 0 M A N N 0 TAT I () N. - Tho Romans expressed their numbers by various repetitions and combinations of seven letters in the alphabet; as, - | #F# - 500 = D, or LO. . . . . 3 = irr 1,000 = M, or CO. - - --> . 4 == IV. 2,000 = MM, or IIOOO. #EY: 5,000 = V, or L00. i-vii. 6,000 = WI, or MMM ; = Yºr. 10,000 = X, or COO. 1ö–X. 50,000 = L, or L000. 20 = XX. - 60,000 = LX, or MMMO 30 = XXX. ,UUU = LX, or MMI Mij. ; -: ;I. 100,000 = C, or COOO. #E}x. 1,000,000 = MI, or COOOO. 70 = LXX. - 00,000 = MM. Or MMOOO. 80 = LXXX. A º ,00 MM, or MM000 - 90 = XC. ar, thus, – over any number, in-) 1öö–6." creases it 1000 times. - - ExAMPLES.—1872.-MDCCCLXXII. 524,365, DXXIVCCCLXV. An imperfection in the Roman Notation consists in the fact that there is no nification for the cipher, as in the Arabic Notation. Signification of Characters. w = Equality, as 6 = 6, reads 6 is . Nº - equal tº 6. y J Integral, . . . f dy=y. + Plus, Addition, .. 3 + 6==9 dy Differential, . . dy = d2 +x, — Minus, Subtraction, 6–2 = 4 3f4 Fraction, . . . . = #. N × Multiplication, . 834–12 (x) Ship sign, dead flat, —- Ol' : Division, . . 15 : 5 = 3 E: Furnace fire-grate. V Square root, . . . Vº = 3 O Boiler heating-surface. º/ Cube root, . . . W 8 = 2 # Sharp. High. > Greater, . . . . . 824 b Flat. Low. < Less, e e - © tº º º 63 T Periphery. Co Infinite, . . . . . O - Astronomical Characters. IPlanets. . Signs of the Zodiac. The Sun. Ó Conjunction in the ies. . . G) r 10 yº same degree or sign, or GP Aries, Q The Moon. having the same longi- 8 Taurus, . § Mercury tude or Right Ascension. ºf tº & Q Venus y 3- Sextile, when two signs II Gemini, . . º - distant, or differing 60° gº, Cancer, . {B} The Earth. in longitude or lêight of Mars. Ascension. & Leo, . . . > [] Quartile, when three Viror Q Ceres. signs distant, or differ- Iſ! Virgo, . . $2 Pallas. ing 90° in longitude or <> Libra, . . - Right Ascension. - •wn: 8 Opposition, when six Iſl Scorpio, . signs distant, or differ- Sagittarius." ing 180° in Longitude 1 Sag e y Q or Right Ascension. MP Capricornus, Q Ascending Node. *Nº. ** {} Descending Node. § Aquarius, . R . A. Right Ascension. 3é Pisces, . . HEAT EXCHANGERS Dynamic Response of Heat Exchangers Having Internal Heat Sources --2, W.S. RPACI, J. A. CLARK. Aſ Soc Mech *nors--paper n 57-HT-6 for meeting Aug 11-15 1957 9 p. Consideration of dynamic response of temperature of he at transfer surface and f | uid surface temperature differeneo in he at exchanger having internal he at sources which is subjected to sudden change in rate of he at generation; re- sults aro obtained by direct mathematical attack on govern- ing differential equations; he at exchangers to which so I u- tions apply include heterogeneous nuclear reactor. 160-206-298 No. 58-- 4960 Printed in U.S. à. C Engineer ing Index Service & .. <<>" . . . . . **** *** HEAT EXCHANGERS Does Fou ling of Heat-Exchanger Surfaces Always Hean Poorer Performance? Brit Chem Eng v 2 n 12 Dec 1957 p. 67 4-6. Proposition that dirt deposits invariably mean poorer heat transfer in heat exchangers is discussed; conditions under which reverse is case are out lined for fou ſing inside ex- changer tubes and on plates of plate exchangers. 160 No. 58 - -20532 Printed in U.S.A. O Engineering Index Service