FHE am. ARCHITECT'S X SHIfeyiLOEii ■ POCKET-BOOK ^ CLEMENT MACKROW,M.I.S.A. ' .^pio'^^: JuTnen CROSBY liDCiCWOOD ^ Ci^il r'Jj /■'!■■■ r^ :/;//>^;.':/.2^ ^-v/.:^ 7i> Digitized by tine Internet Arciiive in 2008 witii funding from IVIicrosoft Corporation littp://www.arcli ive.org/details/arcliinavaltectssOOmackricli <£ THs: NAVAL ARCHITECT'S AND SHiPBUiLDER'S POCKET-BOOK OP AND MARINE ENGINEER'S AND SURVEYOR'S HANDY BOOK OF REFERENCE BY CLEMENT MACKROW • NAVAL DRAtGHTSMAK MEMBER OF THE INSTITCTIOX OF KAVAl ARCHITECTS Cnbirli coition, acfaii^ctr gpiol]^ LONDON CROSBY LOCKWOOD AXD CO. 7 feTATIOKERS'-HALL COURT, LUDGATE HILL 1884 c i_i . - - L0XD05 : PniKTED B7 jPOTTi.suOOUE AND CO.. KKW-StnKliT SQCAUE AMJ I'ARLIAJIEST STUEKT PEEFACE The object of this work is to supply tibe great ^^ant which has long been experienced by nearly all who are connected professionally with shipbuilding, of a Pocket- Book which should contain all the ordinary Formula?, Kules, and Tables recpiired when working out necessary calculations, which up to the present time, as far as the Author is aware, have never been collected and put into so convenient a form, but have remained scattered through a number of large works, entailing, even in referring to the most commonly used Formulae, mucli waste of time and trouble. An effort has here been made to gather all this valuable material, and to con- dense it into as compact a form as possible, so that the Naval Architect or the Shipbuilder may always have ready to his hand reliable data from which he can solve the numerous problems which daily come before him. How far this object has been attained may best be judged by those who have felt the need of such a work. Several elementary subjects have been treated more fully than may seem consistent with the character of the book. This, however, has been done foi' the benefit of those who have received a practical rather than a theo- retical training, and to whom such a book as this would be but of small service weie they not first enabled to gather a few elementary principles, by which means they may learn to use and understand these Formulae. ^^' PREFACE. In justice to those authors whose works have beer consulted, it must be added that most of the Rules and Formulae here given are not original, although perhaps appearing in\ new shape with a view to makinf' them simpler. There are many into whose hands this work will Ml who are well able to criticise it, both as to the usefulness and the accuracy of the matter it contains. From such critics the Author invites any corrections or fresh mate- rial which may be useful for future editions. CLEMENT MACKROW. London : Jul// 187i). NOTE TO THE THIRD EDITION. The rapid sale of the iirst and second editions of this work has shown that the efforts made to supply a much felt want have in some measure succeeded, and the present opportunity has been taken of thoroughly revising it, so as to make it more worthy of the confidence it has received. Many strangers to the Author have taken a generous interest in the book by making suggestions, &c., which have, where possible, been carried out ; and it is hoped that the same kindly interest in it will continue to be shown. CLEMENT MACKROW. London : AjjHI 1884. CONTENTS. PAGE Algebraical Signs and Symbols 1-3 Deci>lal Fractions . 4-6 Practical Geometry 7-19 Trigonometry 20-28 Tables of Circular Measure of Angles. . . . 29-31 Mensuration of Superficies 32-41 Mensuration of Solids 41— iG Mensuration of the Surfaces of Solids .... 47 Circumferences and Areas of Circles .... 48-66 Areas of Segments of Circles 67-09 Ce>'tres and Moments of Figures 70-81 Ton'nage Rules and Tables 82-102 Board of Trade Regulations for Ships .... 103-114 English Weights and Measures 115-122 Metrical Weights and Measures 122-130 Decimal Equivalents of English Weights and Mea- sures 1.37-140 Foreign Money, Weights, and Measures . . . 141-142 Mechanical Principles 143-145 Centre of Gravity of Bodies 146 Laws of Motion 147-151 Displacement of Ships 151-158 Centre of Gravity- of Ships 159-161 Stability of Ships 102-177 Waves 178-181 VI CONTEXTS. Rolling Propulsion of Vessels . . , Distances down Courses of Rivers, etc. Steering . . « . . ^ Squares, Cubes, Roots, and Reciprocals of Numbers . Evolution Weight and Strength of Materials . . . . Values of Whitworth's Gauges, etc. . . • . . Useful Numbers often used in Calculations Riveting as employed in H.M.S. 'Hercules' . , . Sines, Tangents, Secants, etc Masting and Rigging Ships French and English Vocabulary . . . , . Hyperbolic Logarithms Wages and Percentage Tables, etc Logarithms of Numbers Strength of Materials, etc Hydraulic and Miscellaneous Formul.k . . . . Conic Sections, Catenary, etc Mechanical Powers, Work, etc • Board of Trade Regulations for Marine Boilers, etc. Particulars, Weights, etc., of Marine Engines . . Seasoning .vnd Preserving Timber Timber Measures Bricklaying, Plastering, Painting, f:Tc. Varnishes, Lacquers, Dipping Acids, Cements, etc. . . Miscellaneous Recipes, Tables, etc. . . • i PAGK 182-186 186-209 210-219 220-221 222-2G6 267-268 269-305 305-310 311-319 321-824 325-332 833-360 361-376 377-381 382-389 S 90-426 427-451 452-456 457-461 462-467 467-478 479-483 484-485 486-488 488-490 490-495 495-512 INDEX 513 MACKEOWS POCKET BOOK OF FOEMUL^, EriES, AXD TABLES FOE KAVAL ARCHITECTS AND SHIP-BUILDEES. SIGNS AND SYMBOLS. The following are some of the signs and symbols commonly used in algebraical expressions : — = This is the sign of equality. It denotes that the quantities so connected are equal to one another ; thus, 3 feet = 1 yard. + This is the sign of addition, and signifies plus or more ; thus, 4 + 3 = 7. — This is the sign of subtraction, and signifies minus or less ; thus, 4-3 = 1. X This is the sign of multiplication, and signifies multiplied by or into ; thus, 4x3 = 12. -^ This is the sign of division, and signifies divided by ; thus, 4-f-2 = 2. {} [] These signs are called brackets, and denote that the quantities between them are to be treated as one quantity; thus, 5{3(4 + 2)-6(3-2)}=5(18-6) = 60. This sign is called the bar or vinculum, and is sometimes used instead of the brackets ; thus, 3(4 + 2) — 6(3 — 2) x 5 = 60. Letters are often used to shorten or simplify a formula. Thus, supposing we wish to express length x breadth x depth, we might put the initial letters only, thus, lxl> x d, oi, as is usual when algebraical symbols are employed, leave out the sign x between the factors and write the formula l.b.d. When it is wished to express division in a simple form ths divisor is written under the dividend; thus, (a? + y) -h- = -• Z SIGNS AND SYMBOLS. !,!*., ', , These aye signs of proportion ; the sign : = is to, the sign : : = as; thus, 1:3 : : 3 : 9, 1 is to 3 as 3 is to 9. < This sign denotes less than ; thus 2 < 4 signifies 2 is less than 4. > This sign denotes more than : thus 4 > 2 signifies 4 is more than 2. *.* This sign signifies because. .*. This sign signifies therefore. JEj.'.: '.' 9 is the square of 3 ,', 3 is the root of 9. /^ This sign denotes difference, and is placed between two quantities when it is not known which is the greater ; thus (a? 'v^ y) signifies the difference between .r and ij. 1 , These signs are ui?ed to express certain angles in degrees, minutes, and seconds ; thus 25 degrees 4 minutes 21 seconds would be expressed 2.5° 4' 21". Note. — The two latter signs are often used to express feet and inches ; thus 2 feet 6 inches maj' be written 2' 6". \/ This sign is called the radical sien, and placed before a quantity indicates that some root of it is to be taken, and a small figure placed over the sign, called the exponent of the root, shows what root is to be extracted. Thus :2/a or Va means the square root of a. ^a „ cube „ ^a ,, fourth „ — — This denotes that the square root of a has to be taken and divided by h. ■ This denotes that }) has to be divided by the square root of a. y This denotes that the square root of a+ J has to be a-v d divided by the square root of a-^d. It may also be written 'fl + ft ^a + h thus, / , or . V a-\rd ^a + d cc This is anotlier sign of proportion. Ex.: accb; that is, a varies as or is proportional to b. oo This sig-n expresses infinity; that is, it denotes a quantity greater than any finite quantity. This sign denotes a quantity infinitely small, nought. Z This sign denotes an angle. Ex. : l ab would be written? the anffle ab. SIGNS AND SYMBOLS. 3 L This sign denotes a right angle. _L This sign denotes a perpendicular; as, ah L cd, i.e. ab is perpendicular to cd. A This sign denotes a triangle; thus, Aalc^ i.e. the triangle €i,hc. II This sign denotes parallel to. Ex.: ah \\ cd would be written, ah is parallel to cd. f or F These express a function ; as, a =fx ; that is, a is a function of x or equals x. f This is the sign of integration ; that is, it indicates that the expression before which it is placed is to be integrated. When the expression has to be integrated twice or three times the sign is repeated {thns, //,///)', but if more than three times an index is placed above it (thus,y'"). D oxd These are the signs of differentiation ; an index placed -above the sign (thus, r/-) indicates the result of the repetition of the process denoted b}^ that sign. 2 This sign (the Greek letter sigma) is used to denote tliat the algebraical sum of a quantity is to be taken. It is com- monly used to indicate the sum of finite differences, in nearly the same manner as the symbol/. [jl This sign is sometimes used instead of tt, being a modifi- cation of the letter C, for circumference. ii] This sign is sometimes used instead of e, being a modifi- cation of the letter B, for base. g This sign is used to denote the force of gravity at any given latitude. TT The Greek letter pi is invariably used to denote 3-14159 ; that is, the ratio borne by the diameter of a circle to its circum- ference. As the letters of the Greek alpliabet are of constant recur- rence in mathematical formulse it has been deemed advisable to -append the following table : — A a Alpha. I { Iota. P p Eho. B ;8 Beta. K K Kappa. 2 o-s Sigma. r 7 Gamma. A K Lambda. T T Tau. A S Delta. M IX Mu. T V Upsilon E 6 Epsilon. N V Nu. *

a as dZ> and T>e ; join Ef, cb, and Kb ; through E, the intersection of Be and cb, draw JDEF meeting A& in F ; join BF and pro- duce it till it meets T>a in a: then ab will be equal to ab, the distance required. Fig. 15. 15. Another method. (Fig. 15.) Produce ab to any point d ; draw the line T>d at any angle to the line ab ; bisect the line Dd in c, through whicli draw the line B&,, and make cb equal to BC ; join AC and dby and produce them till they meet at a : then ba will equal ba, the distance required. GEO^rETKT. 11 Fig. 16. 16. To meamre the digtance between two objects, both i)i accessible-. (Fig". 16.) Let it be required to find the distance between the points A and B, both being in- accessible. From any point c draw any line cc, and bisect it in d ; produce Ac and bc, and prolong them to E and F ; take the point E in the prolongation of A^. and draw the line bbc, making De equal to de. In like manner take the point F in the prolongation of Be, and make D/ equal to df ; produce ad and ec till they meet in a, and also produce bd and /c till they meet in b : then the distance between the points a and b equals the distance between the inaccessible points A and B. 17. To inscribe any regular polygon in a given circle. (Fig-. 17.) Divide any diameter ab of the circle abd into an many equal parts as the polygon is required to have sides ; from A and B as centres, with a radius equal to the diameter, describe arcs cutting each other in c ; draw the line CD through the second point of divi- sion on the diameter ab, and a line drawn from D to A .is equal to one side of the poly- gon required. 18. To cut a beam of the strongest section from any round piece of timber. (Fig. 18.) Divide any diameter CB of the circle into three equal parts ; from d or e, the two points of division in CB, erect a perjDendicular cutting the circumference of the circle in D or A ; draw CD and DB, also AC equal to db and AB equal to CD : the rectangle abcd will be the section of the beam required. 19. To descHbe the lyroper form of a flat plate by which to construct any given fmstum of a cone. (Fig. 19.) Let abcd represent the required frustum of a cone ; continue the lines AC and bd till they meet in E ; from E as a centre, with ed as radius, describe the arc dh, and from the same centre, with EB as radius, describe the arc Bi ; make bi equal in length to twice AGB, equal to the circumference of the base of the cone ; draw the line ei : then bdhi is the form of the plate required. being Fig. 19. 12 GEOMETRY. 20. To find t-Jic development of the fruxtum of a right cone jvhen cut hy an amjle inclined to the base. (Fig. 20.) Let ABCD represent the required frustum of the cone ; continue the lines AC and BD till they meet in e ; divide the base of the cone into any number of equal parts — say, 12— in the points 1, 2, 3, vN:c. ; join these points to E ; next find the development of the base of the cone, as shown in the preceding example, and on it set off the same number of points — viz. 12 — and draw lines from them to E ; then from E as a centre measure the distance down to the top of the sectional plane CD at each point of intersection with the lines 1, 2, >fcc., and set them off on the corresponding numbers (measuring from E) in the development : a line drawn through these points will give the curve of the top of the section, as required. 21, To find the derelopment of the frustum of a cylinder rvheii cut by a plane inelified to the base. (Figs. 21 and 22.) 1 Fig. 21. Fig. 22. Let ABCD represent the required frus- tum of a cy- linder ; divide the base into any number of equal par'^s — say, 12 — and draw lines through those points on the cylinder parallel to AC and BD; draw a line efg equal in length to the circumference of the cylinder, and divide it into the same number of parts ; on each point of division set up perpendiculars to it, making eh and gk equal in length to bd. and make Fi equal in length to AC ; tlien take tlie height at 1 and set it up on the corresponding number on eacli side of Fl, and so on witli each number : a line traced through the points thus obtained will be the curve of the required development. GEOMETRY. 13 22. To find the development of any ffii-en poi'tion of a seginent of a sphere. Let ABC (fig-. 23) be the middle section of the segment, and cfg in the plan (fig. 2-i) the portion to be developed ; Fig. 25. bisect AB (fig. 23) in E, and c set M^ the jDerpendicular EC ; divide the arc AC into any given number of equal parts — saj', four — and through the points of division draw the lines 1 1, 2 2, »fec., parallel to AB ; on the plan (fig. 24) from C as a centre, with the radius 1 1 taken from fig. 23, draw the arcs 1 1 cutting FC and CG in 1 and 1, and so on with 2 2 and 3 3 ; draw any line BC (fig. 25), making it equal in length to BC (fig. 23), and on it set off the same number of equal parts; at each point of division draw lines perpendicular to BC, and number them the same as on fig. 23. Measure the length of the arc 1 1 in fig. 24, and set off half of it on each side of BC on line 1 1 , and so on with each arc, includ- ing FG; a line traced through the points thus obtained will give the curve of the sides of the given portion of the segment when it is developed. To describe the curve at the bottom of the figure, take one-fourth of the circumference of the base as a radius, and from F and G as centres describe arcs cutting bc in S ; then from s as centre, with the same radius, describe the arc FBG, which will be the curve of the bottom of the figiu'e, as required. Should the top of the figure be cut off at the line 1 1 (fig. 23), from s as a centre in fig. 25 describe the arc 1h1, which will be the curve of the top of the figure, as required. 23. To find the development of any (jvren poHion of ajmraholoid. (Figs. 26, 27, and 28.) The development is found in the same manner as that of a portion of a segment of a sphere, as described in the last example (No. 22), with but one exception — that is, the length of the radius for describing the bottom curve of the figure, which instead 3-/- j ^3 of being equal to one-fourth of the circumference, as in example No. 22, is equal to one-half the length of the arc ACB (fig. 26) in this example. Fig. 26. u GEOMETRY. FIC 24. To find tlie dcvelojyment of an entablature plate. Let fig. 29 be the side elevation, fig. 30 the front elevation^ fig. 31 the plan, and fig. 32 the development of the figure; divide ADC (tig. 30) into eight equal parts, and from the points of inter- section draw lines pa- rallel to ABC, cutting CD (tig. 29) in the points 1, 2, &c. ; on bd (tig. 29) erect a perpendicular EC, and from the points on CD draw lines parallel to BED. From fig. 30 take the points 1,2, &c., on K BC and set them off on AFC (fig. 31), and erect perpendi culars from AFC at these points. D From C (fig. 29) along CE measure the points C, 1, c, 2, &c., and set them off on their corre- sponding lines from AFC in fig. 31 ; draw a line through those points, then measure it with its divisions and set it off in fig. 32 as a straight line aec, and at the points of division erect perpendiculars, continuing them either side of the line aec ; measure the distances 1, 1 ; 2, 2, &c. (fig. 29), on either side of CE, and set them off from AEC (fig. 32) on their corresponding lines, and on their respective sides of AEC. These will give' the development. 25. To describe a cycloid, the generating circle being qiven. (Fig. 33.) Let b6 be the generating circle ; draw a line abc, equal to the circumference of the generating circle, by dividing the Fio. 33. circle into any number of given parts, as 1, 2, 3, .^ &c., and setting off half \^ that number of jDarts on ""•^^ each side of b ; draw xo)L .'((ID lines from the inter- sections of the circle B C 1,2,3, &c., 7,8,9, &c., ))arallel to AC; set off one division of the circle outwards on the first lines .5 and 7, two divisions on the next lines 4 and 8, , tlien three on the next, and so on: then the intersection of i those points on the lines 1, 2, 3, kc, will be points in the curve. GEOMETRY. 15 Fig 26. To dtscj'ihe a jjrolate cycloid, the generating circle and the jjosition of the generating i^oint heing given. (Fig. 34.) Let oB be the generating circle, and p the generating point ; draw the base line 010 equal in length to the circumference of the circle ; divide the circle into any number of equal parts— say, 10 — and draw the radii 1, 2, 3, ice. ; from each of these points in the circle draw lines parallel to 010 ; as in the ^ cycloid, mark off one divi- q sion on the lines 1 and 9, ° two divisions on the lines 2 and 8, three on the next, and so on ; at the end of each line draw a line parallel to tlie radi-as from which it springs, and set on it the distance bp : a line traced through the points so obtained will be the curve required. 27. To drayr a cuytate cycloid, the generating circle and posi- tion of the generating jjoint heing given. (Fig. 35.) Let AB be the generating circle, and P the generating point wdthout ; draw the base line FF' equal to the circumference of the circle ab. divide the circumference into any number of equal parts — say, 10 — and draw the radii 1, 2, 3, ka. ; from each of these jDoints in the circle draw lines parallel ^\^{^ _ ' _.f__.^___'*^__l. to the base line ff' : g"" "" P' ^ . also draw the line GG' pai'allel to it, and at the same distance from it as the generating point is from the circle ; as in the cycloid, mark off one division on the Fig. 36. first line, two on the se- D cond. and so on ; from the ends of the lines thus found draw lines paral- lel to the radius from which the line springs, and set off on them the distance BP: a line traced through the jx)ints thus found will be the curve required. 28.- To desc-nhe an epicyclmdy the generating circle and the directing circle being given.{Fig.d6.) 16 GEOMETRY Let BD be the generating circle, and ab the directing circle ;. divide the generating circle into any number of equal parts — say, 10— as 1, 2, 3, &c., and set off the same distances round the directing circle ; draw radial lines from A through these last points, and produce them to an arc drawn with A as centre and AE as radius, as shown by cccc and c'c'c'c' on the diagram ; draw concentric arcs also through all the points on the generating circle, with A as centre ; then taking c, c, c, c and e', c', c\ c' as centres, and be as radius, describe arcs cutting the concentric circles at 1', 2', ice. : the points thus fotmd will be points in the required curve. A 29. To descTibe a hypo- cycloid, tJie f/eneratinff circle andtlie directiny circle heing given. (Fig. 37.) Proceed as in the epi- cycloid, the exception being that the construction lines are dra\\-n within the di- recting circle instead of outside, as in the epicy- cloid. 30. To draiv an arc of a parabola which shall pa^s through tivo given points, touch a Una at one of those points, and ivhose aarts shall be in a given direction. (Fig. 38.) Let A and C be the two points, ab the given tangent, and bc a line parallel to the given direction of the axis of the para- bola, cutting the given tangent in b ; diNide ab into any number of equal parts, and through the points of division draw lines parallel to BC ; divide BC into the same nimaber of parts, and ilirough the points of division draw lines to A : the points of intersection of 1 and 1'. 2 and 2', thus found, will be points in the required curve. Fig. 89. 3L To draw a tangent to any jmnf in a parabola. (Fig. 3t).) From the vertex A of the parabola draw AC perpendicular to AB, and make it equal to half BD ; through the points c and D draw a line, which will be the tangent required. GEOMETRY. 17 \ \ I \ \ \ i3 1 \ \ I V \ \i FiQ. 41. 32. :7J' describe a Jiijptrhola, the diameter^ abscissa, and double 4)rdiiiate being given. (Fig. 40.) Let AB be the diameter, bc its abscissa, and de its double ordinate ; then through B draw GF parallel and equal to DE ; draw also DG and EF parallel to the abscissa BC. Divide DC and CE into the same number of equal parts, as 1, 2, &c., and from the points of division draw lines meeting in A. Divide GD and ep each into the same number of parts as DC or CE, ^\/' and from the points of division 1', 2', ..^'C, draw lines meeting in B. The points of intersection of the lines 1 and 1', 2 and 2', ice, thus found, will be points in the required 'Z_i-_L_ J_J._l_V._\- curve. .S3. To construct a nemd curve, the length, extreme lialf -breadth^ and approximate fineness being given. (Fig. il.) Let BC be the extreme half- breadtl}, and CA the length. In CA take cx equal to CA X #, co-efficient of fineness, and at x set up the ordinate XD equal to \ of BC. About B and through D describe the circular arc fde, cutting CB produced, in E. About E through A describe the circular arc af, cutting the former arc in F, which will be the focus. Tlirough F draw FG parallel to BC. Join FB and fe, and draw fh, making the angle bfh equal to the angle bfg, and cutting bc, produced if necessary, in h ; divide the angle hfe (equal to | of bfg) into a convenient number of equal parts by lines diverging from f and cutting he m a series ot points, such as h. The points h, b, and e will be three of the points required. About the series of the points thus found describe circular arcs through the focus f. Divide bc into the same number of parts as the angle hfe, and through the points of division draw straight lines parallel to CA. The points, such as k, where these lines cut the arcs re- c 18 GEOMETRY. spectively corresponding to them, will be points in the required curve. 31. To construct an harmonic curve. (Fig. 42.) Fig. 42. A LB C Let AC be the base, CK the greaTest ordinate, and bd a balance ordinate midway between AC (the length of this ordinate varies according to the degree of fineness required in the curve, but it should not be greater than |, nor less than |, of CK) ; then through D parallel to AC draw de, cutting CK in E ; bisect CK in"j, through which point draw .JF parallel to AC : about E with the radius ek describe a circular arc, cutting .jf in F ; join fe and produce it, and at right angles to it draw kg. Bisect KG- in h, and from H erect a perpendicular to kg,- cutting CK in o, from which as a centre describe the arc kmg ; divide the base CA into any number of equal parts, and also divide the arc KMG into the same number of equal parts ; through each point of di\dsion of the arc, as M, draw lines paral- lel to AC, and through each point of division of the base, as L» draw perpendiculars cutting the lines parallel to the base : the points of intersection of the lines will be points required in the curve, as x. 35. To describe the invohite of a circle. (Fig. 43.) Fig. 43. Let AB be the given circle, which divide into any equal number of parts — say, I'J —as I, 2, 3, .ice. ; from the centre draw GEOMETRY. 19 xadii to these points ; then draw lines (tangents) at right angles to these radii. On the tangent to radius No. 1 set oft" from the circle a distance equal to one part, and on each of the tangents set off the number of parts corresponding to the number of its radius, so that No. 12 has twelve divisions set off on it (that is, equal to the circumference of the circle) : a line traced through the ends of these lines will be the ctirve required. 36. To describe a cissoid. (Fig. 44.) Draw any line ab, and drop a perpendicular CD f rotn it ; on CD describe a circle ; from the extremity D of the diameter draw any number of lines, any distance apart, passing through Fig. 44. the circle and meeting the line ab in a, b, c, d, and e ; take the length from D to 5, and set it off on the same line on each side from e, as eT>' : set off the length d4 from d, as r/E. Proceed thus with all the lines, and trace the curve through the points so obtained. 37. To describe a conchoid, the asymptote, pole, and diameter heinr/ ffiven. (Fig. 45.) Let AB be the asymptote, P the pole, and c the diameter ; draw CD at right angles to ab ; on each side of D set off any number of equal parts, as 1, 2, 3, &c. ; from P draw lines passing through the points 1, 2, 3, kc. ; then from each of these points with radius CD describe arcs cutting these lines in a, b, c, &;c. : the points of intersection will be points in the curve. The curve above the asymptote is called the superior conchoid, and the curve obtained by setting off the same lengths under the asymptote is called the inferior conchoid. C2 20 TEIGONOMETRICAL RATIOS. TKIGONOMETRY. The complement of an angle is its defect from a right angle ; tluis if A denote the number of degrees contained in any angle, 90^ — A is the number of degrees contained in the complement of that angle. The supplement of an ang-le is its defect from two right angles ; thus 180" — a is the number of degrees contained in the supplement of that angle. Fig. 46. Trigonometrical Ratios. All the different functions of an angle, or of the arc subtending that anofle, are expressed in a ratio to the radius of the circle which describes the arc. Thus in lig. 46 — r sine A = GL = GL T _ GL _ AD GA AK = 1 cosec A co-sine a = FG = AL f _ AL _ AB AG AH = 1 sec A tangent A = HB = HB 1 _ HB _ AD AB DK = 1 cot an A co-tangent A = DK = DK _ DK _ AB ~ DA ~ HB = 1 tan A secant a = AH =' AH 1 AH _ AG AB AL = 1 cos A co-secant a = AK = AK 1 _ AK AG AD LG = 1 sin A versed sine a = LB = AB- - AL = 1 —COS A co-versed sine A = FD = AD- -GL = 1— sin A. Fig. 47. Note. — The lines dropped upon the radii are pei^pendicular to those radii. It is more convenient to define the sine, co- sine, c II, Base = v/hj^otenuse^ — perpendicular^ in. Perpendicular = s/ hypotenuse- — base- The three angles of every triangle are equal to two right-angled triangles ; thus,A + b -f c = 180°. Of the six elements which compose a triangle — viz. the three angles and the three sides — three must be known in order that the others may be determined, and one of them must be a side — 1st. Wlien two sides (J, c) and an angle (c) are given. A 6 I. o-—h- = a-, from which a can be found. II. - = sin A, from which A can be found. c III. 90° — A, from which b can be found. 24 SOLUTION OF TRIANGLES. 2nd. When two angles (A, c) and a side (c) are given. I. -=sin A, from which we can find a. c II. - = cos A, from which we can find b. c III. 90° — A, from which we can find b, Ex. 1. Taking the first of tlie above cases, let J = 5 C = 90°. I. s/ c- - J- = a/ 169 - 25 = \/ 144 = 12 = a. II. - == lr_ = -9230769 = sine a. c 13 From a table of sines we find •9230769 = 67^ 22' 48"-5. III. 180°-(A + c)=180=-157° 22' 48 "-5 = 22° 37' ll"-5, or 90°-a = 90°- 67° 22' 48''-5 = 22° 37' ll"-5. -Ex. 2. Taking the second of the above cases let c = 2o A = 60° c = 90°. L =sin A, .*. - - = -^— . c 25 2 =^a = 21-6o. 9 II. -=cos A, .*.— = -, .-. -•^.= 77=12-5. c 25 2 2 in. 180°-(A + c) = 180°-150 = B =.30^ Oblique-angled Tnangles. (Fig. 52.) 1. AVlien the three sides a, b, c are given. I. Sin nn - = / 2 V /•(.s--70(.9-c)-l L he / 2 V I hi.' S III. Tan ^= /|(£:-A)_(ir:0]. In the aljove formula s denotes half tlie sum of the sides. Aiwther Mi^hod.—The angles may be found by dividing the triangle, when the sides are given, into two right-angled triangles. In the above figure we have — cd- = ca2— ad2, and also equals cii--DB-; therefore CA^ — cb2 = ad- — db-, therefore (ca + cb) (ca-cb) = (ad-i- db) (ad-db). MEASUREMENT OF HEIGHTS AND DISTANCES. 2o From this we can find ad-db, and then, since AJD + db is known, we can find ad and db ; then AD COS A = CA DB COS B = . CB Thus A and b are determined. 2. Wnien two angles (A, c) and a side (h) are given (fig. 52). L B = 180°-(A + C), from which we can find b. 11. '"^^^J:, from which we can find a. h sin B in ^ =, ^HL2, from which we can find c. b sin B 3. When the two sides a. h and the angle c are given (fig. 52). I ^2 = a- + b--2ab . cos c, from which we can find c. n. ?HLj^ = ?, from which we can find A. sin c c in. 180-(A + c), from which we can find b. Expressions for the Area of Triangles. (See fig. 48, 'Properties of Triangles.') I. Area of triangle = |bc . ad. jii^d AD = ab . sin B : therefore area of triangle = ^« . <^ . sm b. II. Area of triangle = ^^ {s{s - a){s - b){s - c). &2 . sin A . sin c III. Area of triangle = 2 si'nB ' Measurement of Heights and Distances. 1. To find the height of an accessible Fig. 53. object. (Fig. 53.) Let BC be the object and ab a line measured horizontally, a = AB, and 9 = the angle of ele%-ation, then bc = a . tan = height required. '26 MEASUEEMENT OF HEIGHTS AND DISTANCES. Ftg. 54. 2. Toll fid the height of an ina^ces^ihle object on a horizontal plane. (Fisr. 54.) Measure a convenient distance ab in the straisrht line bd, produced, and let a = AB : tiien CD /sin sin <^\ Fig. 55. Fig. 56. 3. To find the height of an inaccemhle ohject Tvhen it is not convenient to measure any distance in a line n-itJi the hase of the ohject. (Figr. .55.) Measure the length AB in any direc- tion from A : at A observe the angles DAC and DAB, 2aid at b observe the angle DBA ; then DC = ABfHi^^^-!iL7. sm ()3 -i- 7; 4. To find the distance hetn-een t.tvo visible but inaccemble t)bjects. (Fig. 56.) Let A and b be the objects ; measure a line CD, and suppose A, B, c, D to be in one plane ; then ob- serve the angles acd and adc, and AC can be found ; again observe the angles BCD and bdc, from which BC can be found : thus knowing AC and BC, and the included angle ACB, AB can be determined. 5. To find the distance of a sJiij} from the shore. ' (Fig. 57.) Let s be the position of the ship ; measure AB, a straight line between two points on the shore ; then sin SBA AS = AB 'sin (^.SAB-f- sba) Areas of Triangles, Polygons, and Circles. Fig. 58. 1. 77ie area of any qvadrilateral figure, ABCD (tig. 58), equals ^AC (be + dp). MEASUREMENT OF AREAS BY TRIGONOMETRY. 27 Fig. 59. 2. The area of any q^iiadri- lateral figure (fig. 59), ABCD, two of whose sides, ad and BG,a7'e jmrallel, equals ^(bc + ad)ae, or ^ (sum of parallel sides) x (perpen- /_ dicular distance between them). ^ 3. The area of any quadi'ilateral figure, abcf (fig. 59), equals ^(bc x AE) + |(CE x FC). 4. The a/rta of any triangle, abc (figs. 60 and 61), Fig. 60. Fig. 61. c c A c 1) B equals ^ ab . cd = | ab . ac . sin a = i^^ . Z* . sin A. 5. To find the radii of the inscribed and drcumscinbed circles H)f a regular polygon. (Fig. 62.) Let ab be the side of a regular polygon of n sides ; let O be the centre of the circles, OD tlie radius of the inscribed and OA the radius of the circumscribed circle. Let ab = a, AO = R, OD = r, then Fig. 62, R = ir 2 sm _ n 2tan'^ 6. To find the area of a regular polygon in terms of its sides, (Fig. 63.) Let EA, AB, bf be three conse- cutive sides of a regular polygon of e n sides, and let each side = a. Bisect the angles EAB and ABF by the lines OA, ob, meeting in o. Draw OR at right angles to ab. Then area of polygon = -—.cot-. 4 'tl/ 28 MEASUREMENT OF AREAS BY TRIGONOMETRY. Fig. 64. 7. To find the area of a regular i)olygon. in" scribed in a circle. (Fig. 64.) Let o be the centre of the circle, r the radiusu and AB a side of the polygon. Then area of polygon = -^-. sin 'l'L. 2 n 8. To find the area of a regular polygon de- scribed about a circle. (Fig. 65.) Let o be the centre of the circle, r tlK radius, and AB a side of the polygon. Then area of polygon = nr"- . tan -. n 9. To Jiiul the dip of the hoHzon. (Fig. 66.) Fig. tj6. Let o denote the centre of the earth, pb a tangent from the eye of an observer looking from a height AP to the earth's surface at b ; then b is a point on the horizon : draw PC at right angles to PO ; then the angle bpc is called the dip of the horizon. Let op cut the earth's surface at a, and let a fhe anofle bpc be denoted bv Q : then pb = AP . cot -. Table giving the Signs axd Values of the Trigonometrical Ratios for Certain Angles. Ratios 0= Signs 1 30= Signs 1 45'^ Signs 60° Signs -1- + 4- 4- + 4- 90° 1 » X 1 Signs 4- 4- 120° ^'1 •J 1 •> n'3 1 ,/3 2 •> 7:i Sine Co-sine Tangent Co-tangent Secant Co-secant 1 + ! 4- + + 1 2 2 1 V3 V3 2 1 + : 1 i + i 1 + iV2 + -t- -h -f + 73 2 I 2 v/3 1 V3 2 2 /3 Ratios Signs -1- -f- 1.35° i^ignsjl50°| Signs 180° Signs 270° Sicrns + 4- 360° 1 i Sine Co-sine Tangent Co-tangent Secant Co-spcant 1 V2 1 V2 1 1 V2 + -f 1 h 2 1 v/3 2 2 4- 4- + -1- 1 X 1 \ - 4- 00 -{- 1 - 1 oo 1 TABLE OF CIECULAE MEASURE. 29 Table of the Clrcular Measure, or Lexgth of Cir- cular Arc subtexdln'g A^'T Angle, Eadius being unity. To calculate the circular measure of any angle, see ' Tri- gonometry ' (pp. 21 and 22). Use of the Table.— ^o?. : Required to find the lengtli of the circular arc subtending an angle of 40° 11' 15" on a circle of 560 feet radius. Tabular No. for 40° = -698131701 „ ., 11' =-003199770 „ 15'' = -0000 72722 Length of arc = (560 x -701404193) = 392-78634808 ft. Seconds. 4 5 6 7 8 9 10 11 12 13 14 15 •0000048481 '•0000096963 ^•0000145444 [•0000193925 •0000242407 ■0000290888 •0000339369 •0000387850 •0000436332 •0000484814 •0000533295 •0000581776 •0000630258 •0000678739 •0000727221 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 -0000775701 •0000824183 •0000872665 i^0000921146 •0000969627 •0001018109 •0001066591 •0001115071 •0001163553 •0001212034 •0001260516 •0001308997 -0001357478 •0001405960 •0001454441 Sec' Circ. Meas. Sec. Circ. Meas. Sec' Circ. Meas 9 31 32 33 I 34! 35! 36 I 37' 38 1 39 I 40 41 I 42 43 ' 44! 45 I 0001502922 0001551404 0001599885 0001648367 0001696848 0001745329 0001793811 0001842291 0001890773 0001939255 0001987736 0002036217 0002084699 0002133180 0002181662 Seel Circ. Mca?. 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I-000223UU i -000227862 '•0002327 lOG i-0002375587 •0002424068 •0002472550 •0002521031 •0002.569513 •0002617994 •0002666475 -0002714957 -0002763437 •0002811919 -0002860401 •0002908882 Minutes. M. T 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Circ. Meas. AI. I Circ. Meas. M. I Circ. Meas, i-0002908882 •0005817764 >0008726646 •0011635528 '•0014544410 i-0017453293 1-0020362175 •0023271057 ^•0026179939 •0029088821 -0031997703 •0034906585 •0037815467 •0040724349 •0043633231 16 17 18 19 20 21 22 23 24 i 25 I 26 i 27 j 28 j 29 30 1-0046542113 •0049450995 •0052359878 •0055268760 •0058177642 •0061086524 •0063995406 •0066904288 •0069813170 •0072722052 •0075630934 •0078539816 ■0081448698 0084357581 0087266463 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [•0090175345I [•009308422 •0095993109148 •0098901991 •0101810873 -0104719755 •0107628637 •0110537519 •0113446401 •0116355283 •0119264166 •0122173048 •0125081921 •0127990812 •0130899694 il.l Circ. Meas. 49 50 51 52 53 54: 00 56; 57! 58: 59 60' ;-0133808576 ;^0136717458 j^01396263401 •0142535222 '•0145444104 1^0148352986 1-0151261869 •0154170751 •0157079633 •0159988515 •0162897397 •0165806279 •0168715161 •0171624043 •0174.5.32925 30 TABLE OF CIRCULAR MEASURE. |Takle or TiiE riiifjULAR Mesaure of any Axole (c.jntinued).j DEGREES. j Deff. 1 Circ. Meas. •017453293 46 1 Circ. Meas. Deg.| Circ. Meas. Deg. 91 1^.588249619 136 i Circ. Meas. '" •802851456 12-373647783 2 •0349065S5 47; •820304748 92 1-605702912 137 ■2-391101075 3 •052359878 48 837758041 93 1-623156204 138 2-408554368 4 •069813170 49 855211333 94 1-640609497 139 2-426007660 5 •087266463 50 872664626 95 1-6.58062789 140 2-443460953 (5 •104719755 51 : 890117919 96 1-675516082 141 2-460914245 7 •122173048 52 i 907571211 97 1-692969374 142 2-478367538 8 •139626340 53 1 925024504 98 1-710422667 143 2-495820830 'J •157079633 54! 942477796 99 1-727875959 144 2-513274123 10 •174532925 55 1 959931089 100 1-745329252 145 2-530727415 11 •19198G218 56; 977384381 101 1-762782.545 146 2^548180708 12 •209439510 57 1 994837674 102 1-780235837 147 2^56563400 la •226892803 58 jl 012290966 103 1-797689130 148 2^583087293 U •244346095 59,1 029744259 104 1-815142422 149 2-600540585 15 •261799388 60,1 047197551 105 1-832595715 150 2-617993878 16 •279252680 61 ir 064650844 106 1-850049007 151 2-635447170 17 •296705973 6211 082104136 107 1-867502300 152 2-652900463 18 •314159265 63 11 099557429 108 1-88495.5.592 153 2-670353756 19 •331612558 64 |1 117010721 109 1-902408885 154 2-687807048 20 •349065850 65,1 134464014 110 1-919862177 155 2-705260340 21 •366519143 6611 151917306 111 1-937315470 156 2-722713633 22 •383972435 67 1 169370599 112:1^954768762| 157 2-740166926 23 •401425728 68 1 186823891 113 1-972222055 158 2-757620218 24 •418879020 69 1 204277184 114 1-989675347 159 2-775073511 25 •436332313 70 1 221730476 115 2^007128640 160 2-792526803 26 •453785606 71 1 239183769 116 2^024581932 161 2-809980096 27 •471238898 72 1 256637061 117 2-042035225 162 2-827433388 28 •488692191 73 1 274090354 118 2-059488517 163 2-844886681 21) •506145483 74 1 291543646 119,2^076941810 164 2-862339973 30 •523598776 75 1 308996939 120 2-094.395102 165 2-879793266 31 •541052068 76 il 326450232 121;2-111848395 166 2-8972465.58 32 •558505361 77 il 343903524 122 2^1 2930 1687 167 2-914699851 33 •575958653 78 11 361356817 123 2^146754980 168 2-932153143 2-949606436 34 •593411946 79 |l 378810109 124 2^164208272 169 35 •610865238 8011 396263402 125 2^181661565 170 2-967059728 36 •628318531 81 11 413716694 126 2^199114858 171 2-984513021 37 •645771823 82 !l 431169!)87 127 2^216568150 172 3-001966313 38 •663225116 83 :1 448623279 128 2^234021443 173 3-019419606 3'.» •680678408 84,1 466076572 129 2^251474735 174 3-036872898 40 •698131701 85 il 483529864 130 2^268928028 175 3-054326191 41 •715584993 86 1 .500983157 131 2^286381320 176 3-071779484 42 •733038286 87 ;i 518436449 132 2^303834613 177 3-089232776 43 •750491578 88 il 535889742 133 2^321287905 178 3-106686069 44 •767944871 S'.'il 5533430:54 134 2-338741 lti8 179 3-124139361 45 •785398163 •.iO(r570796:527 135 2-356194490 180 3-1415926.54 TABLE OF CIRCULAR MEASURE. 31 Table of the Circular Measure of any Angle (concluded>| DEG-REES. Ueg.i r81'3-15i>04:5946 182 3-176499239 183 3-193952531 184 3-21 U05824 185 3-228859116 186 3-246312409 187 3-263765701 188'3-281218994 I89i3-298672286 190 3-316125579 ] 9113-333578871 192'3-351032164 193!3-368485456 Jl94i3-385938749 1953-403392041 19613-420845334 197'3-438298626 198'3-455751919 199|3-473205211 200 3-490658504 01,3-508111797 202'3-525565089 203'3-543018382 20413-560471674 205|3-577924967 206 3-595378259 207 3-612831552 208 3-630284844 209;3-64773813 210|3-665191429 211:3-682644722 700098014 717551307 735004599 752457892 769911184 787364477 21813-804817769 21913-822271062 Peg. I Circ. Meas, 226I3-944444I10 227'3-961897402 228'3-979350695 229 3-996803987 230 4-014257280|2 231 4-031710572 232 4-049163865 233 4-06661715 234 4-084070450 235 4-101523742 236^4-118977035 237 4-136430327 238 4-153883620 09960 291 23130 29 239 4-171336912 240 4-188790205 2414-206243497 242'4 -223696790 243 4-241150082 244 4-258603375 2454 -276056667 246 4-2935 247 4-310963252 248 4-328416545 249-4-345869837 250 4-36332 25l|4-38077642^ 252'4-398229715 253 4-415683008 2544-4331363 255:4-450589593 256,4-468042885 25714-485496178 258^4-502949470 259;4-520402763 260:4-537856055 261 14-555309348 262 4-572762640 2634-590215933 264|4-607669225 220:3-839724354 265 4-625122518 1212' 213' 2l4i 215 3- 21»;|3- 21713- 221^3-857177647 222:3-874630939 223:3-892084232 268 224:3-9095375241269 5;3-9269908r 266'4-642575810 267 4-660029103 4-677482395 4-694935688 270 4-712388980 31 Dep.' 2714- 272 4- 273 4- 274 4- 75 4- 2764- 2774- 2784- 2794' 280 4' 2814' 282 4' 283 4' 284 4' 285,4' 286 4 287'5 2885 2895 290'5 5 292'5 293 5 2945 295 5 5 2975 2985 5 005 301 5 B02 5 303 5 304 5' 305 5' 306 o' 307 5" 308 5' 309 5- 10 5' 115' 312 5' 313 5' 1314|5' 5i5 00 299 729842273 747295565 764748858 782202150 799655443 817108736 834562028 852015321 869468613 886921906 904375198 921828491 939281783 956735076 974188368 991641661 009094953 026548246 044001538 061454831 078908123 096361416 113814708 131268001 148721293 166174586 ■183627878 2010811 •218534463 235987756 253441049 270894341 288347633 305800926 323254219 340707511 358160804 375614096 393067389 410520681 316 5-515240436 317 318 319 320 99 Des.l r?. 427973974 356 445427266 462880559 480333851 497787144 o 5- 5- 5" 5' 5- 5- o 5' 324 5' o25 5' 326 5 rl27 5 ^28 5 329 5 330 5 331 5 3325 333 5 334 5 335 5 336 5 '.375 3385 339 5 340 5 3415 342 5 3435 344' 6 3456 3466 347 6 3486 349 6 3506 35116 352;6 353; 6 354|6 355 6 357 358 359 360 532693729 550147021 567600314 585053606 602506899 619960191 637413484 654866776 672320069 689773362 707226654 •724679947 74213323r 759586532 •777039824 •794493117 ■811946409 •829399702 •846852994 •864306287 •881759579 •899212872 •916666164 •934119457 •951572749 -969026042 •986479334 •00393262 -021385919 -038839212 •056292504 •073745797 •091199089 •108652382 •126105675 •143558967 •161012260 •178465552 •195918845 •213372137 •230825430 •248278722 •265732015 •28318530 32 MENSURATION OF SUPERFICIES. MENSURATION. I. Mensuration of Superficies. PROBLEMS. 1. To find tlie area of any parallelogram. (Fig. 67.) Fig. 67. < -a fiULE. — Multiply the length hy the perpendicular height, and the pro- duct will be the area. Thus if a = the area, ^i = the length, and & = tlie perpendicular height, then A = ah. Fig. 68. b - 2. To find the area of a trapezoid. (Fig. 68.) Rule. — Multiply the sum of the parallel sides by the perpendicular distance between them ; half the product will be the area. Thus if A = the area, h and a = the parallel sides, and c = the perpen- dicular distance between them, then a = ^ ^. 3. To find the area of ant/ triangle. (Fig. 69.) F:g. 69. ^. Rule. — Multiply the ba.se by the perpen- dicular height ; half the product will be the area. Thus if A = the area, & = the base, and a = the perpendicular height, then A = — . Fig. 70. 4. To find the third side of a right-angled tHangle, two being jjiven. (Fig. 70.) (I.) ^Vhen the base and perpendicular are given, to tind the hypotenuse, or longest side. Rule. — To the square of the base add the I square of the perpendicular ; the square root of I the sum will equal the hj-potenuse. e ?• (II.) When the hypotenuse and one side are '"■ given, to find a third side. Rule. — Multiply the sum of the hypotenuse and one side by their difference ; the square root of the product will be the other side. If & = the base, c = the perpendicular, and a = the hypote- Jiuse, then a = v/ h- + c^ b^-s/{a + e) (fl - r) = s/a'-c^ ■-=s/la +>) (»-*)= \^a' - b^ MENSURATION OF SUPERFICIES. 5. To fiiid the area of any regular jJohjgoti. (Fig Rule. — Multiply the sum of its sides by a perpendicular drawn from the centre of the poly- gon to one of its sides ; half the product will be the area. Thus if A = the area, c = the number ■of sides, J = the length of one side, and « = the perpendicular, then a=- . * 33 J- -> Table of Polygons. A = the angle contained between any two sides. E = the radius of the circumscribed circle. r = the radius of the inscribed circle. s = the side of the polygon. • -= Name A R = SX /• = SX j S=RX i ,'2 s=rx Area = s 1 ^ 3 : Trigon 60° •57735 •28868'l^73205 3^46410 •43.301 4 Tetragon . 90° •70711 •50000 1-41421 2-00000 1-00000 .5 Pentagon . 108° •85065 •68819 1-17557,1-45309 1-72048 6 Hexagon . 120° :l-00000 •86603 1-00000 1-15470 2-59808 7 Heptaeon . 128P '1-152381-03826 •867771 ^96315 3-63391 8 Octagon 135° 1-30656 1-20711 •765371 •82843 4-82843 9 Xonao-on . 1 140° 1-46190 1-37374 •68404-72794 6-18182 10 Decagon . 144° l-61803'l-5.3884| -61803, -64984 7-69421 11 Undecason 147^° 1-77473 1-70284, •56347! •.58725 9-36.564 12 Duodecaeon 1.50^ 1-93185 1-86603; -51764! -53.590 11-19615 6. To find the area of a trapezium. (Fig. 72.) PiULE. — Multiply the diagonal d by the sum of the two perpendiculars a and t let fall upon it from the opposite angles ; half the product will be the area. Thus if A = the area, a and h = the perpendiculars, and . -7854. 17. To find the area of a cycloid. (Fig. 33.) Rule. — Multiply the area of its generating circle by 3. 18. To find the area of a parabola. Rule.— Multiply the base by § of the height. (Fig. 40.) 19. To find the area of a common parabola, or a parabola of the second order. (Fig. 79.) Rule. — To the sum of the two endmost ordinates add four times the intermediate !/s ordinate; multiply the linal sum by i of the common interval between the ordinates. The J} result will be the area. Thus if y„ y._., and ^3 be Fig. 79. jr._— I jy-^""^ y, Vz LT. lx MENSURATION OF CURVILINEAR AREAS. 37 the ordinates, A a? the common interval, and fydx the area, then Remark. — The parabolic curve is said to be of the second order, the third order, iScc, according to the exponent of the highest power of the abscissa. Thus a parabola of the tirst order is a straight line ; a common parabola is a parabola of the second order, and so on. 20. To find the area of a parabola of the third order. (Fig. 80.) EuLE. — To the sum of the two end- most ordinates add three times the intermediate ordinates ; multiply the final sum by | of the common interval between the ordinates : the result will be the area. Thus ii fydx ^ the area, then 3Aa! fydx = - g- (y, + 3^2 + 3^3 + y^). Fig. 80. y^ y-7 y* Table Showing the Multipliers for the Foregoing AND SOME other RULES. Vv Vii Vsy -^c. = the ordinates, and Ax = the common interval or abscissa between the ordinates. 1. Trapezoidal rule, . Ax ^^^^ = -2 (y. + y.) 2. Eule for parabola of the second order, Area =^^1^(^^ + 42/2 + ^3) t 3. Rule for parabola of the third order, Area =-^^(2^^ + 3^2 + 3^3 + 3/4) 4. Rule for parabola of the fourth order, ^^'^^ = ^'f{('yi + 32y., + 121/3 + 32y, 4 7y,) o. Rule for parabola of the fifth order, ^^^* = ^2tl^(l%> + '^^-^ + ^^» + ^^y^ + ^-^^5 + 1%6) 6. Rule for parabola of the sixth order, Area = A^ (Uy, + 2167j, + 27 y, + 272y, + 27y, + 216y, + Uy,) 38 MENSUEATION OF CURVILINEAR AREAS. 21. To measure any curmlinear area hy means of the tra- pezoidal rule. Rule. — To the sum of half xv.q two endmost ordinates add all the other ordinates, and multiply the sum by the common interval ; the result will be the area. Thus (?/, + y„ ^-9 "^ y- + ^3 • • • • + Vn-x)- Hemarli. — In ship-building work it is very often convenient to perform the additions in the above rule mechanically, by measuring off the ordinates continuously on a long strip of paper, and measuring the total length on the proper scale. This rule is only approximate, but it is especially useful for getting the areas of the transverse sections in the first rough calculations of trim, displacement, i:c. 22. To measure any curvilinear area hy means of the paraholie j'ule of the second order, or Sinijpson's frst rule. Rule. — To the sum of the first and last ordinates add four times the intermediate ordinates and twice all the dividing ordinates; multiply the final sum by |, the common interval : the result will be the area. Thus A Of* /yj the perpendicular height, and the result by -3927; the product will be the soli- dity. Fig. 97. 44 Fig. us. MENSURATION OF SOLIDS. 16. To find the solidity of the frustum of cu ^paraboloid when its ends are perpendicular to its axis. (Fig. 98.) Rule, — Multiply the sum of the squares of the diameters of the two ends by the height of the frustum ; the product multiplied by "3927 will be the solidity. 17. To find th^ solid'itij of a hyperboloid. (Fig. 99.) Fig. 99. Rule. — To the square of the radius of the base add the square of the diameter at the middle between the base and the vertex ; this sum mul- tiplied by the altitude, and the product by '5236, will be the solidit3^ 18. To find the solidity of the frustum of a hypcrholoid. CFig. 100.) Rule. -^ To the sum of the squares of the semi-diameters of the two ends add the square of the middle diameter; this sum multiplied by the altitude, and the result by -5236, will be the soli- dity. Fig. 100. 19. To measure the volume of a solid hounded on one si-de by a, cw'ved surface. (I.) To measure the volume in slices. Rule. — Take one of the plane surfaces as the base, and divide the mass into slices parallel to that base and sufficiently thin as to be able either to neglect or account separately for the curvature. Then take the volume of each slice separately, and add them together for the whole volume, taking account of the curvature in this addition if necessary. (II.) To measure the volume by the rules applicable to the areor of a plane cxirrc. (Fig. 101.) ^'"'- ''^'- Rule. — Take a straight line in the figure as a base line, or line of abscissa, and divide the figure along that line into any number of equal parts, and measure the areas of the l^lane sections at those points of division by the rules applicable to tlie area of a plane curve. Then treat the areas thus found as if thev were the ordinates MENSUKATION OF SOLIDS. 45 of a plane curve of the same length as the figuie, and the result will be the volume of the solid. Example. (See fig. 101.) 2so. of Sections Areas of Sections Multipliers Products 1 2 3 4 5 5 feet 10 feet 15 feet 20 feet 25 feet 1 4 2 4 1 5 40 30 80 25 Ax 180 = 2 Area = 360 feet (III.) To (Fig. 102.) measure the voluvie hy Dr. Woolley^s method. EuLE. — Take a straight line in the figure as a base line, and divide the figure along that line by an odd number of parallel and equidistant planes perpendicular to the base. Then divide the figure horizontally in the same way by a number of plane sections parallel to the base. Then take ordinates at the inter- sections of the horizontal with the vertical plane sections in their consecutive order, and treat them as follows : — (1) Neglect absolutely all ordinates which are odd in toth planes of section. (2) Neglecting the outside rows of ordinates, double every ordinate which is even in either or toth planes of section, and .add them together. (3) Add to this the simple sum of all the even ordinates in the outside rows. (4) Multiply this final sum by f of the common vertical interval, by the common horizontal interval, and the result will be the volume. Ex. In the accompanying figure the multiplier for each ordinate is shown above it, so that if s = the sum of the products of the ordinates by their respective multipliers, v = the volume, and Aj-' = the common vertical interval, and A.z? = the common horizontal interval, then product of Fig. 102. the 2(8 X A,r;' x Ar) 46 MENSURATION OF SOLIDS. 20. To measnre the rohime of a ')vedfie-shaj)ed solid howided on one ftide hy a curved- mi'face. (Fig. 10.3.) KuLE. — Divide the figure longitudinally by a number of planes radiating from the edge at equal angular intervals, and also divide the length of tigure into a number of equal intervals for or- dinates, and treat each of the radiat- ing planes as follows : — (I.) Measure the ordinates as if for taking the areas of the several planes, but instead of the ordinates them- selves compute their half-squares, and treat them as if they were the ordinates of a plane curve of the same length as the ligure. The result of this calculation is called the moment of the radiating plane. (II.) Treat the moments of the radiating planes as if they were the ordinates of a curve, but taking the common angular interval in circular measure. Example. (See fig. 103.) No. of Planes Moments of the Radiating Planes 105 110 115 120 125 Multipliers Pi'oducts Q _ angular interval 3 8 Volume = 40-1580 21. To find the mean sectional area of a solid. KuLE. — Divide the volume of the' solid by its length ; the result will be th.e mean sectional area. 22. To set off the correct form of a mean cross-section. Rule. — Divide the figure longitudinally by a number of horizontal planes; take the mean breadtli of each of the horizontal planes and set them off perpendicular to a fixed straight line, and at the same height as their corresponding planes in the solid : a line passing through the ends of these mean breadths will bo the correct form of the mean sectional area of the solid. Kote. — The mean breadth of a plane curve is found by dividing the area of the curve by its length. MENSURATION OF SUKFACES OF SOLIDS. 47 ni. Mensuration of the Surfaces of Solids PROBLEMS. 1. To find the slant surface of a cone m^ pyramid. Rule. — Multiply the perimeter of the base by the slant height ; half the product will be the convex surface. 2. To find the convex surface of the frustum of a cone or yyramid. Rule. — Multiply the sum of the perimeters of the two ends by the slant height; half the product will be the convex surface. 3. To find the convex muface of a sphere. Rule.— Multiply the circumference by the diameter, or square the diameter and multiply the product by 3-1416 ; either result will be the convex surface. 4. To find the convex surface of the segment of a sphere. Rule. — Multiply the circumference of the whole sphere bj^ the height of the segment ; the product will be the convex surface. 5. To find the convex surface of the zone of a sphere. Rule. — Multiply the circumference of the whole sphere by the height of the zone ; the result will be the convex surface. 6. To find the convex surface of a cylindrical ring. Rule. — Multiply the sum of the thickness of the ring and the inner diameter, by the thickness of the ring, and that pro- duct by 9'861>6 ; the result will be the convex surface. 7. To find, the mean curved giHh of the convex surface of an irregular solid. Rule. — Divide the figure into an even number of equal imrts, and at the points of division measure girths at right angles to the length of the solid ; multiply these girths by a proper set of multipliers, applicable to the area of a plane curve ; divide the sum of these resillts by 3, and that quotient by the number of intervals : the last result will be the mean girth. 8. To find the convex s^irface of an irt'cgular figure. Rule 1. — Multiply the length of the solid by the mean girth. Rule 2. — Measure the curved girths as if for finding the mean girth ; treat those girths as if tlicy were ordinates of a plane curve of the same length as the figure : the result will be the curved surface. PROPOSITION. If any plane figure revolve about an axis lying in its oivn plane, the surface of the solid generated is equal in area to the rectangle whose sides are the length of the peHmeter of the geH4i- rating figure, and the length of the path of the centre of gravity of the perimeter. 48 CIECUMFEEEXCES AXD AEEAS OF CIECLES. •amuTQ '^ - M cc -r u- -^ l~ X — — — Cl cc 1'^'~ 1- — -H •aiuijTQ 1 -^ ^_ ■M — 1^ — CI ^ -^ — ~ X n Cl X — . t- - X Cl •3 l^ ^ t- 3C b- b- or. X iC X o cc iC cc I- cc t^ o; M c •> < '>\ ^ *~ '— c^j cc "^ tc l~ — " Cl Cl cc w* T— -M t^ C4 t^ 'O- -f X CI ^^ ^— iC c^ cc t^ ^- • •^ /^ «^ . J- -# ^- C^ ^M CC -*i iS 1- X ■^ ^~ cc -:^ LC I- X o ^_ •"^ v^ 2 CE 1 :- t^ a; P C^l iC cc — --. (^ — ^ 1^ ^ .-- .- o; Cl y H X ' w .M u.' — '"' ■" — CI /I •M .. 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C5 ■ri O ^^ rc co c; t- 1-- :c «i :£; cc I- i-- c» o; --H ^1 -*! ^ t^ 05 >-< <^^ ^ cooiO'— •J^JC-^iui^t^CiO'— •(Mi'i-^i'^t— coos 6l ^ 1 i 3 a U c-lx I - ; O Ci '-"^ t^ X i^ 'M c^i X) CO ^^ ip ^ ^1 ^ 7^ 'it* ^ XCsOrH'MCrt-^iCCOt^aOOi— "lCC-#tOt^COCl -< ^ ^ ^ ^ :>^ C^j :m -M C<1 C<1 CM C>^ 7^^ > 9 -f -* r^^ 00 -^ 9 T"' -f- ^. 9 t^ rH cb :c ;m ^1 :^ L^ '>! oo i i t^ 0; -^ i^ ^ ! OOCiOi— IC^CC-*»^^l'^000;0(MCC-^»CCCGO-'. 3 3 -1^ Hx 3 C5 j-1 oo rlj »h 00 -^ -^ i^ c^ t-^ cr: cr c: c^i »c c: ^ ^ ^ ^ ^^ jvj C<4 C^l CM C<1 !M 3— CCi^l^C^^ 00 en ^ !M r: -+I ir: I- X cr. -^ cc -*i L-: t:; I- cr. (MCMrttccrcn^cccc^trttrccC'*-*^-*'-^-*'^-^ c:02 i; ci cci i f: '>! 'c: oD XcriCJCiOOO-H — '-iCMCM.xo — rcici-O^ 00 Ci .-1 CM ^t -* I-': -^ I- 00 en -^ cc -*i -w b- 2 |iCMCMw:cciC(Xccrcrcic:cccicM7^ccco-#-#-* it: •JUIBIQ — "M cc -*H u-: tr t~ X r: — CM r^ -:f u': -^ t- X Ci | . nnmn -^ :c ->c --c •-;; •-::; --^ "-^ "-^ >-- l- i- i- l- i- l- i- t- i^ 1 •I'^^'^KJ £ 2 52 CIRCUMFERENCES AND AREAS OF CIRCLES. Table of the Circumferences and Areas of Circles, advancing by 8tus (concluded). •jinPTrr O »-( iM fC -M »r: -^ t^ GO CTi O r— ^1 -^ --t< ic '■^ r^ -/- '- t-|x T f ""* "-y- ^ ^- ^ ^- i:~ 7"

! cc uo «2 t- rs O C^ cc -:t^ --r .- ^: 5 T^ cr S O CO -5 o o 1.-: o lo o o to -o o CO --r --r --ji iX 1-, r^ fi f:i p: | ^ 1 1 S o ^< •p-< O 1— ('>)-^u.ja;ooai-HC^-.!*HiOwCcr:i— I'lccictocc i '• »ciO<£>otocot^t^t^ooco«Dr3r:r:oco-=^:^ jj c<» (M c<] (M !M 3) CM CM (M o -* GO CO -^ «p c>i ip -Tf< (x; CC' cc ic ri ibcM-HCM^Hcbccocbcbootac^'crj^'Mibajio S orc--0;J5c^c^o;^5^ocoo^c^^CGO^toorooo«Cl ^ 1— icMrc-rticot^GOO'— i:o-*<»i^t^GOO-Hro-+icct^ 0 O O »0 O •'I lO CO CO CO CO CO CO CO t- t— t- t~ t^ t>. 1 ":lx CC -;+^ CO I- — . O rp Cp -* CO I- CO C --^ CC -*! CO t- CO O' cbcbcj:(Mibc^(fqibciorH-+it^r— -^i^Occcbcicr^ >oioiocococci>-t-t^oOQ030ctriCiOOOO^ CM C<) CM -M CM C>CJ CM 'M CM CM CM iM CM CM •>> CO CO CO rO CO !i -^1 cr:co:poaito»0(?— 1^ io»ooio»o««»OiOcoo^D«£>o:r>t^t-t^t^t^t^ < i r^cpcMco»ocoi^a:0!Mcoiocot^^O» -M CM .t:^t-t-. a -< a iOCOC30a5T-(CM-*»CCOCO(3i'-b ci rl-i lb cab A( -^ b- cb CO t^ <6 CO CO cji ifi o c; CM Oiciocococot— t^t^coooGoricric^nooO'— 1 (M'MCMC^'MCMCM'M^MfMCMCMCMCMCMCMCOCOCOCO 3 -I't OC5COCOGOC;cOC5t>-r-^i— it>-OOiOI>-COOOlCCO cJo-^cbcb-^b-^-Tcbcbcbb-cr. cbf:cbibcb(»r^cb S oco^'+ii^O-^i^.-iiOcricocaocMt^'Mt^cMcoco *- 0'-ico-+i'Ot^coci.-i?Mcoococoj;r-(CM'*»ot- !•< io lo o >o "O »o »o io CO CO CO CO CO CO CO r^ t^ r^ t^ t^ s ■i-H r^CO-^'Ol;-C!OOr-|(M'#iCt~-GOO>— l!^l-*lOt^00 ._•! (MibcJor^-rHb-r^'+ib-ocbcbi-. cbcbciCMibcaOr^ =! «o»o»f5cococot-t^t--coooGOGcrs;riCiOOO'— i^ • iM (M •>! 'M !M (M CM "M C^ (M CM (M CM CM ■>! (M CO CO CO CO --' — -f Hoo "M -ro i^ CO :o 1— ( lo ^H ct GO d ^H ic^ — 00 CO I— 00 co o CO m Oi— tc^i'rtocoaoci'— ii?ci-i^»oi>, >0 »0 iO »0 >0 O >0 lO CO CO CO to CO CO CO t— r- t^ ^- b- -< -+c s o 3 l--:^p'7HCO-*COt-CiO'7lCO-+CO^-cpp.-ICO'*l <— 1 -ti CO -H -^ i^ O CO CO o CO CO c. ■M b cJo »^ IC2 lo lO CO CO CO t^ r- t^ 00 GO CO 00 en Ci n o O O — 1 M 'M C^l C>1 CM ^1 •>! CM -M (M CM ! ^1 C^l iM CO CO CO CO 3 O »b cp

i o lo o "O CO CO CO «- t^ t- t^ CO c» CO rs rj o o o o -- ■- (M fM !>« Oi Ol '7-i 1>l (M -M C^l -M C<> (M 'M Ol iM CO CO CO CO | ^ o .itimtQ 1 O —1 C^l CO -H lO CO t^ 00 Ci O — 1 CT CO -H iC CO «:- CO C. ajcoooc»cOGOoocoooooc5C5r;ctcr. cscsoio: CI JlUGt(] 1 CIRCUMFERENCES OF CIRCLES. 63 w H O 1— 1 o ^ t> n crT M o o CQ a O H H 2^ O pa MTOBia O r-( (?ci CO -* u: tOt'OOCiOr-iiMCO'^OCDt^QOCi •JtaBiQ; 1 3 o MTn 9 -* o «C C<1 X -* t- 05 O CM CO UO C^ CO 1— 1 »C ~ CO :» g; -7^ CM CO LO jq lb c; CM lb :Jo OCOC^IX-^OCCOIX-^OCOIMX t^XO'-iCO-TfCOXcrip-CO'^iCCl^ I— I— .CCO^XlMCOOl^C^COl^'— 1 CCXOir^C^COlCCOXC-. piMcoo rH-*b-rH-*i^Ocb-Ocfi:bjbc:cM :^l(M(MCOCOCO-«r-*-*-*iiOlOiOCO 9 1 S o 00 2-5133 5-6548 8-7964 11-9380 15-0796 18-2212 X':*'0«0(MX':fO«OcMX'9— 't^cocnio XO'-^OOiOCDOOC:!— lfM-^wt-C5 -*OiCOl— i—iiOw^COXCOCCO-^X 0>-iCO^CCb-xpfHCp^cpi;-Gp 'H^b-OcbcbcncbcbcncMibdi^ C^(M(MC0C0C0C0^'^^iOlO»O«O b- o OiC— it-COniO.— 't-COCilQi-^t^COCSiar-lt^CO lOccX'Ci — M-!+— ( OCOCOG;(MCOC5(MlOOOr-l-*X<— 1 CMCMCM "f r- •9425 4-0840 7-2256 10-3672 13-5088 l(*)-()504 OCOCMXOOCOCMX-^OCOO'IX ^■J CO lO CO X -^ " CO -t' CO X 31 — ^ -M CTiCOt^— iiCO'*XCMCOO'+iC:CO ,..- I;- cr: p CM CO ip CO t_- c: p C-1 CO ^ CO • jtCMcboiCT'llbcX^^-XrH-^l^cb r-<^^(^^:McococOTt(^^lOlOlOco CM CO Oi »0 1— 1 b- CO OiO»OCO CM I— 1 -3142 3-4557 6-5973 9-7389 12-8805 160221 t— COOilOi— ib-COCllCi— ib-COOllO coocoooO"CO-^coxc". .— lc^^':t^ to O -* OO CO t^ rH UO cr. CO I- CM CO o T-iCOTt1 -^ ip I;- CO «b Ci CN lb ^^ l-T! '-- Ot-H C^ CO Tf UO tocMX-+iocco;iX'*ioccoqx-^ cr. rH (M ^ CO t- cr. o c n o M o o m 1— 1 O w H P^ O W H ■jinrrrr 1 O ^ (M cc -*- u: -^ t- oo r. O -« -M rc 'rt- u': -^ t- cc c: jLUbi^i , -,j 1-^^ ^j (M n c^4 :m ?) 7-1 -M cc « cc ro M re re cc cc M • jmBtQ 1 o a 1 9 CS^'MQOlOt— CT. O^1CCiOr>-CiO^t~t^t^COOOOOCiCiCiOOOOi-H,-Hi-i;<4'M CO b- ■A E a B d qo (M CO -*< O tC -M 00 '^^ O •-£ 'M CO '*co'M?c^ieriret— '^tco^ooo^r — locioe rp Tf- :p b- ?5 O -7" :c -* ;p t- — . O r-, re -*- --f b- 30 p ibcbi^'+it>'^-+'t>.o?^=c^-ceic^'7^ibxirl-ie i-ii>ro5iia^b-re:ri»Oi-(t- -^■M^»ob--r:o^ireict^ococ^ic»ct^QOON ret^--»ocirecc-M:co^oorot^"ioc;ceco-COC00055aiaiOiOOO»-i,-l.-iCo ^cxjO — re-H--r>30C^ — -^^-*'tccor:^re-*ltD^- -^».eO'^co7'^^^o-^r:ret ^lecr. -^-oc-Mtco b- CO p -y- ^1 -^ le b- » c: ry- ^1 -+ »p ?r a> c; ^ •>! '^ -4Hb-i^Tfb-ore:bi:>ii>d:'>i»hcb'^^!i)r^^ COcOt^b-t^COGCCOCO". c;~ooo^i— ^I^1?l 1— lr-l?--i— Ir— li— II— 1^^ ■■f »p CO':HO«0(MaO-fO'0-MCO-^ CM ^ ^ b- c; o M -^ ^ t^ CO o ?-i -* ue b- ao o 'M ce O'^oocM^ — »c~ret^"--co-*co-Mio-H»ec; -t<»o:oco?ioo'Mre»etpx)r;--i>ireo;pqpcip '+' i> O re tb o re tb r; ->! lb i 'M >b CO 1^ -4t< b- o -^ ?£>«Ob-t^t^aoaoooxir;r;cr. ooOi-HrHr-fM-M ip ;«oOao-*rccocMcO'*< V2 C -t< '+I -M 1^ — iQ r; re t- -^ O ?^i re o ^ b- r: O ^1 re ip tp b- r; o ri re -* tr: b- 4< t^ O CO cb J: '>! 'O c^ -f-1 ic i) rl-i 4i X '>! -^ b- o re jM --f re -« :o to CM CO 'f o to n c» -*- O ^.-Ht^C50?i-f»ct^coocM'*iot^cooci »b do -^ -^ b- o -^ b- o re totot^b^t^t^cOQOcocinrvooo — ^1— c^ic<) V V CCC^iOi-Ht^CCOlO-^t-CCOi 1 0'-ireiotoao-H^re-^;£!t^O'-'b-'* O -ti CO ei to e-i le cr. re b- — ( to o 'M CO -M to o >o -^ tp t- » O r--4 to -+i ip b- CO O 7^ re '+ »e b- CO p -71 ^i Mtb^3l<^5tbci'^1»bc«f^-*oo1^'+lb-oreltbo^b !0«o<:ot^b-.t^aoaoaocic;c5000>-i— i^c^i-M »— 1 -HOro3;o-^c;rec^>e-^b- to t^ rs (M -f ~ r- CO rj re >e b- rs N cc i.e t^ -+i CO c-i t 1 re t^ M tc -f CO :m t^ .-1 »o Ci re r-i e-i -t< ip t- CO re — •>! -f ip b- qo ?i 7- 'M '^t^ »p tp 00 M tb Ci (fi lb 00 fi 'b do -^ -^ b- ^^ b- M «b c: e-i tOtOtOl^b-t-cocoooc^CiCiOOO^r-ir-i,— >;<) r o OtOlMOO'i'O^O'MCO-fOtO ojco»o«oaoo-^re-+itoooc:r-o cr. ce OD (M 30 -— 1 e-i re ip to 00 cr; r— -M re >p tp CO c: p ?^ re »o C'l »b c; c-i »b c« f^ -+i b- r^ -^ b- re tb c; re tb ci :m t0t0«0t^t-b^000000C5aiC:OOOO— i.-t.-i'M c juiCTQ r^^ ^, ^, -,, ^, ^, -,,-,,-,,-,, ^ j^ j^ 1^ -^ j^ j^ „ j^ j^ jureiQ CIRCUMFERENCES OP CIRCLES. 55 a o o H H o 1—1 M 12; -«^ t* o t^COCi •JUIBTa 1 to a a d -. Ci cc i^ — i-o — . -*i X) M o o -* Ji CO t^ ^ lo crs -ti CO '^Ot^CwCpr-iM-rt0'7l ctot£it>>i--t^GOcoco C5 o o o S-c r-(i-l ,-,.r-.«r-l ^ >-(i-li-li-l,-( r-1 ^ l^OiO- Ci O Ol <* OO'*'00t'5t--^iOCir0t^CM-OO'^t-^!M00r-i»0 ooO'7'<^)'?'»p^;~^^'7^^i'*''Pi;-co?*T'^'*»p (MCOCOCO-*-*l'^^tilO»CiC:r>OOt-t^t^COCOOO t;- ''P -*i3irot^'-t»ocicococ>i^O-+"Cirot^.-iiocico u5-x>oo9.7-ic-<:o»pooocir--i(^irpip;paoc50c;j t^Ocb'iocbocii^oao;r^ibdo-H-*ib-0'*b- CMC0I0C0':t<'*i'*i'*OOt^t^t^C000C0 »f5 K) CO i^ ^^ o O -^ CO T1 ^ o lO ri ro t^ .— 1 »o O -* 00 M CO "?* iM "X> O -^ CO •>! t^ .— 1 lO Ci CO l^ 3^1 ?r> O -+< 00 iM X> 1— 1 C50 0'1?p-+lOl;-C:)pT— iCO'+iOb-aiO'TtCO-^tp C^COCOCOCO-+i-*-:tt-cr50J «bc5>iibr;^iibcb'^'*iciorl-t-+it^ocb«bocb«b t>1 C^'M?0«>.0iO'M-*iab-G0O-*'C0lO«O00O'— ICO C5co»>..— i»oO'*ioo'M«r>o»ocicot^<— iioo-*oo <^^'5H^p^-qop'7^3^l-*|»pt.^aop'H(^^'*•»p^-ooo od5^iibcb(fiibd0'^-*t^Oebb-ocb«bcic.oooo T— t t— (MCOO-^OOCMOi— ciOCSCOt^— lOO-^OOfMCO C5r^0^^ip?pa0^T^(?^cp«pOG035r-lcqc0i0cp »oci(M»ocoi— i-rHt^T— (•rtai:b«bocoO'*oocot^.-iio «pOOC5p^COOcpt--Cip<>^CO»0«Ot>.050CqcO •ooo— HiooOl— i-^t-ocbt^ocb«bc5i?*»b6i(>i»b s^7-'»cococO'+i'*'*-i>.i--ooco •joinia: 56 CIRCUMFERENCES OF CIRCLES. •junJia; O C5 «0 CO O CO ?0 C5 O (^ t- t- t- t^ l^ t- l- t- t- t- -linTJIQ CO CClCt^OOOi— 'C0O?O0005i-<0C-<*C0l>.0ir-l<>J-*< G<)tOO^OiCC(>-i— lOC5COCO(M«00'*0:>CCt^^ cc-^ot-ooOfTirC'+iot-ooO'— (cc^iot-ooo c o o W o 1—1 O M o o < t— 1 O c o Cm O 03 0) s y 1— l-^t^OCCt^OCOCOCTiCMOOi'MlOOOr-.'^t^^ OiClCiOOO-— 1.— (1— 1.— ('>l!MC^1COCOCC-*-*'*iiO r-(f-l.-('MC^lC^I>lS.QOOJ G^l CM !M (M !N Ol <>» r^3>"lCp>ptOCOCir-H(>lC:t Ocb?bOMcb55(>i»bao'Miba)rl(-*b-o-*i>-o ^ ,_l t-H I^ CM (M C^ (M 3^1 CM 3^1 Cq 31 0>j Ov) t^ r .-l«-:fC0t-C:--l'M^»0t-.aiOCMC0>0t~C»O.-^ GOfMCOO'+'COCOt-i— lO:J;COCOCMCOO-*ODCCl^ co»ococociOc>iccocpt--aicpcM:p>p«ot~-aio ._-, OMOS^cMcbcJi:>i»bcbr^-*c»-lt-+ih-cb:bi>o ! • ,H.-(i-i.-l.-MioO'*coc^itoo-*'55cr:t^.— 'ic pc^^CC-*Ol;-^iO'M^C'*!pt^-C^p^-lCO'*^«Ot^ j_^ Occcbc;cM»bdoc>iibt3oAi-+iiV.o-^t^Ocb«b:r. i • OiC^CiCiOOOf-i'-i'— I'M'MCMCCCCrO-*'*'^-*' ^ r_i ^ r-t CM CM 3v| CM Cq 3^ ?^ ^ CM 'M !M CM -^ CO'^tOb-Oi'— (G>1-+il0t^criO30t>.Q0O'— IM iOOicct^i— icoo-*'i?Oi— iioc?irci^i— ioo-t< t^C»P'-ICp'*tpt--COCDr-HCC'Tt<«pi--CX)pr-(rp-r^ dl1'^ipb-aOC?lf--( Ci(M»boodiiba)i^4tHi^Ocbr^ocbcba5(M»bc5 cx)OlOia^ooo^'-|^-lo.-^^^lc<^cococooc-*^'*|'* i-( — ( rH i-H 3^J ^1 3^ 3^1 CM 3^ ^ O'l 3^ 3^ 3<1 3-c»O^>OCOG001>— i3qcC»ClDa)C5r-iC^CO»0«OQO 3irc»p 1—1 p H ^ ,»( r-i f-^ 2^1 -M !M 'M >! S^l -M >l W 3^ 3^ !M 30 t- QO O f?5 CO »0 ?0' ' n CO t^ 'M ^ O -H CO ■>! t^ 1— 1 >-0 r: CO t^ ■n to o -+< CO n ' -* i-l r-t i-( !M C>1 S^J 3>l C^J 0>1 -M 3^ S^J 'M <>J CN '*»0'y3t--QOa50'-< CM Cv< CM CM C-l C^4 C-A <>t CM CM CM CM CO CO CO CO CO ' t- Oi ocq^iot-oooco CO >0 CD 00 O I— 1 CO CD CD rn ; cq «o T— 1 IC OS CO t- 1— 1 CD 00 TtH CO Cv, CD 1—1 »o a CO t^ l—t t^ •? «o oo Oi o c.-*COt~Cii— liM-^iOt-OJOCMCO ^lOOSCOl— CMCOO-*00500^Cpt-a5pl^lCO'+icpi;-aip'^CO-*CDl;-Ci l^^cb65C^^bc»c<^lcoOr^-^l^O'*r^-ococbaiC^^ ioio»ocDcocDt^t-i--coooGoa:ciaiOOOO— I IC^*qCMCMCMCMCMCMCMCMCMCMCMCqCMCOCOCOCOCO 3iO(M-<*<»ot-ooocqco»ocoooO'-ico-*coooa: 3it*JiociDr-H OlOkQCOCDCDl^l— t-COCOCOOOCtOiCSOOO^- CMCMCMCMC^CMC^lCMCMC^CMCMCMCq-a;cMCM-*»oi— i lOOi->*(OOCMCDOOOaOCOt^i— llOOiCOt— CMCOO'^ ojptNcoJOCDccc; pc^coiocpb-c;pc^cp»pcp -^^ 1— 'OOOr^'J^ib-Ocbl^-OCOcbasOqOCiCM'OOOi— ; • OlOlQCOCOCDl^t^r-ODCOOOCOOiOiO^OOO— ' ! C^CMCMCqCqCM(MCMCqcqCMCMCMCMCMCMCOCOCOCO , -*'*ioi>-a50c-ico»ob-ooOr-ico»Cicoooa5i— ICO C^OOCqCDOiOC^COl-^1— .ICO'^OOCMCDO-^CSCO cot>.cipcqco-:t— (■^b-r^->*it>.ocbcba;cocboic^»ooot— iioooi— I oioiocDcocDt— t^i-~t^oooooocia;a;000'-' C^CMCMCMCMC^CMCMCMCqCMCMCMCMCMCMCOCOCOCO QOO.-ICO-*COOOOii— l(>^"*COt^aiOCM'<*bOC0t^00C; j-iucii^L ^ 30 00 00 00 CO 00 00 00 CO GO r: C. C w: C5 Cr. Ci C5 Oi C-. •jxn'ETa 58 AREAS OF CIRCLES. 1 I 02 rt H O t— ( tH c Q < o M o o CO H O ■jxnvid *-* '"' c^ico'^t'otot'cjocso— i?ico-tii-otot— cori '-jro^jfT 1 75 r; 1-1 ^1 IC CO :0 CO 'ricr5-:t.coiocotoioo tbi^iX)b-b-ciT'i-o:b'^0-^-^cb-+iTlHOFl-( rHi-l'MCO'^tOt^Cl— ICOlOt^Ci-MlQOO— 1 T— 1 --H 1— 1 r-l f—l 7^1 IM O) CO 30 < 00 CO CO o -* ip »p »Or-ltOOOa)t^COOOOOriOO CS O ^^ -*H t> C5 M -rH t>- O I— 1 I— 1 1— 1 1— 1 I— I CI !M 'M CO b- ■*ri0^:r^t-o;0wcoocot^^i>-:o0i>>to"o ro ->! t^ t;- cp o '^l f -*< =p s^ fi '-p ^ t;- o O O to CO ,r-l 1-1 C^ CC Th »0 t^ X) O -M -*i to Ci --1 'tH t^ O i-ii-ii-ii-i,-i^i?ic^ro t— •o t^ to 00 1—1 C>1 o COt^O— lOt^— '-*<»a'^— 'lOoOCiOOiOCl-M noociOM-*'oo-Mt^coot^iO'5M'*iiotoci Ot^ — cOT-i-ooDoo-:*! to to ^1 -*l i-l -^ -CI t-- t- ootb-*i"+'ibcc-fic»ib-?Hvbb-Aitbcb'H'^ i-l^'MCO':t-— HCOCOl— 't>.i-to -Iti c: o :b CO -^ to O -i CO ^1 CO o i) M o oo CO 1— lUCO-^lOt-OOOM'^tOOOi— I'^tOCS —1 1-1 1-1 1-1 1-1 n -M 'M CM 10 • -+f to CO (M CO I— ( C5'Mco'MC:koooc:cc»Oi— i-*oti— ^t>.o— 1 CO Ci uo :m Ci X t- t- CO O CO to O o 1— ! t- lO CO n t- O O -O O — 1 C5 -rH I- to -M to to rt^ 00 O C5 iOp7-ip-7«p'^COppb-pC»'>1'7Ht;'0»0 "jfi Ci ib 'fi ">) CO ;»< ro to CO O r- O CO I— 1 t-0000»OC5'M'MOt----*i-^C1COCOtOtO -+I CO M r-i -M CO to C^ -M t- CO Ci -OJ -T*- C<1 CI CI CO lO lO CI to t- lO O CI CO 00 CI CI O LO b- to CI lO -H o o p "T- O) -1 ~. CI CI 00 Ci to CO -o p p o •^ob-^ci-^r^-*it~cbo:bi)Ocbi)ih)cbci r-lCOCOC; — -^t— '-*'MC5'^tO»OC1iO»OC) 00 p » CI -- t- CO ^ b' O CO CO CO "^ -^ CO r-H ip CO ci> CO -^ O o CI to "^ i) to -o i) -^ to CI o c; ^HdCOrtiOtOGOCli— iCOKOCOOCOtOCO 1— 1 r-l 1— t 1— ( CI CJ CI CI CJ 1 • •0078 •9503 tOtO>OClt-OOC5tO-^-*''*COOOOOGOt- C0t-C100-tU0C:C000-:t<-H to -+I O •M M ~ CO CO -^ to C5 CO -^ t— 00 »0 O CI -+I lO CI -+I CI iO »p p -71 t- c: t;- -;- p ip to CO ip cbt-cboc:c;-^i'ootb-^-^'OCicbc^t^to 1— 1 C< CI CO >0 to 00 Ci •— 1 CO »0 t— CI >0 CO rH i-( 1— 1 ,-( CI CI CI CI 9 O -H to to -f o -t< to to -*< o -^ to to -^ o -t< to to -M O 'O 1— 1 c» 'O <0 -f -*< »0 t^ O CO r^ ■M :c >o M 30 rs Ci Ooo-titotocor-aoto— i'+0 CO ,— I r-l 1— 1 .— 1 >^ C<1 C^l M p MOI «!a o ^ CI CO I** 10 -o I- 00 Ci 1-1 'M CO »:*< »o to t>- 00 c; ■jnnnQ AREAS OF CIRCLES. 59 H O llOiDt0CC30T-HC<53.0000iOO cc:Ci-Hcc;r>tCGOi— tiCC^lOiCiO'MtOfMSiaoccc «rc-*i'+i'*iu5U5?c^»>t^i>aociCiOOi-(i— ?^ C^ iC CC OO O — . O X) -*i ?C tC !>! ;C t^ iC C -M r- t- ^-l b- c^T :^^ qp O b- rH :^ '^ -* O (^j a; c>l ,-, ix: --c C^l cc --I cc?tOiit?c-*0'M'Mc;rc-:j''Mt-c;co-faox--c re 20 t^ T— — t^ en to c; t^ 'M ■>! GO r; ?c c^ X iM c1 b- (>• !>1 -* -H '^ (M to y? C^l cc O re C^l — ce tb -^ t^ ue "* »c oD ! o o — le'-oo^ct^.— (joci'*xrf5ooccoo-«*ic;'0»-^r^re X I— ( t^ I— 1 re re ,— , h- 1—1 re re f— 1 1^ r— 1 re 00 -^ ^^ X i^ ^ tc r^ X) ^^ ^ X re X «C :M Ci Ci Ci r5 ;^' re re ~*t »c -* wt ,-^ ^M «^ ,-^ C^ ^, -+i cc ».e '^J c c^ ■-0 t^ -i- I— '.e CI -* ^c re IC -*< X X re -i- ^^ -rf le re t^ re — ^ ^-i re r^ ce Ci c^ ^H -+I c. tc -^i -* le ce -0 * re t~- >fi ci re X2 t- -M X re X — 4« ■M M re ro -*! -* U5 »o 10 CC t^ t^ CO 00 CJ w« 1—1 T— I ri I— t .— i 1— I i-H ot^rctct^-sOoocfic^ifOiMasiooooico 7qrHe'ireuex^to(noow;iM»-HOO>-i'*i!X)Ci >eixxoci055-*i>>«£>reb-oo«£).— iccJ'iQO^ x^pp'et-:pJi-*OX)oo ^uer:ro-oo-*ixret>(Mi>-?'i»>>c<)x-*iCiie rorere-Ttt^t^oox)CiC:00'-i -*i.-irHxxtocei^ooxioc5(Mueo lOXiOLeotcreocnxt^oociS^i— icr. reciiO'M tpreioret^t-e^reO'MO-^cociOtcdt^OO « -i O re -fi ?e >h ci 'ii "^ Oi Ci o '^ x ^+1 M e-i ce c^i o c; M -o o -* X !M t^ -M !£) I— 1 t^ (M t~ ce Ci ^e — rerere-^^ioiOLe^cct-t-ooxciCiOOi— 1^:1 re to— iicvc%s-*ic:ire'*'*e<cio-^reioci-+'c;io?^oxt--i>. t^ r: t- re -o '^o (M i^ X «5 1— I re ro c; re -rhi o^i io X t-. -+oretc«pr-<'>ixpx o^it^>iCixc^r^4-rH4— («Ot— lt~3^X-^0 rorere':t^-*|'*^lOo;J^«cl>t^coxciCiOO^— •>! |'*c^c<)c«oOwi«o 1 iCit^t^t^OOClC^OCi'^OyS'+iS^OO'— iT^iCt- ^_^ jM^»opT7-i00pX)rHOiC^^-*^1^»-Cppt~- ' * 't-f^.rboio-^ocbOibT^CiSiOMb-re-^oo , — <-*x.— !ioc;ret^eqt£)i— i>oOiO'— iiO'^JX-^o cc^5re-*'^'^»o>o«ocot>t^xxC5C; 00 — '?^ ^- ^j 1— I ^-^ i_ I o -* «c to '*< o~H^"co «o -^ o -^"^ «o -^ !;> ~ I Oi— iretooiQOtoroi— ioaiviO?>i«5xrcxri tocoeet^r^t^reio»ei(Mtoto-*OT>"i.— it^-i— ii— ire I O |'7^^'7l'^xx)c;>eb-ipx^^!^^re^:'7^x^^';-. >c ' -* tb o »b >i o o 7-1 »b o i -^ -^ 10 b- ^1 b- 'b -jf* -th i ^H'^ixi— lloc^ret^^— ijooioo'oo^'— t^MCi coceoO'*'*i-+iio»Q;o:£)t>.t^QOcowi3iOO^'— i I I— I i-H ^^ 1— t I JLUenJ^ I ^-J ^-| ^ Jy-j J;, ^ jq -q 5q J,-^ jf; j^ j^ j^ j^ j^ j^ j^ jr; j^ .llUL.l(J^ 60 AREAS OF CIRCLES. -rtl-<*4-^'*-*-^-^-*'*i-ctilOlOlO»0»OOlOOir:0 H o (X)00->tOc»t>.(N(>^cpC5i--O OOOC^-*— ir-cOOCiOCOt^t^COC^O^"''MCO-rt*lC?Ot^OO ^_|r-(i-l,-l,-ti-H,-HrH^3vj?I.MCMtM!MlM.o-*-*iaeo r^ lb <>i Oi Oi o i'l t^- "f 1 O oo j; -^ 4)H ci -^ -^ sb c; o«ocoaio-^r-Hoocr>'+'— iC^cosO'rt^ccc^ii— 103^ CCCC-^-^iiOOt^l-^OOC^OOi— '^<^(X-^OOl^t- ,— (r-H.— Ii— I,— (,— (1— ii— 1.— li— i3>4CvIC^1(nC<4C^CO->!ti-*«OCOl--t^OOCVOO — 'MCC^lOCOOt^ ^ ^ ^ ,-1 ,-1 r-l .— i —j « .— -M '>1 'M -M C^l C^ C^l !M C^ C-1 (MiOOSCCcO-*'— iC;i:^eOCOl^Ci^-*00CCC:iO(M U5 m o^ t~ 00 i^ cc ir: ic 'M w t^ ic ^ CO (M CI c^i CO <-H 00»C— lOOOlMC-. l^'+i'MOCC^'r^CO.— lOOiOOOO O-lCO-^^lOOOl-^OOCiOO'— (MCC-^OlOOl^ rH i-( r-H i-< r-i ^ 1—1 ,—1 I— ( ^ 'M '^ J ?M 7-1 3>1 -M J.^ (M Cq C^ o Em o H pq H QO'*''-Ct— oOrHC;cccccoCiw>cocpOCi»poch OS >b ij^ -^ r^ CO i'-- c^i i) t^ cb ao r^ o '^ c: OD a: Ai i^coot^"*— iGOioccooonC:-*ai?rCO-— lOlOOCiOC^llOOlCOOO-rH COCOl^-'^COOOO'Jfl^t^CCt^OOl^I^-^CCOCC'^ (>J.— ICDb-CC«pCOt--CCi— lCl'7H«pi>-CO>p c;cbao>h-*"^tb6^'*iAa;aooi<]t^:oo6iOOi 'OCOCiCCCCOt t<(MOl^«^'*'MOC;00«CCC»C 'McorC'^»n«oot^ooc;050rHC^^cccc-^iocoi^ r-lr-Hi-lrHl— |i-If-Ii-It-Ii-Hi-I— I CO 1^ -^ CO 00 30 »p 1-- -^ CO t~ C>1 C-l w li (>> OD t^ i^ c; ?^i i^ ^7 -^ C "^ -*< 00 '^ f^ O r^ CO <© o to i^ oj oo ci o ^ 'M ^0!M^IOOC>^CO— icooot^ COW-^C^IlO-^iOVO^lOb-tOaOt^iTHC^GOOb-OCi CO o >b i?-i o o '^ "+< oi >b 00 o-i CO CO o o CO Ai (f^ CO >oc^coo'MC:cDcooooco-f<'>40ait~cDic-*ioo 'M CO CC -* »Q "O CO t^ OO CO Ci O >— ' 'M "M CO -fi O CO b- ,_,_,,_( ^ rH r-H .—I ,—1 r-H r-i --I -M 'M !M •M C^l •M -M (M C^l •JIUTJIQ O ^ (M CO -^ O CO 1^ QO C; O — ' 'M CO -^ »0 CO t- CO Oi {•uuvirr ^ -+i -^ -t< -+i -^ -t< -^ -+i -}. IC «o "O to IC »-0 «c o to O I -'""^•iU. AKEAS OF CIRCLES. 61 H O o jiuBia 183 3-3 '^'O^^t^aOOSO— <100-+<'MOC5C5O-H os'+'trio *-Tf— .ot-io— .jcico^^fNooo ' C0rCCC05— iOO — 0^-^0 7-105(M— 110»0— 130 S^05t^oa)0»b'^30b-aoorbo5cb'^'*«bc5-* . ^^OOOO— '— i?^>irc-^tr;t^oo07^'*<«ooo— * ■ 05 0— 'MC0^O->0b-00 0;O— 'M^iOtOb-OOO 1 1 05 ^ "^ t 00 2908-841 2999-630 8097-492 3196-924 3297-926 8400-499 8504-648 8610-858 ^flOb-iO-*'^+1COO;CO-*ioo5q5-*^ro^-*C5oo-* b-tctbooi>ib-'^^(rqco«b'^ — i:>ico-*iccb-o5— ^coiot^o b-00050— I'MCOO^b-OOO COCCCO'*-<*<^-+i'et<^r*4-HO 00 b- 2893-798 2989-981 8087-634 8186-910 3287-755 8890-171 8494-164 8599-716 iC-^'^Ot^OiCOt— C<|00'^— ( -*-+!— iu;^-*iOCOCOoO(Mro • 30»p0O^OO-~0b-'*lCOi ' iibuib-^i^i^iOo-^'+icb ^ O-^^-ICOlOwOOO'M'^5000 1 t-30050— '^ICOiO^Ot^OOOi COCOCO'*i^^'*Tf-*-rt*'^'*i •6 2884-262 2980-247 8077-794 8176-912 8277-600 8379-859 8488-689 8589-090 toco:iJ03<10b--*!noO-^ OO— 30 p:pb-^*^-#QOaoroTti— i-5j< -o-^-*i'-bo5-?«oooaoo5Mi) *? 05 0— lMCOiQl--aOOC<^lf^l- •000050-^COCO'*l^£>b-0005 Cr5CCC0-*'-*-*'*i-*-*i--H-^-*i •5 2874-760 2970-579 3067-969 8166-929 3267-460 8869-562 8478-285 8578-479 COOO-tf— i05t^;O«Ot^C5— J^ 05t^COtOiO'MOt^»OOCOC^I >1?pO— 17-105— iq5JOC00005 ib:b:r:iooocN05?b!i>b-C5cv5 00050— ■M-:?Db-050 — c^co^iot^ooo: C0COC^'*|-+i■*'*--^-^■^■rJ^-t< i 1 1 < "? I0t^05:0t ^t Ji— '05 00t^050!^^OQOrOOO':f O?0Oi0rC300500-*l-~0-O?^b-05t-CMlO'*-^b-ib-^»bb-.^ ;CtDiOOlC»OtCOt^0005 0— 'rC--+itC00O!MJO 00050— i=.-aD05 0— (rc»rvb-05-^co ^t^0OO5— (->T-f^,+,ir>^Q005 cocococo^^::^:?^^;^^ 1 !>! — l05t^O--0b-CC— i (N ^ OO l^ CO --o tC -*i C0:p»00— il^pb- tb T^ 30 b- b- OO .^ -i -^-!iHco:ococo-+i-TH 30050— I^ICO-T'O (MC:t0050cMi>?5o;i)dbo-+' -+itaob-3005 — o^-*i«Oic:— 1 tOb-oOOsO— 'CO'+iiO^t^O: COCOCOCO'+i^-^-^i^-^H-TH-^ri —1 ' 9 2827-440 2922-473 3019-078 3117-258 3216-998 8818-815 3421-202 3525-661 0050— i-*t^O>OOb--^— 1 0500iriO-H05lOb-t^COI^OO oob-— COCO-*U2t^OOO— iCOlOb-O tDt-00050— ICO-^lOCTJt^OS COCOCOCO'+l'+l'*'*'*!'*'^'^ P •.iniBi(i O^^S>lCO-#»OiOt-C0050— ICi p; M o ;«! o ci 1— ( w El, W -»; f^ Cm — ieoc.t^t— CO t- 03isoo^oo-Mioai'**Ci»Q30 O — ''MCCiCtOt^aiO-^IC'+'tOt^OOO^CCrHtOOO t;- o c0'^>ia:'7^a;cc:pcoaitpcoccc:;0»o:oc^ ! i-i OS i a: -^ 4< o -i la 0 .— 1 -f O -M ->C r^ to . H -t- O -* L': »pco:opqi-^jp;?^-^H'>i;o>pcp-^c»Oco->j~to :r;-oibtbt~-'^tbcc-^'^hotbchccccioib X)^-*il^O-*t- — lOCSJCt^'MtC'-^tO'-tO^Jt- O^lIC'+i-Xit-OOO— '^-^JOl-COO^CC-^tOt^ iO lO lO «o »c »o to --0 --o to to to --o to t^ l^ t- t^ t- l^ O "f lO?iaOt^l^OCO'>100-fQOl^-#?M— ^'— I'-'OCm lO-Mt-Oioo-^coooioo-^OOcccotoaiOaocc aicptoootopai'*i»p'>-j-^ritp'*ia:Otp3-. top to 4tH ■f) '>! -+i rSD •>) 3t t^ l^ C» ^ >b f^ O) CO OO o 4- o I— O cc to :j; c^) to cr. cc t^ --I to O »c :7i -^ :j: m o to 0>irC'#»ot^ooc:r-('M-t-ioi-ooa:.-i'M^tot- »oiaioo»n»Qio>ototototctototot^t~t^t— !>• -* « o-^-*'*b-^^toa5t>--*'Mr-ito^^'— cctoO'^Ci CC m '^ O CC -*i -— 1 to l^ to 'M 55 ^ 0 ^ CC ^1 CO ^J l^ C» -*! to -# l_-- to r-l i>1 CO p GO r^- O "O to <>) -*i "^-^a; dir^-i0>0»CtOtOtOtOtOtCtOl-l-t-l^b- '■ 9 -*i-Hooto>n>o"0-«*o.-HGOO-#c(tcit'* CC':t<'-tOCOt^CCtOtO-^OOOCO-+'tOtOCOI^aOtO --t;-o^tocrji»ccc«air^(»i>ipioip'7-icopcp'>i a: lb cb cb "# t^ <> I do lb lb >b cb '>i b- -^ cb w: ih 00 n5 CCtOO". CJ'!t •bcb'^O'^^cC'^'fq-^.l'cb^-i^-TCioocbJiWb- 'MOOOr-i-#t^O'#GO'MtOO-^C5rCQOCCGO-*a5 O ,_( !M ^ lO to CO o; O 'M cc "O to 1- c; o fM cc »rj to OiO«OiCiOiO»C»OtOtOtCtOtOtOtOl^l-~l^t^b- p •JIU'BIQ o— ''Mcn-+<>ratot>-coc50'-«otot^ooa5 CO QC GO CO o) QO oo CO CO GO a; Ci C5 c. r; Ci cr; cr; -r. 05 •jniBifi CIRCUMFERENCES OF CIRCLES. G-:i jTU'Bia: Oi— i2^fc-*ocrt-aoc;o>— t(Mcit->#»c«ot>'Q005 • jratJiQ Si O 3 '-'1'-' {»o--^rC'*iOt^<»0--i<^^'*»p^-GOOT— c-i'^ip M|?! c c en M cm 1 MM 1 OCOOqoO^O«OC^OO'*'OCC+i'*'»o»cio«c -12 M -*Oi:0^COO'*OC ^12 >■ P O PS M o P3 R H C pi |:0(MG0-*O«£>(M00^O^-< ic Oi CO oc iM i, n ■1 I— It— ii— ((McMfMCCCCCCCC-^-^-^OOJOCC laO^O^iMt-^»coicct^i— i«c 1" o b- QO Ci T-H (M -^ o b- CO qi'r-H (T^ ^.la 'pep ^^ T' T' =^12 "l^ 0^(Mao-*oo(MoO'*'0«ocqoO'*iOO(MOO-+iO?CC^OO-*O^C^CO-^0 (Nil CCCOO'^'O-. CCt-r-iOCiCCQOCNCCO-^COCCt-1-l '"^ c(»c;0l!>1C0CCC0'*<^^»O»Cin<:r> ri'-' ^1;] O^OCCC<)C0'#Ot0(M00'*iOO(M00-*O«r:?-l GOCClOOCOOli— 'C0'TtocpQoair--(C^ccipcpcoo-^c^cc»p«pQpci rnl-' o OO(Ma0'*O«C(MC0^OC0(M00^OO1 I »;: ■*Occc. toooc:i^cq-*y5i>-CiO(M-*(M«ooor:cci--— ( C-1 O CO C-1 »0 OO — -Tf ^l;^' -*oorc--OCiWiooooOOr-^r-'-iMC"i „;!?. '*lCO(M«nO'*G5CCt^^H10a5COC5.— '?q-+ii5;ooo ■4t< t^ o CO tb ci !?q tb ci ->! lb oo o o o — ■ -^ -- ri T'T (MC0-rHOt0^la0-+iOt0'N0000O'MCCOt20CO ^O'M'+iOt>.C0O!MCCi0t0»>.'MtDO-*'00r)t^ Jg t Dt--t>-t^ b->G OO0CO -t-t-cooooociwiCi>— I,— I— ^ — I r-l 0«OCOO-H«'*iO iot^coooicoictoaoo»-tccocscooo(M«oo-+ioocob-— '0>— ifCfs^rir-cso I O ■-r^b-t-l>.OOGOOOC5a5Cir--H,— ,— ,— 11— lr-1,-1 'co-t)co-*otoiMoo-*co'Otoocai'-'«-* iM'Ct— oDO'— i«oicococ5i— incot—.— i»^O-*i00 -[?< Ci CO t^ — I to O -^ 00 O^ to O »0 1^ C5 O !M X "O CO t^ '"^ . O ■M CO »0 to 00 C; O (M CO »o to cotos;'M'Ooo»-i»oc» tototct^t^i^oooooo O CO h- O CO to Ci •>! '^t^OOO — — — i^-J^l C531r— 11— 1^^^-^-— li— I.— ( IOt0J CO Aj ^ .V -i; ^ <^ -V ^-j ! •>! i'o Ci 0-1 >b OO F^ -^ h- 1^ -^ b- o c5 O O r-i ^^ ^ i'l 'tOtOtOt^h-t^OOOOQOClCSCSr-lr-l— Jr-"i— Ir-f-Hr- .ri OOC5'-'CO-+ltOt~C5'-H'M-+llOaO?qt^ — iCCiCOt- oo'>it>--H»0 5;cot— '^^;oo-*'co01— ico-^^t^QO ^cp^--pb-cpO-7^cp'*tpt>-^^^_^^, ^^jj, iCcOr- i"rt -H „I(N ' g ^K n''"' "l-< jm"irr O— 'O-ICO-HiOtOt— COCiO>-icqcO-HiCtor^aori '•uiirirr •*'" ■!'! 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Pt .-C :rt Ci^ en M « CO ■^^-■^■ IQ ' ' 1 AEEAS OF SEGMENTS OF CIRCLES. 67 Table or the Areas of the Segt^lents of a Circle, THE Diameter beln^g Unity. To find the area of the segment of any circle from the following tables. Rule. — Divide the height of the segment by the diameter, take out the corresponding tabular area, which multiply by the square of the diameter for the result. Ai-ea •001 •002 •003 •004 •005 •006 •007 •008 •009 •010 •Oil •012 •013 •014 •015 •016 •017 •018 •019 •020 •021 •022 •023 •024 •025 •026 •027 •028 •029 •030 •031 032 033 034 035 036 037 •000042 •000119 •000219 •000337 •000470 •000618 •000779 •000951 •001135 •001329 •001533 •001746 •001968 •002199 •002438 •002685 •002940 •003202 •003471 •003748 •004031 •004322 •004618 •004921 •005230 •005546 •005867 •006194 •006527 •006865 •007209 •007558 •007913 •008273 •008638 •009008 •009383 -— Ajea •038 •039 "•040 •041 •042 •043 •044 •045 •046 •047 •048 •049 •050 •051 •052 •053 •054 •055 ■056 •057 •058 •059 •060 •061 •062 •063 •064 •065 066 •067 068 069 070 071 072 073 074 •009763 •010148 •010537 •010931 •0113.30 •011734 •012142 •012554 •012971 •013392 •013818 •014247 •014681 •015119 •015561 •016007 •016457 •016911 •017369 •017831 •018296 •018766 •019239 •019716 •020196 •020680 •021168 •021659 •0221.54 •022652 •023154 •023659 •024168 •024680 •025195 •025714 •026236 •075 •076 •077 •078 •079 •080 •081 •082 •083 •084 •085 •086 •087 •088 •089 •090 •091 •092 •093 •094 •095 •096 •097 •098 •099 •100 •101 •102 •103 ■104 •105 106 •107 108 109 110 HI Area •026761 •027289 •027821 •028356 •028894 •029435 •029979 •030526 •031076 •031629 •032186 •032745 •033307 •033872 •034441 ■035011 •035585 •036162 •036741 •037323 •037909 •038496 •039087 •039680 •040276 •040875 •041476 •042080 •042687 •043296 •043908 •044522 ■045139 045759 046381 047005 047632 [ Area •112 •048262 •113 •048894 •114 •049528 •115 •050165 •116 •050804 •117 •051446 •118 •052090 •119 •052736 •120 •053385 •121 •054036 •122 •054689 •123 •055345 •124 •056003 •125 •056663 •126 •057326 •127 •057991 •128 •058658 •129 •059327 •130 •059999 •131 •060672 •132 •061348 •133 •062026 •134 •062707 •135 •063389 •136 •064074 137 •064760 138 •065449 139 •066140 140 •066833 141 •067528 142 •068225 143 •068924 144 •069625 145 •070328 146 •071033 147 •071741 148 •072450 F 2 68 AEEAS OF SEGMENTS OF CIRCLES. Table of the Areas of the Si 5GMENTS DF A Circle, THE Diameter being U^ 'ITY (continued) H Area H Ai'ea H I) 1 Area H 17 U I) Area •149 •073161 •193 •106261 •237 •142387 •281 •180918 •150 •073874 •194 •107051 •238 •143238 •282 •181817 •151 •074589 •195 •107842 •239 •144091 •283 •182718 •152 •075306 •196 •108636 •240 •144944 •284 •183619 •153 •076026 •197 •109430 •241 •145799 •285 : -184521 •154 •076747 •198 •110226 •242 ' -146655 ■286 : -185425 •155 •077469 •199 •111024 •243 ! ^147512 •287 •186329 •156 •078194 •200 •111823 •244 •148371 •288 •187234 •157 •078921 •201 •112624 •245 •149230 •289 •188140 •158 •079649 •202 •113426 •246 •150091 •290 •189047 •159 •080380 •203 •114230 •247 •150953 •291 •189955 •160 •081112 •204 •115035 •248 •151816 •292 •190864 •161 •081846 •205 •115842 •249 •152680 •293 •191775 •162 •082582 •206 •116650 •250 •153546 •294 •192684 •163 •083320 •207 •117460 •251 i ^154412 •295 •193596 •164 •084059 •208 •118271 •252 •155280 •296 •194509 •165 •084801 •209 •119083 •253 •156149 •297 •195422 •166 •085544 •210 •119897 •254 •157019 •298 •196337 •167 •086289 •211 •120712 •255 •157890 •299 •197252 •168 •087036 •212 •121529 •256 •158762 •300 •198168 •169 •087785 •213 •122347 •257 •159636 •301 •199085 •170 •088535 •214 •123167 •258 •160510 •302 •200003 •171 •089287 •215 •123988 •259 •161386 •303 •200922 •172 •090041 •216 •124810 •260 •162263 •304 •201841 •173 •090797 •217 •125634 •261 •163140 •305 •202761 •174 •091554 •218 •126459 •262 •164019 •306 •203683 •175 •092313 •219 •127285 •263 •164899 •307 •204605 •176 •093074 •220 -.128113 •264 •165780 ■308 •205527 •177 •093836 •221 •128942 •265 •166663 •309 •206451 •178 •094601 •222 •129773 •266 •167546 •310 •207376 •179 •095366 •223 •130605 •267 •168430 •311 •208301 •180 •096134 •224 •131438 •268 •169315 •312 •209227 •181 •096903 •225 •132272 •269 •170202 •313 •210154 •182 •097674 •226 •133108 •270 •171089 •314 •211082 •183 •098447 •227 •133945 •271 •171978 •315 •212011 •184 ■ •099221 •228 •134784 •272 •172867 •316 •212940 •185 •099997 •229 •135624 •273 •173758 •317 •213871 •186 •100774 •230 •136465 •274 •174649 •318 •214802 •187 •101553 •231 •137307 •275 •175542 •319 •215733 •188 •102334 •232 •138150 •276 •176435 •320 •216666 •189 •103116 •233 •1389!»5 •277 •177330 •321 ■217599 •100 •103900 •234 •139841 •278 •178225 •322 •218533 •191 •104685 •235 •140688 •279 •179122 •323 •219468 •192 •105472 •236 •141537 •280 1 •180019 •324 •220404 AREAS OF SEGMENTS OF CIRCLES. 69 Table op the Areas of the Segments of a Circle, | THE Diameter being Unity (concluded). H D Area H D Area H 1) Area H iJ Aj-ea •325 •221340 •369 •263213 •413 •306140 •457 •349752 •326 •222277 •370 •264178 •414 •307125 •458 •350748 •327 •223215 ■371 •265144 •415 •308110 •459 •351745 •328 •224154 •372 •266111 •416 •309095 •460 •352742 •329 •225093 •373 •267078 •417 •310081 •461 •353739 •330 •226033 •374 •268045 •418 •311068 •462 •354736 •331 •226974 •375 •269013 •419 •312054 •463 •355732 •332 •227915 •376 •269982 •420 •313041 •464 •356730 •333 •228858 •377 •270951 •421 •314029 •465 •357727 •334 •229801 •378 •271920 •422 •315016 •466 •358725 •335 •230745 •379 •272890 •423 •316004 •467 •359723 •336 •231689 •380 •273861 •424 •316992 •468 •360721 •337 •232634 •381 •274832 •425 •317981 •469 •361719 •338 •233580 •382 •275803 •426 •318970 •470 •362717 •339 •234526 •383 •276775 •427 •319959 •471 •363715 •340 •235473 •384 •277748 •428 •320948 •472 •364713 •341 •236421 •385 •278721 •429 •321938 •473 •365712 •342 •237369 •386 •279694 •430 •322928 •474 •366710 •343 •238318 •387 •280668 •431 •323918 •475 •367709 •344 •239268 •388 •281642 •432 •324909 •476 •368708 •345 •240218 •389 •282617 •433 •325900 •477 •369707 •346 •241169 •390 •283592 •434 •326892 •478 •370706 •347 •242121 •391 •284568 •435 •327882 •479 •371705 •348 •243074 •392 •285544 •436 •328874 •480 •372704 •349 •244026 •393 •286521 •437 •329866 •481 •373703 •350 •244980 •394 •287498 •438 •330858 •482 •374702 •351 •245934 •395 •288476 •439 •331850 •483 •375702 •352 •246889 •396 •289453 •440 •332843 •484 •376702 •353 •247845 •397 •290432 •441 •333836 •485 •377701 •354 •248801 •398 •291411 •442 •334829 •486 •378701 •355 •249757 •399 •292390 •443 •335822 ■487 •379700 •356 •250715 •400 •293369 •444 •336816 •488 •380700 •357 •251673 •401 •294349 •445 •337810 •489 •381699 •358 •252631 •402 •295330 •446 •338804 •490 •382699 •359 •253590 •403 •296311 •447 •339798 •491 •383699 •360 •254550 •404 •297292 •448 •340793 •492 •384699 •361 •255510 •405 •298273 •449 •341787 •493 •385699 •362 •256471 •406 •299255 •450 •342782 •494 •386699 •363 •257433 •407 •300238 •451 •343777 •495 •387699 •364 •258395 •408 •301220 •452 •344772 •496 •388699 •365 •259357 •409 •302203 •453 •345768 •497 •389699 •366 •260320 •410 •303187 •454 •346764 •498 •390699 •367 •261284 •411 •304171 •455 •347759 •499 •391699 •368 •262248 •412 •305155 •456 •348755 ■500 •392699 70 CENTRES OF FIGURES. CENTRES AND MOMENTS OF FIGURES. To Find the Centres of a pew Special Figures. 1. Triangle. (Fig. 104.) Fig. 104. ^ EuLE. — From the middle points of any two sides draw lines to the opposite angle ; the point of intersection d of these lines is the required centre. 2. Trapezoid. Fig. 105. AF. 3 __ (Fig. 105.) Rule. — Bisect ab in e and cd in p, and join ef. Produce AB beyond B to --^.-..2 ^' Daaking bh^^CD, and produce CD beyond C to i, making 01 = A3; then join HI, and where this line intersects EF is the centre of gra\'ity G. 3. Trapezium. Fig. 106. (Fig. 106.) Rule. — Draw the diagonals ad and CB intersecting in e ; along CB set ofE CF equal to eb, and join fa and fd : the centre of the triangle afd will be the centre of the tra- pezium. 4. Circular arc. Fig. 107. (Fi 107.) Rule. — Let adb be the circular arc and c the centre of the circle of which it is a part (to find c see p. 7); bisect the arc AB in d, and join do and AB; multi- ply tlie radius CD by the chord AB, and divide by the length of the arcADB:lay off ^^' the quotient CE upon CD, then E is the centre required. 5. Yery fiat curved line (^approximate'). (Fig. 108.) Rule. — Let adb be the arc ; draw the chord ab, and bisect it in C ; draw CD perpendicular to AB ; make CE cciual to 5 of CD : then E Avill be the centre required. no.) Fig. 110. CENTKE& OF FIGURES. 71 6. Sector of a circle. (Fig, 109.) EULE. — Let ABC be the sector, E its centre ; multiply the chord ab by | of the radius CA; divide the product by the length of the arc : the quotient equals the distance CE set along the line CD, D being at the bisec- tion of the arc ab. 7. Sector of a playje circular ring. (Fig. EuLE. — Let CA be the outer and CE the inner radius of the ring : divide twice the difference of the cubes of the inner and outer radii by three times the diffe- rence of their squares ; the quotient will l>e an intermediate radius cf, with which (Inscribe the arc ff, subtending the same angle with the sector: the centre H of the circular arc ff, found I'V Eule 4, will be the centre required. 8. Circular segment. (Fig. 111.) EuLE. — Let c be the centre of the circle of which it is a part ; bisect the arc AB in D, and join CD ; divide the cube of lialf the chord ab by three times the area of half the segment adb : set off the quo- tient OE along CD, and E will be the centre required. 9. Parabolic half-segment. (Fig. 112.) Eule. — Let abd be a half -segment of a arabola, bd being part of a diameter parallel TO the axis and AD an ordinate conjugate to that diameter — that is, parallel to a tangent at B. Make be equal to f bd, and draw ef l/arallel to AD and equal to | AD. Then F will l)e the centre of the half -segment. 10. Height of centre of semicircle from its base. Eule. — Multiply the diameter of the semicrrcle by 4, and divide the product by Stt. 11. Height of centre of jyaralfola from its base. Eule. — Multiply its vertical height by 2, and divide the product by 5. 12. Height of centre of ellijytic segment from the lesser diameter tf the ellijjse of which it is a imrt. Eule. — Take the square of half the greater diameter of the ellipse, and divide the product by the square of half the lesser diameter; multiply that result by the cube of half the length of the base of the segment, and divide the result by three times its lialf-area. '72, MOMENTS AND CENTRES OP FIGUEES. JLx. : Let d = greater diameter of ellipse, and d = lesser diam. B = base uf segment, and A = area of segment. H = height of centre from lesser diameter of ellipse. .-©■■(!)■ "(4)"" 13. PHsm or cylinder with plane 2ycirallel ends. Rule. — Find the centres of the ends ; a straight line joining them will be the axis of the prism or cylinder, and the middle point of that line will be the centre required. 14. Cone or j^yt'omid. Rule. — Find the centre of tlie base, from which draw a line to the summit ; this will be the axis of the cone or pyramid, and the point at \ from the base along that line will be the centre. 15. Hemispliere or Jiemi-ellipsaid. Rule. — The distance of the centre from the circular or elliptic base is | of the radiiis of the sphere, or of that semi-axis of the ellipsoid which is perpendicular to the base. 16. Paraboloid. Rule. — The distance of its centre from the base along its axis is I of the height from the base. ^iG. 113. 17, To fnd the centre of r/rarity of any continuous curved line. (Fig. 113.) Ex. : Let abc be the given curve ; bisect it at B ; join ab and BC, and bisect those chords at the points D and e respectively; set off fd per- pendicular to ab, and eg perpendicu- lar to BC; make fh = |df and gk = fGE, and join hk; bisect HK at the point L, which will be a close approx- imation to the position of the centre ^^ of gravity of the curved line ABC. rt7lbs for fixding the moments and centres of Figures. Thd geometrical moment of a fgure, whether a line, am, area, or a solid, relatively to a given jilane or axis is the product of tlie magnitude of that figure, into the perpendicular distance of its centre from the given plane or axis, and is equal to the sum of the moments of all its parts relatively to the same pla)w. The centre of an area is determined when its distance from two axes in the plane of the ligure is known. The centre of a figure of three dimensions is determined MOME^'TS AXD CENTRES OF FIGURES. 73 ■^hen its distance from tliree planes not parallel to one another is known. 1. To find the moment of an iii'egular figure relatively to a fjiven plane m' axi». Rule. — Divide the figure into parts whose centres are known ; multiply the magnitude of each of its parts into the perpendi- cular distance of its centre from the given plane or axis ; dis- tinguish the moments into positive and negative, according as the centres of the parts lie to one side or the other of the plane : the difference of the two sums will be the resultant moment of the figure relatively to the given plane or axis, and is to be regarded as positive or negative, according as the sum of the positive or negative moments is the greater. 2. To find tJie perpendicular distance of the centre of an irre- gular figure from a given plane or axis. Rule. — Divide the moment of that figvire relatively to the given plane or axis by its magnitude ; the quotient will be the jDerpendicular distance of its centre from the given plane or axis. 3. To find the centre of a figure consisting of two parts whose centres are hnoivn. (Fig. 114.) Rule. — MultijDly the distance between the two known cen- tres by the magnitude of either of the parts, and divide the product by the magnitude of the whole figure ; the quotient will be the distance of the centre of the whole figure from the centre of the other part, the centre of the whole figure being in the straight line joining the centres of the two parts. Ex. : Let abcd be such a figure, m and m the magnitude of its two respective parts, M + 7« the magnitude of the whole figure, D the dis- tance between the centres ]5i and m of the two parts, and c the centre of the whole figure. 3IC = m X D M+ ni mc = : M X D M + 7)1 Fig. 115. 4. To find the centre of any plane area hy means of ordinates. (Fig. 115.) Let ABC, the quadrant of a circle, be such an area ; cb the base line, di\'ided into a number of equal parts by ordinates ; AC the transverse axis traversing its origin. \st. Betermiyie the perpendicular distance of the centre of the quadrant from the trans- verse axis in the following manner: — Rule. — Multiply each ordinate by its dis- tance from the transverse axis ; consider the products as ordinates of a new curve of the same length as the given figure : the area of that ciu-ve, found by the proper rule, will be the moment of the figure relatively to the transverse ziaiii'^ MOMENTS AND CENTRES OF FIGURES. axis ; this moment, divided by the whole area of the figure, will give the perpendicular distance of its centre from the transverse axis. In algebraical symbols the moment of a plane figure rela- tively to its transverse axis, and found by the above rule, is expressed thus : — fxydx. Note. — In practice it is better to proceed as follows : — Multiply the ordinates first by their multipliers, and then those products by the number of intervals from the origin ; take the sum of those products and multiply it by ^rd of a whole interval squared, if Simpson's first rule is used, by |ths of a whole inter- val squared, if Simpson's second rule is used, and so on for the other rules. Exaviple. No. of Intervals Ordinates Multi- pliere Products Products X ^'o. of Intervals from Origin 16-0000 1 16-0000 •00000 1 15-4919 4 61-9676 61-9676 2 13-8564 1^ 20-7846 41-5692 2i 12-4900 2 24-9800 62-4500 3 10-5830 3 4 7-93725 23-81175 ^ 9-3274 1 9-3274 30-31405 H 7-7460 i 2 3-87301 13-5555 n 5-5678 1 5-5678^ 20-87925 4 0-0000 i Interval 0-0000 •00000 150-43765 4 Interval 2.54-54735 3 3 3 3 Ap proxima te area = 200-58353 Approx. moment = 1357*585 Moment 1357-585 ^ a.-aa /"approximate perpendicular distance Area 200-5835 ™ ' \ of centre from the transverse axis. Ind. Find tlie perpendicnlar distance of its centre from the base line. Rule. — Square each ordinate, and take the half -.squares as ordinates for a new curve of the same length as the iigure ; the area of that curve, found by the proper rule, will be the moment of the figure relatively to the base line : this moment, divided by the whole area of the figure, will give the perpendicular distance of its centre from the base line. In algebraical .symbols the moment of a plane figure rela- tively to its base line, found by the above rule, is expressed thus : — / H'-d^. MOMENTS AND CENTRES OF FIGUEES. Example. 75 No. of Intervals Ordiaates Half-squares Multipliers Pro'lucts 1 2 3 4 16-0000 15-4919 13-8564 10-5830 » 0-0000 128-0000 1 119-9995 4 95-9999 2 55-9999 4 00000 1 Interval 3 Approximate moment 128-0000 479-9980 191-9998 223-9996 0-0000 1023-9974 _ * 3 = 1365-3298 Moment 1 365-3298 _ c.^ac f approximate perpendicular dis- Area 201-0624 ~ \ tance of centre from base. Actual moment = 1365-3 Actu^ararea ^~2'0]70624 5. To Jind the centre of a plane area hoinided by a cnrre ayid two radii by means of polar co-ordinates. (See fig. 82.) 1st. J^termine the peipejidicular distance of its centre from a plane trd^ersiiig the j^ole and at right angles to one of the bound- ing radii, called the first radivs, in the folloiving manner : — EuLE. — Divide the angle subtended by the arc into a conve- nient number of equiangular intervals by means of radii ; mea- sure the lengths of the radii from the pole to the arc, and multiply the third part of the cube of each of them by the cosine of the angle which they respectively make with the first radiiLS ; treat these products by one of the rules applicable to finding the area of a plane curve (the only difference being that the common interval is taken in circular measure) ; the result will be the moment of the figure relatively to the plane tra- versing the pole : this moment, divided by the area of the figure, will give the perpendicular distance of its centre from the plane traversing the pole. Exampli'. ■1 ^^ 5 No. ^ of Radii Radii a 1 12 2 12 3 12 4 12 r 12 Cubes of Radii 576 576 576 576 576 Angles First Radius 0° 5° 10° 15° 20° Cosines! Products 1-0000 576-0000 -9962 573-8112 •9848567-2448 •9659 556-3584 •9397 541-2672 I Simpson's Multi- pliers 1 4 2 4 Products Interval in circular measure 576-0000 2295-2448 1134-4896 2225-4336 541-2672 6772^4352" -0291 Moment relatively to plane traversing pole= 197-077864 76 MOMENTS AND CENTRES OP FIGURES. Homent 197 '0778 64: _ ^.g^j^ /perpendicular distance of centre Area 25' 1327 \. from plane traversing pole. In algebraical symbols the moment, as here found, is ex- pressed thus : — / cos 6dQ. 2iid. Determine tlie moment of the fgure relaiivehj to the first radius precise li/ i/i the same jvay as in the foregoing rule, with tlie exceptiou that sines must he used i?i the place of cosines ; this moment, divided by the area of the figure, mill give the perpen- dicular distance of its centre from the first radius. Note. — It is usual, in practice, to defer the division of the cubes of the radii by 3 until after the addition of the products. Example. No. of Radii Radii 1 12 2 12 3 12 4 12 5 12 'Angles Cubes of Radii with First Radius 576 576 576 576 576 0° 5° 10° 1.5° 20° Sines of Angles Products •0000 •0872 •1736 •2588 •3420 •0000 50-2272 99-9936 149-0688 196-9920 Simpson's Multi- pUera ProdiTcts •0000 200-9088 199-9972 596-2752 196-9920 Area 25-1327 Interval in circular measure _ Moment relatively to first radius : = 1-38 1194-1732 -0291 34-750440 Moment 34-75044 . i„ f perpendicular distance of centre from \ first radius. In algebraical symbols the moment as liere found is ex- pressed thus : — / sin ede. 6. To find the perpendicular distance of the centre of a solid, hounded on one side by a curved surface (figs. 101 and 102), from a plane perpendicular to a given axis at a given point. Rule. — Proceed as in liule 4, p. 73, to find the moment relatively to the plane, substituting sectional areas for breadths; then divide the moment by the volume (a<3 found by Rule 2, p. 44): the quotient will be the required distance. To determine the centre completely, find its distance from three planes no two of wluch are parallel. MOMENTS AND CENTRES OF FIGURES. n Fig. 116. w. 7. Having the moment and centre of a figure relatively to a given plane, to find the neiv vioment and centre of the figure rela- tively to the same plane ivhen a part of the figure is shifted. (Fig. 116.) In the figure wlk let c be its centre, and zz' a plane with respect to which the moment of the figure is known : suppose the part WSM to be transferred to the new position SNL, so as to alter the shape of the figure from wlk to mnk ; let i be the original and h the new cen- tre of the shifted part : then the moment of the figure mnk relatively to the plane zz' t^ found as follon-s : — EuLE. — Measure the distance, perpendicular to the plane of moments, between the centres of the original and new position of the shifted part, as hd, and multiply it by the magnitude of the shifted part ; the product will be the moment required. The new position of tlie entire figure is then found hy the following rule : — EuLE. — Multiply the distance between the centres of the original and new position of the shifted part by the magnitude of that part ; that product, divided by the magnitude of the whole figure, will give the distance the centre has traversed in the direction in which the part has been shifted, and in a plane parallel to a line joining the centres of the original and new position of the shifted part, as from c to c' in fig. 116. 8. To find the centre of a wedge-shaped solid (fig. 117) hy means of polar co-ordinates. \st. Determine the perpendicular dista.ice of its centre rela- tively to a transverse sectional 2^ lane, as pab. EuLE. — Divide the Fig. 117. solid by a number of parallel and equidistant planes, as pab, PjAjB,, P2A0B2, &c. ; then mul- tiph" each sectional area by its distance from the plane pab ; treat the products as though thej* were the ordinates of a curve of the same length as the figure ; the area of that curve, found by the proper rule, will be the moment of the figure relatively to the plane pab : that moment, divided by the volume of the figure, will be the distance required. £d -e* 78 MOMENTS OF INERTIA. 2nd. Determine the perpendicular distance of its centre re- latively to a lonyitudinaL plane passing through its edge, as MPM, perpendicular to the first radius, pb. Rule. — Divide the figure by a nnuiber of longitudinal planes radiating from the edge MPM at equiangular intervals (as PP^AA^, PP4CC^, PP4BB4) ; also divide the length of the figure into a number of equal intervals by ordinates, and treat each of the longitudinal planes as follows : — Measure its ordinates, take the third part of their cubes, and treat those quanti- ties as if they were ordinates of a new curve ; that is, find its area by one of Simpson's rules : the area of that new curve is termed the moment of inertia of the longitudinal plane in question. Then multiply each moment of inertia of the several planes by the cosine of the angle made by the plane to which it belongs with the plane pb, and treat these products by a proper set of Simpson's multipliers ; add together the products, and multiply the sum by | of the common angular interval in cir- cular measure if Simpson's first rule is used, and by | if Simp- son's second rule is \ised. The result will be the moment of the figure relatively to the plane mpm. This moment, divided by the volume of the figure, will be the distance required. The algebraical expression for the moment as found in this rule is // ^ cos QdxdQ. 37'd. Determine the perpendicular distance of its centre re- latively to a longitudinal plane passing through its edge, and a radius as pp^be*, by the foregoing rule, with the exception of multiplying by sines instead of cosines. Note. — In practice it is usual to defer the division of the cubes of the radii by 3- until after the addition of the products. Moments of Inertia and Radii of Gyration. 1. To find the moment of inertia of a body about a given uxis. Rule. — Conceive the body to be divided into an indefinitely great number of small parts ; multiply the mass (or weight) of each of these small parts into the square of its perpendicular distance from the given axis : the sum of all these products as obtained will be the moment of the body about the given axis. 2. To find the squaa'e of the radius of gyratiott of a body about a given axis. Rule. — Divide the moment of inertia of the body rela- tively to the given axis by the mass (or weight) of the body. MOMENTS OF INEETIA. 79 3. (rive/i the moment of inertia of a body about an axis traversing/ its centre of gravity in a given direction, to find its moment of ineHia about another axis parallel to the first. Rule. — Multiply the mass (or weight) of the body by the square of the perpendicular distance between the two axes, and to the product add the given moment of inertia. 4. Given the separate moments of inertia of a set of bodies about jjarallel axes traversing their several centres of gravity, to find the moment of ineHia of these bodies about a common axis parallel to their separate axes. Rule. — Multiply the mass (or weight) of each body by the square of the perpendicular distance of its centre of gravity from the common axis ; the sum of all these products, together with all the separate moments of inertia, will be the combined moment of inertia. 5. Given the square of the radius of gyration of a body about an axis traversing its centre in a given direction, to find the square of the radius of gyration about another axis parallel to the first. Rule. — Square the perpendicular distance between the two axes, and add the product to the given square of the radius of gyration. 6. To find the moment of inertia of a plane area, bounded on one side by a curve (see tig. 115), relatively to its base line. Rule. — Divide the base line into a suitable number of eqiial intervals, and measure ordinates at the points of division ; take the third part of the cube of each of these ordinates, and treat those quantities so computed as the ordinates of a new curve : the area of that new curve, found by the proper rule, will be the moment of inertia required. In algebraical symbols the above rule is expressed thus ''tdx. 3 / JS'ote. — When the moment of inertia is required as a whole, and not in separate parts, it is usual to postpone the division of the cubes till the end of the calculation. 7. To find the moment of inertia of a piano area, bounded on one side by a cuj've, relatively to one of its ordinates. Rule. — Multiply each ordinate by its proper mtdtiplier, ac- cording to one of the rules for finding the area of such figures ; then multiply each of the products by the square of the number of whole intervals that the ordinate in question is distant from the 80 MOMENTS OF INERTIA. ordinate taken as the axis of moments : the sum of these pro- ducts, multiplied by ^ or | the cube of a whole interval, accord- ing as Simpson's first or second ride is used, will be the moment of inertia required. In algebraical sj-mbols this rule is expressed thus : — Example I. Calculation of Moment op Inertia of the Quadrant OF A Circle Relatively to the Base Line. Cubes of Ordinates :.o. of Intervals Ordinates 3 Multipliers Products ICvOO 1365-.33 1 1365-33 1 15-40 1238-89 4 4955-56 2 13-86 887-50 u 1331-25 n 12-49 649-48 2 1298-96 3 10-58 394-76 3 4 296-07 H 9-33 270-72 1 270-72 3| 7-75 155-16 1 2 77-58 3f 5-57 57-29 1 57-29 4 0-00 0-00 1 4 Interval 3 0-00 9652-76 4 — 3 12870-34 Example II. Calculation of the Moment of Inertia of the Quadra.nt OF A Circle Eelatively to the Endmost Ordinate. No. of Intervals Ordinates ilultiplier:? Products squares of Nos. of Intervals Products 160000 1 16-0000 0-00 000 1 15-4919 4 61-9676 1-00 61-9679 2 13-8564 1,V 20-7846 4-00 83-1384 H 12-4900 2 24-9800 6-25 156-1250 3 10-5830 3 4 7-93725 9-00 71-4353 H 9-3274 1 9-3274 10-5625 98-5207 H 7-7460 a 3-8730 12-2500 47-4443 3| 5-5678 1 5-5678 14-0625 78-2972 4 0-0000 I 4 0-0000 16-0000 0-0000 Approximate momen Interval' 3 t of inertia = , 596-9288 _ 64 ~ 3 12734-4810 SQUARES OF RADII OF GYRATION. 81 Table of Squares of Radii of Gyration of Special Figures. A few Body 1. Kectangle ; sides a and h 2. Triangle ; sides a^b, c, heights a', V , c' 3. Circle or ellipse ; dia- meters a, b 4. Common parabola ; height a, base b 5. Sphere ; radius r 6. Spheroid of revolution ; polar semi -axis a, equa- torial radius r Axis side a side a diameter a base h diameter polar axis axis 2a / diameter > longitudinal axis 2a transverse diameter } Radius -= 7. Ellipsoid; semi-axes a, 5,e 8. Spherical shell; external radius r, internal ra- dius r' 9. Circular cylinder; length 2a., radius r 10. Circular cylinder ; length 2a^ radius r 11. Hollow circidar cylin- der ; length 2a, exter- nal radius ■;•, internal radius r' 12. Hollow circular cylin- der ; length 2a, exter- nal radius o; internal radius r' 13. Elliptic c)' Under ; length 2a, transverse semi- axes b,c i J 14. Elliptic cylinder; length jj 2a, transverse semi- I transverse axis 2b axes b,c |J 15. Rectangular prism ; di- |\ mensions 2a, 2b, 2c j / 16. Ehombic prism; length H 2a, diagonals 2b, 2e ! / 17. Rhombic prism ; length 1 1 2a, diagonals 2b, 2e \] Moment of inertia = square of radius of gyration (or weight) of the figure. longitudinal axis 2a transverse diameter longitudinal axis 2a axis 2a axis 2a diagonal 2b •6 6 16 8a' 35 5 5 2rr'— /•") 5(r'— r") '1 4+T 7-* + H 2~ ~4 "*■ 3 4 c- a- -4+3- 3 c' a- X the mass 82 TJEGISTER TONNAGE. TONNAGE. Kegister oe New Measukement Tonnage. The gross register tonnage of a ship expresses her entire cubical capacity in tons of 100 cubic feet each, and may be found approximately by the following formula : — L=the inside length on upper deck from plank at stem to plank at stern. B = the inside main breadth from ceiling to ceiling. D = the inside midship depth from ujjper deck to ceiling at limber strake. Ree:ister tonnage = c. ^ ° 100 Sailing ships Steam vessels and clippers Yachts Value of c. r cotton and sugar ships, old full form \ ships of the present usual foiTU r ships of two decks . \ ships of three decks {above sixty tons under sixty tons •7 •65 •68 •5 •45 To Calculate the Gross Register Tomiage. The tonnage deck is the upper deck in all vessels under three decks, in all other vessels the second deck from below. Measurements to be expressed in feet and the decimals of a foot. The length for register tonnage is taken from inside of plank at stem to inside of midship stern timber, or plank there, as the case may be, and is taken on the tonnage deck ; the length so taken (having made deductions for the rake of stem and stern, if any, in the thickness of the deck, and one-third of the round of the beam) is to be divided into the prescribed number of equal parts, according to the length, as follows : — Length. No. of Intervals. Not exceeding 50 feet and under , , . 4 Exceeding 50 feet and not exceeding 120 feet. . 6 Exceeding 120 feet and not exceeding 180 feet. . 8 Exceeding 180 feet and not exceeding 225 feet . . 10 Exceeding 225 feet . , . . , . 12 Transverse sections are tlicn measured at each of the points of division, as follows : — REGISTER TONNAGE. 83 The total depths of the transverse sections are measured from the vmder side of the tonnage deck to the ceiling at the inner edge of limber strake, deducting one -third of the round of the beam. The depths so taken are to be divided into four equal parts, if midship depth should not exceed sixteen feet ; otherwise into six equal parts. The breadths are measured horizontally at the points of division, and also at the upper and lower points of each depth, each measurement extending to the average thickness of that part of the ceiling which is between the points of measurement. The areas of the transverse sections are then computed by Simpson's first rule (p. 38), and then the capacity of the ship is computed by the same rule (Rule 2, p. 44) — that is, the areas are treated as the ordinates of a new curve of the same length as the vessel ; and the area of that new curve, found by Simpson's first rrde, will be the capacity of the vessel in cubic feet, which being divided by 100 gives the gross register tonnage under tonnage deck. The capacity of tlie poop, deck houses, and other permanently enclosed spaces available for cargo or passengers is to be mea- sured and included in the register tonnage, but the following deductions are allowed, the remainder then being deemed the register tonnage of the ship. Deductions Allowed frmn the Gross Tonnage. — (1) Buildings for the shelter of passengers only. (2) Space allotted _to crew (for crew space see p. 114). (3) Propelling space. Screw steamers : if the cubic content is 13 and under 20 per cent, of the gross tonnage, deduct 32 per cent. ; if the space is smaller than 1 3 and larger than 20 per cent., deduct either 32 per cent, or the cubic content multiplied by I'To. Paddle steamers : if the cubic content is 20 and under 30 per cent, of the gross tonnage, deduct 37 per cent., and if the space is smaller than 20 or larger than 30 per cent., deduct either 37 per cent, or the cubic content multiplied by l-o. ! Factors for Measurement and Dead-weight Cargoes. 1. To ascertain approximately foi' an average length of voyage the measurement cargo, at \0 feet to the ton, which a ship can carry. Rule. — Multiply the number of register tons by the factor 1*875, and the product will be the approximate measurement cargo. 2. To ascertain approximately the dead-weight cargo in tons n'Mch a ship can carry on an average length of voyage. Rule. — Multiply the number of register tons by 1-5, and the product will be the approximate dead-weight cargo required. With regard to the cargoes of coasters and colliers as as- certained above, about 10 per cent, may be added to the said results, while about 10 per cent, may be deducted in the cases of larger vessels going longer voyages. In the case of measurement cargoes of steam vessels the spaces g2 84 builder's and yacht tonnage. occupied by tlie machinery, fuel, and passenger cabins under the deck must be deducted from the space or tonnage under the deck before the application of the measurement factor thereto. In the case of dead-weight cargoes, the weight of machinery, water in the boilers, and fuel must be deducted from the whole deadweight, as ascertained above by the application of the dead- weight factor. The deductions necessary to be made for provisions, stores. kc, are allowed for in the selection of the two factors. Buildee's Tonnage, or Old Measurement Tonnage. To Compiite the Builder's Tojinage. Ktjle. — Measure the length of the vessel along the rabbet of the keel from the back of the main stern-post to a perpendicular line let fall from the fore part of the main stem imder the bowsprit ; measure also the extreme breadth to the outside planking, exclusive of doubling planks. Three-fifths of that breadth is to be subtracted from the length ; the remainder is called the length of keel for tonnage. Multiply the length of keel for tonnage by the breadth, that product by the half- breadth, and divide by 94 ; the quotient will be the tonnage. If L = length, B = breadth, then Tonnage (B.O.M.)= ^^~^^^ ^ ^ "^ ^^ ° ^ ^ 94 Measurement of Yachts for Tonnage. Royal TJtames Yacht Club. Eule. — Measure the length of the yacht in a straight line at the deck from the fore parr of the stem to the after part of the stern-post, from which deduct the extreme breadth, which is measured from the outside of the outside planking ; the remain- der is the length for tonnage. Multiply the length for tonnage by the extreme breadth, that product by half the extreme breadth, and divide the result by 94 ; the quotient will be the tonnage. If any part of the stem or stern-post i:)rojcct beyond the length as taken above, such projection or projections shall, for the purpose of finding tlie tonnage, be added to the length taken as before mentioned. All fractional parts of a ton shall be con- sidered as a ton. The measurement to be taken either above or below the main-wale. If l = length, B = breadth, then _, (L — B)xBxlB Tonnage = ^^ '-— ^— . 94 TONNAGE TABLES. 85 Table of the Tonnages of Vessels according to Builder's Measureitent. In the following tables tonnages are only given for vessels whose lengths are multiples of 10, except at the head of each group, where the tonnage for each extra foot of length tip to 9 feet is given, in order that the tonnages of vessels whose lengths are not given in these tables may be fomid by a simple addition, as per example. Fx. — Required the tonnage of a vessel and 22-5 feet beam. Tonnage for 200 feet length = 502 Tonnage for extra 7 feet length = 18 Tonnage for 207 feet length =521 207 feet long i|-813 i|-875 ^-688 JVote. — In the tables the ninety-fourths of a from the tons by a dash ; thus, 126i|-125 - 126- ton are divided -18-125. If^J TONS ^t^^, TONS \l^'^,i TONS Kj TONS 10 feet beam 1 0-50 5 2-62 3-18 3-68 4-24 9 30 40 50 4-74 12-72 18-8" 23-38 60 70 80 90 28-68 34-4 39-34 44-64 3 4 1-6 1-56 2-12 6 7 8 10-5 feet beam I 1 2 8 0-55-125 1-16-25 1-71-375 5 6 7 8 2-87-625 3-48-75 4-9-875 9 30 40 5-26-125 13-84-463 19-71-713 60 70 80 31-46-213 37-33-463 43-20-713 4 2-32-5 4-65-0 50 2.5-58-963 90 49-7-963 ' 11 FEET BEAM | 1 2 3 4 5 0-60-5 1-27-0 1-87-5 2-54-0 3-20-5 6 7 8 9 30 3-81-0 4-47-5 . 5-14-0 5-74-5 15-5-7 40 50 60 70 80 21-46-7 27-87-7 34-34-7 40-75-7 47-22-7 90 100 110 120 130 53-63-7 ^ 60-10-7 66-51-7 72-92-7 79-39-7 11-5 FEET BEAM | 1 2 3 4 0-66-125 1-38-25 2-10-375 2-76-5 6 7 8 9 4-20-75 4-86-875 5-59-0 6-31-125 40 50 60 70 80 23-26-738 30-29-988 37-33-238 44-36-488 90 100 110 120 58-42-988 65-46-238 72-49-488 79-52-738 o 3-48-625 30 16-23-488 51-39-738 130 86-5.5-988 | 12 FEET BEAM | 1 2 3 4 5 0-72 1-50 2-28 3-6 .3-78 6 7 8 9 40 4-56 5-34 6-12 6-84 25-11 50 60 70 80 90 32-73 40-41 48-9 55-71 63-39 100 110 120 1.30 140 71-7 78-69 86-37 94-5 101-67 86 TONNAGE TABLES. |in Ft. in Fi.l ILgth.i linFV TONS ILgth. |in Ft.' 1 2 3 4 t 5 12-5 FEET BEAM 0-78-125 6 1-62-25 7 2-46-375 8 3-30-5 9 4-14-625 40 4-92-75 50 5-76-875 60 6-61-0 70 7-45-125 80 27-1-063 90 35-30-313 43-59-563 51-88-813 60-24-063 68-53-313 100, 110 120 130 1401 76-82-563 85-17-813 93-47-063 101-76-313 110-11-563 13 FEET BEAM 0-84-5 1-75-0 2-65-5 3-56-0 4-46-5 6 7 8 9 40 5-37-0 6-27-5 7-18-0 8-8-5 28-88-9 50 60 70 80 90 37-87-9 46-86-9 55-85-9 64-84-9 73-83-9 100 110 120 130 140 82-82-9 91-81-9 100-80-9 109-79-9 118-78-9 0-91-125 1-88-25 2-85-375 3-82-5 4-79-625 6 7 8 9 40 5-76-75 6-73-875 7-71-0 8-68-125 30-86-888 40-58-138 50-29-388 60-(J-638 89-65-888 79-37-138 1001 89-8-388 110} 98-73-638 120; 108-44-888 130| 118-16-138 140 127-81-388 14 FEET BEAjM 1-4 6 6-24 50 43-34 2-8 7 7-28 60 53-74 3-12 8 8-32 70 64-20 4-J6 9 9-36 80 74-60 5-20 40 32-88 90 85-6 100 110 120 130 140 9;>-46 105-86 116-32 126-72 137-18 14-5 FEET BEAM 1-11-125 2-22-25 3-33-375 4-44-5 5-55-625 6-66-75 7-77-875 8-89-0 10-6-125 46-17-663 60 57-34-913 70i 38-52-163 80' 79-69-413 90J 90-86-663 100 102-9-913 1101 113-27-163 120 124-44-413 1301 135-61-663 140 146-78-913 150i 158-2-163 15 FEET BEAM 1-18-5 6 7-17-0 2-37-0 7 8-35-5 3-55-5 8 9-54-0 4-74-0 9 10-72-5 5-92-5 50 49-6-5 60 70 80 90 100 61-3-5 73-0-5 84-91-5 96-88-5 108-85-5 110 120 1301 140 150' 120-82-5 132-79-5 144-76-5 156-73-5 168-70-5 15-5 FEET BEAM 8-88 10-21 11-47 52-1-088 64-74-338 77-53-588 801 90-32-838 90 103-1 2088 lOOJ 115-85-338 no: 128-64-588 120 141-43-838 130i 154-23-088 140 167-2-338 150 179-75-588 160' 192-54-838 170 205-34-088 180,218-13-338 190 230-86-588 TONNAGE TABLES. 87 Lffth in Ft TONS [Lgrth JinFt 1 TONS \lf^,:\ TONS f^Tp'Jj TONS 1 16 FEET BEAM | 1 1-34 7 9-50 80 95-81 140 177-53 2 2-68 8 10-84 90 109-45 150 191-17 3 4-8 9 12-24 100 123-9 160 204-75 4 5-42 50 55-1 110 136-67 170 218-39 5 6-76 60 68-59 120 150-31 180 232-3 6 8-16 70 82-23 130 163-89 190 245-61 16-5 FEET BEA:\I | 1 1-42-125 7 10-12-875 80 101-48-363 140 188-37-863 2 2-84-25 8 11-550 90 115-93-613 loO 202-83-113 3 4-32-375 9 13-3-125 100 130-44-863 160 217-34-363 4 5-74-5 50 58-6-613 110 144-90-113 170 231-79-613 5 7-28-625 60 72-51-863 120 159-41-363 180 246-30-863 6 8-64-75 70 87-3-113 130 173-86-613 190i 260-76-113 1 17 FEET BEAM | 1 1-50-5 7 10-71-5 80 107-28-1 140 199-50-1 2 3 3-7-0 8 9 12-28-0 90 122-63-1 150 160 214-85-1 230-26-1 4-57-5 13-78-5 100 138-4-1 4 6-14-0 50 61-17-1 110 153-39-1 170| 245-61-1 5 7-64-5 60 76-52-1 120 168-74-1 180| 261-2-1 6 9-21-0 70 91-87-1 130 184-15-1 190| 276-37-1 17-5 FEET BEAM | 1 1-59-125 7 11-37-875 80 113-20-188 1401 210-89-6881 2 3-24-25 8 13-3-0 90 129-47-438 150 227-22-938 3 4-83-375 9 14-62-125 100 145-74-688 160 243-50-188 4 6-48-5 50 64-32-438 110 162-7-938 170 259-77-438 5 8-13-625 60 80-59-688 120 178-35-188 180 276-10-688 1 6 9-72-75 70 96-86-938 130 194-62-438 190 292-37-938 | 18 FEET BEAM | 1 1-68 7 12-6 80 119-24 140 222-62 2 3-42 8 13-74 90 136-46 150 239-84 • 3 5-16 9 15-48 100 153-68 160 257-12 4 6-84 50 67-52 110 170-90 170 274-34 5 8-58 60 84-74 120 188-18 180 291-56 6 1 n-;^-:> 70 102-2 130 9A.^ in 190 308-78 18-5 FEET BEAM | 1 1-77-125 8 14-53-0 100 161-79-013 1701 289-25-763 2 3-60-25 9 16-36-125 110 180-4-263 180! 307-45-013 3 . 5-43-375 50 70-76-763 120 198-23-513 190| 325-64-263 4 7-26-5 60 89-2-013 130 216-42-763 200; 343-83-513 5 9-9-625 70 107-21-263 140 234-62-013 210 362-8-763 6 10-86-75 80 125-40-513 150 252-81-263 220, 380-28-013 7 12-69-873 90 143-59-763 160 271-6-513 2301 398-47-263 TONNAGE TABLES. Lgth.i in Ft.l ILgth. lin Ft.l |Lgih.| |in Ft.l TONS lin Ft.1 19 FEET BEAM 1 2 3 4 5 1-86-5 3-790 5-71-5 7-64-0 9-56-5 11-49-0 13-41-5 9 50 60 70 80 90 15-34-0 17-26-5 74-11-3 93-30-3 112-49-3 131-68-3 150-87-3 lOOj 110 120 130 140: 150; 160i 170-12-3 170 304-51-3 189-31-3 180 323-70-3 208-50-3 190 342-89-3 227-69-3 200 362-14-3 246-88-3 210 381-33-3 266-13-3 220 400-52-3 285-32-3 230 419-71*3 19-5 FEET BEAM 2-2-125 9 4-4-25 50 6-6-375 60 8-8-5 70 10-10-625 80 12-12-75 90 14-14-875 100 16-17-0 110 18-19-125 77-43-788 97-65-038 117-86-288 138-13-538 158-34-788 178-56-038 198-77-288 120 130 140 150 160 170 180 190 219- 239- 259- 279- 299- 320- 340- 360- 4-538 25-788 47-038 68-288 89-538 16-788 38-038 59-288 200 210 380- 401- 220! 421- 230 441- 240 250 260 270 461- 481- 502- 522- 80-538 7-788 29-038 50-288 71-538 ■92-788 20-038 41-288 20 FEET BEAM 2-12 4-24 6-36 8-48 10-60 12-72 14-84 9 60 70 80 90 100 110 120 19-14 102-12 123-38 144-64 165-90 187-22 208-48 229-74 130 140 150 160 170 180 190 200 251-6 272-32 293-58 314-84 336-16 357-42 378-68 400-0 210 220 230 240 250 260 270 280 421-26 442-52 463-78 485-10 506-36 527-62 548-88 570-20 20-5 FEET BEAM 2-22-125 4-44-25 6-66-375 8-88-5 11-16-625 13-38-75 15-60-875 17-83-0 91 20 60 106 70 128 80 151 90 173 100 196 110 218 120 240 -11-125 58-963 92-213 31-463 -64-713 3-963 -37-213 70-463 130 140 150 160 170 180 190 263- 285- 307- 330- 352- 374- 397- 200 419- -9-713 -42-963 -76-213 -15-463 -48-713 -81-963 -21-213 -54-463 210 220 230 240 250 260 270 280 441-87-713 464-26-963 486-60-213 508-93-463 531-32-713 553-65-963 576-5-213 598-38-463 2-32-5 4-650 7-3-5 9-360 11-68-5 1 1-7-0 16-39-5 18-72-0 9 60 70 80 90 100 110 120 21 FEET BEAl^ 275- 21-10-5 111-17-7 134-60-7 158-9-7 181-52-7 205-1-7 228-44-7 251-87-7 130 140 150 160 170 180 190 200 298- 322- 345- 369- 392- 416- 43!»- 36-7 79-7 28-7 71-7 20-7 63-7 12-7 55-7 210 220 230 240 250 260 270 280 463-4-7 486-47-7 509-90-7 533-39-7 556-82-7 580-31-7 603-74-7 627-23-7 TONNAGE TABLES. 89 Le-th in Ft 1 TONS tr^. TONS ^^l TONS IKJ TONS r 21-5 FEET BEAM 2-43-125 9 22-12-125 130 287-86-738 210 484-58-738 2 4-86-25 60 115-75-988 140 312-47-988 220 509-19-988 3 7-35-375 70 140-37-238 150 337-9-238 230 533-75-238 4 9-78-5 80 164-92-488 160 361-64-488 240 558-36-488 5 12-27-625 90 189-53-738 170 386-25-738 250 582-91-738 6 14-70-75 100 214-14-988 180 410-80-988 260 607-52-988 7 17-19-875 110 238-70-238 190 1 435-42-238 270 632-14-238 8 19-63-0 120 263-31-488 200 ' 460-3-488 280 656-69-488 22 FEET BEAM j 1 2-54 9 23-16 130 300-65 210 506-61 2 5-14 60 120-45 140 326-41 220 532-37 3 7-68 70 146-21 150 352-17 230 558-13 4 10-28 80 171-91 160 377-87 240 583-83 5 12-82 90 197-67 170 403-63 250 609-59 6 15-42 100 223-43 180 429-39 260 635-35 7 18-2 110 249-19 190 455-15 270 661-11 _8_ 20-56 120 274-89 200 480-85 280 686-81 22-5 FEET BEAM j 1 2-65-125 91 24-22'i25 130 313-67-063 210 529-13-063 2 5-36-25 60 125-20-313 140 340-60-313 220 556-6-313 3 8-7-375 70| 152-13-563 150 367-53-563 230 582-93-563 4 10-72-5 80| 179-6-813 160 394-46-813 240 609-86-813 5 13-43-625 90l 206-0-063 170 421-40-063 250 636-80-063 6 16-14-75 lOO; 232-87-313 180 448-33-313 260 663-73-313 7 18-79-875 no! 259-80-563 190 475-26-563 270 690-66-563 8 21-51-0 120' 286-73-813 200 502-19-813 280 717-59-813 23 FEET BEAM | 1 2-76-5 9 25-30-5 130 326-90-9 210 552-6-9 2 5-590 60 129-93-9 140 355-9-9 220 580-19-9 3 8-41-5 70 158-12-9 150 383-22-9 230 608-32-9 4 11-24-0 80 186-25-9 160 411-35-9 240 636-45-9 5 14-6-5 90 214-38-9 170 439-48-9 250 664-58-9 6 16-83-0 100 242-51-9 180 467-61-9 260 692-71-9 7 19-65-5 110 270-64-9 190 495-74-9 270 720-84-9 8 22-480 120 298-77-9 200 523-87-9 280 749-3-9 1 23-5 FEET BEAM | 1 2-88125 9 26-41-125 140 369-78-138 220j 604-78-138 2 5-82-25 70 164-19-388 150 399-19-388 230 634-19-388 3 8-76-375 80 193-54-638 160 428-54-638 240 663-54-638 4 11-70-5 90 222-89-888 170 457-89-888 250 692-89-888 5 14-64-625 100 252-31-138 180 487-31-138 260i 722-31-138 6 17-58-75 110 281-66-388 190 516-66-388 270 751-66-388 7 20-52-875 120 311-7-638 200 546-7-638 280, 781-7-638 8 23-47-0 130 340-42-888 210 575-42-888 290 810-42-888 m TONNAGE TABLES. l.lfth in Ft. TONS 1 ,^„^rc: TONS KJ, TONS Kl TONS 24 FEET BEAM 1 3-6 9 27-54 140 384-76 1 220 629-86 2 6-12 70 170-32 150 415-42 230 660-52 3 9-18 80 200-92 160 446-8 240 691-18 4 12-24 90 231-58 170 476-68 250 721-78 5 15-30 100 262-24 180 507-34 260 752-44 6 18-36 110 292-84 190 538-0 270 783-10 7 21-42 120 323-50 200 568-60 280 813-70 8 24-48 130 354-16 210 599-26 290 844-36 24-5 FEET BEAM | 1 3-18125 9 28-69-125 140 400-5-663 220 655-45-663 2 6-36-25 70 176-52-913 150 431-92-913 230 687-38-913 3 9-54-375 80 208-46-163 160 463-86-163 240 719-32-163 4 12-72-5 90 240-39-413 170 495-79-413 250 751-25-413 5 15-90-625 100 272-32-663 180 527-72-663 260 783-18-663 6 19-14-75 110 304-25-913 190 559-65-913 270 815-11-913 7 22-32-875 120 336-19163 200 591-59-163 280 847-5-163 8 25-51-0 130 368-12-413 210 623-52-413 290 878-92-413 25 FEET BEAM j 1 3-30-5 91 29-86-5 140 415-52-5 220 681-48-5 2 6-61-0 70' 182-79-5 150 448-75-5 230 714-71-5 3 9-91-5 801 216-8-5 160 482-4-5 240 748-0-5 4 13-28-0 90i 249-31-5 170 515-27-5 250 781-23-5 5 16-58-5 100 282-54-5 180 548-50-5 260 814-46-5 6 19-89-0 110 315-77-5 190 581-73-5 270 847-69-5 7 23-25-5 120l 349-6-5 200 615-2-5 280 880-92-5 8 26-56-0 I30i 382-29-5 210 648-25-5 290 914-21-5 T^ 3-43-125 9 25-5 FEE T B EAM 1 31-12-125 140 431-29-088 220i 708-1-088 2 6-86-25 70! 189-18-338 150 465-84-338 230i 742-56-338 3 10-35-375 80 223-73-588 160 500-45-588 2401 777-17-588 4 13-78-5 90 258-34-838 170 535-6-838 250! 811-72-838 5 17-27-625 100 292-90-088 180 569-62-088 260 846-34-088 6 20-70-75 110 327-51-338 190 604-23-338 270 880-89-338 7 24-19-875 120| 362-12-588 200 638-78-588 2801 915-50-588 8 27-63-0 1 30, 396-67-838 210 673-39-838 2901950-11-838 26 FEET BEAM | 1 3-56 91 32-34 140 447-29 220 734-91 2 7-18 70! 195-57 150 483-25 230 770-87 3 10-74 801 231-53 160 519-21 240 806-83 4 14-36 90 267-49 170 555-17 250 842-79 | 5 17-92 100 303-45 180 591-13 260 878-75 6 21-54 1 10^ 339-41 190 627-9 270 914-71 7 25-16 120! 375-37 200 663-5 280 950-67 8 28-72 130' 411-33 210 699-1 290 986-63 1 TONNAGE TABLES. 91 EgttT iu Ft TONS tn'ft: TONS K;| TONS \tr,1:i TONS 1 26-5 FEET BEAil 3-69-125 9 33-58-125 140| 463-52-613 220 762-36-613 2 7-44-25 70 202-7-863 150 500-85-863 230 799-69-863 3 11-19-375 80 239-41-113 160; 538-25-113 240 837-9-113 4 14-88-5 90 276-74-363 170 575-58-363 250 874-42-363 5 18-63-625 100 1314-13-613 180 612-91-613 260 911-75-613 6 22-38-75 110 351-46-863 190 650-30-863 270 949-14-863 7 26-13-875 120 388-80-113 200 687-64-113 280 986-48-113 8 29-830 130, 426-19-363 210] 725-3-363 290 1023-81-363 27 FEET BEAM | 1 3-82-5 9 34-84-5 140! 480-5-1 220 790-25-1 2 7-71-0 70 208-58-1 150i 518-78-1 230 829-4-1 3 11-59-5 80 247-37-1 160 557-57-1 240 867-77-1 4 15-48-0 90 286-16-1 170 596-36-1 250 906-56-1 5 19-36-5 100 324-89-1 180 635-15-1 260 945-35-1 6 23-25-0 110 363-68-1 190 673-88-1 270 984-14-1 7 27-13-5 120 402-47-1 200 712-67-1 280 1022-87-1 8 31-32-0 130 441-26-1 210 751-46-1 290 1061-66-1 27-5 FEET BEAM | 1 4-2-125 9 36-19-125 150: 537-1-688 230 858-77-688 2 8-4-25 80 255-40-938 160 577-22-938 240 899-4-938 3 12-6-375 90 295-62-188 170 617-44-188 250 939-26-188 4 16-8-5 100 335-83-438 180 657-65-438 260 979-47-438 5 20-10-625 110 376-10-688 190' 697-86-688 270 1019-68-688 6 24-12-75 120 416-31-938 200! 738-13-938 280 1059-89-938 28-14-875 130 456-53-188 210 778-35-188 290 1100-17-188 8 32-17-0 140 496-74-438 220; 818-56-438 300,1140-38-438! 28 FEET BEAM | 1 4-16 91 37-50 150 555-44 230 889-8 2 8-32 80 263-52 160 597-16 240 930-74 3 12-48 90 305-24 170 638-82 250 972-46 4 16-64 100 346-90 180 680-54 260 1014-18 5 20-80 110 388-62 190 722-26 270 1055-84 6 25-2 120| 430-34 200 763-92 280 1097-56 7 29-18 130, 472-6 210 805-64 290 1139-28 8 33-34 140; 513-72 220 847-36 1 300. 1181-0 1 28-5 FEET BEAM | 1~ 4-30-125 9 38-83-125 150 574-18-013 230 919-78-013 2 8-60-25 SO 271-71-263 160 617-37-263 240 963-3-263 3 12-90-375 90 314-90-513 170 660-56-513 25011006-22-513 4 17-26-5 100 358-15-763 180 703-75-763 260 1049-41-763 5 21-56-625 110 401-35-013 190 747-1-013 270 1092-61-013 6 25-86-75 120 444-54-263 200 790-20-263 280 1135-80-263 7 30-22-87-5 130 487-73-513 210 833-39-513 290 1179-5-513 8 34-53-0 140 530-92-763 220 876-58-763 | 300 1222-24-763 V2 TONNAGE TABLES. Lsth. in Ft. TONS jLgth.| |in Ft.| TONS jLgth.i liu Ft.l [in Ft.l 29 FEET BEAM 4-44 8-89' 18-39' 17-84' 22-34' 26-79- 31-29- 35-74- 9 80 90 100 110 120 130 140 40-24-5 280-3-3 324-72-3 369-47-3 414-22-3 458-91-3 503-66-3 548-41-3 150 160 170 180 190 200 210 220 593-16-3 637-85-3 682-60-3 727-35-3 772-10-3 816-79-3 861-54-3 906-29-3 230 240 250 260 270 280 290 300 951-4-3 995-73-3 1040-48-3 1085-23-3 1129-92-3 1174-67-3 1219-42-3 1264-17-3 29-5 FEET BEAM 4-59-125 9-24-25 13-83-375 18-48-5 23-13-625 27-72-75 32-37-875 37-3-0 91 41-62-125 801 288-36-288 901 334-63-538 100! 380-90-788 llOj 427-24-038 120 473-51-288 130 519-78-538 140 566-11-788 150| 612- 160, 658- 170, 704- 180 751- 190 797- 200 843- 210 890- 220 936- 39-038 66-288 93-538 26-788 54-038 81-288 14-538 41-788 230; 982-69-038 240'l029-2-288 2501075-29-538 2601121-56-7 2701167-84-038 2801214-17-2 290 1260-44-538 300,1306-71-788 30 FEET BEAM 4-74 9-54 14-34 19-14 23-88 28-68 33-48 38-28 9 90 100 110 120 130 140 150 43-8 344-64 392-52 440-40 488-28 536-16 584-4 631-86 160 170 180 190 200 210 220 230 679-74 727-62 775-50 823-38 871-26 919-14 967-2 1014-84 240 250 260 270 280 290 300 310 1062-72 1110-60 1158-48 1206-36 1254-24 1302-12 1350-0 1397-82 30-5 FEET BEAM 4-89-125 9-84-25 14-79-375 19-74-5 24-69-625 29-64-75 34-59-875 39-55-0 9 90 100 110 120 130 140 44-50-125 354-73-463 404-24-713 453-69-963 503-21-213 552-66-463 602-17-713 150; 651-62-963 160 170 180 190 200 210 220 230 701- 750- 800- 849- 899- 948- 998- 1047- -14-213 -59-463 -10-713 -55-963 7-213 52-463 3-713 48-963 24011097-0-213 2501146-45-463 2601195-90-713 2701245-41-963 2801294-87-213 290;i344-38-463 3001393-83-713 3101443-34-963 5-10-5 10-21-0 15-31-5 20-42-0 25-52-5 30-63-0 35-73-5 40-84-0 9, 90 lOOl no. 120 130 140 150 31 FEET BEAM l60 46-0-5 364-91-7 416-8-7 467-19-7 518-30-7 569-41-7 620-52-7 671-63-7 170 180 190 200 210 220 230 722- 773- 825- 876- 927- 978- 1026- 1 080- 74-7 24( 85-7 25( 2-7 26f 13-7 270 24-7 280 35-7 290 49-7 300 57-7 310 1131- 1182- 1233- 1285- 1336- 1387- 1438- 1489- .68-7 -79-7 -90-7 7-7 ■ 18-7 29-7 40-7 TONNAGE TABLES. '93 LfTth.. in Ft.l TONS l^;. bn Ft. ILcth. in Ft., TONS 31-5 FEET BEAM 5-26-125 10-52-2O 15-78'375 21-10-5 26-36-625 31-62-75 36-88-875 42-21-0 9 47-47-125 100 428-3-738 110 480-76-988 120 533-56-238 130 586-35-488 140 639-14-738 150 691-87-988 160 744-67-238 170j 797-46-488 25011219-68-488 180' 850-25-738'260! 1272-47-738 190 903-4-988 270ll325-26-988 200 955-78-238:2801378-6-238 210'1008-57-488;290|l430-79-488 220 1061-36-738 3001483-58-738 2301114-15-988 310!l536-37-988 240,1166-89-238 3201589-17-238 3-2 FEET BEA3I 5-42 10-84 16-32 21-74 27-22 32-64 38-12 43-54 49-2 100 110 120 130 140 150 160 170 180 440-9 494-53 549-3 603-47 657-91 712-41 766-85 821-35 875-79 190 200 210 220 230 240 250 260 270 930-29 984-73 1039-23 1093-67 1148-17 1202-61 1257-11 1311-55 1366-5 280] 290 300 310 320 330 340 350 360 1240-49 1474-93 1529-43 1583-87 1638-37 1692-81 1747-31 1801-75 1856-25 32-5 FEET BEAM 5-58-125 11-22-25 16-80-375 22-44-5 28-8-625 33-66-75 39-30-875 44-89-0 50-53-125 100 452-26-063 110' 508-43-313 120 564-60-563 130 620-77-813 140 677-1-063 150 733-18-313 160 789-35-563 170 845-52-813 180' 901-70-063 190 200 957 1014 21011070 220|1126 23011182 2401238 250 1295 2601351 270140 280 1463-54-563 2901519-71-813 3001575-89-063 3101632-12-313 3201688-29-563 3301744-46-813 340!l800-64-063 3501856-81-313 -37-313 3601913-4-563 -87-313 -10-563 -27-813 -45-063 -62-313 -79-563 -2-813 -20-063 -37 33 FEET BEA3I 5-74-5 11-55-0 17-35-5 23-160 28-90-5 34-71-0 4f)-51-5 46-32-0 52-12-5 100, 110: 120 130; 140; 150 160 170; 180 464-52-9 522-45-9 580-38-9 638-31-9 696-24-9 754-1 7-9 812-10-9 870-3-9 927-90-9 190 200 210 [220 230 240 ,250 !260': :270 985- 1043- 1101- 1159- 1217- 1275- 1333- 1391- 1449- 83-9 76-9 69-9 62-9 55-9 ■48-9 41-9 -34-9 -27-9 2801 290 300! 310' 320 330' 340 .350 360 1507-20-9 1565-13-9 1623-6-9 1680-93-9 1738-86-9 1796-79-9 1854-72-9 1912-65-9 1970-58-9 33 -S FEET BEAM 5-91-125 11-88-25 17-85-375 23-82-5 29-79-625 61 100: 35-76-75 41-73-875 47-71-0 53-68-125 476-89-888 110: 536-61-138 120; 596-32-388 130 656-3-638 140 715-68-888 150 775-40-138 160 835-11-388 170! 894-76-638 180' 954-47-888 1901014-19-138 200 1073-S4-388 94 TONNAGE TABLES. iLfrth. lin Ft.! Lgtiv n FtJ iLgtli. JinFt. TONS iLgth. lin Ft. TONS 33-5 FEET BEAIM (concluded) 210ill33-o5-638 220ill93-26-888 230J1252-92-138 240'l312-63-388 i!50 1372-34-638 260J432-5-888 2701491-71-138 2801551-42-388 290,1611-13-638 3001670-78-888 3101730-50-138 3201790-21-388 3301849-86-638 3401909-57-888 3501969-29-138 360 2029-0-388 34 FEET BE.AM 6-14 12-28 18-42 24-56 30-70 36-84 43-4 49-18 55-32 100 110 120 130 140 150 160 170 180 489-42 550-88 612-40 673-86 735-38 796-84 858-36 919-82 981-34 190 200 210 220 230 240 250 260 270 1042-80 1104-32 1165-78 1227-30 1288-76 1350-28 1411-74 1473-26 1534-72 280 290' 300 310 320 330l 3401 350' 360 1596-24 1657-70 1719-22 1780-68 1842-20 1903-66 1965-18 2026-64 2088-16 34-5 FEET BEAM 6-31-125 100 12-62-25 110 18-93-375 120 25-30-5 130 31-61-625 140 37-92-75 150 44-29-875 160 50-61-0 170 56-92125 180 502-5-413 565-34-663 628-63-913 691-93-16.- 755-28-413 818-57-663 881-86-913 945-22-163 1008-51-413 1901071- 200'll35- 2101198- 2201261- 2301325- 2401388- 2501451- 260|1515- 270|1578- 80-663 15-913 45-163 74-413 9-663 38-913 68-163 3-413 I 32-663 2801641-6] -913 2901704-91-163 3001768-26-413 3101831-55-663 3201894-84-913 330il958-20-163 340 2021-49-413 350 2084-78-663 360 2148-13-913 35 FEET BEAM 6-48-5 13-3-0 19-51-5 26-6-0 32-54-5 39-90 45-57-5 52-12-0 58-60-5 100 110 120 130 140 150 160 170 180 514-71-5 190 579-86-5 200 645—7-5 210 7-10-22-5 220 775-37-5 230 840-52-5 240 905-67-5 250 970-82-5 260 1036-3-5 270 1101- 1166- 1231- 1296- 1361- 1426- 1492- 1557- 1622- 18-5 280; 33-5 290 48-5 300 63-5 310 78-5 320 93-5 330! 14-5 3401 29-5 350| 44-5 360 1687-59-5 1752-74-5 1817-89-5 1883-10-5 1948-25-5 2013-40-5 2078-55-5 2143-70-5 2208-85-5 35-5 FEET BEA:M 6-66-125 1 3-38-25 20-10-375 26-76-5 33-48-625 40-20-75 46-86-875 53-59-0 60-31-125 100 110 120, 130' 140 150 160 170 180 527-52-838 594-56-088 661-59-338 728-62-588 795-65-838 862-69-088 929-72-338 996-75-588 1063-78-838 1901130- 2001197- 210,1264- 220,1331- 2301399- 2401466- 2501533- 2601600- 270;1667- 82-088 85-338 ■88-588 91-838 1-088 4-338 7-588 10-838 14-088 2801734-17-338 290;i 801-20-5 300.1868-23-838 310,1935-27-088 320 2002-30-338 330i2069-33-588 340|2136-36-838 350|2203-40088 360'2270-43-338 TONNAGE TABLES. 95 TgtT in Ft. TONS IFI 1 TONS {^^^) TONS iH^y;: tons ( 36 FEET BEAM | 1 6-84 110 609-37 210 1298-71 310 1988-11 2 13-74 120 678-31 220 1367-65 320 2057-5 3 20-64 130 747-25 230 1436-59 330 2125-93 4 27-54 140 816-19 240 1505-53 340 2194-87 5 34-44 150 885-13 250 1574-47 350 2263-81 6 41-34 160 954-7 260 1643-41 360 2332-75 7 48-24 170 1023-1 270 1712-35 370 2401-69 8 55-14 180 1091-89 •280 1781-29 380 2470-63 9 62-4 190 1160-83 290 1850-23 390! 2539-57 100 540-43 200 1229-77 300 1919-17 too 2608-51 36-5 FEET BEAM | 1 7-8-125 110 624-29-613 210.1332-90-113 310 2041-56-613 2 14-16-25 120 695-16-863 220,1403-77-363 320 2112-43-863 3 21-24-375 130 766-4-113 230.1474-64-613 330 2183-31-113 4 28-32-5 140 836-85-363 240,1545-51-863 340 2254-18-363 5 35-40-625 150 907-72-613 250;i616-39-113 350 2325-5-613 6 42^48-75 160 978-69-863 26011687-26-363 36012395-86-863 7 49-56-875 170 1049-47-113 2701758-13-613 370|2466-74-113 8 56-65-0 180 1120-34-363 28011829-0-863 380 2537-61-363 9 63-73-125 190 1191-21-613 29011899-82-113 390 2608-48-613 100 553-42-363 200 1262-8-863 300,1970-69-363 400 2679-35-863 37 FEET BEA3I | 1 7-26-5 120 712-16-1 220 1440-34-1 3201 2168-52-1 330 2241-35-1 2 14-530 130 784-93-1 230 1513-17-1 3 21-79-5 140 857-76-1 240 1586-0-1 340 2314-18-1 4 29-12-0 150 930-59-1 250 1658-77-1 350 2387-1-1 5 36-38-5 160 1003-42-1 260j 1731-60-1 360 2459-78-1 6 43-65-0 170 1076-25-1 270| 1804-43-1 370 2532-61-1 7 50-91-5 180 1149-8-1 280| 1877-26-1 380 2605-44-1 8 58-24-0 190 1221-85-1 290 1950-9-1 390 2678-27-1 9 65-50-5 200 1294-68-1 300 2022-86-1 400 2751-10-1 110 639-33-1 210 1367-51-1 310 2095-69-1 410' 2823-87-1 37-5 FEET BEAM | 1 7-45-125 120 729-28-688 220 1477-29-188 320,2225-29-688 2 14-90-25 13a 804-9-938 230 1552-10-438 33012300-10 938 3 22-41-375 140 878-85-188 240 1626-85-688 340 2374-86-188 4 29-86-5 150 953-66-438 250 1701-66-938 350|2449-67-438 5 37-37-625 160 1028-47-688 260 1776-48-188 360 2524-48-688 6 44-82-75 170 1103-28-938 270 1851-29-438 37012599-29-938 7 52-33-875 180 1178-10-188 280 1926-10-688 380:2674-11-188 8 59-79-0 190 1252-85-438 290 2000-85-938 390 2748-86-438 9 67-80-125 200 1327-66-688 300 2075-67-188 400 2823-67-688 110 654-4'~-'438 210 1402-47-938 310 215O-48-438'410l2898-48-938| 96 TOXXAGE TABLES. tr^,\ TONS j^f^W TONS |-«-;| TONS K;, TONS 38 FEET BEAM 1 7-64 120 746-54 220: 1514-62 320 2282-70 2 15-34 130 823-36 230! 1591-44 330 2359-52 3 23-4 140 900-18 240| 1668-26 340 2436-34 4 30-68 150 977-0 250 1745-8 350 2513-16 6 38-38 160 1053-76 260 1821-84 360 2589-92 6 46-8 170 1130-58 270 1898-66 370 2666-74 7 53-72 180 1207-40 280 1975-48 380 2743-56 8 61-42 190 1284-22 290 2052-30 390 2820-38 9 69-12 200 1361-4 3001 2129-12 400 2897-20 110 669-62 210 1437-80 310] 2205-88 410 2974-2 38-5 FEET BEAM 1 1 7-83125 1201 763-93-013 220 1552-39-513 320 2340-80013 2 15-72-25 130: 842-78-263 230 1631-24-763 330 2419-65-263 3 23-61-375 140i 921-63-513 240 1710-10013 340l2498-50-513 4 31-50-5 1501000-48-763 250 1788-89-263 350 2577-35-763 5 39-39-625 1601079-34-013 260 1867-74-513 360 2656-21-013 6 47-28-75 170'll58-19-263 270 1946-59-763 370 2735-6-263 7 55-17-875 180'l237-4-513 280 2025-45-013 380 2813-85-513 8 63-7-0 190'l315-83-763 290 2104-30-263 390 2892-70-763 9 70-90-125 2001394-69-013 300 2183-15-513 400 2971-56-013 110 685-13-763 2101473-54-263 310 2262-0-763 410'30o0-41-263l 39 FEET BEAM t 1 8-8-5 120, 781-50-3 220 1590-54-3 320 2399-58-3 2 16-170 130 862-41-3 230 1671-45-3 330 2480-49-3 3 24-25-5 140 943-32-3 240 1752-36-3 340 2561-40-3 4 32-34-0 150 1024-23-3 250 1833-27-3 350 2642-31-3 5 40-42-5 160 1105-14-3 260 1914-18-3 360 2723-22-3 6 48-51-0 170 1186-5-3 270 1995-9-3 370 2804-13-3 7 56-59-5 180 1266-90-3 280 2076-0-3 380 2885-4-3 8 64-68-0 190 1347-81-3 290 2156-85-3 390 2965-89-3 9 72-76-5 200 1428-72-3 300 ^2237-76-3 400 3046-80-3 110 700-59-3 210' 1509-63-3 310 2318-67-3 410 3127-71-3 39-5 FEET BEAM | 1 8-28-125 120 799-20038 22011629-12-538 320 2459-5-038 2 16-56-25 130 882-19-288 2301712-11-788 330 2542-4-288 3 24-84-375 I40i 965-18-538 2401795-11-038 340 2625-3-538 4 33-18-5 150!l048-l 7-788 250 1878-10-288 350 2708-2-788 5 41-46-625 1601131-17-038 260 1961-9-538 360 2791-2-038 G 49-74-75 1701214-16-288 270 2044-8-788 370 2874-1-288 7 58-8-875 1801297-15-538 280 2127-8-038 380 2957-0-538 8 66-370 190il380-14-788 29012210-7-288 390 3039-93-788 9 74-65-125 2001463-14-038 300*2293-6-538 400 3122-93-038 110 716-20-788 2101546-13-288 3102376-5-788 410^ 2205-92-288 TONNAGE TABLES. 97 Lfrrh. in Ft.l fin Ft. ILeth. in Ft.' ILgth. hn Ft., 40 FEET BEAJtf li 2J 31 4 5 6 7 8 9 120 8-48 130 902-12 230 17-2 140 987-22 240 25-50 150 1072-32 250 34-4 160 1157-42 260 42-52 170 1242-52 270 51-6 180 1327-62 280 59-54 190 1412-72 290 68-8 200 1497-82 300 76-56 210 1582-92, 310 817-2 220 1668-8 320 1753-18 1838-28 1923-38 2008-48 2093-58 2178-68 2263-78 2348-88 2434-4 2519-14 3301 340 350! 360! 370 380; 390 400 410 420 2604-24 2689-34 2774-44 2859-54 2944-64 3029-74 3114-84 3200-0 3285-10 3370-20 40 -J FEET BEAit 1; 8-68-125 21 17-42-25 3| 26-16-375 4l 34-84-5 51 43-58-625 6! 52-32-75 71 61-6-875 8! 69-75-0 9' 78-49-125 120 834-89-963 130 922-19 1401009-42 150'1096-65 160'll83-88 213 463 713 963 260 1701271-18-213|27O 1801358-41 1901445-64 200'l532-87 210:i620-17 2201707-40 463 137 963 213 463 230 240 250 280 290 300 310 320 1794-63 1881-86 1969-16 2056-39 2143-62 2230-85 2318-15 2405-38 2492-61 2579-84 713 963 213 463 713 963 213 463 713 963 330 340 350 360 370 380 390 400 410 420 2667-14213 2754-37-463 2841-60-713 2928-83-963 3016-13-213 3103-36-463 3190-59-713 3277-82-963 3365-12-213 3452-35-463 41 FEET BEAM 1 2 3 4 5 6 7 8 9 120 8-8«-5 17-83-0 26-77-5 35-72-0 44-66-5 53-61-0 62-55-5 71-50-0 80-44-5 853-1-7 130 140 150 160 170 180 190 200 210 220 942- 1031- 1121- 1210- 130O 1389- 1478- 1568 1657 1747 -40-7 -79-7 -24-7 -63-7 -8-7 -47-7 -86-7 -31-7 -7a-7 -15-7 .^30 240; 250' 260| 270 280 290 300 3101 320' 1836-54-7 1925-93-7 2015-38-7 2104-77-7 2194-22-7 2283-61-7 2373-6-7 2462-45-7 2551-84-7 2641-29-7 330 340 350 360 370| 380 390' 400; 410! 420' 2730-68-7 2820-13-7 2909-52-7 2998-91-7 3088-36-7 3177-75-7 3267-20-7 3356-59-7 3446-4-7 3535-43-7 41-. 5 FEET BEA3I 9- 18- 27- 36- 45- 54- 64- 73- 9| 82- 120871- 15-125 30-25 45-375 ■60-5 75-625 90-75 •11-875 -27-0 ■42-125 ■18-988 130i 962-76-238 140*1054-39-488 150 1146-2-738 160 170 180 190 1237-59-988 1329-23-238 1420-80-488 1512-43-738 20011604-6-988 210il695-64-238 2201787-27-488 2301878-84 2401970-47 250 2062-11 260^2153-68 2702245-31 2802336-88 2902428-52 3002520-15 310!2611-72 3202703-35 ■738 988 238 488 738 988 238 488 '738 330 340 350 360 370 380 390 400 410 988 420 2794-93-238 2886-56-488 2978-19-738 3069-76-988 3161-40-238 3253-3-488 3344-60-73 3436-23-988 3527-81-238 3619-44-488 98 TONNAGE TABLES. ILgth.l lin Ft.| LfthX lu Ft.1 K.1 |in Ft.' 42 FEET BEAM 1 9-36 2 18-72 3 28-14 4 37-50 5 46-86 6 56-28 7 65-64 8 75-6 9 84-42 120 889-47 130 983-31 140j loO! 1601 170| 180 190 200 210 220 230 240 1077-15 1170-93 1264-77 1358-61 1452-45 1546-29 1640-13 1733-91 1827-75 1921-59 2015-43 250 260 270 280 290 300 310 320 330 340 350 2109-27 2203-11 2296-89 2390-73 2484-57 2578-41 2672-25 2766-9 2859-87 2953-71 3047-55 360 370 380 390 400 410i 4201 430 440| 450 460i 3141-39 3235-23 3329-7 3422-85 3516-69 3610-53 3704-37 3798-21 3892-5 3985-83 4079-67 42-6 FEET BEAM 1 2 3 4 5 6 7 8 9 120 130 9- 19- 28- 38- 48- 57- 67- 76- 86- 907- 1004- 57-125 20-25 77-375 40-5 3-625 60-75 23-875 ■81-0 44-125 87-313 0-563 140|1100-7-813 150;il96-15-063 1601292-22-313 170'l388-29-563 180,1484-36-813 ]90j 1580-44 -063 200jl676-51-313 2101772-58-563 220ll868-65-813 2301964-73-063 240 2060-80-313 250|2156-87-563 260 2253-0-813 270!2349-8-063 280 2445-15-313 29012541-22-563 300 2637-29-813 310 2733-37-063 320 2829-44-313 330 2925-51-563 340 3021-58-813 350 3117-66-063 360 370 380 390 400 410 420 430 440 450 460 3213-73-313 3309-80-563 3405-87-813 3502-1-063 3598-8-313 3694-15-563 3790-22-813 3886-30-063 3982-37-313 4078-44-563 4174-51-813 43 FEET BEAM 9-78-5 19-63-0 29-47-5 39-32-0 49-16-5 59-10 68-79-5 78-64-0 88-48-5 926-43-9 9 120 130'l 024-76-9 140 150 160| 170, 180 190, 200i 210, 220' 230' 240' 1123-15-9 1221-48-9 1319-81-9 1418-20-9 1516-53-9 1614-86-9 1713-25-9 1811-58-9 1909-91-9 2008-30-9 2106-63-9 250 260 270 280 290 300 310 320 330 340 350 2205-2-9 2303-35-9 2401-68-9 2500-7-9 2598-40-9 2696-73-9 2795-12-9 2893-45-9 2991-78-9 3090-17-9 3188-50-9 360 370 380 390 4oo: 410 420 430 440 450 460 3286- 3385- 3483- 3581- 3680- 3778- 3876- 3975- 4073- 4172- 4270- -83-9 22-9 55-9 88-9 27-9 60-9 93-9 32-9 65-9 4-9 37-9 43-5 FEET BEAM 10-6-125 , home trade passenger steamers. Form survey 3 (excursion) is given for steamers plying along the coast during daylight between any of the places mentioned in column 1 of the following table of limits and the places set opposite to them in column 4 of the same table. Form survey 4 (river) is given for steamers plying between any of the places mentioned in column 1 of the table and the places set opposite to them in column 3. [ Form survey 5 (rivers and lakes) is given for steamers plying in the smooth-water limits lying between the places mentioned in column 1 and the places set opposite to them in column 2. Table of Plying Limits for Excursion, River, and PARTIALLY SMOOTH WATER CERTIFICATES. Form Survey 5. Form Survey 4. ^^^.^ g^^^g^ g^ Col. 1. Col. 2. COL, 3. qql 4 Name of Port Smooth Water PartiaUy Smooth Excursion Limits Limits Water Lmiits Aberdeen . All withinAber- deen Nil Nil Bristol Portishead TheHohnes . Tenby or Ilfra- combe BO^VT^ESS Anywhere on Nil Nil the Lake 1 1 Boston . Above the El- The Lvnn Well Grimsby or WeUs | bow Buov Light Ship Berwick (N.) Nil . ' . Within a line from Berwick to An- struther See Leith Belfast Hoh-wood Within Carrick- Larne and the fergus and Bangor, South Rock and to Grooms- Lighthouse point Barrow Walney Islands Places within More- Liverpool | cambe and Lan- caster Bays Brigg (HuLL)jSame as Hull . Nil Nil 104 BOAED OF TRADE. EEGULATIONS FOR SHIPS. Table of Plying Limits for Excursion, River, and PARTIALLY SMOOTH WaTER CERTIFICATES (continued). Col. 1. Form Survey 5. Col. 2. Form Survey 4. COL. .3. Form Survey 3 CoL. 4. Excursion Limits Name of Port Smooth "SVater Limits Partially Smooth "Water Limits Carlisle Above Carlisle Dumfries and Whitehaven or Southerness Kirkcudbright Cardiff Penarth . The Holmes . Tenby Carnarvon . Inside Carnar- von Bar and Priestholm Island Conway . Nil Conway On the river Any place between Liverpool Conway Priestholm Island and Carnarvon Bar Cork A line from A line from Cork Youghal or Kin- Camden to Head to Poor sale Carlisle Forts Head Campbeltown In the Harbour Nil See Glasgow ^Dartmouth . River Dart Nil ... Inside a line from Start Point to s Portland Bill (Dover . Nil . . . Nil . , . Rye or Margate Dundee New Railway Broughty Castle . Xlontrose or Fife- Bridge at Dun- ness dee Drogheda . Nil . Nil ... Dundalk and Bal- briggan DUNDALK Nil . Nil . . . Drogheda and Kilkeel Dublin . Nil . Kingstown . Howth or Wick- low Douglas (I.M.) Nil . .. Nil See Liverpool Fleetwood . All above the Places within More- Liverpool Upper Light- cambe and Lan- house caster Bays Falmouth . A line from Zose For special St. 4a Lizard or Start Point to Pen- Declarations, Black Points dennis Point Head or Gull Rock Folkestone . Nil . Nil Rye or Margate Galway Kinvarra Kilkerrin or Lis- cannon Bays, inside the Arran Isles Gloucester. ; Sharpness Pointl Bristol, Newport, or Tenby or Ufra- any place abt)ve combe the Holmes Gaixs- Bop.oiGH (see Hull) BOARP OF TRADE REGULATIONS FOR SHIPS. 105 Table of Plylxg Lnrrrs for Excfrsiox, Rn"ER, a:n"d | PARTIALLY Smooth Water Certificates (continued). 1 Col. 1. Form Survey 5. Col. 2. Fonn Survey 4. p^rrn^ Snryey .3. Name of Port Smooth Water Limits Partially Smooth Water Limits Excursion Limits Glasgow Dunoon . Cumbray and Skip- Inverness per ness Crinan and Cale- donian Canals, and to a line from Ayr to Campbeltown, inside the Island of Arran Grimsby Hull and New Grimsby Scarborough or Holland Lvnn GOOLE . Hull Grimsby . . See Hull Hartlepool . Nil . Nil . . . jNewcastle or Scar- borough Hull Hull and New Grimsby . . Lynn or Scar- Holland borough IXYERXESS . Inwards to Fort Outwards to Lough Nil William Cromarty, Naini, and Three Kings ; inwards to South End of Loch Linnhe and West End of Sound ot MuU Ipswich Languard Fort Walton-on-the-Naze Orford Ness and Walton-on-the- Naze Lancaster . Lancaster Har- Places within More- Whitehaven or bour cambe and Lan- Liverpool caster Bays Leith . Queensferry . North Berwick and Fife Ness to St. Anstruther Abb's Head Limerick Fojnes . Kilcradine Light- iLoophead and house Tralee LlTTLEHAMP- River Amn. Nil Nil TOX above Little- hampton Pier LOXDOXDERRY MoviUe . Nil Port Rush and Malinhead Lowestoft . Nil . Nil ... 'Cromer or Ald- 1 borough The Bell Buoy andjAnv place within Liverpool . The Rock Lighthouse N. W. Light Ship the Menai Straits or to Fleetwood LOXDOX . Gravesend A line from St. Osyth Harwich or Dover Point to Fore Ness 106 BOARD OF TRADE REGULATIONS FOR SHIPS. Table of Plying Limits for Excursion, River, and | PARTIALLY Smooth Water Certificates (continued J. 1 Col. 1. Form Suryey 5. COL. 2. Form Survey 4. CoL. 3. Form Survey 3. COL. 4. Excursion Limits Name of Port Smooth Water Limits Partially Smooth Water Limits MlLFOKD Dale Bay . St. Anne's Light- Swansea or St. house I Da^'id's Head Norwich Yarmouth . Nil . . . Nil Neath . Nil . . . Swansea . . Tenby Newry . Warren Point Carlingford and Dundalk and Kil- Whitehouse Point 1 keel Newcastle, T3-nemouth Bar Nil , . . 'Berwick or Scar-| North anu borough South Shields Padstow Padstow Har- A line from Stepper bour, above a' Point to Trebethe- line from Gun rick Point Point to Brea Hill Penzance For special St. 4a'Cape Cornwall or Declarations, a linel Falmouth drawn from Mouse- hole to the Eastern Point of St. Mi- chael's Mount Portsmouth . Inside Ports- St. Helen's and the Weymouth West mouth Har- Needles, within the to Brighton East bour Isle of Wight, and to Langston Har- bour. For small launches not carry- ing boats : In sum- mer, a line from Ryde to Langston Harbour, inside the Isle of Wight, to Hurst Castle; in winter, Spithead Preston Lytham . . Nil ... Barrow or Liver- pool Poole . In the Harbour Nil 1 Weymouth or Portsmouth Plymouth . Inside Drake's The Breakwater . Lizard or Start Island Points Rochester . Sheerness andThe Nore and Mar- Dover or Ilar-j to Whitstable,' gate (s<;e London) wich inside Sheppey Swansea Nil . Neath . Ilfracombeor Mil- ford BOARD OF TRADE REGULATIONS FOR SHIPS. 107 Table of Plying Limits for Excursion-, River, and PARTIALLY SMOOTH WaTER CERTIFICATES (concluded). Col. 1. Name of Port Form Survey 5. Col. 2. Smooth. Water Limits Form Survey 4. COL. 3. Partially Smooth Water Tiimits Form Survey 3. COL. 4. Excursion Limits Sunderland . Stockton Southampton Sunderland Bar Nil . Calshot . Nil Nil ... St. Helen's and the Needles, inside the Isle of Wight, and to Langston Har- Scarborough or Berwick Bridlington or Newcastle Weymouth or Brighton ♦ bour See Port.smouth for limits for small scarborouoh Teignmouth . Waterford . Wigtown Nil . Teignmouth Harbour Passage . Nil *. launches Nil NH ... Dunmore Within Wigtown Bav Newcastle or Hull Portland Bill or Start Point Dungarvan and Cringley Mull of Galloway or Southerness WiSBEACH (see Boston) Weymouth . White V Nil . Nil . Portland Harbour . Nil Portsmouth or Start Point Bridlington or Newcastle Examination of Hulls. Passenger vessels are to be surveyed once a year. New steamships are to be surveyed before the hull is com- plete, and before the paint and cement are put on, as well as when complete. Collision water-tight bulkheads must be fitted in all sea- going steamers. Screw tunnels of all iron passenger steam vessels should be made of iron and made water-tight. A water-tight door should be fitted at the fore end of the tunnel, arrangements being made so that it can be opened from the upper or main deck ; and if there are man-holes in the floor they must be made water-tight, and proper arrange- ments made so as to let the water off the floor of the tuimel. The maximum period for which a steamer's certificate of registry is granted is 12 months. 108 BOARD OF TRADE REGULATIONS FOR SHIPS. Boats. Sea-going ships are to be provided, according to their ton- nage, with boats, duly supplied with all requisites for use, and not fewer in number nor less in cubical contents than the boats — the number and cubical contents of which are specified in the following table — for the class to which the ship belongs. Sea-going ships carrying more than 10 passengers must be provided, in addition to the boats hereinbefore mentioned, with a life boat, unless one of the boats heretofore required is ren- dered buoyant after the manner of a life boat. Table of the Dimexsiozs's of Boats REQrrRED to ee CAERIEB BY PaSSEXGER StEAMEES. in s *^ r Either 1 c*-l O Dimensions 2 Dimeusious 1 •n ! .2 5 , .0 c , 66 3 i 5 5 a- 1 -a 1 and upwards 1 2 1 ft. in. 18 24 27 ft. in. 5 6 5 6 8 6 ft. in. 2 3 2 6 3 8 cub. ft. 13.3-7 396-0 504-9 1,034-6 1 2 2 ft.in.ift.in. 18 5 6 24 5 6 22 5 6 ft. in. 2 3 2 6 2 6 cub. ft. ' 133-7 396-0 363-0 892-7 999-6 *" 2 Life 28 8 6' 3 6 999-6 2 Life 28 8 6 I j$ e 6 Boats of ... 2,034-2 7 Boats of . . 1,892-3| 00 -^ 1 ,18 5 6 2 2Q 0, 6 6 1 Life 26 0| 8 2 3 133-7 2 8 540-8 3 8 457-6 1 118 01 5 6 2 24 5 6 2 |22 OJ 5 6 2 3 2 6 2 6 133-7 540-8 363-0 4 Boats of . . 1,132-1 5 Boats of . . 1,037-0! o o CO o -l-l o o i-0 1 !18 Oio 6 2 124 0; 5 6 1 Life(26 0} 8 2 3 1 133-7 2 6 ! 396-0 3 8 ! 457-6 1 2 2 18 24 22 5 6 2 3j 133-7 5 6 i 2 6 1 396-0 5 6 2 61 3630 4 Boats of . . 987-3 5 Boats of . . 892-7 | o o o CO 1 16 5 6 2 24 5 6 1 Life 25 7 2 3| 118-8 2 6 396-0 3 6 367-5 1 il6 2 124 2 :22 5 6 5 6 5 6 2 3 2 6 2 6 118-8 396-0 363-0 4 Boats of . . 882-3 | 5 Boats of . . 877-8 j BOAKD OF TEADE REGULATIONS FOR SHIPS. 10C> Table of the Dimensions of Boats required to be CARRIED BY PASSENGER StEAMERS (concluded). II o Either ■2 0-- ft. in. 1 16 1 22 1 Life 22 Dimensioas pq ft. in. 5 6 5 6 6 6 ft. in.; cub. ft. 2 3: 118-8 2 5 I 175-4 3 3 278-9 3 Boats of . 573-1 o •^ o 1 14 5 2 2 . 91-0 1 Life'20 6 013 0! 216-0 Or tz;m Dimensions « ft. in. ft. in, 16 5 6 22 0}5 6 22 5 6 Q ft. in. 2 3 2 5 2 6 c: S 65 cub. ft. 118-8 175-4 363-0 4 Boats of 657-2 2 Boats of 307-0 o ^1 o ' 1 il4 0|5 0|2 2 1 Life 16 5 612 9 91-0 145-2 2 Boats of 236-2 14 0. 5 '22 5 6 2 2 2 6 91-0 363-0 3 Boats of 454-0 14 5 0|2 2 18 5 6 2 4 91-0 277-2 3 Boats of 368-2 'S .9 I 1 Life 14 0'; 5 r2 2 \' 91-0 If the nunibei- of boats in tliis column are carried, one of them must be a launch of at least the capa- city named. No steam life- boat will be permitted. If the number of boats in this column are carried, their cubical contents (equal in the aggregate to the cubical contents required) may be spread in any way over the whole number of boats. The life boat or life boats must be the largest boats. If owners wish to carry a fewer number of boats, or wish to substitute rafts, &c., application must be made to the Board of Trade. To ascertain the cubical contents of a boat, take the length and breadth outside and the depth inside, multiply them into each other, and then that product by the factor -6. The result will be assumed to be the cubical contents. An efficient life boat is deemed capable of carrying one adult for every 10 cubic feet of lier capacity. A life boat must have at least 1^ cubic feet of air-tight compartments for every 10 feet of her cubical contents. Zinc must not be used in the construction of a life boat. 110 BOARD OF TRADE REGULATIONS FOR SHIPS. Life Buoys. A life jacket or belt to be supplied for each of the oarsmen, and one for the coxswain, of each life boat. Every life jacket or belt must be capable of floating in water for 21 hours with 23 lbs. of iron suspended from it ; and each life jacket, in which the cork must be exposed and have a can- vas back and straps only, should weigh 5 lbs. when dry. All cork life buoys should be built of solid cork, and must be capable of floating for 24 hours in water with 32 lbs. of iron suspended from them. If not made of cork they must be capable of floating in water for 24 hours with 40 lbs. of iron suspended from them. Xo contrivance will be passed as a life buoy that requires inflation before use. Pumps, Sluice Valves, !Steerixg Geae, etc. There must be in each compartment a pump of sufficient size which can be worked from the upper deck. There must be a valve or cock fitted at the bottom of each water-tight bulkhead, which can be opened from the upper deck, and also a sounding tuVje to each compartment. Pipes connected with pumps, worked by the engines, are also to be carried through the bulkheads into the compartments fore and aft of the engine room ; so that each compartment can be pumped out separately by the engines as well as by the deck pumps. A spare tiller, relieving tackle, &c., should be carried in all sea-going steamers. Rudder pendants should also be secured to the back of the rudder. A deep-sea lead-line of at least 120 fathoms, a lead of at least 28 lbs. weight and a suitable reel, together with at least two hand lead-lines of 25 fathoms each, and leads of at least 7 lbs. each, should be supplied to all foreign-going steamers. In home-trade steamers two hand lead-lines of 25 fathoms each, and leads of 7 lbs, each, must be supplied. For a flrst-class certificate of registry (ie. 12 months) double . the number of leads and lines must be supplied. Lead lines are usually marked as follows: — At 2 fathoms a piece of leather split into two strips. „ 3 „ 7 ., „ three strips, white bunting, red bunting. „ 10 leather with a hole. „ 13 blue bunting. „ 15 white bunting. » 17 „ 20 „ red bunting. a strand with two knots tied in it. BOARD OF TRADE EEGULATIONS FOR SHIPS. Ill Distress Signals. The signals required are 12 blue lights (or 6 blue lights and 6 of Holmes's patent storm and danger signal lights), 12 rockets, each containing 16 ozs. of composition, and one gun of at least 3^ ins. the bore, or one mortar of 5^ ins., with ammunition for 12 charges, or, in the case of foreign sea-going passenger ships, 24 charges. Each charge must contain 16 ozs. of pebble or bean powder in a flannel bag. An air-tight copper magazine, rammers, sponges, wads, priming wires, friction tubes, powder flasks, with fine powder for priming, and means for firing and withdrawing charges, should be provided. Rocket lockers should not be air-tight. Fire Hose. A fire hose adapted for extinguishing fire in any part of the ship, and capable of being connected with the engines of the ship, or with the donkey engine if it can be worked from the main boiler, should be supplied. Passenger Acco:!iiioDATioN. Passengers in Foo'eign-going Steamers. The upper weather deck, and the upper surface of the poop, forecastle, and spar deck, are never to be included in the measure- ments for passengers ; nor are the poop, round house, or deck house to be measured for passengers, unless they form part of the permanent structure of the vessel. Foreign-going steamships carrying passengers are to be measured as follows : — Saloon or 1st Class. — The number of fixed berths or sofas that are fitted determine the number of passengers to be allowed. 2nd Class, — The number is determined in the same way as the 1st class. 3rd Class. — The number may be determined in like manner if berths are fitted ; if not, the net area of the deck, multiplied by the height between decks and the product divided by 72, gives the number to be allowed. The breadth of the deck is taken inside the water-way, or at the greatest tumble-home of the side, if there is any. When cargo, stores, &c., are carried in the space measured for passengers, one passenger is to be deducted for every 12 superficial feet of deck space so occupied. 112 boai:d of trade regulations for ships. Passengers in Home-Trade Sea-going Steamers. Fore-cabin passengers include all passengers except those entered as after-cabin or saloon passengers in the way bill. The number of passengers allowed to be carried in sea-going home-trade steamers is ascertained as follows : — The clear area of the deck in square feet is divided by nine ; the quotient is the number allowed to be carried on deck in summer. Passengers in home-trade steamers are allowed to be carried on the main and lower decks only. The breadths of the deck are taken from inside the gutter water-way, or the inside edge of the raised covering-board, or inside edge of the rail, if the bulwarks tumble home farther than the inside edge of the water-way or covering-board. In cases where adequate shelter is not provided for deck passengers the whole nvimber of passengers must not exceed one-fourth of the number representing the gross tonnage, with the addition of the number of after-cabin passengers, calculated as before. Wliere cargo, cattle, &c., are carried in the space measured for passengers in home-trade passenger steamers, the following deductions are to be made : — For every square yard of space measured for passengers occu- pied by cattle or other animals, or by cargo or other articles, one passenger is to be deducted. If, however, the whole number so to be deducted on account of cattle or cargo carried on deck eqimls or exceeds the original number of passengers due to the deck space, so that no pas- sengers are carried on deck, it may be covered with cattle or cargo, without any reduction on that account in the number of passengers carried in the cabins. Between the 31st of October and the 1st of April the number of passengers which, according to the preceding rules, are allowed to be carried on deck in summer are to be reduced one-third, unless there is accommodation below, or in properly constructed cabins on deck, for half the full complement of passengers. This re- duction not to be made in the case of foreign-going steamships. One-third, however, of the space on deck measured for pas- sengers may be occupied by cargo and cattle, without any reduction of the winter number of passengers. The number of passengers to be carried in the after-cabins is determined by the number of berths or sofas ; to which add the number due to the space on deck appropriated to the saloon passengers, and tlie sum will be the total number of after-cabin passengers allowed to be carried. BOARD OF TRADE REGULATIONS FOR SHIPS. 113 The floor space of saloons, cabins, state-rooms, and passages must not be measured, unless in saloons and cabins in which berths are not fitted ; then the clear available space is to be measured, and one passenger allowed for every 9 square feet. "When sofas or seats are fitted the measurements are to be taken from the backs of the said sofas or seats. The number of fore-cabin passengers is obtained in the same way as the after- cabin number. The total number of passengers must not exceed the number denoting the gross register tonnage of the vessel. ^Mien there are deck-houses, and only narrow spaces between the sides of the deck-houses and the bulwarks, such narrow spaces are not to be measured for passengers. Passengers in Excui'sioyi Steamers. For steamers used in excursions the rules for calculating the number of passengers are the same as in sea-going home- trade steamers, except that if application is made for an ex- cursion certificate for short distances along the coast during day- light, the number, originally calculated at 9 superficial feet to each passenger, should it exceed the gross tonnage of the vessel, need not be diminished so as to bring it down to that number. Where cargo, cattle, Arc, are carried in the space measured for passengers in excursion steamers, one passenger is to be deducted for every square yard of space, measured for passengers, occupied by cattle, ca^go, ka. Passengers in River Steamers. The measurements are to be made in the same manner as in home-trade sea-going steamers, except that after-saloons only are to be included. There will be no distinction between fore- and after-cabiu passengers. River steamers are divided into those which ply on waters part of which only are smooth, and those which ply exclusively on smooth water. Taking this division — For steamers which ply in partially smooth water, divide the number of superficial feet on deck, obtained as before, by six, and the clear space in the after-saloon by nine, and the sum of these quotients will be the number of passengers allowed. In the last-mentioned class of steamers one and a half pas- senger is to be deMiicted for every square yard of space measured for passengers occupie'd-^by cattle, cargo, kc. I 114 BOARD OF TRADE EEGULATIOXS FOR SHIPS. A reduction is to be made during tlie winter months, in precisely the same manner as in home-trade sea-going steamers. These vessels are to be provided with a fore-sail and jib bent, a suitable anchor and cable, a compass, a regulation life-boat, one dozen life buoys, and two safety valves on ea^h boiler. For smooth-water steamers divide the number of superficial feet on deck, obtained as before, by three, and the clear space in the after-saloon by nine, and the sum of these quotients is the number of passengers allowed. Three passengers are to be deducted for every square yard of space measured for passengers occupied by cattle, cargo, &c. Kg reduction to be made in winter months. Crew Space. Every space occupied by the crew shall contain 72 cubic feet, and 12 superficial feet of surface for each seaman. For every 20 men there should be two privies. In measiu-ing the clear area of deck in crew space, beds, bunks, or sleeping berths are not to be deducted as encum- I a-ances, but in cabins there should not be less than 12 square i'eet per man exclusive of the bunk. To compute the cubic capacity of the crew space, multiply the clear area of the floor space by the height from deck to deck at the middle line ; the product will be the cubic capacity of the crew space. Divide the cubic capacity thus obtained by 72, and the quotient will be the number of men the place is to accommodate, provided that there is sufficient area of deck, as before computed. Under the Merchant Shipping Act of 1867 the tonnage of all the places for the berthing of seamen and apprentices, and approjjriated to their use, may be deducted from the register tonnage of the ship, provided that the number the crew space will accommodate is cut in or painted on or over the door or hatchway leading to such place ; and also cut in on one of the beams in the inside of such crew space. Mitiimum Dimensions of Ships'' Lanterns. The back and sides must not be less than 9 ins., and the height inside not less than 11 ins. The lens must not be less than 5 ins. in height, and if it is to be used as a side light the lens must not be less than I of a circle, the chord of the arc made by the lens not being less than 8 ins. ENGLISH WEIGHTS AXD ilEASUEES. 115 ENGLISH WEIGHTS AND MEASURES. Avoirdupois Weight. Drams Ozs. Lbs. Qrs. i Cwts. ! Ton Gramnies 1-.'^ 1 •0625 •0039063 •0001395 •0000349-00000174 1-771846 ■■-: 16 = 1 •0625 ^0022321 -000558 -00002790 28-34954 ' 256 16 = 1 -0357143 -0089285 -00044643 453-5927 7168 448 28 =1-25 -0125 12700-59 28672 1792 112 4 =1-05 .50802-38 573440 35840 2240 80' 20: =1 1016048 A stone of iron, coal, Szc. = 14 lbs. Troy Weight. Avoir. Drs. Grains Lwts. 32 -^'875 = 1 -0416667 768 -^875 24 = 1 17 + (97 ■=-17 5) 480 20 210 + (114 ^1 75) 5760 240 Ozs. j Lbs. I Grammes •0020833 -0001736 I -0648 •05 I-0041667 I l^o552 = 1 (•0833333 ! 3M035 12! =i:373^2420 175 lbs. Troy = 144 lbs. Avoir. Avoir, lbs. x 1-2 1527 = lbs. Tro}-. 175 oz. Trov = 192 oz. Avoir. Trov lbs. X -823 .= Avoir, lbs. LiXEAL Measure. Inches 1 12 36 72 198 7920 63360 Feet 08333 = 1 3 6 16^ 660 5280 Yards 02778 33333 = 1 2 ^1 02 220 1760 Fauis. i'iiies -013889-005051 •166667-060606 •5 •181818 •363636 = 1 40 320 = ] no! PIfO Furls. •000126 •001515 •004545 •009091 •025 = 1 :Mile 000016 •000189 •000568 •001136 Metres •0254 •304797 •914392 1-82878 -003125 502915 •125 201-166 = 1 1609-33 The palm = 3 in. The hand = 4 in. The span = 9 in. The cubit = 18 in. The common military pace = 30 in. An itinerary pace = 5 feet. A cable' s length = 120 fathoms. A league =3 miles. Laxd Measure (Lineal). IncLes Links Feet Yards Chains IMile Metres B 1 •1261261 -08.3.3333 -0277778 -0012626 -0000158 -0254 m = 1 •6666667 -2222222 -01 -000125 -201166 12 m- = 1 -3333333 -0151515-0001894 -304797 86 4ii 3 = 1 ^0454545 -0005682 , -914392 792 100 66 22 J =lj-012o 20-1166 63360 8000 5280 1760 80' =li 1609-33 . 116 ENGLISH WEIGHTS AND MEASL'ESS, Square Measijee. Inc'i: 3 Feet Yards Perches ^ Roods Acre Sq. Metres 1 -0069444 •0007716 -0000255 -00000064 -00000016 -0006452 144 = 1 •1111111 -0036731 -0000918 -000023 -0929013 1296 9 = 1 -0330579-0008264 -0002066 -836112 ■■ 39204 2721 3ui =1-025 -00625 25292 1568160 10890 1210 40 =1-25 1011-696 6272640 43560 4840 160 4 =14046-782 Acres x -0015625 = sq. miles. ^3q. yards x -000000323 ^sq. miles. Land Measure (SarARE), _ Links 1 625 10000 25000 100000 Perch -0016 Chains ;-oooi = 1 -0625 16 =1 40 n 1,60 10" Eoois -00004 •025 •4 Acre -00001 -00625 •1 9- •ZJ = 1 Sq. iletres -04046 25-292 404-6782 1011-696 4046-782 A liidt; of land = 100 acres. A yard of land A chain wide = 8 acres per mile. Ctjbic Measure. 30 acres. Imperial Gallons , Cub. Ins. •003606540822 = 1 6-232102541168 1728 168-266768641554 46656 Cub. Feet -0005788 = 1 27 Cub. Yis. Cub. Metre -00000214 -000016387 -0370370 -0283161 = 1 -764534 A cubic yard of earth = 1 load. A barrel Vjulk = 5 cub. ft. Ton of displacement of a ship = 35 cub. ft. = -99106:^i cub. metre. A\'ixE Measure. Cub. Ins. - 'A 5 a £ r. ^ ■r. 4- 5 .a '-3 '^ < 1 S •S c 8-664}p =1 34^6.59J 4 =1 69-31 Si 8 2 = 1 277-274 2772-740 32 8 4 = 1 320 80 40 10 = 1 4990-932 576 144 72 18 1-? = 1 8734-131 1008 252 126 3U S:^ n = 1 11645-508 1344 336 168 42 4i 9i H = 1 1 7468-262 23291-016 2016 i>688 504 67;^ 2.52 336 63 i 8* 6^ 3I 2 4i % 1^ 2 = 1 = 1 34936-.524 4032 1008 504 ,126 1211 7 4 3 2 n = 1 fi9H73-048 8064 20 IC > 1008 252 1 25i 14 8 6 4 3 2=.j -v^^^^'C^ ^^t^C english weights and 3ieasures. Ale axd Beer Measuee. \i: Cub. lus. V3 34-659i -1 6'J-318:^ 9 t 277-274 8| 241IO-466 72 41tl»0-L>32 144 9981-864 288 1 14072-796 432 19963-728 576 29945-592 864 59891-184 1728 119782-368 3456 m I ^ = 1 4 36 72 144 216 288 432 864 1728 o = 1 9 18 36 54 72 108 216 432 = 1 2 4 6 8 12 24 48 = 1 2 3 4 6 12 ""■^ t; ce o X (1/ o o no -»^ s f-n (5 ' J = 1 ■ u = 1 2 n -1 3 '> H = 1 6 4 3 2 = 1 12 8 6 4 2 = 1 CORX AM . De r Measure. Cnb. Ins. 'X: to t tc o 02 Q "3 If. o tc O ci 34-659i = 1 69-318^ 2 = J. 138-637 277-274 4 8 .; 1 4 L' _ 1 554-548 16 8, 4 2 -1 2218-192 64 32 16 8 4=1 4436-384 128 64 32 16 8 2 = 1 8872-768 2.56 128 64 32 16 4 2 = 1 17745-536 512 256 128 64 32 8 4 2 = 1 i 1 88727-680 2560 1280. 640 320 160, 40 20 10 o = 1, 177455-360 5120 2560 1280 640 320 80 40 20 10 V^ c OAL Measure. Cub. In-. Heaped ileasnre 'v. r. >^2 4 c y CQ ■~1 Ml-! 703-872 181 = 1 2815-487 7ir 4 =1 8446-461 224| 12 3 -1 25339-383 6721 36 9 3 = 1 101357-532 2688 144 36 12 4 = 1 196380-2 18i 5208 279 69f 23i 7f llf = 1 1571041-746 41664, 2232 558| 186 62 15A 8 = 1 2128508-172 56448 3024 7561 252 84 21 io;;| lii = 1 31420834-92 833280 44640 11160'3720|l240 310 160 20 I4if = 1 118 EXGLTSH WEIGHTS AND MEASURES. AVOOL AVEirjHT. Pounds . Cloves Stones Tods ' Weys Packs Sacks Last I i 7 ' 14 = 1 .2 = 1 ' [ 28 ' 4 2 -1 ^ 182 ; 26 13 6h = 1 240 364 34| 52 1'^ 26 8| 13 • 129 : 2 = 1 131 = 1 t 4368 624 312 156 24 18^ 12 = 1 1 Measure of Time. Seconds Minutes Hours Days AVeeks Months Calend. Year Julian Year Leap Y'ear 60 = 1 3600 86400 60 1440 = 1 24 = 1 604800 2419200 10080 168 7 28 4 = 1 40320 672 B1536000 525600 8760 365 521 13-^- = 1 31557600 31622400 525960 8766 5270401 8784 365^ 366 m o2f 13^, 13^ 1_JL_ ■*-1460 Ilk = 1 1 1 = 1 AxgelaPv Me a sere. The Geogi^aplucal Divisiim of any Line round tLs Circumference of tlie Earth jjiurnai ^lotion of the Earth re_iinals metres ot an Inch metres of an Inch metres of an Inch 39 1-535447 78 3-070894 1 0-0S9370 40 1-574817 7i» 3-110-264 •) 0-078741 41 1-614188 80 3-149635 3 0-118111 42 1-653558 J-1 3-189005 4 0-157482 43 1-692929 82 3-228375 5 0-196852 44 1-73-2-299 y-6 3-267746 6 0-236223 45 1-771669 84 3-.S07116 7 0-275593 46 1-811040 85 3-346487 8 0-314963 47 1-850410 86 3-385857 9 0-354334 48 1-889781 87 3-4-25228 10 0-393704 49 1-929151 88 3-464598 11 0-433075 50 1-9685-22 89 3-503968 12 - 0-47-2445 51 2-007892 90 3-543339 i:l 0-511816 52 2-047262 91 3-58-2709 14 0-551186 63 2-0J-'6633 92 3-622080 15 0-590556 54 2-126003 93 3-661450 IG 0-6-29927 55 2-165374 94 3-700821 17 0-669297 66 2-204744 95 3-740191 18 0-708668 67 2-244115 96 3-779561 19 0-748038 68 2-283485 97 3-818932 20 0-787409 69 2-322855 9'S 3-858302 21 0-8-26779 60 2-362226 99 3*897673 22 0-866149 61 2-401596 100 3-937043 23 0-905520 62 2-440967 101 3-976414 24 0-944890 63 2-480337 102 4-0157X4 25 0-984261 64 2-519708 1U3 4-055155 2G 1-023631 05 2-559078 104 4 094525 27 1-063002 66 2-59.S448 1U5 4-133895 28 1-10-2372 67 2-637819 106 4-173266 29 1-141742 68 2-677189 lti7 4-21-2636 SO 1-181113 69 2-716560 l(i8 4-252007 31 1-2-20483 70 2-755930 11 '9 4-291377 32 1-259854 71 2-795301 110 4 -.33074 8 33 1-299-2-24 72 2-834671 111 4-37011« 34 1-338595 73 2-874041 1 ] 2 4-409488 35 1-377965 74 2-913412 113 4-448859 36 1-417335 75 2-95-2782 114 4-48S2-29 . 37 1-456706 76 2-992153 ]]5 4-527(500 38 1-496076 77 3-0315-23 116 4-566970 MILLBIETRES TO INCHES. 125 MiUi- metres Inches and Decimals of an Inch Milli- metres 165 Inches and Decimals of an Inch MiUi- metres Inches and Decimals of an Inch 117 4-606341 6-496121 213 8-385902 118 4-645711 166 6-535492 2!4 8-425272 119 4-685081 167 6-574862 ■215 8-464643 120 4-724452 168 6-614-233 216 8-504013 121 4-763822 169 6-653603 217 8-543384 122 4-803193 170 6-692973 218 8-582754 123 4-84-2563 171 6 732344 219 8-622125 124 4-»81934 172 6-771714 2-20 8-661495 125 4-921304 173 6-811085 221 8-700866 126 " 4-960674 174 6-850455 222 8-740236 127 5-000045 175 6-8898-26 2-23 8-779606 128 5-039415 176 6-929196 224 8-818977 129 5-078786 177 6-968567 225 8-858347 130 6-118156 178 7-007937 2-26 8-89/718 131 5-157527 179 7-047307 2-27 8-937088 132 5-196897 180 7-086678 228 8-976459 133 5-236267 181 7-126048 229 9-0158-29 134 .6-275638 182 7-165419 230 9-055199 135 5-315008 183 7-204789 231 9-094570 136 5-354379 184 7-244160 232 9-133940 137 5-393749 185 7-283530 233 9173311 138 5-4331-20 186 7-32-2900 234 9-21-2681 139 5-472490 187 7-362271 235 9-25-2052 140 6-511861 188 7-401041 236 9-291422 141 5-551231 189 7-441012 237 9-330792 142 5-590601 190 7-480382 238 9-370163 143 5-6-29972 191 7-519753 239 9-409533 144 5-669342 192 7-5591-23 240 9-448904 145 5-708713 193 7-598493 •241 9-48«274 146 5-748083 194 7-637864 •242 9-5-27645 147 5-787454 195 7-677234 243 9-567015 148 6-826824 196 7-716605 244 9-606385 149 5-866194 197 7-755975 245 t 9-645756 150 5-905565 198 7-795346 246 j 9-6851-26 151 5-944935 199 7-834716 247 I 9-724497 152 5-984306 200 7-874086 •248 i 9-763867 153 6-0-23676 201 7-913457 •249 ■■ 9-803238 154 6-063047 ■202 7-95-2827 250 ' 9-842608 155 6-10-2417 203 7-992198 •251 ■ 9-881978 156 6-141787 204 8-031568 252 : 9-921349 157 6-181158 205 8-070939 253 ! 9-960719 158 6-2-205-28 206 8-110309 254 10-000090 : 159 6-259899 207 8-149679 255 10-039460 160 6-299269 208 8-189050 256 ! 10-078831 161 6-338640 209 8-228420 257 1 10-118201 162 6-378010 210 8-267791 258 1 10-157571 i 163 6-417380 211 8-307161 259 ! 10-196942 164 6-456751 212 8-346532 260 1 10-136312 1 l-lh MILLIMETRES TO INCHES. Milli- metres Inches and Decimals of an Inch Milli- metres Inches and Decimals of an Inch Milli- metres Inches and Decimals of an Inch 261 10-275683 309 12-165464 357 14-055244 262 10-315053 310 12-204834 358 14-094615 263 10 -3 54 4-24 311 12-244204 359 14-13o9 :iIILLDIETRES TO IXCHES. MiUi- metres Inches and JliUi- metres 1 Inches and :»Iilli- meires \ Inches and Decimals of an Inch Decimals of an Inch Decimals of an Inch 549 ! 21-614367 597 23-504148 645 25-3939-29 550 ' 21-653738 598 23-543518 646 25-433299 551 ! 21-693] 08 599 23-582889 647 25-472670 552 21-732478 600 23-622-259 648 25-5^2040 553 21-771849 601 23-661630 649 25-551410 554 21-811219 602 23-701000 650 25-590781 555 21-850590 603 23-740371 651 •25-630151 556 21-889960 604 23-779741 652 25-669522 557 21-9-29331 605 23-819111 653 25-708892 558 21-968701 606 23-858482 654 25-748263 559 22-008072 607 23-.S97852 655 25-787633 560 2-2-047442 608 23-9372-23 656 25-827003 561 2-2-08C812 609 23-976593 657 25-866374 562 22-1-26183 610 24-015964 658 25-905744 563 ■2-2-165553 611 24-055334 659 -25-945115 564 22-204924 612 24-094704 660 25-984486 26-023856 565 2-2-244294 613 24-134075 661 566 2-2-283665 614 24-173445 662 26-063-226 567 22-323035 615 24-21-2816 663 26-10-2596 h()S 2-2-36-2405 616 24-252186 664 26-141967 569 2-2-401776 617 24-291557 665 26-181337 570 2-2-441146 618 24-330927 666 26-2-20708 571 2-2-480517 619 24-370-297 667 26-260078 572 22-519.S87 620 24-409668 668 26-299449 573 22-559928 621 24-449038 669 26-338819 574 22-598628 622 24-488409 670 26-378189 575 2-2-637998 6-23 24-527779 671 26-417560 576 2-2-677369 6-24 24-567150 672 26-456930 577 22-716739 6-25 24-606520 673 26-496301 578 2-2-756110 626 -24-645890 074 26-535671 579 22-795480 - 627 24-6)S5261 675 26-575042 580 22-834851 6-28 ■24-7-24631 676 26-614412 581 2-2-874221 629 24-764002 677 26-653782 582 2-2-913591 630 24-803372 678 26-693153 583 2-2-952962 631 24-842743 679 26-73-25-23 584 22-992332 632 24-882113 680 26-771894 585 22-031703 633 24-921483 681 26-811-264 586 23-071073 634 24-960854 682 26-850635 587 23-110444 635 25-0002-24 683 26-890005 588 •23-149814 636 25-039595 , 684 26-929376 589 23-189184 637 25-078965 685 26-9;';8746 590 23-228555 638 25-118336 686 •27-008116 591 23-267925 639 25-157706 687 •27-047487 592 23-307296 640 , 25-197077 688 27-086857 593 23-346666 641 ' 25-236447 689 27-1-262-28 594 23-386037 642 i 25-275S17 690 27-165598 595 23-425407 643 25-315188 691 27-204969 596 23-464778 644 1 -25-354558 1 692 27-244339 illLLDIETEES TO INCHES. 129 Milli- Inches and Milli- Inches and Milli- me:re3 Inches and metres Decimals of an Inch metres 741 Decimals of an Inch Decimals of an Inch 693 27-283709 29-173490 789 31-003271 694 27-323080 742 29-212861 790 31-102041 695 27-362450 743 29-252231 791 31-14-2012 696 27-401 821 744 29-291601 792 31-181382 097 27-441191 745 29-330972 793 31-2-20752 698 27-480562 746 29-370342 794 31-200123 699 27-519932 747 29-409713 795 31-299493 700 27-559302 748 29-449083 796 31-338.'S64 701 27-598673 749 29-488454 707 31-378-234 702 27-638043 750 29-527824 798 31-417004 703 27-677414 751 29-567194 799 31-450975 704 27-716784 752 29-606565 800 31-490346 705 27-756155 753 29-045935 801 31-535716 706 27-795525 754 29-685306 802 31-575086 707 27-834895 755 29-7-24676 803 31-614457 708 27-874266 756 29-764047 804 31-653827 709 27-913636 757 29-803417 805 31-693198 710 27-953007 758 29-842787 806 31-73-2568 711 27-99-2377 759 29-882158 807 31-771938 712 28-031748 760 29-9215-28 808 31-811309 713 28-071118 761 29-900899 809 31-850679 714 28-110488 762 30-000269 810 31-^90050 715 28-149859 763 30-039640 811 31-929420 716 28- 1892-29 764 30-079010 812 31-968791 717 28-2-28600 765 30-118380 813 32-008161 718 28-267970 766 80-157751 814 32-047532 719 28-307341 767 30-1971-21 815 32-0«0902 720 28-346711 768 30-236492 816 32-120272 721 28-386081 769 30-275862 817 32-165043 7->2 28-425452 770 30-315-233 818 32-205013 723 28-464822 771 30-354603 819 32-244384 724 28-504193 772 30-393973 8-:o 32-283754 725 28-543563 773 30-433344 821 3 -2-3 -23 1-25 726 28-582934 774 30-472714 822 32-36-24.95 727 28-622304 775 30-5 1-2085 8-23 3-2-401866 728 28-661675 776 30-551455 824 32-441236 " 729 28-701045 777 30-590825 8-25 32-480606 730 28-740415 778 30-630196 826 32-519977 731 28-779786 779 30-669566 8-27 32-559347 732 28-819156 780 30-708937 8-28 32-598718 733 28-8585-27 781 30-748307 829 32-638088 73 i 28-897897 782 30-787078 830 3-2-077459 735 28-937268 783 30-827048 831 3-2-7 16?<29 736 28-976638 784 30-800419 832 32-750199 737 29-016008 785 30-905789 833 32-795570 738 29-055379 786 30-945159 834 3-2-834940 739 29-094749 787 30-984530 835 32-874311 740 29-134120 788 31-0-23900 836 3-2-913681 130 MILLTMETEES TO INCHES. ililU- metres Inches and Decimals of au Inch Milli- metres Indies and IjeciniaLs of an Inch MiUi- ; metres j 1 Inches and iJecimals of an Inch 837 32-953052 885 34-842832 933 36-732613 ms 32-992422 886 34-882-203 934 36-771984 839 33-031792 8«7 34-921573 935 36-811354 840 33-071163 888 34-960944 936 36-8507-24 841 33-110533 8?<9 35-000314 937 36-890095 842 33-149904 890 35-039684 938 3G-9-29465 843 33-189274 891 35-079055 939 36-968836 844 33-228645 «92 35-1184-25 940 37-008206 845 33-268015 893 35-157796 941 37-047576 846 33-307385 894 35-197166 942 37-086947 847 33-346756 895 35-236536 943 37-126317 848 33-3861-26 S96 35-275907 944 37-165688 849 33-4-25497 897 35-315277 945 37-205058 850 33-464867 898 35-354648 946 37-2444-29 8.^1 33-r;04238 899 35-394018 947 37-283799 852 83-543608 900 35-433389 948 37-323170 853 33-582979 901 35-472759 949 37-362540 854 33-022349 902 35-512130 950 37-410910 855 33-6G1719 903 35-551500 951 37-4412.'^l 856 33-701090 904 35-590971 952 37-4^0651 857 33-740460 905 35-630241 953 37-520022 858 33-779831 906 35-669611 954 37-559392 859 33-819201 907 35-708982 955 37-598765 860 33-858572 908 35-748352 956 37-63^135 861 33-897942 909 35-787723 957 37-677503 862 33-937312 910 35-827093 958 37-716874 863 33-976683 911 35-866464 959 37-756244 864 34-016053 912 35-905834 960 37-795615 865 34-055424 913 35-945204 961 37-834985 866 34-094794 914 35-984575 962 37-874356 867 34-134165 915 36-023945 963 37-913726 868 34-173535 916 36-063316 964 37-953096 869 34-212905 917 36-1026S6 965 37-99-2467 870 34-252276 918 36-142U57 966 38-031837 871 34-291646 919 36-1S1427 967 38-071208 872 34-331017 !)20 36-220797 968 38-110578 873 34-370387 921 36-260168 969 38-149949 874 34-409758 922 36-299538 970 38-189319 875 34-449128 923 36-338909 971 38-228689 S76 34-448498 924 36-378279 972 38-268060 4 1 12 8 6-945250 16 1 32 ... 1 128 15-279550 1 4 1 3'* 7-143686 16 1 32 1 6 4 ... 15-477986 1 4 r ... 1 128 7-342122 16 _]_ 1 6 4 1 128 15-676422 1 4 I 4 T(T ■'r 32 1 32 1 (•■4 1 64 1 128 7-540557 7-738993 7-937429 1 ::: 1 64 1 128 1.5-874858 16-07.3293 16-271729 16 ... 1 12H 8-135865 ^ I 6 i 1 128 16-470165 IT. 1 6 + 8-334300 j_ 16-668600 EQUIVALEXTS OF ENGLISH AND METEICAL MEASURES. Divisions of the Inch 128 Millimetres Divisions of the Inch 1 1 ~ 128 ilillimetres "21-232622' # 1 32 ... 16-867036 13 16 ... 1 04 8 1 32 1 64 17-06.5472 1 •. 16 1 32 21-431058 F 1 32 1 G4 1 12S 17-263908 13 16 1 32 ... 128 21-629493 11 16 ... ... 17-162343 13 16 1 32 1 04 21-827929 11 ]6 ... ... 1 12S 17-660779 13 16 1 32 1 04 1 1^8 22-026365 11 16 1 64 17-859215 f ... ... 22-224801 11 lil4-2(y2ol 16 S12^8.S.^(i.S2N3 2 101-6047.3410 7 o.55-6166.S9.d6 12 609-6-285-2462 17 86.3^f;40400>;8 3 15-i-407LS116 8 406-41901642 1.3 660-43090168 18 014-44278694 4 •20.S-20!».)08-21 9 457-22]. S9.S47 14 711-2o.S27><7.S 10 ;965^24516.S09 5 •2.54-nl]s,s52fi 10 5()S-O2."!770o2 ]5 i762-0.S56557>: 20 101 6^04 754 11 Table giving the Equivalents of Tons in French Kilograms. Tom 2 3 4 o 6 7 8 9 10 Kilograms 1016-04754 2032-09508 :;048' 14262 4064-19016 .5080-23771 6096-28525 71 12^33279 8128-38033 9144^42787 10160-4754 Tons ~20 30 40 50 60 70 80 90 100 200 Kilograms 20320-9.508 30481-4262 40641-9016 50802-3771 60962-8525 71123-3279 81283-8033 91444-2787 101604-754 203209-.508 Toni- 300 400 500 600 700 800 900 1000 Kilograms 304814-262 406419-016 .508023-771 609628-525 711233^279 812838-033 914442-787 1016047-54 UOO'l 11 7652-30 1200 1219257-05 Tons L30() 1400 1500 1600 1700 1800 1900 2000 3000 4000 Kilograms 1320861-80 1422466-56 1 52407 P31 1625676-07 1727280-82 1828885^57 1930490^33 2032095-08 3048142-62 4064190-16 KILOGRAMS TO LBS. AND TONS. 135 X •^ X o ■M -^ tr cc c c^ i~ i^ T. — r* '~ i^ r; — rt ut i^ n — re i~ r; ^t oo ri ^ O -f r. re I- — «o c; T^ cc "M --r — < lo r: re t^ "M ^ C ~. 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'O n:^ K5 lO O iC o ue IC to »c w ■-C ::; w to ■-^ w w ^ l^ i^ t~ t^ t^ l^ ^mm ^^^ ^^ ^^ __ ^^ ue :r^ t~ i^ X X r: C: C ei ri re -+' -^ le o t- r; — re »e X o "M -H — X c: -M -Tf tr X C e^i -" t;^ X c re ~ re i^ tr o -f X re 1^ ' le ~ -^ x ri t^ o '-e r: re r vr c X i~ i-e -^ ei — r: r~ "-C -T^ re — t. x v^ le re n c; x i^ to -rr tt — te o le le 1^ le -*' -Tfi -+1 -H -+i -t" re re re re re re re n ei -m -m n -m lo tc t- X r: O — ' ei re -t^ le "^ i^ X ~. o — ei re -^ je :r t^ X r; •MT^i-M-M-Mrererererererererere — -^-^i-T^i-^-r-!:"-*^-*--- ooooooooooooooooooooooooo ue t^ O e-i '^ CO X o n 't' t^ Ci ^ re »o t^ Ci — -* tc oo o ^T -* tr — t— -+< o tc "M X to -H t^ re ~ to ei X -+( o t^ re r: lo ■>! X -+• o O -# c: -ri X re I— -M t-- — -^ O o O -^ r: 't^ X re t^ "M t^ 1— ^ — "M M -M re re -^ -^ iO lO "^ ;r i^ t^ X X X r: r: o O — — ^ ~i ei re root-ciiT^roiot-c^-^reiOt^c:— ire»ot^OiM'*'!r)XO'M ^^i^^seicbo'^1■*^^C5•^rb>'oc»o:^^^t^cfl — reioxc; OiOiDCOOiCt-t^t-t~-t^XOOXXCr5C5C;CiCiOC:00 — 2S •x> t^ OO c^ o -^ •M CO ^ lO !0 t^ CO cs o — • 5>1 ? e "^ »o ?c; t^ X ^ -^ i2 M e-1 :^^ eM •■. re "• ... re re cc re rcco-^-^-^-^-^-^i* ^ ^ Id- to — — ' M c-i re -*> -T« to le — t-~ r^ X X r; ct o — ' -^ M e-i re -^ -T^ 'O e-1 -* -o X o •>! -^ — X o •>! -^ to X o "M to t^ ~ — re »o t^ r: — T* X ■M '-c ^- lO r: re t^ "M ^ O -^ X re t^ — to c; -f X •>! '— 3 'O X o to re e-i o X i^ to -e ej — r: t^ ^c -f re — r; X — to re •>! o r: ct r; r: ct r; X X X X X X i^ t^ t^ i^ t-- I-- 'o to vr to '-C to *o O — n re -*- to -o t^ X ~ O — ^1 re -^ »o to t^ X r; o — M re -*■ S I ■>! -f to c^ 1— re >o t^ o: — re >o X o ri -+i to X O re lo t^ o: ^ re • :o^ix-^-Jt^reo:io>Txioot^rec:io-JX-#otoe^icxio ~ -f — rexret^eitr — tro'0 0-t;r:rex)ret^^ii_;- — tro'O ■§ >i ^ -,o X o 5^1 -^ to X o ^"i -;^ to X o >i ■* to X o 55 5 t^ r; ^ > ■fl -it< to X ~ re lb t^ o: fi '^ to do o re lb t^ c^ — 4t< to X o ei ib <1 — -^i-<— i^j^ie^ri-Mrerere"" ■^ -* "# ^' to to to re -^ to vD t^ X o; o ■M re -r lo to 1:^ X r: o — ' (M re -^ to 1— ' " ^ I-' r- — — — -M :^^ (M (^^ (>i c^ 136 KrLOGRAMS TO LBS. AND TONS. ^^" _u »'^ i~ .« ._ t^ X X. ^^ — ^■"" TT CI ri .^ _J. iC J- .- I- I- X X "■1 *-^ t^ X :^ ■M ^T- v^ X C^ cc 1-C 1^ ^ -M CO cc l^ c ^- CO jC l^ — — ^— ^ ^ -*■ X M 1^ — i-C z~~ c* X 7 1 'V^ o — ^ ~ CO l^ cc — — *- X CI cc — ":3 C "^I O X 1^ o ■-r^ CI ^_^ _ , l- ^ -^ cc »— ■ ^, X '— 'O CO CI ■^^^ X I- cc -^ ,- -M "M ^- ^— -^ ^- ^— — :^ :^ o o ^ :^ ^. C- c^ r^ Ci * ~ X X X X r^ C^ *-w 'O -o 1^ X * :^ — CI CO —^ lO t^ w I- X r: o . — \ CI CO -f CO c^ c r— ! l^ T-H — 1 t- t- X X X X X X X X XXX r r r r r?T r -H .^ X ^ ^ UO t^ -~ _ ^- IC t^ o CT _14 --.r X O CI ^ r— Ci — CO CO *- CC * IC C^l X •-^ •^ t^ CO * \ 1^ — tc ^^ iC o ^ Ct *^ X CO x:, ci r ' cc ^^ lO O IC Ci "^ r— ' — ■>\ 0) CO CC "T*- "T*^ »c lO lO » w I— t- XX—. ^ c~^ ^, — — — ■ CI X ■- O M T vi X o CI -t" tc X o CI "^ ?C X C: cq -Tf cc C". — X z. X, 3 M -Tt> t3 — _, -^ LC t^ O CI '^ w X — CO o r^ -^ CI -* ci X O w' < X' ^■^ c^ Ci * * o o o ,— * ^^ ^— . — ' — - CI CI CI CI CI CO CO 'O '^** ^ cc CO cc CO CO CO -* ^ 'Th -^l -*l -f -* ■^ "* -:- •rt^ ':}' '^ -* '^J* -^ -^ -* -T^ •i — 1^ X — o _ CI -.K -+I IC t3 t- X — -~ — CI CO _« »c .- i^ X c; — ^ -5 I- l^ I- t^ X X X X X X X X X X ~ Ci d c; Ci Ci Ci Ci Ci Ci c < X - "■" "^ *" '^ '"' """ *"■ '^ "" "" *"" ^^ ^" "■" '"" -H — 1 . "" ' "^ "^ — ' — CI — — , ■M Ol .-» — _u KC ,~ .- -^ t^ X X — T. '- CT CI CO -" — -^ rc o t^ ^ — cc lO l^ c^ 1 — ' cc lO t— CI — ^ CC c^ X t^ CI 'C C- ct I- ^^ tc o -T^ X C! t- — cc ^ cc X CI tr — cc * CO t~^ — CC ^ — «~ c^ X ;^ '*-0 CO C4 '^ X t^ IC ^ CI ^ ^^ 1'^— w *^ CO ,— '^ X cc lO CO — ,- :^ o >o LO L.0 lO LC LC -* -tH -^ -^ — *- -<:« CO CO CO CO CO cc CI CI CI CI CI A ~ X * o ^— ■M CO -™ UC t^ 1- X c~. ^ ^-^ CI CO *^ C!; cc r- X X T-l 1— 1 I— 1 1— 1 lO o o 1—1 1-H iO I— 1 ^- I— 1 i-H 1—1 s s s r ^ r cc t- t- i^ c _, CO lO I— — -M -f -^ X) o ci -e- ,- — _^ CO lO t^ — _, ~. ■^ m -z. CI — X ;;^ x^ d ^ iC — ^ r^ ■^ ■^ vl." CI X CO -- t- cc * t^ CI ^ 1 t^ 'M r-- — 1C :^ IC c*x •^^ ex -^ X CO 1~- ca ;~^ IC — -+ Ci — X ^ ^ ^ :3 .— — :ri CI re CO ce -^ -rr *c cc t^ iC t^ t- X X * « ;h X ^- CO iC l^ ex — re iO t^ Ci — 1 CO to «>- Ci '— CO cc t- Ci ' — ^ CO cc X — z -M 1^ l^ C^ 1 ^y^ t^ T^ ^ ei — u 1^ — __ -^ CO X o CT — u ^ Ci — CO C~ ^ < cc CO CO rc -* "^ ■^ .^M lO ic o >o IC tc t^ ^ tc l^ 1^ 1— r^ t^ X X X ^ cc ?c CO c^ '^ 'V* *^ /-■^ *»-^ ^^^ r«^ ^v^ ^^ ^o '^ CO CO CO r^y ^^ *^^ <-^ , ■r. ^^ !>J ^ _« »o cr> t^ X cr. ^ ^ CT ~. -M lO tc t- X — ,-^ __ CI CO -:f CO z .-^ i^ lO lO >o LO ^ iO lO iC ■^ tc ir — •-e: "C: "-C — — cc l^ t- l^ t^ t- l^ X - o lO ■^ — t^ r^ X — — o -^ CI e» ^ CO -^ cc »0 ._ — t- X X r. r^ c^ y~~^ CO LO r^ * — — re IC X O CI -4^ w X C: C3 -* tc :/•; ^ CI -r cc X < ^ * '^ X Ol tc * ic ex cc r^ CI t^ :^ — u X CO c^ — cc ~ — *- X CI CC o ^ o ;^ t^ ^ — *- CO ^— ex X w CO CO CI o X t^ cc -;f CI o ex t— cc -+ cc 7-- 5h ^^ * * * ex ex a; X Of. X X 'Jj a; 1^ t- t^ l~ r^ 1^ w cc cc cc cc X — *- -^ 1~ tc t^ X ex ^ -— C^l CO — ^ cc te; t- X —. o — ■ CI ce •"^ i^ cc t^ •V" I-l "M n c^^ :m CN r r r -H . — 1 r r r r?r — -;*- ^^ -^ ^ -:h -Tf X O -M -+I to X 1— -^ re lO t^ C^ '— ' CO CO X O CI -*! ^ X o CI CO t- Ci - ">) lO ,_■ r^ ^^ ■^ te^ CI X -*! — 1 t^ CO ex cc CI X '^ -^ 1^ CO c; CO — -M ^ .^ tc ^ i^O ^. '™' CO X CO (^ 'M ^ -H O O O o Ci CO X CC CT) X * * ^^, o ^~ ... — CI CI CO CO *H ^— J lo »o cc tC t- t- t- X X r. ;- t^ :m -T^ w X :^j -if tc X o CI ^ tr; X o ci -^ cc X c 1^ — o^ ~** ^ X ^-^ re UC r^ Z-. CI '^l CC X O CO CO r^ Ci _i — " "~ X ^ <1 r— r- X X X X ^ ex * ^ * o ^ t^ :^ ' -H ^^ ^^ — ^ CJ CI CI CI CO ^ :m M ^< >4 c^l r^ >i CI ■>\ CI CO CO ce re CO CO CO ce CO CO VJ cc t^ X — O _ ri ^ _- IC ^ t^ X — o — ' CI ct -^ lO cc t^ X Ci o ^ -M "M '>4 ■M ''!*' ^^ ^^s *>^ ^.^ *^ '^ ^ CO ■^ -T?- -*! 'T' -^ 't' -* -*' -* -tM CO ■^ ' ' ' ' ' ' ■ ■ ( ' 1 > > • _i X O "M ri " 1 X r^. ei -* iC X 5 C! I- cc X ~ CO ~ I- CI •^ CO CO t^ IC ^^ —w — CO 1— _^ LO ■^ ■.^ X CI ^ — ^ CO ^ re L- C< --C ■^ — ^ '/: CO t^ — • cc P5 .^ X 1^ »o — - 71 — • ex I^ t^ '-*- CO — • ;^ cc cc LO CO CI — X l^ CO -^ CI H -+i CO CO rc CC re ei C) CI CI CI CI — ^^ ^^ — ^^ ^-i ■^ — o o o — k^ * .■^ »^ ->> *^ — ^ *e ^ r~- X r: o r— < CI ce -:^ cc cc t^ r/) Ci H r-. o o o c; ^ o O o — ^ — — ' '-' — —' — CI CI CI CI 55 T' >o t^ -- ^ ^ • o t^ — CI _« •~z X o CI -f t^ c; — 1 CO CO h- ^. u -^ ^- a 1^ cc *^ t^ "^1 X —f" — r^ re r-. io CI X *rH — cc CO *";, CO f— I t- -r?- O cc ^ '^ to ,.^ '0 ^ ^r\ :7; -*■ ex cc X CI t^ CI ^ ^ cc o *o Ci ■r** ex CO X C7 I- "— < ^ I~ r~ X -r X * * ^ ^ _^ ^^ CI CI CC CO "** —^ -r' 1^ je "C cc 1 ^ 1 ^ W ;-. tC X c^l ^^ X o CO *o I— n --I lO t— ~ — CO CC t- Ci — CO »c ^^ o -M ■.« I-- c^ ^^ -^ • c cc 3 CI -T« O — ; ~-l cc CO l^ o CI -M w X — cc 1" t> 'M 'M -M -M *.^ CO CO CO -+| ~M -*< -^ -* >o CO iC cc cc cc CC sC cc t^ I- l- 5 ^ C^ '>\ Cvi ^ M C^J C4 CM >« c^ ca CM M C) Cq CI CI c^ c^ CM C1 CI CI CI •/•■ ^_ "M — _4, 1^ .- 1^ X — -^ — Cl ^ -^ IC •o r^ X — ^ „_ CI C- -+• iC ■ > — — — — — — — — — — — — — — — — DECniAL EQUIYALENTS OF LBS. IX TONS. 13: Table of the Decimal Eqi'Ivalexts of Paets of A To>-. Lbs. 1 Dtciiuais of a T '11 Lbs. Decimals cf a Ton •165179 Lbs. Decimals of a Ton Lbs. Decimals of a Ton •000446 370 1 820 ' •366071 1270 •566964 2 •000893 380 1 ■169643 830 •370536 1280 571429 3 •001339 390 •174107 840 •37.5000 1290 575893 4 •001786 400 •178571 8.50 •379464 1300 •580357 Q •002232 410 •183036 860 •38.3929 1310 584821 6 •002679 420 •187500 870 •388393 1320 589286 7 •003125 430 •191964 880 •392857 1330 593750 8 •003571 440 •196429 890 •397321 1340 598214 9 •004018 450 •200893 900 •401786 13.50 602679 10 •004464 460 •205357 910 •4062.50 1360 607143 20 •008929 470 •209821 920 •410714 1370 611607 30 •013393 480 •214286 930 •415179 1380 616071 40 •017851 490 i •218750 940 •419643 1390 620536 50 60 70 80 •022321 •026786 •031250 •035714 500 510 520 530 •223214 •227679 •232143 •236607 950 960 1 970 980 •424107 1400 625000 629464 633929 •428571 •433036 •437500 1410 1420 i 1430 < 638393 90 •040179 540 •241071 990 •441964 1440 642857 100 •044643 5.50 •245536 1000 •446429 14.50 647321 110 •049107 560 •250000 1010 •450893 1460 651786 120 •053571 570 •254464 1020 •455357 1470 6562.50 130 •058036 580 •258929 1030 •4.59821 1480 660714 140 •062.500 5:o •263393 1040 •464286 1490 665179 1.50 •066964 600 •267857 1050 •468750 1500 669643 160 •071429 610 •272321 1060 •473214 1510 674107 170 •075893 620 •276786 1070 •477679 1.520 678571 180 •080357 630 •281250 1080 •482143 1530 683036 190 •084821 640 •285714 1090 •486607 1.540 687500 200 •089286 650 •290179 1100 •491071 1550 691964 210 •093750 660 •294643 1110 •495536 1560 696429 220 •098214 670 •299107 1120 •500000 1570 700893 230 •102679 680 •303571 1130 •504464 1580 705357 240 •107143 690 •308036 1140 •508929 1.590 709821 250 •111607 700 •312500 1150 •513393 1600 714286 2G0 •116071 710 720 •316964 •321429 1160 •517857 1610 718750 270 •120536 1170 •522321 1620 723214 280 •125000 730 •325893 1180 •526786 1630 ' 727679 2 WO •129464 740 •3.30357 1190 •5312.50 1640 732143 300 •133929 750 •334821 1200 •.535714 16.50 1 736607 310 •138393 760 •339286 1210 •540179 1660 1 741071 320 •142857 770 •.3437.50 1220 •.544643 1670 1 745536 330 •147321 780 •348214 1230 •549107 1680 ' 750000 340 •151786 790 •3.52679 1240 •553571 1690; 754464 3.50 •1562.50 800 •357143 12.50 •5.58036 1700 • 758929 350 •160714 810 •361607 1260 •562500 1710, 763393 DECIMAL EQUIVALENTS OF EXGLI:^H ^-VEIGHT- Table of IHE DECIiTAL EariTALEXTS 01 Paris of | A Tox (concluded). Lbs. Decimals of a Tou Lis. Decimals of a Ton •825893 Lbs. , Decimals of a Ton Lbs. Decimals of a Ton 1720 •767857 1850 1980 ! •883929 2110 -941964 1730 •772321 1860 •830357 1990; •888393 2120 -946429 1740 •776786 1870 •834821 2000 ■ •892857 21.30 •950893 1750 •781250 1880 •839286 2010 ' •897.821 2140 •955357 1760 •785714 1890 •843750 2020: •901786 2150 •959821 1770 •790179 1900 ; •848214 2030 ; •9062.50 2160 •964286 1780 •794643 1910 1 •852679 2040 -910714 2170 •968750 1790 •799107 1920 : •857143 2050 -915179 2180 •973214 1800 •803571 1930 i •861607 2060 •919643 2190 •977679 1810 •808036 1940 [ •866071 2070 •924107 2200 •982143 1820 •812500 1950 1 •870536 2080, •928571 2210 •986607 1830 •816964 I960! •875000 2090 ! •933036 2220 •991071 1840 -821429 1970 -879464 2100 •937500 2230 •995536 1 2240 lbs. = 1 ton \ \ 0Z3. Decimals of a Lb. Ozs. Decimals of a Lb. Ozs. Decimals of a Lb. Ozs. Decimals of a Lb. •765625 1 4 •015625 4i •265625 8i •515625 121 i 2 •0312.50 4i •281250 8,^ •531250 m •781250 3 4 •046875 4f •296875 8f •546875 •796875 1 •062500 5 •312.500 9 •562500 13 •812500 H •078125 H •328125 H •578125 13i •828125 U •093750 5i •343750 n •593750 13^ •843750 If •109375 5f •359375 9f •609375 13f •859375 2 •125000 6 ■375000 10 •625000 14 •87500O 9i ~4 •140625 6i •390625 lOi •640625 ^^ •890625 91 "1 •1562.50 6* i •406250 lOi •6562.50 14^ •906250 93 -"4 •171875 6f •421875 lOf '671875 lif •921875 3 •187500 7 -437500 11 •687500 15 •937500 3i •203125 n \ -453125 lU •703125 m •953125 3* •218750 1^ ' 2 •468750 lU •718750 15i •96875(1 3J •234375 7f •484375 llf -734375 m •984375 4 •250000 8 •500000 12 -75n(i()0 16 1-000000 f^ , Decimals '■^^- : Of a Ton Qrs. Decimals of a Ton ^ Decimals '^^- of a Ton 3 i -037500 ^ ■ Decimals '^^=- of a Ton 1 i -012500 2 -025000 4 1 -050000 Cw-ts. Deciniiil.- of a Tor •050 •100 •1.50 •2(H) Cwts. 5 6 7 Decimal.- of a Ton ^Too" •300 •350 •400 wts. 9 10 11 ]2 [decimal- of a Toi •4.50 •500 •550 •600 •%rts. 13 14 15 If. Decinial of a Tm. •6.50 •700 •7.50 -800 :'\vts. 17 18 19 20 Deciii.ol.- of a T. ai -850 -900 •950 1-000 1 2 3 4 DECDIAL EQUIVALENTS OF PARTS OF THE FOOT. 139 H O o w H o OQ o M CO l-l > Q H H ■< •—I o M Or-(Coec-*u5«ot-cociOt-i ?— 1 T— I a H C > O m H S^ > ►—1 < Q P3 a H o <1 O rH ?q 'J rlll-l — 'inco— imco— 'oco^^mx OO-M-fiOD-^-^OO-^'-fCO^-rti t^ :^ -:f^ :m — c: t- vr -!f 7-1 -^ c^ O';^(7^tC'T^-*O^lr~00ClCi -1- 1 1 o n rj p^iN ! in oo -M «H 1 O m C- Sp O CI 1 -ID -1- Hoc C: rctc:c:^t:oricnoc:rno 5qtOC;7i:cci(MOCi'MccC5 t^iar'5'MOcot-mcncMOco 0"C?qcc-*HTi^io?ot-co Gic;) Hi LI la CO .-H ^ t- — -rjl t^ -^ -H CM CO CI CIS! 1 t-0'fit^O-*t-0'+t-0'* 2b ', o ic cc — ' C CO o o rc — O CO !:b h— ' T ^^ ^!^ 1*^ "f' "P 'r' "T" "^ 9^ -? o m t^ .1;, '^ TT: rc. -H m CO — 1 i^ »o Cl -12 w|^ incO'Miococ^icaoiMmoocN (MiOCi(MOCiO^KlC5iM»Ori >£> -r« oq —1 Ci t^ 5C, -+ >i — ' r; t^ O-— i!MreCC'*»O«pb-C0C0w^ rtlH m 'Tt^ (M o r: t^ o -t- c-1 o r: i~~. O"— iST IC CO (M O OO lOCC— 'OCOOOcncrtOGO?© Or- iT ic r; -M o ~ "M »r: ~ m ^Irti — ~co!X)'+icc-HC:cOiC-hrc jpcpr-i^icp-+ioocpb-aoc-. -IS pOr-l(>)cp-^iO»OtOt^OOCi rx> —> ~v 9 ? 9 ID 1- nco — ICCO-^ H IOCOt>-lCcniMOCOt^mCC!?-1 OOr^jqcp^jpiotot^coci o S £3 -^ O 1 omb-orct-o«b-ocot- ^ Ocotcont — ococcmcn— ' ^~' O cp -^ ->! rp -^ ip »p :c t- CO Ci 1 c 1— 1 — H M O -H d 140 DECIMAL EQUIVALENTS OF PARTS OF THE INCH. Table of the Fractional Parts of the Inch, "with thlir c0rresp0nl>ixg decimals. Decimals Fractions 0078125 01.56250 0234375 0312500; 0390625 0468750 0546875 0625000 0703125]^ 0781250!io •.. 0859375!!^ ••• 0937o00iTis ^ 1015625:^ A 1093750,1^ i. 1171875|J« i^ 1250000 i ••• 1328125 i ... 1406250 ^ ... 1484375 ^ ... 1562500 i s'o 1640625 ^ ^ 1718750 i sV 1796875 i Jo '1875000t| ... ■1953125Ji='6 ••• •2031250,^6 ••• •2109375lil; ... •2187500'i|, ^ •2265625|l| i •23437.50 i^ ^ ■2421875 •2500000 •2578125 26562,50 •2734375 •2812.500 •2890625 •2968750j i •3046875; i 31 2.5000' y\ 3203125 ^^ ;-2s! 1250^1^ 1 ••• 12S J_ 6i •• • J 1_ G-i- li2.s R4 ••• J, 1_ G4 12b JL l_ 6-i 1-24 _1 1-2- J 1 G-i V2- ... 1^, _1_ 64 ••• 1 1 64 1-J-J ••• lliS J._ 64 ••• JL J._ 64 12-< 1 64 '•• 1 1 (i4 12S 1_ ••• 1-2S JL 64 ••• J^ _1_ 64 128 ••• 12rt _1_ 64 ••• _1 1__ 64 12S 64 ••• J 2_ 64 12« _1_ ... ^,_,s Decimals Fractions 1 32 32 JL_ 32 •3359375;i% •3437o00'i> •35 15625' j^6 ■3593750 1^ 3671875 j| ■3750000; f 3828125' f 39062.50 t 3984375; f 4062500, t 35 4140625! ^ 3^ 4218750| I sV 4296875! f 3^- 4375000 T^ ... 4453125 1^ ... 4531250^ ... 460;:»375,^'^ ... 4687.500^ ^\ 4765625,^ i 4843750|Y^^ -3^ ■4921875!^ ^ ■5000000; i ., •.5078125 1 ... •51562.50' ' .5234375 •.5312.500 •5390625 .54687.50 •5546875 •5625000;^;^ •5703125;^ -.57812.50,^ •5859375^ •.5937.500!^ •6015625; f^ •6093750|f^ •6171875 fs •6250000 •6328125 •64062.50 •6484375 •6562500 •66406-2.5 JL 1 64 12K 1 ••■ 128 J 1 64 12 8 1 . . • . • . t-A!^ 1 ••• 64 1 1 ••' 64 12 8 _1 ... io,s _1^ P4 -.. J 1_ 64 12H 64 •• _1 I 64 12; 1 128 _l 1_ 64 1-J8 2 .. . 64 1 1 1 2 64 ^•?^^ 1 I 'A 3?, . . • 1 1 1 ■A 3?, v?~ 1 1 1 ■^ 3-,', 64 1 1 i 32 64 128 J. 1 6 4 12^ 64 ••• JL _1_ 64 128 64 • • • J 1_ 64 128 Decimals Fractions 6718750' f 6796875: f 687500011 6953125^ 70312.50 iw 710937511 7187500ii 7265625;i| 734375011 7421875iH 7500000! f 7578125 f 7656250! f 7734375 7812500 7890625' 79687.50, 8046875 812.5000!^ 8203125 82812.50'il 8359375!i| 8437500'|f 8515625!f 8593750|^ 8671875|f 8750000 8828125 8906250 8984375 9062500 9140625j 92187.50 9296875! 9375000 if 94.53125|}| 9.5312.50^1 9609375 if 9687.50011 976562511 984375011 992187511 OOOOOOO 1 128 1 . . • 64 1 1 64 128 1 39 • . ■ i 1 3-:> -\->P i 1 32 64 3 1 1 4 64 1 -.' 8 :i 1 4 39 ■ • • 3 1 ■] 4 32 128 3 1 1 4 39 64 3 I 1 1 4 3 9 64 198 LI* L3 1 6 ... l-'s 3 1 16 64 3 1 1 6 3 '> 64 ... 3 1 J ■> 8 32 64 128 8 ■/ 1 8 • • • *• . T'8 / 1 S • • • 64 ■1 1 1 H 64 198 7 1 H 39 . . . 1 1 1 8 39 ^■?H 7 1 1 « 39 64 7 i 1 \ S 32 64 128 _1 64 FOBEIGN MOXKT, WEIGHTS, ASD MEASURES. 141 Table of Foreign Money. Weights, and Measures, WITH THEIR English Value. MONEY Countries Gold Coins Value Silver Coins Talue Silver Coius Value £ s. (!. s. cl. S. (1. Austria 8 florins 15 10 2 florins 3 Hi I florin H Bombay Mohur 19 2 Rupee 1 lOi I rupee 4 China — — Tael 6 S" 3Iace 7 Denmark 20krondaler I 1 Hi 4krondaler 4 5J Krondaler 1 li France' 20 francs 15 10 n fi-ancs 3 11 Franc 9i Germany 20 reichs- , 5 reichs- 20 pfennige 2| mark 10 mark 5 Greece 20 drachma 15 10 5 drachma 3 10 Drachma 9i Holland Ryder 1 5 1 1 Guilfier 1 8 25 cents 5' Madras Molinr 1 9 2 i Rupee 1 lOi \ rupee 5i Portugal 5 milreas 13 4. 500 reas 2 2' 50 reas 2| Russia )0 roubles 1 12 2| Rouble 3 \l 25 copecs ^\ Spain 20 pesetas 15 10 5 pesetas 3 Hi Peseta 9i Sweden 2it krondaler, 1 1 1 Hi 4 krundaler 4 5i Daler H ^ ' " 1 Countries • LENG TH _ Measure Length Measure Length Measure Length Inches Feet Miles Au-tria Fuss 12-445 Klafter 6-2226 Meile 4-7142 Bombay Hath 18 Guz 2-25 China Chik 14-1 Yau 117-5 Li •3458 Denmark Fod 12-357 Abi 2-0595 Miil 4-6807 France Metre 39-3704 Decimetre 32-809 -Myriametre 6-2138 Germany Fuss 12-357 Ruthe 1-2-357 Postmeile 4-68U7 Greece Attic foot 12-10 Stadium 600 — Holland Palm 3-93704 Elle 3-2809 :,Iijle -6214 Madras Covid 18-6 — — Portugal Palmo 8-656 Vara 3-6067 ilil 1-2786 Russia Archine 26 >ac line 7 Verst -6629 Spaiu lie 11-128 Vara 2-782 Legua 4-2152 Sweden Fot H-Gy04 Famu 5 8452 Mil 6-6413 LIQUID CA .PACITY Countries Measures GaUous Measures Gallons Measures GaUons Austria Kanne -1557 Viertel 3-1143 Eimer 12-4572 Bomba}' Adtnilie 1-515 Para 24-24 Camlv 193-92 China 1 thingtsong •12 Tau 1-2 Hwuh 12 Denmark jPott -2126 Viertel 1-7008 Anker 8-2914 France Litre •2201 Iifcahtre 2-2009 Hectolitre 22-0097 Geruiany Quartier -252 Anker 7-559 Eimer 15^118 Gi-eece — — Metretes 8-488 Hcilland Kan •2201 — — Vat 22^0097 Madras Puddy •338 Marcal 2^704 Parah 13-52 Portugal Canada •3034 Pote 1-8-202 Almude 3-6405 Russia i Vedro 2-7049 Anker 81147 Sarokowaja 324-588 Spain Qimrtillo •1105 Azumbre •4422 Arroba 3-5380 Sweden •■ Stop 1 -2878 Kauna •575G Tunna 27-6288 ' France, Italy, Belgimu, and Switzerland have perfect reciprocity in their currency. 142 FOREIGN MONEY, WEIGHTS, AND MEASURES. Table OF Foreign Monet, Weights, and MEAsrREs, | WITH THEIR English Yalpe ^concluded). Countries DRY CAPACITY 1 Measiire Contents Measure Contents Measure Contents Quarters BusIjcIs Bushels Austria Viertel •4i'3o ^letze 1-6918 Muth 6-3442 Bombay Adoulie •1893 Parah 3^03 Candy 3-3 China Shimrtsong •02 Tau •2 Hwuh •25 Denmark Fjerding •&5G7 Tonne 3-82G8 Last 10-5235 France l)ecalitre •2751 Hectolitre 2-7511 Kilolitre 3-5(4 Germany Tiertel •3780 Sclieffel 1-5121 Wispel 3-4022 (Treece Bachel •753 Kila •9152 IStaro •28l4 Holland Schcpel •2751 Mudde 2-7511 Last 10-317 T^la.iras Puddy •0423 l\irali 1-69 Garce 16-9 Portugal Alqueire •372 FauL^a 1-4878 Moio 2-79 Russia Pajak P4426 Osmin 2^8852 Tschfctwert •7213 Sjiaiu Almude ■12!)2 Fanega 1-5503 Caliiz 2-:)-2r4 Sweilen Kaniia •0720 Spanu 2-015 'I'nnna -'(.:-;:,T Countries WEIGHT Name Weiglit Name Weight Name Wcigl.t Tons Lbs. Lbs. ^Vusti-ia Pfund 1-2352 Centner 12-352 — — Bombay Seer •7 Maund 28 Cand^- •25 <""nina Tael •0833 Cattv 1-333 Pecul •05r5 Denmark Mark •5514 Fund 11029 skippund •1575 France Kiliigramme 2^2()46 Quintal 220-46 Tonne •f842 Germany Pfund 1-0311 Centner 113 426 Schiffi)fiind •15l!t G reece Pf.und •8811 Uke 2^8 Cantaro •05 Holland Pon-1 2^2046 — — . — r^ladras Sejr •G25 Maund 25 Candy -2232 Portugal Arratel 10119 Arroba 32-3795 *,)uintal •051.-; Russia Fnnt •90-?64 Pud 36-l()56 Piicken •48 3 U •045:^ Spain Marco ■5072 Libra 1-0144 Quintal Sweden Skalpund . •'J376 Lispnnd 18-752 Skeppund ■K,:. J ENGLISH COINS. Pound Steeling. Pure g-old in sovereign = llo-QOl Tror grains. Copper alloy in sovereign = 10-273 ,, Fineness of sovereign =22 carats = '^161. Total weight of sovereign = 123-278 Troy grains. Silver. Weight of pure silver in haJf-crown = 201*8 Troy grains. „ „ sliilling = 80-7 „ ,, „ sixpence = 40-8 ,, Total weight of shilling = 87-273 „ A pound Avoirdupois of copper is coined in 24 pence or 48 halfpennies. MECHANICAL PRIXCIPLES. 143 MECHANICAL PRINCIPLES. Resultant and Eesolutiox of Force.s. 1. To find the re^jiUant of t7vo fin'ces acting tln'oiKjli one point hut not in the .tame direction. (Fig. 118.) Let AB, AC represent the two Fig. 118. forces P and Q acting through the point a; complete the parallelo- gram ABCD : then its diagonal ad will represent in magnitude and direction the resultant of the two forces p and Q. '" Q E = resultant. = angle P makes with q. a = angle R makes with Q. /3=aniiie R makes with p. R = ^/p- + Q- + 2.p,Q . cos ( sni 0- R sin a = sin v - Fig. 110. 2. To find the resnJtant of any iiinvher of force,^ actinr/ in the sam^ plane and titrough one point hut not in the same direction. (Fig. 119.) . Let p, p,, Pn, P3 be the forces acting through the point of application o ; commence at o and construct a cbain of lines OP, PA, AB, BC, representing the forces in magnitude and paral- lel to them ; let c be the end of the chain : then a line R joining oc will represent in magnitude and direction the resultant of the forces P, Pj, Po, and P3. Kote. — Tliis geometrical pro- blem is true whether the forces act in the same or in different planes. E = resultant. 9 = angle made by e with a fixed axis ox. a. «„ a„, Sec. = angles made by the forces p, P,, p.„ Sec, with OX. 2x = sum of the series of p . cos o + p, , cos a^ +'p., . cos «.,, &c. 2r = sum of the series of P . sin « -t- Pj . sin «, + p.f. sin a.,, Sec. E . cos e^Sx. R=. ^/(5x)M^(5v)^ H . sin e = 2y. tan e= — 2y 2x cos e= —i R sm e=^-. R 144 MECHANICAL PRINCIPLES. Fig. 120. 3. To find the re,mlta7it of three forces acting throwpi one point and mailing right angles with one another. (Fig. 120.) Let OA, OB, oc represent in magnitiida and direction the forces x, y, z acting through one point o ; complete the rectangular solid AEFB : then its diagonal OG will represent in magnitude and direction the resultant of the forces x, Y, z. R = resultant. a, j3, 7 = the angles R makes with x, Y, Z, respectively. Y = R . cos j8. z = R . cos 7. R= A/'X2 + Y- + Z2 X = R . COS a. 4. To find the resultant of any nuniher of forces acting through one point in different directions and not in the same plane. Let P, P], p.,, ^Ncc, he the forces a, /3,7 ; Oj, jS,, 7, ; a,, 3,, 70, the angles their directions make u'ith three axes passing through the point of application and making right angles with one another. 33 R = resultant CJD c 2x = p cos a + Pj . cos a + Po . COS a., + &C. 2y = p COS /8 f Pj . cos )3 + P, . COS )3o + ice. 2z = p cos 7 ■1- pj . cos R = cos a = COS /3 = 7 a/ + P, . cos 72 + &c. c 2 m ^ !Z5 :2x)-^ 2x R 2Y R + (2Y/-r(2Z)^ , cos 7 = 2Z r' Fig. P 121. Parallel Forces. ^ coxiple consists of two equal forces, as P and Q (see fig. 121), acting in parallel and opposite directions to one another, and is tei-med a right- or left-handed couj^le, according to whether the forces tend to turn the rigid body in a right- or left-handed direction. The moment of a couple is the product of either of the forces into the perpendicular distance ab between the lines of direction of the forces. The distance ab t is termed the arm or lever of the couple. a 5. To find the resultant moment of any numher of couples acting upon a body in the same or parallel planes. KuLE.— Add together the moments of the right- and left- MECHANICAL PRINCIPLES. 145 handed couples separately; the difference between the two sums will be the resultant moment, which will be right- or left-handed, according to which sum is the greater. 6. To find the resultant of two parallel forces. (Figs. 123 and 123.) The magnitude of the resultant of two parallel forces is their sum or ditference, according to whether they act in the same or contrary directions. Let fig. 122 represent a case in which the two forces act in ^ig. 122. Fig. 123. the same direction, and fig. 123 a case in which the components act in opposite directions. Let AB and CD represent the two forces ; join ad and cb, cutting each other in e ; in DA (produced in fig. 123) take DP = BA; through F draw a line parallel to the components; this will be the line of the resultant, and if two lines dg and AH be drawn parallel to EC, cutting the line of action of the resultant in G and h, gh will represent the magnitude of the resultant. DC . AD _ AB . AD AF = GH DF = GH 7. To find the resnltant of any number of parallel forces. Rule. — Take the sum of all those forces which act in one direction, and distinguish them as positive ; then take the sum of all the other forces which act in the contrary direction, and distinguish them as negative. The direction of the resultant (positive or negative) will be in that of the greater of these two sums, and its magnitude will be the difference between them. 8. To find the jJosition of the resultant of any nxiniber of parallel forces when they act in two contrary direotioiis. EuLE. — 1st. Multiply each force by its perpendicular distance from an assumed axis in a plane perpendicular to the lines of action of the forces ; distinguish those moments into right- and left-handed, and take their resultant, which divide by the result- ant force : the quotient will be the perpendictilar distance of that force from the assumed axis. 2nd. Find by a similar process the perpendiciilar distance of the resultant force from another axis perpendicular to the first and in the same plane. 146 CENTRE OF GRAVITY OF BODIES. CENTRE OF GRAVITY. 1. To find the moment of a hodi/n weight relatively to a given l)l/ine. Rule. — Multiply the weight of the body by the perpen- dicular distance of its centre of gravity from the given plane, 2. To find the common centre of gravity of a set of detached bodies relatively to a given plane. Rule. — Find their several moments relatively to a fixed- plane ; take the algebraical sum or resultant of those moments and divide it by the total sum of all the vi^eights : the quotient will be the perpendicular distance of the common centre of gravity from the given plane. Note. — When the moments of some of the weights lie on one side of the plane, and some on the other, they must be dis- tinguished into positive and negative moments, according to the side of the plane on which they lie, and the difference between Ihe two sums of the positive and negative moments will be the resultant moment. The sign of the resultant will show on which side the common centre of gravity lies. Let 7V, w\ w"^, &c. = the several weights. d, d\ d\ &c. = the several perpendicular distances of the centres of gravity of w, n-\ iv^, &c., from the plane of m.oments. D = the perpendicular distance of their common centre of gravity from the plane of moments. n-d + w^d^ + rv^-d"^ + &c. Ji fV+7V^ + 7V'-^ + &:c. 3. H) find the centre of gravity of a body consisting of parts of unequal heaviness. Rule. — Find separately the centre of gravity of these several parts, and then treat them as detached weights by the foregoing rule. 4. To fnd the distance through ovhich the common centre of iiravity of a set of detached weights moves when one of those tveights is shifted, into a new position. Rule.— multiply the weight moved by the distance through which its centre of gravity is shifted ; divide the product by the sum total of the weights : the quotient will be the distance through which the common centre of gravity has moved in a line parallel to that in which the weight was shifted. Let w — weight shifted. <^ = distance through which w was moved. W = sum total of weights. D = distance through which the common centre of gravity has moved in a line parallel to that in which the shifted weight was moved. ^ wd J DW W W LAWS OF MOTION. 147 LAWS OF MOTION. Iinpidse is the product of a force into the time during which it acts. Momentum is the product of the mass of a body into lis velocity. The mass of a body is equal to its weig-ht divided by the veln- ciTv which that weight produces during one second of unresisted fall. Geavity. g = force of gravity in feet per second. I = latitude of the place. //. = height above the level of the sea. r =radms of earth in feet = 20,900,000 feet. ^ = 32-1695 {1 --00284(003 2Z);Cl-^^'Y If 21 be obtuse, then . ^ = 32-lG9o[l + -002S4(cos 180-20](l- ~^'\ Unifoem Acceleeatixg Force. w = weight of body. M = mass of boch'. F = accelerating force, orunbalanced effort. I = impulse exerted by F. E = energy exerted by F. t =time during which F acts in seconds. d = distance through which F acts in feet. V = original velocity. v' = increased velocity. g = force of gra\'ity = 32-2 nearly. m = mean velocity. ^x(r' — !•') 1 = F^ = ii(f' — v) — — ^^ = increase of momentum. r7 ■E = Fd = Ftm = — ^^ '' = ■ ^ ^ » 2 2g UXIFOEM PiETAEDIXG FOBCE. The foregoing formula will apply in this case, with the excep- tion that r — v' must be used instead of v' — v, and V' — v'"^ instead r'- — v-, F denoting the retarding force and E denoting the work performed. l2 148 LAWS OF MOTION. Velocity of Falling Bodies. Ji = height or depth of fall in feet. t =1 ime of fall in seconds. V = velocity acquired at end of time t. /J = accelerating force of gravity = 32-2 nearly. r = ^i = y=y2^A; 7i= ^--^X^^^' ^- \/ g~g~v' The velocity acquired by a body falling down an incline is equal to that which it would acquire in falling down its perpen- dicular altitude (see fig. 124). t =time falling from b to A in seconds. Fig. 124. I = length of incline ba in feet. h = altitude of incline bg in feet. g = accelerating force of gravity = 32-2 nearly. Wgh' Fig. 125. If a chord bc be drawn from either extremity of a vertical diameter ab of a circle, the time of descent of a body falling down the chord bc will equal the time of descent down the diameter ab (see fig. 125). KoTATioN Accelerated axd Retarded. Accelerated. W = weight of body in lbs. M = moment of accelerating force in foot lbs, E = energy exerted. V —original angular velocity. v'' = increased angular velocity. 6 =the circular motion during the action of the force in circular measure. n = original speed of circular motion in turns per second. %' = increased speed of circular motion in turns per second. r = length of arm at the end of which w revolves in feet. t =time during which M acts in seconds. g = force of gravit3' = 32-2 nearly. „^ v>'r-(v' — v) 27rW/'-C?/ — w) Mc = ^= — = ^^ ~, 9 9 2 2g 2g LAWS OF MOTION. 149 Retarded. Use the same notation as for acceleration, but substituting moment of retarding farce tor moTnem. of accelerating forre, di- mi/iutiofi for increase of velocity and its squsire, and worli jjtr- formed for energy exerted. Moment of Ixertia of Weight and Radius of Gyration. (See also pp. 78-81.) m, m', vi", &c. = weight of indefinitely small particles com- posing the body. d, d', d", «fcc. = respective distances of 7n, m', m", ko,., from a fixed axis. W = weight of whole body = m + m' + m " + .tc. I = moment of inertia of w about a fixed axis. - R = radius of gyration. K = ^/ 1 . I = vid- + wVZ'2 + m"d"- + &c. Impulse on a Free Solid Body. A single impulse acting on a body through its centre of gravity impresses a motion of translation in the direction of the impulse. v = velocity of translation in ft. per second. P = force applied. ^ = time during which F acts in ft. per second. ff = accelerating force of gravity = 32-2 nearly. w = weight of bodv. 7V gt The imimlse of a conple impresses on a body a motion of rotation about its centre of gravity. A = angular velocity in circular measure. L = linear velocity produced by one of two impulses. F = force applied. w = weight of body. I = moment of inertia of w. E- = square of radius of gyration. Z = length of arm of couple. - m = moment of couple. f = time during which F acts. g = accelerating force of gravity = 32*2 nearly, _ mtg _Yltg _'lI ~ I WR2 e^* uo LAWS OF MOTION. Fig. 126. IXSTAXTAXEOUS AXIS. If P (fig. 126) be the point of application of a single impulse (produced by a force f) acting through a line PA, not traversing the centre of gravity of the rigid body, and x be the position of the instantaneous axis, the body will rotate round x instead of round its centre of gTavity g. ^7 = perpendicular distance of G from pa. V = velocity of translation produced by a single impulse acting through G in a line GE parallel to PA, and equal to the single impulse acting through P (see foregoing formulas). A = angular velocity of rotation around G or x, produced by the impulse of a couple of the force f and arm d (see foregoing formulae). D = distance of x from g, measured perpendicular to PA. R- = GC = square of radius of gyration of body set off perpen- dicular to BG. D = I = ^. A d Table Giving the Lengths of Pendulums IN Inches 1 THAT Vibrate Seconds IN Yaeious Latitudes. | Sierra Leone 39-01997 New York 39-10120 Trinidad 39-01888 Bordeaux 39-11296 ]\Iadras 39-02630 Paris 39-12877 Jamaica 39-03.503 London 39-13907 Eio Janeiro 39-013.50 Edinburgh 39-1 .5510 Simple Pendulum. L=length of pendulum in feet. T=time of one vibration in seconds. K= number of vibrations per minute. 5r=force of gravity=32-2 nearly. 71=3-14:16 nearly. j._60yT^i08:36__ L=5r(iy = -326T\ T=7rv/-=-554VL. & The length of a pendulum vibrating seconds at 4-5° latitude equals 39-11346 ins. nearly. In latitudes less than 9(P the length equal? 39-11 34H [l--«t^84 (cos -2 lat.)]. In latitudes exceeding 90° the length equals 39-11346 [1 + -U02a4 (C03 •130='-2 lat.)]. DISPLACEMENT. 151 Deviating axd Centeifugal Foece. F= deviating force of body revolving in a circle at a uni- form speed. w = weight of body. N = number of revolutions per minute. n = number of revolutions per second. V = linear velocity in feet per second. a = anerular velocity in circular measure per second. r = radius of circle in feet. ^ = accelerating force of gravity = 32-2 nearly. gr g g "Sioi 2935 Centrifugal force is exactly equal and opposite to the deviating force. Revolving Pexdulum (Fig. 127). F == de\"iating force. TV = weight of bob. N = number of revolutions per minute. n = number of revolutions per second. Ji = height of pendulum in feet. r = radius of circle in feet. g — accelerating force of gravity = 32-2 nearly, \\r _ q _-815J:_ 2935 Fig. 127. // // TV 1 / ( ^tt ..^"^ ^<^ ^ Computation of a Ship's Displacement. This consists in computing the volume of the body of the vessel below the water-plane, up to which it is required to know her displacement, by one of the rules used for finding the volume of solids bounded on one side by a curved surface (see pp. 44, 45). Two processes are generally made use of in computing a vessel's displacement, as the calculations in each process are required to determine the position of the centre of gravity of displacement,, or centre of buoyancy, and also because the two results are a check on the correctness of the calculations. One process consists in dividing the length of the ship on the load water-line by a number of equidistant vertical sec- lions, computing their several areas by one of Simpson's rules, and then treating them as if they were the ordinates of a new curve, the base of which is the load water-line. lo2 DISPLACEMENT. The other process consists in dividing the depth of the vessel belov/ the load waier-line bv a number of equidistant longitu- dinal planes parallel to the load water-line ; the areas of their several planes are then computed by one of Simpson's rules, and those areas are treated as if they were the ordinates of a new curve, the base of which is the vertical distance between the load water-line and lirst lowest longitudinal plane. As the vessel generally consists of two symmetrical halves, the volume of only half the vessel, below the load water-line, is calculated, the ordinates all being measured from a longitudinal vertical plane at the middle of the ship. For example of displacement papers see pp. 155 and 156. Detehmixatiox of a Ship's Cextee of Buoyancy for THE Upright Position. The centre of buoyancy is also termed the centre of gravity of displacement, as it occupies the same point as the centre of gravity of the volume of water displaced by the vessel, and its position is determined by the rules used for fixiding the centre of gravity of solids, bounded on one side by a curved surface (see rules, pp. 76 and 77), with the exception that its position need only be determined for its vertical distance from a horizontal plane, and its horizontal distance from a vertical plane ; for the ship consisting of two symmetrical halves, it mttst necessarily lay in the longitudinal vertical plane in the middle of the ship. Calculation of the centre of buoyancy is generally performed on the displacement paper (see pp. 155 and 156). Vertical Height of Transverse Metacentre above Centre of Buoyajs^cy for Upright Position. The transverse metacentre of vessel for all angles of heel always lies in a longitudinal vertical plane bisecting the ship, and vertically over its corresponding centre of buoyancy ; its vertical height above the centre of buoyancy for its upright position is found by dividing the moment of inertia of the load water-plane relatively to the middle line of the vessel by the volume of displacement (see pp. 165 and 175). This calctilation is also generally performed ujDon the displacement paper (see p. 155). Curve of Areas of Midship Section. This curve (see fig. 12S) is used to determine the area of the immersed part of the midship section of a vessel at any given draught of water. Method of Construction. — Compute the areas of the midship section from the keel up to the several lougittidinal water-planes CURVES OF CAPACITY. Fig. 128. Ktrale- oj areas Fig. 129. 153 which are used for calculating the displacement ; set these areas off along a base line as ordinates, in their consecutive order, the abscissae of which re- present to scale the respective distances between the longi- titdinal water-planes : a curve bent through the extremities of these ordinates will form the required curve. Curve of Displacement. This curve is used to determine the displacement a vessel has at any draught of water parallel to the load water-line (see fig. 129). Method of Construe- tiofi. — This curve is constructed in a simi- lar manner to the fore- going curve, with the exception that the or- dinates represent the several volumes of dis- z?" placement (in tons of $caie of Torvs 35 cubic feet for Salt water, and 36 cubic feet for fresh water) up to their respective longitudinal water- planes. Curve of Toxs per Inch of Immeesiox. ^^<^- 120. This curve (see fig. 130) is used to deter- mine the number of tons required to im- merse a vessel one inch at any draught of water parallel to the load water-plane. To find the dis- -^-^ — y-= placement per inch in ScaJe. cf Jons 1 • J- r i cubic feet at any water-plane, divide the area of that plane by 12 ; and if the dis- placement per inch is required in tons, divide by 35 or 36, as the case may be. A=:rarea of longitudinal water-plane in square feet. T = tons per inch of immersion at that water-plane. T = 12x35 for salt water : T = 12x36 for fresh water. 154 COEFFICIENTS OF FINENES; Jlethfld of Co nsto' notion. — This curve is also constructed in a similar manner to the two foregoing curves, with the exception that the ordinates represent to scale the tons per inch of im- mersion at the respective water-planes. COEFFICIEXTS OF FINENESS. Tff^ coefficient of fineness of displacement of a vessel is the ratio that the volume of the displacement bears to the paral- lelopipedon circumscribing the immersed body. V = volume of displacement in cubic feet. L = length of vessel at load water-line in feet. B = extreme immersed breadth in feet. D = draught of water in feet. K = coetficient of fineness. K: L X B X D The coeficient of fneness of a midsldp section, or of a water- plane, is the ratio which their respective areas bear to that of their circumscribing rectangle. To determine the mean coefficient of all the nater-plane^ of a ship. Rule. — Multiply the immersed area of the midship section by the length of the load water-line, and divide the volume of displacement by the product. Table of Coefficients of Fineness. ■>:3 S-l ■s. Class of Sliip SB a o O o aj .a. 07 o Feet Feet c-r Or- OS cl Feet Fast steamer, H.^NI. Royal Yacht m^y) 40-27 14-0 -414 -711 -711 Swift steam ' H.M.S. • Inconstant' . 337 3 50-28 22-75 -483 •787 -614 cniisers 1 H.M.S. 'Yoiage' 27J-0 42-0 19-0 •497 "792 -628 T> ^„i „,„;i ( National Line . Royal mail \ peninsular and Oriental . 385 42-0 22-0 -659 -880 -800 3.-i8-27 42-5 35-0 18-71 21-0 -516 -687 -812 •850 -635 -840 steamers ( ^^.|,^,. Li^e . . . 35U-0 T,.«-,T^=v„•^= f H.M.«. -Serapis' Troopships 1 H.M.S. 'Himalaya' . 360-0 49-12 23-5 -470 ■674 •700 340-5 46-13 15-75 •400 -680 582 Modern rigged ironol., H.M.S. • Hercules' 32-5 59-0 24-75 •640 •810 -710 Modern nia^tless ( H.M.S. • Devastation' ironclads t H.M.S. ' Cvclops ' 285-0 62-25 26-5 -684 -809 •767 •225 450 15-0 -715 •932 -755 ,^ ^ . !,„ 4.^ ' H.M.S. ' Ariel ' . 12.5-0 23 80 -536 •870 -61 1^ Composite gun boats -, jj.m.s. ' Sappho- 160-0 31-33 12-0 -466 •745 •6n3 Small merchant vessels ] ^"^"^ 220-0 00-0 27-0 17 5 80 4-0 -702 -637 -912 914 -742 -704 DISPLACEMENT SHEET. 155 Table showixg Method of Computixg a Ship\s Dis- PLACEMEXT, THE POSITION OF HER CENTRE OF BUOY- ANCY, ETC., WHEN WHOLE INTERVALS ARE USED. Waterl AVater Line 1| Line 2 Water Line 3 Water Line 4 Water Line 5 Siiipsox's Multipliers 11 TFiT •2 -4hI-J^'— 23 9-:; "6^ iTs 25;6 99 39-6 39-6 12-3 2rti W-> 13-5 54-0 _o£n 13^ 2r6ll6^ .-m; 25 •« 54-8 128 51-2 10642-4 42-41 25K 1-9 7-6 7-6 2 •8 2 •4 4 2 16-ii 8-4 9-2 IS^ J8j4 13 1 52-4 26 ■2i -2 56224 22"4^ 43-: _ 14-3 57- 57-2 15 3 306 160 320 16-2!32 3n- K' 16^6 m)|i6^ 66^ 32-0 16 5 33^0 66-0 32-0l 13-7, 54-8 16 640 16 5 660 165 32 15 731-4 31-4 14-8 59-2 29-6 6 4'12-8 11 5 230 14-5 29-0 15 6 230 5-4 21-6 108 ~^ •4 6-8i27-; 6-sj iOS 2r6|ir9i2?S 11-9I 15-0 "jO-0 15-0 Vertical Sections 47-0 _99-i l3a-2 66-0 16 4 3F8|l6l[ 65-6 16 1641 64-4 5-<-0 9 3 37- 37-2 •2 • j62 _ 16-5|66- 16- 165]33-0 16-5 ;36-0 16-5 i60-2 j6^1 163 16-3 15-6 31- 12 6150-4 2. 7-0 1 3U-4 4 413-8 4620 4 488-4 169-9 169-5 164-8 152-9 122-4 68-4 2-4 188-0 198-6 552-8 320-4 x^B 674-8 339-8 678-0 329-6 611-6 24£8 273-6 Metacextre '■J =^ -3 Z.~ 1685-1, 3375 00 4251-52 2-4 112 , 28-8 1492-12 4492- 1; 4410-94 1330-74 ll'57-72 1350000 850304 17968-48 8984-24 17968-46 8«2r88 17322-9t S796-41| 7o92-Si 200037 ~00 8001-48 •00 4416-6=7-0 + 1245-6 + 8276 + 1848-0 + 4 3 _ 2 1 7268-0=28-0 + 3736-8 + 1655-2 + 1848-0 + 4416-6^80 1-64 Vert. Int. 3-812 6-2517 4416-6 )2731 l-fi^ 6'^]83 Long. Int. 13-6 102638 4416-6 V73ir6 ) 113291-4 i Vert Int. r2708\--^Long.Int.l6-6 '4 5612-61.')23 I 0[ Long. Int. 166 ■6 3)93169-413648- 31056 -47 ^~ 2SL' ■ 3 ) 188063 "-2' *62112-9)626.«79^ns 1009 Cub. ft. in a ton 35) 62112-94 * 1774-65 L C. of Buoy, bihi^ w:l. sLc. of B.iS7nrbkiro5j^«;S^'^^"^« 'l^^-^^. of Buoy. 3 )2018 6-727 Metacentre ,.r- , ' 488-4 / !s^ ertjnt.l69-9_;|iang. In t. 166 / 29304 Function _lj27oi\ 114372 ! 114372 76248 I 12708 29304 4«.84 3)8107-44 311-4X1= 311-4 413-8X4=1655-2 4620X1= 462-0 2428-6 7-0X1= 70 311-4X4=1245-6 413-8X1= 4138 1660-4 J Vert. In t. 1-27 1 Vert. Int. 1 -27 3084-322. Long. Int. 16-6 3 ;51199-745 17066-58 2 7 ^4133^ 6 5)4876-16 at W. L1ne_No. 5j —97543 ^sptrio^. L. No. 4 1195-76 , 7-0X1= 7-0 311-4X3= 934-2 413-8x3=1241-4 462-0X1= 41)2-0 2ti44-6 2116-328 i Vert. In t. 1-43 Long.In t. lH-6 37'*r778 3 )35131-0418 Long.I nt. 16-6 11710-36 2 7)23420^70 . 5 )3.34.V8 1 669-16 Dispt. to \ 3)e277ri48 20925-838 7)41851;676 5)5978-81 1195-76 220331 W. Line No. s) Dispt. to Dispt. to W. Line No. 2 .'W. Line No. 4. Long. Interval between OrcLuiates=16-fift. Vert. Interval between Water-Lines=3-8125ft. ♦V ^'-'B- The dark figures are the ordinates ; the light fi.nire^ under them and nUo '-. their right are the products of the ordinate? by their relpecti^e Simpson" niitwD""er. wh-'ch Srorf, of'h/L'^i^'^^.r*^ "^'^ *£ ^^^ l^f* "f^^'^ table; and if each^rownndoo umn onh^.e product.be added together, and the results integrated bv the same n.ultipliers as were used 156 DISPLACEMENT SHEET. ^" ^" >• r 1 1 1 1 1 1 ; 9 1 p i; 1 1 1 1 1 rzz 7) X o ^ C c o c c c c o cl c 2 c! oo::"i) =i S ,9 c tr tc to pox « Oi >1 to — --^ -' ^ 7" X s t^ X I-, '*0 ' ^^ '-C ■^ r- X r ^ >■ 3^' 2"*" -" cc X CO OS 1 t-; IN Sl^S2 — CO 1< w in N in IN ■"3* i, ^ 3-( <; "silM ^ > s| 1 1 1 1 33 II ° ,_ ^ « T •,o t^ X OS o »— ' 1 C»l I C^ 2 ■-- 3 ^^ ^l ^ 1? X ' 1 1 ; G -3^ ^ /^ ■/j ^ o c c c o o ! O C 1 c o o o o S '-~ in , 3 sj 5 a c ii— u T o re « « O O ■ CO C>1 t^ 1 c p PI X O (T. 1 C>1 CO CO CO X ^^15 • 1 s ^ cji o N S 1 CO Cj O I — OS IM X to S; l-ri L- 3 ;^ >— 1 1— « s O CO ] to "1 ^12 ^'^ ■"Its 1 ,H~ •^j e«i "* 1 e«j ■* * -V ^ O CO t'* -T- !■-. ' -^ to 1 X I>» <^ -< ^ rc — "^ ci o c>i o ' 7^ ; o -'!--'-! 1 5! 13 ISl i3 > ^.5 cc |o= Oo O - !o c O - io c: io .- Io o O uo 1 ^ = |0 o 'O c |0 c |0 o O o ' ' ■"^ ^1—1 -< ^Oc ^2 'Jt i i(M X CD ?5 '^ = ;0 :^i 'LO Ti m 71 1^ - j o iONIO-t. o H p ^r ?< ■p O X -J ?1 Uo X h2 j C>1 Uo O 1 — t^ -T M ^o ^ -*<> >j i CO to o M -J: Jo Ko « o 1 io Ui« CD 1 ■ CD CO CO o=^ «5 ■a" N o N ■n |M oTlo- O |N 1^ |_ 1 [ ? X Si O o O - Io e |o = fo o 'O - Io o Oo'O-lo^lOolo,-, lOr- nI :j ^ ■" X OO -^^^:r^^^|05x ^LOH (M-i?:-2 m5 coe 'JJ- oo-iic5-2 im-|os|rHSi »-' V. jj ? 1 1 " It-I •" irH -' rH -Vr-I •' rH ■' r-l •' rH ■' rS ■' rH •' ~' \ \ o "^■" "* C ^ 1 = = 1 5 - 1 o 1 — - ' - -^ O O O IO 1 o "^ OCOjO ^ X If! c>) [O M X 1 5 X 1 fi C; O X X l-< o o ■ o a _^ ' -r* 1 O 'r~t '-M -- o -- o 1 — -r* o b "^ 1 ' b b I — X o *- o CO ico — ' ; x ~ _^ P= Oc OopoPo'O- p-|0- O -!o- 0_i0 = i0c Pc 0-' (r-*'— ' locic^fciOS Mrtps loox 35- ';3?. :sif. cox oaioei'j^stcq?; H = ; — |" ■ '03:-: r~t-'6= i:cci >j'6= ib^~ Iih^ ' ' + + -^ i 1 1 1 |rH""r-lr-H r-l~rHrHrHrH~fl, 1 I c ;o = o o o o = hn -t* ■p ?) •o o o o ■M O o p p 7^ CO pco rl -•' -^1 1^ CO 5? S' i r^ ^ ■1^ 1^ U- b tc ,Lt ■^ X ! = | -!■ N CO M Ho - 2; ' ■'■■■•je-?(lcr3irti ^ c iOclOc D^l0=;|0-|0~l0-P = l=i-|0-i0- 0= O = l0c|0o > ^3 ,^ pSIgjx >^kx 05 OpIO = ;;3-;;!:p= 10= Ockfix rHf> •' ■ ■ c^ o i 2 p 2 '-^ ;; '^ :^ ':<) r 01 -, N r -H ^; ' 6 ■- c~ £ n - " ' * ' "^ ^ ^^~ o c- o !3 c 1= i~ ,o = j = 1 o 1 c o c: 2 o-^rMS"^! o >i 7? ' p c ri jo ] ^> X j — ' o 1 ^ -^ M o p 1 = 1 c 1 ■ ,^ i- — ^ O c: CO i^ 1 J —' -r j ri -^ ^ — < 1 ~" ' I r; ■^ i _ '"■ . — ^.»^^H_ :- _ 1— P' _ 1 _l __ _ to ^ X ' 21 1 ^ i?rH Io = 'O ..o Io u- Io o (O o Io >- Io o Io = Io = !0 = Io >.-: lO o lO ^ lO >.o |0 o -c,l6 ? jw r |p- r^ 115 r l^'l>"'l>'^ C-"'"5'^' C h h" 1-^ '•' 1-^ h '■=' 1^' '-* H ij i'f i| tiUi !1^-">S 1— In j« 1^ luT |-j5 It^ jx l^-. io \~- l« jrj K 1^ " -^ J^ — - «H J ►i- c ^— — ^- DISPLACEMENT SHEET. 157 otsrs ,«— — . ■J>''i |NVB 21^1 S X r5 ■ — s X; 3 W fci's ?ja 5i II Mil s •■ n'-c i'ld OX'- 1 XXX a "3 ' X . I ..- jq , C 5-1 li>.Tir-: — ^ X TlXt^ 3-. |X tg CN = xrM« c ;, - -■- o 1 = •+» . — rt £=; -i -T 1 c; °= iiTiiiTiii |i^ ..- X o ^ ?t« IS ^ Hf;i V; X t-^ c ii ;; « ^ c^ "■ 2 s s 1 '3t- X s 11 II || II II II a'""-g S'S|S5§i 2 c. cili ?1 iC ci Yi •^ 1 "^ " --•S -s & ;= o >B 2 t^ t^ o S > i; QC"^ s- 2 2 c o >- o '-^ ^ 158 EXPLANATION OF DISPLACEMENT SHEET. Explanation of Displacement Sheet. (See pp. 15G and 157.) The length of the ship at water-line 5 is divided into 14 equal intervals, and the depth or draught of water* into 4 equal intervals, the lower two being subdivided into half-intervals (for multipliers for sub- divided intervals see pp. 39 and 40). The ordinates, or half-breadths, at the intersections of the vertical cross sections with the horizontal sections are measured off in feet, and set down in dark figures in rows opposite their respective cross sections and under their respective hori- zontal sections, thus forming the numbers into columns. Each of the ordinates in the several columns are then multiplied by the ' Simpson's multiplier ' at the head of their column, the products being set immediately below in lighter ^gure^, and their sums taken in rows and placed to the right in the column headed 'functions of areas.' Each of these 'functions of areas' is then multiplied by the ' Simp- son's multiplier ' proper to its roiv, the products being placed to the right in the column headed ' multiples of functions,' and their sum taken.f Then, as a check upon the last result, it is usual to multiply each of the ordinates in the several 7'ows by the ' Simpson's multiplier ' to the left of their respective rows, the products being set in the adjoining column in lighter tigures, and their sums taken in columns and placed below in the roiv of ■ functions of areas.' f Each of these ' functions of areas ' is then multiplied by the ' Simp- son's multiplier' proper to its column, the products being placed below in the row of ' multiples of functions.' The sum total of these ' mul- tiples of functions ' should then exactly correspond to the sura total of the column of ' multiples of functions,' thus proving the correctness of the calculations thus far. The latter sura is then multiplied by ^ of the vei'tical interval, and this again by 5 of the horizontal interval between the ordinates. This last product is then multiplied by 2 for both sides of the ship, and the result divided by 35 (that being the number of cubic feet of salt water in a ton), which gives the total displacement of the ship in tons to water-line 5. The horizontal distance of the 'centra of buoyancy ' abaft the stem, or No. 1 section, is then f amd by multiph'ing each of the products in the column headed 'multiples of fvmcl ions' by its multiplier for leverage (that lieing the number of intervals the cross section is distant from Xo. 1 section), the products being placed in the column headed * products for moments.' The sum total of these divided by the sum of the column of ' multiples of functions.' and the quotient multiplied by the horizontal interval, will give the distance of the centre of buo^-ancy abaft Xo. 1 section in feet. The vertical distance of the ' centre of buoyancy' below water-line 5 is found by multip lyiug each of the products in the row of ' multiples of functions ' by its multiplier for leA'erage (that being the number of intervals the horizontal section is from Avater-linc 5), the products being placed below in the row of 'products fur moments.' The sum total of these divided by the i-um of the row of multiy^les of areas, and the quotient multiplied l)y the vertical interval, will give the vertical distance of the centre of buoy- ancy below water-line 5 in feet. * Should tlie vessel have a bar keel, the depth should be taken from top of keeL t These numbers are only proporffonal to the areas of the vertical or hori- zontal sectidn* ; but to find the absolute values of the areas of any of these sections the numbers must be multiplied by J the distance between the ordinates, aud that product by 2 for both sides. CENTRE OF GEAYITT OF SHTP's HULL. 159 To Calculate the Position op the Centre of Gravity OF A Ship's Hull. To find the centre of gravity of a shij/s hull relatively to any fixed plane (see p. 161). Rule. — Find the moments of the component parts of the ship's hull relatively to the given plane by multiplying the weight of each part by the perpendicular distance of its centre of gravity from that plane ; then find the resultant of those moments by adding together separately the positive and negative moments (or right- and left-handed moments), and taking the difference between the two sums ; the resultant will be positive or negative, according to which moments are the greater. Divide the resttlt thus found by the total weight of the hull of the ship; the product will be the perpendiculai- distance of the centre of gravity from the given fixed plane. As the centre of gravity of the hull of a ship is generally in the middle line, it is only necessary, as a rule, to determine its position relatively to two fixed planes, one being a transverse vertical plane and the other a horizontal plane, the midship transverse section and the load water-plane being generally taken as the two respective planes. To determine the position of the centre of qramty of the bottom plating of a shijrs hull when of a nniform thickness throughout. 1. Determine its longitudinal position from a transverse vertical 2)lane as follows (see p. 160) : — Rule.— Measure the half -girths of the plating at equidistant stations, as if for measttring its area ; integrate by means of a set of Simpson's multipliers, and add the results together ; then multiply each of those fitnctions of the halt-girths "in their con- secutive order by the figure representing the number of inter- vals it is from the plane of moments. Find the resultant of those moments and divide it by the sum of the fttnctions of the half-girths, and multiply the product by the common interval between the stations. The restilt will be the perpendicular distance of the centre of gravity from the given fixed plane. 2. Determine its perpendicular distance from a fixed horizontal plane by the following rule, providing that all the centres of gra- vity of the half-giHhs are below the jflane of moments (see p. 160) : — Rule. — Measm-e the half -girths as before ; integrate them by means of the same set of Simpson's multipliers,"and add the results together ; then multiply each of those functions of the half-girths in their consecutive order by the respective distance of its centre of gravity from the given plane ; add together the products and divide the result by the sum of the functions of the half-girths: the result will be the perpendicular distance of the centre of gravity from the horizontal plane. N.B. AVhen the frames of a ship are of a uniform character, and are placed at equidistant intervals, their common centre of gravity may be determined in the same way by means of the two foregoing rules. 160 CENTRE OF GRAVITY OF SHIP 3 HULL. Table showing Method of Calculating the Longi- tudinal Position of the Centre of Gravity of THE Bottom Plating of a Ship"s Hull. No. of Half- Statious- o'irths j Simpson's Functious of Mults. for Products for I Mults. Half-girths Moments Moments 1(38-0 7G1-G oG9-(3 G920 310-4 498-0 170-4 17G-0 •0 17G-0 173 2 5u5-2 322-4 7G-2-0 432-0 980-0 256-0 1830-6 ) 460-8 •246 1.5 Distance of C. of Orav. toward^ Xo. 17 from Xo. 9 Station 8-G90 1 2 3 4 5 6 7 8 9 10 11 12 IS 14 15 16 17 21-0 27-2 30-8 34-6 88-8 41-5 42-6 44-0 44-0 44-0 4^-3 4-21 40-3 3S-1 3G0 35 32-0 21 -0 108-8 61-6 138-4 77-6 16G-0 8.5-2 176-0 88-0 176-0 86 6 168-4 80-6 152-4 72-0 140-0 32-0 No. of Stations 1 2 3 4 5 6 7 8 9 10 11 12 lo 14 15 16 17 Sum of functions of hali-girtlis 1830-6 Table showing Method of Calculating the Vertical Position of the Centre of Gravity of the Bottom Plating of a Ship's Hull. No. of 1 Stations; Simpsou"^ Functiems of Mults. for [ Mults. I Half-girths I Moments | Products for i Moments l No. of Stations 9 10 U 12 13 14 15 16 17 Distance of Centre > 21-0 •60 12-60 1 108-8 1-25 136 00 2 61-6 1-80 110-88 o 138-4 2-10 290-64 4 77-6 2-25 174-60 166-0 2-30 381-80 6 8.5-2 2-35 200-22 / 176-0 2-40 4-i2-40 8 .S8-0 2-41 21208 9 176 2-41 4-24-16 10 86-6 2-40 207-84 11 1G8-4 2-35 395-74 12 8U-6 2-30 185-38 13 15-2-4 2-2o 342-90 14 72-0 205 147G0 15 140-0 1-.50 210-00 16 320 •75 24-00 17 18.S0-6 18.30- fi) 3878-H4 eloAv Lon fritiidinal Plane 2-118 CE^-TRE OF GRAVITY OF SHIP S HULL. 161 O m '3 Y, o t^ ^ o I « O i'l 3-. « X S^ -^ 5 r ^« 1^ T^l -X O •^ 3^ ^ rc t>. -^ s ?; j^ I at I -J I 1 =sk?-5 I 1 ^:^ 1 Ss?J ! I I 1 k I 1 I I I S2 2, ^ >Q I I OOMO r OOI'-■^ — ^ ^1 w r ;i; — -i. »-. — I I ^ *+- ■ -: O O o 1- 'n« — — — — -> S o » ajo 2^ o Ch H S E^ (1. & 02 o o o Q o W o K 02 H m I 1 I 1 1 ( 13 ||| I Ip I l| I I 25 CTpppp«0 p pp X « — -O Cl 00 I oc I I « o ? ? ? ? ? ? ? Sii I o b o ot 35 o CO — qc si c-g = 5|S I 1 I I m ■ o-> — — X 12 ^ N «,«^x _ -J -T C -r rcooppppp^xr'f^ ibnbbb9>t-;B;ac T ; '-^ .5 eS J" ♦^ = s-=!e^ £•2 .5 '- ■3 a .^ °.-Sj= C fe b =» a o 2 2= ^ £ - £'""-52l^.-S^ 2=2=2 ---S-i: o u o e» ,- o ^ o S ° 5.® 162 STABILITY. STABILITY. Statical Stability. Statical stability is defined to he the moment offoi'ce hy ivhich a floating hody endeavours to gain its nprigld xmsition, or position if equilibrium, after Jiaving Fig. 131. iecn deflected from it. Fig. 131 is a trans- versa section of a ship heeled over through a certain angle B. w'l' is the water-line for the in- clined position, and WL is the water-line for the upright position. These two planes intersect each other in a longitudinal direction, and bound two wedges l'sl and wsw' equal in volume to each other, provided tiie displacement remains the same. The wedges are called respectively the wedges of immersion and emersion, or the in, and out wedges. G is the centre of gravity of the ship, and b' her centre of gravity of displacement, or centre of buoyancy. The weight of the ship then acts vertically dowmwards through G, and the resultant pressure of the water acts vertically upwards throiTgh B^, these two forces forming a i-ighting couple, the arm of which is GZ — that is, the perpendicular distance between the lines of action of the two forces. The moment of this couple — that is, the weight of the ship, or its displacement, multiplied by the length of the arm GZ — is the moment of statical stability of the ship at the given angle of inclination 6. This moment is generally expressed in foot tons — that is, the weight of the ship in tons multiplied by the length of the arm GZ in feet. B is the centre of buoyancy of the ship when upright ; s is the point of intersection of the two water-lines, i the point where the verti- cal b'm cuts the plane of flotation ; g and g' are the centres of gravity of the emerged and immersed wedges respectively, gJi and g'h' being perpendiculars dropped to g and g' from the plane of flotation w'l'. The point m, where the vertical line BM, drawn through the centre of buoyancy B when the ship is in an upright position, cuts the vertical line b'm, drawn through the centre of buoyancy b' for the inclined position, is termed the transverse vietacentre when the ship is inclined through an in- definitely small angle, and also when the point of intersection is the same for all angles of heel. I When the position varies for the different angles of heel, it is termed a shifting metacentre. When the ship is inclined longitudinally, it is called the ; longitudinal metacentre. I. STABILITY. 163 During the inclination of the ship the centre of buoyancy moved from B to b', and b' lies in a plane j^arallel to a line joining g and r/'. The distance bb' can be found from the fol- lowing expression: — bb' = ^^< D where d = volume of displacement and v = volume of either of the wedges ; BR = L^_, where br is perpendicular to b'm ; D ^' X lilt' and GZ =- BR - BG . sin 6 = BG . sm 0, D whence Atwood's formula for expressing the moment of statical staMlity at any angle is M = (V X hit') - (D X BG . sin e) Id J The moment of statical surface staJnlity at any angle 6 is BR X D, being what the righting moment would be, supposing the centre of gravity of the ship coincided with b. The angle of heel in fig. 131 is bmb' = lsl', and its sine is equal to — = -^^ — . ^ B-M GM The coefficient of a ship's stability a,t any angle of l.eel is expressed when the displacement is multiplied by tlie vertical height of the metacentre for the given angle of heel above the centre of gravity. That is, the coefficient of a ship's stability at any angle = D X GM = D(BM — EG) BM=-i^iM:. D. sin d BR is said to be the lever of statical surface stahility. When M lies above G the vessel is stable ; if too high, the vessel is uneasy ; when below, the vessel is unstable ; and when it coincides with G, the equilibrium is said to be neutral. The point M in vessels of the common type is usually calcu- lated for the upright position, as it generally remains a tixed point for the first 10 or 15 degrees of heel, when it is useful for comparing the initial surface stability of different vessels. To calculate the height of the metacentre above the centre of buoyancy see pp. 155 and 1 75. Dyxamical Stability. Dynamical stability w defined to be the amo7tnt of mechanical iKorli necessary to canse a body to deHate from its ii^inght ;position, ■ or jjosition of equilibrium. m2 164 STABILITY. Dynamical stability is expressed as a moment by multiply- ing the sum of the vertical distances through which the centre of gravity of the ship ascends and the centre of buoj'ancy^ descends, in moving from the upright to the inclined position, by the weight of the ship, or displacement. In tig. 131 during the inclination of the ship through the angle 6, the centre of gravity has been moved through a vertical height GH — GO, and the centre of buoyancy has been lowered through a vertical distance b'i — bh, and the whole work to do this, or her moment of d^Tiamical stability for the given angle d, is = d{(gh — go) + (b'i — BH)} = d(b'z — bg) = d(b'e — BG . vers 6) = d/_^i^_ — -I — ^_BG . vers e) ; whence Moseley's formula for the moment of dynamical sta- bility at any angle 6 is = \{gh-¥g'h') — (p x bg . vers B). The djmamical stability of a ship at any angle is the in- tegral of its statical stability at the given angle — that is, if ]M = the statical stability and u the dynamical stability, then U =fMd9, where dO is a very small angle of heel. The moment of dynamical mrface stahility is expressed by multiplying the weight of the shij^, or displacement, by the de- pression of the centre of buoyancy diu'ing the inclination — thai is, for the angle Q U = D(b'I — BH). EULES CONNECTED WITH STABILITY. 1. To fnd approximately the onomeyit of statical surface- stahility per foot of lenyth of a vessel at any small angle of lieel.- KULE. — Cube the half -breadth of the vessel and multiply it by the sine of tlie angle of heel ; two-thirds of the product will be the required result. This result is expressed as follows when b = half -breadth of vessel : — Kb^ X sin 0). 2. To find approximately the surface stahility of a vessel for- any small angle of heel. Rule. — Divide the moment of inertia of the plane of flotation for the upright position relatively to the middle line by the volume of displacement ; the quotient multiplied by the sine of the angle of heel will be the required result. Or it may he expressed, morefvlly as follows: — Divide the length of the plane of flotation, or water-line, for the upright position into a number of equal intervals;^ CrEVES OP STABILITY. 165 and measure the half -breadths at tlie points of division ; cube those half -breadths and treat them as if they were ordinates of a new ciirve of the same length as the plane of flotation : two- thirds of the area of the new curve, found by a proper rule, will be the moment of inertia of the plane of flotation relatively to the middle line. This moment of inertia multiplied by the sine of the angle of heel will be the required result. It is usually expressed in algebraical symbols thus : — 2 sin 0/- o 7 — ^ffdx. Note. — The two foregoing rules are exact for any angle of heel if the metacentre remains fixed for the different angles, and there- fore remains also true for any angle of heel when the moment of in- ertia of the plane of flotation due to the angle of heel can be found. 3. To find ike height of the metacentre above the centre of buoyancy for the uprxyht positwn. Rule. — Divide the moment of inertia of the plane of flotation relatively to the middle line by the volume of the displacement. In algebraical symbols it is expressed as follows : — Note. — For moment of inertia see Rule 2, p. 164, also p. 79. 4. To find approximately the dynamical stahility of a vessel ■at any given angle of heel. Rule 1. — Multiply the displacement by the height of the metacentre above the centre of gravity, and that product by the versed sine of the angle of heel. Rule 2. — Multiply the statical stability for the given angle by the tangent of one-half of the angle of heel. Curves of Stability. The JletacentHc Curve, or Curve of Metacentres, is a curve used to determine approximately the initial statical sta-face stability Fig. 132. a vessel has at any draught ^ of water parallel to her con- ^ ^ structed load draught. Method of Construction. — Calculate the height of the ship's metacentre from the under side of keel for several successive draughts of water parallel to her constructed load draught ; set those heights off as ordinates (see flsr. 132) from a base line the abscissas of which represent to scale the respective draughts of water : a curve bent through the -extremities of these ordinates will form the metacentric curve. 10 12 J4- Scale, ej .DrawgHtsefyrorCer 166 CURVES OF STABILITY. The Curve of Statical StaMlity is a ciirve used to determine the exact statical stability of a vessel at any given angle of heel. Fig. 133. curve of statical stabilitt of ax iro>'clad with high freeboard. SCALE CF DECREZS E3R AIISLE 3Iethod of Const7'Hctio?K — Calculate the length of the arm of the righting couple, or GZ (see lig. 131), for several successive angles of heel taken between the upright position and that at which the length of the arm becomes zero ; set the lengths of these arms off as ordinates (see fig. 133) from a base line the abscissEe of which represent to scale the respective angles of heel : a curve bent through the extremities of these ordinates will form a curve of statical stability. The CuT^ve of Dijnamical Stability is constructed in a similar manner to that of the curve of statical stability-, with the ex- ception that the various lengths of the arm (b'z — bg) = (b'r — B& vers Q), (see fig. 131), are taken as ordinates instead of GZ. Fig. 134. curve of dtxajucal st.vbility of ax iroxclad with high freeboard. i' m" 15» za" 25° so'as"" -^o" ^s" so" ss" so" ss" la' 7S ao as so SCALE or BiCCBEES FOR ANCLE OF HEEL Curves of Statical and Dj/namical Surface Stability are also constructed in a similar manner to the foregoing cm'ves, the lengths of the arms br and b'i-bh (see fig. 131) being taken as ordinates for the respective curves. To Calculate the Statical and Dyxamicil Stabilities OF A Vessel at Successive Angles of Heel. 1. Body Plan (fig. 136).— Prepare a body plan in which all the sections are taken perpendicular to the load water-line, and at equal distances apart. In constructing it the sections should be made fair continuous curves, any irregularities STABILITY. 167 2. Angular Interval.- FiG. 136. which might be caused by embrasures, &c., being left out Fig. 135. (as shown in full lines in fig. 135, where the dotted lines show the actual section 1 of vessel), they being treated ^ separately afterwards as ai)pe)ida(jes. When there are appendages it is also necessary to have correct sheer and half-breadth draughts, in order to cal- culate their volume, &c. The body plan has now to be crossed by a number of lines, . » o 6 radiating from the middle point of the load water-plane, and at equiangular intervals, taking care that ont' passes through the edg( of the upper continuou - deck amidships. The equiangular in- terval is determined as follows : — Divide the angle which the radiat - ing line, passing through the edge of the upper deck, makes witli the load water-line, into such a number of equiangular intervals that the line passing through the edge of the uj)per deck becomes a stop-point in the integration to which these radiat- ing lines will be afterwards treated. If Simpson's first rule is used the number of intervals must be even ; if his second rule, a multiple of three must be used, and so on. The angular interval should not be more than 10° or less than 3°. It is usual to introduce an intermediate radiating line at half an interval after the edge of the deck has been passed, in order to reduce the error caused by applying Simpson's rule to so irregular a surface as the upper deck. 3. Measuring the Ordinates. — The ordinates of the immersed and emerged sides of the various inclined longitudinal water- planes are measured ofE right fore and aft for each successive angle of heel from the middle line of the ship, and entered upon a set of tables, styled preliminary tables, under their proper heading. One of these tables is necessary for each separate angle of heel. 4. Preliminao'y Tables (see p. 176). — Three operations are performed upon the ordinates entered in these tables. Firstly, they are affected by a set of Simpson's multipliers, in order 168 STABILITY. to find a function for the area of tlie immersed and emerged sides of the respective radial planes. .Secondly, the squares of the ordinates are affected by the same set of multipliers in order to find a function for the moment of the immersed and emerged sides of the respective radial planes. Thirdly, the cubes of the ordinates are affected by the same set of multi- pliers in order to find a function for the moment of ineHia of the immersed and emerged sides of the various radial planes about the middle line of ship. 5. Combination Tables (see p. 177). — The results obtained in the preliminary tables are made use of in these tables to determine — (1st) The area of the various inclined water-planes, together with their centres of gravity. (2nd) The volumes of the assumed wedges of immersion and emersion. (3rd) The position of the true water-planes at the different angles of heel. (4th) The moments of the corrected wedges of immer- sion and emersion, 6. Areas of the Inclined Watet'-jjJanes. — The area of an inclined water-i^lane is easily found for an}^ angle of heel by adding together the sums of the functions of the ordinates for the immersed and emerged sides of the respective water-planes, and multiplying the result by ^ the longitudinal interval if ISimpson"s first rule is used,* 7. Centre of Gravity of the Inclined Water-planes. — To find the distance of the centre of gravity of any inclined water-plane relatively to the middle line of the ship, proceed as follows : — Take the difference between the sums of the functions of the squares of the ordinates for the immersed and emerged sides of the water-plane ; divide the result by 2 and multiply the quotient by ^ the longitudinal distance between the ordinates, if Simpson's first rule is used. That jDroduct divided by the area of the water-plane will give the distance of its centre of gravity from the middle line. 8. Volumes of Assumed Wedf/es. — Take the sums of tlie func- tions of the squares of the ordinates for both sides of each of the radial planes contained in the wedges of immersion and emersion, and enter them in their proper column in the com- bination table, and affect them by a proper set of multipliers; add their results together, subtract the lesser sum from the greater, and divide the result by 2. The quotient multiplied by ^ the longitudinal distance between the ordinates, if Simp- son's first rule is used (tliis division by 3 is generally done in the preliminary tables) : this final product multiplied by | of the equi- angular interval in circular measure, if Simpson's first rule is again * Note.— The division by 3 is general!}' done iu ttie preliminary tables. STABILITY. 169 used, will give the difference between the volumes of the assumed wedges of immersion and emersion. If there are any appendages the necessary additions or deductions are made here. 9. Correcting Layer. — If the volume of the assumed wedge of immersion exceeds that of the wedge of emersion, it shows that the displacement up to the radial plane is too great, and that to find the true water-jalane a parallel layer must be taken away from the assumed wedges ; but if the wedge of emersion •exceeds that of immersion, a parallel layer must be added to the ^^edges. The thickness of this layer is found by dividing the dif- ference between the volumes of the two assumed wedges by the area of the proper radial water-plane, having made any addi- tions or deductions in the case of appendages. 10. JItvnents of Wedges for Statical Stalnlity. — The sums of the functions of the cubes of the ordinates for both the im- mersed and emerged w'edges are placed in the proper column in the combination table, and are alfected by the same set of multijDliers as were determined for the sums of the functions of the squares ; the products are multiplied by the various cosines of the angles of inclination made by the radial planes with the load water-line ; the products are then added together and the sum divided by 3 ; the quotient is then multiplied by ^ the angular interval, and that product by ^ the longitudinal interval, between the ordinates, if Simpson's first rule has been used (this division b}' 3 is generally done in the preliminary tables) : the final restilt will be the moment of the wedges about a line perpendicular to the radial j^lane, and passing through the middle point of the load water-plane. The corrections for the moments of the appendages must now be added or sub- tracted, as the case may be, also the correction for the layer, if any, must be done here, its moment being foimd by multi- plying its volume by the distance of the centre of gravity of its water plane from the middle point of the load water-plane. If the centre of gravity of the layer lies towards that side for which the assumed wedge is the greater, the correction must be deducted ; if it lies towards the opposite side, it must be added. This final restilt, being divided by the total volume of displace- ment, will give the length of the arm be (see fig. 131). Multiply the height of the centre of gravity above the centre of buoyancy by the sine of the angle of heel, and subtract the product from BR: the remainder will be the length of the arm of the righting couple GZ ; GZ multiplied by the displacement in tons will give the righting moment, or statical stability, of the ship for the given angle of heel. 11. Moments oftlie Wedges for DynamicalStalility. — This result is determined in a manner somewhat similar to that pursued for the statical stability, the only difference being that the 170 STABILITY. sums of the functions of the cubes are multiplied by the sines of the various angles of inclination instead of the cosines ; the sum of the products so obtained being divided and multiplied by the same numbers as were used for the statical stability, in order to find the moment of the wedges uncorrected relatively to the respective radial planes. The corrections for the appendages are then made, that for the correcting layer being subtracted in all cases. The moment for the correcting layer is found by multiplying its volume by half its thickness, that being about the vertical height of its centre of gravity from its radial plane. This final result divided by the total volume of displacement will give the length of the arm b' e, from which if BG . vers 6 be deducted, the remainder will equal the length of the arm for the dynamical stability, or the vertical height through which the centre of gravity of the ship has been lifted and the centre of buoyancy depressed. 12. Geometyncal Mode of Calculatinr/ Dynamical StaUlity. — The dynamical stability of a vessel at any given angle of heel is the sum of the moments of the statical stability taken at indefinitely small equiangular intervals up to the given angle of heel, and is therefore'equal to the area of the curve of sta- tical stability included between the origin of the curve and the angle in question. " It must be noticed that the abscissae of a curve of statical stability is given in angles, and therefore the longitudinal interval is taken in circular measure. But, as the lengths of the arms for statical stability are generallv used to construct a curve instead of the moments of stabilitv, the area, as above f oimd by the rule from such a curve, will necessarily give the length of the arm for djTiamical stability and not the moment. Example (see fig. 133).— To find the length of the arm for dvnamical stability, at an angle of 30^ inclination. Ansrles of Heel degrees 5 „ 10 „ 15 „ 20" „ 25 „ 30 „ Lengths of Statical Simpson's Products Levers GZ Multipliers •0 1 •0 •2 4 •8 •42 2 •84 •68 4 2-72 •9T 2 1-94 1^30 4 5-20 1^66 1 1-66 13-16 i of angular interval in circular measure = '0291 1316 11844 2632 Dynamical lever for 30° = '382956 STABILITY AT LIGHT DRAUGHT. 171 13. Curve of Stahility for Light Bravglit. — The lengths of the arms for this curve can readil}- be approximated from the results obtained for the curve in the load condition. Fig. 137. In fig. 137 WL is the load water-line, and nl the light water-line, for the upright position of the vessel. If the vessel is inclined through an angle Q, and w'l' is the true position of the in- clined water-plane for the load condition, then the true position of the water- plane for the light condi- tion will run parallel to it, as n-'T . To determine its perpendicular distance from w'l', divide the volume of the layer contained between the light and load water-planes by the area of the assumed inclined water- plane hli', w^hich was foimd for the inclined load condition. Let B be the centre of buoyancy for the upright load con- dition, b' for the inclined load condition, and b- for the inclined light condition, be is perpendicular to the vertical b'm, and be' is perpendicular to the vertical b-m'. Let D equal volume of light displacement. „ ^ = volume of displacement contained between the light and load water-planes. „ (? = distance of centre of gravity of assumed inclined water-plane from the vertical through A. „ GZ and G'z' = the lengths of the arms of the righting couples for the load and light condition respectively. Then EE'= ^f ^_+ (^^ BA^nJ)} 3^,, ^ ^^ ^ ^^^ D and g'z' = BE' — BG'. sin Q. Sfiirface of Flotation. — If a ship be inclined through an unlimited number of indefinitely small angles in every possible direction, a curved surface touching all the planes of flotation thus made is called a surface of flotation, and the point of its contact with any water-plane is the cesitre of gravity of that plane. Axis of Level 3Iotioy). — "Wlien the transverse section of a surface of flotation is a circle, the centre of that circle is termed the axis of level motion. This axis lies parallel to the load water-line, and is in the longitudinal middle-line vertical plane of the ship for the upright position, and is so placed as to 172 LONGITUDINAL METACENTRE. keep the same position, when the vessel is heeled over to any .angle, as when she was upright. To determine approxi'mately the height of the axis of level motion above the plane of Jiotatioji. EuLE. — Measure the angles of inclination of the several cross sections to the vertical between wind and water, and find their tangents, distinguishing those tangents respectively into posi- tive and negative, according as the side of the section inclines outward or inward (that is, having any flare or tumble-home) ; multiply the tangents by the squares of the half -breadths of the cross sections to which they belong, and the products by a set of Simpson's multipliers in their consecutive order ; take the dif- ference between the sums of the positive and negative products, and multiply the difference by ~ the longitudinal interval (if Simpson's first rule is used), and divide the product by half tlae area of the water-plane : the quotient will be the required result. Longitudinal Metacentre and Alteration of Trim. To determine the vertical height of the longitudinal meta- centre above ths centre of buoyancy. Rule. — Divide the moment of inertia of the load water-plane, relatively to a transverse axis passing through the centre of the plane of flotation, by the volume of displacement. (For example of calculation see p. 174.) The following method will generally be found in practice to be the simplest for finding the moment of inertia of the plane of flotation relatively to the transverse axis tlirough the plane of flotation : — First determine the moment of inertia of the given plane relatively to one of its ordinates as a transverse . axis (see Eule 7, p. 79) ; then from the result subtract the area of the plane multiplied by the square of the distance of its centre from the given axis. Pig. 138. Approximate formulae for centre (J. A. Normand, L = length on lwl in feet. I = breadth amidships in ft. D = displacement in cu. feet. a = area of lwl in sq. ft. height of longitudinal meta- M.I.N.A.). H = height of longitudinal me- tacentre above C. of Buoy. H •0735 ^. /D Moment to Alter the Trim of a Vessel. — In fig. 138 let WL be the original load water-line, w'l' the load-line to which it is TO FIND CENTRE OF GRAVITY BY EXPERIMENT. 173 required to trim the vessel, c the centre of flotation and the point at which the two load-lines intersect each other. The total alteration of trim = ww'+ll'. Let G be the position of the centre of gravity, b the centre of buoyancy, for the upright position, G' and b' the altered positions of the centres due to the alteration in trim, and m the longitudinal metacentre ; let p = the weight on board that has to be moved, d = the horizontal distance through which the weight has to be moved to produce the required trim, and d = the displacement of the ship in tons : then WL „„^ (ww' + ll') GM -Pxd GG 5 WL D also ww' = ^'^ CP X ^) , ^^^^^c(L^\ and ww' + ll'^ ^^^^^ (^ ^ ^> GM X D GM X D GM XD Moment to alter trim one inch = — x — ^ . 12 WL Moment to alter trim w. inches = « x — x ^~. 12 WL JVbte. — All the measurements are taken in feet. To Determine the Vertical Position of a Ship's Centre OF Gravity by Experiment. In fig. 139 let mz be the upright axis of a ship ; her centre of gravity then lies somewhere in that axis. M is the metacentre, and gm its vertical height above the centre of gravity g. If a weight p be moved transversely through a dis- tance FP'=d, it will heel the vessel over through an angle 6, and her centre of gravity will then shift in a direction gg' parallel to that m which the centre of gravity of the weight has been shifted. Let MT be parallel to gg' and tg' parallel to gm ; let P = weight shifted m tons, and D = displacement of ship in tons : then MT = gg' "^ ^ d J , ^ Y X d ■■ ; and gm = gg cotan 6 = tJiJL cotan 0. D D Note.—\i several weights are shifted the total sum of each of the moments must be taken. LONGITUDINAL METACEXTEE. Calculation of Height of Longitudinal Metacentre abote Centre of Buoyancy, and Moment to alter Trim one Inch, .2 ^ T 9 3 4 5 6 7 8 9 10 11 iH 12 12i 13 Ordinates •1 3-6 7-1 9-5 11-6 13-7 14-3 14-4 14-4 14-4 14-2 13-8 13-4 11-1 8-4 4-4 •9 Sg 2 1 2 4 2 4 2 4 2 4 2 1 2 Products for Area ^ Lens:. Interval •05 7-20 7-10 19-00 17-40 54-80 28-60 57-60 28-80 57-60 28-40 55-20 20-10 22-20 8-40 8-80 •10 421-35- 5-7 24ul-bUo :: c 1 li 2 3 4 5 6 7 8 9 10 lOi 11 lU 12" Products for Moments •0 3-60 7-10 28^50 34-80 164-40 114-40 288-00 172-80 403-20 227-20 496-80 201-00 233-10 92-40 101-20 1-20 2 o ^ Products I for Momeuts of Inertia ' ^t 2569-70 17-1 43941-87 1 2 1 H 2" 3 4 5 6 7 8 9 10 \0h 11" lU ]2 Long. Long •0 1-80 7-10 42-75 69-60 493-20 457-60 1440-00 1036-80 2822-40 1817-60 4471-20 2010-00 2447-55 1016-40 1163-80 14-40 19312-20 I nt. 17-1 30238-62 Int. 17-1 1 U 2 oi — -7 3" 4 5 6 7 8 9 10 11 lU 12 12i 13 Cu. ft. in a ton 35) 4su;J-oi> 12) 1 37-239 Dispt. per inch 11-436 Element of Inertia about Xo. 1 Ordinate Area of Load Water-plane x (104-29)- 5647080-402 Long. Int. 17-1 3) 96565074-8'/ 42 32188358-2914 ■ 2 64376716-5828 52243610-6899 Volume of Displacement in cub. feet 18270 )12133105-8929 Height of Long. Metacentre above Centre of Buoy. 664-1 Height of C. of Grav. of ship above Centre of Buoy. 2 73 Height of Long. Metacentre above C. of G. of ship. 661-37 421-35) 43941-8 7 Distance of C. of Flotation from No. 1 Ordinate 104-29 Moment to alter trim one inch = - ^^-^ x ~~= 140-34 foot tons, 20.0* 12 * Length of ship at' L. \\. Line = 205 ft. t Dis}>t. of ship in tous = 522 CALCULATION OF HEIGHT OF TRANSVERSE METACENTEE. 175 « o ^ !5 o > Ph l-H -eq P* K « « fCfig o 2 a fc o" ;?^ w >H ^ u w ^ IrH <^ P^ >^ o O p 55 pq ^ E^ O O P3 H W. !^ *— 1 y as o H < N cc ■>s> >n tc « Z.^ oi-^copc-^c^Tfxipao-^c l-» O O CO 00 ^ x^ c^ c^ ^ ' o^* — 0:£XO5^3^c;o c^rooocicic; xco»jOo ^- « t C 0 2 «=' >^x ^ -x^x I-- » t>* =5 !ox ■N II II II '^ NK-* XXX -j-r c ■X I ■- - j>. x.5.« a X N ^ O ^ ** ?^ ■* N -^ N -^ W ■^ M -^ fm i3 ^ X »>. ic Ci oi cr. ^ CO a; o g ^ U X ^ r; — f^ O -r — -rr: i -J— S> ^►^ ^ ^ to >--' i^t 'i *i to ^ ic ^ ^ N 5 ft : o X ; ^ X • X r-^ X — X X T) -?• O C C I = — ^-rX^r^^C ^"^[c ^ X — ^^ t^ w ■ — (_ (N — i- t- ID S?0 s '& jO |< s (CO -^> ^~- -^S^T. 2 'x-:^ >-<'<*'5^-^5^'*S^'«9'5^'9'S^'*^^ O C^ W 5^ -^ X C o — — — o; 2s p c^ < X -" ' -^ o C X -* s^ ^ Oi ^>. •— ' b tt^^ ) X c: C. '^ X -X l>- ^1 CO o ~ n .^^ J^ ^ o S55c >^ ''-^ CO •^ u -^Tj'3^TfS^'«a'W^»-« OXXOStt^OCOCTlCr. t^^DO pppC^OpCOXi^Xyr-O xx'^cotoipas'— c»^' : ft (Ti "-*■ CI 1.0 • I-» f X C35 ^. C . N .o o o o c - -^ -^ « CO ^ ^ ^ S3S2?!5- = sSS§S -^Nxc^-roxft'xs^s^ O X t^ X « ?1 — t^ o CO CO cr- uo c -v -^ -2 '-'■^S^-^Cl'^^'^N^N'^^^ ^ Q 1 00 -T< -x h; "=" '-' '-'5 -^ 1 ?« -s" •J5 C ^0 rf» CT. [*■**- l=K^ i>> e XXX ^ 7^ o ^ o w O'X-TasO4^r^'J0i0^-'^»nO C--.--r3<»XCOCCO^C*-'Xp i' e« « o b b X ^t^-^ « to ' ^ -.o i^ o tr- 'M !>. g: ft *x N ft X -r ic ^ o ^ c^ « 5 "5 ? 2E S3 ^yicO"<#*otcwxo;c<— 'Nco 5 a «^ 52 ^ ?i c; a» '^ • >^s i, i, O O =. ', U.S :^ ;j 176 PRELIMIXAKY TABLE FOR STABILITY. PRELIMDfAEY TaBLE FOR STABILITY AT 30"^ AXGLE OF HeEL. Ordi- "2, nates j '-n IS i a-. Squares! . rc 3^ C C — — ^ 3i j^ = '£ X :s — ^ r^ ■fT' 1^ ;:; .. A 1— * ^l^^l^r^ac'O' .occ pji^odjccl :r;2*^nS9li;^ iS^ E Q^ > ^sTiilfi c.s- s •^'>^ c >-i;' p s £^5^ SISS Chilean ..► >> 2r^ cw a = c: X ,x r; o K t^ =; oc »« o .- :« g 1 2 = rj s X 5 o 9 C^I^r^xo^c^ ■^ ?^ t ^^,,^^,,_ i^' n r. T ») ■n r^ 1^ ';.-3-f -? - — jj li X =; — 2U S j3~;ii ■-'^■r : c ci 111 . o^ •r =^ ^ £ ■= s; c £n '^ N • II c c". cr. :r. i n c 1 -^ sjaqd -nihH -« -O" »» -e « -T ■- iiiiiil c I u S :^^iT,?> I P ^ z» ^ ^ ^- -■ « O V sii^i^' r v; a ^ St' > £ 2 '' '< Cr'^ 2 ^Sk EH St aj 5 i r o o ^' £s < -J 1 1^ o ?^ c- ] n • 5r5e-power Cube of speed in knots 2959 Augmented surface 36979 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Half- girths. Feet imps. Mults. 21-0 27-2 30-8 34-6 38^8 41^5 42-6 44^0 44^0 44-0 43^3 42^1 40^3 38^1 36-0 35^0 32-0 Product-; 1 4 2 4 2 4 2 4 2 4 2 4 2 4 •2 4 1 21^0 108-8 61^6 138-4 77^6 166^0 85^2 176^0 88^0 176-0 86-6 168^4 80-6 152-4 72^0 140-0 32^0 Coefe. of prop. 20000)109420861 Indicated horse-power 5471^0430 Divide by 3 )1830^6 Half Xo. of Int. 8 ) 610-2 Mean girth 76-3 Length of ship 380 Product 28994 Coetf. of aug. 1-275 Augment, surface"! qro-o m square feet | Peopulsiox of Vessels. {Scott liu^scU.) = area of iiumersed part of midship section in square feet. = coefficient of form of water-lines (see table, p. 189). = head resistance to midship section in lbs. = area of wet surface in square feet. = coefficient of skin resistance (see table, p. 189). = total of skin resistance in lbs. = velocity of ship in knots per hour. = velocity of shp in knots per hour. : horse-power required to propel vessel at v speed. H = 2^852346cAv2 s = ;^wv2 p = '.^l^^^^^iXlJ^lKlill). Note.—Thes^ fonnulee must not be trusted implicitly for high speeds. A: C: H: W: k-. S- \- V- PEOPULSIOX OF VESSELS. 189 Table of Coefficients of Kesistaxce for yaeious KixDS op Skix. Kiud of Skin Coeff. "FTind of Skiu Coeff. Kind of Skin Coeff. Clean copper sheets Smooth paint •007 •010 Common iron, skin Smooth-sawn plank •ou •016 ^Moderately foul Barnacled •019 •055 Table of Coefficients of Form for various Kinds of Water-lines. Kind of Water-line Wedge z 41°; Wedse z35°; Wedge Z18°, Convex arcs ! Z35= I Kind of Water-line o Convex arcs Z 25°: -31 Convex arcs z 15° -20 Wave form 5 to 1-15 Wave form 6 to 1 -077 Kind of "Water-line Wave form 7 to 1 -050 Wave form 8 to 1-043 Waveform 9 10 1/034: Wave form 10 to l'-028 Table of Eesistance in lbs. to ONE Square Foot of 1 Flat-fronted Vessel, and Horse- power required to J Propel it at VARIOUS Speeds. Ft. per S:^ H-e- Miles an Resistance | ^^^^^' Knots an Resistance Horse- Sec. 1 auce P«"^^^^ Horn- 1 power Horn- power 1 0-00182 2-15111I 0-00574 1 2-8.5235 0-00876 <■> 4 0-01455 2 8-60444 0-04589 2 11-40938 0-07007 3 9 0-04909 3 19^36000: 0-1.5488 3 25-67111 0-23649 4 IG 0-11G36 4 34-41778: 0^36712 4 45-63754 0-56056 o 25 0-22727 S) 53^77778i 0^71704 5 71-30865 1-09484 G 36 0-39273 6 77-44000; 1^23904 6 102-684i5 ]-S9188 / 49 0-62364 7 105-40444' 1-96755 7 139-76495 3-00424 8 64 0-93091 8 137^67111| 2-93698 8 182-55014 4-48446 9 81 ,1-32545 9 174-24000, 4-18176 9 231-04003 6-38511 10 100 1-81818 10 21.5-11111; 5-73629 10 285-23460 8-75872 11 121 2-42000 11 260-28444 7-63163 11 1345-13387 11-65786 12 144 3-14182 12 309-76000 9-91231 12 1410-73780, 15-13506 13 169 3-99450 13 363-53778'] 2-60263 13 1482-046471 19-24290 14 196 4-98909 14 ! 421-61778 15-74038 14 1559-05980 24-03392 15 TZ6 jt)-13636| 16 ! 484-00000 19-35998 15 '641-7178o'29-5606S 190 COEFFICIENTS QF PERFORMANCE OF STEAM TESSELS. Speed Formulae as generally used for Steam Vessels. T^velocity in knots per hour. H^indicated horse-power for V speed. D^displacement in tons. /;=sectional coefficient of performance. V zranv other velocity- in knots per hour. H. vindicated horse-power for V, speed. X=:area of midship section in square feet. x— displacement coefficient of performance. r=^ uxt H=i— ^-i H= _V'XI>J H H H H. Note.— These formulae may be taken as sufficiently accurate up to 12 knots speed, when from 12 knots and upwards \* and even T* may be substituted at high speeds for v^. In the followin? tables let — vzrvelocity in knots per hour. S— slip in knotd per hour. H=indicated horse-power. Dindisplacement in tons. S=area of midship section in square feet. L=:len?th of vessel in feet. B=breadth of vessel in feet. Table of Coefficients OF Performance, etc., of some i OF Her Majesty's S CREW Vessels. v->xu- L H 1 H ■ T*XX Name of Vessel L 400 i B 6-73 ^ D H 907116867 V 15-433 S Neg. X I>§ H H Agincourt 1185 5-79 15-79 634-3 2328 11 400 ; 673 1198 91.i.2 .5971 13-879 Neg. 4-99 13-65 5361 195-9 •1 • 400 i 673 1198 9152 3001 10-998 Neg. 2-51 6-86 530-8 194-0 Minotaur 400 ' 6-69 1158 8800,6336 14-779 Neg. 5-47 14-87 590-0 217-2 »i • 400 6-69 ll.'>8 8800 3451 12-387 Neg. 2-98 8-10 ' 6377 2ai-7 .1 • 400 6G9 1313 10185 3497, 11-842 Neg. 2-66 7-44 623-6 223-1 Achilles . 380 6-52 1120 7895 503.5 14-358 Neg. 4-.50 12-7 658-5 2331 330 6-52 1293 i 9362 4818, 13-349 Neg.: 3-73 10-85 638-4 219-3 380 1 6-.i2 law 1 9487 3205 12'a49 Neg. 2-45 715 7137 244 6 Warrior . 3^0 380 6 -52 6-55 1284 ; 92.58 20? 1 11-132 14-3.56 1 Neg. 1 1705 2-09 6-08 660-8 226-9 231-5 1219 ' 88.52'. 5469' 4-49 12-78 659-4 3*) 6-55 1260 1 :.214:.5092 13-936 ; 1-636 ■ 4-01 ir.59 ' 669-7 233-6 11 380 6'ob 1219 8<52 2%7 12-174 1-000 1 2-35 ! 6-70 1 7671 269-3 ;; ; 380 380 6o5 6-.^ 1219 S?52 lt«SS. 11040 10-415 -210 i 1-63 1 4-65 1 824-9 2-21 6-33 ! 510-6 289-6 178-4 12.i5 9i^0'2777 1-371 Euphrates 360 7a3 814 5~9820^: 11-523 -331 1 2-56 6-39 1 597-5 239-6 1. • 3fi0 733 841 5109 UH2 10-600 -586 ; 214 5-39 555-8 220-8 Serapis . 3)50 7-33 778 .5600 :»45 14-059 2-645 ' 5-07 12-51 548-0 2221 „ ... :5H0 ■•33 S'H 58163698 13-378 1 Neg. 1 4-60 11-43 520-5 209-4 ..... 360 r:?3 778 .560IJ2613 12-554 1 1-616 3-36 8-29 1 .589-1 238-8 Inconstant . 337 4 671 900 .53287361 16-513 1 1-188 8-18 24-13 .5.50-46 186-6 «. • 337 4 6n 900 , 5328;}.53I 13-701 -498 3-92 ! 11-57 i 6.55-61 222-2 Sultai . 32a 551 1320 8714 8629 14134 2-864 6-54 i 20-38 1 431-9 138-6 Captiin » 320 6-01 1176 7672'5:'90 14-239 1-665 5-09 15-40 i .566-8 187-5 320 6-01 1174 76.55 290B: 11-697 -693 2-47 7-49 1 646-0 213-7 Belierophon . 300 a-36 1(65 6372 .5966 14-227 Neg. 5-60 17-36 .514-1 165-9 «< *J0 536 1018 5700: 470? 13-646 Neg. 4-63 14-75 549-2 172-2 11 300 536 1065 6372 3119 12-103 •172 2-93 9-06 605-3 195-3 ... . . 300 5-36 1134 685 r 2984 11-780 -13:} 2-63 ft-27 621-3 197-7 Orontes . .300 1 672 644 3400 1323 10-89C' 1-631 2-05 5-80 628-6 2-20-7 1. ... 30O 1 672 781 4249 10*1 9-755 1-779 1-38 4-12 670-6 2-253 ..... 300 1 672 796 4321 i 775 8-719 1-519 -97 2-92 6810 226-* Raleigh . . . 298 614 851 46176518 15-51M 3-945 7-24 22-11 515-0 168 5 298 614 851 464713414 13-4.57 1-940 401 12-26 607-5 198-8 Devajtatlon * ■iSi) 4-58 1472 9190 6652 13-840 -994 4-52 1516 586-6 174-8 .. ,^b 4-.58 1472 91903399 11-909 •200 2-31 7-75 ; 7:n-4 218-0 Adventure . 12S2 10 777 467 2432:1227 11-447 1-796 2-62 6-78 1 571-0 221 1 >i 1282 10 777 474 2470 1053 10-617 1-945 9-'W 5-76 5386 207-6 11 1292 10 777 467 2432 637 9-256 -948 1-36 3-52 ' .581-7 ' 225-2 282 10 777 436 2248 517 8-507 1-.5.34 119 301 519-5 , 204-5 Audacious* . 280 518 997 .5.594 4*35 13-401 -295 4-85 15-34 ! 496-3 1 1.56-9 •1 280 518 1087 6I70 1021 12-829 -401 3-70 11-95 5708 j 176-6 . 280 51* 997 5.594 2946 10-811 3017 2-96 9-35 427-6 1 135-1 1-280 5-18 997 .5.594 17<13 10091 .Vee. 1-71 ; 5-40 601-6 ! 19(>-2 Active . ;270 6-43 632 3057 4015 14-9.56 1-931 6-35 1906 527-5 1 175-- ., 270 6-43 628 3033 H.^W 14-877 2-650 5.54 16-61 694-5 198-3 ^^ l270 6-43 632 3067 2046 12-295 -773 3-24 9-71 .573-9 191-3 270 1 6-43 628 3033 169:} 11 -76.1 1-241 2-68 8-03 608-0 202-8 flepul^e 1232 1 427 1170 6010 :»47 12-284 ' 3-917 2-86 1012 648 ' IN?,-! » Twin screw vessels. COEFFICIENTS OF PERFORMANCE OF STEAM VESSELS. 191 Table 01 ^ Coefficients of Performance, etc., OF SOME 1 OF Her Majesty's f^ CREW Vessels (concluded). L H H V^X X V-- X 1 ) • Name of Vessel L 77 X D H V s B ♦ X iyi H H Repulse . 252 4-27 1170 6010 1871 10-687 2-308 1-60 5-66 763-2 215-6 Olatton * 245 4-54 918 4900 2868 12-109 Ncg. 3-12 9-94 568-2 178-6 .1 2450 4-54 918 4900 1434 9-872 Neg. 1-66 4-97 615-9 193-5 Hotspur * 235 4-70 839 3980 3497 12-651 2-293 4-17 13-92 485-8 145-4 ,, 2,3o0 4-70 839 3980119^54 10-601 1-666 2-34 7-82 508-9 1.523 ^, 2;}5 470 839 3980; 2650 10-070 1 5-601 3-16 10-65 323-2 ^^6-8 Victor Emmanuel 230 416 788 3578 '2123 12-009 1 2-446 2-69 9-07 ()43-0 190-9 ,^ 230 4-16 794 361412219 11-713 1 1-365 2-79 9-42 675-0 170-6 230 4-16 1065 6106:2424 10-874 i 3-774 2-28 8-17 564-9 1573 2:30 4-16 1065 _ 51061274' 9-072 2-/37 1-20 4-30 &>i-2 K.'J-H Abyssinia* . 225 5-36 555' 7 2816 1 949 9-595 1-417 1-71 4-76 517-5 LV6-7 ^^ 226 5-36 5567 2816! 662 7-327 1-356 1-01 2-82 389-2 139-7 Cyclops * 225 5-00 639-0 3100 1660 11 027 Neg. 2-60 7-81 616-0 171-7 225 5-00 639-0 31(tO| 746 8-720 Neg. 1-17 3-51 567-6 lK(i-(H Magdala * 225 500 589 2997:1436 10-666 Neg. 2-44 6-91 497-6 17.V6 225 5-00 589 29971 816 8-848 Neg. 139 3-93 499-9 176-4 Amethyst 220 5-94 476 1978 2144 13-244 2-211 4-50 13-61 5167 220 5-94 476 1978 1989 12-920 2-167 4-18 12-62 5161 17'i- _^ 220 5-94 476 1978 1034 11-083 1-127 2-17 6-56 626-5 207-4 Briton . 220 6-11 436 1860 2149! 13-126 1-916 4-93 14-21 45&-9 16i!-2 220 6-11 436 186O-J019i 127(;6 ' 1-863 4-62 13-32 450-4 Vib--J ,, 220 6-11 436 1860 933 1 11-026 1 -892 2- 14 6-17 626-3 217-1' 220 6-11 413 1768 11 00 lO-OOO ] l■^93 2-67 7-56 376-4 131-4 ^^ 220 6-11 413 1768 665 7-920 | 1-685 1-37 3-88 .363-3 ll'fl' Modeste 220 5-95 479 1993 2177 12-791 i 2051 4-54 13-76 460-4 ];)2-2 ,^ ■220 5-95 479 1993 1108 10-066 1 1-004 2-31 7-00 624-6 173-4 Algiers 2IH7 3-65 819 3562 2518 12-191 2-644 3-08 10-80 689-2 167-8 ^^ 218 7 3-65 814 3650 1362 10-487 . r799 1-67 5-85 647-6 166-6 ^^ 21S7 3-65 la53 4730 1117 9-000 1 -545 1-06 3-^-6 687-2 IKI^ Euryalus 212 4-23 704 3126 1262 10-038 1-868 1-70 6-90 664-2 171-3 „ 212 4-23 750 3356 1162 9-47 2-036 1-66 5-18 6480 163-s Sirius 212 5-89 377 1654 2302 13-263 1-800 6-11 17-16 382-1 1360 ,, 212 5-89 425 1746 1118 11-283 -723 2-63 7-71 646-3 186-4 „ 212 5-89 377 1564 1070 10-897 -857 2-84 7-98 465-8 162-2 Albion 204 3-39 688 2912;1836 10-986 1-270 2-67 9-00 497-3 147-4 „ 204 3-39 688 2912 1017 8-S><8 -704 1-48 4-99 475-1 140-8 Lion 192 3-37 635 2540 1771110-911 1-133 2-79 9-61 466-7 136-5 „ 1^2 3-37 870 3580J032 9-529 -877 1-19 4-41 729-6 196-2 ,, 193 3-37 768 3120 925 8-334 1 1-342 1-20 4-33 480-5 133-7 Dromedary 189 7-11 247 905 j 430 9-084 2-798 1-74 4-60 430-5 1031 189 7-11 247 905 223 7-520 1-876 0-90 2 -.38 471-6 17f-6 Dryad . 187 5-19 434 1546 1464 11-963 -804 3-37 10-95 507-6 1.56-4 187 5-19 434 1546: 839 10-117 -293 1-93 6-28 635-6 165-0 Myrmidon \^bO 6-53 236 776 782 10-338 4-176 3-32 9-27 3333 119-2 ,, 165 6-53 265 886: 671 9-838 2-850 2-63 7-28 376-9 130-7 „ 185 6-53 236 776, 404 8-763 2-627 1-71 4-79 392-8 140-4 „ 185 6-53 253 836' 219 6-641 1-981 •86 2-46 339-0 lis--* Lapwing * 170 5 '86 228 7691 882 10-847 2-767 3-87 10-51 329-8 121-4 „ 170 5-86 229 774 605 9-H25 1 2-189 2-64 7-18 337-7 124 -2 „ 170 5-86 228 769, 839 8-718 1-442 1-49 4-04 446-4 1640 „ 170 5-86 229 774! 276 7-634 1 1-439 1-20 3-18 369-1 135-7 Egeria . 160 5-11 320 9491011 11-302 '3-406 3-16 10-47 45G-8 l?7-8 Sappho . 160 5-11 280 800' 936 11-191 1 3-lh4 3-34 10-86 419-6 129-1 Beacon *. 155 620 182 690, 677 9-375 1 2-948 3-17 8-10 260-0 100-6 Flirt* . 155 6-20 164 521' 584 10-091 1 5-571 3-66 9-03 288-3 1I3-S 11 155 6-20 159 501 421 9-037 4-463 2-66 6 -67 279-0 110-7 Ariel 125 5-43 160 352' .540 10-802 1-340 3-38 10-83 373-4 116-4 „ 125 5-43 160 362' 278 9-231 -646 1-74 h-^.H 462-4 1410 Coquette 125 5-56 178 408, 406 9-656 1 Neg. 2-28 7-38 394-9 122-0 „ 125 5-56 179 411i 193 7-9.W Neg. rr8 3-48 468-4 144-7 „ 125 5-56 178 405, 168 7-2O6 Neg. -94 3-06 397-4 121' -2 Mosquito 125 6-56 184 4241 501 10-397 1-973 2-72 8-88 412-4 126-6 „ 125 5-56 178 4081 364 9-6.'« 1-547 2-04 661 437-0 135-2 125 5-56 184 424 1 226 8-571 •f.'87 1 -23 4-00 612-8 157-3 Elizabeth 115 5-23 163 365 244 8-916 4-982 1-50 4-78 473-7 148-4 Ant -* a5o 3-25 146 254 213 8-461 1-936 1-46 5-31 416-6 114-1 Pickle* . a5 3-25 146 264 2&8 8-693 2-720 1-H4 6-69 357-5 flS-2 Snake * . 85 3-25 141 244 225 8-646 2-499 1-60 6-77 390-7 108-2 Scoursre * 85 3-25 141 244 263 8-560 2-842 1-79 648 .349-6 96-8 Plucky * 80 3-18 125 198! 224] 8-667 3-463 1 1-79 660 349-4 94-9 Staunch * 75 300 116-8 164 134 7-664 1-864 ] 1-16 4-48 , S87-3 100-1 * T-wiu screw vessels. 192 EFFECTIVE AND INDICATED HOESE-POWEE. Eatio of Effective to Indicated Horse-power. {Froude.) Indicated Thrust. I = indicated thrust. M = mean piston-pressure. T = total piston-travel per revolution. P = pitch of propeller. N = number of revolutions. IHP = indicated horse-power. ^_ MxT^ . 33000 XIHP P P X N * Indicated thrust is resolved into the following six elements : — Xo. 1. The ship's nett resistance, or useful thrust. No. 2. Augment of resistance due to negative pressure created about the ship's stern by the action of tlie screw. This is nearly proportional to the useful thrust. No. 3. Water friction of screw. This is also nearly propor- tional to the useful thrust. No. 4. Constant friction, or friction of engine without external load. This may also be taken as nearly proportional to the useful thrust. No. 5. Friction due to external load. This may be taken as constant at all speeds. No. 6. Air-pump and feed-pump resistance. This may be taken as nearly proportional to the square of the number of revolutions. The above six elements are force factors, and when multiplied the speed of ship in feet per minute , , ^ , . . by q^OOO — ' ' C'^^stitute the ships horse-power as fundamentally due to her progress. Let EHP = effective horse-power — that is, the power due to the nett resistance of the ship. SHP = ship's horse-power. IHP = indicated horse-power. Tlien the ship's horse-power due to the several elements is as follows : — iShi]j"s horse-power due to No. 1 = ehp. No. 2^-4 EHP. No. 3 = -l EHP. No. 4 = '] 43 SHP. No. 5 = -143 SHP. No, 6 = -075 SHP. Or in combination shp == 1 -a ehp + •361 shp. So that -639 shp = 1-5 ehp; EFFECTIVE AND INDICATED HORSE-POWER. 193 or, SHP = ^^'^ EHP = 2-347 ehp. To this must be added — Slip = "l shp, making ihp = 1-1 shp. Thus IHP = 2-582 ehp = -^^ ehp : 38-7 or, EHP= -387 IHP. To convert the formula from one adapted to high speed only to one adapted to all speeds it is necessary to keep the term involving constant friction separate from the rest, for it represents simply the effect of a constant resistance operating with the existing speed of the engine. In shaping the formttla the coefficient 2' 7, derived from rather broad experience, will be adhered to, instead of the co- efficient 2-582, as the latter is built up from somewhat hypothe- tical data, assuming, however, that the constant friction is equal throughout to the one-seventh of the maximum load. Of the 2-7 EHP which make up the ihp at the maximum speed y, one-seventh part, or -385, is the part due to constant friction, leaving 2-315 as due to the other sources of expenditure of power. And to express the IHP due to constant friction at any other speed v, the coefficient must be altered in the direct V ratio of the speed, so that the term becomes — x -385 x ehp at designed maximum speed. Thus the formula for IHP at any speed V is as follows : — IHP = 2-315 EHP + -385 -' X (EHP due to v) ; or, if the useful is finally severed from the collateral expenditure of power, it stands thus : — IHP = EHP + 1-315 EHP + -385 - X (EHP due to v). V To Determine the Initial and Constant Friction of A- Marine Engine. (Fronde.) Construct a thrust curve (see fig. 146) by setting up ordinates j/, y\ if', y^, &c., which represent to scale indicated thrusts taken at various speeds. The ordinates being set off at distances along the base line, commencing from the origin, so as to represent to scale the various speeds at which the thrust was taken, a curve bent through the ends of the ordinates will form part of a thrust curve. Let^^ be the lowest point found for the curve ; at the point p draw the tangent pj)' ; draw the vertical at h so as to cut the space oy into segments, making oy = l-87o^; draw a line 194 SPEED TRIALS. parallel to the base through the point c, where Fig. 146. A=curTe of indicated horse-power. B = cvirve of indicated thrust. c= curve of slip. D= constant friction. ^aHer ofj^peed'.inKnalrS^ the vertical h cuts the tangent jjj?' : the vertical height D be- tween the parallel line and the base will repre- sent the constant friction of the engine, and it will also be the height of the vertex of the thrust curve at the origin of the speed scale, which can thus be completed from the point j:;. Mte.—The heights of the ordinates above the line of constant friction are proportional to th(r ship's true resistance. SPEED TRIALS. Measured Mile. Tfl deterynine the true mean speed of a vessel when the ?'nns are taken on the measured mile, Jialf the yiumher of runs heing taken nith the tide and half against the tide. Rule. — Find the means of consecutive speeds continually found until only one remains. ExamjiU. Runs j Knots I 1st Means |2nd Means .3rd Means i 4th Means : Mean of Means 1st 2nd 3rd 4th 5th 6th 1 } 15-4 10-1 14-3 11-0 13-2 6i7o-8__ 12-633 Ordinary mean speed. 12-75| 12-20l 12-65.- 12'10 12-50* 12-475 12-425 12-375 12-300 12-45 . 12-40 I 12-3375^ 12-425 12-36875 4 49-575 12-396875 True mean speed. 12-39375 Ordinary mean of second means. Note. — The ordinary mean of second means is generally taken as sufficiently accurate. Speed op the Current. To find the .speeds of the cnr?'ent in the line of the shijf's covrse during her speed trials. Rule. — Find tlie differences between the real speed of the 8bip and her observed speeds on the mile during the several runs. SPEED TRIALS. 195 ExamjjJe. „,,^, \ Observed ^^= ! Speed i Real Speed Differences 1st 2nd 3rd 4th 5th 6th 15-4 10-1 14-3 11-0 13-2 11-8 12-397 12-397 12-397 12-397 12-397 12-367 3-003 2-297 1-903 1-397 •803 •597 Knots with the ship „ against ,, ,, with „ „ against „ „ with „ „ against ,, Sea Trials. To determine tlie true mean speed of a vessel 7vhen the distance run is great. Rule 1st. — Calculate the apparent speed of each ran as nsual, by dividing the distance by the time, and group them in sets of three ; for example, 1, 2, 3 ; 2, 3, 4 ; 3, 4, 5 ; &c. 2xD. — Each set of three is to be treated as follows : — Find the two intervals of time between the middle instants of the first and second, and of the second and third runs of the set ; reduce those intervals to the corresponding angular intervals by the following proportion : — As 12*^ 24" (the duration of a tide) : is to a given interval of time : : so is 360^ .* to the corresponding angular interval. 3rd. — Multiply the first apparent speed by the co-secant of the Jirst axignlai interval, the second apparent speed by the sum of the co-tangents of the two angular intervals, the tMrd apparent speed b}- the co-secant of the second angular interval. 4th. — Add together the products and divide their sum by the sum of the before-mentioned multipliers : the quotient will be a speed from which tidal effects have been eliminated. oTH. — Add together the velocities deduced from the sets of three runs, and divide by their number for a final mean. jVote. — When an interval elapses of more than a quarter of a tide, or 3** 6™, between the middle instants of the two runs of a set, certain multipliers and products must be stihtractcd. The following example will determine whether these certain multipliers are to be taken as positive or negative. Exainjjle. Angles. / Between 0° \ and 90° r Between \ and {Between and }r Between \ and 02 Time. Between C and S^^ Between 3** and e'' Between and Between and S^^ 6 3>' 6™ "\ 9h igm 9h |8« 12'' 24™ } } Co-secants. . Positive Co-tangents Positive. 90° \ 180° / 180° 1 270° / 270° 1 360° / Positive Negative. Negative Positive. •Negative Negative. 193 SAILING. SAILING. Centre of Lateeal Resistance. Tlie centre of lateral resistance is the centre of application of resistance of the water; and as this varies in position with the speed of the ship, ice, it is not determinate, but a point is generally taken at the centre of tl e immersed lonaitndinal vertical middle plane of the vessel as sulficiently accurate. Centee of Effort. The point in the longitudinal vertical middle plane of a vessel which is traversed by the resultant of the pressure of the wind on the sails is termed the centre of effort ; its position varies according to the Cjuantity of sail spread, kc, but its position is determined approximately for purposes connected with design- ing the sails, all plain sail only being taken — that is, the sails that are more commonly used, and which can be carried with safetj^ in a fresh breeze (see table, p. 200). They are as follows : — In square-rigged vessels : the fore and main courses, fore, main, and mizen topsails, fore, main, and mizen topgallant sails, driver, jib, and sometimes the fore topmast staysail. In fore and aft rigged vessels : the main sail, fore sail, and sometimes the second or third jib. In calculating the position of the centre of effort by the following rules the sails are taken braced right fore and aft. To find tlie perpendicnlar Jieujld of the centre of effort above the centre of lateral resUtance. Rule. — Multiply the area of each sail by the height of its centre of gravity above the ceni re of lateral resistance: take the sum of those products (or moments) and divide it by the total area of sail : the quotient will be the required result. To find the lateral position of the centre of effoH relativelij to the centre of lateral rcaixtaiirc. Rule. — Multiply the area of each sail whose centre lies to one side of a vertical axis passing through the centre of lateral resistance by the perpendicular distance of its centre from that axis, and add the products (or moments) together. Treat the other sails whose centres lie to the other side of the axis of moments in the same way as before, and add their products together. The difference between the two sums divided by the total area of sail, -^^ll give the perpendicular distance of the centre of effort from the given axis. Xotc. — The centre of effort will lie to that side which has the greatest moment of sail. The following table shows the method in which the centre of effort is c-alculated. SAILIXa. 197 Table showing Method of Calculating the Position OF the Centre of Effort relatively to the Centre of Lateral Eesistance. Distances -3 c; Name of Sail Areas of Centre of Sails Moments Vertical MomentE Before 'Abaft Before Abaft t c Jib . 2040 138 281520 _ 87-8 17S092 Fore course 4050 78 — olo900 — 56-0 226S(Hi „ topsail 4330 78 — 337740 — 109-5 474135 „ topgallant sail 1500 78 117000 158-8 238200 Main course 5488 — 12-5 — 68600 58-3 319950 „ topsail 5440 — 14-0 — 76160 1173 636112 ,, topgallant sail 1881 15-5 29155-5 172-0 3-23532 Driver 2831-5 — jlOO-o — 284565-7 62-5 1769t'.8-7 Mizen topsail . 2645 — 78-0 — 200310 99-5 263177-5 „ topgallant sail 902 — 79-5 71709 136-0 122672 2961639-0 31107-5 1052160 736500-2 Hght. of Centre of Effort above) _ -moment 29CAC,n9-C,__ Centre of Lateral Resistance i^ area 31107-5 ^ Dist. of Centre of Effort before { __ mom ants 1052160-730500 Centreof Lateral Resistance ^ area oll07'5 ~-=io-i Ardency. Ardency is the tendency a ship has to fly np to the wind, thns showing that the position of her centre of effort is abaft the centre of lateral resistance. Slackness. Slackness is the tendency a ship has to fall off from the wind, thus showing that the position of her centre of effort is before the centre of lateral resistance. 198 SAILING. Kelative Position of Centre of Effort and Centre of Lateral Resistance. D = distance of centre of effort before centre of lateral re- sistance. D| = distance of centre of efEort above centre of lateral re- sistance. L = length of load water-line. A = area of load water-line. «^ = distance of centre of buoyancy of ship below load water- line. di = distance of centre of lateral resistance abaft the middle of the load water-line. <^2 = distance of centre of buoyancy before the middle of the load water-line. D = --4t^-— ,' for square-riffged vessels. 10(di + do) ^ = ,"777^ r- for cutter and fore and aft ringed vessels. 10(«i 4- do) T. 4a Note. — The centre of effort of the sails, to produce the best effect, must be higher or lower according as the ship is more or less full at the load water-line compared with the fulness of the body at the extremities below the water. Ships that are full at the load water-line and clean below at the extremities require the higher masts. Real and Apparent IMotion of the Wind. By the real motion of the wind is meant its motion relatively to the earth, and by its apparent motion its motion relatively to the ship when she is sailing. The apparent motion being the resultant of the real motion of the wind and of a motion equal and directly opposite to that of the ship. Fio. 147. In tig. 147 let AB represent in magnitude and direction the /__ a ^^^^r\ ^^^^ ^^^"^^ ^on of the wind, and v^.^__ \\~~I^>r~ ^^ ^'^^ direction and velocity of the motion of the siiip; throucrh b draw bd parallel •^^ ^^B andeciual to AC ; join da: then DA will represent in magnitude and direction the apparent motion of the wind. SAILING. 199 In algebraical symbols let — flw= angle adb made by the point from which the apparent wind blows with the course of the ship. K = supplement of abd, the corresponding angle for the real wind. r = — = ratio of velocity of apparent wind to that of the DB ship. r, = — = ratio of velocity of real wind to that of the ship. DB r={ V(?',- — 1 + cos^ a) + cos a] . When a is obtuse, r= { 'Jir^— 1 + cos^ a) - cos a}. r= -v/(l + r,- + 2r, .cos k). When K is obtuse, ?•= ^/\\-\-t^-—'1i\ cos K). r, = V(\ + ?'2 — 2r . cos a. When a is obtuse, r= a/(1 + r2 + 2r . cos a). Sin K = ^ — sm a. Sm a — — ■' sm K. Effective Impulse op Wind. D = direct impulse of wind on sails = area x pressure in lbs. E = effective impulse of wind on sails in lbs. C = component of effective impulse which produces leeway and tends to heel the ship over. Cj = component of effective . impulse which moves the ship ahead. Q = angle made by direction of apparent motion of wind with the plane of the sails (see fig. 148). a = angle made by the plane of the sails with the ship's course (see fig. 148). ' E = Dsin2 0. c = Ecosa. Ci==Esina. In fig. 148 let PC represent in magnitude Fig. 148. and direction the pressure of the apparent wind on the sail AB ; through P draw PR parallel to ab ; through c draw CR per- pendicular to PR and cutting PR in r: then RC is the effective pressure of the wind on the sail ab, and RX perpendicular to KM is the component of RC which pro- duces heel and leeway, while NC is the component of RC which propels the ship along. 200 railing; Table of Direct Impulse of Winds ix Lbs. per Square Foot, axd Sails commonly set by the Wind, Velocity iii Kuots per Hoitr 1 2 3 4 5 6 7 8 9 10 11 12 13 U 15 16 17 18 19 20 22 24 26 28 30 32 34 36 38 40 45 50 60 70 80 90 100 Impulse Name of Wind in lbs. I •0067 •027 •060 •107 •167 •240 •327 •427 •540 •667 •807 •960 1-13 1-31 1-50 1-71 1-93 2-16 2^41 2^67 3-23 3-84 4-51 5-23 6-00 6-83 7-71 8-64 9-63 10-7 13-5 16-7 24-0 32-7 42-7 54-0 66-7 Licrht air Light wind Light breeze Moderate breeze Fresh breeze Sails commonly set by the Wind Strong breeze Moderate gale Fresh c-ale Strong gale ^ Heavy gale Storm Hurricane Courses,- topsails, topgal- lant sails, royals, spanker, jib, flying jib, and all light sails. Royals and flying jib taken in in a sea way to two reefs in the topsails. Single-reefed topsails and tojagallant sails in much sea, two reefs in the top- sails to taking in topgal- lant sails. Double-reefed topsails to treble-reeled topsails, reefed spanker and jib. Close-reefed topsails, reefed courses to taking in span- ker, jib, fore and mizen topsails. Reefed courses, close-reefed main topsail, fore sta}'- sail, mizen topsail to tak- ing in the main sail. Close-reefed main topsail to storm staysails, or close- reefed main topsail only. SAILIXa. 201 Impulse of Wind. V = velocity of wind in knots per hour. D = direct impulse in lbs. on one square foot. D= "^'^ =v2-006667. 150 Speed of Similar Vessels under Sail. V = velocity of ship. X=:area of midship section. c and c, = constants depending upon form below water. D = displacement of ship. A = area of sails. V = A _ /a C-D3 "" \/ J|X A = cDn'- = (;iV*x. c = Cy = D?V- XV- Table of the Eatio of a Ships Speed under Sail TO Speed of Eeal Wind. Ratio oi Area of Sails to Aug- ; meiited Surface Relation between Course and Wind Probai'leJiatio of Speed of Ship to :Sp. of Real Wind H J Course 5 points near wind (^ Wind 2 points abaft beam r Course 6 points near wind s Wind abeam . t , Wind astern . J Course 5 points near wind (^ : Wind 2 points abaft beam / Course about Q\ points near wind \ Wind on quarter . . . . Table of the Ratio of the Probable Speed of Vessels under Steam and Canva s to those under Steam. Speed under can- vas •¥ speed under steam Probable speed under steam and canvas h- speed Speed under can- vas -f- speed under Probable speed under steam and canvas ~- speed under steam under steam •4 1-02 1-3 1-47 •o 1-Oi 1-4 l"o5 •6 1-07 1-5 1-64 •7 I-IO 1-6 1-72 •8 l-lo 1-7 1-81 •9 1-20 1-8 1-90 1-0 1-26 1-9 1-99 11 1-33 2-0 208 1-2 1-40 — — 202 SAILING. Heeling Moment of Sails. E =etfective impulse of wind on sails in lbs. (see p. 199). D = displacemient of vessel in lbs. c = height of centre of effort above centre of lateral resistance. G —height metacentre above centre of gravity. L = length of arm of righting couple at a given angle of heel. M = heeling moment of sails. a = angle made by plane of sails with course of ship (see fig. 148). e = angle of heel of vessel. M = c . E . cos a . cos 0. The steady angle of heel of a vessel due to M will be that at which M = D . G . sin 6 (for small angles of heel), M = L . D (for any angle of heel). In the two last formulae the reduction in the effective heeling power of the wind due to the sails being inclined from the upright position has been neglected, but if necessary the dimi- nution of the effective pressure of the wind may be taken to vary as the sine squared of the angle of incidence of the wind with the plane of the ship's sails, or as the cosine squared of the angle of heel. ^ Note. — In a general sense the moment of sail is usually understood to be the product of the area of all plain sail into the height of the centre of effort above the centre of lateral resistance, as the pressure of wind is generally taken as one pound on the square foot ; and the product of the weight of the ship in lbs. into the height of the metacentre above the centre of gravity, divided by the moment of sail, is taken as a measure of her efficiency to resist inclination under canvas. Area of Sail. To determine accurately the quantity of sail siiitable for any vessel to carry, make the moment of sail equal to the moment of stability at a definite angle of heel ; but the following rule may generally be taken as sufficiently approximate : — A = quantity of sail suitable to a given vessel. r> = displacement of vessel in lbs. M = height of metacentre above centre of gravity. H = height of centre of effort above centre of lateral resistance. = angle of heel in circular measure suitable to given vessel taken from the following table. D X M X SAILING. 203 Table of Angle of Steady Heel for Different Classes of Vessels. Class of Vessel Angle of Heel Circular ileasure Frigates and large merchant ships Corvettes Schooners and cutters . Yachts 4° 5^ 6° to 9° •070 •087 •105 •105 to^l57 Table of the Area and Moment of Sails of some of Her Majesty's Screw Vessels. Achilles . Bellerophon Favourite Hercules-. Inconstant Iron Duke Monarch Minotaur Penelope Prince Consort Sultan Swift sure Valiant . Vixen "Warrior . 30133 |23792 |l6206 '28882 26034 25054 ,27700 132377 117168 22459 128258 '25095 J21426 I 7860 128809 22-6 |3 !l9-343 '20-62'5 21-62 3 J27-57-4 ,23-92'4 ,22-52 3 '24-233 22-32 3' 18-853 20-42 3- 21-95 3- 17-493- 22-98 6- 23-07 3- D -11 95 •15; 85 •01105 ■26 118 •61.147 25 128 35,129 10' 74 93i 84 36| 43 07112 82 116 18! 59 391 74 16' 63 99!23^33 26 64 26-1 4 '21-2 42 18-9 15-1 17-3 17-3 -50 30-0 -35|26-6 •2851-7 -8419-8 •0 19-3 •10 37-9 •6729-9 •2235-4 G 24 21 24 23 22 24 26 |16 25 |26 :24 '25 111 26 10 1 4 1' Of'l' ■ ^M ■ 8iil- 6iil- 5 j2- 12- 10A|1- 8"'l- 2i! . H 517i3-088 03 !3^28 j3-40 12-69 |2-80 ,3-012 (2-37 999,3-879 35 3-52 ,6-01 2-64 3-05 |4-61 14-21 5^2-285 4-678 In the above table — A = area of plain sail in square feet. B = proportion of sail to one foot of midship section at load draught. c = proportion of sail to one ton of displacement at load draught. D = moment of sail about centre of lateral resistance divided by displacement in tons into the distance between the meta- centre and the centre of gravity in feet. E = weight of the ship in lbs. multiplied by the distance between the metacentre and the centre of gra\dty, and the product divided by the moment of sail about the centre of lateral resistance. 204 SAILING. Xote. — This is a measure of the power of a ship to resist inclination under her canvas. F =mean load-draught of water in feet and inches. G = distance of centre of gravity below load water-line in feet. H = height of metacentre above centre of gravity in feet. Effect of Gust of Wind on a Ship's Sails. The effect of a sudden gust of wind upon a ship's sails is, as a rule, to heel her over to an extreme angle of heel of about twice the steady angle at which the same constant pressure of wind would keep her. In fig. 149 let abc be the ship's curve of statical stability, and DE her curve of varying moments of sail — that is, the ordinates which express the moment of sail at the different angles vary as the cosine - of the angle of heel. Fig. 149. i.oO If the wind is steadily applied the ship will remain inclined at a steady angle of heel of 20°, determined by dropping an ordinate at the point of intersection E of the two curves ; but in the case of the same pressure of wind being suddenly applied she will heel over beyond the steady angle of heel, and she will oscillate for a time about that angle, the reason being that an amoimt of mechanical work has been done in heeling her over to 20°, which is represented by tlie area adeh, whereas the work absorbed is only equal in area to aeh ; hence mechani- cal work has been accumulated equal to the area AED, The ship will therefore continue to heel over till this work lias been absorbed; this will occur at 40°, when the area ekl is equal to the area AED, or, in other words, when tlie area alm — the d}Tiamical stability at 40° — is equal to the area adkm she will commence a retui-n oscillation under the influence of a righting moment, represented by ml. ~ _• ' •- - u- c L~ cu: .-- u- c c - - -^ - "T" X "Z ^ DC t^ l^ ^ tC i-!^ *-^ ""^ T Ct ^ '^^ ^1 "" ' — CT- C^ *— - >) >) :>j ri ?i M >i 7^1 ?-i 3-1 :>! ri Ti ri ?i ri :n :^^ — — 1 ■ ;: ^ - , -ir-4h-^-^'^'^-^'*-*i4<'+''*i"^-t"'^-^-5^^'^'^ '■ - _• L- r: — X t-1 -c; -* cc r^ t- — L-; -T- X ^: t ■■s> ~ ■ •7^ — ~ — . oc oc I- -^ --r -* -^ rf: -M :m , *^ ^ tO!^wL::ou;ooo>poo»p»o»p»p»p»p»oi-i 2 _• ^T U- - C-l L- X — U- 00 — CO -^ CC — _• ^ '— 5 r-I 30 T- I- --i S L-; -^ c^ rt ?i — ^ d CJ — 00 ti t- *z ^ ;:: . Ci r; c^ -. r: -. Ci n r; Ci ci ri ws ci 00 op CO 00 00 « ^ c^ , ^^^^^4^,JjH^^JK4H'^'?t<'*<'*'*l4i'*i-^4t i<« _• irt ti 3D c; -M -^ L-: t^ r: — re t^ r; — r; CO ?i ^ c^ •;: "zr > 3 "" x X X ^ ibo>boo>b»oib>b>b»bib»b»h»bu:ihu-;ihih „• _• •^ r; 3; i^ tr ' -: — r- 7i — c ~ i r- — >c -r- r: oi — '^ H^ S 5 r: C-. r; -. c~. -. -. r. 9 r: 30 3D X 30 a; 3D X CO ao - 13 i ih lb «i >i »h L-; ih ih ih ih »b ih to »b o o t^ -* :m C X vr r: — r: r- '--i -r -M X -o u-i r: :m 3 _: > 3 •^ i~ ^ re — r; X •— -?< re "M -' ~. X I- tc u^ — ,zz . rt J^' H -sC --o -w --r --c --^ i--: ue u: ue -f* '^ -*< -r '^ — " •••••• ~ s _ 3 C5 •^ -o -o to ts -,0 -0 -0 -0 -00-0000 -0 -o ~ X "5 _2 <; ■~ , -" ri re X re t^ ei i^ ei t^ 'M t^ ei t^ ^: X ~: c: -r< ^ ^ -ii^^ .g X -0 i-e re :m — . t- -f re — X t^ --rn 3>j — .~ 2 i-e -rt- -TjH -^ -rr -Tj- -T- re re re re re re re ^1 -n :^^ •>! 71 ei 5 •= z- 50 t^ t-. L-- l--- L^ i--- t^ l^ t>- t^ t- t^ 1-- t^ t>- t>- t- t- t~- t^ X , — . ei-Mrere-^ue-ot-x— "^ 3 -^ ; •H t- re — r: t^ '3 re — r; i^ i.e re — cr; t- re — .~ -^ '3 lO »-e -;?< -^ -*< -^ -]t" re re re re re :^1 ->■« ri ri ?i ei — s ill t- xxxxxxxxxxxxxxxxxxxx r— -< M >-e I- c re tc — re -f t -r — . -- r; •"-■■ '*> iz »-^ — t- -r — —. re C; x le re c t- <-e :-i C: c- '- ei — . — «=^ Tt ~ r; r; ~ X X X X t^ t^ t^ i- -o le te ue — ■^ ^ = - » 6 r; ri cr. i-. =-. c^ fi r; =-. f; f; r. Ci r; i: r; r. r; c; ■^ r: c c c; M re e'e-*-*-r-:fOieoooi?-x~ • ^ •:^ _; •;:; X !>. — re 71 — r; X t;- -0 -^ re 71 — 9 *E 1 « d c^ c'. n f: f^ r^ n f". f: X X X X X cb x> 30 x> 30 r« ' 71 X X ^1 -- - -o 1- t- 71 - re X -o t- -1 t-- t- r^ s .S lie o re c; i^ -^ 71 r; X i-^ t- t- X r: ^— ^ 'S c t;- «-e 71 X -e re -7^ cp -f 71 X -^ 71 X ^ " 6 r: r: d r. X X X x i- i^ t- t- t- -o -o -0 -o -0 o Eh re 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 •A a o — 7ire-*i-e"ot^xciO'— iMre-e'o-ot-xri Z .9 o o The number in this table corresponding to the time in which a vessel passes over the measured knot is her rate in knots per liour. 1 ,_ ./ 1 -T z; _ — o o 5 " —t '-^ ^ T-i r- O I-H '*-^'+t-*'*l-*'*-^-+!'^-*-tlH-* T^ -^H ^^ -^Jl -^^ ^:f^ tJH c a 1— 1 O 'rf CTi CO » (M -O — lO O "+1 C5 M CO Cr; t^ -M O ^ O S i-l o CO '^^ o ri 'M -^ ri c^ o O -* I- ^ '^ co c-i cc c; rt ccoui'+iccrc-M — — ooc^cccot^-occo — -« XcocpaoooxxcoaoccaDt^i^t-t^t^t-t-t^t^ d a a a f-H c: oo t^ t^ — »c -f -** c^ •M — ^ O ci cm cm cm cm cm ue i-~ o CM lo CO o — CO X I- -^ ir; 13 ^ IC010»0»OIQ»CICOOIOOO ic3 lO lO »C O »0 lO 3 o o t- CO CO c: o — CM -M ^t 'i- i,-: ic Ocscot^OwiaTt-rccM-^ocn CO «:- t- t- t- t^ t- I- i^ L- I- i^ -^ t^ X en cT: — re -^ ?2 ^^ 'p '— H: t •'' o »ooioio»oioioioiaioo»o»a id m lo >o »o lo o d a Clt^tO-^CC-MOC^COt^tCiO-^H -M — 1 O cn GO t- --C -+ re; -M — o ~ -*-*-*trc;cccpr;:ccc^:^r^rccM re 'C>T ^ CD rs X X X t- to ue re -M r-i CM CM CM M CM CM (M a ^ -o -o -o ;c ->c tc -^ o -o ;^ -o o a 00 ocot^o-*''M-^Ocot^o-+icc: X -t< o t- re o tr> — cr> r; r- -o le re O O -. ~ r; c; ci t- t?- o i o o o 00 < H O a a 'MCC»0«CC0O oo -*> -:f 'J' JC CC rO re C:^ CM CM CM CM — o t^ -f r- X »c re v:; re ^ r: ic '-t- (M -1 r- r-, d; O O O d a c^ cr. c. c: c; c. Ci cn c. c; c: c; ci Ti ~. C-. a a a en a 11-250 11-216 11-180 11-146 11-111 11-077 11-04:5 11-009 10-976 10-942 10-909 10-876 10-84:5 — X !^ '^ 'M .-- c; — ( t^ ^ ~ X ue ^ CO t^ t-- t^ CD CO O 6 6 6 6 6 6 6 _d «cccOao«Dic-^rc:crerc-^K'^ '*- cp ;p u-; o -^ -^ rp re c^^ c^^ t^ cTi -^ re CO o re X re 2 -« ci »c o CO CO CO M 0^ iN Cv| a „^^^^^„_^^^^^_______ d a C>0'M-*r^-H«o— ico»crO(Mr-i O— i-MrO-HtOI-CiO'tM-^i^OGO p r; oo t- -^ lO >* re; cp C4 'T' O ri cjo i'^ t- ^- t^ b- b- h- t- h- b- b- -i -— I • O -* CO o -M -*< CO c^ ^ re c^ X t- CO ue ic -rti •b CO CO CO CO CO CO d d a -:^CM-M^OOCOcOO-*'— 'O— ''+» — 1 re «o »~ o c<» lo 3i -M --C' o -t< CO l^iQCp-ylOCpcp'^Cer-iOXtp lb »"o JO >b lb -** 'Tt* -*i -*" -rH -^ re rb !M » w (M (M :M C ^ C^ C<1 -4 oooooooo ai cosoo"^-r;-t-b't;-b-b-epcp «0 c: !>1 ic Ci M 't C ^1 w O -1 t^ — u:t Ci c^ t- 1— i «0 c»Xt~-t--t^t~-t^t-^wcp:po^O»pipOO»p Or^C^StCtGCC»Xc»qp»>.t-t--l^--pcptpOO»0 c:i 00 00 00 00 00 00 cbccooxdooocbxicbcboocbao oioo-^'*''*''*|^r<^-fi'^i'^i<^'7^'7^'7-T'Ppo OOOOOOOOOOOOOOOOOOOO xxt^t^co^ip»p»p-^^opco?-i'r5^r-i"^pp 5^ ;j; ^_i _i4 t- o ! o CC-MiM-— lOpC5X^^-^>•«p^O>p-*:pCO'^1'7H'7

.»>»a»(MOC0OOO'-HO2OC<> 208 KNOTS 10 MILES AND MILES TO KNOTS, Table of Comparison of Admiralty Knots and Statute Miles, Kiii^t.-; Miles Knots 2>Iiles Knots Miles Knots Miles | Knots Mii.s •0606 ■848o •6364 •9242 •2121 •5000 •7879 ■0758 •3636 •6515 •0394 •22 1 o •5152 •8030 0909 '3788 6667 9545 2424 5303 8182 1061 3939 6818 969 2576 5455 8333 1212 4091 6970 9848 2727 5606 8485 1364 16-00 18- 16-25 18- 16^50 19- 16-75 19- 17-00 19- 17-25 19- 17-50 20- 17-75 20- 18 00 20- 18-25 21- 18-50 21- 18-75 21- 19-00 21- 19-25 22- 19-50 22- 19-75 22- 20-00 23- 20-25 23- 20-50 23- 20-75 23- 00 24' 25 24 50 24' 122 9-> 22-50 25 ZZ'I O zo 23- •?3' 4242:21 7121121 0000*21 2879121-75 25 5758 22-00 25 8636J22 1515 4394 7273 0152 3030 5909 8788 1667 4545 7424 0303 3182 6061 8939 23 23-75 27 24-00 27 24-25 27 24 Y2i |25 9 00 26 25 26 50 27 25- 25- 50 28 75 28 00 28 25 29 50 29' 75 29 1818 4697 7576 0455 3333 62 IL^ 90; »1 1970 484s 7727 0606 348.^ 6364 924: 2121 5000 7879 0758 3636 65 1; Miles Knots 1-00 1-25 1-50 1-75 2-00 2-25 2-50 2-75 3-00 3-25 3-50 3-75 4-00 4-25 4-50 4-75 5-00 5-25 5-50 5-75 i -8684 1-0855 1-3026 1-5197 1-7368 1-9539 21711 2-3882 2-6053 2-8224 3-0395 3-256(. 3-4737 3-6908 3-90 4-12 4-342] 4-5592 4-7763 4-9934 Miles Knots Miles Knots 6-00 6-25 6-50 6-75 7-00 7-25 7-50 7-75 8-00 8-25 8-50 8-75 9-00 9-25 9-50 9^75 10-00 10-25 10-50 10-75 5-2105 5-4276 5-6447 5-8618 6-0789 6-2961 6-5132 6-7303 6-9474 7-1645 7-3816 7-5987 7-8158 8-')329 8-2500 .'^-4671 8-6842 8-9013 9-1184 9-3355 11-00 11-25 11-50 11-75 12-00 12^25 12-50 12-75 >00 13-25 13-50: 13-75 14-00 14-25 14-50 14-75 15-00 15-25 -50 15-75 9-5526 9-769 9-9868 10-2039 10-4211 10-6382 10-8553 11-0724 11-2895 11-5066 11-7237 11-9408 12-1579 12-3750 12-5921 12- 801 '2 1 3-0263 13-2434 13-4605 13-677«"^ rallies , Knots Miles Knots 1600 13 16-25 14 16-5014 16-75 14' 17-00 14' 17-25 14- 17-5015' 17-75 15' 18^00 15- 1 8^25 15' 18^50 16^ 1 8^75 1 6- 19^00 16' 19-25 16- 19-50 16- 19-75 17' 20-00 17- 20-25 17' 20-50 1 7- 20^75 1 8- 8947 1118 3289 5461 7632 9803 1974 4145 6316 8487 0658 2829 5000 7171 [»342 24 1513 3684 5855|: 8026 0197 00 18 25 18 5018 •75 18 00 19 25 19 50 19 75 19 0019 25 20 50 20 75 20 00 20- 2521' 50 21' 75 21- 00,21- 25121- 50 22 75 22 2368 4531 ■6711 •8882 1053 3224 5395 7566 9737 1908 4071 6250 8421 0592 2763 4:134 7105 927r. 1447 :i61S N.i3. Tlie Admiralrv knot =6,080 ft. ; 1 statute mile = 5.2.^o fr. KILOMETRES TO KNOTS AND KNOTS TO KILOMETRES. 209 1 Table of Kilometiies to Admiralty Knots and Admi- 1 ralty Knots to Kilometres. j Kilos. Knots Kilos. Knots Kilos. Knots Kilos. Knots Kilos. Knots 1-0 •540 "^-o' 4-317 15^-0' "8^094 2"2-0" 11-872 29-0 15-64~9 1-25 •675 8-25 4-4.52 15-25 8-229 22-25 12-006 29-25 15-784 1-5 •809 8-5 4-587 15-5 8-864 22-5 12-141 29-5 15-919 l-7o •1)44 8-75 4-722 15-75 8-499 22-75 12-276 29-75 16-054 2-0 1-07'J 9 4-857 16 8-634 23-0 12-411 30-0 16-18S 2-25 1-214 9-25 4-991 16-25 8-769 23-25 12-546 30-25 16-328 2-5 1 -849 9-5 5-126 16-5 8-904 28 5 12-681 30-5 16-458 2-7o 1-484 9-75 5-261 16-75 9-039 28-75 12-816 80-75 12-951 81-0 13086 81-25 16-598 8-0 1-619 100 5-396 17-0 9-173 24-0 16-72S 8-25 1-754 10-25 5-531 17-25 9-308 24-25 16-863 8-5 1-889 10-5 5-666 17-5 9-448 24-5 13 221 81-5 16-998 8-75 2-024 10-75 5-801 17-75 9-578 24-75 18 856 31-75 17-138 4-0 2-158 11-0 5-93t; 1 8-0 9-718 25-0 13-490 [32-0 17-2()8 4-25 2-298 11-25 6-071 18-25 9-848 25-25 13-625 5:32-25 17-403 4-5 2-428 11-5 6-206 18-5 9-983 25-5 ' 13-760,32-5 17 -53s 4-75 2-568 11-75 6-840 1 18-75 10-118 25-75: 18-895 [82-75 17-672 FrO 2-698' 12-0 6-4751 19-0 10-258 260 14-080 '380 17-807 5-25 2-888 12-25' 6-610 19-25 10-388 26-25 14-165 33-25 17-942 5-5 ' 2-968 12-5 6-745 19-5 10-523 26-5 14-300 38-5 18-077 5-75 8-108 12-75 6-8S0i 19-75 10-657 26-75 14-485 ^83-75 18-212 6-0 8-288 18-0 7-015 20-0 10-792 27-0 14-570^31-0 18-847 r)-25 8-878 13-25 7-150 20-25 10-927 27-25 14-705 34-25 18-482 34-5 18-617 6-.5 3-508 13-5 7-285 20-5 11-062 27-5 14-839 r.-75 3-642 13-75 7-420 20-75 11-197 27-75 14-974 34-75 18-752 7-0 8-777 140 7-555 21-0 11-332 28-0 15-109 85-001 18-887 7-25 8-l)12 14-2o' 7-r.;»0 21-25 11-467 28-25 15-244 [35-25 19-021 7-5 4-047 14-0 { 7-824 21-5 11-602 28-5 15-379 f35-5 19-156 7-75 4-182 14-75 7-;>5:t 21-75 11-787 '2S-75 in-51 4^85-75! 19-291 | Knots 1-0 Kilos. 1^858 Knots; Kilos. 4-75 8-808 Knots Kilos. Kn'its Kilos. 12-25 22-701 Knots 16-0" Kilos. 8-5" 15-752 29-651 1-25 2-316 5-0 9-266 8-75 16-215 12-5 23-165 16-25 30-114 1-5 2-780 5-25 9-729 9-0 16-679 12-75 28-628 16-5 80-577 1-75 3-248 5-5 10-192 9-25 17-142 18-0 24-091 16-75 31-041 2-0 8-706 5-75 10-656 9-5 17-605 18-25 24-554 17-0 31-504 2-25 4-170 6-0 11-119 9-75 18-068 13-5 25-018 17-25 31-967 2-5 4-683 6-25 11-582 10-0 18-5.32 13-75 25-481 17-5 32-480 2-75 5-096 6-5 12-046 10-25 18-995 14-0 25-944 17-75 82-894 8-0 5-560 6-75 12-509 10-5 19-458 14-25 26-408 180 83-357 8-25 6-023 7-0 12-972 10-75 19-922 14-5 26-871 18-25 38-820 8-5 6-486 7-25 13-435 11-0 20-885 14-75 27-884 18-5 34-284 3-75 6-949 7-5 13-899 11-25 20-848 15-0 27-798 18-75, 34-747 4-0 7-418 7-75 14-362 11-5 21-311 15-25: 28-261 19-0 135-210 4-25 7-876 8-0 14-825 11-75 21-775 15-5 '28-724 19-25' 85-678 4-5 8-889 8-25 15-289|l2-0 22-288 15-75 2'.t-187 19-5 86-137 210 DISTANCES DOWN THE THAMES. ^~~" ^1 cm^Iiejox jx;j >. aeij p"3^^^^03 JO ^SBejqy ^r 1 'Z ~ 1 _ I « ^ in r5 iona :J^^l:^unB,l | g i o>p ^j < f S ' ? o ol (=5 o 9snoH uoA'BH H^^S ; "• "^" *? 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Xi t^ ! y- o ~ oc 1 _ J 1 t *"■ — ^- ' »— ' >* "o"= s-s P 1 c. iO:"fC;'i o 1 ~"i -.r ,c ^ jamL\i s.soa TioiAviooAi ^:;i 5!S|^|g;^|s!^|r|^|S!g ^ GO 1 rc is d 1 1 I 1 1— ,— 1— i_ ■1 .t5.' « to 1 rt t-r c^-TOicc TC'fc 7T cci?: ^ pua :>su3 ' 181J IlUA^HOUia gjV iO , CO . i-t C-l , OC i; — j C. t^ ' PT 1 3S 1 r-i X ■«)< L~ is -^ i 1 1 r- , — r~ r-'egir^lC^ CS ' C i.l O O u-t ' O O i L5 ' L-I "C 1 L-3 H IM 0> ,>^ CO^ Tf '>* CC « TJI C-. X i t^ -?-jO JO ^siJAV 'qaiAiuaajg |,1h -^ o t- 30'O| -^i-^ ; O i) f— cc 7^ cc rf Li ' L-j'l nisiio: C 1 ■M ' O 1 "."S 1 >o IC ' «* lO [ 1(5 : i-I C : m C: o 'L-r 1 c- o t~ ra 1 IT. ( CO ' Ci ^ j 00 L-i ; i-i . c; >0 K — — '^ n ' .. 1 'A lau.Cjjaoa 1 ^joj^dOQ ^ "^ TT CO! t^ f C5 |0 O OlO CIO C^ 1 •.-3 O ^_f) -M ^, n r is i j "V Ol h- =rsTo~ i/^ r-- . L* 1 — ^ 1 — -^ — ." I ?" T '.""l"?'''^!*?* T'^!^!? '^l?' o: oci>-t r-'?.?- ? v\ X ■^ irsoOiOj — 'mI-'J'IOiQC'O ■«^ i'^ ; t^ O ! L- 1 l^ — L-3 1 'O (M C^ ; d « ct cc r" ^ i ^ ' 1 1 1 1 1 1 J 1 L R < ■ ' ' "1 '. " ■ ' * J t: K +3 1 1 c 37 5 5 ^im'K 'sc 'C: 3 >■ 5 5 o ^ x'>-,< i^ 11 DISTANCES DOWN THE CLYDE. 21i 2: ^ C 5 c^uIOJ piB.\iox -• »c s^i — I9TJ5[00Jtt0£)|^l^ _ J9T(J :jtoonG9J£) ^ «5 I ->£ — ^1 O »f: ! re , c^j jq -* o oc aqSlT; A10JoSB^£) %10^ 5>i CO 00 1 re 00 00 re ! r-H 9psi33 noii^qninQ; l^'SS — O It- »P ;(N | — r^ ' t- i'e I— ! ^H |C<) I9TJ; ^BAv^iBy; SuTtAiog Ki^ {2 -^ ir O ic ^ if.TJ9 J Ai9IJU9y; Ci'OOl-J'CC rs CI -^ I Ki 00 re, 10 re cc coitc.ie ^;t«;^- -*< tc QC -f c-1 ipUT9;ii[jV\. — ' o c; 1 3- ^ 00 t- g c; — t^ 3<1 re ^l1 5^1 ce 00 2 ii: t- ^ <>* CC '^ t- : •M 00 10 00 s re ce 00 ci 00 t- 00 :$ C^ C: 1 I— 1 re ■* 01 T— 1 I— 1 10 cc s CN ?^ l^ c i.2 c .£.•■ -^ i^ ;^, CI ?/-:: ^ Tr:> ^ tci "S ^ r=- , S ? rf p !^ i? -1^, uJ r- :« C'T'dJ c :::i-o g.S'-.O ? 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( ' -^ z asouSod JO ;s'e8iqY 'jh >1 C5 -M b- 1 CC ^ T T '^ "^T ci) u: ih c ^ X -^ — 1 3^ St l-~ o' -+I Ir-^ .-1 1 (M "M 1 -* iO CD 2^ '~' ^ooH i^a cb ^ b- f 1 b- CO 60 re; cc 1 c: cr. t^'i^ cf^i ~ il ^^j_^^^ >i ?t i~ CM -"i [ :m " -+ ~^^~ . : ^ ^ -; s Aona 3[0Bia: »h 6 GC' GO : ^ : ^ ^ <: — ' 1 CM iCN rc ' ic .—1 X 5 bj. X 218 DISTAXCE> FROM SOUTHAMPTON ROUND THE ISLE OF WIGHT. ioua paa ao^saAV M ■r: X ^oqsi^O X 1— 1 1 .1. ^ — »h 7-1 ^ 5 ^« T— ^ 5i re S 5 7* le ;q°iT; c^oqsiBO ~ T. X 7^ f>1 r 1^ X fi -^ ^ 1 — ?i r^) re ^ \Z tr pua ;s-Ba r. jr; c"; ^ -^ 1^ - tr t^ ■>^ ^ '[B^idsoH if8[i8X ■T' X ^• :c l; z; ?i 7^1 h: X ue -^ >b iona: pay: no;saA\ 9 l^ ic u; i~ C t^ 7- »e X e-1 X »e i^ r" ^^- — — ~ — — 5i re — ^ ^r tc - S C^l X X t— i^ cc cr^i X ■>! r: t^ -^ T C M Z P ^ ■fl w J_ ^1 i ^ X t- i-i »h Ci •— t'-- X o 5 1 — — "^ I— ( ?^ -* ^ rz. 1 .. ^. 1 ' — ^' 1 • i—i > • . .1 1 ^ ^ m ^ ■5 ■5 ^•2 i 1 i ? ^ i^ p •0 j^ ' rv -i— i 1 r~ ^ .2. • • • ■ • • . ,_ ^ . ., . .| s — < s =5 < — .-3 S . . 't^ ^ > ;:3 M IS ! K . . . . 1 ■- ^ r' t:' Txo •^ ■r9 to — — ■ S ^ It I • r^ • X X • X ^ — < 'Z 9 c • ~ • "x ^ • • . s,'^. S Ji • • • < - IS S ■r. > J il i Sj El 3 ^ ^t rH l—i — — _JV ' ir. 'I 1 1 1— 1 ■r' Tc =5 M-i |! ■r. •-? X s5,l ^5 2: 1 '^ 1 II 1 1' • tu ^ e t - X X ^ S e :^ X DISTANCES FROM DEVONPORT TO PLYMOUTH. 219 - :^^to^I ;ib;s 9 X o S3 r^ CO C^ 1 CC c^^IOJ in^ij Ot 1 CC 1 • ^ CO 1 i— i ■^ 1^ ' o la O CO CO ■<5 pB9H ^pa 1 o CO CO 10 CO i> '^ C^ as ■< 1 10 :p g S lo lo CO ' CO -^ 5 S 9no;sM9i\r ai^^iT 1 »? i f QO GO 1 "* 1 1- O 00 C "S ;o pita :;s'B3 'uoo^ag; o ' 9 O O JO »p r-^ 1— ( CO i CO CO 1 CO C^ ' O ^OTifi p9i9nb8q3 I;- 1 1 1 1 1 1 8;F{AV P^i^ P9vr O -* 1 ■ ^ o , o 1 (^qSiT; j8;'B.W5['B8Ja -* ] r- 1 1 1 1 1 -U* ^ ^ ^ •If C si ^ o u: O p o o , CO CO _3 C ifona o^iqAV ^isy o -ti 1 t-^ '•p F— 1 1—1 i -tH J -rH •1 CO i cr. — ( cc CC r^ ' -^ <>i = >> H ^ -# — ;n ' 6 « b- ' lb i—i ' lO ■ C<1 !M CM t- '^ . . . d) . . . . . . "ii o i.^ 22 1 ~^3 w ~ * * ^ s " * 1—^ * > --- ^^ 'to o u w J2 C S S . • J o • ^ ■7? • • • ^ r- go H be pq • 3 1—1 a O c3 d O 1—1 +3 ;z5 1^' a I— I 9 'So S g r-i c c:; S O P, o > Q o r— 1 1 3 o u C5 d) +-> CO Ki o Q C O ! > Name of Ship ll Longit I Socti Foot Op ■Sg2 ^J 2 = 2 i-sg^ Jfe cS S rt '-P 2 c2 -H "^ ? S 1 =^ '-iS g ?-£^ OS ij i. 3 fe > ^ t. — :: »H o* t. O o* <'Z -X <^;^ <-J. \ <>V2 <,P,!r. Achilles 166 9792 59-0 Glatton 163 4579 28-0 Arethusa 114 5359 47-0 Inconstant 191 7640 40-0 BelleroiDhon 248 7301 29-4 ^Minotaur 198 10367 52-4 Blonde 203 7455 36-7 Monarch 231 i 7652 33-1 Canopus 127 4592 36-1 Raleigh 109 ' 38.54 I 35-3 Cyclops 95 3613 38-1 Himalaya 105 6290 60-0 Devastation 165 7615 46-] Warrior ISO 1 9271 ' 51-5 TO MEASrEE A SHIP S CIECLE. 2U A Practical :\Iethod of Measuring the Circle Described BY A Ship. (F. Martin, M.I.X.A.) Fis-. 150 shows the small portable fittings to be used on the Fig. 150. -.s occasion, a is a quadrant with the degrees carefully marked on a piece of wood which is temporarily secured on the ship's rail, with its inner edge ab kept parallel to the middle line of the ship ; c is a batten about -i feet long and 3 inches broad, with two upright wire sights S, s, one in each end, about 8 inches long. The batten is placed on the quadrant, with the centre of one end coinciding with the centre of the quadrant, and tixed with a pin through the centre, so that it can revolve. A ba.se ( AB, fi g. 1 5 1 ) is set off in a fore and aft dh-ection, of any conve- nient length, and at its foremost extremity a straight batten d is fixed vertically to the ship's side, extending a few feet above the rail. The same arrangement is carried out on each side of the ship, and a line joining the edges of the battens D, d must be at right angles to the middle line of the ship. These are all the fittings necessary. "When the helm is Imrd over, and the ship has fairly commenced her circular coiuse, throw overboard a rough wood box about a foot Square and painted black : as the ship moves onwards the box remains nearly stationary on the water, till presently the ship has described a semicircle, which is known by the two battens D, D and the box coming into the same straight line. At that instant the batten c is made to revo-ve till the two wire sights s, s and the box are in the same straight line ; the angle A (tig. 151) is then known, being denoted by the batten c on the quadrant. The angle B is a right angle, and the base AB being known, then DO = tangent A x BA, to which must be added twice the breadth of the ship for the greatest space occupied by her in describing the circle. — Ex. : If the angle A = 80^ 1.5', and the base ba = 0O feet, and the breadth of the vessel = 40 feet, then the greatest space occupied by her in de- scribing the circle is = ('JO x 5-81905) + C2 x iO) = 603-i"6S feet. 222 SQUARES, CUBES, EOOTS, AND EECIPEOCALS. Table OF SQrAREs, Cubes, Square Eoots, Cube Roots, axd 1 Reciprocals of ALL Ixte&er Numbers from 1 TO 2200. No. Square Cube Square Root Cube Root Reciprocal / 1 1 1 1-0000000 1-0000000 L-000000000 2 •4 8 ' 1-4142136 1-2599210 •500000000 3 9 27 1-7320508 1-4422496 -333333333 ' 4 16 64 2-0000000 1-5874011 -250000000 ' 5 25 125 2-2360680 1-7099759 •200000000 6 36 216 2-4494897 1-8171206 ■166666667 i 7 49 343 2-6457513 1-9129312 -142857143 i 8 64 512 2-8284271 2-0000000 -125000000 f 9 81 729 3-0000000 2-0800837 •111111111 ■■ 10 100 1000 3-1622777 2-1544347 -100000000 11 121 1331 3-3166248 2-2-239801 -090909091 4.12 lo 144 1728> 3-4641016 2-2894286 -083.333333 169 2197 3-6055513 2-3513347 •076023077 14 196 2744 3-7416574 2-4101422 •071428571 — 15 225 3375 3-8729833 2-4662121 •066666667 — IG 256 4096 - 4-0000000 2-5198421 •06-2500000 17 289 4913 4-1231056 2-571-2816 •058823529 .--18 324 6832 ^ 4-2426407 2-6207414 •055555556 19 361 6859 4-3588989 2-6684016 •052631579 20 400 8000 4-4721360 2-7144177 •050000000 — 21 441 9261 - 4-5825757 2-7589243 -047619048 •2'2 484 10648 4-6904158 2-80-20393 -04.54.^4545 23 5-29 12167 4-/ 958315 2-8438670 -043478261 24 576 13824 4-8989795 2-8844991 •041666667 2o 625 15625 5-0000000 2-9240177 •040000000 2('. 676 17576 5-0990195 2-9624960 •038461538 729 19683 - 5-19615-24 3-0000000 •037037037 .28 7N4 21952 5-2915026 3-0365889 •035714-286 29 841 " 24389 5-3851648 3-0723168 -034482759 30 900 27000 5-4772256 3-1072325 -033333333 31 961 29791 5-5677644 3-1413806 -032258065 32 1024 32768 5-6568542 3-1748021 -031-250000 -33 1080 35937 - 5-74456-26 3-2075343 •030303030 34 1 1 56 39304 5-8309519 3-2396118 •029411765 35 ] 225 42875 5-9160798 3-2710663 •0-28571429 36 1296 46656 6-0000000 3-3019272 •027777778 37 1369 50653 •6-08-27625 3-3322218 •027U27027 3« 1444 54872 6-1644140 3-3619754 •0263157?<9 — oO 1521 59319- 6-2449980 3-391-2114 •0256410-6 40 1600 64000 6-3245553 3-4199519 -025000000 41 1 681 68921 6-4031242 3-4482172 -024390244 1764 74088 - 6-4807407 3-4760266 -0-23809.V24 43 1819 79507 6-5574385 3-5033981 •0-23-25.5814 44 1936 85184 6-6332496 3-5303483 -022727273 2025 91125 . 6-708-2039 3-5568933 -022222222 SQUARES, CUBES, ROOTS, AXD EECIPROCALS. 223 Xo. Square Cube Square Root ' Cube Root Reciprocal 46 2116 97336 6-7823300 ' 3-5830479 -021739130 47 2209 10S823 6-8556546 ' 3-6088-261 '0212766(1(1 ^48 2304 i 110592 6-9282032 3-634-2411 -020833333 49 2401 ; 117649 7-00001)00 ■ 3-6593057 •0-20408163 50 2500 I 125000 7-0710678 3 6840314 -020000000 — 51 26(11 , 132651 ' 7-1414284 3-7084298 •019607843 52 2704 i 140608 7-2111026 ' 3-73-25111 •019-230769 53 2809 14.S877 7-280 ! 099 3-7562858 •018867925 54 2916 157464 7-34.^4692 3-7797631 •018518519 55 3025 166375 7-4161985 3-80295-25 •018181818 56 3136 175616 7-4833148 3-82586-24 •017857143 «57 3249 185193 " 7-5498344 3-8485011 •01 7543860 68 3364 195112 7-6157731 3-8708766 •01 7-24 13 79 59 3481 205379 7-6811457 3-8929965 •016949153 60 ■ 3600 216000 7-7459667 3-9148676 ■016666667 61 3721 2-26981 7-8102497 3-9364972 •016393443 62 3844 238328 7-8740079 3-9578915 •0161-^9032 —63 3969 250047 - 7-937-2539 3-9790571 •015873016 64 4096 262144 8-0000000 4-0000000 •015625000 65 4225 274625 8-0622577 4-0207-256 •015384615 66 4356 287496 8-1240384 4-041-2401 •015151515 67 4489 300763 8-1853528 4-0615480 •0149-25373 68 46-24 314432 8-2462113 4-0816551 •014705882 69 4761 328509 8-3066-239 4-1015661 -014492754 70 4900 343000 8-3666003 4-121-2x53 •014-285714 71 5041 357911 8-4261498 4-1408178 •014(184507 72 5184 373248 8-485-2814 4-1601676 •013888889 73 5329 389017 8-5440037 4-1793392 •01369863(1 74 5476 405224 8-6023253 4-1983364 •013513514 75 5625 421875 8-6602540 4-2171633 "013333333 76 5776 438976 8-7177979 4-2358236 •01S157N:f5 77 5929 456533 8-7749644 4-2543210 •012987(113 78 6084 474552 8-8317609 4-2726586 •012820513 79 6241 493039 8-8881944 4-2908404 •01-2658228 80 6400 512000 8-9442719 4-3088695 -012500(100 81 6361 531441 9-0000000* 4-3267487 •01-2345679 82 6724 551368 9-0553851 4-3444815 •012195122 83 6889 671/87 9-1104336 4-3620707 •012048193 84 7056 592704 9-1651514 4-3795191 •0119(j4762 85 7225 614125 9-2195445 4-3968296 •011764706 ^6 7396 636056 9-2736185 4-4140U49 •011627907 ~87 7569 658503 9-3273791 4-4310476 •011494-253 88 7744 681472 9-3808315 4-4479602 •011363636 89 79-21 704969 9-4339811 4-4647451 •011235955 90 8100 729000 9-4868330 4-4814047 •011111111 91 8281 753571 9-6393920 4-4979414 •010989011 92 8464 778688 9-6916630 ' 4-5143574 •010869565 93 8649 804357 9-6436508 4-5306549 •010752688 94 8836 830584 9-6953597 4-5468359 •010638298 224 SQUARES, CUBES, EOOIS, AXD EECIPKOCALS. No. Square Cube Square Root Cube Root Reciprocal 95 9025 857375 9-7467943 4-56-29026 -010526316 96 9216 884736 9-7979590 4-5788570 -010416667 97 9409 912673 9-8488578 4-5947009 -010309278 98 9604 941192 9-8994919 4-6104303 •0102040S-2 99 9801 970299 9-9498744 4-6-260650 -OlOlOlolO 100 lOiJOO 1000000 10-0000000 4-6415888 -OlOOOOi 00 101 10201 1030301 10-0498756 4-6570095 •009900990 102 10404 1061208 10-0995049 4-6723-287 -009.^039 --'2 103 10609 1092727 10-1488916 4-6875482 -009708738 104 10816 1124864 10-1980390 4-7026694 -00961 5385 -105 11025 > 1157625 10-2469508 4-7176940 -0095-23810 106 . 11236 1191016 10-29563U1 4-7326235 -009433962 107 11449 1225043 l0-r3440804 4-7474594 -009345794 108 11664 1259712 l(i-392304.s 4-7t;2-_'032 -009-259259 109 11881 1295029 10-4403065 4-7768562 -009174312 110 12100 1331000 10-4880885 4-7914199 •009090909 111 12321 1367631 10-5356538 4-8058955 -009O09U09 112 12544 1404928 10-5830052 4-8202845 -008928571 113 12769 1442897 10-6301458 4-8345S81 ■00NS49.'58 114 12996 1481544 10-6770783 4-8488076 -008771930 115 13225 1520875 10-7238053 4-8629442 -008695652 116 13456 1560896 10-7703296 4-8769990 -008620690 117 13689 1601613 10-8166538 4-^^909732 -0085470i'9 118 13924 1643032 10-8627805 4-9048681 -00S474576 119 14161 1685159 10 9087121 4-9186847 -008403361 120 14400 172^000 10-9544512 4-9324-242 ■008333333 121 14641 1771561 11-0000000 4-946(1874 -008264463 122 14884 1815848 11-0453610 4-9596757 •008196721 123 15129 1860867 11-0905365 4-9731898 -008130081 124 15376 1906624 11-1355287 4-9866310 -008064516 125 15625 1953125 11-1803399 5-0000000 -O0800U000 126 15876 .2000376 11-2-249722 5-0132979 -0079365O8 127 16129 2048383 1 1-2694-277 5-0265257 -007874016 128 16384 2097152 11-3137085 5-0396842 -007812500 129 16641 2146689 11-3578167 5-0527743 •007751938 130 16000 2197000 1 1-4017543 5-0657i'70 -007692308 131 17161 2248091 11-4455-231 5-0787531 •0076335><8 132 17424 2299968 11-4891-253 5-0:'164;i4 -007575758 133 ] 7689 2352637 11-53256-26 5-10446S7 -007518797 134 17i»56 2406104 11-5758369 5-1172299 -0074()26^<7 135 18-_'25 2460375 11-6189500 5-1299-278 -007407407 136 18496 2515456 11-6619038 5-14-25632 ■007352941 137 18769 2571353 11-7046999 5-1551367 -007299270 13S 19014 2628072 11-7473401 5-1676493 -007246377 139 1 9321 2685619 11-7H98-261 5-180lnlo -007191245 140 19600 2744000 1]-8:]21596 5-1924941 •007142857 141 19881 28(t3221 11-87434-22 5-2048279 -007O921« '13-6381817 5-7082675 -005376344 187 34969 6539203 13-6747943 6-7184791 •005347594 188 35344 6644672 13-7113092 6-7286543 -005819149 189 35721 6751269 13-7477271 5- 7387936 •005291005 190 36100 6859000 13-7840488 5-7488971 •005263158 191 36481 6967871 13-8202750 5-7.589652 •0052356U2 192 36864 7077888 13-8564065 5-7689982 -005208333 226 SQUAEES, CUBES, EOOTS, AND EECIPEOCALS. Xo. Square Cute Square Root ' Cube Root Reciprocal 193 ■: 37249 7189057 13-8924440 ' 5-7789966 -005181347 194 \ 87636 7301384 lo-92838s3 ' 5-7889604 -005154639 -195 ! 38025) 7414875 13-9642400 ' 5-7988900 -00512S-205 196 1 38416 7529536 14-0000000 ' 5-8087857 -005102041 197 38809 7645373 14-0356688 ' 5-81b6479 -005076142 198 39204 7762392 14-071 2473 ' 5-8-284767 -005050505 199 39601 7880599 14-1067360 5-83827-25 -0050251-26 200 40000 8000000 14-1421356 1 5-8480355 •005000000 201 40401 8120601 14-1774469' 5-8577660 -004975124 202 40804 8242408 14-2126704 ! 6-8674643 -004950495 203 41209 8365427 14-2478068 ' 5-8771307 -004926108 204 41616 8489664 1 4-2828569 ' 5-8867653 -004901961 205 42025 8615125 14-317^^211 i 5-8963685 -004878049 206 j 42436 8741S16 14-3527001 5-9059406 ■004854369 207 ; 42849 8b69743 14-3874946 6-9154817 -004830918 208 43264 8998912 14-4222051 5-9249921 -004eO7692 209 43681 9129329 14-456^323 5-9344721 -004784689 210 44100 9261000 14-4913767 5-9439220 -004761905 211 44521 9393931 14-5258390 5-9533418 -004739336 212 44944 9528128 14-56U2198 5-9627320 -00471G981 213 45369 9663597 14-5945195 5-97209-26 •004C94^36 214 45796 9800314 14-6287388 5-9814-240 -004672897 215 4G225 9938375 14-6628783 5-9907-264 -004651163 216 46656 1(077696 14-69G9385 6-0000000 -004629630 217 47089 10218313 14-7309199 6-L'O92450 -00460-^295 218 47524 10S60232 14-7648231 6-01^4617 -0045^7156 219 47961 10503459 14-7986486 6-0276502 -004566210 220 48400 10648000 14-83-23970 6-036.^107 -004545455 221 48841 1U793861 14-8660687 ' 6-0459435 -004524887 222 49284 10941048 14-8996644 6-0550489 -OU4504505 223 49729 11089567 14-9331845 ■■ 6-0641270 -004484305 224 50176 11239424 14-9666295 6-0731779 -004464-286 225 60625 11390625 15-0000000 6-08-2-2020 •004444444 226 51076 11543176 15-0332964 6-0911994 -0044-24779 227 51529 11697083 15-0665192 6-1001702 -004405-286 228 61984 11852352 15-0996689 ' 6-1091147 •004385965 229 62441 12008989 15-1327460 : 6-1180332 -004366812 230 62900 12167000 15-1657509 6-1269-257 •0043478-26 231 53361 12326391 15-1986842 6-13579-24 -004329U04 232 53824 12487168 15-2315462 6-1446337 -004310345 233 64289 12649337 15-2643375 6-1534495 •004291845 234 64756 12812904 15-2970585 6-1622401 -004273504 235 55225 12977875 15-3-297097 6-1710058 •004255319 236 65696 13144256 15-362-2915 6-1797466 •004237288 237 56169 13312053 15-394S043 6-18846-28 •004219409 238 56644 13481272 15-427-24S6 6-1971544 •004201681 239 67121 13651919 15-4596248 6-205S218 •004184100 240 67600 13824000 15-4919334 6-2144650 •004166667 241 68081 13997521 15-5241747 G-2230843 •004149378 SQUARES, CUBES, EOOTS, AND EECIPEOCALS. 227 Xo. Square Cube Square Eoot Cube Root Reciprocal 242 58564 14172488 15-5563492 6-2316797 -004132-231 243 59049 14b48907 16-5884573 6-240-2515 •0041152-26 244 59536 14526784 15-6204994 6-2487998 -004098361 245 60025 14706125 15-6524758 6-2573-248 -004081633 246 60516 14886936 15-6843871 6-2658266 •004065041 247 61009 15069223 15-7162336 6-2743054 •004048583 248 61504 15252992 15-7480157 6-2827613 -00403-2-258 249 62001 15438249 15*7797338 6-2911946 -004016064 250 62500 15625000 15-8113883 6-2996053 -004000000 251 63001 15813251 15-8429795 6-3079935 -003984064 252 63504 16003008 15-8745079 6-3163596 -003968254 253 64009 16194277 15-9059737 6-3247035 •00395-2569 254 64516 16387064 15-9373775 6-3330256 -003937008 ^ 255 650251 16581375 15-9687194 6-3413-257 -003921569 256 65536 16777216 16-0000000 6-3496042 -003906-250 257 66049 16974593 16-0312195 6-3578611 •003891051 258 66564 17173512 16-06-23784 6-3660968 •003875969 259 67081 17373979 16-0934769 6-3743111 •003861004 260 67600 17576000 16-1245155 6-3825043 •003846164 261 68121 17779581 16-1554944 6-3906765 •003831418 262 68644 17984728 16-1864141 6-3988279 •003816794 263 69169 18191447 16-2172747 6-4069585 •003802281 264 69696 18399744 16-2480768 6-4150687 •003787879 265 70225 18609625 16-2788-206 6-4-231583 -003773585 266 70756 18821096 16-3095064 6-431-2276 •003769398 267 71289 19034163 16-3401346 6-4392767 •003745318 268 71824 19248832 16-3707055 6-4473057 •003731343 269 72361 19465109 16-4012195 6-4553148 •003717472 270 72900 19683U00 16-4316767 6-4633041 -003703704 271 73441 19902511 16-4620776 6-4712736 •003690037 272 73984 20123648 16-4924225 6-4792-236 -003676471 273 74529 20346417 16-5227116 6-4871541 •003663004 274 75076 20570824 16-5529454 6-4950653 -003649635 275 75625 20796875 16-5831-240 6-50-29572 •003636364 276 76176 21024576 16-6132477 6-5108300 •003623188 277 76729 21253933 16-6433170 6-5186839 •003610108 278 77284 21484952 16-6733320 6-5265189 -0035971-22 279 77841 21717639 16-7032931 6-5343351 •003584229 280 78400 21952000 16-7332005 6-5421326 •003571429 281 78961 22188041 16-7630546 6-5499116 -003568719 282 79524 22425768 16-7928556 6-5576722 -003546099 283 80089 22665187 16-8226038 6-5654144 -003533569 284 80656 22906304 16-8522995 6-5731385 •0035211-27 285 81225 23149125 16-8819430 6-5808443 •003508772 286 81796 23393656 16-9115345 6-58853-23 •003496503 287 82369 23639903 16-9410743 6-59620-23 -003484321 288 82944 23887872 16-97056-27 6-6038545 -003472222 289 83521 24137569 17-0000000 6-6114890 -003460-208 290 84100 24389000 17-0293864 6-6191060 -003448276 Q2 228 SQUARES, CUBES, EOOTS, AXD EECIPEOCALS. ^'o. Square Cube Square Root Cube Root Reciprocal 291 ,,, 84681 24642171 17-0587221 6-6267054 -003436426 292 85264 24897088 17-0880075 6-6342S74 -0034-24658 293 85849 25153757 17-1172428 6-6418522 -003412969 294 86436 25412184 17-1464282 6-6493998 •003401361 295 87025 25672375 17-1755640 6-6569302 -003389831 296 87616 25934336 17-2046505 6-6644437 •0U3378378 297 88209 26198073 17-2336879 6-6719403 -0(13367003 298 88804 26463592 17-2626765 6-6794-200 •003355705 299 89401 26730899 17-2916165 6-6868831 -003344482 300 90000 27000000 17-3205081 6-6943295 •003333£33 301 90601 27270901 17-3493516 6-7017593 -0(J3322259 302 91204 27543608 17-3781472 6-70917-29 -003311258 303 91809 27818127 17-4068952 6-7165-00 -003300330 304 92416 28094464 17-4355958 6-7-239518 •0(,3i89474 305 93025 28372625 17-464-2492 6-/313155 -003--'. 8689 30G 93636 28652616 17-.1928557 6-73«6641 -003-207974 307 94249 28934443 17-5214155 6-7459967 -U03257329 308 94864 29218112 17-54992«8 6-7533134 -003246753 309 95481 29503629 17'5783958 6-7C06143 •003230246 310 96100 29791000 17-6068169 6-7678995 •003225806 311 96721 30080231 17-6351921 6-7751690 -003215434 312 97344 30371328 17-6635217 6-7824229 -003205128 313 97969 30664297 17-6918060 6-7896613 -0u31 94888 314 98596 30959144 17-7200451 6-7968844 -063184713 315 99225 31255875 17-7482393 6-8040921 -003174603 316 99856 31554496 17-77638«c; 6-8112847 -003164557 317 100489 31855013 17-iM:)44938 6-8184620 -003154574 318 101124 32157432 1/ •<"-i3i'5545 6-8-256-242 -003144654 319 101761 32461759 17-.^6U5711 6-8327714 -003134796 320 102400 32768000 17-88«5438 6-8399!;37 -0(.31250OO 321 103041 33076161 17-9164729 6-8470213 •003115265 322 103684 333^6248 17-9443584 6-8541240 •003105590 323 104329 3369<5267 17-9722008 6-8612120 •003095975 324 104976 34012224 18-0000000 6-8682855 •003086420 325 105625 34328125 18-0277564 6-8753443 •0l!3(;7(.923 326 106276 34645976 1S-05547U1 6-8823888 •003067485 327 106929 34965783 18-0831413 0-8894188 •003058104 328 107584 352tt7552 18-1107703 6-8964345 •U03048780 329 10«241 3561 12»9 18-1383571 6-9034359 •003039514 330 10«900 35937000 18-1659021 6-9104-232 -003030303 331 1<;9561 36264691 1 .VI 934054 6-9173964 -0(J3021148 332 110224 36594368 18-2208672 6-9243556 -003012048 333 110889 36926037 18-248-2876 6-9313008 -003003003 334 111556 37259704 18-2756669 6-938-2321 •002994012 335 1 12225 37595375 18-3030052 6-9451496 •002985075 336 112896 37933056 18-3303(128 6-95-20533 •002976190 337 113569 3?<272753 l 41063625 18-5741756! 7-0135791 -002898551 34G 119716 41421736 18-6010752 1 7-02O3490 -00-2890173 347 120409 41781923 18-6279360 ■ 7-02/1058 -002881844 348 121104 42144192 18-6547581 : 7-0338497 -002873563 349 121801 42508549 18-6815147 i 7-0405806 -002865330 sr.o 122500 42875000 18-708-2869 : 7-0472987 •002857143 351 123201 43243551 18-7349940 ' 7-0540041 -002819003 352 123904 43614208 18-7616630 7-0606967 •002840909 353 124609 43986977 18-7882942 7-0673767 •002832861 354 125316 44361864 18-8148877 7-0740440 -U028-24859 355 126025 44738875 18-8414437 ' 7-0806988 -002816901 356 126736 45118016 18-8679623 7-0j<73411 -002808989 357 127449 45499293 18-8944436 7-0939709 -00-2801120 358 128164 45882712 18-9208879 7-1005885 •002793296 359 128881 46268279 18-947-2953 7-1071937 -002785515 3ri0 129600 46656000 18-9736660 7-1137866 -0O277777X 3G1 130321 47045881 19-0000000 7-1-203674 -002770083 3G2 131044 47437928 19-0262976 7-1269360 -00276-2431 363 131769 47832147 19-0525589 7-1334925 -002754821 364 132496 48228544 19-0787840 7-1400370 •0027472";3 365 133225 48627125 19-1049732 7-1465695 -002739726 366 133956 49027896 19-1311265 7-1530901 -002732240 367 134689 49430863 19-1572441 7-1595988 -0027-24796 368 135424 49836032 19-1833-261 7-1660957 -002717391 369 136161 50243409 19-20937-27 7-1725809 •002710027 370 136900 506530(i() 19-2353841 7-1790544 •002702703 37 L 137641 51064811 19-2613603 7-1855162 •002695418 372 138384 . 51478848 19-2873015 7-1919663 •002688172 373 139129 51895117 19-3132079 7-1984050 •002680965 374 139878 52313624 19-3390796 7-2048322 •002673797 375 140625 52734375 19-3649167 7-211-2479 -002666667 376 141376 53157376 19-3907194 7-2176522 •002059574 377 142129 53582633 19-4164878 7-2-240450 -002652520 378 142884 54010152 19-4422221 7-2304268 -002645503 379 143641 54439939 19-4679223 7-2367972 •002638522 380 144400 54872000 19-4935887 7-2431565 -002631579 3^1 145161 55306341 19-5192213 7-2495045 -0026-24672 382 145924 55742968 19-5448203 7-2558415 •002617801 383 146689 561818x7 19-5703858 7-2621675 -002610966 384 147456 56623 l(t4 19-5959179 j 7-26848-24 -002604167 385 148225 1 67066625 19-6214169 7-2747864 -002597403 ?,y-] 14X996 i 57512456 19-6468827 7-2810794 •002590674 3.^7 149769 ! 57960603 19-67-23156 7-2873617 -002583979 b'-^H 15U.'.44 ■ 58411072 19-6977156 7-2936330 •002577320 230 SQUARES, CUBES, EOOTS, AXD RECIPROCALS. No. Square Cube Square Root Cube Root Reciprocal 389 151321 58863869 19-7230829 7-2998936 •002570694 390 152100 59319000 19-7484177 i 7-3061436 -002564103 391 152881 59776471 19-7737199' 7-31238-28 -002557545 392 153664 60236288 19-7989899 1 7-3186114 -O02.">51020 393 154449 60098457 19-8242276 \ 7-3-248295 -002544529 394 155236 61162984 19-8494332 i 7-3310369 -002538071 395 156025 61629875 19-8746069 7-337-2339 -002531C.46 396 156816 62099136 19-8997487 7-34.34205 -002525253 397 157609 6257U773 19-9-248588 7-3495966 •002518892 398 158404 63044792 19-9499373 7-35576-24 -00251-2563 399 159201 63521199 19-9749844 7-3619178 •0O2506266 400 160000 64000000 20-0000000 7-3680630 •002500000 401 160801 644812€1 20-0249844 7-3741979 -00-2493766 402 161604 64964808 20-0499377 7-3803227 -00-2487562 403 162409 65450827 20-0748599 7-3864373 -00-2481390 404 163216 659392G4 20-0997512 7-3925418 -002475248 -405 164025^ 66430125 20-1-246118 7-3986363 -002469136 406 164836 66923416 20-1494417 7-4047206 -002463054 407 165649 67419143 20-1742410 7-4107950 •002457002 408 166464 67917312 20-1990099 7-4168595 -00-2450980 409 167281 68417929 20-2237484 7-4229142 -002444988 410 168100 68921000 20-2484567 7-4289589 -002439024 411 168921 69426531 20-2731349 7-4349938 -00-2433090 412 169744 69934528 •20-2977831 7-4410189 -00-2427184 413 170569 70444997 203224014 7-4470342 -00-2421308 414 171396 70957944 20-3469899 7-4530399 •00-2415459 415 172225 71473375 20-3715488 7-4590359 -0024U9639 416 173056 71991296 •20-3960781 7-4650223 -002403846 417 173889 72511713 20-4-2O5779 7-4709991 -002398082 418 174724 73034632 * 20-4450483 7-4769664 -002392344 419 175561 73560059 20-4694895 7-4829-242 -002386635 420 176400 74088000 20-4939015 7-48S87-24 -002;i80952 421 177241 74618461 20-51«2845 7-4948113 •002375297 422 178084 75151448 20-5426386 7-5007406 •002309668 423 178929 75686967 20-5669638 7-5066607 -002364066 424 179776 76225024 20-5912603 7-5125715 -002358491 425 180625 76765625 20-6 155-281 1 7-5184730 -002352941 426 181476 77308776 •20-0397674 ; 7-5-243652 -002347418 427 1^52329 77«54483 -20-6639 7.S3 7-5302482 -002341920 428 183184 78402752 20-6881 6U9 7-5361221 -002336449 429 184041 78953589 20-71-23152 7-5419867 -002331002 430 184900 795O700U 20-7364414 7-5478423 -002325581 431 185761 80062991 20-7605395 7-5536888 -0023-20186 432 18G624 80621568 20-7846097 7-5595263 •00231481.') 433 187489 811 82737 20-.^086520 7-5653548 -002309469 434 188356 81746.:>04 20-8326667 7-5711743 •00-2304147 435 189225 82312875 20-8566536 1 7-5769849 •{)0229>S:'\ 436 190096 82881856 20-88061.30 j 7-58278G5 •(>0229.357?< 437 190969 83458453 20-9045450 1 7-5885793 -0022'^No.0u SQUARES, CUBES, ROOTS, AND RECIPROCALS. 231 No. Square Cube Square Root Cube Root Reciprocal 438 191844 84027672 20-9284495 7-5943633 -002-283105 439 192721 84604519 20-9523268 7-60U1385 -002277004 440 193600 85184000 20-9761770 7-6059049 •002272727 441 194481 85766121 21-0000000 7-6116626 •002267574 442 195364 86350888 21-0237960 7-6174116 -002262443 443 196249 86938307 21-04756i2 7-6-231519 -002-257336 444 197136 87528384 21-0713075 7-6288837 -002-252252 445 198025 88121125 21-0950231 7-6346067 -002-247191 446 198916 88716536 21-1187121 7-6403213 -002242152 447 199809 89314623 21-1423745 7-6460272 •002237136 448 200704 89915392 21-1660105 7-6517247 -002-232143 449 201601 90518849 21-189 V201 7-6574138 -002227171 450 202500 91125000 21-213-2034 7-6630943 •002222222 451 203401 91733851 21-2367606 7-6687665 -00-2217295 452 204304 92345408 21-2602916 7-6744303 -00221-2389 453 205209 92959677 21-2837967 7-6800857 -002207506 454 206116 93576664 21-3072758 7-6857328 -002202643 455 207025 94196375 21-3307290 7-6913717 •002197802 456 207936 94818816 21-3541565 7-6970023 •002192982 457 208849 9">443993 21-3775583 7-70-26-246 -002188184 458 209764 96071912 21-4009346 7-7082388 -002183406 459 210681 96702579 2 1-42428 "^3 7-7138448 •002178649 460 211600 97336000 21-4476106 7-7194426 -002173913 461 212521 97972181 21-4709106 7-72503-25 -002169197 462 213444 98611128 21-4941853 7-7306141 -002 164." 02 463 214369 99252847 21-5174348 7-7361877 -002159827 464 215293 99897344 21-5406592 7-7417532 •002155172 465 216225 100544G25 21-563«587 7-7473109 -002150538 466 217156 101194696 21-5870331 7-75-28606 -002145923 467 218089 101847563 21-61018-28 7-7584023 -002141328 468 219024 102503232 21-6333077 7-7639361 •00213f37:)2 469 219961 103161709 21-6564078 7-7694620 -002132196 470 220900 103823000 21-6794834 7-7749801 -002127660 471 221841 104487111 21-7025344 7-7804904 -002123142 472 222784 105154048 21-7-255610 7-7859928 -002118644 473 223729 105823817 21-7485632 7-7914875 -002114165 474 224676 106496424 21-7715411 7-7969745 -002109705 475 225625 107171875 21-7944947 7-80-24538 •002105263 476 226576 107850176 21-8174242 7-8079254 -002100840 477 227529 108531333 21-8403297 7-8133892 -002096436 478 228484 109215352 21-8632111 7-8188456 -002092050 479 229441 109902239 21-8860686 7-8242942 -002087683 480 230400 110".92000 21-90890-23 7-8297353 -00-2083333 481 231361 1112S4641 21-93171-22 7-8351688 -002079002 482 232324 1119801G8 21-9544984 7-8405949 -002074689 483 233289 112678587 21-977-2610 7-8460134 -002070393 484 234256 113379904 22-0000000 7-8514-244 -002066116 485 235225 114084125 22-0227155 7-8568281 •00206 1.S56 486 236196 114791256 22-0454077 7-8622-242 002057613 232 PQUAP.E-, CUBE?, EOOT.-, AND EKCIPEOCALS. No. Square Cube Square Root Cube Root Reciprocal 487 237169 115501303 22-0680765 7-8076130 -002053388 ■iSS 238144 116214272 22-0907220 7-8729944 -002049180 489 239121 116930169 22-1133444 7-87836X4 -002044990 490 240100 117649000 22-1359436 7-8837352 •002040X16 491 241081 11S370771 2-2-1585198 7-8890946 -002036660 492 242064 119095488 22-1810730 7-8944468 -002032520 493 243049 119823157 22-2036033 7-8997917 -002028398 494 244036 120553784 22-2261108 7-9051294 -002024291 -495 245025^^ 1212S7375 22-2485955 7-9104599 •002020202 49(1 246016 122023936 2-2-2710575 7-9157832 •00-2016129 497 247009 122763473 22-2934968 7-9210994 •00201-2072 498 248004 123505992 22-3159136 7-92640X5 •002008032 499 249001 124251499 22-3383079 7-9317104 -002004008 oOO 250000 125000000 22-3606798 7-937O0.5S -002000000 501 251001 125751501 22-3830293 7-94-22931 •001996008 502 252004 126506008 22-4053565 7-9475739 -00199-2032 503 253009 127263527 22-4276615 7-9528477 •0019X8072 504 254016 128024064 22-4499443- 7-9581144 •001984127 505 255025 128787625 2-2-4722051 7-9633743 ■001980198 506 256036 129554216 22-4944438 7-9686271 -001976285 507 257049 130323S43 22-5166605 7-973X731 •00197-2387 508 258064 131096512 2-2-5388553 7-9791122 •001968504 509 259081 131872229 2-2-5610283 7-9X43444 •001964637 510 260100 132651000 2-2-5831796 7-9895697 •001960784 511 261121 133432831 22-6053091 7-9947XX3 •001956947 512 262144 134217728 22-0274170 8-0000000 •001953125 513 263169 135005697 22-6495033 8-0052049 -001949318 514 264196 135796744 22-6715681 8-0104032 -001945525 515 265225 136590X75 22-69361 14 8-0155946 •001941748 516 266256 1373X8096 22-7156334 8-0207794 •001937984 517 267289 13X1S8413 22-7376340 8-0259574 -001934236 518 268324 13X991X32 22-7596134 8-0311287 -001930502 519 269361 139798359 22-7815715 8-0362935 •001926782 520 270400 140608000 22-8035085 8-0414515 •001923077 521 271441 141420761 22-8254244 8-0466030 •001919386 522 272484 142236648 •22-8473193 8-0517479 •001915709 523 273529 143055667 22-8691933 8-0568862 •001912046 524 274576 143877824 22-8910463 8-06-20180 -001908397 525 275625 144703125 22-912X785 8-0671432 -001904762 526 276676 145531576 22-9346899 8-0722(;-20 •001901141 527 277729 1463631X3 22-9 ")6J 806 8-0773743 -001X97533 528 27.S784 147197952 22-97X2506 8-0824X00 -001X93939 529 279S41 1480;?:;ss9 -23-ooonooo 8-0875794 •00hs90359 530 2X0900 14XS770(IO 23-0217289 8-0926723 •Oitlxx6792 531 281961 149721291 23-0434372 8-0977589 -0018x3-23y 532 283024 15056X76X •23-0651252 8-10-2X390 •001X79699 533 1 284089 151419437 •23-0X679^28 8-1079128 •001X76173 534 1 285156 152273304 23-1 0X4400 8-ir.'9X03 •07 310249 172808693 23-6008474 8-2278-2.54 •001795332 558 311364 173741112 23-6220-236 8-2327463 -001792115 559 312481 174676'S79 23-6431808 8-2376614 •001 78X909 560 313600 175616000 ■23-6643191 8-24-25706 •001785714 561 314721 176558481 23-6854386 8-2474740 -00178-2531 562 315844 177504328 •23-7065392 8-2V23715 -001779359 563 316969 178453547 23- 7^2762 10 82572633 -001776199 564 318096 179406144 23-7486842 8-26-21492 -001773050 565 319225 180362125 23-7697-286 8-2670294 -001769912 566 320356 181321406 •23-7907r.45 8-2719039 •001766784 567 321489 182284263 •23-8117618 8-2767726 •001763668 568 322624 1832504-;2 23-8327506 8-2816355 •001760563 569 323761 1842200(»9 23-8537-209 8-2864928 •001757469 570 324900 185193000 23-8746728 8-2913444 •001754386 571 326041 186169411 ■23-89'6063 8-2961903 •001751313 572 327184 187149248 •23-916.^215 8-3010304 •001748-252 573 328329 188132517 •23-9374184 8-30586:". 1 •001 745201 574 329476 189119224 23-9'>82971 8-3106941 •001742160 575 330625 19010937. 23-9791576 8-3155175 •001739130 576 331776 191102976 24-0000000 8-3203353 •001736111 ."77 332929 192100033 24-0208243 8-3251475 •001733102 578 334084 1931005r.2 -24-0416306 8-3299542 •001730104 579 335241 194104539 ■24-06-24188 8-3347553 •001727116 580 G36400 195112000 •24-0831891 8-3395509 •0017-24138 581 337561 196122941 . 24-1039416 8-3443410 -001721170 582 338724 197137368 24-1-246762 8-3491256 •001718213 583 339889 198155287 •24-1453929 8-3539047 •001715-266 584 341056 199176704 24-1660919 8-3586784 •00171-2329 234 SQUARES, CUBES, EOOTS, AND EECIPROCALS. ^'0. Square Cube Square Root Cube Root Reciprocal 585 342225 200201625 24-1867732 8-3634466 •001709402 5«6 343396 201230056 24-2074369 8-3682095 -001706485 587 3445G9 2U22620O3 24-2280829 8-3729668 •(101703578 5'^8 345744 203297472 24-2487113 8-3777188 •(iO57O(i()80 5.S9 34C921 20433G469 24-2693222 8-38-24653 •001697793 5'JO 348100 205379000 24-2899156 8-3872065 •001694915 591 < 349281 206425071 24-3104916 8-3919423 •001692047 592 ' 350464 207474^88 24-3310501 8-3966729 •001689189 593 351649 2085 2 < 8.'; 7 24-3515913 8-4013981 •001(;86341 594 352836 209584584 24-3721152 8-4061180 -001683502 595 354025 210644875 24-3926218 8-4108326 -00:680672 596 355216 2117U8736 24-4131112 8-4155419 -001677852 597 356409 212776173 24-4335i534 8-420-2460 •001675042 598 357604 213847192 24-4540385 8-4249448 •001672-241 599 358801 214921709 24-4744765 8-4296383 •001669449 600 360000 216000000 24-4948974 8-4343267 •00166(^667 601 361201 217081801 24-5153013 8-4390098 •001663894 •001661130 602 362404 218167208 24-5356883 8-4436877 603 363609 219256227 24-5560583 8-4483605 -001658375 604 364816 220348864 24-5764115 8-4530-281 •001655629 605 366025 221445125 24-5967478 8-4576906 •00165-2893 606 367236 222545016 24-6170673 8-46-23479 •001650165 607 368449 223648543 24-6373700 8-4670000 •001647446 608 369664 224755712 24-0576560 8-4716471 •001644737 609 370881 225866529 24-6779254 8-4762892 • -001642O36 610 372100 226981000 24-6981781 8-480926 1 •001639344 611 373321 228099131 24-7184142 8-4855579 •001636661 612 374544 229220928 24-7386338 8-4901848 •0016339.S7 613 375769 230346397 24-75«8368 8-4948065 •001631321 614 376996 231475544 24-7790234 8-4994233 -0016-28664 615 378225 232608375 •24-7991935 8-5040350 •001626016 616 379456 233744896 24-8193473 8-5086417 •0016-23377 617 380689 234885113 24-8394847 8-513-2435 •001620746 618 381924 236029032 24-8596058 8-5178403 -0016181-23 619 383161 237176659 •24-8797106 8-5224321 •001615509 620 384400 238328000 •24-8997992 8-5270189 •001612903 621 385641 239483061 24-9198716 8-5316009 •001610306 622 386884 240641848 24-9399278 8-5361780 •0O1607717 i 623 388129 241804367 24-9599679 8-5407501 -001605136 i 624 389376 242970624 24-9799920 8-5453173 •001602564 1 625 390625 244140625 25-0000000 8-.U98797 •001600000 626 391876 245314376 25-0 1999-20 8-5544372 -0015ii7444 627 393129 246491883 25-0399681 8-5589899 •001594N90 628 394384 247673152 25-0599282 8-5635377 •001592357 629 395641 248858189 •25-07987-24 8-5680807 •0015898-25 630 396900 2500470(10 25-0998008 8-:.726189 -001587300 631 398101 251239:.91 25-1197134 8-5771523 •00i5S478(; 632 399424 252435968 2:)- 1396 102 8-5816809 •0O15S227.S 633 400689 253636137 •25-1594913 ' 8-586-2047 •001579779 SQUARES, CUBES, EOOTS, AND RECIPEOCALS. 235 Xo. Square Cube Square Root Cube Root Reciprocal 634 401956 254840104 25-1793566 8-5907-238 •001577287 635 403225 256047875 25-1992063 8-5952380 •001574803 636 404496 257259456 25-2190404 8-5997476 •001572327 637 i 405769 258474853 25-2388589 8-6042525 -001569859 638 1 407044 259694072 25-2586619 8-6087526 •001567398 639 408321 260917119 25-2784493 8-6132480 •001564945 640 ' 409600 262144000 25-2982213 8-6177388 •001562500 641 410881 263374721 25-3179778 8-6222248 -001560062 642 412164 264609288 25-3377189 8-6267063 •001557632 643 413449 265847707 25-3574447 8-6311830 •001555210 644 414736 267089984 25-3771551 8-6356551 •001552795 645 416025 268336125 25-3968502 8-6401226 •001550388 646 417316 269586136 25-4165301 8-6445855 •001547988 647 418609 27084r>023 25-4361947 8-6490437 •001545595 648 419904 272097792 25-4558441 8-6534974 -001543210 649 421201 273359449 25-4754784 8-6579465 -001540832 650 422500 274625000 25-4950976 8-6623911 -001538462 651 423801 275894451 25-5147016 8-6668310 •001536098 652 425104 277167808 25-5342907 8-6712665 •001533742 653 426409 278445077 25-5538647 8-6756974 •001531394 654 427716 279726264 25-5734237 8-6801-237 •001529052 655 429025 281011375 25-5929678 8-6845456 •001526718 656 430336 282300416 25-6124969 8-6889630 •001524390 657 431649 283593393 25-6320112 8-6933759 •001522070 658 432964 284890312 25-6515107 8-6977843 •001519757 659 434281 286191179 25-6709953 8-7021882 •001517451 6:;o 435600 287496000 25-6904652 8-7065877 •001515152 661 436921 288804781 25-7099203 8-7109827 •001512859 662 438244 290117528 25-7-293607 8-7153734 •001510574 663 439569 291434247 25-7487864 8-7197596 •001508296 664 440896 292754944 25-7681975 8-7-241414 •001506024 665 442225 294079625 25-7875939 8-7285187 •001503759 666 443556 295408296 25-8069758 8-7328918 •001501502 667 444889 296740963 25-8263431 8-7372604 •001499250 668 446224 298077632 25-8456960 8-7416246 •001497006 669 447561 299418309 25-8650343 8-7459846 •001494768 670 448900 300763000 25-8843582 8-7503401 •001492537 671 450241 302111711 25-rt< 136677 8-7546913 •001490313 672 451584 303464448 25-9229628 8-7590383 •001488095 673 1 452929 30482 217 25-9422435 8-7633809 •001485884 674 i 454276 306182024 25-9615100 8-7677192 •001483680 675 455625 307546875 25-9807621 8-7720532 •001481481 676 456976 308915776 26-0000000 8-7763830 •001479290 677 i 4.58329 310288733 26-0192237 8-7807084 •001477105 678 ■ 459684 311665752 26-0384331 8-7850296 •001474926 679 401041 313046839 26-0576-284 8-7893466 •001472754 680 462400 314432000 26-0768096 8-7936593 •001470588 681 463761 315821241 26-0959767 8-7979679 •001468429 682 465124 317214568 26-1151297 8-8022721 •001466276 236 SQUARES, CUBES, ROOTS, AND RECIPROCALS. Xo. Square Cube Square Root Cube Root Reciprocal 683 466489 318611987 26-1342087 8-8065722 -001404129 084 : 467856 320013504 26- 1533937 8-810^081 -001401988 685 469225 321419125 26-1725047 8-8151598 -001459!554 686 470596 322828856 26-1916017 8-8194474 •001457726 687 471969 324242703 26-2106848 8-8237307 •001455004 ess 473344 325660672 20-2297541 8-8-280099 •001453488 689 474721 327082769 26-2488095 8-8322850 •001451379 690 476100 328509000 20-2078511 8-8365559 •001449275 691 477481 329939371 20-2808789 8-8408227 •001447178 692 478864 331373888 20-305S929 8-8450854 •001445087 693 480249 332812557 20-3248932 8-84f>3440 •001443001 694 481636 334255384 26-3438797 8-8535985 •0014409-22 695 483025 335702375 26-3628527 8-8578489 •001438849 696 484416 337153536 26-3818119 8-86-20952 •001436782 697 485809 338608873 26-4007576 8-8663375 •001434720 698 487204 340068392 20-4190896 8-8705757 •001432605 699 488601 341532099 26-4386081 8-8748099 •001430615 7(J0 490000 343000000 20-4575131 8-8790400 •001428571 701 491401 344472101 20-4704046 8-8832661 •001426534 702 492804 345948408 20-4952826 8-8874882 •0014-24501 703 494209 347428927 •26-5141472 8-8917063 •001422475 704 495616 348913664 26-5329983 8-8959204 •001420455 /05 497025 350402625 20-55l!-<301 8-9001304 •001418440 706 498436 351895816 20-5700005 8-9043366 •001410431 707 499849 353393243 20-5894716 8-9085387 •001414427 708 501264 354894912 20-0082094 8-9127369 •00141 •24-29 709 5(/26Sl 356400829 20-0270539 8-9169311 -001410437 710 604100 357911000 20-0458252 8-9211214 •001408451 711 605521 359425431 26-6645833 8-9253078 •001406470 712 506944 360944128 26-6833281 8-9294902 •001404494 713 608369 362467097 26-7020598 8-9336687 •0014025-25 714 609796 363994344 26-7207784 8-9378433 •001400500 715 511225 365525875 20-7394839 8-9420140 •00139S001 716 512656 367061096 •20-7581703 8-9461809 •001390048 717 514089 368601813 26-7768557 8-9503438 •0013947(10 718 515524 370146232 26-7955220 8-9545029 •001392758 719 516961 371694959 26-8141754 8-95^6581 -001390821 720 518400 373248000 26-83-28157 8-9028095 •001388f<89 721 519841 374805361 26-8514432 8-900957O •001380963 722 521284 376367048 26-8700577 8-9711007 •001385042 723 522729 377933067 20-.'^880593 8-9752400 •001383126 724 524176 879503424 20-9072181 8-9793706 •001381215 725 525625 38107S125 20-925S240 8-9835089 •001379310 726 527076 382057 1 7(i 20-94-13872 8-9876373 •001377410 727 528529 384240583 20-9029375 8-99176-20 •001375516 728 529984 385S28352 26-9814751 8-9958H29 •001373026 729 531441 387420489 27-01 lOOOOO 9-0000000 •001371742 730 532900 389017000 27-0185122 9-0041134 •001369803 731 531361 390017^91 27-0370117 9-00822-29 •001307989 SQUARES, CUBES, ROOTS, AND RECIPROCALS. 23: No. Square Cube Square Root Cube Root Reciprocal 732 535824 392223168 27-0554985 9-01-23288 •001366120 733 537289 393S32837 27-0739727 9-0164309 -001,364-256 734 538756 395446904 27-0924344 9-0205293 •001362398 735 540225 397065375 27-1108834 9-0-246239 •001360544 736 541G96 398688256 27-1293199 9-0287149 •001358696 737 543169 400315553 27-1477439 9-0328021 •001356852 738 544644 401947272 27-1661554 9-0368857 •001355014 739 546121 403583419 27-1845544 9-0409655 -001353180 740 547600 405224000 27-2029410 9-0450417 •001.351.351 741 5490S1 406869021 27-2213152 9-0491142 •001.349528 742 550564 408518488 27-2396769 9-0531831 •001,347709 743 552049 410172407 27-25.S0263 9-057-24X2 •001345895 744 553536 411830784 27-2763634 9-0613098 ■001344086 745 555025 413493625 27-2946881 9-0653677 -001342-282 746 556516 415160936 27-3130006 9-0694220 -001340483 747 558009 416832723 27-3313007 9-0734726 -001338688 748 559504 418508992 27-3495887 9-0775197 •001336898 749 561001 420189749 27-3678644 9-0815631 -001335113 750 562500 421875000 27-3861279 9-0856030 •00133,33.33 751 564001 423564751 27-4043792 9-0896392 •001.331558 752 565504 425259008 27-42-26184 9-0936719 •001329787 753 567009 426957777 27-4408455 9-0977010 •001 328021 754 56X516 428661064 27-4590604 9-1017265 •0O132(;26O 755 570025 430368875 27-4772633 9-1057485 •001324503 756 571536 432081216 27-4954542 9-1097669 •001322751 757 573049 433798093 27-5136330 9-1137818 •001321004 758 574564 435519512 27-5317998 9-1177931 •00131 9-2<51 759 576081 437245479 27-5499546 9-1218010 •001317523 760 577600 438976000 27-5680975 9-1258053 •001315789 761 579121 440711081 27-5862284 9-1-298061 •001314060 762 580644 442450728 27-6043475 9-1338034 •001312336 763 582169 444194947 27-6224546 9-1377971 •001310616 764 583696 445943744 27-6405499 9-1417874 -001308901 765 585225 447697125 27-6586334 9-1457742 -001.307190 766 586756 449455096 27-6767050 9-1497576 -001.3054X3 767 588289 451217663 27-6947648 9-1537375 •001303781 768 589824 452984832 27-712^129 9-1577139 •00 30-2083 769 591361 454756609 27-7308492 9-1616869 •001300390 770 592900 456533000 27-7488739 9-1656565 •00129X701 771 694441 458314011 27-7668868 9-16962-25 •001297017 772 595984 460099648 27-7848880 9-1735852 •001295337 773 597529 461889917 27-8028775 9-! 775445 •001293661 774 599076 463684824 27-8208555 9-1815003 •001291990 775 600625 465484375 27-838X218 9-1854527 •0012'.to3-23 776 602176 467288576 27-8567766 9-1894018 •0012x8660 777 603729 469097433 27-8747197 9-193.3474 •001287001 778 605284 470910952 27-8926514 9-1972897 •001285.347 779 606841 472729139 27-9105715 9-2012-286 ■001283697 780 608400 474552000 27-9284801 9-2051641 •001282051 238 SQUAEES, CrBES, EOOTS, AND EECIPEOCALS. No. j Square Cube Square Hoot Cube Boot Reciprocal 781 609961 476379541 27-9463772 9-2090962 •001280410 782 ! 611524 478211768 27-9642629 9-2130250 -001278772 7j<3 i 613089 480048687 27-9821372 9-2169505 -001277139 784 ' 6146 '6 481890304 28-0000000 9-2208726 001275510 785 ' 616225 483736625 28-0178515 9-2247914 -0U1273886 786 i 617796 485587656 28-0356915 9-2287068 •00 1 272265 787 1 619369 487443403 28-0535203 9-2326189 -001270648 788 1 620944 489303872 28-0713377 9-2365277 -001269036 789 622521 491169069 28-0891438 9-2404333 -001267427 790 624100 493039000 28-1069386 9-2443355 •001265823 791 626681 494913671 28-1247222 9-2482344 •001264223 792 ' 627264 496793088 28-1424946 9-2521300 •001262626 793 628849 498677257 28-1602557 9-2560224 •001261034 794 , 630436 500566184 28-1 780056 9-2599114 •001259446 795 632025 502459875 28-1957444 9-2637973 •001257862 796 633616 504358336 28-2134720 9-2676798 -001256281 797 635209 506261573 28-2311884 9-2715592 -001254705 798 636804 508169592 28-24x8938 9-2754352 -001253133 799 638401 610082399 28-2665881 9-2793081 -001251564 800 640000 512000000 28-2842712 9-2831777 -001250000 801 641601 513922401 28-3019434 9-2870440 •001248439 802 643204 515849608 28-3196045 9-2909072 •001246883 803 644809 617781627 28-3372546 9-2947671 •001245330 804 646416 619718464 28-3548938 9-2986239 •001243781 805 648025 521660125 28-3725219 9-3024775 •001242236 806 649636 523606616 28-3901391 9-3063278 •001240695 807 651249 525557943 28-4077454 9-3101750 •001239157 808 652864 527514112 28-4253408 9-3140190 •001237624 809 654481 529475129 28-4429253 9-3178599 •001236094 810 656100 531441000 28-4604989 9-3216975 •001234568 811 657721 533411731 28-4780617 9-3255320 •001233046 812 659344 535387328 28-4956137 9-3293634 •001231527 813 660969 537367797 28-5131549 9-3331916 •001230012 814 662596 639353144 28-5306852 9-3370167 •001 228501 815 664225 541343375 28-5482048 9-6408386 •001226994 816 665856 543338496 28-5657137 9-3446575 •001225490 817 667489 645338513 28-5832119 9-3484731 •001223990 818 669124 647343432 28-6006993 9-3522857 •001222494 819 670761 549353259 28-6181760 9-3560952 •001221001 820 672400 651368000 28-6356421 9-3599016 •001219512 821 674041 653387661 28-6530976 9-3637049 •001218027 822 675684 555412248 28-6705424 9-3675051 •001216545 823 677329 557441767 28-6879766 1 9-3713022 •001215067 824 678976 559476224 2s-7o:)4002 9-3750963 •001213592 825 680625 561515625 28-7228132 1 9-3788873 •001212121 826 682276 563659976 28-7402157 i 9-3826752 •001210654 827 683929 565609283 28-7576077 9-3864600 •001209190 828 685584 567663552 28-7749891 9-3902419 •001207729 829 687241 569722789 28-7923601 9-3940206 •001206272 SQUAEES, CUBES, EOOTS, AND RECIPROCALS. 239 No. Square Cube Square Root Cube Root Reciprocal 830 688900 571787000 28-8097206 9-3977964 •001204819 831 690561 573856191 28-8270706 9-4015691 -001203369 832 692224 575930368 28-8444102 9-4053387 •001201923 833 693889 578009537 28-8617394 9-4091054 •001200480 834 695556 580093704 28-8790582 9-4128690 •001199041 835 697225 582182875 28-8963666 9-4166297 -001197605 836 698896 584277056 28-9136646 9-4203873 •001196172 837 700569 586376253 28-9309523 9-4-241420 ■001194743 838 702244 588480472 28-9482297 9-4278936 •001193317 839 703921 590589719 28-9654967 9-43 164-23 •001191895 840 705600 692704000 28-9827535 9-43538x0 •001190476 841 707281 694823321 29-0000000 9-4391307 •001189061 842 708964 596947688 29-0172363 9-4428704 •001187648 843 710649 599077107 29-0344(i23 9-4466072 •001186-240 844 712336 601211584 29-0516781 9-4503410 •001184834 845 714025 603351125 29-0688837 9-4540719 •0011834S2 846 715716 605495736 29-0860791 9-4577999 •001182033 847 717409 607645423 29-1032644 9-4615-249 •001180638 848 71 9 i 04 609800192 29-1204396 9-4652470 •001179245 849 720801 611960049 29-1376046 9-4689661 •001177856 850 722500 614125000 29-1547595 9-4726824 •001176471 851 724201 616295051 29-1719043 9-4763957 •001175088 852 725904 618470208 29-1890390 9-4801061 •001173709 853 727609 620650477 29-2061637 9-4838136 •001172333 854 729316 622835864 29-2-232784 9-487ol82 •001170960 855 731025 625026375 29-2403830 9-4912200 •001169591 85G 732736 627222016 29-2574777 9-4949188 •001168224 857 734449 629422793 29-2745623 9-4986147 •001166861 858 736164 631628712 29-2916370 9-5023078 •001165501 859 737881 633839779 29-3087018 9-6069980 •001164144 860 739600 636056000 29-3257566 9-5096854 •001162791 861 741321 638277381 29-3428015 9-5133699 •001161440 862 743044 640503928 29-3598365 9-5170515 •001160093 863 744769 642735647 29-3768616 9-5207303 •001158749 864 746496 644972544 29-3938769 9-5-244063 •001157407 865 748225 647214625 29-4108823 9-5280794 •001156069 866 749956 649461896 29-4278779 9-5317497 •001154734 867 751689 651714363 29-4448637 9-5354172 •001153403 868 753424 653972032 29-4618397 9-5390818 •001152074 869 755161 656234909 29-4788059 9-5427437 •001150748 870 756900 658503000 29-4957624 9-5464027 •001149425 871 758641 660776311 29-5127091 9-5500589 •001148106 872 760384 663064848 29-5296461 9-5537123 •001146789 873 762129 665338617 29-5465734 9-5573630 •001145475 874 763876 667627624 29-5634910 9-5610108 •001144165 875 765625 669921875 29-5803989 9-5646559 •001142857 876 767376 672221376 29-5972972 9-5682982 •001141553 877 769129 674526133 •29-6141858 9-5719377 •001140251 878 770884 676836152 29-6310648 9-5755745 •001138952 240 SQUAKES, CUBES, ROOTS, AXD EECIPEOCALS. Xo. Square Cube Square Root Cube Root Reciprocal 879 772641 679151439 29-6479342 9-579-2085 •001137656 880 774400 681472000 29-6647939 9-5828397 -001186364 881 776161 683797841 29-6816442 9-5864682 •001135074 882 777924 686128968 29-6984848 9-59(»(i939 •001183787 883 779689 688465387 29-7153159 9-5937169 •0011825(13 884 781456 690807104 29-7321375 9-5973373 •001131222 885 783225 693154125 29-74S9496 9-6009548 •001129944 886 784996 695506456 29-7657521 9-(;045696 •001128668 887 786769 697864103 29-7825452 9-6081817 •001127396 888 788544 700227072 29-7993289 9-6117911 •001126126 889 790321 702595369 29-8161030 9-6153977 •0011-24859 890 792100 704969000 29-8328678 9-619(Xil7 •001123596 891 793881 707347971 29-8496231 9-6226030 •001122334 892 795664 709732288 29-8663690 9-6262016 •001121076 893 797449 712121957 29'8831056 9-6297975 -001119821 894 799236 714516984 29-8998328 9-6333907 •001118568 895 801025 716917375 29-9165506 9-(]3(;9812 •001117318 896 802816 719328136 29-9332591 9-6405690 •001116071 897 804609 721734273 29-9499583 9-6441542 •0011148-27 898 806404 724150792 29-9666481 9-(>477867 •001113586 899 808201 726572699 29-9833287 9-6518166 •00111-2347 900 810000 729000000 80-0000000 9-6548938 -001111111 901 811801 731432701 30-0166620 9-6584684 •0(11109878 902 813604 733870808 30-0333148 9-662(J403 •001108647 903 815409 736314327 30-0499584 9-6656096 •0011074-20 904 817216 738763264 30-0665928 9-6691 762 •001106195 906 819025 741217625 30-0832179 9-6727403 •001104972 906 820836 74367741.6 30-0998339 9-6763017 •001103753 907 822649 746142643 30-1164407 9-6798604 •00110-2536 908 824464 748613312 30-1330383 9-6834166 •001101322 909 820281 751089429 30-1496269 9-6869701 •001100110 910 82.SI00 753571000 30-1662063 9-6905211 •001O9s9in 911 82992 L 756058031 30-1827765 9-6940694 •001(197695 912 831744 758550528 30-1993377 9-6976151 •001096491 9:3 833569 761048497 30-2158899 9-7011583 •001(195290 914 835396 763551944 30-23-24329 9-70469.S9 •001(194092 915 837225 766060875 30-2489669 9-7082369 •001092^96 9!6 839056 768575296 30-2654919 9-7117723 •0010917(13 917 840889 771095213 30-2820079 9-7153051 •0(11090513 9!8 842724 773620(532 30-2985148 9-7188354 •001(1^9325 919 844561 776151559 30-31501-28 9-72-23631 -0OlOS«139 92(1 846400 7786S8000 30-3315018 9-72588S3 •00108(1957 921 S48241 781229961 30-3479818 9-7294109 •001085776 922 850084 783777448 30-3644529 9-7329309 -001084599 923 851929 78633()4(i7 30-3S(l9151 9-7364484 •0010834-24 924 853776 788889024 39-3973683 9-7399634 •001082251 925 855625 791453125 30-4 138 127 9-74347:)8 •001 OK 1081 926 857476 79402277(i 3()-43(»-2481 9-7469857 •001079!* 11 927 859329 796597983 30-4466747 9-7504930 •00107^719 SQUAEES, CUBES, ROOTS, AST) RECIPROCALS. 241 No. Square Cube Square Root Cube Root Reciprocal 928 861184 799178752 30-4630924 9-7539979 •001077586 929 863041 801765089 30-4795013 9-7575002 •001076426 930 1 864900 804357000 30-4959014 9-7610001 •001075269 931 ! 866761 806954491 30-5122926 9-7644974 •001074114 932 868624 809557568 30-5286750 9-7679922 •001072961 9dB i 870489 812166237 30-5450487 9-7714845 •00107 811 934 i 87235(5 814780504 30-5614136 9-7749743 •001070664 935 874225 817400375 30-5777697 9-7784616 •001069519 936 876096 820025856 30-5941171 9-7819466 •001068376 937 877969 822656953 30-6 104557 9-7854288 •001067286 938 879844 825293672 30-6267857 9-7889087 •001066098 939 881721 827936019 30-6431069 9-7923861 •001004963 940 883600 830584000 30-6594194 9-7958611 •001063830 941 885481 833237621 30-6757233 > 9-7993336 -001062699 942 887364 835896888 30-6920185 9-8028036 •001061571 943 1 889249 838561807 80-7083051 ' 9-8062711 •001060445 944 891136 841232384 30-7-245830 9-8097362 •001059322 945 893025 843908625 30-7408523 9-8131989 •001058201 94(3 894916 846590536 30-7571130 i 9-8166591 •001057082 947 896809 849278123 80-7733651 9-8201169 •001055966 948 898704 851971892 30-7896086 \ 9-8235728 •001054852 949 900601 854670349 30-8058436 9-8270252 •00:053741 950 902500 85737500U 30-8220700 9-8304757 •001052632 951 904401 860085351 30-8382879 9-8339-238 •001051525 952 906304 862801408 80-8544972 9-8373695 •001050420 953 908209 865523177 30-8706981 9-8408127 •001049318 954 910116 868250(;ti4 30-8868904 9-8442536 -001048218 955 912025 870983875 30-9030743 9-8476920 •0010471-20 956 913936 873722816 30 9-9933289 -001002004 999 998001 997002999 31-0069613 9-9966656 -001001001 lOOO lOuOOOO 10(JU000000 3l-i.227766 10-0000000 -OOIOOOOOGO lUUL 1002U01 1003003001 31-6385840 10-0033322 -0009990010 1002 1004004 10060121:08 31-6543836 ] 0-00(36622 •00099>0040 10"3 Iij0t3009 1009027U27 31-6701752 10-0099899 •C00997o09o 1004 100801(3 1012O48064 31-6859590 10-0133155 -0009960159 1005 1010025 1015075125 31-7017349 10-0166389 •0009950249 1006 10 1 203(3 10l«Ut8216 3 1-7 17.: 030 10-0199t01 -00<'99403.'8 1007 1014049 1021147343 31-7o32633 10-0232791 •0009930487 :008 101(3<>04 I(.24i925l2 31-7490157 10-0265958 •0009920635 1%'y 1018081 1027243729 31-7647603 10-0299104 •0009910803 1010 1020100 103t'30l0<.0 31-7.^04972 10-0332228 •0009900990 1011 1022121 1033364331 31-796-22IJ2 10-0365330 •0009-91197 1012 1024144 1036433728 31-8119474 10-0398410 •000988 14 -JS 10 i 3 102(31(39 10395091'97 3l-?;276609 10-0431469 •00098716C8 1014 1028 1 96 1042590744 31-8433666 10-0-164506 •000986 19r3 i015 • 1030225 1045678375 31-51590646 100497521 •00098522! 7 101(5 : 1((32256 1048772096 31-8747549 10-0.= 30514 •or 09842520 10 7 1034289 1051871913 31-8904374 10-0563485 •000983-2842 1018 1036324 10.54977832 31-90611-23 10-0." 96435 -0< 09--<231f3 1019 1038361 1058089859 31-9217794 10 0(i293«34 -0009813543 1020 ■ 104O400 1061208000 31-93743.'«'8 10-0662271 -0009^03922 1021 1042411 1064332261 31-9.3.30906 10-0695 1.6 •O(.t097943;'j 1022 ■ 1044484 ■ 1067462(348 31-9687347 lo-07-2H( 20 -O0097?<47o(; 1023 : 1046529 107(J5991(;7 31-9843712 10-07L08(i3 -0009775171 1024 1018576 ' 1073741)^24 32-0000000 10-0793684 -0-(;343377 10-2121347 -0009389671 11166 1 1(^6356 1211355496 32-6-496554 10-2153300 -0009380863 1067 1138489 1214767763 3-2-6649659 10-2185233 •0009372071 !i0911 32-7261363 10-2312766 •00093370^8 1072 '1149184 1231925248 32-74141I1 32-7566787 10-2344599 •()00932835iy 1073 1151329 1235376017 10-2376413 •0009319664 1074 ! 1153476 1 1238833224 32-7719392 10-2408207 •0009310987 r2 244 SQUARES, CUBES, ROOTS, AND RECIPROCALS. No. 1075 1076 ](i77 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 lll(i 1117 1118 1119 1120 1121 1122 1123 Square Cube 1155625 1167776 1159929 1162084 1164241 1166400 1168561 1170724 1172889 1175056 1177225 1179396 1181669 1183744 1185921 1188100 1190281 1192464 1194649 1196836 1199025 1201216 1203409 1205604 1207801 1210000 12122011 1214404 1216609 1218816 1221025 1223236 1225449 1227664 1229881, 1232100 1234321 j 1236544 ' 1238769 1240996 1243225 1 245456 1247689 1249924 1252161 1254400 1256641 j 1258884 : 1261129 1242296875 1245766976 1249243533 1252726552 1256216039 1259712000 1263214441 12667233(i8 1270238787 1273760704 1277289125 1280824056 1284365503 1287913472 1291467969 1295029000 1298596571 1302170688 1305751357 1309338584 1312932375 1316532736 1320139673 1323753192 1327373299 1331000000 1334633301 1338273208 1341919727 1345572864 1349232(;25 1352899016 1356572043 1360251712 13639o.-<029 1367631000 1371330631 1375036928 1378749S97 1382469544 1386195N75 138992^<896 1393668613 1397415032 1401168159 1404928000 140^694561 141246784S 1416247867 Square Root j Cube Root 32-7871926 32-8024389 32-8176782 32-8329103 32-8481354 32-8633535 32-8785644 32-8937684 32-9089653 32-9241553 32-9393382 32-9545141 32-9696830 32-9848450 33-0000000 33-0151480 33-0302891 33-0454233 33-0605505 33-0756708 33-0907842 33-1058907 33-1209903 33-1360830 33-1611689 33-1662479 33-1813200 33-1963853 33-2114438 33-2264965 33-2415403 33-2565783 33-2716095 33-2866339 33-3016516 33-3166ti25 33-3316666 33-3466640 83-3616546 33-3766385 33-3916157 33-4065862 33-4215499 33-4365070 33-4514573 33-4664011 33-4813381 33-4962684 33-5111921 10-2439981 10-2471735 10-2503470 10-2535186 10-2566881 10-2598557 10-2630213 10-2661850 10-2693467 10-2725065 10-2756644 10-2788203 10-2819743 10-2851264 10-2882765 10-29 14-247 10-2945709 10-2977153 10-3008577 10-3039982 10-3071368 10-3102735 10-3134083 10-3165411 10-3196721 10-32-2<'^012 10-3259284 10-3290537 10-3321770 10-3352985 10-3384181 10-3415358 10-3446517 10-347765/ l(i-350s778 10-3539880 10-3570964 I0-3(i02029 10-363307(! 10-3664103 10-3695113 10-372til03 10-3757076 10-3788030 10-3818965 10-3849882 10-3880781 10-3911661 10-3942523 Reciprocal -0009302326 •0oO9293(i80 •00(192^5051 -0(Ki927()438 •0009267841 -0009259259 •0009250694 •0009242144 -0009233610 •0009225092 •0009216590 •0009208103 •0009199632 •0009191176 -0009182736 •0009174312 •0009165903 •0009167509 •0009149131 •0009140768 •0009132420 •0009124088 •0009115770 •0009107468 •0009099181 •0009090909 •00090^2652 •0009074410 •1(009066183 •0009057971 •0009049774 •0009041591 •0009033424 •0009025271 •0009017133 •0009009009 •0009000900 •0008992806 ■0008984726 •0008976661 •0008968610 •0008960573 ■0008952561 ■0008944544 •0008936650 •U00892f<571 ■UO0S920t;o7 '00<>8912i;5b U008904720 SQUARES, CUBES, ROOTS, AND RECIPROCALS. 245 No. Square Cube Square Root Cube Root Reciprocal 1124 1263376 1420034624 33-5261092 10-3973366 •0008896797 1125 1265625 1423828125 33-5410196 10-4004192 •0008888^89 1126 1267876 1427628376 33-5559234 10-4034999 •00(18880995 1127 1270129 1431435383 33-5708206 10-4065787 •0008873114 1128 1272384 1435249152 33-5857112 10-4096557 •O00S865248 1129 1274641 1439069689 33-6005952 10-4127310 •0008857396 1130 1276900 1442897000 33-6154726 10-4158044 -0008849568 1131 1279161 1446731091 33-6303434 10-4188760 -0008841733 1132 1281424 1450571968 33-6452077 10-421945H -0UU8833922 1133 1283689 1454419637 33-6600653 10-4-250138 -0008826125 1134 1285956 1458274104 33-6749165 10-4280800 -0008818342 1135 1288225 1462135375 33-6897610 10-4311443 •0008810573 1136 1290i96 1466003456 33-7045991 10-4342069 -0U0880-2.S17 1137 1 1292769 1469878353 33-7194306 10-4372677 •0008795U75 1138 j 1295044 1473760072 33-7342556 10-4403267 -0008787.346 1139 i '297321 1477648619 33-7490741 10-4433839 -0008779631 114U 1 1299600 1481544000 33-7638860 10-4464393 -0008771930 1141 1301881 1485446221 33-7786915 10-4494929 -0008764242 1142 1304164 1489355288 33-7934905 10-4525448 •0008756567 1143 1306449 1493271207 33-8082830 10-4555948 -000874890o 1144 1308736 1497193984 33-8230691 10-4586431 -0008741259 1145 1311025 1501123625 33-8378486 10-4616896 -0008733624 1146 1313316 1505060136 33-8526218 10-4647343 -0008726003 1147 1315609 1509003523 33-.S673884 10-4677773 -0UU8718396 1148 ' 1317904 1512953792 33-8821487 10-47081«5 •0008710801 1149 1320201 1516910949 33-89690-25 10-4738579 -00U8703220 1150 1322500 1520875000 33-9116499 10-4768955 -000869.3652 1151 1324801 1524845951 33-9263909 10-4799314 •0008688097 1152 1327104 1528823808 33-9411255 10-4829656 -0008680556 1153 1329409 1532808577 33-9558537 10-4859980 -0008673027 1154 1331716 1536800264 33-9705755 10-4890286 -0008665.)il 1155 1334025 1540798875 33-9852910 10-4920575 -0008658009 1156 1336336 1544804416 34-0000000 10-4950847 -0008650519 1157 1338649 1548816893 340147027 10-4981101 -0008643042 1158 1340964 1552836312 34-0293990 10-5011337 •0008635579 1159 1343281 1556862679 34-0440890 10-5041556 •0008628128 1160 1345600 1560896000 34-0587727 10-5071757 •0008620;90 1161 1347921 1564936281 34-0734501 10-5101942 -0008613264 1162 1350244 1568983528 34-0881211 10-5132109 -0008605852 1163 1352569 1573037747 34-1027858 10-5162-259 -00085984-2 1164 1354896 1577098944 34-1174442 10-519-2391 -0008591065 1165 1357225 1581167125 34-1320963 10-5222506 -0008583691 1166 1359556 1585242296 34-1467422 10-5-252604 -0008576.329 1167 ! 1361889 1589324463 34-1613817 10-528-2685 -00085689811 1168 1364224 1593413632 34-1760150 10-5312749 •0008561644 1169 1366561 1597509809 34-19064-20 10-5342795 -0008554.3^0 1170 1368900 1601613000 34-2052627 10-5372825 •00085471 (09 1171 1371241 1605723211 34-2198773 10-5402837 •0008539710 1172 1373584 1609840448 34-2344855 10-5432832 -0008532423 246 SQUAEE?, CrBES, ROOTS, AXD RECIPROCALS. Xo. 1 Square Cube Square Root Cube Root Reciprocal 1173 1 1375929 1613964717 34-2490875 ' 10-5462«10 •0008525149 1174 ■ 1378276 1618096024 34-2636834 10-5492771 -0008517888 1175; 1380625 1622234375 34-27^2730 10-5522715 •0008510638 1176 1382976 1626379776 34-29-28564 ; 10-5552642 -000^^503401 1177! 1385329 1630532233 34-3074336 10-5582552 •0008496177 117)^I13S7684 1634691752 34-o220046 10-561-2445 -00084^8964 1179 : 1390041 1638s583o9 34-3365694 10-5642322 -0008481764 1180 1392400 1643032000 34-35112.^1 10-5672181 •0008474576 1181 1394761 1647212741 34-o656805 1 10-5702024 -0008467401 11.S2 i 1397124 1651400568 34-3802268 I 10-5731849 •0008460237 1183 1 1399489 1655595487 34-3947670 10-5761658 •0(108453085 1184 1401856 1659797504 34-4093011 i 10-5791449 •un08445946 ll.><5 1 1404225 1664006625 34-42382>^9 1.10-5821225 -0008438819 1186 ' 1406596 1668222856 34-4383507 i 0-5850983 -0008431703 1187 1 1408969 1672446203 34-452^663 10-5><80725 •00084-24600 1188 { 1411344 1676676672 34-4673759 10-5910450 •0008417508 1189 ! 1413721 1680914269 34-4818793 10-5940158 •0008410429 1190; 1416100 1685159000 34-4963766 ' 10-5969850 •0008403361 1191 141848 L 1689410871 34-5108678 i 10-5999525 •0008396306 1192 1420864 1693669888 34-5253530 1 10-6029184 •0008389-262 1193 '1423249 1697936057 34-5398321 1 10-6058826 •0008382230 1194 1425636 1702209384 34-5.543051 10-6088451 •0008375209 1195 1428025 1706489875 34-5687720 10-6118060 •0008368-201 1196 ; 1430416 1710777536 34-5s;{2329 10-6147652 •0008361204 1197 1432809 1715072373 34-597t,s79 10-6177228 •0008354219 1198 1435204 1719374392 34-6121366 10-620G7»8 •0008347245 1199 1437601 1723683599 34-6265794 10-6236331 •0008340284 1200 1440000 1728000000 34-6410162 10-6265857 •000833333'; 1201 ; 1442401 1732323601 34-6554469 10-6295367 •0008326395 1202 ' 1444804 1736654408 34-6698716 10-6324860 •000831946X ■203 1447209 1740992427 34-6842904 10-6354338 •0008312552 1204 1449616 1745337064 34-6987031 10-6383799 •000830564S 1205 1452025 1749690125 34-7131099 10-6413244 •000829^755 1206 1454436 1754049816 34-7275107 10-6442672 •0008291874 1207 1456849 175841 6713 34-7419055 10-6472085 •0008285004 1208 1459264 1762790912 34-7562944 10-6501480 •0008278146 1209 1461681 176772329 34-7706773 10-65o0>'60 •0008271-299 1210 1464100 1771561000 34-7850543 10-6560223 -0008264463 1211 1466521 1775956931 34-7994253 10-6589570 •0008257638 1212 1468944 178036012X 34-8137904 10-6618902 -0008250825 1213 1471369 1784770597 34-8281495 10-6648217 -0008244023 1214 1473796 1789188344 34-84-25028 10-6()77516 -0008-237232 1215 1476225 1793613375 34-8568501 10-6706799 -0008230453 1216 1478656 1798045696 34-^711915 10-6736066 -00082-236.^4 1217 1481089 1802485313 34-8855271 '. 10-67; 5317 -0008216927 1218 1483524 1806932232 34-8998567 10-6794552 -00082 10 lis 1 1219 1485961 18113^6459 34-9141805 10-6'<23771 -000820.3445 1220 1488400 1815X48000 34-9284984 10-6852973 •0008196721 1221 1490841 1^20;316s6i 34-:)428l04 10-6882160 -0008190008 SQUARES, CUBES, ROOTS, A^'D RECIPROCALS. 247 No. Square Cube Square Root Cube Root Reciprocal 1222 1493284 1824793048 34-9571 1(;6 10-6911.331 •0008183306 1223 1495729 1829276567 34-9714169 10-6940486 •0008176615 1221 149s 1 76 1833767424 34-9857114 10-69696-25 •0008169935 1225 1500625 1838265625 35-0000000 10-6998748 •0008163265 1226 1503076 1842771176 35-0142.S28 10-7027855 •0008156607 1227 1505529 1847284083 35•02^^5598 10-7056947 •0008149959 122'S 1507984 1851804352 35-042>!3()9 10-7086023 •0008143322 1229 15104 il 185633 19S9 35-05709; ;3 10-7I150S;} •0008136696 1280 1512900 1860867000 35-0713558 10-7144127 •0008130081 1231 1515361 1865409391 35-08a(i096 10-7173155 •0008123477 1232 1517824 1869959168 35-0998575 10-7202168 -0008116883 1233 15202^9 1874516337 35-1140997 10-7231165 •0008110300 1234 1522756 1879080904 35-1-283.361 10-7260146 •0008103728 1235 1525225 1883652875 35- 14-25668 10-7-289112 •0008097166 1236 1527696 1888232256 35-1567917 10-7318062 •O0O.S090615 1237 1530169 1892819053 35-1710108 10-7346997 •0008084074 1238 1532644 189741.3272 35-1852-242 10-7375916 •0O0S077544 1239 1535121 1902014919 35-1994318 10-7404S19 •000^071025 1240 1537600 1906624000 35-213(;o37 10-7433707 •0008064516 1241 15400'^1 1911240521 35-2278-299 10-7462579 •0008058018 1242 1542564 1915864488 35-24-20204 10-7491436 •0008051530 1243 1545049 1920495907 35-2562051 10-75-20277 •0008045052 1244 1547536 19251.347>^4 35-2703842 10-S54.9103 •0008038585 1245 1550025 1929781125 35-2845575 10-7577913 -0008032129 1246 1552516 1934434936 35-2987252 10-7606708 -0008025682 1247 1555009 1939096223 35-31-28872 10-7635488 •0008019246 1248 1557504 1943764992 35-3270435 10-7664252 -0008012821 1249 1560001 1948441249 .35-.34 11941 10-7693001 10-7721735 •0008006405 1250 1562500 1953125000 .3.5-3553391 •0008000000 1251 1565001 1957816251 35-3694784 10-7750453 •0007993605 1252 1567504 1962515008 .35-3836120 10-7779156 •0007987220 1253 1570009 1967221277 35-3977400 10-7807843 •0007980846 1254 1572516 1971935064 35-41 1S(;24 10-78.36516 •0007974482 1255 1575025 1976656375 35-4250792 10-7865173 •0007968127 1256 1577536 1981385216 35-4400903 10-7893815 •0007961783 12.57 1580049 1986121593 35-4541958 10-792-2441 -0007955449 1258 1582564 1990865512 35-4682957 10-7951053 •0007949126 1259 1585081 1995616979 35-4823900 10-7979649 •0007942812 1260 1587600 2000376000 35-4964787 10-8008230 •0007936508 1261 1590121 20051425^1 35-5105618 10-8036797 •0007930214 1262 1592644 2009916728 35-5246393 10-8065348 •0007923930 1263 1595169 201469><447 35-5387113 10-8093S84 •0007917656 1264 1597696 2019487744 35-.5527777 10-812-2404 •0007911392 1265 1600225 2024284625 35-5668385 10-8150909 •0007905138 1266 1602756 2029089096 35-5808937 10-8179400 •0007898894 1267 1605289 2033901163 35-5949434 10-8207876 •0007892660 1268 1607824 2038720832 35-6089876 10-8-236.336 •0007886435 1269 1610361 2043548109 35-62,30262 10-8264782 •0007880221 1270 1612900 2048383000 35-6370593 10-8293213 •0007874016 248 SQUAKES, CUBES, EOOTS, AND RECIPROCALS. Xo. Square Cube Square Root | Cube Root Reciprocal Xo. Square 1271 1615441 1-272 1617984 1273 . 620529 i 1274 1623076 , 1275 1625G25 1276 1628176 1277 1630729 1278 1633284 1279 1635841 1280 1638400 1281 1640961 1282 1643524 1283 1646089 1284 1648656 1285 1651225 1286 1653796 12^7 1656369 1288 1658944 1289 1661521 1290 1664100 1231 1666681 1292 1669264 1293 1671849 1294 1674436 1295 1677025 1296 1679616 1297 1682209 1298 1684804 1299 1687401 1300 1690000 1301 1692601 1302 1695204 1303 1697809 1304 1700416 1305 1703025 1306 1 705636 1307 1708249 1308 1710864 1309 1713481 13.0 1716100 1311 1718721 1312 1721344 1313 1723969 1314 1726596 1315 1 729225 1316 1731856 1317 1734489 1318 1737124 1319 1739761 2053225511 2058075648 2062933417 206779««24 2072671875 2077552576 2082440933 2087336952 2092240639 2097152000 2102071041 2106997768 2111932187 2116874304 2121824125 2126781656 2131746903 2136719872 2141700569 2146689000 2151685171 2156689088 2:61700757 2166720184 2171747375 2176782336 218182.")073 2186875592 2191933899 2197000000 2202073901 2207155608 2212245127 2217342464 2222447625 22275606 1 6 2232681443 2237810112 2242946629 2248091000 2253243231 2258403328 2263571297 226.S747144 2273930875 2279122496 2284322013 2289529432 2294744709 35-6510869 35-6651090 35-6791255 35-6931366 35-7071421 35-7211422 35-7351367 35-7491258 35-7631095 35-77708/6 35-7910603 35-8050276 35-8189894 35-8329457 35-8468966 35-8608421 35-8747822 o5-«887169 35-9026461 35-9165699 35-9304884 35-9444015 35-9583092 35-9722115 35-9861084 36-0000000 36-0138862 36-0277671 36-0416426 36-0555128 36-0693776 36-0832371 36-0970913 36-1109402 30-1247837 36-13862-20 36-1524550 36-1662826 36-1801050 36-1939221 36-2077340 36-2215406 36-2353419 36-2491379 36-2629287 36-2767143 36-2904946 36-3042697 36-3180396 10-8321629 10-8850030 10-8378416 10-8406788 10-8435144 10-8463485 10-8491812 10-85-20125 10-8548422 10-8576704 10-860^972 10-8633225 10-8661464 10-8689687 10-8717897 10-8746091 ] 0-8774271 10-880-2436 10-8830587 10-8858723 10-8886845 10-8914952 10-8943044 10-8971123 10-8999186 10-9027235 10-9055269 10-9083290 10-9111296 10-9 139287 10-9167-265 10-9195228 10-9-223177 10-9-251111 10-9279031 10-9306937 10-9334829 10-9362706 10-9390569 10-9418418 10-9446-253 10-9474074 10-9501880 10-9529673 10-9557451 10-9585215 10-9612965 10-9640701 10-96684-23 -0007867821 -0007861635 -OUO78554U0 -0007849294 -0007843137 •00078369^11 -0007830854 -0007824726 •0007818608 -0007812500 -0007806401 -0007800312 -0007794232 -0007788162 •ti007782101 -0007776050 -0007770008 •0007763975 •0007757952 -00(j7751938 •0007745933 •0007739938 •0007733952 •0007727975 •000772-2008 •0007716049 •0007710100 •0007704160 •0007698229 -0007(;92308 -0007686395 •0007680492 •0007674597 •0007668712 •000766-2835 -0007656968 •0007651109 •0007645260 •0007639419 •0007633588 •0007627765 -0007621961 -0007616146 -0007610350 -0007604563 •0007598784 •0007593014 •0007587253 •0007581501 SQUARES, CUBES, ROOTS, AND RECIPEOCALS. 249 No. Square \ Cube Square Eoot ' Cube Root Reciprocal 1320 1742400 j 2299968000 36-3318042 '■ 10-9696131 •0007575758 1321 1745041 , 2305199161 36-3455637 10-9723825 -0007570023 1322 1747684 ; 2310438248 36-3593179 10-9751505 -0007564297 1323 1750329 ' 2315685267 36-3730670 10-9779171 •000755^579 1324 1752976 I 2320940224 36-3868108 10-9806823 -0007552870 1325 1755625 1 2326203125 36-4005494 10-9834462 -0007547170 1326 1758276 2331473976 36-4142829 10-9862086 •0007541478 1327 1760929 j 2336752783 36-42S(ill2 10-9889096 •0007535795 1328 1763584 ' 2342039552 36-4417343 10-9917293 •0007530120 1329 1766241 , 2347334289 S6-45645-23 10-9944876 •0007524454 1330 1768900 2352637UU0 36-4691650 10-9972445 -0007518797 1331 1771561 2357947691 36-4828727 11-0000000 -0007513148 1332 1774224 2363266368 36-4965752 11-0027541 •0007507508 1333 1776889 2368593(137 36-6102725 11-0055069 •0007501875 1334 1779556 2373927704 36-5-239647 11-008-2683 -0007496252 1335 1782225 2379270375 36-5376518 11-0110082 -0007490637 1336 1784896 2384621056 36-5513338 11-0137569 •0007485(130 1337 1787569 2389979753 36-5650106 11-0165041 •0007479432 1338 179C^244 2395346472 36-5786823 11-019-2500 -0007473842 1339 1792921 2400721219 36-5923489 11-0219945 -0007468260 1340 1795600 2406104000 30-6060104 11-0-247377 -0('074(:2687 1341 1798281 2411494821 36-6196668 11-0274795 -0007457122 1342 1800964 2416893688 36-6333181 11-0302199 -0007451565 1343 1803649 2422300607 36-6469644 11-0329590 •0007446016 1344 1806336 2427715584 36-6606056 11-0356967 •000744047t; 1345 1809025 2433138(125 36-6742416 110384330 -0007434944 1346 1811716 2438569736 36-6878726 11-0411680 •0007429421 1347 1814409 2444008923 36-7014986 11-0439017 -00074239(15 1348 1817104 2449456192 36-7151195 11 •0466339 -0007418398 1349 1819801 24549115-^9 36-7287353 110493649 •0007412898 1350 1822500 2460375000 36-7423461 11-0520945 -0007407407 1351 1825201 2465846551 36-7559519 11-0548227 -00074019-24 1352 1827904 2471326208 3 -7695526 11-0575497 •0007396450 1353 1830609 2476.S13977 36-7831483 ll-06n2752 i -0007390983 1354 1833316 2482309864 36-7967390 11-0629994 i •00073,^5524 1355 1836025 2487813875 36-8103-246 11-0657222 •000738( 074 1356 1838736 2493326016 36-8239053 11-0684437 •0007374631 1357 1841449 2498846293 36-8374809 11-0711689 •0007369197 1358 1844164 2504374712 36-8510515 11-07388-28 •0007363770 1359 1846881 2509911279 36-8646172 n -0766003 -0007358352 1360 1849600 2516456000 36-8781778 11-0793165 -0007352941 1361 1852321 2521008881 36-8917335 ll-0^;-20314 •0007347539 1362 1855044 2626569928 36-9052842 11-0^47449 •0007342144 1363 1857769 2532139147 36-91 8S299 11-0874571 •0007336757 1364 1860496 2537716544 36-93-23706 11-0901679 •0007331378 1365 1863225 2543302125 36-9459064 11-0928775 -000732*. 007 1366 1865956 2548895896 36-9594372 11-0955857 -0007320644 1367 1868689 2554497863 36-9729631 11-0982926 -0007315-289 1368 i 1871424 1 2560108032 36-9864840 11-1009982 •0007309942 250 SQUARES, CUBES, ROOTS, AXD RECIPROCALS. No. Square Cube Sqiiare Root Cube Pvoot Reciprocal 13G9 1874161 2565726409 37-0000000 11-1037025 -0007304602 1370 1876900 2571353000 37-0135110 11-10(54054 •O0O729927O 1871 1879fi41 257(5987811 37-0270172 11-1091070 -0007293946 1372 1882384 25S2(53084X 37-04051x4 11 -1118073 -0007288630 1373 1885129 258«2s2n7 37-0540146 11-1145064 -00072X3321 1374 1887876 2593941624 37-0(;750":0 11-1172041 •0007278020 1375 1890625 2599(509375 37-0809924 11-1199004 •0007272727 1376 1893376 2(50.i2s5376 37-0944740 11-1 -225955 -0007267442 1377 1896129 2610069(533 37-1079506 11-125-2X93 -0007262164 1378 1898884 26i(;(;(;2i52 37-1214224 11-1279X17 -0007256894 1379 1901641 26223<;2939 37-1 34X893 11-13067-29 •0007-251632 1380 1904400 262s( 172000 37-1483512 11-1333628 •0007-246377 1381 1907161 2G337S9341 37-1618084 11- 13600 14 -0007-241130 1382 1909924 26395 149!58 37-1752606 11-1387386 •0007-235890 1383 1912689 2645248887 37-1887079 11-1414-246 •0007-230658 1384 1915456 2650991104 37-2021505 11-1441093 •0007225434 1385 1918225 2G5(5741(;25 37-2155881 11-1467926 •0007220217 138G 1920996 2662500456 37-2290209 37-24244X9 37-255X720 11-1494747 •0007215007 1387 1923769 26(5s2676o3 11-1521555 •0007209805 1388 1926544 2674043072 11-154X350 •0007204611 I 1389 1929321 2679826869 37-2692903 11-1575133 -00071994-24 I 1390 1932100 2:185619000 2(591419471 37-2X27037 11-1601903 -0007194-245 1391 1934881 37-2961124 11-1(528659 -0007189073 1392 1937664 2-597228288 37-3095162 11-1655403 •0007183908 1393 1940449 2703045457 37-3229152 11-16x2134 •0007178751 1394 1943236 270SS70084 37-3363094 n-170XX.52 -0007173601 1395 1946025 27! 4704875 37-34969X8 11-1735558 -0007168459 13913 1918816 2720547136 37-3630834 11-1762250 •0007163324 1397 1951(109 27215397773 37-3764632 11-1788930 •0007158196 1308 1954404 273225(5792 37-389X3X2 11-1815598 •0007153076 1399 1957201 273S121199 37-40320x4 11-1X42-252 •0007147963 1400 1960(100 2744000000 37-41(55738 11-18(58X94 •000714-2857 1401 1962.S01 2749.XS4201 37-4299345 11 ■18955-23 •0(^07137759 1402 19(55604 2755776808 37-4432904 11-1922139 -0007132668 1403 1968409 2761677X27 37-456(5416 11-194X743 •0O07 127584 1404 1971216 27675872' 14 37-4699880 11-1975334 0007122507 1405 1974025 2773505125 37-4X33296 11-2001913 •0007117438 140(1 1976836 2779431416 37-49666(55 11-20-28479 •000711-2376 1407 1979649 278536(5143 37-n099987 11-2055032 •0007107321 i40.S 19824(54 279I3(J9312 37-r)2332(il 11-20X1573 •0007102273 1409 1985281 2797260929 37-536(5487 11-2108101 -0007097232 1410 1988100 280322 lOoO 37-5499(567 11-2134617 •0007092 99 1411 1990921 2.S09189531 37-5(^32799 11-21(51 1-20 •00070x7172 1412 1993744 2.S 1 5 1 66528 37-5765X8.) 11-2187611 -0(»07ox-2i5;; 1413 i 996569 2S21 151997 37-5X98922 11-221 4ns9 -0007077! ! i 1 1414 199939G 2S27 145944 37-(5031913 11-2240554 •0007072 '3r, 1415 2002225 283314H375 37-6164x57 11-22(57007 •0007(16, '3x 1 141(5 2005056 2.S39 159296 37-6297754 11-2293448 •0007OH21 17 1117 2007889 2.S4517.S713 37-6430(504 ll-231i)X76 -00070571(53 SQUARES, CUBES, ROOTS, AND RECIPROCALS. 251 No. Square Cube Square Root Cube Root Reciprocal Ul^7563 •0006261741 1598 2553604 40i5O659192 39-9749922 11-6911955 -0006-2.57822 1599 255680 1 408.S324799 39-9-^74980 11-693(3337 •(3006253909 1600 2560000 4096000000 40-0000000 11-6960709 -0006-250000 1601 2563201 41036X4801 40-01-24980 11-6985071 -0006246096 1602 2566404 4111379208 40-02499-22 11-7009422 -0006242197 16o3 2569609 41190832-7 40-03 74«-24 11-7033764 -0006238303 1604 2572816 4126796864 40-0499688 11-7058095 -00062.34414 1605 2576025 4134520125 4(1-0624512 11-7082417 -0006230530 16o6 2579236 4142253016 400749298 11-71067-28 -00062266.50 1607 2582449 414!)99->543 40-0.«!74O45 11-7131029 -0006222775 1608 2585664 4157747712 40-0998753 11-71553-20 •0006218905 1609 2588881 4165509529 40-11 •234-23 11-7179(301 -000621504O 1610 2592100 4173281000 40-1-248053 11-7203S72 •0006211180 1611 2595321 4181062131 40-137-2645 11-7228133 •0006-207325 1612 2598544 418885292.S 40-1497198 11 -7-25 -.'384 •0006203474 16i3 2601769 4196653397 40-1621713 11-7276625 •000619962« SQUARES, CUBES, BOOTS, AND EECIPROCALS. 255 No. 1G14 Square | Cube Square Root Cube Root Reciprocal 2604996 4204463544 40-1746188 11-7300855 -0006195787 1G15 26082-25 . 421-2283375 40-1870626 ; 11-73-25076 -0006191950 1616 2611456 42-20112896 40-1995025 11-734 92.St; •00061^8119 1617 26146«9 4227952113 40-2119385 j 11-7373487 -0-06184292 1618 2617924 4235801032 40-2243707 11-7397677 -0006180470 1619 ■2621161 4243659659 40-2367990 11-7421 S58 -1/006176652 16-20' 2624400 4251528000 40-2492-236 11-74460-29 -0006172840 16-21 ' -2627641 4259406061 40-2616443 11-7470190 -0006169031 16-22 ; 26308S4 i 4-267293848 40-2740611 11-7494341 -0006165228 162o 2634129 j 4-275191367 40-2864742 11-7518482 •00061614-29 16-24 26373 < 6 4283098624 40-2988834 11-754-2613 -0006157635 hvi:, 2640(525 i 4291015625 40-3112888 11-7566734 -0006153846 1626 2643.H76 4-29894-2376 40-3236903 11-7590846 -0006150062 1627 2647129 43068788.S3 40-3360881 11-7614947 •0006146282 16-2.S 2650384 4314825152 40-34848-20 11-7639039 -0006142506 1629 26536-1 1 43-2-278 11N9 40-3608721 11-7663121 •0006138735 1630 26569n0' 4330747000 40-3732585 11-7687193 •0006134969 1631 2660161 43387-2-2591 40-3856410 11-7711255 -0006131-208 1632 266,!4-2.4 4346707968 40-3980198 11-7735306 •0006127451 1633 2666689 4354703137 40-4103947 11-7759349 •00061-23609 1634 2669956 436270H104 40-4-227658 11-77833«1 •0006119951 163o 2673225 4370722875 40-4351332 11-7807404 •0006116208 16.>6 2676496 4378747456 40-4474968 11-7831417 •0006112469 1637 2679769 43S6781853 40-4598566 11-7^55420 •0006108735 1638 2683044 43948-26072 40-4722127 11-7879414 •0006105006 1639 2686321 440-2880119 40-4845649 11-7903397 •00061 012M 1640 2689600 4410944000 40-4969135 11-7927371 •0006097561 1641 26928S1 4419017721 40-5092582 11-7951335 ■0006093845 1642 2696164 4427101288 40-5215992 11-7975289 •1)006090134 1643 2699449 4435194707 40-5339364 11-7999-234 •0006086427 1644 270273G 4443297984 40-546-2699 11-8023] 69 •0006082725 1645 27060-25 4451411125 40-5585996 11-8047094 •0000079027 1646 2709316 4459534136 40-570,.i-255 11-8071010 •0t)06075;.34 . 1647 2712(;09 44676670-23 40-583-2477 11-8094916 •0006071645 1648 2715904 4475809792 40-5955663 11-8118812 •0006067961 1649 2719-201 4483962449 40-6078810 11-8142698 •0006064281 1650 27-22500 4492125000 40-62019-20 11-8166576 •0006060606 1651 27-25801 4500297451 40-6324993 11-8190443 •0006056935 165-2 2729104 4508479808 40'64480-29 11-8-214301 •0006053-269 1653 273-2409 4516672077 40-65710-27 11-8238149 •0006049607 1654 2735716 4524874-264 40-6693988 11-8261987 -0006045949 1655 2739025 4533086375 40-6816912 11-8285816 •0006042296 1656 274-2336 4541308416 40-6939799 11-8309634 •0006038647 1657 2745649 ' 4549540393 40-7062648 11-8333444 •0006035003 1658 2748964 4557782312 40-7185461 11-8357244 •0006031363 1659 275-2281 ! 4566034179 40-7308-237 11-8381034 •0006027728 1660 2755600 i 4574296000 40-7430976 11-8404815 •0006024096 1661 2758921 j 45»-2567781 40-7553677 11-84-285.N6 •0006020470 1662 ! 2762244 1 4590849528 1 40-7676342 11-845-2348 •0006016847 256 SQUAPwES, CUBES, ROOTS, AND RECIPROCALS. No. 1663 Square Cube Square Root Cube Root Reciprocal 2765569 4599141247 40-7798970 11-8476100 -0006013229 16G-4 2768896 4607442994 40-7921561 11-8499843 •0006009615 1665 2772225 4615754G25 40-8044115 11-8523576 •0006006006 1666 27/5556 462407629(^. 40-8166633 11-8547299 •000600-24 01 1667 2778889 4632407963 40-8289113 11-8571014 •0O0599'^8O0 1668 2782224 4(540749632 40-8411557 11-8594719 •00O5995204 1669 2785561 4649101309 40-8533964 11-8618414 •0005991612 1670 2788900 4657463000 40-8656335 11-8642100 •0005988024 1671 2792241 4665834711 40-^77S669 11-8665776 •0005984440 1672 2795584 4674216448 40-8900966 1 1-8689443 •0005980861 1673 2798929 4682608217 40-9023227 11-8713100 •0005977286 1674 2802276 4691010024 40-9145451 11-8736748 •0005973716 1675 2805625 4699421875 40-9267638 11-8760387 •0005970149 1676 2808976 4707843776 40-9389790 11-8784016 •0005966587 1677 2812329 4716275733 40-9511905 11-8807636 •0005963029 1678 2815684 4724717752 40-9633983 11-8831246 •0005959476 1679 2819041 4733169839 40-9756025 11-8854847 •0005!)55926 168u 28224O0 474163200O 40-9878031 11-8878439 •0005 95-2381 1681 282576 1 4750104241 41-0000000 11-890-2022 •0005948840 1(82 2829124 4758586568 41-0121933 11-8925595 •0005945303 1683 2832489 ^4767078987 41-0243830 11-8949159 -0005941771 1684 2835856 4775581504 41-0365691 11-8972713 •0005938-242 16.^5 2839225 4784094125 41-0487515 ll-899';258 •0005934718 168t; 2842596 4792616856 41-0609303 11-9019793 ■000593)198 1687 28459o9 4801149703 4] -0731 055 11-9043319 •0005927682 1688 2849344 4809692672 41-0852772 11-9066836 ■00059-24171 1689 2852721 4818245769 41-0974452 11-9090344 -0005920663 1690 2856100 4826809000 41-1096096 11-9113843 -0005917160 1691 2859481 4835382371 41-1217704 11-9137332 •0005913661 1692 2862864 4843965888 41-1339276 11-9160812 -0005910165 1693 2866249 4852559557 41-1460812 11-9184283 •00059o6(w5 1694 2869636 48611(^338; 41-1582313 11-9207744 ■0005903188 1690 2873025 4869777375 41-1703777 11-9231196 -0O0.'8'.»97(i5 1696 2876416 487-^401536 41-1825206 11-9-254639 •0005896226 1697 2879.S09 4887035873 41-1946599 11 -9-278073 -0005892752 1698 2883204 48956.-«i0392 41 -2067956 11-9301497 -00O58S9282 1699 2886601 4901335099 41-2189-277 11-93-24913 -0005885815 1700 2890000 4913000000 41-23 0563 11-9348319 -0005882353 1701 2893401 4921675101 41-24318 2 11-9371; 16 ■0005878895 1702 2S96804 493036040S 41-2553027 11-9395104 •0005875441 1703 2900209 4939055927 41-2674205 11-9418482 •0005871991 1704 2903616 4947761664 41 -2795349 11-9441852 •0005868545 1 705 2907025 4956)477625 41-2916456 11-9465213 ■0005S(:5103 1706 2910436 4965203816 41-3037529 11-9488564 •000580 665 1707 2913S49, 4973940243 41-3158565 11-9511906 •00058.- 8-231 170S 2! "17264 49826S691 2 41-3-279566 11-9535-239 -0005854 01 1709 29201 ;81 4991443S29 41-3400532 11-9558563 •0005851375 1710 2921100 5000211000 41-3521463 11-9581878 •0005847953 1711 21)27521 500^988431 41-364-2358 11-9605184 0005844535 SQUARES, CUBES, ROOTS, AND RECIPROCALS. 257 No. Square Cube Square Root Cube Root Reciprocal 1712 2930944 5017776128 41-3763217 11-9628481 -0005841121 1713 2934369 5026574097 41-3884042 11-9651768 -0005837712 1714 2937796 50353^2344 41-4004S31 11-9675047 •000583-1306 1715 2941225 5044200875 41-4125585 11-9698317 •0(105830904 1716 2944656 5053029696 41-4246304 11-9721577 •0005827506 1717 2948089 5061868813 41-4366987 11-9744829 •0005824112 1718 2951524 6070718232 41-4487636 11-9768071 -0005820722 1719 2954961 5079577959 41-4608-249 11-9791304 -0005S 17336 1720 2958400 6088448000 41-47-28827 11-9814528 -0005813953 1721 2961841 6097328361 41-4849370 11-9S37744 -0005810575 1722 2965284 5106219048 41-4969878 11-9860950 -0005807-201 1723 2968729 5115120067 41-5090o51 11-9884148 •0005803831 1724 2972176 5124031424 41-5210790 11-9907336 •0005800464 1725 2975625 5132953125 41-5331193 11-9930516 •0005797101 1726 2979076 5141885176 41-5451561 11-9953686 •0005793743 1727 2982529 6150827583 41-5571895 11-9976848 •0005790388 1728 2985984 5159780352 41-6692194 12-0000000 -0005787037 1729 2989441 6168743489 41-6812457 12-00-23144 -0005783690 1730 2992900 6177717000 41-5932686 12-0046278 -0005780847 1731 2996361 6186700891 41-6052^sl 12-0069404 •0005777008 1732 2999824 5195695168 41-6173041 12-0092521 •0005773672 1733 3003289 5204699837 41-6-293160 12-0115629 -0005770340 1734 3006756 5213714904 41-6413256 12-01387-28 -0005767013 1735 3010225 5222740375 41-6533312 12-0161818 •0005763689 1736 3013696 5231776256 41 •6653333 12-0184900 •0005760369 1737 301 7169 5240>«22553 41-6773319 12-0207973 -0005757052 1738 3020644 5249879272 41-6.^93271 12-0231037 -0005753740 1739 3024121 5258946419 41-7013189 12-0254092 •0005750431 1740 3027600 5268024000 41-7133072 12-0277138 •0005747126 1741 303lO;si 6277112021 41-7252921 12-0300175 -0005743825 1742 3034564 5286210488 41-7372735 12-03-23204 -0005740528 1743 3038049 6295319407 41-7492515 12-0346223 -0005737235 1744 3041536 530443>!784 41-761-2260 12-0369233 -0005733945 1745 3045025 5313568625 41-7731971 12-0392235 -0005730659 1746 3048516 5322708936 41-7851648 12-04152-29 -0005727377 1747 3052009 5331859723 41-7971291 12-0438213 •00057-24098 1748 3055504 5341020992 41-8090899 12-0461189 •00057208-24 1749 3059001 5350192749 41-8210473 12-0484156 •0005717553 1750 3062509 6359375000 41-^330013 1-2-0507114 •0005714286 1761 3066001 6368567751 41-8449519 12-0530063 •0005711022 1752 3069504 5377771008 41-8568991 12-0553003 •0005707763 1753 3073009 5386984777 41-8688428 12-0575935 ■0005704507 1754 3076516 6396209064 41-8807832 12-0598859 •0005701-254 1755 3080025 5405443875 41-8927-201 12-0621773 -0005698006 1756 3083536 5414689216 41-9046537 12-0644679 -0005694761 1757 3087049 5423945093 41-916583.S 12-0667576 -0005691520 1758 3090564 5433211512 41-9285106 12-0690464 •0O056S8282 1759 3094081 5442488479 41-9404339 12-0713344 -0005685048 1760 3097600 6451776000 41-9523539 12-0736215 •0005681818 258 SQUAT^ES, CUBES, EOOTS, AND EECIPEOCAL?. No. 1761 Square Cube Square Root Cube Root Reciprocal 3101121 5461074081 41-9642705 12-0759077 -0005678592 1762 3104644 5470382728 41-9761837 12-0781930 -0005675369 17G3 3108169 5479701947 41-988093.) 1-2-0804775 -0005672150 17Gi 3111696 5489031744 42-0000000 12-0827612 -0005668934 1765 3115225 5498372125 42-0119031 12-0850439 •00U5665722 1766 3118756 5507723096 42-0238028 12-0873-258 -0005662514 1767 3122289 5517084663 42-0356991 12-0896069 •0005659310 1768 3125824 5526456832 42-0475921 12-0918870 •0005656109 1769 3129361 5535839609 42-0594S17 12-0941664 •0005652911 1770 3132900 5545233000 42-0713679 12-0964449 •0005649718 1771 3136441 5554637011 42-083-2508 12-0987226 •0005646527 1772 3139984 5564051648 4-2-0951304 12-1009993 •0005643341 1773 3143529 5573476917 42-1070065 12-1032753 •0005640158 1774 3147076 5582912824 42-1188794 1-2-1055503 •0005636979 1775 3150625 5592359375 42-1307488 12-1078-245 -0005633803 1776 3154176 5601816576 42-1426150 12-1100979 •0005630631 1777 3157729 5611284433 42-1544778 12-11-23704 -0005627462 1778 3161284 5620762952 42-1663373 1-2-11464-20 •00056-24-297 1779 3164841 5630252139 42-1781934 12-1169128 •0005(521135 1780 3168400 5639752000 42-1900462 l-2-1191«-27 •0005617978 1781 3171961 5649262541 42-2018957 1-2-1214518 •0005614823 1782 3175524 5658783768 42-2137418 12-1-237-200 •0005611672 1783 3179089 5668315687 42-2255846 12-1-259874 •0005608525 1784 3182656 5677858304 42-2374-242 1-2-1-28-2539 •0005605381 1785 3186225 5687411625 42-2492603 1-2-1305197 -0005602-241 1786 3189796 5696975656 42-2610932 12-1327845 -0005599104 1787 31933 •9 5706550403 42-2729227 12-1350485 •0005595971 1788 3196944 5716135872 42-2847490 12-1373117 •000559-2841 1789 3200521 5725732069 4-2-2905719 1-2-1395740 •0005589715 17 JO 3204100 5735339000 42-3083916 12-1418355 •0005586592 1791 3207681 6744956671 42-3-20-2079 12-1440961 -0005583473 1792 3211264 57545S508S 42-33-20210 12-1463559 •0005580357 1793 3214849 5764224257 42-3438307 12-1486148 •0005577-245 1794 3218436 5773874184 42-3556371 12-1508729 •0005574136 1795 3222025 5783534875 4-2-3674403 12-1531302 •0005571031 1796 3225616 5793206336 42-379-2402 12-1553866 •0005567929 1797 3229209 5802888573 4-2-3910368 12-1576422 •0005564830 1798 3232804 5812581592 42-4028301 12-1598970 •0005561735 1799 3236401 5822285399 42-4146-201 12-1621509 •0005558644 1800 3240000 6832000000 42-4264069 12-1644040 •0005555556 1801 3243601 5841725401 42-4381903 12-1666562 •0005552471 1802 3247204 6851461608 42-4499705 12-1689076 •0005549390 1803 3250809 5861208627 42-4617475 1-2-1711582 •0005.546312 1804 3254416 5870966464 42-4735212 12-1734079 •0005543237 1805 3258025 6880735125 42-485-2916 12-1756569 -0005.540166 1806 3261636 5890514616 42-4970587 12-1779050 •000.5537099 1807 3265249 6900304943 42-5088226 12-1801522 •00055.34034 1808 3268864 5910106112 4-2-5205833 12-1823987 •0005530973 1809 3272481 59199181:^9 42-5323406 12-1846443 -0005527916 SQUARES, CUBES, EOOTS, AND RECIPEOCALS. 259 No. Square 1810 3276100 1811 3279721 1812 3283344 1813 32869G9 1814 3290596 1815 3294225 1816 3297856 1817 3301489 1818 3305124 1819 3308761 18-20 3312400 1821 3316041 1822 3319684 1823 3323329 1824 3326976 1825 3330625 1826 33342/6 1827 3337929 1828 3341584 1829 3345241 1830 3348900 1831 3352561 1832 3356224 1833 3359889 183-1 3363556 1835 3367225 1836 3370896 1837 3374569 1838 3378244 1839 3381921 1810 3385600 1811 3389281 1812 3392964 1843 3366649 1844 3400336 1845 3404025 1846 3407716 1847 3411409 1848 3415104 1849 341«801 1850 3422500 1851 3426201 1852 3429904 1853 3433609 1851 3437316 1855 3441025 1856 3444736 1857 3448449 1858 3452164 Cube 5929741000 6939574731 5949419328 5959274797 5969141144 5979018375 5988906496 5998805513 6008715432 6018636259 6028568000 6038510661 6048464248 6058428767 6068404224 6078390625 6088387976 6098396283 6108415552 6118445789 6128487000 6138539191 6148602368 6158676537 6168761704 6178857875 6188965056 6199083253 6209212472 6219352719 622950400e 6-39666321 6249839688 6260024107 6270219584 6280426125 6290643736 6300872423 6311112192 6321363049 6331625000 6341898051 6352182208 6362477477 6372783864 6383101375 6393430016 6403769793 6414120712 Square Root 42-5440948 42-5558456 42-5675933 42-5793377 42-5910789 42-6028168 42-6145515 42-626-2829 42-6380112 42-6497362 42-6614580 42-6731766 42-6848919 42-6966040 42-7083130 42-7200187 42-7317212 42-7434206 42-7551167 42-7668095 42-7784992 42-7901858 42-8018691 42-8135492 42-8252262 42-8368999 42-8485706 42-^(602380 42-8719022 42-8835633 42-8952212 42-9068759 42-9185275 42-9301759 42-9418211 42-9534632 42-9651021 42-9767379 42-9883705 43-0000000 43-0116263 43-023-2495 43-0348696 43-0464865 43-0581003 43-0697109 43-0813185 43-0929228 43-1045241 S2. Cube Root 12-1868891 12-1891331 1-2-1913762 12-1936185 1-2-1958599 12-1981006 12-2003404 12-2025794 12-2048176 12-2070549 12-209-2915 12-2115-272 12-21376-21 12-2159962 12-218-2295 12-2204620 12-2226936 12-2249244 12-2271544 12-2293836 12-2316120 12-2338396 12-2360663 12-2382923 12-2405174 12-2427418 12-2449653 12-2471880 12-2494099 12-2516310 12-2538513 12-2560708 12-258-2895 12-2605074 12-2627245 12-2649408 12-2671563 12-2693710 12-2715849 12-2737980 12-2760103 12-2 78-22 1 8 12-2804325 12-2826424 12-2848515 12-2^70598 12-2892673 12-2914740 12-2936800 Reciprocal 00055-24862 •0005521811 -0005518764 -00055157-20 -0005512679 •0005509642 •0005506608 •0005503577 •0005500550 •0005497526 •0005494505 •0005491488 •0005488474 •0005485464 •000548-2456 -0005479452 -0005476451 •0005473454 •0005470460 •0005467469 •0005464481 •0005461496 •0005458515 -0005455537 •0005452563 •0005449691 •0005446623 •0005443658 •0005440(]S6 •0005437738 •0005434783 •0005431831 •0005428882 •0005425936 •0005422993 •0005420054 •0005417118 ■0005414185 •0005411255 •0005408329 •0005405405 ■000540-2485 •0005399568 •0005396654 •0005393743 •0005390836 •0005387931 •0005385030 •0005382131 260 SQUARES, CUBES, ROOTS, AND RECIPROCALS, Xo. Sqnare Cube Square Root Cube Root Reciprocal 1859 3455881 6424482779 43-1161223 12-2958851 -0005379-236 18»i0 3459600 6434856000 43-1277173 12-2980S95 -0005376344 1861 34633-21 6445240381 43-1393092 12-3002930 -0005373455 1862 3467044 6455635928 431508980 12-3024958 -00053.0569 1863 3470769 6466042647 43-1624837 12-3046978 •0005367687 1864 3474496 6476460544 43-1710663 12-306N990 -0005364807 1865 3478225 6486889->25 43-1856458 12-3o9u994 -0005361930 1866 3481956 6497329896 43-197-2221 12-3112991 -0005359057 1867 3485689 6507781363 43-2087954 12-3134979 •0005356186 1868 3489424 6518244032 43-2203656 12-3156959 -0005353319 1869 3493161 6528717909 43-2319326 12-3178932 •0005350455 1870 3496900 6539203000 43-2434966 12-3200897 -0005347594 1871 3500641 6549699311 43-2550575 12-3-222854 ■0005344735 1872 3504384 6560206848 43-2666153 12-3244803 -0005341880 1873 3508129 6570725617 43-2781700 12-3261744 •0005339028 1874 3511876 6581255624 43-2897216 12-3288678 •0005336179 1875 3515625 6591796.^75 43-3012702 12-3310604 -0005333333 1876 3519376 6602349376 43-3128157 1-2-3332522 -0005330490 1877 3523129 6612913133 43-3243580 12-3354432 •0005327651 1873 3526884 6623488152 43-o358973 1-2-3376334 -0005324814 1879 3530641 6634074439 43-3474336 1-2-3398229 •0005321980 1880 3534400 6644672000 43-3589668 12-3420116 -0005319149 1881 3538161 6655280841 43-3704969 12-3441995 -0005316321 1882 3541924 6665900968 43-3820239 12-3463866 -0005313496 1883 3545689 6676532387 43-3935479 12-3485730 -0005310674 1884 3549456 6687175104 43-4050688 12-3507586 -0005307856 1885 3553225 6697829125 43-4165867 12-3529434 -0005305040 1886 3556996 6708494456 43-4281015 12-3551274 -000531)2-227 1887 3560769 6719171103 43-4396132 12-3573107 -0005299417 1888 3564544 67298.-9072 43-4511220 12-3594932 -0005296610 18><9 3568321 6740558369 43-4626276 12-3616749 -0005293806 1890 3572100 6751269000 43-4741302 12-3638559 -0005291005 1891 3575881 6761990971 43-4856298 12-36(;0361 -0005288207 1892 3579664 6772724288 43-4971263 12-3682155 •0005285412 1893 3583449 6783468957 43-5086198 12-3703941 -0005282620 1894 3587236 6794224984 43-5-201103 12-3725721 •0005279831 1895 3591025 6804992375 43-53159/7 12-3747492 -0005277045 1896 359-1816 6815771136 43-5430821 12-3769-255 •0005274262 1X97 3598609 6826561273 43-5545635 12-3791011 -0005271481 1898 3602404 6837362792 43-5660418 12-3812759 -0005268704 1899 3606201 6848175699 43-5775171 12-3834500 -0005265929 1900 3610000 6859000000 43-5889894 12-3-^56233 •0005263158 1901 3613801 6869835701 43-6004587 12-3877959 -0005260389 1902 3617604 6880682808 43-61 19-249 12-3899676 -0()05-257624 1903 3621409 6891541327 43-6-23o882 1-2-39213x6 -0005254861 1904 3625216 6902411264 43-634^^85 12-3943(1X9 •0005252101 1905 3629025 6913292625 43-6463057 12-39647x4 -0005-249344 1906 3632836 6924lJ<5416 43-6577599 12-3986471 •0005246590 1907 3636649 6935089043 43-6692111 12-4008151 •0005-243838 SQUARES, CUBES, ROOTS, AXD RECTPROCALS. 2G1 No. 1 Square Cube Square Root Cube Root Reciprocal 1908 3640464 6946005312 43-6806593 12-4029823 -0005241090 1909 3644281 6956932429 43-6921045 1-2-4051488 -0005238345 1910 3648100 6967871000 43-7035467 12-4073145 -0005-235602 1911 3651921 6978821031 43-7149860 12-4094794 -0005-232862 1912 3655744 6989782528 43-7264222 12-4116436 -0005-230126 1913 3659569 7000755497 43-7378554 12-413S070 -0005227392 1914 3663396 7011739944 43-7492857 12-4159697 •0005224660 1915 3667225 7022735875 43-7607129 12-4181316 -0005221932 1916 3671056 7033743296 43-7721373 1-2-4202928 -0005219207 1917 3674889 7044762213 43-7835585 12-42-24533 -0005216484 1918 3678724 7055792632 43-7949768 12-4246129 -00U5213764 1919 3682561 7066834559 43-8068922 12-4267719 -0005211047 1920 3686400 7077888000 43-8178046 12-4289300 -0005208333 1921 3690241 7088952961 43-8292140 12-4310875 -0005205622 1922 3694084 7100029448 43-8406204 12-4332441 -0005-202914 1923 3697929 7111117467 43-8520239 12-4354001 -0005200208 1924 3701776 7122217024 43-8634244 12-4375552 -0005197505 1925 3705625 7133328125 43-8748219 12-4397097 -OoOo 194805 192G 3709476 7144450776 43-8862165 12-4418634 -0005192108 1927 3713329 7155584983 43-8976081 12-4440163 -0005189414 1928 3717184 7166730752 43-9089968 12-4461685 -0005186722 1929 3721041 7177888089 43-92037-25 12-4483-200 -0005184033 1930 3724900 7189057000 43-9317652 12-4504707 -0005181347 1931 3728761 7200237491 43-9431451 12-4526-206 •0005178664 1932 3732624 7211429568 43-9545220 12-4547699 •0005175983 1933 3736489 7222633237 43-9658959 12-4569184 -0005173306 1934 i 3740356 7233848504 43-9772668 12-4590661 •0005170631 1935 1 3744225 7245075375 43-9886349 12-4612131 -0005167959 1936 3748096 7256313856 44-0000000 12-4633594 •0005165289 1937 i, 3751969 7267563953 44-01136-22 12-4655049 -0005162623 1938 : 3755S44 7278825672 44-02-27214 12-4676497 -0005159959 1939 3759721 7290099019 44-0340777 12-4697937 -0005157298 1940 j 3763600 7301384000 44-04543 11 12-4719370 -0005154639 1941 1 3767481 7312680621 44-0567815 12-4740796 •0005151984 1942 j 3771364 7323988888 44-0681291 12-4762214 •0005149331 1943 3775249 7335308807 44-0794737 ] 2-4783(;25 •0005146680 1944 3779136 7346640384 44-0908154 12-4805029 •0005144033 1945 3783025 7357983625 44-1021541 12-4826426 -0005141388 1946 ' 3786916 7369338536 44-1134900 12-4847815 -0005138746 1947 \ 3790809 7380705123 44-1248-229 12-4869197 •0005136107 1948 j 3794704 7392083392 44-1361530 12-4890571 •0005133470 1949 ! 3798601 7403473349 44-1474801 12-4911938 •0005130836 1950 3802500 7414875000 44-1588043 12-4933-298 •0005128-205 1951 3806401 7426288351 44-1701256 12-4954651 •0005125577 1952 , 3810304 7437713408 44-1814441 12-4975995 •0005122951 1953 1 3814209 7449150177 44-1027596 12-4997333 •00051203-28 1954 1 3818116 7460598664 44-2040722 12-5018664 •0005117707 1955 3822025 7472058875 44-2153819 12-5039988 •0005115090 1956 3825936 , 7483530816 i i 44-2-266888 12-5061304 •0005112474 262 SQUAEES, CUBES, EOOTS, AND EECIPEOCALS. No. Square Cube Square Root Cube Eoot Reciprocal 1957 3829849 1 7495014493 44-2379927 12-5082612 -0005109862 1958 3833764 ' 7506509912 44-2492938 1-2-5103914 -0005107252 1959 3837681 7518017079 44-2605919 12-5125208 •0005104645 1960 3841600 7529536000 44-2718872 12-5146495 •0005102041 1961 3845521 7541066681 44-2831797 1-2-5167775 •0005099439 1962 3849144 7552609128 44-2944692 12-5189047 •0005096840 1963 3853369 7564163347 44-3057558 12-5210313 •0005094244 1964 3857296 7575729344 44-3170396 12-5231571 •0005091650 1965 3861225 7587307125 44-3283205 12-5252822 •0005089059 1966 3865156 7598896696 44-3395985 12-5274065 -0005086470 1967 3869089 7610498063 44-3508737 12-529.-.302 -0005083884 1968 3873024 7622111232 44-3621460 12-5316531 -0005081301 19G9 3876961 7633/36209 44-3734155 12-5337753 '-00050787-20 1970 3880900 7645373000 44-3846820 12-5358968 •0005076142 1971 3884841 7657021611 44-3959457 12-5380176 -0005073567 1972 3888784 7668682048 44-4072066 12-5401377 •0005070994 1973 3892729 7680354317 44-4184646 12-5422570 -000506S424 1974 3x96676 7692038424 44-4297198 12-5443757 -0005065856 1975 3900625 77037343 < 5 44-4409720 12-5464936 ■0005063291 1976 3904576 7715442176 44-4522215 12-5486107 -0005060729 1977 3908529 7727161833 44-4634681 12-5507272 -0005058169 1978 3912484 7738893352 44-4747119 1-2-5528430 •0005055612 1979 3916441 77501536/ 39 44-4859528 12-5549580 -0005053057 1980 3920400 7762392000 44-4971909 12-55707-23 •0005050505 1981 3924361 7774159141 44-5084262 12-5591860 •0005047956 1982 3928324 7785938168 44-5196586 12-561-2989 •0005045409 1983 3932289 7797729087 44-5308881 12-5634111 •0005042S64 1984 3936256 7809531904 44-5421149 12-5655226 •00050403-23 1985 3940225 7821346625 44-5533388 12-5676334 •0005037783 1986 3944196 7833173256 44-5645599 12-5697435 •0005035247 1987 3948169 7845011803 44-5757781 1-2-57185-29 •0005032713 1988 3952144 7856862272 44-5869936 12-5739615 •0005030181 1989 3956121 7868724669 44-5982062 12-5760695 •0005027652 1990 3960100 7880599000 44-6094160 12-57S1767 •0005025126 1991 3964081 7892485271 44-6206230 12-580-2832 •0005022602 1992 396^064 7904383488 44-6318272 12-58-23«91 •0005020080 1993 3972049 7916293657 44-6430-286 12-5844942 •0005017561 1994 3976036 7928215784 44-6542271 12-5865987 -0005015045 1995 3980025 7940149875 44-6654228 12-5S87024 -000501-2531 1996 3984016 7952095936 44-6/ 66158 12-590.^054 •0005ol0('20 1997 3988009 7964053973 44-6S78059 12-5929078 -00O50')7511 1998 3992004 79761 123992 44-6989933 12-5950094 •0005005005 1999 5996001 7988005999 44-7101778 12-5971103 -000500-2501 2000 4000000 80O0O00O0O 44-7213596 1-2-5992105 -0005000000 2001 4004001 8012006001 44-7325385 12-6013101 -0004997501 2002 4008004 8024024008 44-7437146 12-6034089 •0004995005 2003 4012009 8036054027 44-7548880 12-6055070 -0004992511 2004 4016016 804809(UHU 44-7660586 12-607(;044 -0004990020 2005 4020025 8060 150 125 44-7772264 12-6097011 •0004987531 SQUAEES, CUBES, EOOTS, AND EECIPROCALS. 26' No. 2006 Square Cube Square Root Cube Root Reciprocal 4024036 8072216216 44-7883913 12-6117971 •0004986046 •20U7 4028049 8084294343 44-7995535 12-613f^y24 5 •0(10498-2561 2UU8 4032064 8096384512 44-8107130 12-6159,s70 -0004980(180 2009 4036081 8108486729 44-8218697 12-6180810 •0004977601 2010 4040100 8120601000 44-8330235 12-6201743 •0004975124 2011 4044121 8132727331 44-8441746 12-6222669 -0004972660 2012 4048144 8144865728 44-8653230 12-6243687 -0004970179 2013 4052169 8157016197 44-8664685 12-6264499 •0004967710 2014 4056196 8169178744 44-8776113 12-6-285404 -0004965243 2015 4060225 8181353375 44-8887614 12-6306301 •0004962779 2016 4064256 8193540096 44-8998886 12-6327192 •0004960317 2017 40682^9 8205738913 44-9110231 12-6348076 •0004967858 2018 4072324 8217949832 44-9221549 12-6368953 •0004956401 2019 4070361 8230172859 44-933-2S39 12-63898-23 •0004952947 2020 4080400 8242408000 44-9444101 1-2-6410687 •0004950495 2021 4084441 8254655261 44-9555336 126431643 •0004948046 2022 4088484 8266914648 44-9666543 12-6452393 •0004946598 2023 4092529 8279186167 44-9777723 12-6473235 -0004943154 2024 4096576 8291469824 44-9888875 12-6494071 -0004940711 2025 4100625 8303765625 45-0000000 12-6514900 -0004938272 2026 4104676 8316073576 45-0111097 12-6536722 •0004935834 2027 4108729 8328393683 45-0222167 1 •2-6556638 •0004933399 2028 4112784 8340725952 45-0333210 12*6577346 •0004930966 2029 4116841 8353070389 45-0444225 12-6598148 •0004926636 2030 4120900 8365427000 45-0555213 12-6618943 -0004926108 2031 4124961 8377795791 45-0666173 12-6639731 •0004923683 2032 4129024 8390176768 45-0777107 1 2-666061 2 -0004921260 2033 4133089 8402569937 45-0«88013 1-2-6681 2 4177936 8539701184 45-2106182 12-6909354 -000489-2368 2045 4182025 8552241125 45-2216762 12-6930047 -0004889976 2046 4186116 8564793336 45-2327315 12-6950733 -0004887686 2047 ! 4190209 8577357823 45-2437841 12-6971412 •0004885198 2048 i 4194304 8589934592 45-2648340 12-6992084 •0004882813 2049 : 4198401 8602523649. 45-2658812 12-7012760 •0004880429 2050 j 4202500 8615126000 45-2769-257 12-7033409 •0004878049 2051 j 4206601 8627738651 45-2879676 12-7054061 •0004875670 2052 4210704 8640364608 45-2990066 12-7074707 -0004873294 2053 i 4214809 8653002877 45-3100430 1-2-7095346 -0004870921 2054 4218916 8665653404 45-3210768 1 12-7115978 -000486«549 2G4 SQUAEES, CUBES, EOOT?, AND EECIPROCALS. Xo. Square Cube Square E,oot Cube Root Reciprocal •2055 4223025 8678316375 45-3321078 1-2-7136603 •0004866180 2056 4-227136 8690991616 45-3431362 1-2-7157-222 •00048r,3813 •2057 4231249 8703679193 45-3541619 12-7177835 •0004861449 •2058 4235364 8716379112 45-3651849 12-7198441 -0004859086 •2059 4239481 872909 J 379 45-3762052 12-7219040 -00048567-27 2060 4243600 8741816000 45-387-22-29 12-7-239632 •0004854369 2061 4-247721 8754552981 45-3982378 1-2-7260218 •0004852014 2062 4251844 876730-2328 45-4092501 12-7-280797 -0004849661 •2063 4-255969 8780064047 45-4202598 12-7301370 •0004847310 2064 4260096 879-2838144 45-4312668 12-7321935 -0004844961 2065 4264225 8805624625 45-4422711 12-734-2494 •0004842615 •2066 4268356 8818423496 45-4532727 12-7363046 0004840271 2067 4272489 8831-234763 45-464-2717 12-7383592 •0004837929 2068 4-27(;6-24 8844058432 45-4752680 12-7404131 •0004835590 2069 4-280761 88o6894509 45-486-2616 12-7424664 -0004833253 2070 4284900 8869743000 45-4972526 12-7445189 •0004830918 2071 4-289041 8882603911 45-508-2410 12-7465709 •0004828585 2072 4293184 8895477248 45-5192267 12-7486222 •0004S26255 •2073 4297329 8908363017 45-5302097 12-7506728 •0004823927 2074 4301476 8921261224 45-5411901 12-75272-27 •0004821601 2075 4305625 8934171875 •15-5521679 12-7547721 •0004819277 2076 4309776 8947094976 45-5631430 12-7568207 •0004816956 •2077 4313929 8960030533 45-5741155 12-7588687 •0004814636 2078 4318084 8972978552 45-5850853 12-7609160 •0004812320 2079 4322-Ml 8985939039 45-59605-25 12-7629627 •0004810005 2080 4326400 899S9 12000 45-6070170 12-7650087 •0004807692 2081 433056 i 9011897441 45-6179789 12-7670540 -0004805382 2082 4334724 90-2-1895; '68 45-6289382 12-7690987 -0004803074 2083 4338889 9037905787 45-6398948 12-7711427 •0004800768 2084 4343056 9050928704 45-6508488 12-7731861 -0004798464 2085 4347225 90639641-25 45-6618002 1 2-7752288 •0004796163 2086 4351396 9077012056 45-6727490 12-7772709 •0004793864 2087 4355569 9090072503 45-6836951 1-2-7793123 •0004791567 2088 4359744 9103145472 45-6946386 12-7818531 •0004789272 2089 4363921 9116230969 45-7055795 12-7833932 •0004786979 2090 4368100 91293290(10 45-7165178 12-7854326 •0004784689 2091 4372281 914-2439571 45-7274534 12-7874714 •0004782401 2092 4376461 9155562688 45-7383865 12-7895096 •0004780115 2093 4380649 9168698357 45-7493169 1-2-7915471 •0004777831 2094 4384836 91S184 584 45-760-2447 12-7936840 •0004775549 2095 4389025 9195007375 45-7711699 12-7956-202 -0004773270 2096 4393216 9-2081 «0736 45-78-209-26 12-7976558 •0004770992 2097 4397409 922136(;(i73 45-7930126 12-799(;907 •0004768717 2098 4401604 9234565192 45-8039299 12-8017-250 •0004766444 2099 4405801 9-247776-299 45-8148447 12-8037586 •0004764173 2100 4410000 9261000000 45-8257569 12-8057916 •0004761905 2101 4414-201 9274236301 45-83666(n5 12-8078239 •00O4759('38 2102 4418404 92S7'1 85208 45-8175735 12 8098556 •0004 757374 2103 4422609 9300746727 45-8584779 12-8118866 •0004 755112 SQUAEES, CUBES, EOOTS, AND RECIPROCALS. 265 No. Square 2104 4426816 2105 4431025 2106 4435236 2107 4439449 2108 4443664 21.09 4447881 2110 4452100 2111 4456321 2112 14460544 2113 4464769 2114 4468996 2115 4473225 2116 4477456 2117 4481689 2118 4485924 2119 4490161 2120 4494400 2121 4498641 2122 4502884 2123 4507129 2124 14511376 2125 4515625 2126 4519876 2127 4524129 2128 4528384 2129 4532641 2130 4536900 2131 4541161 2132 4545424 2133 4549689 2134 4553956 2135 4558225 2136 4562496 2137 4566769 2138 4571044 2139 4575321 2140 4579600 2141 4583881 2142 4588164 2143 4592449 2144 4596736 2145 4601025 2146 4605316 2147 4609609 2148 4613904 2149 4618201 2150 4622500 2151 4626801 2152 4631104 9314020864 9327307625 9340607016 9353919043 9367243712 9380581029 9393931000 9407293631 9420668928 9434056897 9447457544 9460870875 9474296896 94S7735613 95011?>7032 9514651159 9528128000 9541617561 9555119848 9508634867 9582162624 9595703125 9609256376 9622822383 9636401152 9649992689 9663597000 9677214091 9690843968 9704486637 9718142104 9731810375 9745491456 9759185353 9772.S92072 9780611619 9800344000 98140S9221 9827^^47288 9841618207 9855401984 9869198625 9883008136 9896830523 9910665792 9924513949 9938375000 9952248951 9966135808 Square Root 45-8693798 45-8802790 45-8911756 45-9020696 45-9129611 45-9238500 46-9347363 45-9456200 45-9565012 45-9673798 45-9782557 45-9S91291 46-0000000 460108683 46-0217340 46-0325971 46-0434577 46-0543158 46-0051712 46-0760241 46-0868745 46-0977223 46-1 085675 46-1194102 46-1302504 46-1410880 46-1519230 46-1627555 46-1735855 46- 1844130 46-1952378 46-2(160602 46-2168800 46-2276973 46-2385121 46-2493243 46-2601340 46-2709412 46-2817459 46-2925480 46-3033476 46-3141447 46-3249, ;93 46-3357314 46-3465209 46-3573079 46-3680924 4()-3788745 46-3896540 Cube Root 12-8139170 12-8159468 12-8179759 12-8200044 12-8220323 12-8-240595 12-8260861 12-8-281120 12-8301373 12-8321620 12-8341860 12-836-2094 12-838-2321 12-8402542 12-8422756 1 2-8442964 12-8463166 12-8483361 12-8503551 1 2-8523733 12-8543910 12-8564080 ] 2-8584-243 2-8604401 12-8624552 12-8044697 12-8664885 12-8684967 12-8705093 12-8725213 12-8745326 12-8765433 12-8785534 2-8805628 12-8825717 12-8845199 12-8865874 12-8885944 12-8906007 12-8926064 12-8946115 12-8966159 12-8986197 12-^006229 12-9026255 12-9046275 12-90662S8 l-2-9(.862;i5 12-9106296 Reciprocal -0004752852 -0004750594 -0004748338 •0004746084 -0004743833 •0004741584 •0004739336 •0004737091 •0004734848 -0004732608 -0004730369 -0004728132 -0004725898 -00047-23666 -0004721435 -000471 9207 -000471t.;9Sl -0004714757 -0004 712535 -0004710316 -0004708098 •00047058)^2 •0004703669 •0004701457 •0004699248 •0004697041 -0004i. 94836 •0004692633 •0004690432 •0004688233 •0004686036 -0004683841 -0004681648 -0004679457 •0004677268 -0004675082 •0004672897 •0004670715 •0004G68534 •0004066356 -0004664179 •0004662005 •0004659832 •0004657662 •0O04655493 ■0w04653i;27 -(100-1651103 ■01:046490(10 -0004646840 266 SQUARES, CUBES, EOOTS, AXD EECIPEOCALS. No. Square Cube Square Root , Cube Eoot Eeciprocal 21o3 i 4635409 1 9980035577 46-4004310 12-9126291 -0004644682 2154 1 4639716 i 9993948264 46-4112055 12-9146279 -0004642526 2155 4644025 10007873875 46-4219775 1-2-9166262 -0004640371 21513 i 4648336 10021812416 46-4327471 12-9186-238 •0004638219 2157 4652649 10035763893 46-4435141 12-9206-208 -0004636069 2158 4656964 1 1004972^312 46-4542786 1-2-9-226172 -0004633920 2159 4661281 1 10063705679 46-4650406 12-0246129 -0004631774 2160 4665600 10077696000 46-4758(102 12-9-266081 •0004629630 2161 4669921 10091699281 46-4865572 ■ 12-9286027 •0004627487 2162 4674244 1 10105715528 46-4973118 12-9305966 -0004625347 2163 4678569 10119744747 46-5080638 ' 12-9325899 •0004623209 2164 4682896 ' 10133786944 46-5188134 1-2-9345827 -0004621072 21 65 4687225 10147842125 46-5295605 12-9365747 -0004618938 2166 4691556 10161910296 46-5403051 12-9385662 -0004616805 2167 4695889 10175991463 46-5510472 12-9405570 -0004614675 2168 4700224 10190085632 46-5617869 12-9425472 -0004612546 2169 4704561 102041 92S09 46-5725241 12-9445369 -00046104-20 2170 4708900 10218313000 46-583-2588 12-9465-259 -0004608295 2171 4713241 10232446211 46-5939910 1-2-9485143 -0004606172 2172 4717584 10246592448 46-6047208 12-9505021 -0004604052 2173 4721929 10260751717 46-6154481 1-2-95-24893 -0004601933 2174 4726276 10274924024 46-6261729 12-9544759 -0004599816 2175 4730625 ' 10289109375 46-6368953 12-9564618 -0004597701 2176 4734976 10303307776 46-6476152 12-9584472 -0004595588 2177 4739329 ! 10317519233 46-6583326 12-9604319 -0004593477 2178 4743684 ' 10331743752 46-6690476 12-9624161 -0004591368 2179 4748041 10345981339 46-6797601 1-2-9643996 •0004589261 2 ISO 4752400 10360232000 46-6904701 12-9663.^26 •0004587156 2181 4756761 10374495741 46-7011777 12-9683649 -0004585053 2182 4761124 10388772568 46-7118829 12-9703466 -0004582951 2183 4765489 10403062487 46-72-25855 12-9723277 •0004580-^52 2184 4769856 10417365504 46-7332858 12-97430.S2 -0004578755 21>!o 4774225 10431681625 46-7439836 12-9762881 -0004576659 2186 4778596 10446010^56 46-7546789 12-9782674 -0004574565 2187 1 4782969 10460353203 46-/653718 12-9802461 -0004572474 2188 4787344 10474708672 46-7760623 12-9822-242 •0004570384 2189 4791721 10489077269 46-7867503 12-9842017 -0004568296 2190 4796100 10503459000 46-7974358 12-9861786 -0004566210 2191 4800481 1 10517853871 46-8081189 12-9881549 •0004564126 2192 4804864 ! 105322618.^8 46-8187996 1 2-9901306 •0004562044 2193 4809249 ! 10546683057 46-8294779 12-9921057 •0004559964 2194 i 4813636 10561117384 46-8401537 12-9940802 •0004557885 2195 4818025 10575564875 46-8508-271 12-9960540 -0004555809 2196 4822416 10590025536 46-8614981 12-9980273 •0004553734 2197 4826809 10604499373 46-8721666 130000000 •0004551661 2198 4831204 10618986392 46-88-28327 13-0019721 •0004549591 2199 4835601 10633486599 46-8934963 13-0039136 •0004547522 22U0 4840000 10648U00OU0 46-9041576 13-0059145 •0004545455 2201 4844401 10662526601 46-9148164 13-0078848 -0004543389 EVOLUTION. 267 EVOLUTION. To Extract the Square Koot. EULE. — If there be decimals in the given niimber, make them to consist of two, f oui-, six, Sec, places by annexing ciphers to the right hand ; then separate the whole into periods of two figures e'ach, beginning at the right hand, and the left-hand period will consist of one or tw-o figm'es, according as the num- ber of figures in the whole number is odd or even. Find a square number equal to or the next less than the left-hand period, and put the root of it in the quotient ; subtract this square from the left-hand period, and to the remainder bring down the next period for a dividend, and to the left hand of it write double the quotient for a divisor ; then consider what figure if annexed to the divisor and the result multiplied by it the product may be equal to or the next less number than the dividend, and it will be the second figm-e of the root. From the dividend subtract the product, and to'the remainder bring down the next period for a new dividend ; double the figures in the quotient for a divisor, and continue the operation as above till all the periods are used. Fxamjjle 1. Example 2. Extract the sq. root of 10291264. Extract the sq. root of 177746-56. 10291264 9 3208. Ans. 177n6-56 16 421-6. Am. 62 20 129 124 6408 51264 51264 82 2 177 164 841 1 1346 841 8426 50556 50556 To Extract the Square Eoot op a Vulgar Fraction. KULE 1.-- -Multiply the numerator by the denominator, and extract the square root of the product; the numerator of the given fraction, written above this root, or the denominator written below it, will express the root of any fraction when reduced to its lowest terms That is— \l\~ Val) 'Jab h EuLE 2.— Eeduce the given fraction to its lowest terms ; then extract the square root of the numerator for a new numerator, 268 EVOLUTION. and the square root of the denominator for a new denominator. If the fraction will not extract even, reduce it to a decimal and then extract the square root. To Extract the Cube Root. Rule. — If there be decimals in the given number, make them to consist of three, six, nine, ire, places by annexing ciphers to the right hand, if necessary ; then separate the whole into periods of three figures each, beginning at the right hand. The left-hand period may consist of one, two, or three figures. Find the nearest cube to the first period, subtract it therefrom, and put the rcot in the quotient; then thrice the square of this root will be the trial divisor for finding the next figure. Multiply the root figure, or figures already found, by three, and prefix the product to the next new root -figure (which will be seen by the trial divisor) ; then multiply this number by the aforesaid new root-figure, and place the product two figures to the right below the trial diWsor, and add it to the trial divisor : this sum will be the true divisor. Under this divisor \^T:ite the square of the last root-figure, which add to the two sums above, and the result is the next trial divisor ; the true divisor being found as before directed. Example, Extract the cube root of 4088324799. True divisor 1' Trial divisor 1^ 35 x3 = 3 X 5 = 175 True divisor 52 Trial divisor 459 92 Trial divisor 4779 True divisor 475 X 5 = 25 675 x9= 4131 71631 X 9 81 = 75843 x9= 43011 4088324799 1 3088 2375 713324 644679 68645799 = 7627311x9 686457P9 I 1599. Ans. To Extract any Root whatever. If X be any given number whatever whose root is sought, n the index of the power, r the near- est rational root; or r» the nearest rational power to N, whether gTeater or less, and R = the root sought ; then — B = [yx (7?H-l )}-l-{(w-l)xy" } '{N X {71- 1)} + {{11 + 1) X r"} "^ ^' i WEIGHT AlsD STEEXGTH OF MATERIALS. 269 Table op ttte Weight axd Strength of Materials. :metals. 1 ' T He in Tearing Crushing Modulus of Name Specific^^(?X G---t^| Eoot Force Force Elasticity- Lbs. on Lbs. on Lbs, on .■^q. In. £q. In. Sq. In. Aluminum, cast . 2-560, 160-0 — — — „ sheet 2-670 166-9 — — — Antimony, cast . 6-702 418-9 1,053 — — Arsenic 6-763 360-2 — — — Bismuth, cast . 9-822 613-9 2,798 — — Brass, cast 8-396 524-8 18,000 10,300 9,170,000 „ sheet 8-525 532-8 31,360 — — „ wire 8-544 533-0 49,000 — 14,230,000 Bronze 8-222 513-4 — — — Cobalt, cast 7-811 ' 488-2 — — — Copper, bolts . 8-850 : 531-3 36.000 — — ,, cast 8-607 i 537-9 19,000 — — „ sheet . 8-785 549-1 30,000 — — „ wire 8-878 548-6 60,000 — — Gold, pure . 19-258 1203-6 20,400 — — „ hammered 19-362 12101 — — — „ standard . 17-647 1102-9 — — — Gim metal . 8153 ' 509-6 36,000 — 9,873,000 Iron, cast, from . 6-955 434-7 13,400' 82,000 14,000,000 „ „ to . 7-295 455-9 29,000145,000 22,900,000 „ „ aver age 7-125 445-3 16,500112,000 17,000,000 „ wrought, from . 7-560 472-5 50,000 40,300 — >> j> to 7-800 487-5 63,000 32.000 — 5J 5> average 7-680 480-0 60,000 36,000 28,000,000 Lead, cast . , 11-352 709-5 1,792 6,900 — „ sheet . 11-400 712-8 3,328 — 720,000 Mercury, fluid . , 13-068 848-0 — — — „ solid . 15-632 977-0 — — . — Nickel, cast 7-807 487-9 — — — Pewter 11-600 702-5 — — — Platinum, pure . 19-500 1218-8 — — — „ sheet 20-337 1271-0 265,000 — 24,240,000 Silver, ptire 10-474 654-6 42,000 — — „ standard 10-534 658-4 — — — Steel, hard 7-818 488-6| 103,000 ' — 42,000,000 „ soft . 7-834* 489-6 121,700 ' — 29,000,000 Tin, cast . 7-29li 455-7 4,600 ■ 14,600 4,550,000 Tj^De metal 10-4.50 653-1 — — — Zinc, cast . 7-028' 439-3 8,500 — 13,500,000 „ sheet 7-291 455-7 7,111 — 12,650,000 270 WEIGHT AND STRENGTH OF MATERIALS. Table of the Weight AND Strength of ^VKterials (cont.) TlJIBKR. 1 Lbs. bo .So bo ,3 > 33 1 CS — j o>.j-^- =P .1 Name "5 > E a; C Name •■c.-tf i 9 5 : 5 gcc 2-50 o 156 k-3 _ol "-^ Asphalte — Peat, hard . V33I 83 — Alabaster 1'87| 117 — Plumbago 2^27!l39 — Basalt . 2-7-2! 170 16,800 Porcelain, China . 2^38, 149 ] — Brick, common 2-00 125 — Portland stone •2^571161- 6,85G „ red 2-16 134 808 Pumice stone •914i 57 — ,, Welsh fire . 2-40 150 — Purbeck stone 2^60il63 9,16( Cement, Portland . 1-35 84 5,984 Rag stone •2^47il54 — Chalk . 2-77 173 505 Rotten stone. 1^98!l24 — Coal . 1-27 79-4 — Salt 2-13 133 — Coke . •744 46 — Sand, fine pit 1-52 95 — Freestone 2-45 153 6,842 „ coarse pit . 1^61 100 — Gypsum . 2-17 135 — ,, river . 1-88 117 — Granite . 2-70 169 12,800 Slate . 2-62 164 15,000 Grindstone . 2-14 134 — Sugar . 1-61 100 — India rubber •9S4 58^4 Sulphate of soda . 2^20 137 — Lime, quick . •843 53 — Sulphur, native . 2^03 127 — Limestone 2-95 184 9,160 „ fused 1^99 124 — Marble . 2-72 170 9,219 Tallow . •94 59 — Mica 2-79 173 — Tar . . . 1^02 63 — ]\Iortar . 2^48;155 — Tile, common 1^83ill3 ~ Liquids. o >■. o >> •go «.-tf ~ ^ -t^^-t . «=i.-;2 ^ -1-=" -U H^ _. Name 1^ Name II -S c •I'll XO i> ^ ■^ ^ ■JlO 1 '^ ^ .* ^ 1-06 ^?. •915 ^3 a Acetic acid . 66-4 •615 Oil of olives . 57-2 ' -530 Alcohol, proof •916 57 •530 „ turpentine . •87054^9 1 ^508 Ether, acetic •860 54 •501 ,, whale . •923i57-7 1 ^534 „ muriatic . •730 45-6 •422 Oils, average •880'55^0 I ^510 ,, sulphuric . •740 46^3 •428 Petroleum •878;54^8 1 -508 Muriatic acid 1-20 75 •694 Sulphuric acid 1^84!115 i,l- 06G Xitric acid . 1-27 79^4 •736 Vinegar 1^0163^1 j ^585 Oil of aniseed •987 61-6 •570 Water, rain . 1^00:62^5 : -579 „ caraway seed •905 56^6 •524 ,, sea . 1-03 64-4 1 -595 „ hempseed •920 57-8 •536 Wine, champagne •998|62^4 ' -578 „ lavender •894 '>5^9 •517 „ burgundy . •99r82^0 •573 „ linseed . •940 58-8 •544 „ madeira 1^04:65^0 •601 „ rapeseed •913 57-0 •528 „ port . •99762^3 •577 272 CONSTANTS FOR ESTIMATING WEIGHTS. Estimation of Quantities. Tons X 2240= lbs. Tons x 20=cwts. Lbs. x •000446428= tons. Weight of Round or Elliptical Bars. Diameter x diameter X length in feet x constant = weight in lbs. Weif/ht of Sqiiare or Rectangular Bars. Width X thickness x length in feet x constaut= weight in lbs. Weight of Plating or Planhing. Thickness x breadth' in feet x length' in feet x constant = weight in lbs. Values op Constants for Round or Elliptical Bars. Material Brass, sheet Iron, wrought . Lead, sheet Steel, soft . Elm, American . Mahogany, Honduras ,, Spanish Oak, Dantzic ,, English Pine, red . „ yellow T<^ak. Indian Diameters taken in Ins. 2^905980 2-61800 3-88773 2-67036 •261800 •196350 •287980 •261800 ■307615 -196350 •157080 -287980 iln. \ln. ' jln. j A Id- I In. •181624 •163625 •242983 •166898 •016363 •012272 •017999 •016363 •019228 •012272 •009818 •017999 •045406 •040906 •060746 •041724 •004U91 •003068 •004500 -004091 •004807 •003068 •002454 -004500 •011351 •010227 •015186 •010431 •001023 •000767 •001125 •001023 •001202 •000767 -000614 •001125 •002838 •002557 •003797 •002608 -000356 -000192 •000281 •000356 •000300 -000192 •000153 •000281 Values of Constants for Square or Rectangular Bars. Width and Thickness taken in J •V. . -■. Material Ins. iln. iln. Jin. 3^ In. 3\ In. Brass, sheet Iron, wrought . Lead, sheet Steel, soft . Elm, American . Mahogany, Honduras „ Spanish . Oak, Dantzic „ English Pine, red . ,, yellow Teak, Indian 3-70000 3-33333 4^95000 3-40000 •333333 •250000 •366667 •333333 ■391667 -■250000 •20CM)00 •366(567 •925000 •833333 1^23750 •850000 •083333 •062500 •091667 •083333 •097917 •062500 -050000 •091667 •231250 •208333 •309375 •212500 •020833 •015625 •022917 •020833 •024479 •015625 •012500 -02291 7 •057813 •0520.^3 -077344 •053125 •Ot»52<)8 •0039(»6 •005729 -005208 •006120 •003906 -003125 -005729 •014453 -013021 •019336 •013281 •001302 •000977 •001432 •001202 •001530 •000977 •000781 -001432 •003613 -003255 •004834 •0033-20 •000326 •000244 •000358 •000326 •000382 •000244 -0001 95 •00O358 Values of Constants for Plating or Planking. Material Thickness taken in | Ins. ^In. iln. iln. 1^ In. ^ In. 2^7750 1^38750 rh In. •69375 Brass, sheet 44-4 22-2 11-100 5^550 Iron, wrought . 40^0 20^0 10^000 5-000 2^5000 ^•25000 •62500 Lead, sheet 59-4 29^7 14-85 7-425 3-7125 1^85625 •92813 Steel, soft . 40-8 20^4 10^20 5-100 2^5500 1^27500 •637.50 Elm, American . 4-00 2-00 1^000 •5000 •25000 •12500 •62500 Mahogany, Honduras 3M0 1-50 -750 •3750 •18750 •09,375 •04688 ,, Spanish . 4-40 2-20 1-100 •5500 •27500 •13750 •06875 Oak, Dantzic 4-00 2-00 1^000 •5000 •25000 -125000 •06250 ,, English 4-7(> 2-35 1^175 •5875 •29375 •14n88 •07344 Pine, red . . 3 00 1-50 •750 •3750 •18750 ^09375 •04688 yellow 2^40 1-20 •600 •3000 •1-5000 •07500 •037.50 Teak, Indian 440 2-20 1100 •5500 •27500 ^13700 •06875 CONSTANTS FOR ESTIMATING WEIGHTS. 273 Weight of Pipes. w=: weight per lineal foot in lbs. K= constant from below. ■W=(D2 — rf»)K. Values of K for Pipes. Brass =2-9060. Iron, cast =2-4282. Copper=2-9943. „ wrought =2 -6 180. Weight of Angle Iron. D= outside diameter in ins. d= inside „ „ Lead = 3-8877. Steel= 2-6704. "vr=weight in lbs. per lineal foot. s=sum of the widths of flanges in ins. T= thickness of flanges in ins. W=T (S-T) 3-33333. Kelative Weights of Different Substances. TTrought iron=l. Brass, sheet=l-1100. Beech Copper „ =1-1438. Iron, cast = -9275. Lead, sheet =1-48.50. Steel, soft =1-02W. Tin = -9.500. Zinc = -9494. = ■0896. Elm =-1000. Fir, spruce = -0833. Mahogany, Honduras= -0750. Spanish =-1100. Maple =-1021. Oak, Dantzic =-1000. Oak, English =-1175. Pine, red = -0750. „ yellow =-0600. Sycamore =-0308. Teak, Airican = -1146. „ Indian =-1277. Willow =-0521. Weight, &c., of Fresh Water. A cubic foot = -0279 ton =62*39 lbs. =998-18 avd. ozs.= 6-2321 galls. .^ / A cubic inch=-0361 lb. =-5776 avd. oz. =Wm gall.' • .0 o C:? A gallon =-0045 ton =10-000 lbs. =160-15 avd. ozs. = -1315 cu. ft. A ton =35-905 cu. ft. = 2240 lbs. =223-76 galls. Weight of fresh water= weight of salt water x -9740. Weight, &c., of Salt Water. A cubic foot =-0286 ton =64-05 lbs. =1024-80 avd. ozs. = 6-2321 galls. A cubic inch =-0371 lb. ='5930 avd. oz, ='0036 gall. A gallon =-0046 ton =10-276 lbs. =164-41 avd. ozs. ='1315 cu. ft, A ton =34-973 cu. ft. = 2240 lbs. =217-95 galls. Note. — A cubic foot of salt water is usually taken at 35 cu, ft. to the ton and 64 lbs. to the cubic foot, fresh water being taken at 36 cu. ft, to the ton and 62-25 lbs. to the cubic foot. Miscellaneous Factors. A ton tonneau. An avd. lb. A foot A sq. foot A sq. inch metres. A cu. ft. A cubic yard A mile Knot per hour second. Mile per hour A gallon = 1-01605 tonne or =-45359 kilogram. = •304797 metre. = -092901 sq. metre. = 645-148 sq. milli- = •028316 cu. metre. =•764534 cu. metre. = 1-60933 kilometre, = 1 -688 foot per second . = -5144 metre per = 1*467 foot per second. = 4^64102 Utres. A tonne or tonneau = A kilogram = A metre = A sq. metre - A sq. millimetre : A cubic metre - »» ,» ■ A kilometre = Foot per second = hour. Metre per second = hour. Foot per second hour. A litre = •984206 ton. : 2-20462 lbs. :3^2808693 feet. :10-7641 sq. feet. : -00155003 sq. in. :35-3156 cu. feet. : 1-30799 cu. yd. : -621377 mUe. :-592 knot per : 1-944 knot per :-682 mile per : -220315 gaUon. 274 PERCENTAGES FOR BUTT STRAPS, ETC. Table of Percentages to ije addkd to the Calculated Weights of Flush-jointed Plating ON Account of Edge Strips and Butt Straps. i tB c 3 ■So ^^ •.= '3 afq II 11 fee c3.9 -11 ■sg X o CO n cc <-i in CO -^ 00 »n CI SI o m n i^ i^ 0^ ^ ^ Butt Straps on opposite side to Edge Strips, 6^ diameters wide CO -- o o •* n o ci qp -^ O »p 7^ »7» "*! cb CO io (fl c^ T^ 1 Edge Strips 6i diameters wide 1 CO »0 t^ CO Ci lo k: ^ CO CO j^ Double-riveted Butt Straps, when on opposite side to Edge Strips, lU diameters wide r; -*^ o -^ r: uo o -*( t^ p CI '^i t-- o t>. i o lb -* CO CO Single-riveted Edge Strips, 6i diameters wide CO lO t^ CO Ci iM CO CO b- sp ip -* -J" CO O CO O '^l CI O 00 CI — " — -- — 5 o fcC« o cs C "^ o Double-riveted Butt Straps, when on same side as Edge Strips, 11* diameters ^vide lo o o c^ -* «r) t^ CO «p CO Ci IC p o -* -^ -^ CO CO CO CI Double-riveted Butt Straps, when on opposite side to Edge Strips, 11^ diameters wide. 00 -* O -*i Ci lO O ■*! b- p cq ■* t^ O b- i» cb ih "^ ~: ~5 Double-riveted Edge Strips, 11 J diameters wide ?C'*i — 00 IC CO o Cq b- CI -O T-^ w r-l ih -^ CO -^ .^ t^ 4*1 CO r: CI :?j CI r- — ( Plates 16 ft. long. Butt Straps full breadth of Plates Percentages applicable to Vertical Keel, Longitudinals, Stringers, &c. Single-riveted Double or Single Straps. If double, each Strap half the thickness of the plate. Width of Straps, ej diams. CO --^ O O -f CI o C^l 00 ■*! p UO .— t- -* CO M CO CI CI 4- Double-riveted Double Straps,each half the thickness of ttie plate. Width of Straps, llj diams. 00 "* O -*< a: lO O -* !>• O CI -* b- p t» «b «b ih -^ ^ c<^ Treble-riveted Double Straps,each half the thickness of the plate. Width of Straps, 16^ diams. ^ t^ o c? ^ tr c; b- CC :C in "^i M ^ o cj oo b- ?b lb -^t^ ,—1 Treble-riveted Double Straps, each ^ in. more than half the thickness of the plate. Width of Straps. 16i diams. 1 00 »0 CO Cj lO !^ -^ O O O p p T ^ Ca r^ O C> CO b- '^ 1— ( r-l .— i Diameter of Rivets in Inches .^-ioi Hoc wit -nix -IN ^^ 1— 1 f— < Tliir kuf-^- of Plates in Inches t~|ar:!'>r''*c - ?t r:lx Hi- percentages for butt straps, etc. 'liO Table of Percentages to be added to the Calculated Weights of Labped Plating on Account of Labs, Butt Straps, and Liners. d C •2. i a 'S a s a s ■s o o to "a •5* B cs 5 Siuglc-riveted Edges and yingle-rivetcd Butts Liners to Stifieners, 2 ft. 6 ins. apart O M -^f- -O CO — -r< ?) cc '7" ip ep CO ri 4*H xf -^ 4t< -*i -J? -?ri 1 Single-riveted Butt^. 3^ dia- '■ meters wide •TT — O O t^ «r O T- r-. L- -* ri c » Single-riveted Laps, o\ dia- meters %s"ide t- 30 'S- — lO — t- '* rc c^ c^l ^1 >i — c r: i t- '-b ill 4- ,2 fO s cs to § 1-1 1 -S-2 Liners (inclu'ling Wide Liners at Water-tight Frames and Bulkheads) — r: <-■: c L-t — t— ^1 r^ L-: I- DO c — ih >b ih ic l:: '-c "-S Single-riveted Butt Straps, ei diameters A\-ide U-: ^: tc re 1- O ri i^ 'Tt' c; t^- r^ 9 "O iri vt tTi i\ i\ ^ ^ Double-rivetetl Butt Strap?, Hi iliameters wide -t< i:; o ■>! ri n 'O •O O '*' CO — > ip CO i i »h "^ "^ « ^« Single-riveted Laps, 3^ uia- metei-s wide M c M vr -^ t^ r: t^ re — . le ri r: t-- -fi — c; oc t- ih '^ s o CO s es to a o iJ ■u o .-< 1 6 to §>! >.• u go -=» «^ to o J2 Liners (including Wide Liners at Water-tight Frames and Bulkheads) -* r; i-e o Lt — h- , :m re ip t^- XI 9 -7- ' ih »b »h ib •i '-^ Single-i-iveted Butt Straps, 6i diameters wide C t;; fc -o 15 «r i ih -^ ^ ^^ ^? ^1 ^1 Double-riveted Butt Straps, Hi diameters -^vide i-t r^ t^ C^ Ci — ' j CO 'T'l "* »p I:- * 1 oc 00 b- «b »b -^ re J. -Single-riveted Laps, 3^ dia- meters wide i i-i c: e-ir-i3^i3^i'Mcqcerecere'<*<'*'<*<-«**»r;>:2»ooco Breadtli of Plate (ins.) retooret—cret^Oret^Oret-oret^Oret-o x^pxpiere — pxtoierer-pxtpipre — p _S :e'*»hihtbi^doo;oor^-fire-?#^»h»htbb-xo; o'l! 5 1 £ H o|-^ f-H r; to «^ '>! O op >f: ^^ -71 ~. t;- »p ^1 O CO to "jH -;- CV b- l-.i-j "'"■ M «? '^i »b «b i^ t- 00 c^ -^ -fi cc 4- '+' ih b- b- do '"'" -M »o 00 cc --T C-. -M u: 00 ^ -f i^ JC cc — < '^i i-^ t^a- C; CC 1— 1 X l;: M C: t^ '^i ^"1 C: tC -* ^ 00 1:: ic t^ Hx a § ■N re -^ i-i tr; -.:; i^ cc cc — . C — ?i re rt ^ ts tc i- "'^1 ?^ « '*' 4< ib i -i i^ do oC' i: -^ >i ^1 ?e ^ "*! «b P-H lOrei-eooOJeiriacoreieGOOceooocreoooc:' rtjv *? T '^ \^ 9* f '^' ^ V"^ T 'T \^ -^ *-f T^ ?^ 'f T^ 'r V ? »-r Is-. CO-* — oo?occOQCO?iJ:t^-*-^c:tc:reoooiai ■^'"^i rtx 1 5! re cc '^ ^ »h ue to to t^ I- cc 00 — c; c: — — ^1 i ■ Igo-*!— 'cci.e>^0". toreo". toreot— -^1— •so-^'-^xie r.|= P5 1-9 l-H <^ H C o 1 •" o «|-^i X re X ?) I- T^j to — to c le c »-e 0; -* 0; re X re t- -M .-•'?^<>irere-*'*ojletbot-t>.t-dcdoo:c^oo^ -+M t^xO'Mreot^xo^reotr-xO'Mreist'XC! toOLec. ret^r--i»r:c-^xcNcop«cpret---i»pp! _^ r^?q^i^j:eM-^'?f»h»hi5tbtbb-b-t?-dcoco:diO; ,„■ CO ri ~ »e c^i X ue .-1 X -* C t^ re to re C'. to -M ci ic ... t-j;; -* X — le q; >i to c re i_-- r- -* X ?^ C-. •>) to cp re t- j t-,;:; — — !fi7'iJ• ,„l-*'Oto3>ix-*iooreci»a — t^recio — i^reciio! ,„ >-"-i5. X le X re «5 X p^ ?e to 0: -7- -* to 3; c-^ ^ t- C5 « , »S|i j,l^,l4rljrl^'jqj^(?qi'irederece-rt<-Tt''?*'-rt* 3-. »r; ' ,„ rtp' to i^ 0; c: c<» -+i I- X -^ re »e to X O: — c^ ^ ie 1^- . ctin -*x c^^nrerere-r*<•r^^^c^eotocol^^^xxxc;^:00 ■■*i?e'*<»ptob-xq;p'7-c^'?*»p -♦x |i-ito^toiMt-'?^t^cexrexre3'. '*i ^ -13 C3 3 •/ ^-f^M^., ,^-_4pB5lTf -.(.,_4cie*» -«'-<-M«H' -»-- ♦Nrtf ij.= r— — r-. — s^jvic^jc^ceoececc-^-^-^-^i^ieO'Ct^ 1 J i c^i c^i 7^ 2^1 >T ?i 77 71 7T cc re re yt yt r-T c^ re r: -f L-ir O rt O 30 :r "* CN O t- »p ^ 'p ^- «p '^ ^1 O OD « C: O 1 — "M 71 7^ ■M 07 77 T'T 77 71 77 7^1 71 71 re ce re re rc re re reiDri77>occocetcr:7viiccc — -^t^oceco-^c nciw-^'— octrreot-«e77r; t^-* — c;«po»e^ 303Cc;o-^'^'77ye-^'#»bi'i^-30^^O77rei-e — _ _ M ?1 77 71 77 7^7 77 7^7 7^7 77 7) 77 77 77 re re re re y7 0ocir-:t<^r:t~'*'7^7C:3Ci-ere'-^co^c-*c;ieo riw>7riwrer:iSceot^reOt^-*Ot-^i^-'^»-e ib-aocx;c;OC:"7^7r:r:-#»bib«bb-t^cc~ — 77 — ,—. — — .-- 77 77 77 7<1 77 7^7 7<1 77 7^1 77 7^7 77 77 77 Ce r e reieDOOreieccoceiecoOrciccooreicoico to 7^7 30 o ^ I;- re o w 7-7 op ip -7^ t- ce o "-p 77 »-e »;- ^ >i«btbi^303Cc;C:.C:'^'^7-7reye"^oueib-ic —I — — r- r- — ^ 77 77 77 7^7 7-7 7^ 77 7-7 77 71 77 77 77 re 77 Ot — * — ritcre — Gcuireot — r77~tc — leo '1= re — . ^ C; w — t^ ^ r: -# c: '>p ^7 t- re r-. 7* O 77 re 1-e '''^ -*i '^i lb i i t^ t- 30 i f; o O r— -^ 7^7 77 ^e -# le tc t>- ! — ,— i-^r-i^^ — 1 — 7-7 7-1 77 7-7 7M 77 7^ O '=*' c^ '^i CO re cc 7-7 t- :77 ^ «p Le Le ■''^j ^77 77rere-^•4t-olb^bi;i^-■^-302^^^^^7 l7^ ? -:c.'^ ^oo-:*' — t>--^Ot-reOwrer;u:7-7cci-e ;.^ — . -^ oc 77 1-e c: re w O -^ t— >— -?■ CO 77 1^ -. t>. le -*| -* -r- O to tC t^ i_^,iY^_i4ir';ca5r>0-^re'*i»cococ:0'— reirjQOO oo^'^t-orecDOreirc;77icoo^>pcO'7it;~rec: i t^O0O030cr;C5Ci6666'^'^'^, 77 7-7 7-7««j^»C |__t^^^lr^ — t^rec^icr-jco-#c:w7-7co-*«ccco .1-c' iet^0 77^3COreipcp.— retpr:T--'Tt<«pc:'fc:»f "'■^1 ■iit-^^-b-t-C»00007-tb>bebtbo«b-e tc 30 O — re -^ --f t- r: O 77 -^ ue a; -7" L' r; lO — ^ CO c: C ic CO »— Le o-^-H77 7cirerere'*'^io»ci!7?osct^t^cocociO; ;ct-3cr; o — 7^re-*'>csct^ooop';J7^ire«pt-Oj -icn 77 77 77 77 re r ;--;»-^ — tc-J"-r — t~77b-77ccreacreco-*'r:CiC C (j-' rere'*'':fLeue--Owt-t^ccco~r:c:C' ^77-:f»c I • ;- ^H ^ -U ^ ^ tI- ^ rl- ^ ^ tI^ .^ 77 >7 7-7 -77 77 77 77 =^Z ra c 278 WEIGHT OF SHEET METALS PER SQUAEE FOOT. O < p c? 125 M ai m ^; O w H o 1— i < > O GO o H w H sh O a H 1 ■■5 1 O ir. v: -j: _s 1 O X) -f t_-- r-: —. 'J" 1 i 6 -^ >h r; i'- •^ 1 ^ -f -^ -* »r: c^ 5 C % g 3 S 73 [3 CO CO 55 :n ^ -^ (M o^j CM iq CO 0^ it c m CO O — CO -T^ "^i -^ »p CO -*' CO — . X -+ -^ to CO i ir; CO C". CO o U-.H: 1 "r 7* ^ '^ *■? ^ r^i^ i^ cJo r^ (f 1 ib o 1 cc M -:t< '*■ ui a: i2 , 00 -*• o r; oc CO •^ 00 r; c>j c:-! (>i i^ ^ |>T CM CO CO -*i CM «| C — :^ 1 " Hx O o CO u: CO cr: i^ (X O cr. C7J >b »b cio o -^ « CO CO CO -*' to CO ^ CM C~. CO O CO 'O '"_ CO CO ;o cx- r: — 5: 00 —. t-0 <0 l^ ~i- »o CO r>. -Tf »o n\~ "P T' "^ T ?■• 9P ■iH C'l « ?b t- 00 o re CO CO cc '^^ cc " i O 00 CM to -*« f- ■"_ 1 cx ao c^l CO CO :o C' j CO CO -+' -^ UO CO s^ 1 CO -f -t- c^ i^ '^i . i « -Tt- X o; -+' r- £* . 4* -^ -^ 4" CO -+i to tlo I;- ^ 1 O O --H CO (30 1 O O CO CO i-O "* "'" ] 6 6 CO -* -* c» CO CO CO CO -t< (M 25 CO i^ ■M -^ CO CO 1 O CO O ^1 >-^ r-l-^ j »P O O -+I CO ry- '-— o «b 1 71 CM CO CO -*i (M ""_ X ^ ^ ^ ^ o ^ '^ -i- ih »o i^ -?i^ K 1 7. "i '2 , O UO C. ^ CO o o t^ »o — i t— "^'^ ! o ih t- 00 i^ CO i '>l CM C^l (M CO '>\ C* UO to »lo CO t^ to (2j : ;5 M CO t- I;- o rj l~ X ex oc p CO ■ -15 O CC CO CO o iO c^ c: t^ -t^ CO cf'i c>i -^ ilo :o r^ i cp t- ^ oo oo 0-. c~. c^^ t'- o.-. -:t< X CO CO cr>ci c>j CO p •='2 IH io cx) o o I- »0 t--. CX) CO to CO CM -f 1 CO -* ci) r^ 30 ci I- p c ct oo c» O". ci :o oc 52 X — CM CO -M CO CD CM ~f ~f T. ri to p X CO Ht)«' O O '^ '^ CJ P CM -H -^ XI -t- O O "^ r^ -* i^ •* CM — 1 t^ r: CO -f . to i^ to cc ^- C glOCO c i ^ "^ -IS to CO C» CO ^M »0 :o CO t-O ■— ^ t*- b- 30 CJO f^ t- CO t^ O to CO -*■ C« to to C» CO GO S QO — ' CO ■M o: c; . «o t--» CO n -*• to O O CM c^^ i^ Hx 9 r *? «:- f ^r lO lO »o »o I— -^H ^. CO To CO ci oc i-^ j| ^ r^ 00 -M CO 6 3C CO o f^ -M X -^ — CO — 71 r; CO 7C1 cfi fl — CO to -f p 71 cfi CO ri i-o CO ec — t- ^|-^ 'O O l~ CC l^ ?^ ''" c^j ->i '^^ ri ct o^i -- 1 O -+' CM CO ^ o; ^. I cp CM ?t I;- cc CO t^ j Oq CM M CO l^ r^ ^ CMt^ _. CO CO 5^ cr>i 01 ' c -1 i a> 5^ et-^ 3 P i: S o o .= CM 'S' ■(J 1 ll S S. o ^ S i WEIGHT OF ANGLE IRON. 279 3 ^'C 72 Et, c S.g 3 I?; CO Hx r^ rj :m r" o I I 00 O r^ re o to CO oo Oi O O — -^ 3^ M O t- i cp ,-1 ^^-coaori550^•^'^^ 3i iS ^ — . CO ir: 1— I t^ re CO -X ! 1 i 1 CJ — 1 re O t;^ r; — ^ tc- cc C: ' 1 ' 1 ' 1 ih o t- t- do oD j; j^ o c -^ ■Qtx «!S| MM t-rc— looo^ooonc^cocco 1 1 1 1 »p o »f c^ -f- r; ?t oc re t;- ^1 b- :^^ 1 1 ' ' -?t< »h »h ue tb «b t- i^ 30 30 c". c: o c-.'- reu:t-X)02-r^-:iN ?7?t-*4f«Jt-t-b-aoaoJ;; O0--JJ — t^'^Ob-reooe^i~»-e'Mooo^ s^cioret- — -fcoMOCs'TitrOfet-'^ e^■^^^tx:e-^'j^'*'0»hlhiit-t^-b-oo '-i;: ?-ire-titrt-oo~— 1 O op -;^ 'fi b- O -^ t;- O i ! 'M ro r: cc cc cc r^ itt I O re t- C O 00 cc i--; t^ O re i^ O CC ^H o oc "O ic re 00'-i'Mre-*i»o»ecoi:^ccciO !M C^ !M C^^ 3<1 C) M C^l C-1 re iore-^c^t^>crei— lOcocO'+f'MOcocouiee oj t^ lo ei o cc tp -ti ri Ci t^ lo re -^ 00 -o -^ c^ (X) c; o --H ^1 ">! re -^^ >i lb '-b t- cio r; c; o '-' -fi — I 1-H (M 2^1 :>^ c-i c^i •M ^i 3-1 ■>! c^i s^i C'l c^i re re rc lGO^-<-*!£Oi'Mieicoi— i-*it^ore«r>C5'-i-^t^Ofe«c i-o-ti,— ico»oceot^»oc<»cit^'*i-HootoceoooioiM ue «ct-t^oo5^00^^'M'Mre-+i»oic«ci^ocacci ^ ^ ^ ^ ^ C^l Js:) C-T -M (M >T T'T -M C^l "M ^1 •>! n 'M Hx |'MOco«orer-cit^-*''MOt^»ore^co;r!-#-Hc:t^ .._. t-- -^i o t^ '*' >— ' t^ -^i ^- ^ le I— I CO o "M oo le '>! d le ^t '''"UH»h«bsb^-ooccrsoOr-'e^^ire'*'*»ci«bb-oo Lh r-H r-l ^ -™ r-H ,-H r^ M C^l M •M ^1 C>1 >1 J-^l ■>! 3^1 C^l "M (M icaooceiooooreocoore>jecoore»oooore»a t^reo;o'Mooo^^i^reotc^ioo»pT--i(--cec^(>i cc-^ibiitbt^aodoi^oO'— I'-i'J'icb ce-^ibiei ^ ^ _ ^ ^ r_l ^ ^ r-i 1— I C^l C^4 -M -M G^ (M iM !M -M ^ ^ «iTt : o loo-ici^c-*''— iao^reoco>C(MOi>--*i—iCi«ocerH t~reo0'*0«0r-i e^irece-^t^ioib^bsbt-b-oociCiOO'^-^e^cere'+i ^ „ ^ ^ ^ ,-1 ^ ^ r-( 1-1 r-l fH r-l Cet^c;T— i-rt< |t^ -M i^ -M cc ce oo re cC' '^ r; "^ Ci -*< O »c O »e c tp 'y- ''^'>i'fTre«-*i-^»bihi«bb-i>>30Cic;c:O'— '-^'1 •Six koreC;t-~-*^ooO'+ioD»-e>TC^'Xireo?creo— --^ i^o^toO"00-*Ci'^qcrecoo^"^^-:^1t-r--'^pT-Hlpo io-— i'-^>t>ireMre4tJ^ieiecb«bt^t^cbx/c:ciO ^S oooe^ireict^ocOtMreob-coO'Mreict^coO'M »ieO'+'ooc^icoo»ocirei:^^^>oo-*oO(Mtcooj5 ^0'*|-^^•^re0^rec;^0(^^CllO!^■^cOiO-Ht~^^Ot^ -*oo'M»ociretooret^FH'*iooC'i»ociretcO'*t^' X)(»diC:c:oO'-Hr^T^'>iJ'i>irerere-4t<4tt^coci^(Mce •* to i>.coc:'-''Mre-+ic£:«>'OOC: rc«£>ciC^iicci(M»ocoi— i'#t^O"^t^Ore!OCiC'i>c t-t-it-cOOOOOCiCiCiOOO •-^ c^) T'l c^i c-rec:iOi— it-reCi»C'— it-recjio-^t^rec; ^ ^ t^ Ci -M -* I- O ■M »p t- o re o c» o re ?p a; — re hbOssbot-^t^-t^dooooooociCiCiCiOOOO'-''— < 0'-H'Mrere^ic«ct^ooco5;o— 'i^rorC'+iwscct^ 0-wi'— ireiob-cir^ u:5>h>hu5>i5i>«b-oocoooaoooo5 v^»-'^0*-£>^^t^reco-*i0«0i— itoi-iooce i^U-,^, ^/^^-.^lOi^ooOiMre »p<»ooc5-H ''^ « re -^ -* -^ -^ -^ "* "b »c »c lb >'o "b >b cb '|H -^ -C -.£ I ^H-H(Nni4' -H'-^ c 5 i: ~ WEIGHT OF ANGLE IRON. 281 c < > < t^ t~ I;- X cc ~. c; C p C — ~ i>i ^1 ^1 ^^ :^ ■* "^ cq — O r: x t^ tr ir: IT r^ M — C X t^ "-c: ^ -^ ^ -j= cc oc x i- i^ t- I;- t^ t- t;- t;- t;- t;- 5f ^ '-f '^ '~f =f -H -^ GC T^ -^ c: cc t ^ X M •— r: -ix C; X t^ t:^ ""^ i-~ i-'t "j^ '*' ^7 ^1 ^1 — P •"^ !oof;0 — ■^^^^■^L^Or^-xr:C•— ^t t^ — "^i X C ~ ^. X t- tC iC IC IS ^ -4x t^ »0 -e n — -2 , r. X t- -^ IS «r> t- X r^ o r; X — i-~ r: ?T — r; X tc »s re ■^^ O i , r: n — o r: X t- »^ -^ rt ri — O r: ; r-|i — ■r)~t-^-^i-':-ib-xf;0 — ^'^^ '— ' C 'cx'-ror: — C:x--rt--;r; — C:x:risrc-^0 | ^ l»oic'-2t^xr;oo — 'Mc>e-*'>s>c^t^xcic; i ^ L — ir;t^i2re — c;x%r-5-M~x:co^:~cr;t;- re r^ -^ lb i '-b t^ X f: i C — ^1 ^: rt ^ )-': i t^ — . -M u; X — -^ — r: ?T o X — -^ i^ O re — ="- — ^t^-j._i::r;^rrecx'SC'iC;t^'<*^ic:^re — C — ^1 re rt 4*< ib i Or t^ X f: r: o -^ f^ c^i re 'i* rerererererer^rererer^rere-^-^-^-^-*-^ >e X ?- ^ X le re — X cr -t t^ ?:' t- le ^ ^ X •>:: re t- — ^ — ?i ?i ^ 2 — ?i re re "^ le ue '?^ t^ t- X ?^ ^^ ^ X X te ^ >s S 5 re »e X X le ^ le re C re le X O rt X o — ?1 X X ei :e S zr ^ e^i re H -r le le ?. ti t^ X X 15 ei 5 ^ ~ ts re le X le re X t^ le ei c\ r: ic o 1 »e ei le ei X X ei M e^i ?T ^ *^ — (M e>i H :^ X t~ t- ?i X o 1— re X :$ X re ~. ^ te t^ ^- — j; 1 c^i r^ re -u -*i le le vs iC i^ t^ X X c: c^ O c- — e-i — x^ — xieeTc;'-crec:--orec:»^-^--xo I _te L3 ri -r- ^ re X re t^ ri t- ei !^ — «c p »f p 'f Ci ; j.;= "'" ,oo — •— eii"irere'*-*'i»e5bit-t>-xi!X ' ■M ei c-1 ■>! ei "M ei e-i rq e^ :m ei c^^ ei ei e^i yi e^ e^i ■ ' re 'e 1^ X c M re le r- X c ^1 re le t-~ X o ei re , ^.j, . re t m c; -?■ X ei '-c p le r: re t- — f p -f X | ^j^, ' x X c: f: i C: o — — ei en ei re re -*■ -^ »e Le ue i t ; — — — — 71 7-1 :m e^i e-1 ei t^i c ^M ! — ei re "^ -x t- X r; — ei re -e --c t~ X r: — • ei re ,,3. ! r: ei ue X — -^ t- O -*■ t;- p re -^ p ei »e p ei ip re -^ -*i 4t< ih »h le Or Or 0: t^ t^ t- t- X X X r; ~ - if -/ -*r)rt'-f -—-*:)?: - — '— - 282 WEIGHT OF ROUND AND SQUARE IRON. Table of the Weight OF Malleable Round and Square Iro:?^ IN Lbs. per Lineal Foot. *i = Weight in Lbs. Weight in Lbs. ~ CO i= a 8 Weight in Lbs. Round Square Round Square Round 167-53 Square 3 •093 •117 n 36-812 46-875 213-33 i 4 •164 •208 7 g 39-306 50-052 i 172-81 220-05 5 1 « •256 •326 1 4 178-17 226-88 f •368 •469 4 41-884 53-333 183-61 233-80 ' 1 R •501 -638 * 44-542 56-719 2 189-13 240-83 ; 1 2 •654 •833 1 4 47-283 60-208 194-73 247-97 ■ 9 1 R •828 1^055 3 g 50-105 63-802 4 200-42 255-21 1 1-023 1-302 1. 53-009 67-500 7 8 206-19 262-55 : 11 1 R 1-237 1-576 "8 55-995 71-302 3 4 1-473 1^875 3 4 59-062 75-208 9 212-04 270-00 ' 13 1 1^ 1-728 2^201 7 g 62-212 79-219 i 217-97 277-55 Q 2-004 2-552 1 4 223-98 285-21 ■ 15 1 fi 2-300 2-930 5 65-443 83-333 f 230-07 292-97 J.D 1 3 68-756 87-552 i 236-25 300-83 1 2-618 3-333 1 4 72-151 91-875 h. 242-51 308-80 i 3-313 4-219 3 g 75-628 96-302 t 8 248-85 316-88 i 1 4 4-090 5-208 1 79-186 100-83 255-27 325-05 3 g 4-949 6-302 1 82-827 105-47 i 5-890 7-500 3 1 g 86-549 110-21 10 261-77 333-33 4 6-912 8-802 90-353 115-05 I 268-36 341-72 ; 3. 4. 8-017 10-208 X. 275-03 350-21 '■ 7 9-203 11-719 6 94-238 120-00 1 281-77 358-80 O i 98-206 125-05 1 2 288-60 367-50 2 10-471 13-333 1 4 102-26 130-21 295-52 376-30 ■■ i 11-821 15-052 106-39 135-47 4 302-51 385-21 i 1 4 13-252 16-875 ' J. 2 110-60 140-83 7 8 309-59 394-22 : 3 g 14-766 18-802 1 114-89 146-30 \ i 16-361 20-833 3 1 119-27 151-88 11 316-75 403-33 : 1 18-038 22-969 123-73 157-55 A 8 323-99 412-55 : 3 4 19-797 25-208 1 4 331-31 421-88 1 21-637 27-552 7 128-27 163-33 f 338-71 431-30 o i 132-89 169-22 1 2 346-20 440-83 ; 3 23-560 30-000 1 4 137-60 175-21 1 353-76 450-47 ' 1 25-564 32-552 3 g 142-98 181-30 4 361-41 460-21 - 4. 27-650 35-208 1 147-25 187-50 7 8 369-14 470-05 1 29-818 37-969 1 152-20 193-80 1 2 32-067 40-833 3 4 157-23 200-21 12 376-95 480-00 ' 34-399 43-802 7 8 162-34 206-72 is Round Square Round Square Round Square Weight iu Lb?. Weieht in Lbs. Weight in Lbs. WEIGHT OF BULB BEAMS. 283 Table of the Weight of the Butterly Company s Patent Solid Wrought-iron Deck-beams. Depth of Beam (ins.) I Width of I Top Flange I (ins.) Width Average of i Weight Bulb ' per Lineal (ins.) Foot (lbs.) 1 2 3 4 5 6 7 8 9 10 16 H 15 H 14 6i 13 H 12 H 11 H 10 6 9 H 9 H H H H H H 91 H 9 153 to 56 I 52 „ 55 i 50 „ 54 I 49 „ 53 I 47 „ 50 i 43 „ 44 35 „ 37 I 42 to 45 31 „ 33 -. , , ■ Width I^ePt^, of T,""^ Top f.^ .Flange (^^^•) (ins ) o 4 4 3 Width of Bulb (ins.) Average Weight per Lineal Foot (Ibs.j 1^ i27 to 28 22 „ 25 U 19 to 20 14itol6 If 16 to" 17 * These two are bulb ang le-iron ; all the others are bulb T-iron Table OF THE Weight of Solid Wrought-iron Bulb- | PLATE Beams. ~ 1 ^ — ^ 3 • .a o u = IWeight t_ '~: \ per s =5^ Weight per p; ^^ o s7 5 Weight per o i. ° s Lineal ~ o: c :: ■ c X Lineal o i S ■" Lineal 3 15 ? 1 ^J ^ 1^ Foot (lbs.) a .gX 5'-^ Foot (lbs.) S" Thick Web Foot (lbs.) 12 a. 2f 1 39-20 9 _7_ IJL 1(3 IG 16-64 7 5 16 ¥ 9-02 »> o 8 2^ ! 31-40 >j 8 IG 13-55 >> 4 7. 8 6-85 1. i \\ \ 24-09 » 16 H 11-21 6 k if 14-1 11 i 2t 36-70 5> 1 4 8 8-52 55 7 16 il"6 12-06 ^ 1 2^ ! 29-32 8 1 2 If 17-42 55 !¥ 9-80 >» 1 2 If 22-42 » 7 16 ^16 14-98 55 Is ^* 7-98 10 3. 8 2j| 24-92 » 3. 8 ll^ 12-30 55 1 4 ,8, 6-02 >» 9 16 1 li| 23-70 » 5 IB H 10-06 o 1 •3 H 12-42 » 1 2 If 20-76 »? 1 4 7. 8 7-69 55 7 i^ 10-60 »> 7 IG 1^ 17-54 7 1 2 1^ 15-76 55 3 8 i^ 8-55 » 1^ 14-80 >» 7 16 ^16 13-52 55 5 16 1^ 6-94 9 1 2 ! If ' 19-09 >» % 1-^ ^16 11-05 5» 1 4 ¥ 5-19 Table of the Weight of Deck Caulking in Lbs. per Foot Run. Thickness of plank (ins.) Size of seam (ins.) Weight per foot run U 70 -60-50-40-30 -25 '-18 -10 ■284 WEIGHT OF GIRDER AND BEAM IRON. Table giving Weights of Girder and Beam Iron, AS Furnished by Messrs. John Wallace and Co., OF London and Dundee. _ 1 CD 1 1^^ 5 « 2; GQ 1 ■73 ^S"- ■ -»3 . ^ U. '-^'o-~ H 1 o 1 1*^ 3 "3.51 H 1 "o ^ ¥.S GIRDER IRON |— H 1 > «2 ! 20 ! 1 1 7 90 21 57a 8 t 6J 38 1 62a 20 ± n 100 22 14 8 iV 5 29 2 63 19 1 6i 88 22 14a 8 t 5^ 35 2 63a 19 1 7 97 22 15 8 f 4 22 3 64 18 H 77 22 15a 8 * 4| 25 3 64a 18 1 6f 86 23 6 8 •1^ 2i 15 4 65 17 t 6i 70 23 6a 8 7 16 2t 18 4 65a 17 f 6| 77 24 31 7 1 4 20 5 5 54 54a 16 16 -9_ 1 9 58 70 24 24 31a 13 7 7i -9- 16 5 16 4i 3i 25 18 6 53 15 i 5 50 24 13a n i 3| 22 6 53a 15 1 5| 60 23 5 7 t 2i 12 7 29 14 9 IS 5i 55 23 5a 7 2f 15 1 ft 7 29a 14 3 5| 65 25 12 6t 16 ^R 10 8 55 12| 9 5i 45 25 12a 6i i !t 20 8 55a 121 s 5i 53 20 20 4 4a 6i 6i t 2^ 2i 11 14 9 27 12 i 71 80 Te 10 27a 12 1 7i 85 25 19 6 7 16 5 25 11 26 12 9 6 55 25 19a 6 i •5i 30 11 26a 12 f 6i 65 26 60 6 1 4 19 12 10 12 7 5 40 26 60a 6 i 4* 22 12 10a 12 i 5i 50 26 30 6 5 le 3 14 13 58 11 JT_ 5 36 26 30a 6 7 T6 H 18 13 58a 11 i" 'H 43 17 16 54 5 16 l^ 12 14 25 10 1. 5 35 17 16a 5i if 3 15 14 25a 10 a 5i 45 18 3 5i i 2 10 15 9 10 7 4i 30 18 3a 5i 1 n 12 15 9a 10 g" 4i 40 16 18 5 i 4i 22 16 23 ^i # 4i 27 16 18a 5 J 4f 25 16 23a Sir 1 4g 32 19 17 5 J 3 12 17 8 9i i 3i 22 19 17a 5 7 3J 15 17 8a H § 4 30 15 2 4| i 2 8 18 11 H ^ 3^ 20 15 2a 4| 1 2i 10 18 11a H i H 24 14 52 4 i 3 11 21 59 9 1 6 39 14 52a 4 # 3| 13 19 56 9 f 4 26 12 1 4 ^ 2 7 19 56a 1 9 i 4| 29 12 la 4 ■ 5 16 2J 9 20 7 8J ^ n 18 13 H 3 16 2 6^ 20 7a 82 i n 22 13 Oa ^ S T6 2i 8 21 57 8 X 6 35 11 24 3 i 3 9 2 BEAM IRON |i-« 32 32 75 74 12 9 6 5£ 49 36 34 44a 7 i 4t 24 lOi ¥ 34 43 6 g 4i 18 33 73 9 i 5i 30 34 43a 6 i 4f 22 33 72 8 # 5& 25 35 42 5 5 16 3* 12 33 71 7 1 5* 22 35 42a 5 7 la 3| 16^ 33 70 6 5 4J 16 35 41 4 s le 3 9 34 44 7 i 4i 20 35 41a 4 3i 1 16i WEIGHT OF 5IALLEABLE FLAT STEEL. 28^ •£2 Bo OiOOirsOiCOi^OL-tOOOOOiOOiOOOO • "•-.jc I^ CO GO oo t^ t- o cr i^ o :c '^ ir; I- o cc •>*l a: o ^ c .. ^"- M « -* lO CO t^ t^ 00 CI o ^ I— 1 »c '^''' 3q(«-4^i-^ib«b«bt-ooc»s^O'^'^^«cC'4tH»i:»oeo Iir-dcccoo-^QO-^iccicctco-^-oo— lOCicctoo in — < oo -^ ^ 1-- cc O "O C'l c: «p I'J 00 -* -7- t- cp o «p oc i |-*i(Mi— ic;oo«c-*cc-^Ooot^orcc^)Oc;t-ts-rt«oc ' -ic cccniootocqoo-^ocpT-t-cccio-y^cpi^icp^w re CO CO s CM CM CO CC 00 QO CC ?. ^ ^ t^ cc CO c; c CO '^ CO CM C-1 (M CM cc cc 'i- ^+1 10 CO CO l> t^ 00 cr. Ci ^ 1-H CM ^ =■•:';: I— iClt-lOCCOOOCO-f-MJlt^OCC— 'OOCO-^C^lOOO w;CCQOCCOOfCt-CMl^C^CO^-COr--COO>nO»nO"^ f^cb»boooO!M»aooocc»ncoo^i»nQOOcc»ocoo ''i'^c^i»pcicccoO'^QO'-<»p5i T^»^b>h»hcb so t>- t- t- 00 00 00 ^M QOSl—tCCmt-C- — CC-+ — 3: C'M'^COOOwi-HCCO (,lH,li,lH?'Ci!f^(>icccccC"^'^'^>^c»bmcoc;cot— t-t^ mx COCCCSCOCCOSCOS'lCiin-MOOOO^OOia— lOO-*!— '00 OCCiOOOi— iCCCOCir-<'rHt-Cic>l»f5l>-OCCOaO'^CC ^Air^T^iyii^i'icqcc'Mccot-^'^'^'h'bihibcbcb * Ht)-! iCcooooiO"— icC'^>ocoaoc^o-^cc-«*iiacoocc^O oooc-^-^t^c; -Hccuct-cr. 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W '^ ^ t^ '^ O «£ C^ C^ IC ^1 GC »p — t^ o r7 «c ^^ »b i t^ t- 05 c^ c: o ^ — •fi 'fi cr: -^ -*! »h i i i c; o ' ' -— — ' '— 1 -^ »-^ — -^ '>j M -M -M ri 'M -M ri C-) c^i :m c^i -m r; -■•c !p -H t^ CO Ci »0 .— t^ 3^1 CO '^ O iC: •^^ t^ re Ci IS t^ 00 C: '-I-.; "'^ 4f< lb lb «b ! cs r: -* il- -i do '^''^ -H ^ — ^ ^ ^ — -^ — — . -M -M -M -M -M -M :rj -M ri -M 3^1 loo— '-t'CO'-i-*«>-0«tC!r;ro«oc:?>iiaaO'— cc-r'O! ^^ijn! '>! <» v^ <» ^ w^ -* O IS O lo -^ ep ty^ b- '>! b- rs «>t -rhi ip 1 ^ «r:-^-^ibibcbt^t^cbcoc;cr. oO'^--'f-if:4wib| ! — 1 , — — — — — ^ — — -^ .— ^1 M M -M ri -M M :M ! "' -1=- uses — r;b^'^MOoo->crs^r-. t-'*icsooO'+'riis ri-rJ-C^^XISOOrSb-^b-^lcp — CpT—cpOOCiC; F^si'fi?trt4'-Visisebob-b-aocbc;c:o — — (n =e i -^T csisooocsisoooisisoocrtccooorsisoiso «pO-7"CiCSt^ — OO'^OOPtb-— lSC'fCCb-lS-5f' O — — ^'N'>i?^cb4'"^'^ibisiib-b-b-acr;c> -^151 1 -^ Ob^'^>-'co«ofoot^'*'s.ic:n!:ccsooois'Mt2— lis rt(piO-^t,,-Hiocicq«pocY3b- — iccoritsrs»— c» CiC5000'^-^'^!^ii^icb^cb'*4f--*isisib-b- '-is 1 ,i^<»oooociCiCiOOC:-^'^'^i-i>ii''ir:r:-*i'^ib 1-^ — t~'+'Ot^csot>-ccOtccsc;s t^ Ci O ?-) c-t IS tp o rs «f! .-j^ '^rt'*-^'^'^-^-4jHibisisisisiS'i«bi^i'fi^cbcbcbrs^cbcocbM'#'^-^'*-^-*-*ibi te 1'-' rsoocsci-^ciisois — o.-'b-cot^rtoocs-^-tiio rSTS-*'-^>sistCb-t^QCcoc. c; 00»^ — cirs-^HO -!2 1 c - i^ c : ^j - '-"- ir C^ WEIGHT OF ANGLE STEEL. 287 H O O Pi CO P5 1-2 o c|x f -12 , , , , , , , . , .■MOt^»OCC'-ir>SO'*(M r^lCl I 1 j 1 j 1 ] ' ^ o ^ S !p cp ■» c>i ^ 9 S cr. -fM -12 ►-IS «lx O M t- o ro -o cr. re "^ 5i c^1 »p r; ri uo c» KlX -!2 (Mict^occiooo— 'CotDcrir^^t^ci'y-iipt- .l^ ,14 4-1 j-i •f^i f-» fi ic « « Jb ^ '^i ^ ^ «5 o m -IS r^l-^ acoccioooocc»ocooeoiccoocoiCQOC«i-': :ri??ot^cx)0 — o^^c>olX>^^GOO^?-^cc»c«ct-- •o ^ o 5^ ^ t- c -3 M »p b- c: 7- -* -^ CO O ^J -f^ «f Hj^ "IS COt-t-«OiOiC-*i'*.ccr:^^co-#^t^coci«0'McO'*';^ ?.?:Sc;Sco-HriM«-#-#uouo3£t-t-ooci ^ ,1, ,!< J-, A- ^ ,1- ^ ^ ^ CM ^1 'M .-v-*Mr::ri< -'!'»-4n«!-*' -4!-i'-'I«»«l-* ^ Z%-r,i 3<, C^V T^M CO CO CO CO -* '^^ 'f ^ LO IS 1 <*-< a: ■^■M WEIGHT OF AXGLE STEEL. r- aC Oi .„ -<'-tHoi?;l-t -H'-r-'kNrii-)' -h:-)<-i cq c^i (M c^i ! c<4 ■>! ?^ cq iCCfM-— (OC;c0t^?C»0>O'*iCC(M'-tOO00t^«0>0 Cicc«r>o-^ooi— (iaoicc«oo-*co^^»ac;cccco 'MOcoMioco^-*t-Orct^O«i:cic:CqQ0-^OO(M00THO«0^^-M '■■^ H',ei oaoocciCQOOMiacooecocooccioooOM -l*> ""IS '^s hi^ I -'•*-*N~'-f" -I>-4C»«1-J' ^l-tl-Jl'yKcl-j" .-il-ll,-l|C«>5|'# i-ll-4'H0*«l'* a a a '^««£>ot-.»>.t^t>.cocoQOO&CiC5C5CiOOOO o o ^^ S g g WEIGHT OF ANGLE STEEL. 289 5«S I-l'-ir-l'-l'-l'-lr^.— M^ — r-r-. .-^,-l--,-H| -t: "5 c a o ■J. ■r. 3 £ O 3C I;- tp -T- :^^ — C-. X --^ o r;: :m c :r. i- --^ •^ 1 T- -^ o i t^- tx> r: c^ O -- ^1 :^ -^ '^~ lit i r^ ■-1 ICCCCCCICCCCCCCCO'*'*-*''^'*'*'*'-*!-*! 1 .2 1 § o 1 •r. y c L-.-j ' O 3C tc 'r C^< O qo ff ^ ^1 p qc «p -Tf !>) c: oD f^;--! : ^ ^-, ^ ^ j}: 0:; cb b- ( c^ -i- -^ti ole iZ — cctcrci— icp»p:ppoc»prcot-ir:?qcp Hx < Iz: f..„ T-( OD IC ->^ Ci 5f: ->! C; Cf ?t p I;- -^ "7 X.»p r^ 212 -1 2^ 1 ^ CO — 1 -o Ci cc --C c -+ cc — >^ ?; c^ •£ O ^ ! i i t~ X 00 ii Ci O — -^ "T-l C^ ?t -ri >h »h -o ^1 CM ^1 ^1 7-1 ■>! -M re cc re c^ re re cc :^ r: re -,^ — —. t~- '-r: -^ re r-, o X' <:;: ue re e-1 O c: t- ^ ^153 f-^i;p(>-jcC'*ipp(>ii;-c^p>f'7^t-|^cO'^ c-1 c^ (M (M c! i CM M CM CM CM M c^^ CM re re •oix H ^ e-.|^ tc -^ -M o t- le ce — cr. tr -* M C: X tc re — p^p-*iCOceXC^t-(Mt-CMt--';-p-7'P c^ CO o -^ "-H "M 'f-i re c^ -^ -+' 'e o — ti t~ b- — M CM •M CM -M ei CM CM CM - I CM -M M CM M CM -i'i;i ic oc o re j-e CO o re »e 00 C re ic oc o re o ..^ X (?q t^ -- ip p -+i CO ei <:f .^ ue 3; « GO >i -o t-xoociCiC^oO'^ — ci'Niricere-*'-* 1 — 1 ,_! ^ ^ ^ _l ri CN !M e-T CM n en -M CM CM e-i 1 — - X ?c re o »>• ':*' — -i ".r re o t^ i-e^ CM ^ ^ j.to t- O '^ X CM o r; re tr c; -+- X — >e r; m ^C5--reo „i3,''p=<5"^'iXr-i-^i-~ — -^t-prepprep i^t ~7 4- -^ '^ »h lb ih i i 'i t- t- ^ ^ ^ ^'^ y.lx Si »i. •-r-Mc;^e^Xlr^ — xice— 'X-"— it~-7j;rr repx — '^t2pcM-*i>pcopMp r^-^rl-i'vi'MCCICM«~^cb'^-*i-+'^>b»bo -15 P5 Tj^uetcxcr. CO — ce-r-i^^trx , , , -'- z; 11 g § g r II f 2 z: 1 1 1 1 ! "-'?9^^? 1 1 1 1 1 1 1 1 1 1 1 1 j -o t- t- t- t- h rfl-H^lf^iJ^;],). rfl^^^jr^l-. ^lj_ilo>~'J. rfl-H-iteKOl'* — 1 — r-i — s^iiM'McMrererere-rH-^-^-^ic ' i as au G 1 o X 11 II II ^ CC Is "S *2 cS > js ;:; "sE.S ^ ^ o -CC t. T3 t^ ^"^ — ' e:: .Sfo rt g'^ WEIGHT OF ROUND AND SQUARE STEEL. Table op the Weight of Round and Square Bar Steel IN Lbs. per Lineal Foot. Weight in Lbs. Round •042 •094 •167 •261 •375 •511 •667 •845 1-043 1^262 1^502 1^762 2-044 2-347 2-670 3-380 4^172 5-049 6-008 7-051 8-178 9-388 10-681 12-058 13-519 15-062 16-690 18-400 20-195 2^-072 24-033 26-078 28-206 30-417 32-712 Square •053 •120 •213 •332 •478 •651 •850 1-076 1-328 1-607 1-913 2-245 2-603 2-988 3-400 4-303 5-313 6-428 7-650 8-978 10-413 11-953 13-600 15-353 17-213 19-178 21-250 23-428 2.5-713 28-103 30-000 33-203 35-913 38-728 41-650 Weisrht in Lbs. ^.2 I Round I Square 31 35-090 3 4 37-552 1 8 40-097 4 42-726 1. 8 45-438 i 48-233 3 8 51-112 54-075 .57-121 60-2,50 63-463 66-759 70-139 73-602 77-148 80-778 84-492 88-288 92-169 96-133 100-18 104-31 108-52 112-82 117-20 121-67 126-22 130-85 135-56 ! 140-36 1145-24 1.50-21 155-26 160-39 44-678 47-813 51-053 54-400 57-853 61-413 65-078 68-850 72-728 76-713 80-803 85-000 89-303 93-713 98-229 102-85 107-58 112-41 117-35 i 122-40 127-55 |l32-81 il38-18 143-65 149-23 154-91 160-70 166-60 172-60 178-71 184-93 191-25 197-68 204-21 :g mi Round S(juare 5 2 i R"und | Square ^.a Weiglit in Lbs. 10 Weight in Lbs. Weight in Lbs. 11 12 - a Round 165-60 170-90 176-29 181-75 187-30 192-93 198-65 204-45 210-33 216-30 222-35 228-48 234-70 241-00 248-38 253-85 260-40 267-04 273-75 280-55 287-44 294-41 301-46 308-59 31.5-81 323-11 330-50 337-97 345-52 353-15 360-87 368-68 376-56 384^53 square 210-85 217-60 225-25 231-41 238-48 245-65 252-93 260-31 267-80 275-40 283-10 290-91 298-83 .806-85 314-98 323-21 331-55 34000 348-55 357-21 365-98 374-85 383-83 392-91 402-10 411-40 420-80 430-31 439-93 449-65 459-48 469-41 479-45 489-60 Round Square Weiglit in Lbs. WEIGHT OP MALLEABLE IRON PUPES. 291 Table of the AVeight oe Malleable Iron Pipes ix Lbs. per Lln'Eal Foot. B.'i-e (ins.) Thickness in Inches 10 J 4 1 % 11 J 12 I -I. S-'27 3-93 4-58 5-21 5-89 6"55 7-20 7*85 8-51 9-16 9-82 10-47 11-13 11-78 12-43 13-09' 13-74 14-40 15-05 15-71 1G-3G 17-00 17-67 18-33 18-98 19-63 20-28 20-94 21-60 •22-25 22-91 23-56 24-21 24-87 25-52 26-18 26-83 27-48 28-15 28-80 29-45 30-75 32-07 Bore (ins.) F ~5^ 6-38 7-36 8-34 9-33 10-31 11-29 12-27 13-25 14-23 15-22 16-20 17-18 18-16 :9-i4 •20-12 21-11 22-09 •23-08 •24-05 25-03 26-01 27-00 27-98 28-96 29-93 30-92 31-90 32-89 33-87 34-85 35-83 36-81 37-79 38-78 39-75 40-74 41-72 42-71 43-69 44-66 46-62 48-60 F 7-85 9-16 10-47 11-78 13-09 14-40 15-71 16-02 18-32 19-63 20-94 22-25 23-56 24-87 26-18 27-49 28-80 30-11 31-41 32-72 34-03 35-34 36-65 37-96 39-26 40-57 41-88 43-19 44-51 45-81 47-12 48-43 49-73 51-05 52-35 53-66 54-98 56-28 57-60 58-90 60-20 62-82 65-45 10-63 12-27 13-91 15-54 17-18 18-82 20-45 22-09 23-72 25-36 27-00 28-63 30-27 31-90 33-54 35-18 36-82 38-45 40-08 41-72 43-36 44-99 46-63 48-26 49-90 51-53! 53-17: 54-81 j 56-45 I 58-08 ! 59-72 61-35 62-99 64-62 66-26 67-90 69-54 : 71-17' 72-81 74-44 76-07 79-35 82-63 15-71 17-67 19-63 21-60 23-56 25-52 27-49 29-45 31-41 33-38 35-34 37-30 39-27 41-23 43-20 45-16 47-12 49-08 51-05' 53-01 54-97 56-93 58-90 60-86 62-82 64-79 66-75 68-72 70-68 72-64^ 74-61 76-56 78-53 80-50 82-46 84-43 86-38 88-35 90-31 92-27 96-20 100-13 21-76 24-05 26-34 28-63 30-92 33-21 35-50 37-79 40-08 42-37 44-67 46-96 49-25 51-54 53-83, 56-12 58-4l! 6O-70; 62-99 65-27' 67-57: 69-86 72-151 74-44! 76-73 79-02 81-32 83-60 85-90 88-18 90-47 92-77 95-06 97-35 99-64 101-92 104-22 106-51 108-80 113-38 117-15 28-80' 31-41 34-03 36-65 39-27 41-88 45-50 47-12 49-74 52-35 54-98 57-59 60-21 62-83 65-45 68-06 70-68 73-29 75-91: 78-53 81-15' 83-77: 86-38' 89-00 91-62 94-24 96-86 99-47 102-29 104-71 107-33 109-95 112-56 115-18 117-79 120-42 123-04 125-65 130-88 136-13 W Bore \ (ins.) 36-81 39-76 42-70 45-65 48-59 51-64 64-48' or4o; 60-38: 63-32: 66-26: 69-21 72-16: 75-10' 78-04 1 80-98 83-93 86-87 89-82' 92-77' 95-71 98-65 101-60 104-24 107-50 110-43 113-38 116-33 119-27 122-22 1-25-16 128-10 130-05 133-99 136-95 139-89 142-83 147-95 154-61 45-81 49-08^ 52-351 55-63 58-90 62-1 7j 65-45 68-72: 71-99 75-26i 78-54! 81-8 85-081 88-34; 91-621 94-89; 98-161 101-44; 104-7l|- 107-98; 111-25 114-52 117-801 121-07 1-24-34! 127-62] 130-89 134-16' 137-43 140-70' 143-97! 147-25 150-52 153-80 157-07 160-33 166-88 173-43 1 li Thickness in Inches U2 Bore (ins.) 292 WEIGHT OF CAST-IRON PIPES. Table op the Weight of Oast-ikon Pipes in Lbs. PER Lineal Foot. Bore (ius.) Thickness in Inches Bore (ins.) 1 3 i 5 3 7 1 1^ ii 4 5 2 g 7 8 1- ^4 1 1 3-06 5-06 7-36 9-97 — i 3-69 5-98 8-59 11-51 14-73 — — i h 4-29 6-90 9-82 13-04 ' 16-56 20-4 — — 2 i 4-91 7-83 11-05 14-57 18-41 22-55 27-00 — f 2 5-53 8-75 12-27 16-11 20-25 24-7 29-45 34-46 2 i 6-14 9-66 13-50 17-64 22-09 26-84 31-85 37-28 42-95 J. 4 i 6-74 10-58 14-72 19-17 23-92 28-93 34-36 40-03 46-02 1 2 t 7-36 11-50 15-95 20-70 25-71 31-14 36-81 42-80 49-08 f 3 7-98 12-43 17-18 22-19 27-62 33-29 39-28 45-56 52-16 3 k 8-59 13-34 18-35 23-78 29-45 35-44 41-72 48-32 55-22 I 4 h 9-20 14-21 19-64 25-31 31-30 37-58 44-18 51-08 58-29 1 2 1 9'7G 15-19 20-86 26-85 33-13 39-73 46-63 53-84 61 -.36 1 4 10-44 16-11 22-10 •28-38 34-98 41-88 4909 56-61 64-43 4 i 11-10 17-08 23-37 •29-97 36-87 44-08 51-60 59-42 67-55 4 i. 11-66 17-94 •24-54 31-44 38-65 46-17 53-99 6-2-12 70-56 h 1 12-27 18-87 25-77 .32-98 40-50 48-32 56-45 64-89 73-63 i 5 12-88 19-78 26-99 34-51 42-33 50-46 58-90 67-64 76-69 5 i 13-50 20-71 28-23 36-05 44-18 52-62 61-36 70-41 79-77 X 4 i 14-11 21-63 29-45 37-58 46-02 54-76 63-81 73-17 82-84 h 1 14-73 22-55 30-68 39-12 47-86 56-91 66-27 75-94 85-91 i 6 15-34 23-47 31-91 40-65 49-70 59-06 68-73 78-70 88-75 6 i 15-95 24-39 .33-13 4-2-18 51-54 61-21 71-18 81-23 92-04 k i 2 16-57 25-31 .34-36 43-72 53-39 63-36 73-41 84-22 95-10 i ^1 17-18 26-23 35-59 45-26 55-23 65-28 76-09| 86-9/ 98-18 f 7 17-79 27-15 36-82 46-79 56-84 67-65 78-53 89-74 101-2 7 i 18-41 28-08 38-05 48-10 58-91 69-79 81-00 92-50 104-3 i 1 2 19-03 29-00 39-05 49-86 60-74 71-95 83-45, 95-26 107-4 ' i 1 19-64 29-93 40-50 51-38 62-59 74-09 85-90 98-02 110-5 f 8 20-02 30-83 41-71 52-92 64-42 76-23 88-35100-8 113-5 8 1 4 20-86 31-74 42-95 54-45 66-26 78-38 90-8 1)103-5 116-6 1 4 1 o 21-69 32-90 44-40 56-21 68-33 80-76 93-49 106-5 119-9 2 1 22-09 33-59 45-40 57-52 69-95 8-2-68 95-72109-1 12-2-7 1 9 22-71 34-52 46-64 59-07 71-80 84-84^ 98-18111-8 125-8 9 i 23-31 35-43 47-86 60-59 73-63 86-97 100-6 114-6 128-9 i i 23-9.5 .%-36 49-09 1)2-13 75-47 8913 103-1 ,117-4 131-9 I 2 5 24-55 37-28 .50-32 63-66 77-32 91-28 105-5 120-1 135-0 f 10 25-1 (i 38-20 51-54 65-20 79-16 93-42108-0 122-9 138-1 10 i 25-77 39-11 .52-77 66-73 80-99 95-57 11 U-4 1 1-25-6 141-1 i 4 2'i-38 4004 54-00 68-26 82-84 97-71 112-9 ;l-28-4 \ 144-2 i i 27-00 40-96 55-22 69-80 84-67 99-86ili5-4 [131-2 j 147-3 1 11 27-(;2 41-88 5<)-46 71-33 86-52 102 117-8 133-9 150-3 11 1 2 2^-84 ■I3-71 58-90 74-39 90-19 106-3 i 122-7 1139-4 | 156-4 A 12 30-0(i 45-55 , 61-35 77-46 93-60 1 110-6 127-6 1 145-0 1 1^ 162-6 12 Bore (ills. J i ~i~ i t 1 1 h \ I \ U Bore Thickness in Inches 1 (ins.) WEIGHT OF SHIP FITTINGS. 203 Table of tele Weights of Messrs. James Taylor and Co.'s SiEAii Winches axd Cranes. Steam winch to lift, in tons 5 8 21 2 5 10 2| 6 10 3 7 12 5 8 12 6 9 12 88-5 Diameter of cylinder in ins. Length of stroke in ins. Weio-ht in cwts 34'o 35-0 .52 i 57 Steam crane to lift, in tons . . • | 2 2^ 3 i 4 1 Weight with pillar to 'tween decks, in cwts. 73 75 80 84 1 Table of the Weights of Ships' Galleys. | Xo. to cook for • | 12 25 35 1 50 60 25 70 j 90 100 26 i 32 42 125 44 Weight in cwts. . y 11 1 16 ! 20 Xo. to cook for . ! 150 220 5G 250 j 300 1 400 450 500 600 650 1 Weight in cwts. . 47 6(5 1 75 1 82 102; 113 , 120 135 Table OF the Weights of Doeble AND Single Pur- chase Crabs. Single Purchase Double PuiiCHASE 1 Xo. 1 To Lift Weight with Break No. To Lift Weight with Break | Tons Cwts. I Qrs. 1 Lbs. Tons Cwts. Qrs. Lbs. 1 2 14 10 2 3 1 12 2 n 2 1 16 11 3 3 3 14 3 2 3 12 4 5 1 ^2 4 o O 3 2 12 13 6 6 2 8 5 4 4 3 15 14 8 7 3 6 6 5 3 16 15 10 9 3 18 — — — — — 16 12 11 3 20 — — — — — 17 16 16 • Table of the Weight of a Cubic Foot .ind Cebic Inch OF YARiors Metals. c. Iron Cub. ft. in ozs. |7,271 CTub. ft. in lbs. ! 45p4 W. , c. Iron , Copper 7,680 8,788 Copper; Brass 87915 '8,3W Brass 8.525 480-0 549-25 557-19 .524-75 532-8 H. Steel Steel 7,81817.833 488-61489-6 Cub. in. in ozs. !4-208l4-444 5-086 \ 5-159 i 4-859 iJ 4-5244-533 Cttb. in. in lbs. -263 -2777 -3177 -3225 -3037 '-3083'-2828 -2833 50-i YrEIGHT OF LEAD PIPE AND COPPER EODS. Table of the Weight of Lead Pipe ix Lbs. per Lineal Foot, AND Lengths ix which it is usrALLT Manufactuked. iici^ 3 — ' Weight per Foot in Lbs. Wght. per Ft. in Lbs. _j •V-' ^ .3 1 •933 1-07 1-2 l-47,l-73 1-87 2-33 L2 3 9-0 15-1 1 1-2 147 1-67 i-8o: — — — / 2i -^ J. 130 i * 4 1-17 1-60 1-73 1-87I2-13 2-4 300 2i •2 9-6 10-5 12-0 — V 1 1-87 2-4 2-8 300'3-60 3-93 4-20 3 11-6 12-0 13-4 15-0 / n 3-00 3-17 3-50 4-33'5-08 o-2o — 10, 3i 13-5 15-0 16-6 18-4 n 3-50 i-00 4-67 o-osle-oo 7-00 — 4 13-5 16-0 18-4 20-0 124 If 5-83 7-00 7-33 8-00 — — — 4i 200 21-6 23-4 — t 2 7-00 8-00 9-33 — — — — V 23-4 2o-4'28-0 — . 2i 10-5 — — — — — « 33-0 — — , — * Also iu 60-ieet coils. t Also in 36-ifet cnils. Table of the Weight of Round Copper Rod in lbs. per Lineal Foot. Diam. (ius.) Weight D •1892 •2956 •4256 •5794 •7567 •9578 M824 1-4.307 lam. (in?.) Wei gilt Diam. (ins.) H 1^ 1-7027 1-9982 2-3176 2-6605 3-0270 3-4170 3-8312 4-2688 H If ^ -2 If H 11 Weight 4-7298 5-2140 5-7228 6-8109 7-9931 9-2702 10-6420 12-1082 Diam. (ins.) Weis-ht 2^ 2i 91 2i n n ^ 15-3251 17-0750 18-9161 20-8562 22-8913 25-0188 27-2435 Diam. (in~.) •H H H 31 Weight 29-5594 31-9722 34-4815 37-0808 39-7774 42-5680 45-4550 48-4330 Table of THE Weight of Cast-] rpoN B. iLLS. 1 Diam. Wght. Diam. Wght. Diam. Weight Diam 1 Weight Diam. Weight (ins.) (lbs.) (ius.) (lbs.) ( ins.) (lbs.) (ins.) (lbs.) (ius.) (lbs.) — H 12-55 Q^ 35-68 81 84^57 •14 2f 2-86 H 13-62 6't 37-81 8f 92-25 u •20 21 3-27 ^ 14-76 (^'i 4004 9 100-39 1 1 •27 3 3-72 H 15-95 n 42-35 9} 10>l-99 ll •36 31 4-20 5 17-21 H 44-75 91 118-06 li •47 :n 4-73 ^ 18-54 7 47-23 9f 127-63 1 ll •59 H 5-29 •H 19-93 ^ 49-80 10 137-70 •74 3^ 5-90 5t 21-38 7f .52-47 lOi 148-29 li •91 32 6-56 oh 22-91 ^ 55-23 10 \ 159-40 2 MO 3i 7-26 ^ 24^51 7.V 58-09 lOf 171-06 2* 1-32 31 8-01 ^^ 26-18 '1 6004 11 183-28 2f 1-57 4 8-81 4 27-92 T} 64-09 11} 196-06 22 1-84 ^ 9-67 6 29-74 7? 67-24 lU 209-42 9l_ 2-15 4|- 11057 6^ 31-64 8 70-50 11^ 223-38 2§ 2-49 H 11-53 <')f 33-62 81 77-3>> 12 237-94 WEIGHT OF COFFER PIPE, ETC. 295 Shrixkage of Castings. The usual allowance for each foot in lensfth is as follows : — In large cylinders . = ^ inch. In zinc In small „ • = ^ » I^^ ^^^^ In beams and girders = ^ „ In tin In thick brass • =^ j> I^ copper . In thin „ • = ^ v ^ bismuth In cast-iron pipes = ^ inch. = ^ inch. 5 _ _3^ ~ 16 _ 5 ~32 Table op tfe "Weight of Copper Pipe I:^f Lbs. per Li>^EAL Foot. ^ 05 CO Bore of Pipe in Inches A ir.^ 1 4 5 16 f 1 2 1 3 4 1. 8 1 3 ££ 1 32 1 IG t 5 32 3 ].3 7 32 \ •11 •24 •13 •28 •15 •33 •53 •76 1^01 r28 1^57 1^89 •20 •43 •67 •95 r24 1^56 1-90 2-27 •25 •52 •82 M4 1-48 1-84 2-23 2-65 •30 •61 •96 1-32 1^71 £•13 2-57 3-03 •34 •71 1-10 1-51 1-95 2^41 2-90 3-41 •39 •80 1-24 1-70 2-19 2-70 3-23 3-78 1 32 1 16 3 32 JL 8 5 32 3 16 7 32 1 •39 •57 •77 •99 1^24 1-51 •46 •66 •89 M4 1^41 1-70 -^ X :c Bore of Pipe in Inches fi:f7 (in U li 1 3 1| If If ^8 2 1 8 3 16 1 4 •90 1-89 2-98 4-16 •99 2-08 3-26 4-54 1-09 2-27 3-.55 4-91 1-18 2-46 3-83 5-30 1-28 2-65 4-12 5-67 1-37 2-84 4-40 6^05 1-47 303 4-68 6-43 1-56 h 3-22 -^ 4-97 j| 6-81 i 'U oi a: Bore of Pipe in Inches -Ji V. EI 2-S. H 2i 93 2| n 2f 2| 3 1 16 i 3 16 1 4 1-66 3-41 5-2o 7-19 1-75 3-59 5^53 7^57 1-84 3^78 5-82 7-94 1^94 3^98 6-10 8-33 2-04 4-16 6-39 8-70 2-13 4-35 6-67 9-08 2-22 4-54 6-95 9-46 2-32 4-73 7^24 9-84 1 16 3 16 1 4 '- a: a: Bore of Pipe in Inches ■k a: ^ 'I'lli nr (ii H H 3| H 31 3f ^1 ^8 4 3-08 6-25 9-51 12-51 |c-3 r 3 16 1 4 2^41 4^92 7-52 10-22 2^ol 5-11 7-81 10-60 2-60 5-30 8-09 10-97 2-70 5-49 8-37 11-35 2-79 5-68 8-66 11^73 2-89 5-87 8-94 1211 2-98 6-05 9-22 12-49 y 3 16 1 4 296 WEIGHT OF SHEET TIN, ZINC, ETC. Table OF THE Sizes axb Weight of Sheet Try. j ^ c Size in Wei-ht ■". Size in Weight cMi Ins. per Box op3 Ins. per Box Brand Marks Brand Mai'ks II l| 100 bo "g i6f:i2i •5^ ^ 225 13f 10 o T 02 T 1 1 IT lC,orlCoin DX 2C 225 13i 9f 3 21 DXX 100 16fl2l 1 1 7 80 225 12f 91 3 14 DXXX 100 16fl2i 1 2, HC 225 13f 10 7 DXXXX 10016|12| 1 2 21 HX 225 13f 10 1 7 SDC 20015 \\ 1 2! IX 225 131 10 1 8DX 20015 !ll 1 2 21 2X 225 13i 9f 21 SDXX 20015 11 1 3 14 3X 225 12f H 14 SDXXX 20015 11 2 7 IXX 225 ] 3f 10 1 21 S DXXXX 20015 11 2 1 ' IXXX 225 LSf 10 1 2 14 Wasters 225 13f 10 1 14 IXXXX 225 13f 10 1 3 7 TT 450131 10 lioi DC 10016^ 12J;- 1 3 21 1 XTT 450] 3? 10 I 'O 14 Table of the Sizes and Weight of Sheet Zinc. Approxi)nate Weight Per Square Foot 8 9 10 11 12 13 14 15 16 17 18 Lbs. Ozs. 7 8 9 10 13 15 1 3 .0 7 10 14 2 2 Drs. 10 12 13 14 1 4 6 9 12 14 1 6 11 Approximate Weight of Sheets 7 ft. by 2 ft. 8 in Drs. 2 6 Lbs. : Ozs. 8 I 15 10 11 12 15 17 20 22 25 27 30 35 40 3 7 11 4 12 4 13 5 13 6 7 7 10 1 17 13 2 9 15 4 1 13 7 ft. X 3 ft. Lbs. Ozs. 10 1 11 7 12 14 14 5 17 2 20 22 13 25 10 28 8 31 5 34 3 39 13 45 S Drs. 12 7 2 9 16 13 4 10 1 15 12 8 ft. by 3 ft. Lbs. 11 Ozs. 1 8 1 13 1 i 14 11 16 5 19 9 22 13 , 26 1 29 5 32 9 35 13 39 1 45 9 52 15 14 13 12 11 9 8 7 6 4 1 14 Table of the Sizes and A\ eight of Cokrugated Iron Sheets. Tliknss. B.W.G. Size of Sheets I'eet. 16 l6x2to8x3 17 »; 18:6x2 „ 8 ' 3 19 V '>0 r> y 2 .. 8 y 3 |\Vt;ht, fer jpq. Ft Lb. Oz 2:1 800 Thknss, B W.G. V\ght. Size of Sheets per Sq. Ft. Feet. Lb. Oz 21 x22i6x2to 7x2i| 1 7 I 4 1,050[23 X 24!6 x 2 „ 7 x 2'J 1 3 1 2 1 ..^OOl25 y 2r.'f; X 2 ., 7 y 2 V I ' 1,600 1,900 2,250 DDIEXSIONS, ETC., OF SHIPS' GUNS AND SLIDES. 297 Table of the Dimension s AND Weight of Ships' Guns. I bLiDES; AND Pivot Bars. 1 Description of Gun Guus Slides j |5) i3 3 ■5 ^ 3 ^. +3 be j^ 'S ~ ^ l.i rS %. .^ ^ ^ Jz v3^3 " 1 ^ Tons. ¥t. Ins. Ins. Ins. tt.Ins. Ins. ftJns. ri2i-inch 38-00 18 9-50149-4 57-50 _ _ — — 12'' „ 38-00 18 9-50149-4 57-50 — _ — — 12 „ 35-00 15 11-75 122-1 56-00 _ _ — — 12 „ 25-00 14 3 -.50 110-7 53-50 15 6 33^ 6 2 11 „ 25-00 14 2 111-35 53-00 17 3 — ~ — 10 „ 18-00 14 2 108-35 45-00 15 30 5 9 „ 12-00 12 3 90-00 39-00 14 12i 4 3i 8 „ 9-00 11 4-50 87-00 35-50 13 12i 4 04 7 „ 6-50 10 6 81-25 33-50 12 9| 3 8| 0^ 7 „ 4-50 10 450 79^35 26-00 U 9 16i 4 6' 64-pomider . 3-20 9 3^50 70^25 22-75 12 6 — 3 7 64 „ 3-55 9 64-34 23-50 10 6 — 2 10 40 „ 1-70 7 11 62-125 17-75 _ - — — 9 « •40 5 8^50 41-25 9-75 7 6 — 1 5i 9 „ •30 Lbs. 4 10 35-00 9-50 6 10 — 1 5i . 7 „ 200 3 £•9 23-80 6-875 5 10 — 1 5 Tons. f 7-inch 40-pr., screw . 4-10 10 74-70 27-70 — _ — — 1^75 10 1 73-875 16-40 10 6 — 3 7i ^ i-H 40 „ wedge 1^60 8 2 63-80 19-20 10 6 — 3 7l '.-i' 20 „ hea^y •75 5 6* 39-50 13 -.50 7 6 — 1 9 1— 1 20 „ light . 1-65 5 H 40^00 12^50 6 7 11 1 9 12 „ •40 6 38^75 9^75 7 6 — 1 H . 9 „ •30 5 2 36^50 9^40 6 10 — 1 H (Extreme) '100-pr. 6^2510 10-75 75*55 31-50 12 — 3 lU lO-inch 4-3010 8-72 67-20 27-45 14 — 3 7" 8 „ 3-2510 2-72 64-80 23-50 12 — 3 74 8 „ 3^00;10 •86 63-60 22-80 12 — 3 7i 8 „ 2-70l 9 2^75 57-60 22-75 12 — 3 7i 68-pounder . 4-75111 4-55 72-00 22-76 14 — 3 7" 7^ 32 2-90!l0 7-45 68-40 22-60 12 — 3 7| ■ji 32 2-80ilO 5^14 65-10 22-24 12 — 3 7| :^2 2-.50;iO •42 66-15 22-46 10 6 — 3 7h 32 2-25 9 5-96 62-47 22-46 10 6 — 3 7\ 32 „ 2-10 8 11^91 58-80 21-90 12 3 3' 33 „ 1^60 7 5-60 44-58 18-60 10 3 3 .32 „ 1-25 6 8 43-20 17-68 10 3 2 -• f24-pr. •65 5 3-10 32-10 12-80 . _ -,:• 1 12 ,. •30 4 1-.50 10-20 — ^ — — 298 ■VTEIGHT OF SHIPs' GUXS AND AMMUNITION. O O o OQ < M J-H O l-H GQ H w ^^ o n < :>: z; cdoooocboo o o o i 'O o cr. i-i O o o o o o C2 — >c re — M CO — r-ricoccoocO'* ■«*< ^- •* ^< —1 " CC: (M r-t -* Ci to '^i rJH !M 1— i cr. >— o c: c; ip r— 1 »p -* o rc to o a: jq !Jq -^ r^ , 1 O 1 ^ O O O lO 1 1 ao 1 cp O —i cp qs OO OS CC ?^ 'H ' xo CO t^ *?* "i" T' C ;;^ O O la 1 to r: « -fi ^ I— ( T— i t^ O lO b- b- -H O cc O lO O — 1 O Cp rH Jp M -TtH '^ c^ "jq th rli ooooooooo o 9 o o o o ip o c^l ih lb th OO T'l c; «b -^ « CC T^l C<) — 1 -H o »o >o ,-, b- o o o o o o o O O O O iO I -^ " t> " CO cq Ci «o o ■* '-' r-l Ci O o c; CO t> t- -*< r- to V ■TK'H sunS SatA^OAay; san3 apig WEIGHT OF ships' GUNS AND AMMUNITION. 299 1 Table of the Weight of Ships' Guns with Ammunition and Stores Complete (concluded). Shot Shell m •O C o O d (M C<1 (M C<1 t^ t- t^ b- t- C<1 (M (M o ■« CO t- '^i CO cocccccccocccocooocococo o a o H "3 to -a 3 ■3 1-1 10 la 000 "* -* I^J o tn 3 8 o -a ■S o a •*£ [5) 'S tOr-lCii— iCOCiC^JO-— lO CO CplOl^-GO-TiT-lOCCGOOcpip lO CC iM i-l r-l 1—1 rH 202 ^ 000 b- t- »o CO p O Ul 00«r>oroOO^ioo^ic ||i t^ — 1 ni ^ ^^ 00^ 4| |<^| 1 «b CO CO (fi r^ Ai Ai 1 U5 U5 1—1 I— 1 iO 1 -*i N Ins. s •r-< « Ins. 5° i For Ins. Ins. Ins. Lbs. 1 upper 1 lA lower / Wire 7 91 7 H 25 2| 2f no 160 2 upper "1 2a lower J » 6 2| 6 H 20 2i 2i r54 \43 3 upper "1 3a lower j >j 5 2| 5 H 15 If 24^ /30 125 4 upper ^ 4a lower J if 4 1| 4 H 10 1| If ri8 116 5 upper 1 5a lower j » 3 n 3 li 5 li If 1 10 (5 upper 1 6a lower J 5> 2i H 2i ll^a i\ 1 n r^ 7 upper ^ 7a lower / » 2 7 8 2 .3 4 3 3 4 i {I For 2ojj Bach-stayg. For Ins. Ins. Ins. Ins. Tons Ins. Ins. Lbs. 11 upper ~\ llAlowerJ Rope 8| 3 4 u 15 H If /28 116 12 upper "1 12a lower/ J5 7 2| 3i u 10 li H r2i 113 13 ujDper "1 13a lower/ f> 5i 2 3 H 5 H ■■■8 ;-i5 110 14 upper "1 14a lower/ if 4 If 2 3 4 3 3 4 1 8 / 7 I 5 15 upper ~| 15a lower j 5> 21 '^2 1 H t 2 1 3 4 r3-5 1 2-5 302 WEIGHT CF IRON BLOCKS. Table op the Sizes of Lenox's Patent Malleable Casi- IROX Blocks as used in Her Majesty's Duckyards. m o S Hope sheaves Description a Diiiieiisiuus of Shackles 1 t> K ;h In the 1^, 1 1 ^1 To Reeve C3 O Diameter of Rope I, I. 0) 5 i7 of Blocks a2 O c u 11^ li Ins. Clear i. +3 Oi-i +2 ^ s Ins. T. Ins. T. Ins. r Single \ Double t Treble 4 1 2 ll 1\ 8 ?. 1 4 1^ 1 2 1 2 3 1 3 8 1 9 16 li 1 11 8 1 16 H If a 4 H 5 2 1 1 4 7 8 1 2 Single i| 8 If 1^ 3. 4 r Single i| 1 n 1 3 4 2 6 ^ U 3 i 5 1 1 <^ Double i 2i li 16 H 1 13 IB L Treble 3| 4 2 1 J. 8 r Sinoie I Double [Treble 2| 3 4 2^ 1^ i 3 8 3 H 1 6 H 7 8 3 13 IR 2i 13: 15 •■■IGi 16 4^ 1 2f l_3_i 1 -^16 ^ r Single 3 2i ll^'l 4 10 3i 2 1* n H 1 <^ Double L Treble r Single 4 6 4i 1^' 1 911 -1(3 2f 3 1/6; li iflH 5 12 4&4i 3 li&lf 9 If U <^ Double I Treble r Single ■ Double [Treble 6 9 1 H 1^ 3| If Hi i#ii| U? 1 1 6 14 5 3^ If 10 2 li 1 7 U 3^ 113 1 J. |10i If P 3^ 4 liiiH f Single 8^ H 2 !1t^ 7 16 6&7 5| ^^^Ts 12 2i n 'i Double 111 If li 2 1^ o O lb a 2 [Treble 16^ 2 |1| rSino:le 13^ If H 2illf 8 18 H 9 H 13| 2a If <^ Double [Treble r Single !18 27 15f H 4.3 ti *2 4f 2i If 2i li 2i If 9 20 8i&9 10i2|&2| 15 H 2 { Double 21 H 43 2^1 If LTrol^le '3U If To Find the Approximate Weight of Castings or FORGINGS FROM THEIR WOODEN PATTERNS. Description of Pattern. ° - <» -r) i^ V (1) S '-3 t- ■»-' .-5 ■:3 /17-0 13-3 ]3-3 12-8 12-0 11-9 11-8 Vll-0 . ^- / 25-0 I3 S ^ I 20-3 ^"^ -c 20-3 198 18-4 18-0 17-8 UG-8 'S a 5 Weight in 24-0=cast lea^I. 19-3= „ copi>er. 19-3= ,, frun-metal. 18-8= ,, brass. 17-4= Bessemer steel. 17'3 = cast „ 17 1= wrought iron. VlGO=cast ,, I WETGET AITD STRENGTH OF SAIL CAITV^AS AND ROPE. 303 Table of the Weight and Strength of Sail Canvas IN Lbs. per Bolt of 24 Ins^ wide. No. of canvas Length of bolt (yards) Weight of bolt (lbs.) Tenacity in lbs. (weft) Tenacity in lbs. (warp) 0_ 39 2 48 lei 43 39 40 39 39 6 7 39J40 ^i"27 36 133 -— '480 4601440 400 370|350 390 380 — 340 320i300 280 260!250 330 310 Table of the Number of Cubic Feet required to stow 100 Fathoms of Chain Cable. Diam.of chain(ins.) 5 14 11 , .3 13 IG 1 + 16 17 1 20 23 7 8 27 ii 1 31 35 44 55 No. of cubic feet Diam.of chain(ins,) H H If Ifi 1| 2 140 2i 158 2i i 2i 177 1218 No. of cubic feet 66 79 92 107 123 Stowage of Chain Cable. D = diam. of chain in ins.; s = No. of cub. ft. to stow 100 fathoms. B = D-. < 35. Table OF the Weight and Stuength of Flat Hemp 1 AND Wire Rope. Hemp Iron steel Equivalent Strength Size in Inches Weight per Fathom (lbs.) Size in Inches Weight per Fathom (lbs.) Size in Inches Weight per Fathom (lbs.) Work- Break- ingLoad ingLoad in Tons in Tons 4 xl^ 20 21 xi 11 — 2-20 20 5 xli 24 21 xi 13 — — 2-60 23 Six If 26 2f x| 15 — — 300 27 5fxli 28 3 x| 16 2 xi 10 3-20 28 6 xli 30 3ix| 18 ^xl 11 3-60 32 7 xll 36 3|x| 20 21 X 1 -"4 2 12 4-00 36 8|-x2^ 8ix2i 40 45 03 „ 11 4 xii 22 25 91 vi -■3X3 2|x| 13 15 4-40 5-00 40 45 9 x2i 50 41 xf 28 3 xf 16 5-60 50 9ix2f 10 X 2 55 60 4ixf 4txf 32 34 31 x 1 Ql V 3 18 20 6-40 6-80 56 60 304 "WEIGHT AND STRENGTH OF CHAIN AND ROPES. ^ ] t b < ;:^ H 1— t C5 -y -Z:! ^Ie» HlC?l r-'I'M ,-(|Cl r^lCl r^itS -- ^) - '1 ^X) .j5 ' c -M M re re '* '^ »o ui iT: «0 t^ t- CO 00 Ci c; O O ^ »-' -M ^t +^ 53 :5 . sc" ^1 H^ C". -^ 9 9 9 ^ ao 9 ?j t^ .^ CO iH ^ ^'? ! J " " " t^ '^ i^ M 'cH ib b- OO C; -^ Ol -^ i 30 O ^1 4- t'- T'l "*" ■ — ,— , ^ ^ c^j T1 ^1 >i :c -g 5 c (^^ 7< "X) c>i c» -^t- c>T -Tt- o cp >i -^ » O ^D -T^ q-i p O o ^1 3D 3^ ,y rj ■^'ii HiS -l|(JI rHlSl H?M ^l"?* r-(!Cl r-ilO -151 -<:^ 2^ It „ > -^ E 5 '^ "^^ ^ ? '1^ O p i^ t'-^ p ■>! »!: O 5 :>:) 5 p C >i ir: i;^ 5 o ^ ^£ ,5 ' T-^ th cq CO ^^ '^ i b- do o rl- ot >h i"-- d: -^ r: 1*3 « '^ ^ ' '- — — ^ -^ —1 CM -M :M 'M ^t ^ -J- o »c o o o ic o o '-": lit o '-•:»:: o C: 'f 1.-: 13 o o »;: o g 3 a c^ p iC -M p 'C^ ->! ip >^ -M cc •>! u: 13 t^T C — ^ ^ _ „ (Tq -<^ -^^ >i ^ ^ p 1 ./ .-,::i -j'ci »-'lij) ,h|c.i r-i;^] r-^'Cl r-llH — '-) i-ilCl nH-S o o p ^ai . i^ v; :— ic c>T r-. c; r. c. -- u- i^ -m -+ — r: -+• ~ -:- — t- ~ ^C|^ipp>or-cpoxoo^t^;^^r--copcp— .(M'+ — t-c-5 _ r/;i0C0Cc:O'MC:OOOi^Oir;i^»C M^ ,s-*^'7iipc;<:'^opb-oq:coopi^t-cit-cc — ^r; - i*^ — ^ — — — -M -M r: -4. >:*" 1- iC o o o 1-1 -^^t3:s'2^»pci"cppxt--b-p»poaoi.'^cpcoipi3t.':GCLr:cc ^ a-? |!!3' * ■rlHiq?tcc4HOt^oodiOrH:cb-r^«b^cb'i;: >c 't yg C^^-;Hl^Oip^:pO»p»pT-iOCOpQOqOOlC(M'-— f — ' — — n ■^j TO r'. -Tf .2 "5c ^ ^rv ,3 r^ -^ -i c f-1 X -^ t^ ;;^ CI 22 9 *;r !-3 ^ "^"^ r: ^ ~" ■^^ "^ a, CM ^ iC t- r. O O C^ C. t- lO CM ~. O C '^ X C-1 1- t- x 2^..^t^^^r,^OCO«0«Cil^p'7^lCp»0'ydt;-ppp 3 ■ ■ J_ rln M CO CO ■* «5 «b 1^ i; O (J5 r"-: »c -o CO c; T'l i"- rH , _ , ^ — CM -M -M g r -lx"l--i-r'"!2rtiif-|';2„|j,t''-Mi.-.|oo^r;2« i--i-t-li)^H "l^-ii"!-- •e'-' -.-:;j3-i:. "§ ■3 . . -■ 1 . ■ -f X c^i t^ -* OS 00 (M 'O >^ '3 1--: GO CO r. ^ _ 1 .^ >0 — iC 00 00 -T" p 'P >P 7^ ^ ^ ^ T' f^ « «? ?^ 9 S 5 j i^ dc 6 -^ ^t »h oc (^^ cjc '^ t^ '"'5 « 3<> -- ^ •!;; c^ p r: T- ^0 1^ ^ _ ^ „ ^ c^ f>i cc -^ -f '-': ^ i^ X ri c — • c>i 71 r- 2 |j«ix«ISMir-'^ir-i^ j:'^'^:''!!; — '-^ ^^'"^""c^i '^\'^\'i:f%f%f'->~'^^ STRENGTH OF HOPES AND CHAINS. 305 Table op Comparisons of Equivalent Strengths of j Hemp Rope, Iron Wire Rope, and Chain Cable. j ^}'-^^- Circum. (ins.) (ms.) ^ Diani. (ins.) Circum. (ins.) Hiam. (ins.) Circum. (ins.) | Chain Cable H 11 ¥' 13 IG 1 5 1-1^ Hemp Hope Wire Rope Cbain Cable Hemp Rope Wire Kope Cbain Cable Hemp Rope Wire Rope 6i ■ 7" 7h 8 9 9i 11" H 3§ 4i 1* 1-3- i.i U 12 13i 14 Hi 15i 16" 16i 41 5 6^ If n 17 m 18 19 20 22 24 26 6t 7k 7§ 8f 82 Table of the Points of the Compass and their Angles with the Meridian. North N. bv E. XNE. XE. by N. NE. NE. by E. EXE. E. by N. East N. by W. NNW. XW. bv X NW. XW.bvW. WXW. \V. by X, West Points ° 0^ Of 1 H u 1^ 2 *-4 3 4 ^4 4| 5 oh 5f 6 H ^4 7 "i 7l 2 48 45 5 S7 oO 8 26 15 11 15 14 3 45 16 52 30 19 41 15 22 30 25 18 45 28 7 30 30 56 15 33 45 36 33 45 39 22 30 42 11 16 45 47 48 45 50 37 SO 53 26 15 56 15 69 3 45 61 52 30 64 41 15 67 SO 70 18 45 73 7 30 75 56 15 78 45 81 33 45 84 22 30 87 11 15 90 Points Oi Oi Of 1 H H H 2 21 2i 3| 4 4f 5 04 H bi 6 H C>i 8 rOuth S. bv E. SSE. S. bv W. SSW SE. by S. SW. by S. SE. SW. SE. by E. SW. by W. ESE. E. bv S. wsw. W. bv S. East West 306 LOGARITHMIC SINES, ETC., FOR POINTS OF THE COMPASS. Table of the Logaeithmic Sines, Taj^gents, and Secants to every Point and Quaeter-point of the Compass. p. 'iatS' H Oh o! 1 H u If 2 •>i -4 •2i •)1 -4 3 Cosine 4 0-OOOOUO 10-000000 Taugeut | Cotangent i Secant i Cosecant Point 8-.;9079l) 8-991S02! 9-lC)65-20l |9-29023Gi |9-885571 i9-4(5282-4l I9-527488! '9-5828-tO: :9-i;.-^>0992' 9-673387' :9-711050j 9-744739! 9-775027| '9-802359 9-827084 9-849485i 9-9994771 9-997904, 9-9952741 9-991574 9-98(578(5 9-980885 9-973841 9-965G15 9-9561(53 9-945130 9-933350 9-9198461 9-904828! 9-888185; 9-869790 9-849485 0-000000 Infinite 8-691319 11-30868L 8-993398 ll-00.iG02 9-171-247 10-828753 9-298662 10-701338 9-398785 10-601215 9-481939 10-518061 9-553647 10-44(1353 9-6172-24 10-382776; 9-6748-29 10-3-2517L 9-727957 10-27-2043 9-777700 10-2-22300: 9-8-24893 10-175107' 9-87019910-129801 9-914173 10-085827 9-957295 10-042705 10-000000 10-000000 10-000000! 10-000523, 10-00-2096 10-004726 10-0084-26^ 10-013214 10-019115 10-026159 10-034385 10-043837 10-054570 10-066650 10-080154 10-095172 10-111815 10-130210 10-150515 Infinite 11-309-204 11-008698 10-833480 10-709764 10-614429 10-537176 10-47-2512 10-417160 10-369008 10-326613 10-288950 10-255261 10-2-24973 10-197641 10-17-2916 10-150515 P.unt:,! Cosine 6f 6i 6i 6 of H 5i 5 ■if 4* 4i 4 Points Table OP Distances of THE Visible Horizon in Nautical Miles, the Height of the Eye being in | Feet -i^ -ij -i^ ■1^ -tJ "be Dis- "S Dis- tL Dis- ti) Dis- :£ Dis- — Dis- i> tance o tance O tance 5 tance 1, tance o tance T ■n -H 61 y -M 1-06 4-87 41 6-81 ?:!-3 1 81 9-57 101 10-69 2 1-50 22 4-99. 42 6-89 62 8-37 82 9-63 102 10-74 3 1-8-t 23 5-10 43 6-97 63 8-14 83 9-69 103 10-79 4 2-13 24 5-21 44 7-05 64 8-51 84 9-75 104 10-81 5 2-38 25 5-32 45 7-13 65 8-58 85 9-80 105 10-89 (5 2-60 2(5 5-42 46 7-21 GG 8-64 86 9-86 106 10-95 7 2-81 27 5-52 47 7-29 67 8-70 87 9-92 107 11-00 8 3 01 28 5-62 48 7-37 68 8-77 88 9-98 108 11-05 9 3-19 29 5-72 49 7-44 69 8-83 89 10-03 109 11-10 10 3-3G 30 5-82 50 7-52 70 8-89 90 10-09 no 11-15 11 3-53 31 5-92 51 7-59 71 8-96 91 10-14 111 11-20 12 3-68 32 6-01 52 7-67 72 9-02 92 10-20 112 11-25 13 3-83 33 6-U 53 7-74 73 9-09 93 10-25 113 11-30 14 3-98 34 6-20 54 781 74 9-15 91 10-31 114 11-35 15 4-12 35 6-29 55 7-89 75 9-21 95 10-36 115 11-40 16 4-25 3(5 6-38 56 7-96 76 9-27 96 10-42 116 11-45 17 4-38 37 6-47 57 8-03 77 9-33 97 10-47 117 11-50 18 4-51 38 6-5(5 58 8-10 78 9:5i» 9S 10-53 118 11-55 19 4-53 39 6-GI 59 8-17 79 Hi.') 91' 10-58 119 11-60 20 1-76 10 (■•7:! CO 8-24 80 :t-:.i 100 lo-G,-; 120 11-65 VALUES OF THE GAUGES. oU" Table of the Values of the G. ^UGES IN Decimals of j THE IXCH. Bii'minghani Gauge for Iron Wire, and Sheet Iron, and Steel. Mark Size Mark 5 tfize •220" Mark I Size Mark size •032 Mark 29 Size i -013 OUUO •454 13 •095 21 000 •425 6 •203 14 •083 22 •028 30 i •ori 00 •380 7 •180 15 •072 23 ' ^025 31 : •OlO •340 8 •165 16 •065 24 •022 32 ' -009 1 300 9 •148 17 •05« 25 •020 33 •008 2 •284 10 : •134 18 •049 26 •018 34 •007 3 -25i> 11 ' -120 19 •042 27 •016 35 •005 -i •23S 12 -lOit 20 ■03:) 28 •014 36 •004 Birmingham GaiKjefor Sheet . Metals, Brass , Gold Silvei •, .5-^". -Mark ^-ize Alark ^ize Mark Mze Mark size -Mark 29 Size — — — — — — •124 1 •004 8 -016 15 •047 92 •074 30 •l->6 2 •0(»5 9 ^019 16 •051 23 •077 31 •133 3 1 •uus 10 ^024 17 •057 24 •082 32 •143 4 •010 11 -o-^g 18 •061 25 •095 33 •145 •012 12 ^-034 13 ••OSG 19 •064 26 •103 34 •148 6 ■013 20 •067 27 •113 35 •158 7 •015 14 ^041 21 •072 28 •120 36 •167 Lancashire Gauge for Round Steel Wire, and also for Pinion Wire. Mark iSize 80 79 78 1 1 76 75 74 73 72 71 70 69 68 67 (i6 65 64 63 62 61 60 5^) •013 •014 •015 •016 •018 •019 •022 •023 •0-24 •026 ■027 •029 •030 •031 •032 •033 •034 •035 •036 •038 •039 ■U40 Mark Size 58 ■041 57 •042 56 •044 00 •050 54 •055 53 •058 52 •060 51 •064 50 •067 49 •070 48 •073 47 •076 46 •078 45 •080 44 •084 43 •086 42 •091 41 •095 40 •096 39 •098 38 •100 37 •102 36 •105 Mark 35 " 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 •107 •109 •111 •115 •118 •125 •134 •138 •141 •143 •146 •148 •150 •152 •157 •160 •164 •167 •169 •174 •175 •177 •180 Mark 12 Size Mark •185 L, 11 •189 M 10 •190 X 9 •191 o 8 •192 p 7 •195 Q 6 •198 R 5 •201 S 4 ■204 T 3 ■209 U 2 •219 V 1 •227 w A ■234 X B •238 T C -242 Z D •24G A' E •250 Bl F ■257 Cl G •261 Dl H ■266 El I •272 Fl J •277 Gl K ■281 Hi Size '^9"0" •295 •302 •316 •323 •332 •339 •348 •358 •368 •377 •386 •397 •404 •413 -420 -431 -443 •452 •462 •475 •484 •494 x2 308 whitwoeth's wire gauge, and weights of rivets. Table of the Values of AVhiiwoeth's ^^'IEE Gauge IX Decimals of ax Ixch. M ark Size 1 2 3 4 5 6 7 8 9 10 11 12 13 •001 -002 •003 ■004 •005 -006 •007 •008 •000 •010 •Oil •012 •013 Mark 14 15 16 17 18 19 20 22 24 26 28 30 32 Size Mark -014 -015 -016 •017 •018 •019 -020 -022 •024 -026 -028 -030 •032 34 36 38 40 45 50 55 60 65 70 75 80 85 -034 •036 •038 •040 •045 -050 -055 •060 •065 -070 -075 -080 •085 Mark ?ize * Mark Size 90 95 100 110 120 135 150 165 180 200 220 240 260 •090 -095 -100 -no -120 -135 -150 •165 -180 -200 •220 -240 -260 28U oUO 325 350 375 400 425 450 475 500 -280 •300 •325 •350 •375 •400 -425 -450 •475 •500 Table of ihe Weight of 100 Pax-headed Rivets IX Lbs. -■ — . Length under Head in Inches 5 ■•'■- -t i£ 1 5 ' 3 7 8 1 Hi H ^ H If l_ 1-25 1-44 1-62 l-Slj 1-99 2-20 2-35 2-54 2-72 2-90 i 1 , 3-46 3-86 4-27 4-69! 5-09 5-50 5-91 6-43 6-94 7-25 1 1 ! 7-27 8-00 8-73 9-45 10-18 10-50 11-65 12-89 13-12 13-86 i 1 11300 14-2415-37 16-51 17-65 18-80 19-93 21-10 22-20 23-35 i f 120-75 1 32-23 22-95 24-35 26-15 27^22 29-87 31-70 32-70 34-40 36-34 f 34-46 36-69 3891 41^15 43-40 45-64 47-89 50-12 52-35 t 1 46-54 49-44 52-35 55-26 58-17 6M0 64-00 66-93 69-81 72-72 1 - ^ Lenf:rtli under Head in Inches 1 4 ^ .2 i If 2 9i 9i 9 3. -4 -2 1 -4 3 3^ 1 H 3| 4 3-08 3-43 3-80 4-16' 4-53 4-90' 5-26 5-63 5-99 6-35 4 7*55 8-37! 9-19 1001 10-84 11-68 12-49 13^32 14-1414-96 ]l X 14-59 1 5^10 17^50 18-96 20-41 21-86 23-32 24^79i26^24 27-71i i 1 '24-48 26-7529-11 31-29 33-56 35^84 38-11 40-39 42-66 44-94^ ^ i '37-67 40-95;44-23 47^50 51-10 54-06 57-34 60 61 63-90 67-17; ^ ^ 54-59 !59-05'64-51 67-97 72-44 76-89 81-35 85-8190-26 93-70! f 1 75-62 81 -44 87-1 6 93-09 98-91 104-7 110-5 116-4 122-2 128-0 1 Metal Sheathing Is usually made in sheets of the follo\ving weights and sizes : — Length in inches . Breadth „ . . . Thickness „ . . . Weight in lbs. per square foot Weight in lbs. ixt sheet yote.—ODe cwt. of metal nails should be allowed for every 100 sheets. . 48 48 48 . 20 14 14 •025 •038 -044 . 1-1-J.5 1-75 2-00 . 7-50 11-67 13-33 vthitworth's taps, dies, nuts, aisD bolt-heads. 309 Table of AVhiiworih's Sta^'dard Taps A>'D Dies. | ■j: X -x. IP r. a: = x 1.^ i a: ■^ = o 3 V. m 3-000 48 •100 14 — •475 7 H 1^125 3i 40 * •125 12 1 2 •500 7 H 1-250 3^ 34^ 3-25U 32 •150 12 •525 6 1* 1-375 3i 3.^ 3-500 24 — •175 12 •550 6 H 1^500 3 q.5 3-750 24 — •200 12 •575 5 1^ 1-625 3 4 4-000 24 — •225 12 •600 5 If 1^750 91 *"3 H 4^250 20 i •250 11 1 •625 4i 1^ 1-875 91 ->< ^i 4-500 20 •275 11 •650 u 2 2^000 2f if 4-750 18 •300 11 •675 4^ 2^ 2^125 9il ~4. 5-000 18 — •325 11 •700 4 9i *-4 2-250 95 -8 H 5-250 18 — •350 10 3 4 •750 4 2^ 2-375 2* 5i 5-500 16 1 •375 10 •800 4 2^ 2-500 2i 5f 5-750 16 — •400 9 7 •875 4 2^ 2-625 2^ 6 6-000 14 — •425 9 •900 3i 93 2-750 ■ — — — 14 — •450 8 1 1-000 H 91 8 2-875 — — — Xl re.— The ang e of ti- Lread= :55=. Depth c )f tlire ad=| of pitch bore- -that is, de lucting i; for the qu antity rounded off t op and ^off t wttoni. Table of Whiinvorih's Standard Hexagonal Xn and | Bolt-heads. 1 •— I"? c m m .1-5 ill ^ "O -•=J C-1 Ins. 1 i-§ lis Ins. Ins. Ins. Ins. i Ins. Ins. Ins. Ins. IBS. * -338 * •1093 1 -0929 u 2-0483 H 1-0937 1-0670 3 •448 3 16 •1640 1 -1341 H 2-2146 -■-8 1-2031 1^1615 1 4 •525 JL 4 -2187 -1859 u 2-4134 u 1-3125 1-2865 Te •6014 A. le •2734 -2413 1^ 2-576-3 H 1-4218 1-3688 8' •7094 8 •3281 -2949 If 2-7578 If 1-5312 1-4938 7 16 •8204 7 16 -3828 -3460 H 3-0183 H 1-6406 1-5904 i •9191 1 2 •4375 -3932 9 3-1491 2 1-7500 1-7154 9 16 1-0110 9 16 -4921 •4557 2* 3^3370 91 ~8 1-8593 1-8404 0. 8 1-1010 ^ -5468 •5085 2^ 3^5460 2i- -"4 1-9687 1-9298 11 16 1-2011 11 16 -6015 , -5710 2| 3^7500 2t 2-0781 2-0548 4 1-3012 3 4 •6562 •6219 2i 3^8940 2i 2-1875 2-1798 13 16 1-3900 i| -7109 -6844 9A 4-0490 2i 2-2968 2-3048 -L 8 1-4788 ^ •7656 -7327 93 —4 4-1810 2f : 2-4062 2-3840 1.5 IG 1-5745 15 16 -8203 •7952 91 -8 4-3456 91 -8 2-5156 2-5090 1 1-6707 1 -8750 -8399 3 4-5310 3 2-6250 2-6340 H 1-8605 1| -9843 -9420 — — — 1 — — 310 SIZES AXD WEIGHTS OF ADMIRALTY TANKS. Table of the Sizes, AVeights, &c., of Admiralty Tanks. Cdm.mi ix Wat f I .-TANKS. No. Hei^li Ft. In c Width s. Ft. ins. Dejrth Capacity ^\■ei.l;llt : No. Ft. Ins. Gals. Cub. Ft. Cwt. Qrs. Lbs. 1 4 S 4 Oi 6 Oi 600 101 10 2 12 1 lA 4 S 2 Oi 6 Oi 300 51 7 25 lA -t 4 ^ 4 O.i 5 Oi 500 85 9 2 4 1 4a 4 0: r 2 Oi 5 Oi 2.50 42 6 15 4a 1 t 4 0^ r 4 Oi 4 Oi 400 68 7 17 7 i ''^ 4 0^ r 2 Oi 4 Oi 200 35 5 6 7a 10 4 OJ r 3 2i 3 2i 200 35 4 2 17 10 lOA 1 4 0^ : 1 n 3 2i 100 18 3 1 5 10a 12 i 4 0^ : 4 Oi 3 Oi 300 51 6 9 12 1 IB ! 4 0^ - 4 OV 2 Oi 200 34 4 3 23 13 i U 4 03 - 1 2 7i 1 7i 100 17 3 1 14 Bilge-water Tanks. [ Dimensions Capacity Ft A K c D E ^V eight No . Ins. P t. Ins. Ft. Ins. Ft. In 3. Ft. Ins GaLs. Cub. Ft. C\rtQr.Lb 2 4 Oi 5 Oi 5 li 2 4 Oi 575 96 10 22 2 2 A 2 Oi ( 3 Oi 5 li 2 4 Oi 287 49 6 3 15 2a p, 4 Oi ( 3 Oi 4 li 6i 4 0.1 510 87 9 18 3 3 A 2 Oi ( 5 0i 4 li 6i 4 Oi 255 44 6 2 3a 5 4 Oi 1 5 Oi 4 6i 4 OV 475 80 Q O 5 5 o J. 5 A 2 Oi \ 5 0i 4 6i 4 01 237 41 5 3 17 5 a (;a 4 01 . 2 0^ ( 5 0i 5 Oi 3 0.^ 3 Oi 6i 4 Oi 410 68 8 6 6a 6i 4 Oi 205 35 5 1 24 s 4 01 ' tOi 3 1 2 4 01 375 64 6 3 26 8 8 a 2 0^ ^ t Oi 3 1 2 4 01 187 ; 32 4 3 8a <_) 4 O.V ^ tOi 2 Oi 6i 4 01 310 ; 54 6 16 9 9a 2 0\ . t Oi 2 Oi 6i 4 01 155 1 27 4 1 7 9a 11 3 2 ^ } 2i V 8 6i 4 01 110 20 3 1 7 11 BuEAi) "J'AXKs. j Paint Cfstk-ns. Oil Cl'^terxs. | No \ t W'dth . FtT.iiT. Dpth \ V^C N u. iigtlt|\\ dtb t.in. Dpth \V. Ft .ST.! LI. No. i^th ■'t.in. Wdth Ft.in, Dpth Ft.in. Wgt Lb5. Ft.iii. 1 Ji.S. 1 Ft.in. F A 2 3 2 2 6 ': 581 A ; 1 6 1 11 3 l_ A 1 6 3 3 6 275' B 2 3 2 2 Gjc 521 B 1 6 1 8 2 6 _ B 1 6 2 2 2 6 196 C 1 6 2 1 9*1 .52 C 1 3 1 6 2 O!- C I 3 2 3 2 105 D 1 1 3 2 fi'l 17 D 1 O'l 2 0'_ D 1 6 1 9 2 — I'lu. 152. 1 WATCR 1 ./s. BREAD OIL PAINT | - •^ / ■— — ■ 9 1 ^\ ."^ V i \ »:\ \ / / / \ COMMON 1 N^fellk ^1 ^ i \ BILCE '* , \ tankI TAMK. '. ^ V -1.^ ...\^ Il- ia ' 1 %J r^ .11 *■» i^ """. 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X.- -- -i C-. --S Kj 1 - '^^ 313-1114 313-2110 .313-034-1 312-80i»6 257-5264 257-(i328 257-4608 257-2760 /-A t^ 0-9398 7-0602 •2 1 28 230^2559 •21833 224-4214 •1988 2'1G-8G9 •2093 240-618 •5285 92-7395 8-4539 5-79614 99-5617 •49217 — o — -f- XI r- 3-j o 410 X'l-O X> O 00 X --^ 1-- 'X O O CO 3^ 1^ c; cp c; t_-^ CO TT 'N — c^ -Vi CO CO lij lO O ir; 1^ I^ t^ [^ 'M O^l (M CM OJ'M'M'>1(M<»0<13 po CO CO CO ~ Ci r; c:; © 5 4-9570 5-0130 •1520 164-4685 •15595 160-3010 •14-20 176-335 •1495 171-870 •3775 66^2125 6-0385 4-14010 71-1155 35155 S t-- -T^ S »6 Ol — ~. cp i^ -p »p C-, O C-. I-- o ih >h lb c -— i i Ci Ci Oi Oi ;c; --r ^0 '-C •r. 9 .-S -2 -o ;?^ P 2 '-^ « '-^ ^- ^ -^ 5 o -ri : 00 00 X 00 lO O lO »0 -t^^||>*^ «r- '^ ^ S fl ." II '- ' '^ II . S S---5:i^.^^|-^^-=i--sii" Lgth of sees, pendulum in ins., London . „ „ „ Ldinburgb „ „ „ Paris „ „ ., New York Fdrceofgravity* in London, ft. per see. . Edinburgh „ „ „ Paris „ „ „ New York „ o 5c Si O * 320 DISTANCES OF FOEEIGN PORTS FROM LONDON. Table giving Distances of Foreign Ports London in Nautical Miles. FROM Aberdeen Aden . Alexandria Amsterdam Antwerp Archangel Auckland Barbadoes Barcelona Batavia (Java) , Bombay Bordeaitx Boston , Bristol . Buenos Ayres Cadiz . Calcutta Cajiton . Cape of Good Hope Cajie Horn . Cardiff . Charlesto^v^l Colombo (Ceylon) Constantinople Copenhagen . Cork . Dover . Dublin . Dundee. Ferrol . Funchal (Madeira) Gibraltar Glasgow Halifax . Hamhurg Havaima Hobart Town Ilong Kong . Hull . Kingston (Jamaica) Leghorn Leith . Lima Li-;bon . Liverpool 433 / 4,695 ( 9,955 3,095 333 182 2,22ri f 10,916 { 12,120 3,795 1,902 f 8,330 111.270 ] 6.330 1 10.595 6-.0 3,025 534 6,280 1.322 / 7,950 I 11,450 ( 10,468 1 13.553 6.065 7.395 517 4.307 f 6.795 I 10.885 3,085 7(i8 531 87 680 4:0 785 1,397 1,330 735 2.692 418 4,229 (10.291 '1 11.495 f 9.775 "( 12,910 233 3.9t;8 2,258 418 10,655 1 .053 C60 Lizard . Madras , Malacca , JIalta . . Manilla . Mauritius Melbourne . New Orleans New York , New Zealand Ostend . . Otago . Pekin (Gulf) Pernambuco Plymouth Port Jackson Portsmouth . Pulo Penang Quebec . . Rangoon Rio Janeiro . Rotterdam . San Francisco Shanghai Sheemess Shields . Sierra Leone Singapore Southampton St. Hdena . St. lago (Cape Tonlo Is.) St. John (Newfoundland) St. Petersburg Stockholm . Swan River . Sydney . Teneriffe Venice . Washington . Waterford . Yokohama . EIVETING IX H.M.S. ' HERCULES. 321 «- H^T '•-2^ i*^^;:!f '-w-f=H=«-e;:?Ha:t-»:^=- ;jf ;if;jf^ - ^ — 1 H= mh- '-^ r^ .n 1— 1 N.— ' ntx OD " 1 -^_4^-«« u-5 u-5 o O lO «S 1 "c; > ■— "1"' ■^ *5 -i2 -*- o o « o t^t in l:^ w T* Tj< Tj< T^ Ttl IC 1 U^ l^-l '- --^^ m ||a 1 '^ 1 1 ^ 1 1 1 1 1 1 1 1 1 -o-^t)-i^-;i -l:i-tti 1 1 >^ 2 *™' 1 3^ 1 1 ^ 1 1 1 1 1 1 1 1 1 t- '•^ w u; L^ ^ o 1 1 i«^ ^ *^" C 12 c — i_4 ^ -- ^ H- 1 - ? /^^'^'^"S fc « 15 1-5 ; 3 ^ -.^^ « — P P^ ;-■ _;_ '*-'^ = i:o3 o p4 C tT «= »*5- ... ^ <= SL. "■ ^ ^ tS=Mt=S - ^ s = P^ s ,5^ C — cc X 3Q ^7. X ^ X t .^^ j^ . -»J <1 X H ^c^' > CO H /^ cc -^- X _ _ , ^ , ..^^ ^ 1— ( x: ' r *3 ;:; g o fc* Y, • • • • -^^ • • • • 2 • . • 3 O ,^ ti4 cc i^ H PC • • • *•#— I©**'*' • • • C Q ^ £ S c be . " 9 * ^ • • • • o vi • . 03 • • • P-( E: . . • . «^ T^ . . « * -3 ^"^ *" as a: bD S < CK o > • 01 1 .9 fc 1 S g i • as 'ft s o ^1 '^ , o ■73 « 'fcC H S . >»^ ^ c cc s 03 ,1: cc eS £t4 K< H^H O -t-> 53 C 1 ^ « 0: < 2 O 03 5 "S fcD • i i 1 ■=^ P^ H ^^ 822 EIVETTNG IN H.M.S. ' HERCULES.' _: cj s 3 > t— 1 o o o w o H M n m pq O tn ^^^^rrj^^fj^tfi^,:^ coH- ^-W f^l^ H»— m^'- t<*Ti-i cohr C !'J*«|^c<*tt<;WTC|*mlr ^ ^ "* '-'1 M — "- -t^ -*i m ri 4J f— EC 1 5 o '3 1 I I -lt»,^ Ht)^ rt o ■Is « X ^ -in Cl. CO -leq i_«».^ I--^ ^ 5 -KsJ-c^-l:i«H--;e«^ o-lt>-|N^ -K -in-i ^ 3 « X ^3 2 10 -- X ^. -1=' X _0 -|:> -Itl-IN-ln-lN . S 'fcfcrt o 'O m fl &i « hi a; ■^ IH m =2 c; S S c fcb c c ^ £ 1.5 P- 113 c8 3 "-I -ti CO O 3 3 to a Cfi CO eS P^ fcD C C j; j:-*^ c c ■3 '~ EIVETIXG IX H.M.S. ' HEECULES.' 323 M fa o 1^ W H (^ O 5 >^ y^yHr:!<^rtt'^K^Ki '■•tr :^ ?Tt^ cok? :»^ xj^ ;o|T? «;!S "'.IX -r,! — !M -;t< ■ TJrt r^r- 1*^ "^K -cte -f-lx >. 1— l'^ ^ .^ -Tf< rP^.S :~ "^ t> CQ C^ -^ JO l^ to 3^1 t^ ^'•' T4 -<- •!^ c; -In-?)-'; 5 §* ;£ '3 :q --o ^\-- = O '£ c - ;)-:t«-ii^— :rf"io5r«:r;iB - -i^-;; X CO s t- i; ^" 50 3 H^ =" iC ^ t- '^^ r^ S. '-^ y - t t c v: r. /: CO ~ _. 0) I^ be r; :- i-l Zi tXi ,— .t^ ^ " " '^ "^ •~ •■ •~ •" "" *^ •~-^ ** Ic s "E. ^ 5 « C - br r -C „ ^^ .2 •^ ^ be *^ —i -^ ^ 1, f— ^ ^ '""" ^1 »— « ^C -'^ '-' ^ -~ y2 324 eivet:X':. ix h.m.s. ' heecules. X - > — -4;»-4^-<5»— _J",^_J" ,^ — — = ^^^ ^ ->= « X - " ff- ■^ "Ofncotx '■" ■•- -t;i C .-^ — ,;i £-1 >r X X -<:*■ o '^ ~ ? — ttJ ^ -J^ - ■=. '-'• - - ? ?r *:& r- -^ 5 =^ ~ = X 10) - ' ;:^ c 15^ 5-= yT )2 ir^'i^ jiioiiijy :i. c as - 5? 5 3 LOGAEITHMIC SINES, TANGENTS, SECANTS, ETC. 325 Table of Log.4JIithmtc Sines, Tangents, Secants, &c. Deg. 10 Deg. Cosecaut Tangent Cotant;enti iSecant •000000 63982 94084 11693 24186 33875 41792 8-48485 8-54282 8-59395 8-63968 8-68104 8-71880 8*75353 8-78568 8-81560 8-84358 8-86987 8-89464 8-91807 8-94030 8-96143 8-98157 9-00082 9-01923 9-03690 9-05386 9-07018 9-08589 9-10106 9-11570 9-12985 9-14356 9-15683 9-16970 9-18220 9-19433 9-20613 9-21761 9-22878 9-23967 9-25028 9-26063 Cosine Infinite 12-36018 12-05916 11-8830 11-75814 11-66125 lll-58208 ill-51515 '11-45718 11-40605 j 11-36032 11-31896 11-28120 11-24647 !ll-21432 ;il-18440 11-15642 ill-13013 11-10536 11-08193 11-05970 11-03857 11-01843 10-99918 10-98077 10-96310 10-94614 10-92982 10-91411 10-89894 10-88430 10-87015 10-85644 10-84317 10-83030 10-81780 10-80567 10-79387 10-78239 10-77122 10-76033 10-74972 10-73937 Cosine Des. •000000 7-63982 7-94086 8-11696 8-24192 8-33886 8-41807 8-48505 8-54308 8-59428 8-64009 8-68154 8-71940 8-75423 8-78649 8-81653 8-84464 8-87106 8-89598 8-91957 8-94195 8-96325 8-98358 9-00301 9-02162 9-03948 9-05666 9-07320 9-08914 9-10454 9-11943 9-13384 9-14780 9-16135 9-17450 9-18728 9-19971 9-21182 9-22361 9-23510 9-24632 9-25727 9-26797 Infinite 1 10-00000 12-36018;10-00000 12-05914 10-00002 11-88304 10-00004 11-75808 10-00007 11-66114 10-00010 11-58193 10-00015 11-51495 10-00020 11-45692 10-00026 11-40572 10-00034 11-35991 10-00041 11-31846 10-00050 11-2806010-00060: 11-2457710-00070 11-21351 10-00081 11-18347 10-00093! 11-15536 10-001061 11-12894 10-001201 11-10402 10-00134| 11-08043 10-00149 11-05805 10-00166' 11-03675 10-00183 11-0164210-00200 10-99699 10-00219 10-9783810-00239 10-96052 10-00259 10-94334 10-00280 10-92680 10-00302 10-91086 10-00325, 10-89546 10-00349 10-88057 10-00373 10-86616 10-00399 10-85220i 10-00425, 10-83865 10-00452 10-82550 10-81272 10-80029 10-78818 10-77639 10-76490 10-75368 10-74273 10-73203 1— i Secant iCotangent Tangent 10-00480 10-00508> 10-00538 10-00568 10-00600 10-00632 10-00665 10-00699 10-00733 89 88 85 Cosecant 10-00000 90 9-999991 i 9-99998| ^ 9-999961 J 9-99993 9-99990 9-99985 9-99980 9-99974 9-99967 j 9-99959i 9-99950 9-99940: 87 9-99930 9-99919 9-99907 9-99894! 86 9-99880 9-99866i 9-99851; 9-99834 9-99817 9-99800 9-99781 9-99761 9-997411 ^ 9-99720i ^ 9-99698' I 9-99675 83 9-99651 4 9-99627' 4 9-99601 J 9-99575 82 9-995481 4 9-99520; i 9-99492' i 9-99462 9-99432| 9-994001 9-993681 9-99335 9-99301i 4 9-99267 ; 79^ Deg. 84 81 80 Sine o26 LOGAi:iTHiHC SIXES, TANGENTS, SECANTS, ETC. Deg. 11 12 13 14 15 16 17 18 19 Siue , Cosecant Tangent ICotangent Secant | Cosine 20 21 22 Beg. 9-27073 9-28060 9-29024 9-29966 9-30887 9-31788 9-32670 933534 9-34380 9-35209 9-36022 9-36819 9-37600 9-38368 9-39121 9-39860 9-40586 9-41300 9-42001 9-42690 9-43367 9-44034 9-44689 9-45334 9-45969 9-46594 9-47209 9-47814 9-48411 9-48998 9-49577 9-50148 9-50710 9-51264 9-51811 9-52350 9-52881 9-53405 9-53922 9-54433 9-54936 9-55433 9-55923 9-56408 9-56886 9-573^^ Cosine 10-72927 10-71940 10-70976 10-70034 10-69113 10-68212 10-67330 10-66466 10-65620 10-64791 10-63978 10-63181 10-62400 10-61632 10-60879 10-60140 10-59414 10-58700 10-57999 10-57310 10-56633 10-55966 10-55311 10-54666 10-54031 i 10-534061 10-52791 1 10-52186 10-51589 10-51002 10-50423 10-49852 10-49290 10-48736 10-48189 10-47650 10-47119 ! 10-46595 10-46078 10-45507 10-45()f;4 10-44567 10-44077 10-43592 10-43114 10-42642 Secant 9-27842 9-28865 9-29866 9-30846 9-31806 9-32747 9-33670 9-34576 9-35464 9-36336 9-37193 9-38035 9-38863 9-39677 9-40478 9-41266 9-42041 9-42805 9-43558 9-44299 9-45029 9-45750 9-46460 9-47160 9-47852 9-48534 9-49207 9-49872 9-50529 9-51178 9-51819 9-52452 9-53078 9-53697 9-54309 9-54915 9-55514 9-56107 9-56693 9-57274 9-57849 9-58418 9-589ai 9-59540 9-60093 9-60641 10-72158 10-71135' 10-70134 10-69154 10-68194 10-67253 10-66330 10-65424 10-64536 10-63664 10-62807 10-61965 10-61137 10-60323' 10-59522; 10-58734 10-57959 10-57195 10-56442 10-55701 10-54971 10-54250 10-53540 10-52840 10-52148 10-51466 10-50793 10-50128 10-49471 10-48822 10-48181 10-47548 10-46922 10-46303 10-45691 10-45085 10-44486 10-43893 10-43307 10-42726 10-42151 10-41582 10-41019 10-40460 10-39907 10-39359 10-00769 10-00805 10-00843 10-00881 10-00920 10-00960 10-01000 10-01042 10-01084 10-01128 10-01172 10-01217' 10-01263 10-01310' 10-01357! 10-01406| 10-01455* 10-01506; 10-015571 10-01609) 10-01662 10-01716 10-01771 10-01826! 10-01883' 10-01940, 10-01999'i 10-02058 10-02118: 10-02179 10-02241 10-02304 10-02368 10-02433 10-02499; 10-02565' 10-02633; 10-02701 10-02771: 10-02841 10-02913 10-02985 10-03058 10-03132 10-03207 10-03283 9-99231 9-9".n95 9-99157 9-99119 9-99080 9-99040 9-99000 9-98958 9-98916 9-98872 9-98828 9-98783 9-98737 9-98690 9-98643 9-98594 9-98545 9-98494 9-98443 9-98391 9-98338 9-98284 9-98229 9-98174 9-98117 9-98060 9-98001 9-97942 9-97882 9-97821 9-97759 9-97696 9-97632 9-97567 9-97501 9-97435 9-97367 9-97299 9-97229 9-97159 9-97087 9-97015 9-96942 9-96868 9-96793 9-96717 Cotangent Tangent Cosecant Sine loCtARIthmic sines, tangents, secants, etc. 327 Peg. ' 1 ■ 4. |23 Cosecant | Tangent ■ Cotangent, Secant Cosine 24 26 27 4 I i I 29 30 31 32 33 9-57824 9-58284 9-58739 9-59188 9-59632 9-60070 9-60503 ' 9-60931 ' 9-61354 j 9-61773 ! 9-62186' 9-62595 9-62999 9-63398 9-63794 9-64184 9-64571 9-64953 9-65331 9-65705 9-66075 9-66441 9-66803 9-67161 9-67515 9-67866 9-68213 9-68557 9-68897 9-69234 9-69567 9-69897 9-70224 9-70547 9-70867 9-71184 9-71498 9-71809 9-72116 9-72421 9-72723 9-73022 9-73318 9-73611 9-73901 9-74189 10-42176 10-41716 10-41261 10-40812 10-40368 10-39930 10-39497 '10-39069 1 10-38646 110-38227 110-37814 i 10-37405 110-37001 10-36602 10-36206 10-35816 10-35429 ilO-35047 !l0-34669 110-34295 !l0-33925 10-33559 10-33197 ilO-32839 10-32485 '10-32134 :10-31787 ilO-31443 10-31103 ilO-30766 10-30433 10-30103 10-29776 10-29453 10-29133 10-28816 10-28502 10-28191 ;10-27884 10-27579 110-27277 10-26978 10-26682 ^10-26389 10-26099 10-25811 9-61184 9-61722 9-62256 9-62785 9-63310 9-63830 9-64346 9-64858 '9-65366 9-65870 9-66371 9-66867 9-67360 9-67850 9-68336 9-68818 9-69298 9-69774 9-70247 9-70717 ',9-71184 9-71648 9-72109 9-72567 9-73023 9-73476 9-73927 9-74375 9-74821 9-75264 9-75705 9-76144 9-76580 9-77015 9-77447 9-77877 9-78306 9-78732 9-79156 9-79579 '9-80000 19-80419 '9-80836 !9-81252 19-81666 19-82078 Deg 10-38816:10-03360 9-96640 10-38278ilO-03438 9-96562 10-3774410-03517 9-96483 10-37215 10-03597 9-96403 10-36690 10-03678 9-96322 10-36170 10-037601 9-96240 10-35654 10-03843 9-96157 10-35142 10-03927 9-96073 10-34634 10-04012 9-95988 10-34130 10-04098 9-95902 10-33629 10-04185 9-95815 10-33133 10-04272 9-95728 ! 10-32640 10-04361: 9-95639 ' 10-32150 10-044511 9-95549 j 10-31664 10-045421 9-95458 10-31182 10-04634J 9-95366 10-30703 10-04727' 9-95273 10-30226 10-04821 9-95179 10-29753 10-04916 9*95084 10-29283 10-05012 9-94988 10-28816 10-05109 9-94891 10-28352110-05207 9-94793 10-2789l'lO-05306 9-94694 10-27433 10-05407 9-94593 10-26977 10-05508 9-94492 10-2652410-056 10 9-94390 10-26073;10-05714 9-94286 10-25625'l0-05818' 9-94182 10-25179110-059241 9-94076 10-24736 10-06030! 9-93970 10-2429510-06138 9-93862 10-23856,10-06247 9-93753 10-23420'lO-06357 9-93643 10-22985 10-06468! 9-93532 10-22553'l0-06580| 9-93420 10-22123 10-06693 9-93307 10-2169410-06808 9-93192 10-21268' 10-06923 9-93077 10-20844 10-07040' 9-92960 10-20421 10-07158' 9-92842 10-20000 10-07277 9-92723 10-1958110-0739719-92603 10-19164 10-07518| 9-92482 10-1874810-0764119-92359 10-18334 10-07765' 9-92235 10-1792210-07889' 9-92111 67 66 65 64 63 62 61 60 59 58 Deg,| Cosine j Secant Cotangent Tangent Cosecant [ Sine i 57 1 3. i 561 ; Deg. 328 LOGARITHMIC SIXES, TANGENTS, SECANTS, ET( Deg. 1 33f Sine 9-74474 Cosecant Tansent Cotan.L'ent Secant Cosine DeK. 1 10-25526 9-82489 lU-17511 10-08015 9-91985 1 i 34 9-74756 10-25244 9-82899 10-17101 10-08143; 9-91857 [ 56 1 4. 9-75036 10-24964 9-83307 10-16693 10-08271' 9-91729 3 4 i 9-75313 10-24687 9-83713 10-16287110-08401 9-91599 1 ■4 9-75587 10-24413 9-84119 10-15881 10-08531 9-91469 1 4 35 9-75859 10-24141 9-84523 10-15477 10-08664 9-91336 55 1 4 9-76129 10-23871 9-84925 10-15075;10-08797! 9-91203 3 4 1 9-76395 10-23605 9-85327 10-14673110-08931' 9-91069 1 2 4" 9-76660 10-23340 9-85727 10-1427310-090671 9-90933 1 4 36 9-76922 10-23078 9-86126 10-13874 10-09204' 9-90796 54 1. 4 9-77182 10-22819 9-86524 10-13476 10-09343 9-90657 3 4 1 2 9-77439 10-22561 9-86921 10-1307910-09482 9-90518 1 •2 3 9-77694 10-22306 9-87317 10-12683 10-09623 9-90377 1 4 37 9-77946 10-22054 9-87711 10-1228910-09765 9-90235 53 1 4 9-78197 10-21803 9-88105 10-11895 10-09909 9-90091 3 4 1. 9-78445 10-21555 9-88498 10-11502 10-10053 9-89947 1 3 "4 9-78691 10-21309 9-88890 10-1111010-10199 9-89801 1 4 38 9-78934 10-21066 9-89281 10-10719 10-10347 9-89653 52 1 4. 9-79176 10-20824 9-89671 10-10329 10-10496 9-89505 f i 9-79415 10-20585 9-90061 10-09939 10-10646 9-89354 ^ "5" 9-79652 i 10-20348 9-90449 10-09551 10-10797 9-89203 i 39 9-79887 10-20113 9-90837 10-09163 10-10950 9-89050 51 1 J. 9-80120 10-19880 9-91224'lO-08776 10-11104 9-88896 3 4 i 9 9-80351 10-19649 9-91610,10-08390 10-11259 9-88741 1 2 1 4 9-80580 10-19420 , 9-91996| 10-08003 10-11416 9-88584 1 4 40 9-80807 10-19193! 9-9238l[l0-07619 10-11575 9-88425 50 1 4. 9-81032 10-18968: 9-9276610-07234 10-11734 9-88266 3 4 i 9-81254 10-18746 9-9315010-06850 10-11895 9-88105 1 ■2 3. 9-81475 10-185251 9-93533,10-06467 10-12058 9-87942 1 4 4 41 9-81694 10-183061 9-93916,10-06084 10-12222 9-87778 -t'^. 1 J 9-81911 10-18089| 9-94299110-05701 10-12387 9-87613 3 4 9-82126 10-17874i 9-94681110-05319 10-12554 9-87446 1 "X 9-82340 10-17660: 9-95062 10-04938 10-12723 9-87277 1. 4 42' 9-82551 10-17449J 9-9544410-04556 10-12893 9-87107 48 1 9-82761 10-17239 9-95825 10-04175 10-13064 9-86936 3 i i 9-82968 10-17032, 9-96205|l0-03795 10-13237 9-86763 1 2 1 9-83174 10-16826 9-9658610-03414 10-13411 : 9-86589 1 4 43" 9-83378 10-16622' 9-96966 10-03034 10-13587 9-86413 47 1 9-83581 10-16419. 9-9734510-02655 10-13765 [ 9-86235 3 4 i 9-83781 10-16219 9-97725|l 0-02275 10-13944 9-86056 1 1 9-83980 10-16020i 9-98104'lO-01896 10-14124 9-85876 1 4 44* 9-84177 10-15823 9-98484slO-01516 10-14307 , 9-85693 46 1 9-84373 10-15628 9-98863 10-01137 10-14490 9-85510 3 4 1 9-84566 10-15434, 9-99242 10-00758 10-14676 9-85324 1 1 9-84758 10-15242 9-99621 10-00379 10-14863 9-85137 1 4 45' 9-84949 10-1505210-OOOOC 10-0000010-15052 1 9-84949 i Sine 45 Deg. Deg. ropine Secant Cotangent Tangent | Cosecant NATURAL SINES, TANGENTS, SECANTS, ETC. 329 Table of Natural Sines, Tangents, Secants, &c. Deg. Sine Cosecant Tangent Cotangent Secant Cosine I Deg. 10 •000000 •004363 ■008727 •013090 •017452 •021815 •026177 •030539 •034900 •039260 •043619 •047978 •052336 •056693 •061049 •065403 •069757 •074109 •078459 •082808 •087156 •091502 •095846 •100188 •104529 •108867 •113203 •117537 •121869 •126199 •130526 •134851 •139173 •143493 •147809 •152123 •156435 •160743 •165048 •169350 •173648 •177944 •182236 Infinite 229-1839 114-5930 76-39655 57-29869 45-84026 38^20155 32-74554 28^65371 25-47134 22-92559 20-84283 19-10732 17^63893 ;16-38041 15-28979 14^33559 13^49373 12^74550 1207610, 11-47371: 10^92877 10-43343 9-981229 9^566772 9-185531 8-833672; 8-507930! 8-205509' 7-923995' 7-661298; 7-415596 7-185297 6-968999| 6-765469 6-573611| 6-392453 6-221128: 6-058858! 5-904948! 5-75877li 5-619760: 5-487404! •000000 -004363 -008727 -013091 •017455 -021820 -026186 -030553 -034921 •039290 •043661 -048033 •052408 •056784 •061163 -065544 -069927 -074313 •078702 -083094 •087489 •091887 •096289 •100695 -105104 •109518 •113936 •118358 •122785 •127216 •131653 •136094 •140541 •144993 •149451 •153915 •158384 •162860 •167343 •171831 •176327 •180830 ■185339 Infinite 1 229^1817 1 114^5887 1 76^39001 1 o7^28996 1 45-82935 1 38-18846 1 32-730261 28-63625 1 25-451701 22-90377 1 20-81883 1 1908114 1 17-61056 1 16-34986 1 15-25705 1 14-30067,1 13-45663 1 |12-70621 1 '12-03462 1 1 1^43005 1 |10^882921 10-38540 1 9-931009 1 !9-514365 1 9-130935'l '8^776887:i 8-4489571 8-144346 1 7-8606421 7-595754 1 17-347861 7^115370 6-896880! 1 !6-6911561 6-4971041 6-3137521 6^1402301 5-975764:i 5-819657 1 5-6712821 5^530072 1 o^395517!l 000000 000010 000038 000086' 000152 000238 000343 000467 000610 000772 000953 001153 001372 001611' 001869: 002146 002442 0027571 003092 003446; 003820; 004213! 004625 005057 005508 005979! 006470! 006980: 007510. 008060; 008629 009218 009828 010457 011106 011776 012465 013175 013905 014656 015427 016218 017030 Deg. i Cosine i Secant Cotangent Tangent | Cosecant 1^00000 •999991 •999962 •999914 •999848 •999762 -999657 -999534 •999391 •999229 •999048 •998848 •998630 •998392 •998135 •997859 •997564 •997250 •996917 •996566 •996195 -995805 •995396 •994969 •994522 •994056 •993572 •993069 •992546 •992005 •991445 •990866 •99u268 -989651 •989016 •988362 •987688 •986996 •986286 •985556 •984808 •984041 •983255 90 89 87 86 84 83 82 81 80 79i Deg, 330 NATURAL SINES, TANGENTS, SECANTS, ETC. Bes. lOf 11 12 13 14 15 16 17 18 19 20 21 22 186524 190809 195090 199368 203642 207912 212178 216440 220697 ■224951 ■229200 233445 ■237686 ■241922 •246153 •250380 •254602 •258819 •263031 267238 •271440 •275637 •279829 •284015 •288196 •292372 •296542 •300706 •304864 •309017 •313164 •317305 •321440 •32556S •329691 •333807 •337917 •342020 •346117 350207 354291 358368 362438 366501 370557 374607 Cosecant , Tangent : Cotangent -recant Cosine De, 5-361239; !5-240843 5-12583l| 5-015852| 4-910584! 4-809734' 4^713031 14-620226! |4-531090^ ^•445412, I4-362994! i4^283658: :4-207233 i4^133566 4-062509 3-993929' 3-927700 |3-863703 3-801830 !3-741978 :3-684049, i3-627955 3-573611' 13-520937' i3-469858 '3-420304: 3-372208 3-325510: ;3-280148! ;3-236068| 13-193217 |3- 151 545 3-111006 13-071554 i3-033146 2-995744' 2-959309, 2-923804^ 2-889196 2-855451 2-822538 2-7904281 2-759092 2-728504' 2-698637i 2^669467 189856 194380 ■198912 ■203452 ■208000 ■212557 ■217121 ■221695 •226277 •230868 -235469 -240079 •244698 -249328 •253968 -258618 -263278 -267949 •272631 •277325 •282029 •286745 •291473 -296214 -300966 •305731 -310508 •315299 -320103 •324920 -329751 -334595 •339454 -344328 -349216 -354119 -359037 -363970 -368920 •373885 •378866 •383864 •388879 •393911 •398960 •404026 5^267152 0-144554 5-027340 915157 807685 704630 605721 510709 419364 331476 4-246848 4-165300 4-086663 4-010781 3-937509 3-866713 3-798266 3-732051 3-667958 3-605884 3-545733 3-4S7414 3-430845 3-375943 3-322636 3-270353 i3-220526 3-171595 j3-123999 13-077684 |3-032595 12-988685 '2-945905 12-904211 ;2-863560 :2-823913 12-785231 '2-747477 i2-710619 12-674622 [2-639455 12-605089 2-571496 12-538648 12-506520 2^47508; 1-017863, 1-018717 1-019591' 1-020487; 1-021403' 1-022341! 1-023299 1-024280 1-025281 1-026304 1-027349 1-028415 1-029503 1-030614 1-031746 1-032900 1-034077 1-035276 1-036498 1-037742 1^039009 1-040299 1-041613 1-042949 r044309 1^045692 1-047099 : 1^048529 ' 1^049984 1-051462 1-052965 1-054492 1-056044 1-057621 1-059222 1-060849 1-062501 1-064178 1-065881 1-067609 1-069364 1-071145 1-072952 1-074786 1-076647 1-078535 •982450 ■981627 •980785 •979925 •979046 -978148 •977231 -976296 •975342 -974370 •973379 •972370 -971342 -970296 •969231 -968148 -967046 -965926 -964787 •963631 •962455 -961262 •960050 •958820 -957571 -956305 -955020 -953717 ■952396 •951057 •949699 •948324 •946930 •945519 •944089 •942642 •941176 •939693 •938191 •936672 •935135 •933580 •932008 •930418 •928810 •927184 79 78' 77 76 /o 74 73 72' 71 70 69 68 Deg. ; Cosine | Secant Cotangent j Tangent Cosecant; Sine Deg. NATURAL SINES, TANGENTS, SECANTS, ETC. 331. Deg. i Sine 1 Cosecant Tangent | Cotangent j Secant j Cosine | Deg. •378649 2-640971 -409111 2-444326 1-080450 -925541 | 3 4 1 •382683 2-613126' -414214 2-414214ll-082392 •923880 1 ■2 1 •386711 2-585911 -419335 2-3847291-084362 •922201 1 4 4: 23 •39G731 2-559305 •424475 2-355852 1-086360 •920505 67 1 •394744 2-533288' •429634 2-3275631-088387 •918791 3 4 1 •398749 2-507843 •434812 2-299843 1^090441 •917060 1 3. •402747 2^482950 •440011 2-272673 1^092524 •915312 1 4 24* •406737 2^458593 •445229 2-246037 1-094636 •913546 66 i •410719 2^434756 •450467 2-219918 1^096777 •911762 3 4 1 Q •414693 \ 2^411421 •455726 2^194300'l-098948 •909961 1 •2 3. •418660 2^388575 •461006 2-1691681-101148 •908143 1 4 25* •422618 2^366202 •466308 2-1445071-103378 •906308 65 J, •426569 , 2^344288 •471631 2-12030311-105638 •904455 3 4 i •430511 i 2-322821 •476976 2-0965441-107929 -902585 1 2 "X •434445 2-301786 -482343 2-0732151-110250 -900698 1 4 26 •438371 ; 2-281172 -487733 2-050304 1-1 12602| -898794 64 1 4. •442289 '' 2-260967 •493145 2-027799 1-114985 •896873 3 4 i •446198 2-241159 •498582 2-0056901-117400 •894934 1 •450098 2-221736 •504042 1-983964 1-119847 •892979 1 4 27 •453991 2-202689 •509525 1-9626111'-122326 •891007 63 4. •457874 2-184007 •515034 1-9416201-124838 -889017 3 4 i •461749 2-165681 -520567 l-920982'l-127382 •887011 1 2 ~^ •465615 2^147699 -526126 1-900687 1-129959 •884988 1 4 28 •469472 2-130055 •531709 1-880727 1-132570 -882948 62 1 4 •473320 2-112737 -537319 1-861091 1-135215 -880891 3 4 1 •477159 2-095739 -542956 1-841771 1-137893 •878817 1 2 1 4. •480989 2-079051 -548619 1-822759 1-140606 -876727 1 4 29 •484810 2-062665 -554309 1-804048 1-143354 •874620 61 1 4 •488621 2-046575 -560027 l-785629ll-146137 •872496 3 4 i •492424 2-030772 -565773 1-767494 1-148956 •870356 1 2 i 4 •496217 2-015249 •571547 1-749637 1-151810 -868199 1 4 30 •500000 2-000000 -577350 1-732051 1-154701 •866025 60 1 4 •503774 1-985017 -583183 1-714728 1-157628 •863836 3 4 1 9 •507538 1-970294 •589045 1-6976631-160592 -861629 1 2 3 4 •511293 il-955825 •594938 1-6808491-163594 -859406 1 4 31 •515038 11 •941604 •600861 1-6642801-166633 •857167 59 1 4 •518773 ; 1-927624 -606815 1-647949 1-169711 •854912 3 4 1 2 •522499 1^913881 -612801 1-6318521-172828 -852640 1 2 3 4 •526214 r900368 •618819 1-615982 1-175983 -850352 1 4 32 •529919 1-887080 -624869 1-600335 1-179178 -848048 58 1 4 •533615 1-874012 -630953 1-584904 1-182414 •845728 3 4 i •537300 1-861159 -637070 1-569686 1-185689 •843391 1 2 3 4 •540975 1^848516 , -643222 1-554674 1-189006 •841039 1 4 33 •544639 1^836079 i -649408 1-539865 1-192363 •838671 57 4 •548293 b823842 •655629 1-525254 M95763 •836286 3 4 1 2 •551937 1^811801 j •661886 Cotaro-er.t 1^510835 Tangent 1 •I 99205 Cosecant •833886 56i Deg. j Cosine Secant Sine Deg. 332 NATURAL SINES, TANGENTS, SECANTS, ETC. I Peg- 33| 34 Sine Cosecant 35 36 37 38 39 40 41 42 43 44 45 555570 559193 562805 566406 569997 573576 577145 580703 ■584250 ■587785 •591310 ■594823 ■598325 •601815 •605294 •608761 •612217 •615662 •619094 •622515 •625924 •629320 •632705 •636078 •639439 •642788 •646124 •649448 •652760 •656059 •659346 •662620 •665882 •669131 •672367 •675590 •678801 •681998 •685183 •688355 •691513 •694658 •697791 •700909 •704015 •707107 Tano-ent Cotan.eent Secant Cosine i Deg. 1-799952 1-788292 1-776815 1-765517 P754396 il-743447 1-732666 |l-722051 ;1-711597 ;i-701302 1-691161 1^681173i 1^671334j 1^661640l 1^652090' 1-642680 1-633407 1-624269 1-615264 1-606388 1-597639 1-589016 1-5805151 !l-572134| il-56387ll ;i^555724' P5476911 1 •539769; ;i-531957j ;i-524253; 1-516655 1-509161 1-501768 |l-494477 ll-487283 1-480187 1-473186 il -466279 T)eC -N t^ « CI CI c rc o » F^ us 1— i 52 ^- 00 i-< •«ia .=■;; 1 12 1 2 1 1 2 1 i2 1 2" 1 : ss •q^T ^?; ?o C5 ;o ■^ f- »« -»< t— -J ■* ^ ^ ^^ ■^ C^ rH «r ^ c; o ^ :z < us M 1^ C<> .^ t~ ~ - t~- ^ ■ tC 'M X! r^ •■Bia j=2[o|cs| I l2Hl2l2'-l2l --S -q;^-! ;0 O ^ ; ' « o o tc cc o c; I -H b- rt :c -M 00 -H I I I I •B;a =3C I - ! 2 I ■q^^ii ■ «2 — S -.c » O O S - — o ~ c; cs s o s 5<> X -- 00 -- o << •«ia i| 1 2 1 2 1 1 1 S 1 ^ i 2 i 2 1 " 1 '^ 1 •qjs-i ^<:ofCO»-JOOOOC;-.3:rOO^O«>»00 _^0— ■>}< CO — 1— t— «D lO CC r^ ^5- ^S«2^ c ~ - • if t: ;= i e - v: - S — -- " *- .S -^ -- ' ^•;=s s s S c T — ^•^^ C ,3; ^ j£^ " '^ "^ ^ ^ i:/: "^ = C- Ji iiP= > ,«5 S --• r .5 ^ — r- eJ J- D. 3 -2 >> ^ -^ 334 DIMENSIONS OF SHIPS' MASTS AND SPARS. § - ■''SCI lis I 2 1 "^ I -r c; , c~ I <-^ •iI^3T;1~0-^;'-'=2-'-= -^-o -- 12 I I -r r; o i-H c^ T-i «^ ■«< » 1 I c^ •nial =:;; I ^ I 2 "S ",H ' -^1 ^is I— c i-s T^t-t 3C :3 -r ~ rs i ^' -^ .*:m — L- ■M — X ■-:; S :r! O « «o o — . M o — ro c<» ?: o 'M -- ■BIQ I =T?' i^ -qiSi'-E; £.i^=3 s^ •«!ci|i2 Igl^l ig^ 1 :;• I ^ I « I » I ^ •"^l^iia^i.': "T^i— ' ~ :3 •* ?5 ■* I , . , -4:1 , -e> ^^. -qjSi — '^ * ^r^'~ j, ^ j j~ V :o~^^C> CO olcTo O » rs cousoOOOO «0 ■«ra .=2 2 I i s I 2 ! i: I ^' i I I I MM 5 O Sui S<5 -11131 ■'" :o o ic X — — ■* o 'S" -?» o '^« -H •BIQ coo -M - i S' i I S I 2 I - ! - M '/3 < '-3 cr: c. te ~ ,-. -7- C I I I 5 • ^ • "a • -j: -5 • • o 5 s -J — -^ , tr. ■/. •-' — "^ "C . ^xi S 60 uJ H Eh >> — :2 I 1 Si o o o ^4 H - ^ — .ti) 2 i io 3 >:S"~ ^ o- :2 o = y = 5 = 5 -1- - £ "^ 5 i; a DDIENSIONS OF SHIPS MASTS AND SPARS. 335 > z Is II ii •BIQ, =:£ 1^ ,b- 1 |C |0C iS'j J?'| ;o j 1 1 1 1 1 j;c> « 00 e soo 000 — oc c: •qjS'Ji •"« I- 1— o CO oc ffi CO ■<*« w o<> t» -^ cq 1 1 1 1 !;;;-«< rs ^ -^r-eo CM 7-1 CJ '''1. ro < .=g5 1 2 1 ^ 1 I 2 1 S 1 '^ 1 "* 1 ^' 1 i 1 1 1 1 jjjO-nTC (Mi-i II •"la 1 .=1 1 2 1 2 1 1 2 1 « 1 « 1 « 1 ^ 1 1 1 1 1 1 5 - 72 ' ■BICT |i«l2o|o| l^l^l^l'^l^l I^Sl I ^<£>00«CO«0«OCC-*C>.-iC:CC!CnOO COOO o - •^lali- l«l-i l-lEl-l-!-l2i i 1 1 to : Mi. ^ ill! i I.I'? ii^ ■5 s 11 •"!a| s|? I^!-I 1^:121*1=^1^1 i 1 ' 1 ' 9 ' •q^l :f 3 •»}a .Sg? I^ 1;:: 1 1^ 12 1* !'-« 1^1 IgS'l 1 ■■"ooi^oot^coc^MO--— <'t — <^>Xr-<>■^■^^ | rct^coc-^ S c CE 1- H h ft : Si >-l > -( - \ d -1 C <: t- < % JO 3 /J 1^5 -3 ^ -S ^ ^ 'I -i • '1 -1 'i -1 r M-|--1i-l|i'^ 1 ?.|.^^2SH|^^S!5 .111 ! ^- ^a« -^ -S ^ ^- o £2 SB'S I I % 1 1 1 U d 3P5 1 >• •" ^ ^ ^ ^ -*> ^^ rt '^ -^ 33 3 a ^ J '^ '^ fi -3 -r^^ H -= - I— I •5 '*^ -^ "^ * 336 PIMllNSIONS OF BARQUES MASTS AND SPARS. H J 1— t S M r/J < rn J w r/; /'-s O > '^ § 'ZL <( c •Xi >■ ^ 'i^ a < cs WH o p; yj o a o § rK o 5^ r/J Q j^ J o ta^ 1:3 XJl -^ P- ^ 7J 72 Q C Iz; iz^ -< < xn T' < o a o o C3 V? 1— ( C a 02 IS! Iz; <) a E f^ — ( UH Q a K H X m (i !z: hH > ^^ O a I-] P3 < ^ :- "^'■^\iX 1 ^ N S S 00 tll'"^T[ •" u- c^ ■^'Q m^ii' — ?n ' •qi3Ti ■^;a - ! I oo-r 'i-i — rro(^«r»! e CO \\ o o o s ;s «o >-1 O ■>! -O -H lO 1 ■^ ?: CQ 1 1 1 ^ ?f^ •^la .=£ 1 1^ ! If 1 if i f 1 1 Ci 1 t- 1 cx> 1 ?J2 1 ^C;S3c:os Oiscs;j:o OOCi •qj^l — r: -- -.a ;o 'N -^ C^ O C! O I lO o ' ■^fa =- .s-* -qia-il' ■^iQi SS "qja-i;' 3 tl l2 oo ! I I m I t>- r» ' — I •J o ;a o ■?< ' — = I I ' 2 l?52 I . ^- :o I I 00 c^ I !■< -H ■« o M -^ r: O :c O =a ^ 3 . .2 13 r/: • o * .5 c =-?£—''; ■5« ^ s o 6f a> PcS c OS ?^ 2^ o ?^ ^ K.S2 J3 S S _- g^"^ P "5; 3 c 5 -n V s i- - ;; js SD , u = 0^=3 = • S _' 03 Ji ^"S. en — - * 5 — s "O *- j^ as S bo c c Si,itt H « o c y 5 o ^ O o « 2 a> >> £ J; 5 SJ o o; Q. o -CJ. ^ p. o -S •- O v: « 1-5 Q DIMENSIOXS OF BARQUES MASTS AND SPARS. 337 m H c 02 ^ IS- ?q M < e, 02 c HO § n c» rT/ >c & c O l-t m ^ rt j; -o ' 1-1 ^ t* o "BtQ -==3 J- 5 . cs = i^i-qja-i'-- c^ -. sj 3 •'5!a •q*3a I I- I 1^ IH" roooo ooooco ■Bia ^ I I i- I s O CS O ■* X !>« O W © •<* 00 c^ I I ■-I cc — I I o © «o C I ■<* 00 C^ O 1 © O CO © ■^ 00 len o IT B -•J -3 5 1 C d c Yard a ard. WhoU ail yard. W Y nt yard. W Yf rd. Whole 1 Yard ai hole length, joom. Who , exclusive o s ?= >> If 1 alya . W iker sprit oom ^ c p.a IS afc! ft >. SB = &-= c c o o c c; o o c: S. o .-2 1—1 H ^E-1 1-; Ha H ?^ C X C2 H, 338 DIMENSIONS OF BRIGS' MASTS AND SPARS. O S ?•■ 02 2 :-c "fa I 2 1 1- TT - , r^ S ' . »■" o " •qjSrj »0. CO 'r-i ooac^i— cIQiO'cO •■Bia : =» I in I I w .9. I ^ I o I « I ^ I I I I o « I .cj-«iW)i-ic5.-i I CO 'r-i O CC C-J >— I ' ■^ta ^<; -qi^l ---.o-^c^iw 1 r-| ■"ICT et;- ->«'■-< ' -- ; :r iM ~ o PH r, '^ « >* T^ •— SS .'qiSil ••^co- '^\(1 a I CO M«H ■^la 1 >J O -H ' ' L-3 *2 l2 -^ I r- O Tjl LT »C O CO ^ t— O CO ■ t'^^ I I I I I ooooooo oo I I ^JCr l£;5 I ^ I I o I I o I >o I o I t- I -H I c3 ej ;o » o i= ;o o c<> t-- OOOOOOOeo^i^ ■^la S; 00 o I 1 I • o ^ :s ^ OsSOOOOOOOOco-OO oo~ "^•^ jU^ST' •"?■> vOOlt— I-— l-t<5^-"-^-^l-'5 0^'MC:i-liOS^ I ^ -^ .*: i-O r-c o I C-^ ^ CO O ■«< CO CO ' tn .3 to 2 - S 5 =^ d- 2 » ? s :3 o >- - 3 ^sS-^iq = -5 _j s' '■'^ ij 3ii S fe rM 2 C3 - CO 2 c-^ >T3 5 5^ c 2 C CD -T^ ^. V. 'rr. 3 a o o a >. a o O r» •d 75 C H H C5 C3 x:2~c a -3 f. 50 !-< iy O o; to O C3 ::^ DIMENSIONS OF SCHOONERS' MASTS AND SPAES. 339 '"If ^ 3 •«ia ,.=- 1 3' 1 1 1 1 1 1 1 1 1 1 , 5-' 1 =^ 1 1 ! X P5 •qisi 3^ f- ^ 1 1 >= 1 1 1 -^ 1 1 1 00 M 1:2 1^ 1 1- H ■5 J ••cia - =5'- 1 * 1 1 1 ' M ! ' i ! '-ill;: 13 ^ ic -0 s •q;2l;~-;t-- 1 i ''^ 1 1 i "^ 1 i ! 1^ 1 1 1 li e.l •ma ! if 1 ^ 1 1 1 3 ' f- 1 1 ^ ' 1 1 ^ 1 1 cc?:' I OJ m ■-oot-cio !'^2'^~ I~?~' 00 000 1 o» -^ ( CO L- ^ fc^S •qjgl 3 < d -^ •IJIQ =^2 . X j • '-* 1 "^' 1 ^^ 1 c3 s = aO s S o -^ CO m 2 — 5 '^ ?- DC :^ H o ^ 1 — 1 '~ -Jl •71 — § 'f ' .^ ^ 1— < ~ a > i^ •qi^T ^E;;^^ 1 1 "^ 1 1 1 ;5-< 1 1 1 1 I- T-l 00 1 • =; d ■via 1 _2-^ 1 ^' 1 1 1 1 1 1 :^ . 1 1 1 1 =» 1 2 1 ! ■ •q;3^ f ^- = " 1 1 --= 1 i i 2^ 1 1 1 ! i.c' i<( t^ 1 1 -^ ■Bia ; =2 1 2 1 1 i ^ ! 2 : ! '^' . j X 00 00 1 c 1 i -H C^ -) C^ « 1 ^^-^\ Is? ^ Is •''ia i .=1^ ! - ! 1 1 1 ! 1 « 1 i 1 1 i« 1 1 M : ^ s -J 00 1 ' " »^ 1 I 1 == 1 03 01 i1 j'^Jo 'igj 1^ 1 1 i;^; I3 1 t*- 1 1 1 « 1 ! ji- ! T 1 ijJOOOSO »SOO OOCJ 00 s = j 1 C-. c^ 1 -^ - _ -3 +- CO a < rn tc ■< :€ a; 3 5 ^"^ il||iillN ^ Ij C r- f^ 3 T. >5 _ _H ^ 2 5 1 2-^ ^ c z2 £)40 DIMENSIONS OF SCHOONEES' MASTS AND SPARS. o O o a o a o w. ai a -,o S Si *" r^ rf ^ '~' -! 3> M jr O s *i — o «. M 01 n ' c — S ^ 5252 ^2sE = i> ~ o «-^C 2 . 30 t- r— ?: o o O e<5 so w o o CO M 00 I I I -s O CO o o r— t~ cc r: .s.^ ;::i I 1 JO I-H o oo b- M lO S I I |o |o I |C.«. I I OwO OOOOO <0OOOO .COOCO L.- o ■ri ^ C ^ ! c o o«o CO o o © oo I I ■* C^ CO d HN =>* -tT •*> ^ U3 1 => 1 1 <= 1 flO 1 "=> 1 CO 1 1 ® 1 1 I a •- '"' 1 1 ^ o - COOO CO oo o CO OOOOO o o t h tL '>i O »— ' ^ O' ?t ■-*• ,— —l ■-^ i; rf •— C^ '-~ - 1" -' ■- '' ~ 5^ S 3 S o o ■= ;: ~ -= >> \5 Js^^ — Ph :;^ C '^ ^r^r^ 6£ ^>^^5 to - & bo's -.9 ^ '■0 2j S ?- ^ S o C c3 o ^ 7; c ^ v: — o S 3 CJ KELATIYE PROPOKTIOXS OF SHIP.?' MASTS AND SPARS. 341 ^ CD 53 'o X X n re L-t -^ t^ I- -N r: oc -w » c; X — — e- ; ^ a j-r- -tr -^ — i^ v= --r r: X — oc y :3 |— -M Cn -N X r) C: — — l^ — i~ ^ -M 1 1 V ^ ^ 1 1 ^ '-~ 1 1 c; c: T) 1 1 X 1_- s r^ 5 ,^ ^ — C- ' in ^-X. £ - t^ cr. --C cc r^ ^! '-t "M — -M vr --C — L- X — ~ X. X VD c^ - — .- — t- - -M — I- I- r: u- — — I- 1 1 ^ I'- 1 1 — X c ^ Tl ri r: c^i X CM r; — c; — I u- r^ — f^ s^ ,,. 1 ' ? ~ T 1 1 I- 1- ^ ™" 3 5^ •^ ', o _ 1 s > _ C 1 -i- H ii : X X -T' X ^t ~ c: ic u-t -T- -r l--^ c; --r c: --r — s ^ £ £ ;= :s :-M -M cr. Tl X ri - — 9 — t.- — ';-: cc p 1 1 ^ ^ ^- 1 1 X t^ 1 1 c^ c; 1 1 t- I- >ac:- ^'Z, T— — o "^--^ ; |c 1 t^ TJ l- a: re 9 9 ? ' 9 9 X i_~- m H JL ^ "' '•* "' '•' j:- "^ ^ "^ • '^ ~" '^" T— 1 T^ j_^ • • • GO << 1 ':g32x§^'^ = i::§^f.'^^ C-- — r;r ^ 1 1 C: Ot^ 1 ! ^ ^' i^- 5 1? , '~r ■* zi g C '■" «-" — ^^ci — cjZZ'^ c •~ z — ■ Oi ^o-t30^--^c;-c;-cj^ t^^ c; c; ^— C- T ^ CV Ci S tt> -^ 2^ i_^ C^ 2-r- 5 . • ' Z! ~ ZI '^ >~ /; — t£ r as s — S S ?: 5 — 5 ^ 5 .=? 5 — 5 -3 5 c -' 1^ -^ 5 c ^ p^ » |±:-r -'"'::;~'^;=;-:;~-r''-r^ -r :^ TT -^ =^ -r " -^ "^ H -< ss = = =: = s = = = c=;:i-s — ..ll— 1 ,— .-^ .- — -^.-H ,^.— .. -- ^„^ 2 icSoSSJiSSSSSEi ?• ;:; cS - S ^ c ^ "^ X ^^S^c-S-o,:=s^=^^ ^ ^^ = -S :: jz ^ < !l II II II II !l II II 1! II II II II II II II II II 11 U 11 II II II r^ , I™^ • iinded Ic . iiiidcd 02 . . % -0 j t^ ■-t ^ ir; c^ r: X "M ' t"' ? ? 9 '." T^' ' I I ' ' ''.■"??? ^ ?'■? ^ ipi|iiS I I I I i li^iiiiiil * 3 X vr 3 p t- X 1^ -M 2 Tl cr. IC CI 'M O "M -f ~ — ~ i':^pooxoxd;ooxoc^ooot]^oxo * I CO t;^ C: r ^ Ci X CC -f O -+i O w Cl 'M O 'M CC X -^ c. ■r: CO C: Cf: X CC -f 'M O 'M l^ -m — lO O -O O CC r ' r: 9 9 9 I - 9 X o 9 o t;- cp C5 9 o 9 X c: X o xxxxxxxxxxxxxxxxxxxxx r* 'A r- ^ r- >>Zi S * i ■ =: 5 .S S i/.'? •- ? -=3 'P .5 T' .5 ? .5 ? .5 T^ .S "^ .2 '^ .S '3 — ;r^r-, ^. ;:: >->s >, s ?^s >-iS "^-1;= >%s >-iC >-iS >-. • ~j i; « S -rz,-^ -■=. — ;y: ^ • 2 ^ 2 ii- S ^ ■■'' •A a; v. - - S W r; ^ — 2- ^r— J,— O—- ^r— 1, "^ . -_ y o « 1^ &=< »^ F 5 rt 1^ ci o ^ -~ O — ^— 2 ^ 2 2 t- TJ -o 7l r^ • "^ • * • ~^ • • •" "^ •"? ~ 9" ^ — '•.'' ^* T f "^^ '"T r-i ; t^ - 2- 30 t> yq — ^ O -O rJ t^ .- ~ ' •^ :n — i^ u- i^ -:; z: xxxxxxxxxxxxxxxxx X X X X X X X >. a. a, >^ i-. S i^ II !l II II II II II II II X OC ^ -, -Z •— ~ •— .~ "" -.C •• " "^ "^ "^ ■ II H II II II II II il 11 li i~ CJ 7; >-i « « X T^ T ^ w ^ — b. C :« — tS *-^ C3 tp -s ^ 7? g ^ ^ 7; ^ >< ^. X X i •« ^ , >. ^ _' ^ -. g -2 __^ _ _ _ *";h i:^ c> E3 ^ •" O) o ss ^ ^ rr ^ ^ C *^ -» 1- N 05*^^ — ^ »^ ,^ ii: c- c — -^ S ^ S 72 C2 ::: 9 .= s o4:4i PKO PORTIONS OF BARQUEs' AND BRIGS' MASTS AND SPARS. P o (=1 H W H (—1 P5 1^ H B1 Q o H fi P3 O H O -A > S!- ^ -T^ S: '^ I ^ -^ X 00 00 T-^ c^ i^ o t^ c. t^ o .= ^ ^ ;i ?, I- I- -M ~^ -M " --0 V? — --C X O STi — o tc e j'J lO '^^ CI T) Cj 'M -M — t o ^H T^j -M ix> "tHOgalii c2^^pq-S i'-^ ^ ^ ^ 00 X) O X -N t^ o t^ ' ^ * X -t O' -t X w o o o I o cc o r: c3-^, I 'C '>0 -^ « O oST"alcCC^Ci^ r^* Q^ 5< J' ? ^ '-' '-I (M lO O lO lO t^ o t-- O ^^ O 1— ( i-H I>. O t>. CO ^ o vo 'O o o -^ c; '^t^ o -* -^ 55 O !X: -N O O O ->5 X X '-C VD ^ O -f O l^ O X i- t^ -M CO — I 'M Ci ri C: — I X t^ - IX-- o rt t^ o -j< — O --^ CN fN -^ o X 'M c; t^ o t^ o i^ i^ --T 10 o uo "■ — CO CO o -Ti ci i^ t^ 'o o --c 'O w v:; x -m i o i i^ -^ -^ — -f -^ X ■j:> ^1 X CM ci — o -^ o — o o -o 1 o 1 -!- CO O' CO CO W ^- — , ,— 1 .^ ^ * -H — H -r; m ■ P t« 0/ OK? S ,_^ — . -fi t^ •M t- f^ O n O C5 ^ ^ O O O ~ -^ -^ '-C ^^ & -H CO 'M X CO CO O l^ O L^ -M -tH ^ O O O 1 1 1 1 o-j -0 -^ D3 -^ '^^ c%. -M X oi o — o --1 X CO --r .-o ~ .0 1 1 1 1 -M 1^ - -H ^ -' '3 |(M :^D l-^ .0 ^ CO' X I^ ^ . H, . , 5^J — O O l^ 'M O ^ohclX -^CO'CSCOX'NX ists'Si^ -^ O I O -^ Ol y S !^ -+ CO X CO o r^ O -^ l^ t^ CO X TJ Ci 01 X "M O t^ X -+1 X 'M •-* xxxxxxxxxxxxxxxxxxxxxxxx T3 13 nS ^ '-' > c — c — ' 73 '^ 1 cu c- 1^ c w ;^ ^ I- c ;; S.9 ^M i=JDt^ SJDS faC^ fcX« bCg tf-g o2a>_24>'~w_2'yt3"^S^r- 5 ci a jD^ !3,C5 II II II I! 11 II t3 id '73 13 -^ 73 r-J (D eS r— ' i. ^ a ^ jj d OoooSo'c — ■^ ^ .a ^ ^ j3 Cm*^ '3 i a c ^^^^^^ fco D O C ^ '- kH^^ '5 s !- .. N3 .-• tn ;.;j ;;« ^5 v^ i-g r-; ,«S &^ z ■—1 »5 H P O H Q CO p < o -1 Sailing Merchant Vessel, Barque- rigged i^J 00 O X ->£ t^ O t^ 1 t;- 9 9 9 1 1 "9 999 [^999—1 I-T^^-irt-^-^T— 1 -t^! '■ 9 ^ , -s ~- ■-* .H £ .^ ?n "^ ^ 5 aC' 1 1 X 1^ 2 S 1 1 1 iH ip 1 i|g 1 a „• § 5 ' -^1 f^ - f — ' ~ lilf • j_j ~ ~ • ~ ^ • • "•" • ►^ 7 c^ 'o rt - -t * ~ -^ . & . X r-. c t^ 1 i ~~f^~ X 5 § >^ § X ?. 1 ^ X f^ ~ ^' J^ « 3 ^ 1 al« o o c: u-:. * 1 ^ X 9 X s t;;; X 9 X 1 : 1 1 -3 § X X ^ - ?i § ! 1 1 il^liggli < m Q < JO s 5 a X X X X X X X X X X XXXXXXXXXXXX •i-H-i?! II II II II II = yard = main topsail yard , = yard = main topgallant yard = yard )gallant yard fal yard . boom ■ vessel bowsprit . jibboom . water-line 3 ii 3 - •s'H-5'SS^i; t s :: s t II II II 11 II II II II 11 II II 11 Main topgallant yard , Yard .-irms, each , Fore topgallant yard Yard arms, each , Main royal yard . Yard arms, each . Fore royal yard Yard arms, each . Spanker gafU Whole length „ boom „ „ Bowsprit, outside stem . .libboom, housing excluded . Ilou.sing Fore mast before rudder post Mizen ^ = = £ •:S .9 =^ S p !-c s: *^ --? "" 346 KELATIVE PROPORTIONS OF SCHOONERS' MASTS AND SPARS. ^^ ^^~ ^^ M z ="ii 1 "T - X -M ^ X ^ x :^ . . — t^ '^ ^ t^ ^ ^ M •^ ■— X - tSi -- '-^ I- 00 1 <^ ■^ 1 - T* 1 -j: -— u '5.^ ^■t ^^ ?^ T^ T ' ' ^ 'T" 1 :o •7" 1 1 1 -0 1 CO ? T ^ "7 ? « 4 c^>- T^ * -Ah ' -^ — Ai T— ' ' * ' rr\ s CO t^ CO ^2 1 § 1 11 ?i o o 7 i 1 1 1 1 1 1 Mill -< S " ■ ^ ^ f^ ^ ^ m c'^. CO 'M t^ O c: i-O O^l — ' 1 is CO o ■>! o 1 1 w — "^ 1 :: 1 1 1 1 i I 1 1 1 X -J 1 1 1 CO a) 2 tc T^ 1'*^ (>1 ' T^ ■ T^ — W "ti 1 o 5 ^-^ O i^ — ''^ o CO 00 -^*' CO OD ^ a = a »-"^ O '~ ri 1 1 C: 1 1:: 1 ? i ISf 1 1 1 1 1 lO — o 02 2_tc Is Ci i>4 O — 1 1 X 1 l_^ 1 1 1 1 CO Pi s t^ O OO CO X CO o 00 !r> CO OJ CC CO — ^ -+ t^ "O' 1 <= 1 -f CO — 1 1 1 1 1 ^" tC M 5^ -?• '>! O O-l » — oc 1 rp 1 T^ -T" 0^ lO CO 1 1 1 1 1 CO ^ ,-. ^ ^ r^ i'i-^ ^ O iC o t^ X — "o o^ ■" s v 'II CO CO CO CO 1 1 o o o^ o oi 1 1 a; IS 1 ? 1 14^ 1 1 1 1 1 -7 tA ~ V t^ Ai ' A- * ^ ^rr. 1 CO ii t^ C^ >-0 — o i~ — " — i-O lO s 2 c c c > c c ^1^^ J ^ pi S 'fl o'o 2 2 's "r 3 o'o o) S r? g < 3 ci 3 c: S « c o c ci o 0^ -S '*"' o ., E S J, J, . j,sc -^ ~ _ fl -^ ^ 3 "" C - ^ ' c3 .2 <» <; - 5 t - '•c:, ' 'J; 3 H u- ,^ .^ :i «^ ^^ ^ Esj f< i-i ^ ^ EELATIVE PKOPORTIONS OF SCHOONERS' MASTS AND SPARS. 34< 3 o o ^72 02 OC CD s s o ; - « ita I ■ C: if: i~ rf^o'pg.y' J^ • • • • >--t O O O — O X o t- o ^ -r ^t 1.C i.-^ ■M O ^^ O 1 - O O O t- O t— 1 O CO o cc 00 -o o 1— I 'N ir^ -^ -!» t-- t^ CO -~ 30 rc — ^ o ^o "MX) I — Oi I lO — I :>4 O I O ,-1 I l^ 'O . , — o or o -^ c- I l?55 l§?, If^;;: I a -a C2 1^ t- O I I CO I -o o 'O -M O O O .-I O CCC» l^ O (M -f -^ ■>! i-O o -o o c^3 t- — I ic o o t^ ro ^-1 iC "M I - 'M O O to rr -TTi T-11— cCO-^T-iOt^-fi^ ^►.~ o Si, " o o t^ o ic 00 I -O 00 I --0 o 3 li^ — 00 11 cc 10 cc I- o o O :^ CO cr; Ci X X --0 -O uO -O O -* 5 t r^r.-^ T.r:: C4_ 5fc: o =i . to X »> >; o _• ».»-•« S^ ^ -^^ ^^J=^ C « C « ^ fee t; fcp .^^ o '^ '3^ 2 i>: S^ «-=: ^ o §-1^ 2T''i: > ^" i:^ ix -^ ^ ,_: "o -:^ ::3 -s 5 ^'TJ c3 en SI, ,S c; ■z - p •- s a p^- 3}^5 - ci r C •^ ?> .s c so c: .2; ^ J^ *==; «^ 5j c o O S O c5 -3 .2 y^ — — — 348 PROPORTIONATE DIAMETERS OF MASTS AND SPARS. Table of Factors used to Determine the Diameters OF Ships' ;Masts and Spars. Species op Masts a>;d Spars L I Ships', brigs', and barques' . Cutters' Schooners' Luggers' Ships', brigs', and barques' . Cutters' and schooners' Luggers' Topgallant masts ^ I Ships', brigs', and barques'. o i Cutters' and schooners' / Lower 3'ard Topsail yard . . , . ^ Topgallant and royal yard . ^ -< Cross jack yard . . , . kh Cutters' and schooners' square sail I Cutters' and schooners' topsail . ^ Luggers' yards . . . . Driver boom . . . . Main and cutters' booms o ■{ Jibboom O 1 Flying jibboom . Jib and flying jibboom in one ^^ , Ships' and brigs'. '^ J Cutters' and schooners' O (Trysail gaffs Given Diamr. I = Whole Length x End Diameters = Given Diameter •025 to -028 •020 ,, -021 •020 „ ^022 •020 „ ^021 •0-23 „ •0-25 •020 „ -022 •020 „ -021 •020 „ ^021 •040 „ -050 •040 „ -050 •020 „ •025 •017 „ ^020 •017 •0-20 •014 •017 „ -0-20 •018 „ -023 •017 „ ^020 •017 „ ^020 •020 „ •0-25 •017 „ ^020 •022 „ ^026 •018 „ •0-22 •018 „ •0-22 •030 „ •OiO 018 025 017 rheel . < hounds (.head • rheel . K hounds (_head . rheel . < hounds (_head . rheel . < hounds (_head . (hounds ( head . f hounds I head . f hounds \ pole f hounds I skvsail pole fheel . \ head . fheel . \ head . yard arms* (outer end ( inner „ f outer „ ( inner „ \ outer „ I inner „ (outer „ ( inner „ (outer „ \ inner „ outer end •833 •.s(jO •755 •83b •750 •500 ■833 •800 •670 •933 •750 •583 •800 •G90 •771 •500 •771 •500 •771 •500 •833 •GG6 1^000 .G6G •500 •500 •500 •500 •500 •500 •500 •75 •6G •75 •71 •75 1^00 •66 •75 '66 •50 •50 GO •GO yote.— The factors in the above table will apply equally well whether the masts and spars are of wood or of iron. RELATIVE POSITION OF MASTS AND SPAES. 849 Table of Position and Rake op INIasts for Sailing Vessels. Rig axd NAiiES of Masts Friffate Corvette Clipper ship Lugger Barque 3-masted schooner Coramon schooner Bermuda schooner fmain mast . < fore „ i_mizen „ fmain „ . < fore „ [^mizen „ fmain ,, < fore „ l^mizen „ r main „ < fore „ [_mizen „ fmain „ . < fore „ [_mizen ,, r main „ < fore „ [_mizen „ fmain ,, \fore „ {main „ fore „ Brig of war {-^° {main fore Yacht as brig Ketch Revenue cutter /main \^mizen } mam Cutter yacht main Distance from ^Middle of Water Line in Fractions of the Length, of that Line Before •37 to -39 3-72 to -399 •274: •396 •300 •295 •338 •279 to -31 •331 to -323 •323 •11 •13 to -101 •112 to •14 Abaft Rake in Twelve Feet Inches 062 to ^069 341 to •404 •096 to -06 375 to ^356 •047 •309 •04 •396 •067 •34'J •033 •366 •046 ■108 to -084 •147 to -138 •144 •395 6 to 5 2 to 1 10 to 9 6 to lOi 2 to If 10 to lOf 9 3 15 •12 •6 •24 13 11 17 27 24 30 24 15 24 to 33 16 to 36 10 to 9 3 to 2 10 9i " -2 12 18 14 to 13 12 to 15 3-50 WEIGHT OF MASTS, SPAES, EIGGING, CABLES, ETC. Table of the Weight of Vessels' Masts, Spars, Rigging, AND Sails in Tons. Kind of Vessel ^ailiKg Ships Tonnage (B.M.) 2,600 2,200 , 1,700 1,400 I, 'AH) Lower masts aud bowsprit^ Topmasts and yards Spare gear and booms Standing rigging t Running: ,, Blocks to ,, Ship's sails Spare „ 52-6 37-1 16-5 22-0 18-1 12-2 6-9 4-2 .51-9 36-0 16-0 21-2 17-2 11-1 7-3 4-4 36-7 27-5 12-6 20-2 16-9 10-6 6-0 3-7 34-1 26-.5 12-0 19-1 16-3 10-0 6-1 3-4 •21-6 18-6 7*5 11-0 11-4 5-4 3-« 2-3 Kind of Vessel Tonnage (B.M.) Lower masts and bowsprit Topmasts and yards Spare gear and booms Standing rigging t Running ,, Blocks to „ Ship's sails Spare ., Sailinc Ship^ oOO 9-1 8-8 4-1 9-4 6-5 4-2 2-1 1-5 Brigs 380 230 Schooner Cutter ISO 160 7-5 7-2 3-0 3-8 4-5 2-0 1-5 1-2 4-3 6-4 5-8 1-9 2-2 1-2 2-4 1-9 2-8 1-4 1-0 •3 1-2 1-3 •S.5 •8 2-6 1-7 1-3 •2 1-8 •25 * The masts and spars are all of wood. t Standing rigging of wire. Table or the Eelative Proportions of Iron and Hempen Cables, together with their Weight. <- '? 3 S w ti -^ ^ c O o Ins. Ins. f 7i f - 8 i Si 1 1 9i ! 1 (10 1 (lOi , Pl ! 1| ■ 12" [m u (13 (13i ]3 iU (Uh Weight of TOO Fathoms Chain Cwt. Qr. Lb. 29 17 39 1 21 .50 14 65 3 5 81 3 12 94 7 Hemp 5 '^ Cwt. (10 12 (13 (15 117 (19 (21 (23 J 25 127 1 29 (32 135 (37 40 Qr.Lb 2 9 26 3 16 2 25 22 21 20 1 15 1 12 5 o Ins li Ig If n 2 O c lus. 15 1.5i 16" (16i 117' (17i 118" (18i 19" (20 (21 1 "^2 (23 24 (25 Weight of 100 Fathoms Chain Cwt. Qr. Lb. 110 14 136 2 10 1.55 9 180 3 14 190 14 216 Hemn Cwt. f 43 its in j 5S 1 61 ( 65 - 69 i 76 ( 84 1 92 (101 J 110 119 Qr.Lb. 1 3 27 3 24 17 3 1 1 14 2 IG 2 S 2 1 3 2 PROPORTIONS OF TRESTLE AND CROSS TREES. 351 Table Giving the Girths of Hemp and Wire Eope of Equivalent Strengths. Hemp Wire Hemp ' "Wire Hemp Wire Hemp Wire Hemp Wire Hemp Wire (ins.) (ins.) (ins.) i (ins.) (ins.) (ins.) (ins.) (ins.) 1 (ms.) (ms.) (ins.) (ins. 12 H lOi U 8* 3§ 6f i n 5 2 Si ! If IH _ 10 4 8i Sh 6i 2| 4^ — 3 H IH 4h 9| — 8 31 6i i 2| H 1^ 2i i n IH n ^ 7f 3i 6 1 2| H — 2i ! 1 11 4| ^ 7* 3^ 51 ! n 4 H n ' I lOf 9 3f ^ 8 H ; 2i 34 H 2 1 lOi H 8f 7 91 ■H ' - 3* H 11 i I Table Showing from what Numbers of Canvas the Different Sails are Made. No. of Canvas Species of Sails made of the given Number of Canvas Courses, lower staj-sails, trysails. Courses, lower staj'sails, trysails, awnings. Courses, topsails, lower staysails, trysails, spankers, awnings. Courses, topsails, spankers, jibs, lower and topmast staysails. Topsails, topgallant sails, spankers, jibs, topmast staysails. (Topsails, lower and topmast studding sails, spanker.s, jibs, \ upper staysails, gaff topsails. TTopgallant sails, studding sails, jibs, flying jibs, upper stay- < sails, gaff topsails, cutters' and schooners' crossjack sails (^ and square topsails, sails of boats. ("Topgallant sails, studding sails, flying jibs, royal staysails, ( cutters' and schooners' topsails, sails of boats. j Royals, skj'sails, topgallant and royal studding sails, cutters' \ and schooners' topgallant sails, sails of boats. Note. — For the weight of the several numbers of canvas see p. 303. Weight of Ships' Rigging and Blocks. Weight of a ship's running rigging = weight of fKarnuls^^^^^ * wire standing rigging X ( br ricrs 1-534 1-632 1-719 -369 -302 -254 Note. — The above constants must only be taken as rough approximations. Proportions of Trestle and Cross Trees in Ships' Tops. Weight of a ship's blocks = weight of running flailing ships Length of trestle-trees Breadth of ditto Depth of ditto Length of cross-trees Breadth of ditto Depth of ditto hounded length of topmast x = length . . . . X = breadth . . . . x = hounded length of topmast x = breadth of trestle-trees . x = breadth . . . . x Length of lubber's hole = length of trestle-trees . x Length of lid = diameter of lower mast . x -22 •11 -67 -31 1-0 -67 -41 1-.50 352 SIZES AXD ^VEIGHTS OF SHIPS' STANDING RIGGING. Tl — •^ I o : r. ! ?? I ^ I ?r I - ~ f^ < o 'Jl o r^ E o ■— 2; III ^ ^ cu III I I iil^ xlx;sr:r:ol It^-^-jol l-wr^-cil^l?? o t^ I t^ I ^ — r^ ;3 I 1 t^ •- -J o ~! N -r K -r -r r: ': c i - > "■ ■ t^ I -^ -r -■ t^ t ! I ..- t H ? --- x 02 5; -^ =i < ^<^ . X I ac 5i w c • X I X I o a» -j; T- r^ •-"5 T- ui u- rt : t- .; o X X -^ X t>. u" •— t- T- s: . 2, i- |-r~3-slc5«Nr3l«l«««?»— ?J-"NW — ^N«^ 5'=*= I = I — Jq— 'l-i i i « I i N i 12 "" O-^'-' I I I «q n I 1 2»< M C^ ?1 -^ I t^ r^ o X o i- o t 3? X X -i X -i > X X Rjsniu sjsntu Rjsmu ^- — . ' SIZES AND WEIGHTS OF SHIPS' STANDING FJGGING. 353 z o N^ n w "- — KS'^?O'«J«rtC0»O'*F^'^0«*4coC^C0flqfl^'— I*3>i-r*CC-r; ; -^ M 00 o ; ^ ^ ^ ■^"^oor?W50'^tO"*5^^^s^W2^^'^3»as^»r;cco«OM "•" ^ o ec iccw-^w^i — ros^ico^i^ — — ""^'^^'^^^vi^c^^^^ en ■^-t- (^o- xoj:o-v«t^'at^>-'^r5"""'''«^^W'-~^'-"«o-«" oD t» o ., 5 S ? ^ ~ a. z > s < o ^ ^ a; a: ^ '^ ?4 Ei K S is ^^ c |3| -"' « ^ rs :^' ^ -H ri .^ c-i »1- ^'^ -^ ^ 9\^ rj ^ ;*i ^ m « o a 3) -J ci c O -«" oc o X 1.^ M ^5 '-ir r7 ^ « cc ^i r^' r: -J 'cc -J ■ x (^ON o . .= 1 r c -x — 1 -c^rc«cct>»"^t^'^^o cs ff ' =" -• #l:;S FCO'> n « ^ m — « ?i r; ?j r: i^ -^ " r^ W Ot^Cqt^;D-^3". OC^O'V^ -»Ct^*0 2 ^l'*'^''^ §2 p • >> -^ K >> -^ , > = « if" Sc& " 'c £ 1 2^ %:z.~^~j''''Z''' 5 - ^ '^ ^"f i* «^ £ ?, °"^ jj.„ „^ o a o o a ^, c-- c— c;::; c~ 0-- 3— =~1ft! i I sasTini ^ ' ' si?T!iTi pisBtn ^ \- — He .^^ X r- o Ul Ci X c H, u ■n bfi vi p o ^ X ^ c ^ ^ ■;:; Qi >- ^ ::; 3 ~ i: ^ tfl s -^ cS C o - o JJ ■^^ ^ ^ ^ F C- v A P a) c r" JS OS o ^f ^ ^ c c -^ .1- ti ■^ ■= — , -, c t- e; •— a; CJ .' .i C > z. CJ ^ c s s A A 354 PROPORTIONS OF RIGGING TO THE I^IASTS AND SPARS. Table foe Sailing- Ships, showing the Propoetions WHICH the DIAMETEE OF CHAIN ElGGING AND THE GiETHs OF Hemp or wire Kigging should beae TO THE Diameters of the Masts, Yaeds, kc, from WHICH THE KIGGING LEADS.* Parts op Ri&ging Pendants Shroudsf 5 5 "i Stay " ^ I Ratlines. (-Foot-ropes . Stirrups Lifts . Braces . Tacks . Sheets . Clew garnets Bowlines Bridles . Buntlines -i Leechlines . Slabline Fore staysail stay Halliards Sheet? . Tack lashing Downhaul Lower studding sail Halliards . Inner halliards Sheets and tack rShroudsf . I Ratlines I Backstays f I Burton pendants Stays Futtock shrouds ^ I Ratlines I Staysail halliards I Downhaul ^Pendants . . Sheets J . i-~ Topsail tyes ■2 a Halliards .S ^'' Foot-ropes p "5 I Stirrups . Floinisli horses L S %^ § ; Ratios •375 •188 •282 •057 •800 •2oO •250 •250 •300 •300 •200 •250 ■250 •200 •150 •120 •282 •200 •200 •150 •loO •200 •200 •200 '188 ■065 •225 •300 •225 •030 •057 •200 •160 •300 •050 •050 •250 •300 •250 •208 Parts of RiggesG Ratios halliards cr c: -; __.■ r Braces . Lifts Parrel rope Clewlines Buntlines Bowlines Reef tackles ^J Sheets J ? ■ Studding-sail Sheets Tacks Downhaul Boom ji.i^gers =^ I Heel lashing 2 I Boom-brace pendant ^ *- Whip _j -jf / Shrouds f . z g _5 M I Backstays . '"" " ' ■ Stay t Royal stay f Backstay f f Halliards and strapping Foot-ropes . Braces and strapping Lifts . Parrel ropes . Clewlines o'-j Bowlines ^ Bridles Sheets . Studding sail halliards =y Sheets Tacks I. Downhaul . /Halliards . Foot-ropes Braces and strapping Lifts . Parrel lashing - ^ Clewlines & bowlines 'X ^ Sheets * AH tbu riggiiip is of hemp, 'xcopt tlmt marked oth. rwiso, t Wire-rope riggiug. j Chuiii rigging. § Iron rods. PROPORTIONS OF EIGGIXG TO THE MASTS AND SPARS. OOO Table foe Sailing Ships, showing the Proportions of Chain and Hemp and Wire-eope Rigging in Rela- tion TO the Masts and Spars (concluded).* Parts of RiGGixa f ^ Shrouds t . I Burton pendants -: Ratlines . ^ " Stayt V Seizings f . I / Foot-ropes ^^■H Stirrups ,.M::) Lifts . 1^ '^ ' V Braces & strapping Shrouds f . Stay t = - Ratlines . §■ j Backstays t I Futtock shrouds (' Topsail, tyes % Halliards for do Foot-ropes Stirrups Flemish horses Parrel rope Lifts . |--| Braces . Sheets X Clewlines Buntlines Span Bowlines Bridles ^ Reef tackles tf I Shrouds t 5 ^ Backstays f ^ " Stay t Royal stay f \ Backstays f / Foot-ropes Parrel lashing Lifts . Halliards Sheets Clewlines Bowlines Bridles Strapping, ^-blocks ^ ia 11^ Ratios Parts of Rigging Ratios •146 ^ , Foot-ropes. •300 •250 '« I Braces and strapping •250 •069 c r^ 1 Parrel lashing . •lUO •174 ': ?, - Lifts .... •350 •027 i >^ 1 Halliarcls . •400 •300 4 1 Sheets •400 •250 ^Clewlines . •220 •250 2 /Topping lifts •400 •250 o o Falls and strapping •300 •188 Bot)m sheet . •400 •2-25 (U -, Outhauler •400 •056 g Guv pendants •400 •156 ", Falls .... •300 •030 c/2 Strapping to do. . •300 •050 /Throat halliards . •400 •225 Peak halliards •400 •300 ^ Vang pendants •350 •250 ''"'5 ; Falls and strapping •200 •300 1 Peak brails . •200 •333 Throat brails •200 •300 Middle brails •200 •250 VHook brails , •200 •050 •r; /Gammoning J ^ J Shrouds X ' ' ' •028 •300 •028 •250 1 j Bobstays J . . . •033 •250 ^ ^Man-ropes •133 •250 ' Jibstay f . . • •2O0 •250 Guys, single f •200 •250 r; Foot-ropes •250 •225 •225 o Martingale stay f . Martinyale backropes f •250 •175 •225 1-^ Halliards •240 •113 Downhaul •200 •113 Sheets . . , . -240 •300 '"Pendants •321 •231 s /Flving-jib stay f . •175 •350 o o Guysf . ." . . •175 •400 3 Foot-ropes •300 •400 •^ Martingale stayf . •200 •222 Halliards and strapping •250 •222 >-^ Downhaul and strapping •200 •250 r^ Sheets . . . . ••250 •308 ^Heel lashing . •250 * All the rigging is of hemp, except that marked otherwise. t Wii-e-rope rigging. J Chain x-iggiiig. § li-on rods. ffoie Girth of any lanyard= girth of rope set up by it x '5. AA 2 356 DIMENSIONS OF SHIPS BLOCKS. Table of the Dtmexsions of Ships' Blocks (in Inches). bjt-g S o 3 cc -5 J;E: f3S;Hol>2S.So|Ho OipH ■■^ o '"'3 5 3-2 2^ 1^ -'^1 = s --/: c ^ ~ r-io|r-ir^.HHO'r-iOiaj(-i Comviun Single-tJncTi Bloclts. 3 H H ^i If 3 8 8 17 1 4 H 2 H* ^* 1 2 i^ 18| 4 93 2i y| 'H 5 ■2 19 6 H 4^ n f 9 20 7 H ^i 5^ H '< 8 * 21 8 H ^* H 5 1 li 22 9 7 H H 5f H 23 10 n 4* 71 * 8 6* H 3 4 24 il H 5 H 7 H 7 8 25 12 H H H 7| H 1 26 18 lOi H 10 «i H 1 27 14 m 6 10^ 9 ^ 1* 28 15 Hi H iH H H H 29 16 i2| 7 ^n 10 2 H 30 12| 13i 14| 15i 161 16i m 18i 19 19| 20i 21 7* 13 13f 8 14^ 8i 9" 151 m 10 16| 17| 18i lOi 19 lot 19| 20i iH 111 12 ' 2 10^ Hi 12 12i 13 14 14i 18 16^ 17 21^1 173 -^■^8 -^'4 22 18^ 9->i! 12 Common Sin/fle-thin BlocTts. 8 6| 3? 6| ?:3 ^8 7 8 A 8 19 15 9 n 3^ 7* 6 1 4" 20 15| 10 H H 8 6| 1^ 3 4 21 16i 11 9 H 8| H 1* 7 S 22 17i 12 i^^ H 9* H li 7. 8 23 18 13 10^ 5 10| 9 li 7 8 24 18|1 14 iH 5i lU 9f H 1 25 19^1 15 12 5i 12 lOi H 1* 26 20i\ 16 12f 5f 12^ Hi H H 27 21 17 1 13^ 6 13f 12 H H 28 2H 18 1 14i 6 14i 12| i# H — 6i 6f -6| -6| ■6i 6f ■7 7 "7 .7 15 j 13^ 16 1 14 i6i' m 17" 15' 17i 151 ISJ-j 16| ! i'4; iH I 50^1 18^ ! 22A[ 20^ Clumj) BlocM. 3^ n 3* n 7 8 # 13 <) 7 n 6-^ 2^ 1 ^^ H H 3^ 1 ^ 14 . 9^ - 8 lOi n 5f 1 41 4 H 4 H 4 15 10 8^, 10? 8 2f 5f 5 6 4i H 7 8 16 lOj H lU 8i 3 6i H 6i 4-^ '1 i# 7 8 17 loi n- 11-^ 8,i 3 ( H 7^ 2 1 18 11 9 12 9 3 n 6 5f 2^ 1 19 lU 9.\ 12? ^h 31 s 8 6^^ 8'i 6| 2i li 20 12 10 131 10 31 6 Clervline Blocks 4} 6 6?- 4| 6^ 3| •> 1 1^ 8 10 91 11 10? 4 DIMENSIONS OF SHIPS BLOCKS. 357 Table of the DIiIENSIo^^s of Ships' Blocks (in luclies) — continued. t. o I s^- o -x ""x '.i^^ -'^ .2 ? is 5 ^ ^ o UuiMc-tkich Bloclts. 3 2i 2^ ^8 2* 3 ! 4 3i 3 B^ 5 4 3^ 3^ i 6 4f 4| 4| 1 7 5* o 5i ¥ 8 6i 5* (5i ^ 9 7 (^ 7 1 10 7f 7 7f 1* 11 8* 7i «i li 12 "^4 8i 9| 1* 13 10 9 10 U 14 10^ 9^ 10-^ U 15 lU lOf lU Iw irj 12^ 11 12i If 1 2i ^4 5. qa * 5 5f 1 ,^« 7 1^ 711 \\ '4. -^2 9| 10 If 1 17 113 18 ll3| 19 :i4i 20;i5| 21 16 22 ilGf 23 17^ llf 13 12il3| il2fl4i 131 15i 14^ 16 15 16f 2* 2i 2^- ■'4 03 -8 Oil "•8 2^ 24 18^:16 ilSi 25 119 |16|'19 26;i9f!l7 19f 2i 27 :20i 28 21i 29l22 30 |22| 17^20^ 2- 18' 2U 2| 2^ 19"22f'2| 19 18|'22 iioi: 12 13 14 14| 15 151 16i 17 17f 18' 2* 1-^ ^4 21 1^ '24 lA 91 1 -2 -^ 2^ 1 93. 1 91 -8 3 3L ^4 3^ 3| 3| 3| 8 If If It 2 2 2i ':?A Donhlc-tkin Blocks. 8 n 5f 6i 7 8 9 H 6 7* 1 10 84 6^ 8 1^ 11 9 6f 8f 1* 12 n lOh 7 7{ 9i 1* 13 10^ i* 14 lU 8 111 H 15 12 8^ 12 11 16 12| 8f 12| If 17 131 yj 13^ U ^sl 8 6 1 6^ 11 7iili 8||li 9 \H m 1^ lOl: li 1 lii 1^ II41 Is 12 ! \i 8 7. 8 li li U 18 14i 91141 19 15 i 9115 I 20 15f 10fl6 ' 21 161101161: 22 17110117 ; 23 18 IOII71' 24 18|J0il8i 25 191 1 101 191 26 20i|lOi20i 27 21 ]l0i21i TrcMc-t/dcl- Blocks. ;12f|lf'li 14 i 1| li :141 1| li 115 jll'H ^'^■2 Is 14 IQl ll li i"4 Is I4 171 1 1 -11 181 12 IX J- o Q X g J. _! ]C)i 17. 11. 1' 4 ^8 -'^4 911 -4 m.\ 4 4f! 5^1 6i 7 73 811 9f 10 lOf lU 12i 4 5 6 7 7! 81 n 4f 5 6| 7 7| lOfj 81 lU 9f 12 I 10 13 1 lOf m' m 14| 121 8 If 1 91 2 -2 * 3i ^ 3f 4 4i ^ 5 I 5f 1* 6i ii 7 1| 7f H 8i H 9 H H If 10 1 1* ll u ll 1# 17 18 19 20 21 8 22 f 23 24 26 27 li 28 H 29 H I 30 13 141 151 16 16f 17i m 19 19f 201 21i 22 993 151113 16|il3f 171^141 18|15i 192:16 201161 21 171 21|18?; 22119 I 23 19? 24 1211 24^'22 95 ''993 -^ — 4 H H 2 9JL 91 -4 91 -4 9^ 2I 91 95 -8 95 101 2^ lir 111 91 i li 1 9 93 1 1 j 91 91 15 13 2^ I If 14 2f : If 14i2|il| 15 3 I If j X 3 1 qi i 1 7. 161 31 2 17 17f 18i 19 3f 12 3|i2i 3f 21 358 DIMENSIONS OF SHIPS BLOCKS. Table op the Dimexsioxs of Ships' Blocks (iu Inches) — continued. Loiifi-tackle Blocks. i to ^ ^ •=3 |s . «w 'Jl Qi O 't> — o 3 O 1 "3 O .1^ ^ O -^f 2^ 4t 3f 3 4 1 13 ^M 5* 3| 5^ M ^ 1 14 lof! 6i 3^ 6i 5 1 8 15 iHI 7 4i 7 5.^ H 3 4 16 121 7| 4| 7t 6:^ U 3 4 17 13 B| •H 8^ ^'i i# 1 18 13f ••H 5f H n li ^k — — r^ c -^ ^. 61 iOj 6^ lOi n Hi 7h iH n 12^ H 13 — — If -2 5 ; -h:! u_( : — o 8| h\ H 2 9 2i 91 2i 10 24 lOJ 2| — - — I c -1. H Xiiie-jjin Blocks. Thickness of Block = 1 8 51 4 •6i 1 9' 5;^ 41 6f 1 10 61 41 7f :3 g^ 4i 4i 5i sit 61 11 51 12 53 J__ DIMENSIONS OF SHIPS BLOCKS. Taele of the Dimexsio:s's of Ships' Blocxs (iu Inches) — continued. Snatch BUehs. q-i CM o •S 3?8 u to « OQ ^W •s ai tw m c m P ■ZET 7 -1-3 044 a ~3f 5.8 ^1 ^ Ho 1 11 >• .::< ^ ■ H=S 7 8 17 5 9i ' 2 -^ Z. a 5 5-3.;^ .2 5 II Ho i^qg : 7iii /^ it ^ 2t Is - ca 3^ H v-i o.d A 7^2^ 8 H 3| K3 ^8 1^ 34 1 ^ 18 10 7ill^ 2i 7f 2i If y H. M 5^ H y^ 1^ 4 ly 10^ 8 121 2f 8i 2| 1* 10 H 4i H 1^ ■•■8 4 H 3 4 20 11 : 8^131 93 -4 8f 21 U 11 5| 6f If ^? If 7 8 21 iH '■ 9 .14 97 ^8 9i2t U 12 H 5i 7^ 1* •H u 22 12 : 9i 141 3 10 ,2| It 13 'H 5| 8i n 5| It 23 12^ 9^5^ 3* 101' 2^ It 14 n «i 9 2 6i ir |J, 24 13 10 16 3:f lU 3 It 15 8 H n 2* H 1*, 1^ 25 131 10|16| 3^ llf 31 2 16 H. 7 lOh 2i 7 2 1 1 ^4 26 14 lU17i 3* \1\ 31 2 Sinf/le-sister Blocks. j 4-1 4-1 •+-I f^ ;h r"r! t< ^ f-i S 2 ^ i^ ijh S « aj > a) 03 S .0 > O ' 31 1 |cH 1 *j 5-3 si 7 T 11 3 So" ^«M 1 • Ho Ic 1 1 ! H C CP-I 1 1-^ ' 21 ! 3 4 1 2 4:? 4 1 li ; 3 * 8 4 6 3^ 9V 1 7 ' 8 ' # 8 ^ 3^ U 3 4 — ' 1 DouMc-sutcr Bloclis. 1 =4-1 cl«f CM *. o ^> ^ ^ ^ ^ §3 ig u "0^0 10 31 "5 2f Ho 3 4 SI Pro a H c 17 47. *8 ;5m H ^4 I4 t£5 3 4 - ^ H ■8 0.^ 11 •^i H 7. 8 ^ 18 5* 4f 1^ 7. 8 H 7 8 1 8 12 y^ 3 ¥ 3* ^ 8 19 H 4^ 1* 7 8 4 1 5 8 13 n 3i 1 -^ 3^ i ■R 20 5* 4^ 1* 8 4 1 5 8 U H 3^ 1 ■| H * 21 5* 5 1* 1 4* 1 5 8 15 4 Bt 1^ T H 8 i 8" 22 6 5* H 1 4i H 3 4 16 4| 3| H 3 4 H 7. 8 ■ t — Trnclis. \ Diameter of Truck 5 i 6 7 1 8 2 9 2i 10 11 12 , 13 Thickness of Truck 13 i 13 13 ^4 ! ^4 ' ^4 21! 21 2f i 3a 1 1 ! Double Double Xo. of Sheaves . . 3 i 3 3 3 3 3 3 3 3 1 360 DDIEXSIOXS OF SKIPS BLOCKS. Table OF THE DniExsioifs OF Ships' Bloczs (in Inches) — cor icli ided. Dead-eyes. 1 T. ^ a: g .a -^ - - 5 r — £ .q g tfi "s -T iC ■^ -i? u ^ -§ 6C •a e3 Hf? 3 I ^ • — ^ ■„ ^ n 1 ^ M C-- t-l n — 5 5 3 9 i 9 i ^^ 13 13 8 17 17 10 6 6 4 10 10 61 14 14 81 18 18 l^ 7 7 4i 11 ' 11 ! 7" 1 15 15 9 19 ! 19 11 s 8 5 12 12 ^lo8e L 1 16 16 — — — 6 -' ^ a; tc £ if ■ - • 'f ix ,r^ '-ii — ^ ^ ^ c n ^ — — ^J — — £0 ' c3 ~ tc ii = •_ tc ;s :f. ^- 5 •_ -■ o > £i JQ Q 5 , S ;2 1 h j:: z r — — - Kl o H ^ t« H i-:! 1 C ^ 'j; - - - w 5 41 3f 3 4 9 8i 6i u 13 12 8 li 17 153 9^ 2 6 5A 4^ 1 10 9t 6^ li 14 13i U U 18 16f 101 2 7 ; 6f H 1 11 9| 7 ll 15 14185 If — — — — 8 7| SS3 -I 1 12 11 n 1|?16 14f It If — — — — Table of ' THE Value of the Belgian Gauge Decimals of ax Inch. IN Mark Size Mark Size ^fark >\7fi ^'ark Size Mark Size 1 - -004 • 7 •015 13 •034 19 ^067 25 •111 2 ' -006 8 •017 14 •037 20 074 26 •120 3 -008 9 •019 15 •041 21 •OS 2 — 4 1 -009 10 •022 16 •045 22 : •OS;t — — 5 ■ -Oil 11 •026 17 •052 23 •oo: — — 6 -0 1 3 12 •03r» IS •o.-^o •1\ -104 — — VOCABULARY OF TECHNICAL TERMS. 361 VOCABULARY OF TECHNICAL TERMS USED IN SHIPBUILDING. ENGLISH— FREXCH. Abaft, en arriere Aboard, a bord Admiral, amiral Admiralty, amiraute Adze, herminette Afloat, a fiot Aft, arriere, de I'arriere Air pump, pompe h air A-lee, sous le vent Amidships, au milieu du navire Anchor, ancre Angle iron, corniere, f er d'ano-le Apron, radier, centre - etrave, platine Ash, frene Astern, a I'-arriere, de I'arriere Athwart, par le travers Awning, tente Azimuth compass, compas de variation Back of stern-post, contre- etambot Backstay, sralbauban Barge, c;raud canot, allege Bar iron, fer en banes Barque, barque, bateau Barrel of the capstan, mecbe du cabes'an Barrel of the steering wheel, tambour de la roue du gou- vernail Batten, liteau Beam, bau Beech, hetre Bending press, machine k cin- trer les tules Between-decks, entrepont Bevel, angle oblique, angle d'equerrage Bilge, petit fond dun navire Bilge pump, pompe de cale Bilge ways, coittes Binnacle, liabitacle Bitts, bittes Blade of a screw, aile d'helice Blister steel, acier poule Block, poulie, moutie Block and fall, palan Boarding pike, pique dabardage Boat, bateau, canot Boatswain, maitre d"equipage Bohstay, sous-barbe Body plan, plan vertical Boiler maker, cbaudronnier Boiler plate, tole Bollard, corps mort Bolt, clieville, boulon Bolt rope, ralingue Boom, bout dehors, arc-boutant Bow, Tavant d"un vaisseau Bower-anchor, ancre du bossoir Bowsprit, beaupre Brace. l:»ras Bracket, courbaton Brail, careue Bread room, soute au pain Breadth extreme, plus grande lar-eur Breaker, brisant, baril de galore Breast-plate, conscience Brig, brig Brigantine, brigantin Bucket, bailie Builder, constructeur Bulk head, cloison Bunker, soute Bunt-line, cargue-fond Buoy, bouee, balise Buoyant, leger. emerge Burton, petit palan Butt, about, tete.d'un couple Butt cover, plaque de jonction d'ecart de tole Cabin, cabine, chambre, lit Cable, c.tble Cable tier, fosse aux cables 362 YOCABULART OF TECHNICAL TEKMS. English into French (continued). Caisson, bateau-iDorte Cant, oblique, tring-le Cant timbers, couples devoyes Capstan, cabestan Careen, carene Cargo, carjjaison, chargement Carling, entremise, aillure Cast iron, fonte de fer Cast-iron girder, poutre en fonte Cast iron pipe, tuyau en fonte Cast steel, acier fondu Cat-head, bossoir Cat's-paw, fraicheiir, petite brise Caulk (to), calfater Caulker, calfat Chain, cbaine, cable-chaine Cheeks of a mast, jottereaux Clack valve, clapet Clamp, bauquiere, jumelle Cleat, taquet Clew, i^oint decoute Clew garnets, cargues-points Coal bunker, soute alimentaire Coamings, cliambranles Coaster, caboteur Cockpit, theatre Cockswain, patron de chaloupe Cciisr-dam, batardeau Commander, capitaine de fre- £;'ate Companion, capot d'tfchelle Compass, boussole, compas de ruute Copper, cuivre Copper-bottomed, double en cuivre Cordage, cordages Corvette, corvette Counter, grande voute Countersink, fraisure Countersunk head, tete fraisee Course, route, basse- voile Crab, cabestan volant Crab winch, virevaut, treuil Cradle, berceau Craft, petit navire I Crane, grue Crank shaft, arbre a manivelle Cringle, aguiee, ancette Cross-tree, barre de hune Crow-bar, presson, pince Cruiser, croiseur Crutches, fourcats Cutter, cutter, cotre Cutwater, taille-mer, eperon Davit, davier Dead-eyes, caps de mouton Dead-light, faux mantelet Dead-wood, courbes de remplis- sage Deal, bordage mince Deck, pont, tillac Deck planks, bordages des ponts Deck stopper, bosse a bouton Delivery pipe, tuyau d'ecoule- ment Delivery valve, clapet de de- charge Depth of hold, creux de cale Distilling apparatus, appareil distillatoire Dock, bassin, darse Dockyard, arsenal Down-haul, hale-bas Draught of water, tirant d'eau Dredging machine, cure-mole a vapeur, machine a draguer Drill, foret, meche Drilling machine, machine 4 percer Driver, tapecul, paille-en-cul Driving wheel, roue motrice Drum, tambour Dry dock, forme s^che, forme do radoub Dunnage, fardage Ebb, reMux, jiTsant Elevation, elevation, projection vorticale Elm, orm(; Endless chain, chaine sans fin Engine, machine Engine-bearer, carlingue VOCABULARY OF TECHNICAL TEEMS. 363 ExGLiSH IXTO French (continued). Engine room, chambre de la machine Ensign, pavilion de poupe Eye bolt, cheville a ceillet False keel, fausse quille Fathom, brasse Feathering paddle, aube arti- culee Feathering paddle-wheel, roue a aubes articulees Feed pump, pompe alimentaire Fender, defense Ferry, gue Ferry boat, bateau de passage Fid, clef Figured dimension, quote ' File, lime Filling piece, pi^ce de remplis- sage Fir wood, sapin Fire ship, brulot Fish pendant, pantoire de la candelette de Tancre Fish-tackle fall, garant de la candelette Flange, collerette, collet Flange joint, joint a collet Flare, re vers Floating body, corps flottant Floor, fond d"iui navire Floor heads, tetes des varangues Floor timbers, varangues Flukes, oreilles d'une ancre Flush deck, pont entier Flush joint, assemblage bout a bout Flush rivet, rivet a tete fraisee Fly wheel, volant Flying jib, petit foe, clinfoc Flying jibboom, bout dehors de clinfoc Force pump,' pompe foulante Forecastle, gaillard d"avant Fore mast, mat de misaine Fore sheets, ecoutes Fore stay, etai de misaine Fore staysail halliard, drisse de [ la trinquette du petit foe, or du tourmentin Fore topgallant mast, petit mat de perroquet Fore topmast, petit mat de hime Fore topmast stay, etai du petit hunier Fore topsail braces, bras du pe- tit hunier Foundation plate, plaque de fondation Four-way cock, robinet a quatre fins Frame, couple Framing, batis or charpente Funnel, cheminee en tole Furnace, fourneau, foyer Futtock, allonge Graff, pic, corne d'artimon Gaff topsail, flfeche-en-cul Gallant mast, mat de perroquet Gallant sail, voile de perro- quet Galvanised iron, fer galvanise Gammoning, lim-e Garboard strake, virure de ga- bord General drawing, desstn d"en- semble Girder, poutre Girt, ceintre Goose-neck, ecu de cygne Grapnel, grappin Graving dock, forme de radoub Grummet, estrope Gunboat, cbaloupe canonni^re Gun carriage, affut Gun metal, bronze de canon Gun port, sabord Gunwale, plat-bord Guy, cordage de retenue Half-breadth plan, plan hori- zontal Halliard, drisse Hammock, hamac Hand pump, pompe a bras Handspike, anspect 364 VOCABULAET OF TECHNICAL TERMS. English ixto French (continued). Hard a-lee, lof tout Hard a-port, la barre toute a babord Hard a-starboard, la barre a tribord toute Hard a-weatlier, la barre toute au vent Harpings, preceintes renforcees de I'avant Hatch, panneau Hatchway, ecoutille Hawse pipes, plombs des ecu- biers Hawse plug, tampon des ^cubiers Hawser, aussiere Head sails, voiles de Tavant Helm, gouvernail Hemp, chanvre High water, pleine mer Hinge, penture, gond Hog, goret d'un navire Hold, cale Hoop iron, f euillard, f er &,ruban Horse box, wagon-ecurie Horse power, cheval de vapeur Hounds, jottereaux Hulk, ponton, cayenne Hull, corps, coque d"un navire Hydraulic press, presse hydrau- lique Intercostal, entre les cotes Iron, fer Iron frame, couple en fer Iron keel, quille en fer Iron plate, tole Iron rigging, greement en fer Iron ship, navire en fer Iron side, bras de fer Iron wire, fil de fer Iron work, ferrure Jack, eric, eric a vis Jaw of a boom, machoire d'une borne Jib, foe Jolly boat, petit canot Jump joint, bout ^ bout Junk, jonque chinoise Kedge anchor, ancre a jet Keel, quille Keelson, carlingue Kingston's valve, soupape du navire Knee, courbe, genou Knight-heads, bouts des apotres, bittons Knot, noeud, bouton Ladder, eciielle Lap or cover, recouvrement Lapped joint, joint superpose, joint a clin Larboard, babord Lashing, aiguillette, fouet Lateen sail, voile latine Lateen yard, antenne Lathe, tour Launch, avant-cale Lead, plomb Leak, fuite, voie d'eau Leeboard, semelle de derive Lee side, cote sous le vent Leech rope, ralingue de chute Leeward, cute sous le vent Leeway, derive Life buoy, bouee de sauvetage Lighter, allege, barque Limber hole, anguiller Lock chamber, sas k ecluse Lock gate, porte d 'ecluse Locker, equipet Lower rigging, haubans et etais des bas m;its Lower yards, basses vergues Lug sail, voile de lougre Main, grand Main mast, grand mat Main royal mast, grand mat de cacatois Main royal sail, grand cacatois Main sail, grand voile Main sheet, ecoute de la grande Vdilc Main shrouds, grands haubans Main topgallant mast, grand mat de perroquet. VOCABULARY OF TECHNICAL TEEMS. 365 ExG-LiSH INTO French (continued). Main topgallant sail, grand perroquet Main topgallant staysail, voile d'etai de grand perroquet Main topmast, grand mat de hune Main topmast stay, etai du grand mat de hune Main topsail, voile du grand hunier Main topsail yard, vergue du grand hunier Man-hole, trou d'homme Man-of-war, batiment de guerre Man-rope, garde-corps Marine boiler, chaudi^re marine Marine engine, machine a va- peur marine Marine glue, colle marine Master shipwright, premier in- genieur-Gonstructeur d'un port Merchantman, navire de com- merce [chande Merchant service, marine mar- Messenger, tournevire Metacentre, metacentre Midships, milieu du navire Mizen, artimon Mizen mast, mat d'artimon Mizen sail, voile d'artimon Mizen shrouds, haubans d'ar- timon Mizen staysail, benjamine Mizen topgallant mast, mat de perruche Mizen topgallant staysail, voile d'etai de perruche Mizen topmast staysail, dia- blotin Moorings, corps-mort Mould, gabari Mould loft, salle des gabaris Mud hole, trou de sel Nail, clou Neap tide, morte-eau Netting, iilet de bastingage Nut, tenon Oakum, etoupe Oar, aviron Orlop, entrepont Orlop deck, faux pont Outrigger, aiguille de car^ne Paddle beam, bau de force Paddle box, tambour Paddle float, aube Paddle wheel, roue a aubes Palm, patte d'ancre Parrel rope, batard de racage Partner, etambrai Paunch, natte Pendant, flamme, banderole Pig iron, gueuse Pinnace, pinasse, canot Pintle, aiguillot Pitch, poix, brai sec Pitch chain, cha'me a la Vau- canson Plank, bordage, planche Pole, pole, baton Pole mast, mat a pible Pontoon, ponton de carenage Port helm, babord la barre Port lid, mantelet de sabord Port sill, seuillet de sabord Post, poteau Preventer stay, faux etai Propeller, propulseur Propelling screw, helice pro- jxilsive Propelling screw-shaft, arbre d 'helice Pulley, poulie, rouleau Pump, pompe Pump handle, brinqueballe Punt, aeon, pont flottant Quadrant, octant Quarter deck, gaillard d'arri^re Quay, quai Eabbet, rablure Rake, inclinaison Ratchet brace, cliquet a percer Ratline, enflechure Reef, rt'cif, ris Relieving tackles, palans de car^ne 3G6 VOCABULARY OF TECHNICAL TEEMS. English ixto Frexch (continued). Eepair, radoub I Rib, membre, rame ' Riband, lisses des couples "Rig (to), greer Rigging, greement, manoeuvres Ring bolt, clieville a boucle Rivet, rivet Rivet (to), river Rolling mill, laminoir " Rope, corde, cordage Rope yarn, fil de caret Rough-tree rail, lisse de ba- tayoles Rowlocks, toleti&res Royal mast, mat de cacatois Royal sail, cacatois Royal yard, vergue de cacatois Rudder, gouvernail Running rigging, manoeuvres courantes Safety valve, soupape de surete Sail, voile Sail of a lugger, bourcet Sampson's post, epontille Scantlings, echantillons Scarf, ecart, empature Schooner, goelette Screw jack, verin, eric k vis Screw propeller, helice propul- sive Scupper, dalot Scuttle, ecoutille, hublot Seaman, matelot Shackle, manicle Shaft, arbre Sheathing, doublage Sheave, rouet de poulie Sheer, tonturo Sheer draught, plan d'el6vation Sheer-legs, bigue, chevre Sheet anchor, ancre de miseri- cord e Shipwreck, naufrage Shipwright, charpentier de na- viro Shrouds, haubans Side scuttle, hublot Signal flag, pavilion de signal Skin, bordage Skylight, ecoutille \atree ^ Skyscraper, aile Je pigeon Sling of a yard, suspente Smack, semaque Sounding lead, plomb de sonde Sounding line, ligne de sonde Sounding rod, sonde de pompe Spanker, voile d'artimon Spar, espar, matereau, montant Spar deck, pont sur montant Spindle, tige, meche, broche Spirit room, cale au vin . Spirjietting, f euilles bretonnes Splice, epissure "^ Spoke, rayon d'une roue Sprit sail, voile de civadi^re Square sail, voile carree Staging, echafaudage ^ Stanchion, epontille, montant ■ Standard, courbe, verticale Standing rigging, manoeuvres dormantes Stand pipe, tuyau alimentaire k colonne d"eau Staple, crampe de fer Starboard, tribord Starting gear, mise en marche Stay, etai. relache Steam engine, machine h vapeur Steamer, vapeur Steam frigate, fregate a vapeur Steel, acier Steer (to), gouverner Steering wheel, roue du gou- vernail Stem, etrave Step of the mast,carlingue du m^t Stern, ]ooupe, arriore Stern frame, arcasse Stern post, «'tambot Stewards room, cambuse Stock of an anchor, jas d'ancre Stoke hole, parquets des chauf- feurs Store room, soute VOCABULAET OF TECHNICAL TEEMS. 367 English into Frexch (concluded). Stores, provisions Stowage, arrimage Strake, ^drure Strap, chape, courroie, bride Stream anchor, ancre de touee Stuffing box, presse-etoupe Studding sail, bounette Studding sail boom, bout dehors Suction pipe, tuyau d'aspiration Suit of sails, jeu de voile Swab, faubert Swivel, tourniquet en f er Tackle, palan Tarpaulin, bagnolet Tee iron, fer en T Telescope, longiie-vue Tell-tale, axiometre Template, gabari Thimble, cosse en fer [corne Throat halliard, drisse d'une Throttle valve, papillon registre Thwart, banc de nage Tide, maree Tie bar, tirant Tie beam, entrait Tier of a cable, bitture Tiller, barre du gouvernail Tiller rope, drosse du gouvernail Tilt hammer, martinet, mar- teau a bascule Tonnage, tonnage Top, hune [roquet Topgallant mast, mat de per- Topping lift, balancine de gui Top mast, milt de hune Top sail, hunier Topsail yard, vergue de hune Tow rope, grelin Trail board, frise de Teperon Transom, barre d'arcasse Transport, transport Trestle trees, barres des hunes Trim, assiette, allure, arrimage Truck, pomme, roue, cosse Trunnions, tourillons Truss, drosse de racage Try sail, voile de senau Tubular boiler, chaudiere tubu- laire Tug boat, remorqueur Tumble-home, rentree Tun, tonneau Universal joint, joint universel Tipper deck, franc tillac Upper works, oeuvres mortes Uptake, culotte Vane, girouette Vangs, palans de retenue, bras de home Vessel, navire, batiment Victuals, vivres, approvisionne- ments Wake, sillage, eaux, houache Wale, i^receinte Ward room, gTande chambre Warp, cablot, gTelin, touee Warped plank, bordage dejete Wash boards, fargues Water line, ligne d'eau Water tank, caisse a eau Water-tight bulkhead, cloison etanche Water-tight compartment, com- partiment etanche Water-way, gouttiere Wave, vague, lame Weather bow, bossoir du vent Weather braces, bras du vent Weatherly ship, navire bon boulinier Wharf, quai Wheel, roue Whelps of capstan, fiasques du cab est an White lead, blanc de ceruse Winch, moulinet, virevaut Windlass, guindeau, virevaut Windward, au vent Workmanship, main-d"oeuvre Wreck, naufrage Yard, vergue Yard arm, bout de vergue Yarn, fil de caret Yawl, yole, moyen canot 368 YOCABULART OF TECHXICAL TEEMS. VOCABULAEY OF TECHNICAL TEEMS USED IN SHIPBUILDING. FRENCH— ENGLISH. A bord, aboard About, butt, end part Aboutement, scarf Accastillage, upper works Acier, steel Acier fondu, cast steel Acier poule, blister steel Aeon, punt, flat Affut, gun carriage A flot, afloat, floating Aguiee, cringle Aiguillette, lashing, laniard Aile d'helice, blade of a screw Aile de pigeon, skyscraper Ailiure, carling A I'arriere, astern Allege, lighter, barge Allonge, futtock Allure, trim Aman, halliard Amiral, admiral Amiraute, admiralty Ancre, anchor Ancre du bossoir, bower anchor Ancre de misericorde, sheet anchor Angle, quoin Angle oblique, bevel Anguillers, limber holes Anspect, hands^jike Antenne, lateen yard Apostis, gunwale Appareil distillatoire, distilling apj^aratus Approvisionnements, victuals, naval stores Arbre, shaft, mast Arbre a manivelle, crank shaft Arbre d'helice, screw propeller shaft Arcasse, stern frame Arc-boutant, boom Arriere, abaft, aft, stern Arrimage, stowage, trimming Artimon, mizen sail Assemblage, framing, scarfing Assiette, trim Aube, paddle float Aube articulee, feathering paddle Au milieu du navire, amidships Au vent, windward Avant, bow, forward Avant-cale, launch, slip Avant d'un vaisseau, bow of a vessel Aviron, oar Axiometre, tell-tale Azimut, azimuth Babord, larboard Babord la barre, port the helm Bagnolet, tarpaulin Bailie, bucket Baisse, ebb tide Balancine, lift Balancine de corne, topping lift Balancine de gui, topping lift Balaou, schooner Baleiniere, whale boat Banc, seat Banc de nage, thwart Banderole, pendant Barbette, gunwale Barque, barque Barre, helm, tiller, cross-tree Barre d' arcasse, transom Barre de hune, trestle tree Barre du gouvernail, tiller [ Basses vergues, lower yards I Basse voile, course ■ Ba«sin, sliii)ping, dock Batard de racage, parrel rope Batardeau, cofter-dam Bateau, boat, craft, barge Bateau de passage, ferry boat Bateau-porte, caisson YOCABULART OF TECHNICAL TERMS. 369 Feench into ExciLlSH (continued). Batiment, vessel, ship Batiment de guerre, man-of-war Baton, head, mast, pole Bau, beam Bau de force, paddle beam Bauquiere, clamp Beaupre, bowsprit Benj amine, mizen staysail Berceau, cradle Bigue, sheer-legs Bittes, bitts Bittons, knight-heads Blanc de ceruse, white lead Borne, boom Bonnette, studding sail Bordage, plank, skin Bordage dejete, warped plank Bordage mince, deal Bordages des ponts, deck planks Bosse a bouton, deck stopper Bossoir, cat-head Bossoir du vent, weather bow Bouee, buoy Bonee de sauvetage, life buoy Boulon. bolt, pin Bourcet, sail of a lugger Boussole, compass Bout, butt, end Bout a bout, jump joint Bout dehors, studding-sail boom Bout dehors de clinfoc, Hying jibboom Bout de vergue, yard arm Bouts des apotres, knight-heads Brai, pitch Bras, brace, arm Bras de borne, vangs Bras de fer, iron side Bras du petit hunj^r, fore topsail braces Bras du vent, weather braces Brasse, fathom Bride, strap Brig, brig Brigantin, brigantine Brigantine, spanker, driver Brinqueballe, pump handle Brisant, breaker Broche, spindle Bronze, brass Bronze de canon, gun metal Brulot, fire ship Cabane, cabin Cabestan, capstan Cabine, cabin Cable, cable Cablot, warp, painter. Cabotage, coasting trade Cabotier, coasting vessel Cabrion, whelp Cacatois, royal sail Cache-adent, scarf Caillebottis, gi-ating Caisse a eau, water tank Caisson, chest, locker Cale, hold Cale au vin, ^spirit room Calfat, caulker Calfater, to 'caulk Cambuse, steward's room Canonniere, gunboat Canot, boat, yawl Cap de mouton, dead-eye Capitaine de fregate, commander Capon, cat block, cat hook Capot d'echelle, companion Careue, careen Cargaison, cargo Cargue, brail, garnet Cargues-points, clew garnets Carlingue, keelson, engine- bearer Carlingue du mat,step of the mast Carre, square-rigged Carreau, gunwale of a boat Caveau, store room Chaine, chain Chaine a la Vaucanson, pitch chain Chaine sans fin, endless chain Chambre, cabin Chambre de la machine, engine room Chanvre, hemp B B 370 VOCABULARY OF TECHNICAL TERMS. French into English (continued). Chargement, cargo Charpente, framing Charpentier, carpenter, ship- wright Chaudiere marine, marine boiler Chaudiere tubulaire, tubular boiler Chaudronnier, boiler maker Cheval-vapeur, horse power Oheville, bolt Cheville a boucles, ring- bolt Cheville a oeillet, eye bolt Chevre, crane, sheer-legs Clapet, clack valve Clapet de decharge, delivery valve Clinfoc, flying jib Cliquet a percer, ratchet brace Cloche, bell Cloison, bulkhead Cloison etanclie, water-tight bulkhead Clou, nail Coittes, bilge ways Colle marine, marine glue Collerette, flange Collet, flange Compartiment etanclie, water- tight compartment Compas de route, compass Compas de variation, azimuth compass Comput, calculation Constructeur, builder [post Contre-etambot, back of stern Contre-strave, apron Coque d'un navire, hull Cordage, rope, rigging Corde, rope Corne, throat, peak Come d'artimon, gaff Corniere, angle iron Corps, hull Corps flottant, floating body Corps-mort, bollards Corvette, corv^ette. sloop of war Cosse, truck, thimble Cote, side, broadside Cote sous le vent, lee side Cou de cygne, goose-neck Couleurs, ship's flag, colours Couple, frame, timber Couple en fer, iron frame Couples devoyes, cant timbers Courbaton, bracket Courbe, knee, standard Couronnement, tafErail Cours, strake Crampe de fer, iron staple Crapaud, goose-neck Crapaudine, bed plate Creux, depth Creux de cale, depth of hold Cric a vis, screw jack Croiseur, cruiser Cuisine, galley Cuivre, copper Cul, poop, after part, stern Cure-mole, dredging machine Cutter, cutter Dalot, scupper Darse, dock Davier, davit Debarquement, unloading Defense, fender Derive, leeway Dessin d'ensemble, general drawing Diablon, mizen topgallant stay- sail Diablotin, mizen topmast stay- sail Doublage, sheathing Double en cuivre, copper-bot- tomed Drisse, lialliard Drisse d'une corne, throat hal- liard Drisse de la trinquette du petit foe, fore siaysail halliard Drisse de racage, truss Drisse du gouvernail, tiller rope Drisse du grand hunier, main topsail halliard TOCABULARl OF TECHXICAL TEEMS. 371 French into English (continued). Drisse du petit perroquet, fore topgallant sail halliard Dunette, poop Eaux, wake Ebbe, ebb tide Ecart, scarf Ecbafaudage, staging Echantillon, scantling Echarpe, head rail Echelle, ladder Ecluse, dock Ecoute, sheet Ecoute de la grande voile, main sheet Ecoutille, hatchway, scuttle Ecoutille vitree, skylight Egouttoir, grating Elance, flare, projecting Elevation, elevation Elongis, trestle trees Emerge, buoyant Empature, scarf Emplanture, step Enclaver, to mortise Encouture, clinched En-dessous, after part En-dessus, fore part Enflechure, rat line Engraver, to trim, to stow Enseigne, flag, ensign Entrait, tie beam Entre les cotes, intercostal Entremise, carling Entrepont, between-decks, or- lop deck Entretoise, transom, partner Eperon, head, cutwater Epissure, splice Epontille, stanchion, pillar Epontille, Sampsons post Equerre, bevel, movable square Equipet, locker Espars, spars Etai, stay Etai du grand mat de hune, main topmast stay Etai du petit hunier, fore top- mast sta\^ Etai et faux, fore stay Etambot, stern post •Etambrai, partner Etance, Sampson's post Etanche, tight Etancher, to free from water Eioupe, oakum Etrave, stem Fa9ons, run, rising floor Fardage, dunoage Fargues, wash-boards Faubert, swab Fausse quille, false keel Faux baux, orlop beams Faux bras, preventer braces Faux etai, preventer stay Faux mantelet, dead-light Faux pont, orlop deck Fer, iron Fer a ruban, hoop iron Fer d' angle, angle iron Fer en barres, bar iron Fer en T, tee iron Fer galvanise, galvanised iron Ferrure, iron work, hinge Feuillard, hoop iron Feuilles bretonnes, spirketting Fil de caret, rope yarn Fil de fer, iron wire Filet, netting Filet de bastmgage, netting Flamme, pendant Flasques, whelps cheeks Fleche, skyscraper mast, boom, prow Fleche-en-cul, gaff, topsail Flottaison, water line Flottant, afloat Foe, jib Foe d'artimon, mizen staysail Fond, bottom, hold, floor Fonte de fer, cast iron Foret, drill Forme de radoub, dry dock Forme flottante, wet dock bb2 372 VOCABULARY OF TECHNICAL TERMS. French into English (continued). Forme seclie, dry dock Fort, extreme breadth Fortune, fore sail, lug sail Fosse, pit, store room Fosse aux cables, cable tier Fouet, laniard, lashing Fourcats, crutches Fourche, sheers Fourneau, furnace Fraicheur, cat's-paw Frais, breeze, wind Fraisure, countersink Franc tillac, upper deck Fregate a vapeur, steam frigate Frise de I'eperon, trail board, frieze Fune, rigged Fut, cask G-abari, mould, template Gabet, vane Gabcrd, garboard strake Gaffe, boat hook Gaillard d'arriere, quarterdeck Gaillard d'avant, forecastle Galhauban, back stay Gambes, futtock shrouds Garant, fall, running Garant de la candelette, fish- tackle fall Garde-corps, man rope Gatte, manger Genou, knee Girouette, vane Gisole, binnacle Goelette, schooner Gond, hinge Goret, hog Gorgere, cutwater Gournable, tree nail Gouttiere, water-way Gou vernal! , rudder, helm Gouverner, to steer Grand, main Grand cacatois, main royal sail Grand foe, main topmast stay- sail Grand hunier, main topsail Grand mat, main mast Grand mat de cacatois, main royal mast Grand mat de hune, main top- mast Grand mat de perroquet, main topgallant mast Grand perroquet, main top- gallant sail Grande chambre, ward room Grande hune, main top Grande vergue, main yard Grande voile, main sail Grande voile d'ttai,main staysail Grande voute, counter Grands baubans, main shrouds Grappin, grapnel Greage, rigging Greement, rigging Greement en fer, iron rigging Greer, to rig Grelin, warp, tow rope Gros de I'eau, high water Grue, crane, windlass Gue, ferry Guindeau, windlass Guirlande, breast hook Guitran, pitch Habitacle, binnacle Hale-bas, down-haul Hamac, hammock Hampe, handle Haubans, shrouds Haubans et etai des bas mats, i lower rigging Havre, harbour Heaume, tiller Helice propulsive, screw pro- peller Herminette, adze Hetre, beecli Houache, wake, track Hublot, side scuttle Hune, top Hunier, top sail Inclinaison, rake, dip, heeling, stive VOCABULARY OF TECHNICAL TERMS. 373 French into English (continued). Inventaire, inventory Jambe de chien, stem timber Jas d'ancre, anchor stock Jeu de voiles, suit of sails Joint a clin, lapped joint Joint a collet, flange joint Joint superpose, lapped joint Joint universel, universal joint Jottereaux, cheeks, hounds Jouet, iron plate Jusant, ebb tide Kot, awning, canopy- La barre a tribord toute, hard a-starboard La barre toute a babord, hard a- port La barre toute au vent, hard a- weather Lame, wave Laminoir, rolling mill Lanch.e, launch Largeur, breadth Leger, light, buoyant Lest, ballast Levee, swell, surge Liaisons, strengthening pieces Ligne d'eau, water line Ligne de sonde, sounding line Lissedebatayoles,rough-treerail Lisse de fort, extreme breadth line Lisse d'eperon, head rail Lisses des couples, ribands Lisses des fa9ons, rising of the floor Lit, bed, berth Liure, gammoning Livarde, sprit of a shoulder of mutton sail Lof tout, hard a-lee Longue-vue, telescope Lougre, lugger Lumiere, limber hole Lunette, telescope Machine, engine Machine a cintrer les toles, bending press Machine a draguer, dredging machine [chine Machine a percer, drilling ma- Machine a vapeur marine, ma- rine engine Main-d'oeuvre, workmanship Machoire d'une borne, jaw of a boom Maitre d'equipage, boatswain Manicle, shackle Manivelle, handle Manoeuvres, rigging Manoeuvres courantes, running rigging Mantelet de sabord, port lid Marbre, steering-wheel barrel Maree, tide Marguerite, messenger Mariage, lashing Marie-salope, mud barge Marine marchande, merchant service Marsouin, stemson Martinet, peak halliard Martingale, bobstay Mat, mast Mat a pible, pole mast Mat d'artimon, mizen mast Mat de cacatois, royal mast Mat de grand perroquet, main topgallant mast Mat de hune, topmast Mat de misaine, fore mast Mat de perroquet, topgallant mast Mat de perruche, mizen topgal- gant mast Matereau, small mast, spar Mateur, mas-t maker Meche, spindle, barrel Mecbe du cabestan, capstan barrel Metacentre, metacentre Milieu du navire, midships Misaine, fore sail Mise en marche, starting gear Mitraille, case shot 374 VOCABULAEY OF TECHNICAL TEEMS. French into English (continued). Modele, model, mould Moise, cross beam, cross-tree Molle mer, slack water Montant, stanchion Moque, dead-eye, heart Mortaise, mortise Morte eau, neap tide Moulage, moulding Moulinet, winch Moulure, moulding Moustaches, standing lifts Natte, paunch Naufrage, shipwreck Nautique, nautical Naval, naval Navire, vessel, ship Xavire bon boulinier, weatherly ship Navire de commerce, merchant- man Navire en far, iron ship Nocher, boatswain Nceud, hitch, bend, knot Noix, hound Nolis, freight Nuaison, steady wind Oblique, cant, slant Obusier, howitzer Octant, quadrant CEillet, eye, cringle (Euvre, free-board CEuvres mortes, upper works Office, pantry Oreille, fluke Ourse, vang, mizen boom Pagaye, paddle Pailie-en-cul, driver Paillet, paunch Paillot, bread room Palan, tackle, burton, halliard Palans de carene, relieving tackles Palans de retenue, vangs Palme, palm Panneau, hatch cover Pantoire de la candelette de I'ancre, fish pendant Papillon, skyscraper Papillon registre, throttle valve Paradoses, limber boards Par le travers, athwart Parquets des chauffeurs, stoke hole Passager, passenger Passeresse, brail Patron de chaloupe, cockswain Patte, palm, fluke Pavilion, flag, colours Pavilion de detresse, signal flag Pavilion de poupe, ensign Payeur, paymaster Peinture, paint Pene, mop Penture, hinge Perpigner, to set the frames Perroquet, topgallant sail Perroquet de fougue, mizen topsail Perrucbe, mizen topgallant sail Petit, fore top Petit foe, flying jib Petit fond dun navire, bilge of a ship Petit mat de cacatois, fore royal mast Petit mat de hune, fore topmast Petit mat de perroquet, fore top- gallant mast Petite brise, cat's-paw , Pic, peak ' Pied, shoe, forefoot, heel Pinasse, pinnace Pique d'abordage, boarding pike Plan vertical, body plan Plaque d'ecart de tole, butt cover Plaque de fondation, foundation plate Plaque de jonction, butt cover Plastrons, knight-heads Plat-bord, gunwale Pleine mer, liigh water Plus grande largeur, breadth extreme ^€Cr-7^^X^ VOCABULAET OF TECHNICAL TEEMS ^^X^ 375 Frejstch into English (continued). Point d'ecoute, clew Pompe, pump Pompe a air, air pump Pompe a bras, hand pump Pompe alimentaire, feed pump Pompe de cale, bilge pump Pont, deck, stage Pont entier, flush deck Pont principal, weather deck Ponton, pontoon Port, burden, tonnage Porte d'ecluse, lock gate Pouillousse, main staysail Poulie, block, pulley Poutre, girder Preceinte, wall, rail Presse-etoupe, stuffing-box Presse hydraulique, hydraulic press Presson, crow-bar Propulseur, propeller Proue, prow, bow, head Pjrroscaphe, steamer Quai, quay, wharf Quille, keel Quille en far, iron keel Quintelage, ballast '" Raban, earring, gasket Eablure, rabbet Eacage, parrel, truss Eadeau, raft Eadier, apron Eadoub, repair Ealingue, bolt rope Ealingue de chute, leech rope Earns, oar Ease, dismasted Eayon, spoke Eecif, reef, ridge Eeflux, ebb tide Eelache, stay Eemorqueur, tug boat Eemplissage, filling piece Eenflement, bluff Eenfort, lining, binding Eentree, tumble-home Eesistance, resistance Eessac, surf Eetenue, relieving tackle Eevers, flare, hollow Eibord, garboard strake Eide, laniard Eis, reef Eisade, reefing Eisson, grappling Eivet, rivet Eivet a tete fraisee, flush rivet Eiviere, river Eobinet a quatre fins, four- way cock Eoue, wheel Eoue a aubes, paddle wheel Eoue a aubes articulees, feather- ing paddle-wheel Eoue de poulie, sheave Eoue du gouvernail, steering wheel Eouf, canopy Eouleau, jDulley Eoyaux, royal sails Sabord, gun port Sainte-barbe, gun room Salle, loft Salle des gabaris, mould loft Sapin, flr wood Semaque, smack Semelle de derive, leeboard Seuillet de sabord, port sill Sillage, wake, steerage Sonde de pompe, sounding rod Soupape de surete, safety valve Soupape du navire, Kingston's valve Sous-barbe, bobstay Soute, bunker, store room Soute au pain, bread room Stabilite, stability, stiffness Suspente, sling of a yai'd, guy, straps Tableau, after ]3art of a ship Taille-mer, cutwater Taille-vent, main sail of a lugger Talonniere, heel of the rudder 376 VOCABULARY OF TECHNICAL TERMS. French into English (concluded). Tambour, drum, washboard, paddle-box Tambour de la roue du gouver- nail, barrel of the steering wheel Tampon desecubiers, hawse plug Tapecul, ringtail sail, driver Taquet, cleat, clamp Tariere, auger Teck, teak Tenon, tenon, ntit Tente, awning Tete, upper end. head . Tete d'un couple, butt Tete de varangue, floor head Tete fraisee, countersunk head Theatre, cockpit Tiercon, tierce Tige. spindle Tillac, deck Tille, platform Tirant d'eau, draught of water Toile a voiles, sail cloth, canvas Tole, tjoiler plate, iron plate Toletiere, rowlock Ton, mast-head, cop Tonnage, tonnage Tonne, ton, butt, cask Tonneau, tun, 1,000 kilogrammes Tonture, sheer, round up Torpedo, torpedo Touee, warp, tow line' Tourillons, trunnions Tourmentin, fore staysail Tournevire, messenger Tourniquet, roller, swivel Transport, transport Treou. lug sail Treuil, crab winch Tribord, starboard Tringle, cant Trinquet, fore mast Trinquette, fore staysail Trois-mats, three-masted vessel Trois-ponts, three-decker Ti'ou. shelter, harbour Trou de sel, mud-hole Trou d'homme, man-hole Tuyau alimentaire a colonne d'eau, stand-pijDe Tuyau d'aspiration. suction pipe Tuyau d'ecoulement, delivery pipe Tuyau en fonte, cast-iron pipe TJretac, winding tackle Vague, wave, sea Vaigrage, walling, ceiling, lin- ing Vaisseau, ship, vessel Vapeur, steamer Varangue, floor timber Vareuse, sail cloth Vassole, coaming Vent, wind, breeze Ventilateur, wind sail Vergue, yard, peak, boom Vergue de cacatois, royal yard Vergue de hune, topsail yard Vergue du grand hunier, main topsail yard Verin, screw jack Verticale, standard Vindas, windlass Virevaut, crab winch Virure, strake Virure de gabord, garboard strake Voile, sail Voile carree, square sail Voile d'artimon, spanker Voile de civadiere, sprit sail Voile d'etai de grand perroquet, main topgallant staysail Voile d'etai de perruche, mizen topgallant staysail Voile de I'avant, liead sail Voile de senau, try sail Voile latine, lateen sail Voiite, counter Wagon-ecurie, horse box Yole, yawl Youyou, iirig Zinc, zinc HYPEKBOLIC L00AF.ITHM3. Oi ( Table of Hyperbolic LoGAEiTTr\fs. To find the hyperbolic losfarithm of a number multiply the common logarithm of the number by the fierures I 2-30258505299i, "and. the product is the hyperbolic loga- | rithm of that number. Example.— The common lograrithm of 3-7. 5 is -5740313 ; the hv]:)erbolic logarithm is then found b\ ' multiplyingr 2-302.58O by -oriOSlS- 1-3217559 , the hyperbolic loga- | rithm. No. Loircrithm No. 1-35, Loararithm N o. Lotrarithm •5247284 No. 203 Logarithm ^080357 Toi •009^1503 -3O01046 T m 1-02 •019S02*i 1-36: -3074847 1 J. 70 -5306282 204 •7129497 103 •029.5.588 1-37| -3148108 71 -5364933 2-05 -7178399 1-04 •0392207 1-38; -3220833 72 -5423241 2-06 •7227058 1-05 •0487902 l-39i •3293037 73 -5481212 2-07 -7275485 1-06 •0582690 1-40 -3364721 74 •5538850 2-08 •7323678 1-07 •0676586 1-41 -3435895 75 -55961.56 2-09 -7371640 1-OS •0769610 1-42 -3506568 76 •5653138 2-10 •7419373 109 •0861777 1^43! ^3576744 77 •5709795 2-11 •7466880 MO •0953102 1^44; -3646431 78 •5766133 2^12 •7514160 1-11 •1013600 1^45i -3715635 79 •5822156 2-13 •7561219 1-12 •1133285 1-46 -3784365 •80 -5877866 2-14 -7608058 1-13 •1222174 1-47 -3852623 81 -5933268 2-15 •7654680 1-U •13102S4 1-48 -3920420 •82 -5988365 2-16 -7701082 l-lo •1397614 149 -3987762 83 •6043159 2^17 •7747271 1-16 •1484199 l-50i -4054652 •84 •6097653 2-18 •7793248 1-17 •1570038 1-51 -4121094 ■^o •6151855 2-19 •7839014 MS •1655144 1-52 -4187103 -86 -6205763 2^20 -7884573 1-19 •1739534 1'53 -4252675 -87 •6259384 2-21 -7929925 ' 1-20 •1823215 1-54 -4317823 -88 •6312717 222 -7975071 1-21 •1906204 1-55 -43825.50 -89 •6365768 2^23 •8020015 1-22 •1988507 1^56j -4446858 -90 •6418538 2^24 •8064758 1-23 •2070140 1-571 -4510756 •91 -6471033 2-25 -8109303 l-2i ; -2151113 1-58| ^4574247 •92 -6523251 2-26 -8153647 1-25 1 -2231435 1^59 ^4637339 •93 •6575200 2-27 •8197798 1-26 •2311161 1^60 -4700036 •94 •6626879 2-28 •8241754 1-27 ' -2390167 1-61 •4762341 •95 -6678294 2-29 •8285518 1-28 ' ^2468601 1-62 •4824260 -96 -6729445 2-30 •8329089 1-29 •2546422 1-63 •4885801 -97 -6780335 2^31 •8372474 1-30 : ^2623643 1-64 •4946959 •98 -6830968 2^32 •8415671 1-31 ■ ^2700271 1-65 •5007752 1 •99 •6881346 2-33 •8458682 1-32 , ^2776316 1-66 •5068176 2 •00 •6931472 2-34 •8501509 1-33 1 -2851787 1-67 •5128237 2 •01 •6981347 2-:^5 -85441.54 1-31 ^ -2926^96 }l-6S! •51s7v^38 2 02 •7030974 2 36 •S.586616 378 HYrEEBOLTC LOGAEITHMS. No.i 2-371 Lo?aritlim •8628899 2-85| Lo;iarithm s No. Lojri'.ritlim T2O29722 3-81 Lofrarithm 1 l-337629i 1 1-0473189 p-33 2-38 •8G71004 2-86 1-0508215 §3-34 1-2059707 3-82 1-3402504 2-39 •8712933 2-87 1-0543 120^:3-35 1-2089603 3-83! 1-3428648 240 •8754686 2-88 1-0577902' 3-36 1-2119409 3-84 1-3454723 2-41 •8796266 2-89 1-0612564 5 3-37 1-2149127 3-85 1-3480731 2-42 •8837675 2-90 1-0647107 f! 3-38 1-2178757 3-86 1-3506671 2-43 •8878912 2-91 1-0681529 ^3-39 1-2208299 3-87 1-3532544 2-44 •8919980 2-92 1-0715836 13-40 1-2237754 3-88 1-3558351 2-45 •8960879 2-93 1-0750024 1 3-41 1-2267122 3-89 1-3584091 2-4G •9001613 2-94 1-0784095 3-42 1-2296405 3-90 1-3609765 2-47 •9042181 2-95 1-0818051- 3-43 1-2325605 3-91 1-3635373 2-48 •9082585 2-96 1-0851892 3-44 1-2354714 3-92 1-3660916 2-49 •9122826 2-97 1-0885619 13-45 1-2383742 3-93 1-3686395 2-50 •9162907 2-98 1-0919233 1 3-46 1-2412685 3-94 1-3711807 2-51 •9202825 2^99 1-0952733 \ 3-47 1-2441545 3-95 1-3737156 2-52 •9242589 3-00 1-0986124 j 3^48 1-2470322 3-96 1-3762440 2-53 •9282193 301 1-1019400 3-49 1-2499017 3-97 1-3787661 2-54 •9321640 3-02 1-1052568 3-50 1-2527629 3-98 1-3812818 2-55 •9360934 3-03 1-1085626 3-51 1-2556160 3-99 1-3837911 2-56 •9400072 3 04 1-1118575 1 3-52 1-2584609 4-00 1-3862943 2-57 •9439058 3-05 1-1151415 3-53 1-2612978 401 1-3887912 2-58 2-59 •9477893 3^06 1-1184147 i 3-54 1-2641266 402 1-3912818 •9516578 3^07 1-1216775 i 3-55 1-2669475 403 1-3937763 2-60 •9555112 3-08 1-1249295 3-56 1-2697605 4-04 1-3962446 2-61 •9593502 309 1-1281710 3-57 1-2725655 4-05 1-3987168 2-62 •9631743 3-10 1-1314021 3-58 1-2753627 4-06 1-4011828 2-63 •9669838 3-11 1-1346227 3-59 1-2781521 4-07 l-403642r- 2-64 •9707789 3-12 1-13783301 3-60 1-2809338 4-08 1-406096'J 2-65 •9745596 3-13 1-1410330 3-61 1-2837077 4-09 1-4085449 2-66 •97832.59 314 1-1442227 1^1474024 ' 3-62 1-2864740 4 10 1-4109869 2-67 •9820784 315 3-63 1-2892326 4-11 1-4134230 2-68 •9858167 3-16 1-1505718 3-64 1-2919836 4-12 1-415853] 2-69 •9895411 3-17 1-1537315 3-65 1-2947271 4-13 1-4182774 2-70 •9932518 3-18 1-1568811 3-66 1-2974631 4-14 1-4206957 2-71 •9969486 3-19 1-1600209 3-67 1-3001916 4-15 1-4231083 2-72 rO0O6318 3-20 1-1631.508 3-68 1-3029127 4-16 1-4255150 2-73 1-0043015 3-21 1-1662708 3-69 1-3056264 4-17 1-4279161 2-74 1-0079579 3-22 1-1693813 3-70 1-3083328 4-18 1-4303112 2-7o 1-0 1 1600'.) 3-23 1-1724821 3-71 1-3110318 4-19 1-4327007 2.76 r0152306 3-24 1-1755733 3-72 1-3137236 4-20 1-4350844 2.77 1-0188473 3-25 1-1786549 3-73 1-3164082 4-21 1-4374626 2.78 1 •0224509 3-26 1-1817271 3-74 1-3 190856 4-22 ]^43'.i8351 2.71 1-0260415 3-27 1-1847899 3-75 1-3217559 4-23 1-4422020 2-8r 1-0296193 3-28 1-1878434 3-76 1-3244189 4-24 1-4445632 '2-81 1-0331843 3-2'. 1-1908875 1-3270749 4-25 1-446918'.) |2.8!J 1-0367368 3-30 'M939224 3-78 1-3297240 4-26 1^449269 1 2.8:' 1-0402766 3-31 !il- 1969481 1 -33236 (;0 4-27 1 •4516 138 2.!^ 1 l-Oi:i^OM)3:V32 I'.l-199'.irvt7 1 :vsn 1-3:;. ".00 10 1-28 1^4.-.39530 HTPEEBOLIC LOGAETTHMS. 379 No. 1 Jjutrarithni . Xo. LoL'arithin • No. Loiraritlim 5-73 1 Logarithm 1-7457155 4-29 1-4562867 I4-77 -5623462 5-25 1-6582280 4-30 1-4586149 4-78 5644405 5-26 1-6601310 5-74 1-7474591 4-31 l-460".)379 4-79 5665304 5-27 1-6620303 575 1-7491998 4-32i 1-4632553 4-80 5686159 5-28 1-6639260 5-76 1-7509374 4-33 1-4655675 4-81 1 1 5706971 5-29 1-6658182 5-77 1-7526720 4-34 1-4678743 4-82 5727739 5-30 1-6677068 5-78 1-7544036 4-35 1-4701758 4-83 1 5748464 5-31 1-6695918 5-79 1-7561323 4-36 1-4724720 4-84 5769147 5-32 1-6714733 5-80 1-7578579 4-37 1-4747630 4-85 5789787 5-33 1-6733512 5-81 1-7595805 4-38 1-4770487 4-86 5810384 534 1-6752256 5-82 1-7613002 4-39 1-4793292 4-87 5830939 00 1-6770965 5-83 1-7630170 4-40l 1-4816045 4-88 5851452 5-36 1-6789639 5-84 1-7647308 4-41 1-4838746 4-89 5871923 5-37 1-6808278 5-85 1-7664416 4-42 1-4861396 4-90 5892352 5-38 1-6826882 5-86 1-7681496 4-43 1-4883994 4-91 5912739 5-39 1-6845453 5-87 1-7698546 4-44 1-4906543 4-92 5933085 5-40 1-6863989 5-88 1-7715567 4-45 1-4929040 4-93 5953389 5-41 1-6882491 5-89 1-7732559 ; 4-46 1-4951487 14-94 5973653 5-42 1-6900958 5-90 1-7749523 4-47 1-4973883 495 5993875 5-43 1-6919391 5-91 1-7768458 4-48 1-4996230 4-96 6014057 5-44 1-6937790 5-92 1-7783364 4-49 1-5018527 1 4-97 6034198 5-45 1-6956155 5-93 1-7800242 4-50 1-5040773; 4-98 6054298 5-46 1-6974487 5-94 1-7817091 4-51 1-5062971 ! 4-99 6074358 5-47 1-6992786 5-95 1-7833912 4-52 1-5085119 [5-00 6094377 5-48 1-7011051 5-96 1-7850704 4-53 1-5107219 5 5-01 6114359 5-49 1-7029282 5-97 1-7867469 4-54 1-5129269 5-02 6134300 5-50 1-7047481 5-98 1-7884205 4-55 1-5151272 [5-03 6154200 5-51 1-7065646 5-99 1-7900914 4-56 1-5173226 1 5-04 6174060 5-52 1-7083778 6-00 1-7917595 4-57 1-5195132 5-05 6193882 5 5-53 1-7101878 601 1-7934247 4-58 1-5216990' 5-06 6213664 j 5-54 1-7119944 6-02 1-7950872 4-59I 1-5238800 15-07 6233408 5-55 1-7137979 6-03 1-7967470 4-601 1-5260563 5-08 6253112 5-56 1-7155981 604 1-7984040 4-61 1-5282278 5-09 6272778 \ 5-57 1-7173950 605 1-8000582 4-62 1-5303947 5-10 6292405 I 5-58 1-7191887 606 1-8017098 4-63,1-5325568 5-11 6311994^5-59 1-7209792 6-07 1-8033586 4-64! 1-5347143 5-12 6331544 5-60 1-7227660 6-08 1-8050047 4-65 1-5368672 5-13 6351057 5-61 1-7245507 6-09 1-8066481 4-66 1-5390154 5-14 6370530 5-62 1-7263316 6-10 1-8082887 4-67 1-5411590 [5-15 6389967 5-63 1-7281094 6-11 1-8099267 4-68 1-5432981 jo-Ki 6409365 5-64 1-7298840 6-12 1-8115621 4-69 1-5454325 5-17 6428726 5-65 1-7316555 6-13 1-8131947 4-70 1-5475625 1 5-18 6448050 5-66 1-7334238 614 1-8148247 4-71 1-5496879 5-19 6467336 5-67 1-7351891 6-15 1-8164520 4-72 1-5518087 5-20 6486586 5-68 1-7369512 6-16 1-8180767 4-73 1-5539252 5-21 6505798 5-69 1-7387102 6-17 1-8196988 4-74 1-5560371 5-22 6524974 5-70 1-7404661 6-18 1-8213182 4-75 1-5581446 5-23 6544112 5'71 1-7422189 6-19 1-8229351 ,4-76 1-5602476 '5-24 1-6563214 '5-72 1-7439687*6-20 1-8245493 380 HTPEEBOLIC LOGAKITHMS. No. Lo^uritliin j N.p. lAijiaruhin Nil. l.-'i.ii-ini -No. l-o^arithm ti-21 1-82616(»8 6-69 1-9006138 7-17 Ly6vnHJ56 ! '7-65 '2^034705Tr 6-22i 1-8277699 6-70 1-9021075 7-18 1-971 29U3 7-66 2-0360119 6-28 1-8293763 6-71^ 1-9035989 7-19 1-9726911 7-67 20373166 6-24 1-8309801 6-72' 1-9050881 7-20 1-9740810 7-68 20386195 6-25 ; 1-8325814 6-73; 1-9065751 7-21 1-9754689 7-69 2-0399207 6-26| 1-8341801 6-74' 1-9080600 7-22 1-9768549 7-70 2-0412203 6-271 1-8357763 6-75i l-y095425 7-23 1-9782390 7-71 2-0425181 6-28! 1-8373699 6-76! 1-9110228 7-24 1-9796212 7-72 20438143 6-29i 1-8389610 6-77| 1-9125011 7-25 1-9810014 7-73 20451088 6-30: 1-8405496 6-78 1-9139771 1-9154509 7-26 1-9823798 7-74 2-0464016 6-3l! 1-8421356 6-79! 7-27 1-9837562 7-75 2-0476928 6-32i 1-8437191 6-80' 1-9169226 7-28 1-9851308 7-76 2-048:1823 6-33i 1-8453002 6-81 1-9183921 7-29 1-9865035 7-77 20502701 6-31 1-8468787 6-82 1-9198594 7-30 1-9878743 7-78 20515563 6-35 1-8484547 6-83 1-9213247 7-31 1-9892432 7-79 2-0528408 6-36 1-8500283 6-84 1-9227877 7-32 1-9906103 7-80 2-0541237 6-37 1-8515994 6-85! 1-9242486 7-33 1-9919754 7-81 2-0554049 6-38 1-8531680 6-86i 1-9257074 7-34 1-9933387 7-82 2-0566845 6-39 1-8547342 6-87! 1-9271641 7-35 1-9947002 7-83 2-0579624 6-40 1-8562979 6-88' 1-92861S6 7-36 1-9960599 7-84 2-0592388 6-41 1-8578592 6-89i 1-9300710 7-37 1-9974177 7-38 1-9987736 7-85 2-0605135 6-42 1-8594181 6-90! 1-9315214 7-86 2-0617866 6-43 1-8G09745 6-9l| 1-9329696 7-39 2-0001278 7-87 2-0630580 6-44 1-8625285 6-92' 1-9344157 7-40 2-0014800 7-88 2-0643278 6-45 1-8640S01 6-9311-9358598 7-41, 2-0028305 7-89 20655;,Mjl 6-40 1-8656293 6-94 1-9373017 7-42 2-0041790 7-90 2-0668627 6-47 1-8671761 6-95 1-9387416 7-43 20055258 7-91 20681277 6-48 1-8687205 6-96i 1-9401794 7-44^ 2-0068708 7-92 2-06931)11 6-49 1-8702625 6-97! 1-9416152 7-45 2-0082140 7-93 2 0706530 6-50 1-8718021 6-98; 1-9430489 7-46; 2-0095553 7-94 2-0719132 6-51 1-8733394 6-99i 1-9444805 7-47| 2-0108949 7-95 2-0731719 6-52 1-8748743 7-00 1-9459099 7-48i 2-0122327 7-96. 2-0744290 6-53 1-8764069 7-01 1-9473376 7-49i 2-0135687 7-97| 2-0756845 6-54 1-8779371 7-02i 1-9487632 7-50 2-014'.»030 7-98; 2-0769384 6-55 1-8794650 7-03 1-9501866 7-51 2-0162354 7-99' 2-0781907 6-56 1-8809906 7-04 11-9516080 7-52 2-0175661 8-00:20794414 6-57 1-8825138 7-05 : 1-9530275 7-53 2-0188950 8-01i 2-0806907 6-58 1-8840347 7-06 ' 1 -9544449 7-54 2-0202221 8-02 2-0819384 6-59 1-8855533 7-07 1 1-9558604 7-55 2-0215475 8-03 2-0831845 6-60 1-8870697 7-08 1-9572739 7-56 2-0228711 8-04: 2-084429() 6-61 1-8885837 7-09 , 1-9586853 7-57 2-0241929 8-05: 2-0856720 6-62 1-8900954 7-10 ; 1-9600947 7-58 2-0255131 8-06 2-0869135 6-63 1-8916048 7-11 1-9615022 7-59 2-0268315 8-07: 2-0881534 6-641 1-8931119 7-12 1-9629077 7-60 2-0281482 8-08: 2-0893918 6-65! 1-8946168 7-13 1-9643112 7-61 2 0294631 8-09 2-0906287 6-66' 1-8961194 7-14 1-9657127 7-62 2 0307 7 63 810 2-0918640 6-i;7 l-S'.)7619Si 7-15 1-9671123 7-63 2-0320878 8-11' 2-0930!tS4 6-6S 1 •.>'.):(] IT'.t J 7-1 1; 1-9685099 j7-64i 2-0333!t76 8-12 2-o-.t4:i:;o6 HTPEKBOLIC LOGARITHMS. 381 ISo. I Loirarthm No 8-13, 8-141 8-1 5i 8-16i 8-17; 8-18| 8-19I 8-20, 8-211 8-22, 8-23 8-24 8-25i 8-26i 8-27, 8-28: 8-29 8-3o: 8-31i 8-32 8-33 8-34 8-35 8-36 8-37 8-38 8-39 8-40 8-41 8-42 8-43 8-44 5-45: 8-46 8-47 8-48 8-49 8-50 8-51 8-52 8-53 8-54 8-55 8-56 8-57 8-58 8-59 8 GO, Logarithm \ Nu. i Loj^urithm 20955613 2-0967905 2-0980182 2-0992444 2-1004691 2-1016923 2-1029140 2-1041341 2-1053529 2-1065702 2-1077861 2-1089998 2-1102128 2-1114243 2-1126343 2-1138428 2-1150499 2-1162555 2-1174596 2-1186622 2-1198634 2-1210632 2-1222615 2-1234584 21246539 2-1258479 2-1270405 2-1282317 2-1294214 2-1306098 2-1317967 2-1329822 2-1341664 2-1353491 2-1365304 2-1377104 2-1388889 2-1400661 2-1412419 2-1424163 2-1435893 2-1447609 2-1459312 2-1471001 2-1482676 2-1494339 2-1505987 2-1517022 8-61' 2' 8-62| 2 8-631 2 8-64! 2 8-65| 2 8-66i 2 8-6712 8-681 2 8-69| 2 8-70: 2 8-71:2 9 8-73 8-74 8-75 8-76 8-77 8-78 8-79; 2 8-80; 2 8-81i 2 8-82. 2 8-831 2 8-84j 2 8-85; 2 8-86 8-87i 8-88: 8-89: 8-90 8-91 8-92 8-93 8-94 8-95 8-96 8-97 8-98j 2 8-99| 2 9-001 2 9-01! 2 9-02; 2 9-03! 2 9-04i 2 9-05| 2 9-06, 2 9-07i 2 9-08! 2 1529243 1540851 1552445 1564026 1575593 1587147 1598687 1610215 •1621729 ■1633230 •1644718 •1656192 •1667653 •1679101 ■1690536 •1701959 •1713367 -1724763 •1736146 •1747517 -1758874 •1770218 •1781550 •1792868 -1804174 •1815467 •1826747 -1838015 •1849270 •1860512 •1871742 •1882959 1894163 1905355 1916535 1927702 1938856 1949998 196112S •1972245 1983350 1994443 2005523 2016591 2027647 2038691 2049722 2060741 Lo;furithTn 909; 2^2071748 9'10; 2^2082744 9^11i 2^2093727 9-12: 2-2104697 9-13 2-2115656 9-14: 2-2126603 9-15' 2-2137538 9-16 2-2148462 9-17J 2-2159372 9-18! 2-2170272 9-191 2^2181160 9^20, 2^2192034 9-21 1 2-2202898 9-22I 2-2213750 9-231 2-2224590 9-24I 2-2235418 9-25 2-2246235 9-261 2-2257040 9-27 2-2267833 9-28i 2-2278615 9-29; 2-2289385 9-30; 2-2300144 9-31; 2^2310890 9-32; 2-2321626 '.)-33l 2-2332350 9^34| 2-2343062 9-35| 2-2353763 9-361 2^2364452 9^371 2^2375130 9-38, 2^2385786 9^39 2^2396452 9-40: 2-2407096 9-4112-2417729 9-42' 2-2428350 9-43; 2-2438960 9-44| 2-2449559 9-45I 2-2460147 9-46! 2-2470723 9-47i 2-2481288 9-48; 2-2491843 9-49: 2-2502386 9-50 225 12917 9^51 2^2523438 9^52^ 2^2533948 ^•53 2^2544446 9-54 2^2554934 9^55 2-2565411 9-5-J 2-2575877 9^57j2^2586332 9^58!2^2596776 9^o9!2-2607209 9-60|2-2617631 9-61J2-2628042 9-6212-2638442 9-6312-2648832 9-64i2-2659211 9-65|2-2669579 9-6612-2679936 9-6712-2690282 9-68 2-2700618 9-6912-2710944 9-70,2-2721258 9-7112-2731562 9-72i2-2741856 9-73|2-2752138 9-74|2-2762411 9-7512-2772673 9-7612-2782924 9-77i2-2793165 9-78'2^2803395 9^79!2-2813614 9-80,2-2823823 9-8112-2834022 9-82:2-2844211 9^83 2^2854389 9-84|2-2864556 9-8512-2874714 9-86|2^2884861 9-87|2^2894998 9-88'2-2905124 9-89J2-2915241 9-90i2-2925347 9-9112-2935443 9-9212-2945529 9-93|2-2955604 9-94J2-2965670 9-95!2-297o725, 9-96i2-2985770 9-97j2-2995806 9-98|2-3005831 9-99;2-3015846 I0-00|2-3025851 11-00,2-3978952 12-00;2-4849065 1500 2-7080502 20^00:2^9957322 382 READY RECKONER. 1 2 3 4 1 1 5 6 7 » f 9 10 d. s. f/. s. d. s. d. s. d, s. d. s. d. «. d. s. rf 5. f7. 1 4 1 2 1 3 1 3 it 1 2 a 3' 1| 2 4 2} 4i H 5 3 4 1 1^ 2 2f 3 3 4 3| 5 4i 6^ 5i 7 6 8 6| 9 7i '2 10 It 2i 3 3f 4i 5 6 6i 7i 7i 9 81 lOi 10 Hi li 1 o| 1 3 If 2 4 5J 6 7 8 8| 10 lOi 0' o| 2 2 4 3| 6 1 51 1 8 n 4i 5 6! 7i 9 10 Oi H 3" 3f 5i 6 8 8i lOi 1 lOi 2 1 2} 3 5| 6 8i 9 11 If 3 4i 6' 7f 9 2 10 2 2 Of 3 2 31 2 6 ^ 6i 9| 1 4i 7^ lOf 2 2 2 5i 2 8i H 7 1 Oi 2 5i 9 2 0.^ 2 4 2 7^ 2 11 n 7i ' 2 1 H 3 6| 10| 2 n 2 6 2 9| 3 U 4 8 1 4 8 2 2 4 2 8 3 3 4 ^ 8i Of 5 ^i 2 u 2 5! 2 10 3 2i 3 6| 4| 9"^ H 6 lOi 2 3 2 7^ 3 3 4i 3 9 4| 5 ^2 10 2i 3 7 8 2 111 1 2 2 4i 6^ 2 2 91 11 3 3 2 4 3 3 6| 9 3 lli 4 2" 5.^ io| 3| 9 2 2i 2 H 3 01 3 6 3 Hi 4 4i 5| 11 4| 10 2 3i 2 9 3 24 3 8 4 u 4 7 5f 6 iH 5i 6 2 11 2 2 4f 6 2 3 lOi 0" 3 3 4f 6 3 4 10 4 4 3| 6 4 9i 5 0' , H o| i 6ii 2 1 2 n 3 H 3 7| 4 2 4 8t 5 21 6i 1 7^1 2 2 2 8i 3 3 3 9| 4 4 4 lOi 5 5 6f 7 2"^ 1 8i 1 9 2 2 3 4 2 2 9| 11 3 3 4i 6 3 4 Hi 1 4 4 6 8 5 5 Of 3 5 7i 5 10 n 2| 1 9f 2 5 3 0^ 3 7^ 4 2| 4 10 5 5i 6 Oi 7i 3 1 lOi 2 6 3 u 3 9 4 4i 5 5 7i 6 3 7| 8 3| 4 1 111 2 2 2 7 '8 3 3 2f 4 3 4 lOi 4 4 6i 8 5 5 2 4 5 6 9| 6 5i 6 8 8i 4| 2 01 2 9 3 5^ 4 u 4 9| 5 6 6 2} 6 lOi 8| 5 2 H 2 10 3 6 J 4 3" 4 lU 5 8 6 4i 7 1 8f 9 5| 6 2 2£ 2 3 2 3 11 3 3 71 9 4 4 4i 6^ 5 5 1| 3 6 10 6 6 6! 9 7 31 7 6" ^1 6| 2 3f 3 1 3 10} 4 7i 5 4f 6 2 6 Hi 7 8i H 7 2 4i 3 2 3 lU. 4 9 5 6i 6 4 7 u 7 11 n 71- ' 2 2 6i 3 3 4 Of 4 lOi- 5 8f 6 6 7 3f 8 U 10 8 2 6 3 4 4 2 5 5 10 6 8 7 6 8 4' 10} 8^ 2 6f 3 5 4 3f 5 U 5 11^- 6 10 7 Si 8 6:' io.\ 9 2 71 3 6 4 4i 5 3' 6 U 7 7 lOi 8 9" 10| n 2 8f 3 7 4 5f 5 4i 6 3| 7 2 8 o| 8 lU 11 10 2 9 3 8 4 7 5 6" 6 5 7 4 8 3 9 2' in 10| 2 9f 3 9 4 8| 5 7^5 6 H 7 6 8 5i 9 4| 11 \ 11 2 10\ 3 10 4 9^, 5 9' 6 S\ 7 8 8 7i 9 7 11^ Hi 2 Hi 3 11 4 i£I 5 10.\ 6 10} 7 10 8 -£i 9 9| DECIMAL EQUIVALENTS OF MONEY, ETC. 383 Taele of Income, ^VaGES, ok ExpEN:? E3. Per Pel Per , Per Per - Per Pel Per Year Mom £ v." .h Week 1 Day Year Month Week £ Day s. d. £ s. d. £ s. d. & s. d. £~ 6-. d.£ x. d. 1 1 8 41 Of 13 1 8 5 81 1 10 2 ejO 7 1 13 13 2 9 5 3 9 2 3 4 9i li 14 3 4 5 H 91 2 2 3 6 9| li 14 14 4 6 5 8 9f 2 10 4 2 lU If 15 5 5 9 10 3 5 1 If 2 16 15 6 3 6 01 101 3 3 6 3 1 2i 2 16 6 8 6 2" 101 3 10 5 10 1 4|. 2i 16 16 8 6 51 11 4 6 8 1 6i 2f 17 8 4 6 ^ 111 4 4 0- 7 1 7i 2| 17 17 9 9 6 10 111 4 10 7 6 1 8| 3 18 10 6 11 llf 8 4 1 11 3| 18 18 11 6 7 3 1 Oi 5 5 8 9 2 Oi 3i 19 11 8 7 31 •0 1 oi 5 10 9 2 2 11 3| 20 13 4 7 8 1 H 6 10 2 3f 4 30 2 10 0, 11 6 1 "1 6 6 10 6 2 5 41 40 3 6 8 15 H 2 21 6 10 10 10 2 6 41 50 4 3 4 19 3 2 9 7 11 8 2 81 41 60 5 1 3 Of 3 31 7 7 12 3 2 10 4f 70 5 16 8 1 6 11 3 10 7 10 1'2 6 2 101 5 80 6 13 4 1 10 9 4 41 8 13 4 3 1' 51 90 7 10 1 14 71 4 11 8 8 14 3 2f 51 100 8 6 8' 1 18 5 5 6| 8 10 14 2 3 31 51 200 16 13 4 3 16 11 10 111 9 16 3 51 6 300 25 5 15 H 16 51 9 9 15 9 3 71 61 400 33 6 8 7 13 10 1 1 11 10 16 8 3 10' 61 500 41 13 4 9 12 H 1 7 4f 10 10 17 6 4 01 7" 600 50 on 10 9 1 12 101 11 18 4 4 3" 71 700 58 6 813 9 2f 1 18 41 11 11 19 3 4 51 71 800 66 13 415 7 8i 2 3 10 12 1 4 71 8 900 75 017 6 If 2 9 31 12 12 1 1 4 10' 81 1,000 83 6 8 19 4 H 2 14 91 Table of the Decimal Eqfivalents of Pence and | Shillln'gs. Pence ShiUings Pence ShiUines Pence 6^ hilllng-s •5208333 Pence Shillings 1 4 •0208333 '^ •270S333 ■770^333 1 2 •0416666 31 •2916666 61 •5410666 91 •7916666 f •0625000 3f •3125000 6f •5625000 9f •8125000 1 •0833333 4 •3333333 7 •5833333 10 •8333333 H •1041666 ^ •3541666 7i •6041666 101 •8541666 1* •1250000 41 •3750000 71 •6250000 101 •8750000 If •14583B3 4f •39.18333 'f •■6458333 lOf •8958333 2 •1666666 5 •4166666 8 •6666666 11 •9166666 9i -4 •1875000 H •4375000 8^ •6875000 iH •9375000 21 •2083333 51 •45S3333 81 •70f^3333 I'U •9583333 2f •2291666 5f •4791666 8f •7291606 llf •9791666 3 •2500000 6 •5000000 9 •7500000 12 1-0000000 384 DISCOUIsT TABLE. PRICES PER LB., QR., CWT., AND TOX. r: -o o r^: -o o ^: --c c r-. tr o :c o o rr: o o ^aoajGoajaocjcscicscncTiCiOsooooo "^fH 00 O -^ CO c tC t^ 00 Ci C ^» O O O O ' ^X'?- ^Qocooococoo^ascrs^oioicjciooooo ■< ic: o la :o CO :J ^S to — t s -o i^ t- t^ t- i^ t^ t^ i^ F^ oc oc cc cc :m -# to X c; c 00 cr. o — :>5 -^ l^ l^ OC X) 00 X to O — ♦ »C X C iriu;ir5?o:o«0'-ot-t^t-xxxxa;ciC^O _ o o — c^ c^ '^5 -M |"=to--otototo:o--o t^t^t^t-t^c^t^i^xxx '— 'X- 'M>--:'-'X X X X X X J: -rXO-^X C^^XO'^XO-^XO-^XO-JfX'O-^'XO .r:toocc-~oor:--oor<:-oocctoorctoor^tocco:oc I c^ o -^ C 'C '^ O IC .^•C^l-Tf^tOXOO'N-*tOXC;C;C'4-*tOXC:OC^4-<1'tOXC:C c^ o ■— ' c^i ot »' •< ■M r: c7 !X ?^ ?" t- X ~ o 'M c^ -^ '^ to t-- c: o — * '^t ^t -^ to — -1 r-C^ r-l'-:) r.-'Il — d -- ^t to C — < *0 X O ..^•t^rr-t-xxxsi^iCiOiOOO — — ' — — '-^^'^^•^^ecc^^<:'*' i; j; I . ,-'»^i-»^a>-ii"M>nfcDr:'-'-Hx — 'x^-;'r:ix--";i>cixr:-'-Hx .-'x^.-*r:'x-'^i'OfcDr5l-*«S'X £._::~:rtrcc^ccr^rcc^-^-*'-*-^-+'-^-*'-tic:oi::»^oc;oioto o c -^XO-^XO-t^XO-^XO'*'XO"*XC-^XO'*XC; , r: to O re to O rt to o re to o c:^ to o r^ to o fft to c .-^ to c lii— i:»o0050'-<^i-*'iaeot^xc:— 'C>ir:-t-»ct; X c o X 1^- -M ^ to X O O C<1 -* to X C: O S^I to X C: C "M ,-^5^?^'*ot-xo50-^c>i-fir:tot-xo:— '■M?e-t<«-':tox re l^ O 3^ i-e a; o -:f i- £ )_; ~ — — — < ~ — • — — ' — >:i M r^i M -M >i c-1 M re C C 38C WAGES TABLE. Table showing Amount Earned i> \ ANY NU MBER OF Hours | FROil 1 TO 54, AT ALL R A.TES FROM Is TO 15*:. FOR A. ^\ 'EEK OF 54 H OURS. 3 •; Kate of Wages in Shilling s for a Week of 54 Hour s ~> 7 1 8 1 f> _^i_ 10 , 1 12 ; 1: ' 1 1* i 15 "i. A. QIO S. un d. t Earned in given Number of Hours £ s. ;o 40 [0 41 42 4;j|0 44 1 1 45 11 46 il 47 1 4x 1 t;» 1 50 1 51 I 5- 1 5:j I 54 1 9 9 10 10 11 11 12 12 6 12 lU 13 5" 14 lOJ 14 4\ 14 15 15 16 16 17 17 18 18 l||0 7 OilO ^0 5. d. i£ 5f0 lUU 1 5^0 1 11 10 2 5 2 10| 3 4i'0 3 lOijO 4 4 4 9|0 5 UO 5 9||0 6 3 6 9 7 2^!u 7 8h'0 8 2|:o 8 8 |0 9 1||0 9 7iU s. d. £ s. 60 6.0 00 10 U ■H 9 2i H 7 0,^ 6i 18 11^ 19 o\ 19 11 4i lo" 10 7 11 1 11 63 12 oi:o 12 6|:o 13 \U (J n 13 5f 13 lU 14 5| 14 11 15 5 15 105 16 U 16 lOljO 17 4 '0 17 9|;o 18 3i|0 18 9i!0 19 3 1 19 9 1 2,^4 8il 1 2|l 711 1 II (in 1 1 7i:i H 1 11 OAl 6 1 I 1 6 6 6 6 6 6 t 6 6 6 6 6 6 6 6 6 6 6 c 6 6 6 61 o;i 6 1 1 d. \£ .1. ej'o 7 7 8 8 9 9 10 10 11 11 I) 12 12 13 14 14 :o 15 15 16 (» 16 '> 17 17 I 18 i» 18 19 19 1 1 2 2i 8i 2| 9 3 H Sijo 9|0 4 io 9 10110 10 4i'0 10 10|iO 11 5 11 11 12 5J:|0 12 1U:0 13 5i-:U 13 d. \£ s. 6i'0 10 1 71,0 1 l|jO 8i'0 2|,0 9 Io 3i0 10 4ilo 1 5 1 5 6 6 14 15 15 16 16 17 11 17 7J1 18 l||0 18 8 iO 19 2110 19 8i:l 2^;i 9 1 3 1 9i'l 3i'l oil 4 il lOlil U I 10^ 1 5 |1 11 !l 511 lUl rri I 1 11 51 lis 61 Oj 7 U 8 2i 9 31 n 41 10^ 5 Hi 6 OA 7 u 730 17 6 6 7 7 8 8 9 10 10 11 11 12 12 13 13 14 15 15 16 16 17 I) 18 18 9i0 19 4 |1 lOA 1 5';i 1 Ui'l 1 5^,1 2 Oil 2 em 3 1 |l 3 7il 4 2 |1 5 8il 5 3 16 911 1 6 3.n 7 loi'l 7 4ijl 8 11 !l 8 5*1 9 0' 1 10 d. \£ s. 6^10 lliO 8 :o 2m 910 4 |0 103 5^0 io 6^0 110 8 ,0 230 91.0 8 4 lO 8 103 9 510 9 10 6|0 10 110 11 8 jO 12 2|i0 12 91i0 13 4 ;o 13 Vrp) 14 5|0 14 U ju 15 6i 16 11 16 5 U 17 23 17 91:0 18 4 18 10^0 19 5^ 1 10 Gil Ul 8 I 2^:1 911 4 1 103 1 511 1 6|1 111 8 11 7 211 8 9ill 8 4 19 103,1 9 5} I 10 1 11 d. £ 7 13 83 3:^0 luSo 510 010 7 |0 2 9 !0 330 loio 5^0 7 OiO 8 710 8 210 9 9 10 4 10 11 tO 11 530 11 03 ;0 12 7^;0 13 2i 13 910 14 41 14 11 !0 15 6 |0 16 1 iO 16 730 17 23:0 17 9=V'0 18 45:0 18 1110 19 61I1 1 11 8 |1 3 1 9.^1 43 I lUl eiji ii:i 8i;l 3 11 10 |1 5 !1 llfl 63 1 ni 8i,l 3l!l loiii 5 1 d. \£ 7 Io 210 910 4i0 lUU 63]0 1^0 1 12 9 4 11 61 11^0 8Jr:0 3i,0 i03:o 53'0 1 ^0 8 |0 3 loilo 51,0 oilo 7I0 2^:0 9j!0 5 jo 7 Io 210 910 4i0 lllO 63 1 13 1 9 1 4 1 11 1 611 111 8il oil 103 1 53 1 1 ,1 8 J 3 1 1011 511 Oil 7ii 23 1 9.il 5 1 1 91 .s. d. 71 1 23 1 10 2 5 3 V^ 3 8 4 o 4 103 5 6 6 U 6 S, 7 4 7 111 8 C>i 9 9 10 43 11 11 71 12 2^ 12 10 13 14 Of 14 8 15 31 15 103 16 6 17 11 17 83 18 4 18 111 19 ()}• d\ 1 4 2 2 7 3 2.i 3 10 4 51 5 03 5 8 6 3] 6 lOj 7 G SMl 8 83 9 4 9 111 10 f;i 11 2 11 9; 12 4\ 13 WAGES TABLE. 389 Table showing Amount Earned in any Number of Hours FROM 1 TO 54, AT ALL RATES FROM Ms. TO i'Js. FOR A Week of 54 Hours. Rate of Wages in Shillings for a Week of 54 Hoiu s 34 So 38 39 42 Amount Earned in given Number of Horn's £ ■1. d. £ 5. d. £ 5. rf. £ »'. d. .£. s. 1 7iO 7|0 80 8i0 2 1 3 1 3i!0 1 40 1 4*0 1 3 1 lOfO 1 iiiio 2 0:0 2 03 2 4 1) 2 6i0 2 7 2 80 2 9 10 2 5 t.i 3 lio 3 3 40 3 5 3 e 3 9^0 3 103 4 00 4 IJO 4 7 4 5 4 6i,0 4 80 4 9i0 4 8 5 OiO 5 2i0 5 4:0 5 53,0 5 9 5 8'.0 5 10 '0 6 o!o 6 2 6 10 6 3iO 6 530 6 80 6 lOJO 7 11 6 11 7 UO 7 40 7 6h'0 7 12 U 7 C|0 7 9t0 8 00 8 2|0 8 13 I) 8 ■2i0 8 5 lO 8 80 8 11 9 14 U 8 9|0 9 1 10 9 40 9 7 9 ).5U 9 oiO 9 SJIO 10 00 10 330 10 16 10 1 10 Wo 10 8:0 10 11; 11 17 10 8iO 11 0f;0 11 4!0 11 73 11 IS 11 4 11 8 jo 12 12 4 12 19 11 1140 12 3jl0 12 8)0 13 OiO 13 20 iO 12 7 12 ll^'O 13 40 13 8iO 14 21 |0 13 2f 13 7i:0 14 14 43 14 22 13 lOjO 14 3 ]U 14 8 15 1 15 23 ,0 14 5^0 14 11 iO 15 40 15 9 16 24 10 15 HP 15 630 16 O'O 16 51 16 25 ;o 15 9 lO 16 2hV 16 8 17 no 17 26 !0 16 4i 16 lOi'O 17 4,0 17 930 18 27 ;o 17 o'o 17 6 |0 18 00 18 6 19 28 17 7i'0 18 1| 18 8^0 19 2i:0 19 29 18 3"0 18 H 19 4:0 19 lOU 30 31 18 lOAO 19 1 Oil 81 U 6*.! 1 19 6il 1 •* 3 !l 1 32 1 lii 9 1 4!l 1 11 il 2 33 1 911 1 43 2 Oil 2 7^1 3 34:1 1 5 a 2 Oi 2 8:1 3 3J1 3 35 11 2 OJl 2 8i 3 41 3 1131 4 36 ;i 2 8 1 3 4 4 Oil 4 8 15 37 1 3 3i'l 3 111 4 81 5 411 6 38 3 ll'l 4 7* 5 41 6 0*1 6 39 4 6|1 5 3|l 6 oil 6 83 1 7 40 5 2il 5 11 1 6 81 7 5 18 41 5 9|1 6 7 1 7 4!l 8 1 '1 8 42 6 oil 7 21 1 8 Oil 8 911 9 43 7 1 1 7 m 1 8 81 9 oil 10 44|1 7 8^1 8 6il 9 41 10 13 1 11 45 1 8 4 1 9 2 10 Ojl 10 10 1 11 46 1 8 lUl 9 9| 10 8;l 11 6^1 12 47 1 9 7 1 10 5i 11 41 12 2*1 13 48 1 10 23 1 11 li 12 1 12 103 1 13 49 1 10 lOfl 11 9 12 81 13 7 1 14 ■y> 1 11 5^1 12 5 13 41 14 3 1 15 .51 1 12 lil 13 03 14 01 14 Hi 1 15 ■r2 1 12 9 1 13 8A 14 81 15 7^1 16 •53 1 13 4il 14 4| I 15 4 1 15 35^ 1 17 54 1 14 1 15 1 16 V 1 17 1 IS d. £ s. 8*0 5 liO 9J0 6i0 230 11 jO 7^0 4 ojlo 9 10 510 IfO lo|iO 10 63 10 3 11 llj 12 8 13 4*;0 13 1 ,0 14 9i U 15 5J 15 2i'0 16 103,0 17 7 '0 18 3*10 18 10 19 8*'l 5'!l 93|1 6s'l II ll n\i 4 1 0*1 9 il 5^1 8 111 8 lOf'l 9 611 10 3 '1 11 11* 1 11 8 1 12 4*1 13 1 ;i 13 9i:l 14 53 1 15 2il 16 103 1 16 7 1 17 3*1 18 1 19 d. £ 83 5i0 2 lOfO 7iO 4 030 5 9iO 5 6 6 2|0 7 lliO 8 8 '0 8 430 9 lirO 10 10 ,0 11 63 11 3^0 12 13 8^|0 14 5|:0 14 2 15 1030 16 7* 17 4 10 17 03 18 9f 19 d. £ s. 9 :o 5|!o 23'0 11*0 8 o^lO 2ii0 11 ;0 8 5 '0 1|0 103,0 7*0 4^0 10 li,0 11 lOiiO 12 7 12 d. £ s. d. 9 ,0 9i 6i'0 1 6| 3i: 2 4 0*0 3 Ij 9*0 3 103 13 14 9|0 15 O^O 15 3|'0 16 OiO 1 G 1 231 iH'i 8 ll 4|1 lil 3 10 1 4 6|1 5 3^1 5 16 83 1 7 oil 8 2 18 1U| 1 9 7il 10 4 1 11 Ofl 11 9il 12 6 1 13 211 14 llil 14 8 1 15 4|1 16 lil 17 10 I 17 63 1 18 3il 19 u 2 U 9i 6i 3 9 5_ n Hi 18 18 19 2i!l 11 63iO 33.0 1 10 jO 7 7 '0 7 4i|0 8 liO 9 10^0 10 7*0 10 103 430 11 1|0 12 11 13 8 14 5 14 2i 15 lliO 16 8*0 17 li 5^0 17 103 23 18 8 11^0 19 5 9 1 6 1 3 ;i Oiil 9il 6*1 m 031 6 2: 7 7 9 6: 4 23 9^ C; 4 2 1 1 9 2 6 3 4 4 1 4 10 5 8 5 1|1 8 103 1 9 7il 10 4il 11 lil 11 lOil 12 7 il 13 4 1 14 1 1 14 9^1 15 6-3 1 16 3* 1 17 0|l 17 9il 18 6il 19 3 2 2 1 93.1 7 1 4 1 1 !i lOil n H H 7i 1 10 4 4*1 11 1 l| 1 II U't lOl'l 12 8 7^ 1 13 5 5 1 14 2| 2 jl 15 II |l lo H 8il 16 C| oil 17 4 2*1 18 14 III I 18 10 390 LOGARITHMS OF NUMBERS. Table of Logarithms of Numbers from 1 to 10000. Indices of Logarithms. The indfx of the logarithm of a niimber is one less than the number of integral figures used in expressing tliat number. Number i Logarithm Number 1 Logarithm Number Logarithm 4134 3-6163705 41-34 1-6163705 -4134 -1^6163-05 413-4 1 2-6163705 4-134 1 0-6163705 -04134 -2-6163705 To Find the Logarithm of a Number. * Find log. of 837-2468 Find log. of 830465 Log. of 837-2000 = 2-9228292 Log. of 830400=5-9192873 Tab. diff. 519x .468= 243 Tab. diff. 523 x .65= -339 Log. required 5-9193212 Log. required 2-9228535 To Find the Number corresponding to a given Logarithm. Find number of logarithm 2-9228-^35 Find number of logarithm 5-9193212 Logarithm of 837-2000 = 2-9228292 Logarithm of 830400 = 5-9192873 2435 56 7481H80 7f; 1-8S()8136 96 9822712 17 1-2304489 37 1-568-2017 57 7.-).58749 / 1 1-8864907 97 9867717 18 1 •2.552 725 38 1-5797836 58 7634280 78 1-89-20946 98 991-2261 19 1-2787.53(5 39 1-591(1646 59 7708520 79 1-8976271 99 9956352 20 1-3010300 40 1-6O20000 60 1-7781513 8(» 1-9030900 100 2-000(Kl00 1 LOGARITHMS OF NUMBERS. 31)1 . "N f-~ t^ t^ 71 I- X -~ ^ ^■" — IC ■— -^ X t— \ 71 1 — ' — " ' — * :^ ^ ~ — . X x X I- I- I- tc cc cc cc iC IC UC -# ^ ^ •^ -* -* ■-*' -j> rr -^ -* cc cc cc cc CC cc cc cc ■cc cc cc cc cc cc cc cc '"^ "N ^T -^ ic »a c; i- ■^ C5 r- LC .— CI t— ~ -f. I,-; X CI CI cc I— — . cc -+I -^ IC ic IC '-^ I— t— 1 t^ r- -^ c cc cc c; cc "^ ^-J cc cc —1 —1 0^1 Ci 1^ .^ .— t— c: I- 7) 7-4 c^ ^ 71 71 X — ^ — • X ^ 74 C IC I- cc rr ■rl- :C t^ -ji c; ^ o Ci c; X cc O CC O X IC ~« X — *- ^ ic o ic C5 ^» *^ X 74 ^ '.— ' ""^ "-/- CC r- ■^ -*i X 7-4 cc o -—t t^. ^— IC X 71 CC C. cc cc ^ ^^ ^— 71 71 71 cc cc -^t- -*• -r?' IC 4C cc cc cc I— t^ 1— X X X cr. c: — O 9 -r 9 ^' V 9 o 9 ^ 9 9 9 9 9 w ^ o 9 9 9 9 9 9 9 iC cvn _i '+' cc t- cc X c. cc X X — cc ~ cc X cc -^f X ~ cc -f cc CO ^ (^ cc 1^ ^-^ »^ ^— c/ c/. 71 C^ ■ — ' CI 74 — X 71 1.7 cc cc t- X o -* t^ ■.T*^ * * •^ X t- — 71 O cc "^ ^ ■^ "^ O "^ ■r^ — 'C cc ^ Ci 7-1 -H ..^ r^ c^ f— r* '•^ ir^ cc CC CC IC -^ cc »— 1 *, I- -* ^-H -X' Tf "^ cc — t- 7-1 r^ r^ .— < \^ C -* X 71 cc -^ -* X 74 CC c^ cc t- .— « •r^ X 71 UC C-. 71 CC X f^^ *_• ^M Tvl 71 7^1 CC cc ■* ^ rH iC IC iC t^ "^ 1- r— I- X X 00 Ci Ci T -r 9 9 9 9 V ^ 9 9 9 9 9 9 9 9 9 9 9 9 9 \r: O -:t" cc t- o -+ t^ iC cc cc 71 Ci IC ^ -r ~ IC I— 71 cc cc cc tr- ic ^ oc ^ m -t< iO c^ cc t- CC cc C;C O cc * -*■ t— w -r ~ cc -M -M r^ t^ -^ t^ tr .— 7-1 ',_ -*! IC 71 cc cc CC t— r- i-C * O "^ ^f CC CC U" 1- Ci O ^— 71 7-1 7-1 — o Ci I- »-C 7^ O 1— .T+- ^ t- 71 X cc X ^-' t^ ..^ i/^ c: -* X 7-1 CC ■^' -^ X »— 1 LC C- CC I- '"^ -+ X — IC X 7-4 IC -^ ^^ ^« — ■ 7^1 71 CC CC -:*H -.-4. — LC IC >c cc CC 1- t— I- X X 9 9 9 9 9 ^ ^ "^ ■r -r V • • • • '^ r- -r "T' T' T' T' ^ T' ^ r^ t~- ># 00 t~ cr. 7c c (^ X CC '*< cc sC L'^ oc ^— cc IC tr l~ X X cc t- cc l-C cc — ' * cc *^ r- -^ C: IC CC r:i •^ •— 1 IC * cc L— ^- ii: C; cc t- . — 1 >-C ~. 7-1 CC -^ •r^ r— r— -# X 7-1 ic ■^ ■^ — — ^ — 7) 71 C^ cc C^ ~^ :i • c )C IC cc cc r^ t- t~- X X X Ci Ci 9 9 9 9 9 9 9 9 9 O ■■^T? 9 9 9 9999999999 „■ ^ — -^ CC UC tc ir> r^ ^ cc Cl IC r: ic o C. -; -* — O cc — 1 o -* ^ t^ rc r""" • 71 C-. or: C^ -rt^ 71 -r 71 IC IC 71 IC r^ X' r- t^ cc cc t- c: tc ^ -^x T*^ ^ O -* ^ 7-1 1— t CC t- »C Ct o X G-1 ,~^ cc X l~- cc cc cc la — ^ -?*- x^ C^ — 71 CC -r*" -*< -r^ CC 71 f^-( ~ X UC CC I,"* 1^ cc ~ m lO :^^ ^ o -=:?•- c: CC I— , — 1 IC C^ CC t- — - 'rr X 71 CC o cc t— O '^ X — I lO '^ — • ^^ — 71 71 ^^ cc re ^J^ *^ LC iC i~ CC ic t— r— r- X X X r: c^> o ^ -r^ T' 9 9 o 9 9 9 9 9 9 9 -r- O 9 9 o o o 1 r^ 00 O ic tr; CC o o -^ '^ !M- r^ — i 71 C. IC CC ~i ^ X c ic cc X tr- S2 iC tr- -<*< 7-1 '^^ cc X — 1 IC o -+< Ci cc cc :^ 71 O -^ '+' ^ 74 .—1 cc X cc —1 7-1 o ic cc ie ^ 7-1 —1 X —1 7-1 ■^ i^ C!-" r^ t- X) c: X 71 --z o o o C. X l^ ~r^ X 71 T. cc cc o CC 71 t- CC X -*. o c^ ^^ -H — 71 7-1 cc cc cc •^ -t- IC 17 IC CC cc cc 1— 1- X X X c; Ci o 9 9 9 9 9 9 9 T 9 9 9 9 o o ■r -r o 9 o 9 9 9 9 9 — -+i •■£> C^ CC -^ CC t^ >c 71 >C 71 X c; 7-1 cc tr- o t— -« cc X IC — ' -H o Cv o o -^ X cc C5 X ^ IC IC ^ c; cc X -r*^ ^ IC O CC cc ""^ o o t— o X 71 CC C^ 71 71 1— 3: 1- 71 -" 71 t- '^ X -T« 7-1 cc tr- ... ^ io oo ^ 71 -ri^ U7 X 0^1 «£ >o ri, X 71 CC cc 7-1 O X IC — ' IC 71 71 ;-£ 7-1 X O cc -*< c; -f t- O "^ cc ^ -^ X o -^ o ^H — (71 71 ^c *S^ ce •m.^ -mM >c IC IC cc cc CC t^ t^ X X X C5 CJ o o r-> 9 9 9 9 9 9 ? 9 9 9 9 9 9 9 9 9 9 9 9 t^ o c: t— t- t- UC X ~. cc cc X - -u -^ IT —1 cc IC ^ ic cc 7-1 t- CC t- o C» t— O -* -+I r— 7-1 — 1 -^ 71 cc cc 7-1 CC t— t- \^ -*l 7-1 tc 00 lO t^ CC ■ — ( '71 ^ 7-1 71 X o ~u ^^ uc o 71 tr- O O t- — i 7-1 !M 00 ^^ -* cr X O --I .-H 7^4 71 — : — ( Ci X cc ^ 7-1 C^ cc 7-1 C^ IC O CC --1 71 ,*^ CC -+< -f -^ O IC CC cc r— t^ t- X cX) C t- o o -+I r; X o c^ ^ .— , T-l -+' X -H t- O o tr- ee cc r; ic cc X o X T-i ^ 71 CC Ci 7-1 -n X -+l t- X CTi IC 00 7-4 X t^ ^ ^ IC X X O IC — < IC O 7-1 r-^ ■^^ O ro o OO O '-I CC cc -+I -t< CC cc 71 o c^ CC -:t< — t X KC — ( Ir- CC CTi -t^ o o ^^ cc 7^1 t- ^ UC c^ cc t^ — UC C^ cc cc o -* X -H IC es 7-1 CC c; CC o o o ^H —I 71 71 71 cc cc -tH — " •:H IC IC CC cc cc t- t-- r— X X X C5 9 -r 9 T 9 9 9 r- ■T' 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 15 o — ^ o £J CC -+1 ic cc o o o t- X 2 o ^ 7-1 — ( cc -^ ^H 1—1 liC tc .— ( —J t- X ^ O -i 7-1 71 7-1 cc -^ 71 7-1 71 *~* 392 LOGARITHMS OF NUMBERS. d ^ CC iO O ^ 00 3M CO -* Ci to C) — "^ ~* 1— I — CO to .^ iC IC iC iC 1^ tj 'O cc o c/:: lo C-i o I— »C CM O 00 IC cc 1-H Ci t— -ti CI — ^ CO to -tM CI <~> fi *4J -*< -h cc cc cc cc :m CM CM C^l 1—1 1-H 1— ' r— t O c*^ ^ 'r^ CO cc cc cc cc cc cc cc cc cc cc cc cc CO cc cc cc CO CO CO CI CI CM CI ci t- ;£! ic Ci c^i to CO o to Ci lO ^+1 cc CM t- O -+I CI 00 "tH cc 00 CI tr- to o -H O 1 to Wi ^1 iC I- c C) -+I >c l^ CO Ci o o c O O Ci CO CO ^^ ^* ■^^ cc --o o cc to '^ to o cc to Ci CI IC 00 I— ( IC QO p_^ 'rh I- Ci M »c wr r O O -^ rH 1— I 1—1 :m C>1 c^i cc ■ — 1 -H f-H cc r-( -^ — -1 r-H lO T— ( IC IC 1— i 1—1 ^H to to to I- I— ( 1-H r— 1 1— 1 1-H tc cc Ci ~ c~ 1^ -*^ 1—1 -H Ci 00 ~H CM IQ CM l^ CI C^l Ci to IC r-H --H Ci CO o Ti O IC -:fi l^ >c 00 to Ci Ci to Ci Ci l^ CT to CO CO ro 1- to 'tt^ CI C/- ^^ CC ""^ t^~ • _ : r^ Ci lO CO cr. CO ■^ CO O O 1^ CI )C IC CI I— O ^^ CO tc to Ci r^ -* to 00 O CI cc "C to to t- l^ 1— t— I- to to IC CO * cc to c^ cc to Ci cc to Ci CM to Ci C^l IC 00 1—1 -+I I- C-. CO to Ci CI IC 9 O O O -H 1— t — ^ 1—1 :m 1—1 CM CM 1— ( I— 1 cc 1—1 CO CO '^l '^l '^ 1—1 r-H I— ( IC IC iC 1-H r-H to -H to to to b- 1-H r— f—l I— 1 t- I-H cc --0 cr. ic o to CO Ci -* to CO IC Ci lO "^H 1-1 -Ci C CO 1^ to 1-^ IC o oo >c "O O CO o »c »c o — 1 t^ Ci CO CO to to '+I sTi ~i- to -n Ci o o -=. 1-H cc t^ Oi t^ '^f t^ CO c^ cc to t^ to CC I- Ci cr. to CI IC IC cc CO r-H CI t^ *c t 1 O Ci ■^^ »o 00 r-^ cc lO r^ Ci o -—1 CI cc -^' ^ •^u 'i- -+! cc cc CT Ci. .ri -o Ci c CO Ci ~^ r/) — 1 -f tO' CO r—t O) cc ic -^ O •* »c -+I O '^ IC IC to CO 00 t^ 1— ' IC to <^ ^ T^l CO c-i to Ci C4 IC oD O C) -fl ^ I- CO Ci C: r-* ~- 1-H ^- -— O ''^ — to CO c-i irj Ci (M iC Ci -M >C Ci CI >c CO — J -*l t~ -^ I- ^ cc to Ci CI -:t< o O O O -H T-( rH 1— 1 I— 1 l-H r-H T— 1 CM T— 1 (M CM cc T— 1 CO cc r-( ^ ':t4 '^l I— ( I— 1 T— 1 lO I— t IC »c I— 1 I— 1 to T—i to to to t- ■>-H r-H I-H t-H T-H t^ lO Cq ^ CO »iC CO Ci cc cc cc t~- t- 00 CI cc -Tf ^ ^ 00 O to O iC CI cc O O cc Ci ^ IC IC r-l C<1 Ci CI CI Ci -H to to '* 1— I t^ cc l^ CI to 1-H -rH Ci — ' o to . — 1 ) CI C) I— 1 r-( cc I— 1 cc cc -*^ -rfl -rf 1—1 1—1 1—1 IC IC IC i-H l-H IC 1—1 1— 1 1— 1 I-H 1— i t- »o — l-^OCCtO-rtJOCOCCt^'^ 1-- r-H CO 1-^ -* O CM CM '+' 1-H lb Ci to t^ t- Oi lO -* t^ i.C r/^ O Ci QO ^ to to CI t- Ci O Ci t- -+I — 1 i^ cc o Oi -^ to to cc r^ Ci CO iC Ci 1—1 1-H CO co to to '^l 1-^ -*l to to -*< Ci cc IC -+I OJ t 1 IC Ci CI 1'-; en —J cc to CT) c^ 1-1 CI CO -# IC IC IC IC IC -r^H -:f cc -*< cc — i IC CO ^ IC CO lO QO -Tfl r^ -H -^ I- o cc to Ci CI IC CO 1-1 -t<' 9 c o o .-< I— < I— 1 I— 1 I— ( 1—1 I— 1 1—1 t— 1 C-1 CM CM I— 1 1—1 CO 1—1 CO CO T-H -H '+' ^ l-H I— 1 ^H IC I— 1 »C IC r-H r-t r-H to to to t- l-H rH t— 1 I— ( rH ,_^ -*i -* t^ liC ^ r^ CO — 1 O CO Ci »0 CM -H t- CI Ci CI CO to CO t- C^l CO ,—1 cc CO to CO ^ -f O to t— IC O CM -H t- CI -f to to IC ■* CM l-H Ci iC o CM <>i cr. -+I to iC cc t- Ci Ci t^ CI IC >c -*! O •+' to to '+I o -+' IC cc c^ -*^ CO C>1 LC '^ n »c CO O CI -fj to CO Ci O 1—1 CI CJ CI CI CI CI 1-1 o CO 1- --^ -+I CO r-H -* CO — -f CO ^M -^ r- O cc I- o cc to Ci CM IC CO ^H -Tt^ 9 O O O .-H 1 — i r-H CM . — 1 CM CI CO CO cc 1— ( 1— t ^- IC IC IC l-H r-t iC T—l to to to t- T-H l-H 1-H r— 1 CO -tl — 1 o >c ^ m iC CI to t- 1— ( ^ O CM O t^ to O CO to '^ oo CI QO ■rt< Ci l^ CO i>i -H cc 1—1 -+4 CM t^ r^ -^ CO Ci CO -r±( ■Ci cc IC to t- t- CO on (^ IC CO CO to ^H cc o ic c- l^ 'C O CO -+< cc Ci -H to to -^ O -+1 to r- 1^ t- — -* 1- 1— 1 -rfl I- o •"U t^ O cc to Ci CI IC 00 rH -*l 1-- O CO o o c ^ 1—1 r-H Cv) CM CM CO r-H CO 1—1 CO 1—1 -*l -^t^ -f r-H I— 1 r-H 1—1 IC o IC l-H to to to t— rH l-H 1-H -H • cc — 1 to —■ C^l ^ t^ CO ^ 00 CO I— 1 lO b- rH 1— 1 o r-i to O '^ CI t^ 1-H to 1^ •C IC Ci to t^ CI CI QO CO IC CO t^ cc t- c» t^ -^ Ci '^l t- O CI IC t-- >o — -+I -tl >4 1^ O O t- ^1 >c 'C cc Ci CI cc C>l Ci cc to to lO r-H iC 1^ 1—1 n t-- -^ O Ci 'M to C5 ^ "^l to CO O ^ cc -+I IC IC to to to to to IC -n ^^ •^ O -+< 1^- o -+< t- O "^H t- o t- O CO to Ci CI IC CO i-H -;t< t- O CO — 9 O O O r-( r^ ,-H .— 1 -^ 1— ( CM 1—1 CM CM t— I I— 1 cc 1—1 CO CO r-( -f -*4 -H r-H ^H 1— ( 1— 1 O IC 1-H 1—1 KC 1-H to to to t- rH r— 1 i-H i-H ic I- o b- -^ -^ Ci to CO CO Ci to -4 00 o 1—1 cc O IC O Ci cc l-- CO o o cc o Ci cc l-H -H -fH cc CO O Ci -*' CO Ci CO to CI CO CI t— rH to — ( I- O -H CO ^ t- 1^ ic o CO cc C^l I- — ' CI *-H CO cc to to C IC • c to to to t- rH 1-H rH r-1 1—( in to t^ CO Ci o •^ CM cc -H IC to l^ CO Ci O T-* CM cc -^ »c to t— CO Ci 6 1.. ?^ -M '>\ 'M (M C^ cc cc — ( cc cc cc — H — ^ ■^ cc cc —" - ^H -H -H -+4 -+I -t< rfl I^ LOGARITHMS OF :^f^MBEPvS. 593 - CC -X CC' \^ >M :M 3^ ?1 3<1 o H i-^ 5 ~ ■>! o ^ ?i 5] ?i "M c-i C'l ^) X r^ X r: f- -^ ■>i '>^ i^\ '>"\ ri 1 '" ■M o i3 r: Ci : -* t- Ci r; 00 1 t- c^i I- M t- ^ >i ic t- o ^ 1 "• ^: ?t :o -# -f :m c^i >i 3^ 2-1 3S c: ri — r: T-i iC t^ C ?t --c cc -H -f ->C ri h i^ ~ c> o ^1 u; M »~ — cc ^1 >i c~. u: 'r^ u~ re X cc C O ^ X o cc oo t^ -M • i^ c; — — ~ ^- — — ^ "M L"^ — — ^ ^- C L-: -^ t- ic c; t^ O ~. -+• ?>i r: o Ci c c; c. ^ o re -:*■ -f ■^i X CI ^r I- 1^ i" •>( ^ h -^ •* X — le cc •>! c i^ »^ re O t-- -f ^^ 1^ — f O tc ^1 30 -+ ~ le c »e o i^ o re --^ r; >j -f i^ C re le cc T- re vr — ■ f ^ c. '— -r t^ r-. c^ I- X X X X r: r: r: c O O c ' :m ^i >i ^ re re re re -- ^ -^ — — ^ ^ — ^1 ^1 (M M -M e^T M e-T T-i :m :m e-e-^cor;0(>Joocit — j<^^o-rt< t2-Nr^cca0O'^O35O»e7^rei^>ex-*'^*'r;r;'^'+'C;c:ie re — ts o e-i re — X e>-i tc t- t- »e — ! ^i r; ^ 1— ' r: cc e— r^ i^ »e -M o »:— '^ — CO Le — i^ 'f o >e — t^ e-i i^ e CO r; r: c; o O O C ' ^ — m c-i >i cm re re re re -^t- ^ ^ ^ ^ ,_( r-i ^ ,-1 CM (>» 3-1 2>1 (3!1 N e-l S-l 3^1 ^l (M e-1 51 "M e-l 3-1 3-1 30c:O'*'ccC5i^t— e-ire-^ieo — ccierete' »^^xre — >e i^ le Le »e t ^ :r -* >e x -^ re tr 3-i ^ ic ce »c e-i re r: C «~ c; cr -+3-ix3-i-^iereoiexoc;xiCOre»-e>e-:?<^tr — re-f-f •3-1 1— r; X — -^ 31 3 t^ '^ 3-1 r; >e ei r: i;e r- b- re c: ':+• c; r: 31 (X — ' re re — oc re t^ c: c: oo »e — ue oC' c; c; . 3 r; X ■— »e re — r; t — * 3-1 c; -— re ~ tr 3-1 x -* o ^ 3-1 i^ e-i i^ m ce •^ ~ 31 1-3 X '— re tr r; 3-1 -^ t^ O 3-1 le oc o re — X — re iT X — t-t^XG0xc^c;r;c;OOO — — ^^r- '■M3-i3iecc:w'— t^'+'t^— rec:~ — X 0-^C5Le3-i — 3-i«r — o — »ere-:fxt^r:trt~re-+'C;0>-cx) X ^ 3-1 t- o '— ' O t^ re i^ r: c: CO »e o -# — I- ^ -f C : '3 X ~ ~. re3i— 'CioOw'^i-'r^ — reoi^-f'— i--rer:»e — ^1^311^311— t^Cie^-t't^orec^x "i^r:3-i»ei^c:3iLexoreiexo i^t^oocococ;cir:c-. OOOO'-' — — 3-13-13-13-irererere'^ N-. r— ^ 1-1 ,-1 ,- r-l T— ^ 3-1 3-1 3-1 3-1 3-1 3-1 3<1 3-1 3-1 3^ 3-1 3-1 3-1 ei 3-1 31 re~?r!-*■ r: 31 re 3-1 c: -e — X'^^t^ceoo— c;i-eOo o-irr:3-i'#»--0 3-iiet^0 3-iiexO t-»-ooxxciCic:c:OOC:0 — '-1-^3-13-13-13-irecerere-H ^ ^ ^ _ ^^ ^ ,— r-l ^ 5<1 3-1 3-1 3-1 3-1 3-1 3-1 3-1 O-l C-l 3^ 31 3-1 3-1 3-1 31 C:.--3-ire-*iet3ir~xc^o-^3ire-*'ietct-ooc^C:--3-ire i_e te »e »e 1-3 te Le o le le- «o ^ w is :c ^ tc w w ti t- r— t~- I— 304 LOGAEITHMS OF NUMEERS. • ' lT" — r^ -^ — .- re Ci t^ -*( 71 C. to -*< 71 O X t — ^ re — ( — -, X t^ """ ^ t^- t^ •^ rc — O c^ t^- tc »e •T*' 71 -^ o r-. X to »e -^ re 74 — * X t- ^ 2 -f -- ■— « -^ -v^i — *• r^ re re re re re re re 7i 7i 74 71 74 71 74 71 ...^ " :m » ci L-: t- r; 7j -* tc C". — re to X c re i-e i- —. 7i -^ to X ■ -f -+< o o o o le c^ t^ t^ to l^ l^ t^ l- X X X X X Cl Ci Oi 1 'T' ?'"' ■N ?• '^ ?' (>i 7^ ->» 71 7-1 7-1 7-1 C71 71 74 71 7-1 7^1 71 (71 71 (71 7-4 re — ; '^ GO o t- -^ r: 7-1 u; 3 t-- r; to o 71 -* to O X O t- — ~« -»" L7 30 -M t- T-. CC re -o te 71 le to »-e 71 to X X I- re r; 7-1 le ^ to le *^ rc t^ — -~* -^4 t^» — ^ -~. -— .^ 1^ -^ o le X — 7-1 ,^ -^ 30 O -*i fTi le c^ re 1- :3 re "-0 c; 71 le 71 ~M 1" X (>1 ?1 Ci ?i ^ i- n — ' -^ — S i^ i^ ?1 tl 515 7-1 71 71 71 71 71 O l^ — 1— -Ti- to X ^^ ^ ^< ^1 71 71 71 71 71 71 71 71 71 71 7-1 71 71 re 00 >o "^ 'O —1 -M T. >e 71 -. ^. re re c re tr~ -H t^ to O X -* t^ T. — —1 -o r^ -^ CO X -« X Ci ;.^ ,— J -t4 -f — 1 t- c; 1^1 ^" w- ■— ■_.' "T' to l^ X t- CC on -M ^ iC to *M -+i to t- ^ 'e 71 X 71 to -. -Z. C_^ C^ 1^ t- t^ ^1 ?f> -J la r; re t~ — -+I X — 1 -H t^ o re to X — ■ re ue X ^ '"^ re t- -f t^ -n -*< -o r; — -:*< X — i ~: o X o 71 — * L^ Oi — ^ re to X ~ >+ -*< ~r O IC »c 1*^ tC w to t^ t^ l^ t- X X X X X r. r; — i 0; * ^^ ?^ ■>i ^1 ^J >1 -71 71 71 71 71 71 71 71 71 71 71 71 7-1 71 ^1 7-1 71 7-1 re u: t^ ^ 1^ re t^ X X t^ t- O to X t— re C5 le re -+ X r: le -s 71 i.e '^i o cc — -^ t^ le O -71 — ^ X '-I 71 ^ X '71 Lo to le 71 X re to - 2< ^' T' ?' 71 ^1 71 7-) 74 7-1 71 71 71 71 71 74 74 71 71 71 -74 7-1 re — t-^ _H -M >i; -M t= cs — _« -K X re -+i 71 o X t^ o ^ X to ^^ le C5 t T< CO X -^i t^ t^ 71 t^ to re X — — r; o CO O "^ ""i to 71 4- 7-1 I « ^ re ir t^ t^ vc re ^ -:*< t^- O "— o ^ to re X . ^^ to »C j C>^ t^ :::::' — ^ r^ X 7^ "-C * X to o re to X o re '-e t^ ri 71 -*> to X — re le 1-7 »a X — ' re le -i* -^ -Tf i'^ J— ,-» i'^ t^ tc t^ t^ t^ t^ b- X X X X ri r: * ^ Ci V ?* ^' T^ "^^ ?^ 71 71 71 •74 7-4 71 71 >1 71 71 71 71 7-1 7^1 71 71 7-4 71 71 ci -* »e re X re -- r^ o to cn o o; c; r-i o re to la 1—1 t- 71 Ci 00 ^ -f ?^ iZ t- -^ n o c- -^ le to to -^ — 1^ .— 1 le -M ~ »c c^ c^i re re n r; to o -:*< to X X to -^ O O r: 71 -^i ■tW -+' M •ri* o -* ^ re X ri tc o re I^ -« i^ O re to r; 71 -f to r; — re ue t^ -# -*i -^ CO ^-" re * X — re la X O 71 >e 1^ n — -*i to X o re »e t^ n ^ -t- ~^ le le le 1*^ to tc to to t— t^ i^ t- t- X X X X r: oi ^ Oi Ci S<» (M ! Tq 71 71 71 (TJ 71 71 (71 71 7-4 71 (71 71 71 71 71 71 7-4 N (71 1 c; re t^ re re t^ X t^ le ~» -# c: X re to X o re Ci X 71 re _ t^ re 1 ^^ ^j cr — • O o t^ vr; 71 o i-e 71 »- C O CC »e r; —171710 t^ 71 t^ 1 CC IC O le X r". T. X :c 71 >^ — re le o re — t^ re t~ 71 71 >1 ec t- ^1 r^ — ^ *e ^ re i^ — 'e y; 7-1 le X — -*- t- r: 71 — I- r; r—i re le re , " '-5 CO r-^ re *e X o re le t^ C 7-1 -f t^ ~. ^- ce to X 7"^ le t^— 0^ 1 -f -f -*| ue »e le le "-C tr to to t^ I- t^ t^ b- X X X X — Ci ^ Oi * 1 -^ t^ l^ O X 71 -H le to ^ t~ X to r-l »C 0; -^ —1 71 X — t- re —44 l'^ t^ X -t- X X ir: Ci ^^ o; i-e n -H o t- re t- r; r; c: C^ '^ o -+' O -* (71 X -^ ~ re to c; 7i -^ I- 71 -w to =r X 71 -*^ re i^» ^— iC^ "^ 71 on O re >e X O 71 le t- — . 7-1 -¥ to ~. — r7 --o X 71 -^ t^ T. -J< -tl ■^ iC »c »c le to to to to to I- l^ I- l-- OC 'JC' -1/L/ '.Xj w* <«• ^^ ^1 '>) t>i ?' >i ^ 71 71 CN CN ^ 74 71 7-1 (71 7^ (7-1 7^4 71 71 71 71 7-4 71 (71 w -f .-K c. --z l- o J^ re X -^ -*( X X i.e -^ t — H ce «e re to to >e re cr re *e re X ri X re tC to re X — 71 1^ 7-1 »e t^ t^ to -t< — ' OO 1^ t— 1 '■Z T. >— I ^ o X le 1^ -* t^ X r: X to 71 X 71 >e t~- X X b- ^H i "M t- ■M '.^ ^ 'e ^ *^ t^ O re to * 71 <.e t- 71 >*< to X i-^ -r t~ C 71 >.e r- ~ 71 -t- to X — . re le x 71 -+< to e 9^-i r^ — ' -f le to ue re c: c re »e '^ to >e O O O -^ X 71 to o -^^ X — < le X — -f i- re X 71 *^ to X :c i.t t~ o C-) 'e I- O 71 -* t^ r. — -- to X — re le t- c: 7i -^ to X ' ^i ^1 ?^ le le "e ?' ?^ ?' 71 71 ?i to ■^ to t^ t^ l~ t- T^^l 7^^4 7-1 71 71 X X X X C- T. 74 71 7-1 71 71 7-4 71 71 7-1 71 C-1 . ,~ -^ t^ X r. o ^ 71 re -f le to t— X ~ O ^ 71 re '^ »e to h- X - 1 ^ Ul: t^ l~ I- X XXX X X X X X X r. ^ z;z: i. LOGARITHMS OF NUMBEES. 395 t^ t- ?r uc o -* -*l cc cc cc cc cc cc ^ 'f IC CO CO t~ t- Ci o — CO cc '^t^ ^"^* ?5 ' ^ O -^*^ cc iM O Oi CO i- CO o -*< cc C4 i-< O Ci CO t^ t s lO ^+1 CO Jfci ^ |c^ I— .— 1 .— r— t C<) C<1 J t— CM CO o -^ 00 ^1 »c <;_; CO CM ;^ !>■ i-c it; IC t^ O -* Ci t^ CC CO -— 1 -*i AC CO iC re 1—^ i^ cc t~ — cc • c^ -rH ::: CO --H cc la t-~ Oi r— cc lO r— Ci ^H cc iC t- Ci '-^ CO IC t^ Ci 1— 1 o C O O f-: 1—1 1—1 ^^ .-1 CI C^l CI CI CO cc cc CO cc CO -+I ^ -n 'f 'Tt^ IC cc CC CC CC CC cc cc -J. cc cc *:• *;• V cc CO CC CC CO CO CO CC CO CO CO CO ^-v iC -* CO ?o O cr. I- »c ct cc I- O -*H ^ l^ C~J ^—i cc cc cc 'T^ —h l~~ cc CI t^ CO >c Ci CCI cc ■^ Ci CO CO OO O CO IC CO IC Ci CO CO CO lO I— 1 Ci 1 c^ cc ^ ^ CO to cc : CO O CO CO o cc CO — CI cc cc -« IC CO IC o CI -f< IC ic IC cc O CO CO t^ O IC -!f ^ CO CO CO CO CO CO CO CO CO IC !M -r ir OO O cc IC I- Ci ^— cc IC t^ Ci 1— cc »c t- c: — cc IC t- Ci 1-1 O C O C --H r-^. r— i-H -^ CI CI CO CI CI cc cc CO CO CO -fH -+ -*< »+< 'tl IC 1 • cc cc cc cc =r- cc cc cc cc CO cc CO CO CO CO cc CO CO CO CO CO CO CO CO '^ r-i o ^ cc co 1— ( "—I r- O 1— -^ C-. Ci cc CO Ci CO -^ »Q CO t^ O IC CO -*< lO >o -ti cc 1—1 O CO CO -Tfi CI c cc l^ 1^ l^ l^ Ci ^ -# 00 CO O t^ CO t- CN CO Ci F— ' o-l 1— O CO' IC 1—1 »c Ci CO -+I IC ic >c cc O t- CO t- 1-1 ta 1—1 (M -* lO cc CO cr. o 1 — ^— CI cc cc cc -^ '^ ~i* ^ ~^ ^ -^ CO CO CO CO lO C<1 -+I CO CO O c-\ -t, t- cr. — cc IC I— Ci ^ CO »C t^ Ci 1— cc IC l^ Ci ^ O O O O' ^ 1-^ 1— 1 ^^ I—' CA Ct CO CI CCI cc cc cc CO cc -f -:ff '^ ■* -* lO cc cc cc cc cc cc cc cc cc cc cc cc cc CO CO CO CO CO CO CO CO CO CO ^5 CO t^ »a ic cs en -Tf t^ CCi t- t^ t- o »c ^+1 CO t- cc lO CO CO CO cc QO CO Ci i^ c: O O O O C. GO' t- CO UC iC -* -+( '^i tC t^ Ci CO CO r-( t +< CO CO JC i-H O CO O O O a t- -t- o IC Ci CI '*« IC lO »C CO — . t^ cc cc CO ■* ' CO O CM cc '^i CO t^ 00 CO c. O r-H 1— < 1-1 CO CO CO CO CC| CO Cq r-l —1 o o -* 1— 1 -H CO CO O C^i '^i CO CO O cc «c b- c: r-i cc IC t- Ci 1-1 CO lO t- Ci — 1 ^ C O O --1 T— ( 1—1 r—i — C^J CI CI C1 CI cc cc cc CO CO •* -Itl -H -^ -*i lO cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc CO CO CO cc CO CO CO cc CO a CO Ci -* ^ "cT CO cc cc CO cc »c OCiCOO»Ot^t^CO lO ^ lO t^ CO o cc IC t^ CO 55 Ci c; C: C-. OOCi-t CO CO CO CO ec CO CO Ci -^ (M cc ^ IC CO t- cc CO Oi CiOOOOOOOCi Ci CO CO CO I— 1 cc IC CO o CO 'rf^ CO co O CI >+ CO CO 1—1 CO IC »^ Ci — cc -+• CO 00 O ! O O O O r^ T—t 1— ( 1— ' ^ CO CI CO CO CO CO cc cc cc cc -^ -+1 ■* -rfl ^ lO ^ CO cc cc cc cc cc cc cc cc cc cc T- cc cc cc CO cc cc cc CO CO CO CO CO ; — ' O S^l t^ t^ -+1 t- CO t^ t^ t- Ci ~n CO IC CO t- CO »:^ iC CO '-* .-1 CJ CO -*< CO 1— ' cc IC t^ oo c^ O i-< cq cc o l^ Ci GO IC Ci '^l o l^ IC -H 'f IC sO c-i CO cq IC r- CO CO CO CO cc Ci >* 00 r-H ^+1 IC IC IC -^ 1— CO -+I Ci cc -• C^l '+' >C t^ CO n o ^ CI cc -H ^ »c IC CO CO CO CO CO CO CO lO IC -rt< •* 1—1 ^^ cc »c t— c; ,^^ -* CO CO O CO ^ CO CO O CI 'f CO CO O CO -^ CO CO O o o o o o ,_ — CI CI CI CI CO cc cc cc cc cc -+• >+! -+" -i' ^ iO . T- cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc CO cc cc CO CO CO i '==' -^ -* O CO '^~ c^ cc cc cc cc IC o CO CO IC GO t^ IC 1— 1 Ir- cc O Ci O 1 o •O — . CO c cc ^ o cc CO c to t— CO C-. o ^ CO cc -^ >C CO t- CO Ci O 1— ' cq cc "* d •^ "^ o o o o ^ o o o o , — . — ■ w-: ^-1 — • ,—1 ,—1 CI CI cq CO CO ^ C-l 'i "M — X r~ r^ i^ i^ I- to LC -T L- l^ l^ l^ - ClC^OiCiOOCOCOCCCOCO X 00 cc CO cc s<) o ro GC' o Ci i^ lO c^< O — rl -ti — CM CO Ci »c CO CO ic ^^ le Ci re j?o-*i-*icr;rco:r-ore I- o -^ to o to >e t^ .— ( CO CO .-^ to -Tt< rHce-^'+icesMOtoeeco e.f!»00t0t00t0t0l^ i^ t- l^ t^ CO CO CO cc CO CO Ci Ci Ci CTi Ci coeoccccccrccojcecre re ce re ce ce « cc cc ce cc CC CC CC CC CO 1 Cti^l^OOoO'^OiC— t ^' t- ri ce ?i le ce t- t- -rt^ Ci' -M ce^" -t< '" rereretooorere-^30 re -- -^ ^ Ci to to cc re 1-1 C>1 !M -rt* «e o o re I— 1 CO -^ ~. -rt- CC — ce -H >c J-C -^ re — ^ cc -^ O le Ci GO i^toue'rhre'Mi— ir; ooto le re ei o co to -r ei o CO LC cc ^ 00 Le ceiai--c^r-icc»ctocoo 'M -h to cc- Ci — re Lc i^ -c c^ ei -* o i- cc »0 lO o o to to to to to I— t^ t~ I— t^ 1^ CO CC' CO cr cc Ci O Ci ^" cerocorercrererecere re re re re re re re ce ce iOiaocvjo«oo-*<»— 1-1 to ce e-T ':f o — it-COOO'M— lOoO li^ 'i— (rHCCiOr-Ht-tOtCOOCC C5 Qo c; -M CO to to c:i le -* LC Ci to LC CO M»Ocr:tC«C'*iC0 l^ Ci ^cC'ftoccO'— ceo 1^ lO »o o »o to to to to to I- L— I- 1- l^ 1- 00 CO CO CC' CO cCi Ci ^ ^^ Ci ccrcceccrecerecerere re re re ce re cc cc cc cc ce ce CC ce CC »5!M'*l'Mt>-CjOOCiCO "^ r-l to -* o — ( .— t^ O o iC Ci C'l -+i IC trcO'-Htoi^io-^cecctM o — 1 re CO >e »0 t^ I— Ci cz> T—H Z^ f^ t^ o -^itSOOOOOOl^iCCMCO'^ c: re to CO o •— " ^-^ "-^ Ci l^^ LC — »^ iM t^ JO 1 — O C: 00 I- O O -+1 'M -^ Ci CO to -* re — Ci I f CM O CO Lc cc o iC cc >o to 6c O c ^ -h CO O (M ce CC ce CI o t H CTi LC Ci -^noot^^iC'+'re'M— !C; CO '^ "-f* 'VJ ■— ' cr. I- LC re — oc to ce " 00 "* ^-+iOOOOiM-r to CO Ci M ce »c to lO o o le. tc ^ to «o wi «o t^ l^ l~- t^ !- cc cT) c/D cc cc cc Ci Ci Ci Ci «cc«cerececececoce re ce CC ce ce re ce ce ce CO CC CO CO CC cc I'Mto^Ci^CitO'Mi^re a l^ t^ o t^ CO cc -^ '-' le IC -^ — ' t^ ce — <-+--+|-^l^ 72 to d e-i -t^ >e >e »e -+i cm cr. to CM i^ ei cc i-^ to »Q O -H C>! Ci i^ c^t 'N -+i to CO o e^i -)- to i^ r; .-^ ce le t- CO O -M -* to 1^ cr. ^ Ci -*> to o >ie o »o to to to to to to t^ I- 1^ i^ 1- CO CC CC CO CC cc re ce f: cc cr. Ci Ci Ci Ci recccecerceecerececece ce CO ce cc ce cc cc re ce *"'•**•' l^«oceoocCGO'Mir;ci "ce 2 t^ CO ei O re i-i to b- lO O »C CO o CO 'M CO lo Tti ue i^ c-i 00 to t-1 re 1 — t' re o O CO CO — GO 1 -O Ci O — — ' O CO -O •»-+< o oD o c^i re o t^ c^ -*i -M — Ci l^ re o to CO le ce ■— ( Ci i^ »e ei o CO to CM O -M -* le 1^ Ci — ce -*- ;>f5J0>0»0t0t0t0t0t0O t^ (^ i^ 1- t^ CO CO CO CO CO CO Ci Ci Ci Ci cpceMcerocecececerc M cc ce sc CO ce cc cc cc cc ce CC CC CO CO >o«o— •rC'^toci-^ce-^ to o to -H to c-i ei CO O CO ce to r-- CO t^ 'O o t^ »e >o to o le 'M '— 1 "M to -- Ci Ci (M \ — H »e t^ ce r- c>i o cc «- O -- 1 -H ?> I- -ti O O cr. cc "O i^ Ci Ci Ci CO to -*4 -H t^ Cl t^ -^ ; re re ei -H o Ci i^ to lo -+i e^t ""^ '—' i-» j*^ ce ^ Ci 1- ic cc 1-H 00 to CO ^^ it^j-ftocoo— ice»ei^c^ re -H to CO O CM CC le 1^ Ci — 1 CM '^l to o »e >e le to to to to to to 1^ 1^ 1^ 1^ 1^ cc cc cc cc cc CO C^ Ci Ci Ci Icocececerecererecece ce ce re cc re cc re re ce cc cc ce cc cc CO >0-tlCiC0>ftC0OOCiC5 Ci O CO O Ci -C^l O -+< CO QO ,-( i-l O t- CO 'M CO >o -H le t^ ri CO »o "O 1^ (M 00 l^ t^ ^ t^ »C to Ci to >c t^ -^ Ci 00 o ei re re ei — CO »e -^ to — -:t< t-- c; — ' — — 1 O cc to ce c^ ic Ci O — 1 — o C5 cc t- to ~H re e^i o Ci 1- le ce ei o CO to ce .- Ci to -*i •-^ '■... e^i -ti to t- r; — re >e i- r; (M -H to CO c; -M re l^ O t^ '^ O ■X t^ O -^ t^ — i^ OC 7^1 --C C " X — ' ~ ~ ~r y: T^i S " ^J >1 — ' O O Ci CO pc J-. -^ t^ ).2 -^ -r r: r- rM — — • C: c: r: tc X .'^ 1^ n CO i^ CO c^ — ^t — cc — ' u; ^r -f c ?t — I— CO cri O :m i~ C: t^ O X CO p2 C; o -Ti •— ' ">> w -f o o s^ "M X a; >i o "M ::?: r: re iM -f >£ 7-^ ~. ^ ^ ^ ^ ^ '•— ^ -r*^ ~ i-^ ~^ i:^ i^ T^ c^. i^ ^i ^ -^ '^ li — i^ 1.1 1^ ~ O »0 c C^l -* ^S I— ri ^ ! ■>1 C>1 'M C^i re cc re ce n re re'+''^-^-*'^-f-+'^-*'-f-^-*'-H-^|-f-+|-+|-rl-^'*-^-^T1-:r'T^ ' re (M -M i-e Ci i^ c: -f -^ 31 o t ^ — c: tr o -* i^ c: oe o — x i^ -+ire — x -e — i- re o — ei x -t- o o r^i x t- ri o ei -^r' — 1^ ~ — r^i -*• w i^ ~. — • ei -t« :r »-- ri o "M -j< ue t~ x ciooooo^' < ejei>iCMM>irererererere re-rH^^-*-#^-+i-*-*-^-*^-Th'-:t<-*-r"-j<-ti-ti-*-t>-rri-:ri-^-r I- o -^ o CO c: -H 7^1 >e -* t^ t-- re tr i^ lii CM X re X re X le r: re C; re vr i-^ x c^ x i^ 3 re o i-- >i x ^"i to r; ea -f- i^ tc tc 5 i- -^' u; X le ei Ci — re o I — " — cc -t- — i^ — 1< O ^ re crs le ^^ t- re —. le X o ei re Le 1^ r. c: e^i -f te i^ ~ o e-i '^^ ue i- x o ej r- i e -.r x n o O o o O O — — -- — — — ei ei ei 7-1 r^i ei re rt re re re re ,re-H-TH-:^':*<^-f-H-f^«-+l-t<-t<-*l-ri-^-+<^-+i-*'*-h-f , re re '^ c2 ^^ c; o o le o O o CO GO le ~ '"M- -f »e to i^ QO I— 1 e — -rfHiecscct^oxooe-JO^-oreie — o-H — 7>^i^oc-. I--X — -* if re c; X le e^ — -3 ?Ji c: -3 re ^ t:: ?i r^ ie ^ i^ re O -3 ej T-I re X o ei re 'e i- x o c-j re »o i^ co o ei re o i~ x o >a re '~. -^ x ■CiOOOOOO — — >— 1^^— <^-Me^s-e e-i -^i re Ci e-i e-i M o t — h o x tr " x O M i~ le t^ c: X ~ -f CI -«< ~. X o i^ i^ r; c; — tr c. I- r: le to r; — re -H "C -t" re e-4 o t^ re ~. le cr. re I-- ~ ei re — " -* — re m -« — n vo re c i^ " — X -^ — 1 O t- r7 r; to -M X -t< C --o 71 00 o— re ie I- X o 7-1 re >e i-- X o eq re >e tr X t:^ '-I r7 'e tr X c; o O C c: O O ' 1 — , ,— , T-i e-1 ei 7-j ej ei re re re re re re re--*'-^-^-T-)-'-*-r'-^^'T''+'-*-^-^-+|-+i-r-r-r--r-r'-r-+-+' ceto— !t-jei^rHCe-JOree-it-.rioo»er-ieoo.— iret-^-i^iS i~ C5 »e re »e o ci r- to ie i^ re Tsj o ei re X) ;c X o > 71 c I- -^ri -— X ic r) ~. --7 re oj to ei ~ le M X -> 2 "-^ ^' ^ ~^ ^ c; o O O o o O — I— ' i^ T-- -— — eg e^ ei 7-i ei 7-1 re re re re re re re-*'*-*i-*^-*-t"-^-'i<^-*'*-4'-*<-+!-*-p-*i-j<-ri'+c-t--^^ i^ i^ X -^ i^ >e -.^ — ' O -*■ -^^ c; O X -r I— a: c: X t- to >e le t^ 3 retoeiej--jc:7^. ejxi~otor--c;co~-Hretore-i-~x — r; — — X le ej o i~ -? ^ i~^ ■? -^ X ^ ^ i^ ~ o to 51 0} lie — 1% f7 H i ci ct o o o o o — ^ ^ -^ ^ -^ tI ei ei ei ?i 7-1 7^ re re re re re rer7-^'*-f— rH-t'-rr-*i-TH-)"-:t"-t^-+-^-*-T'-+i-i<-f-*'-f-l--+-r. O 1- le «e t^ ei o '-^ 1- CO re le re t- 0-. 3^ to re CO re X re — , — — > w ce o o re o o re c: o re o — »e re ».e r-( r- -*■ ei re m X 7-J O -+■ 1- oeire-^-fre — r;t- — ^oue o-;hx — re>eto to ^ 0^ to -^ — X le ei o; to 7-1 Ci to re ~ to 7-i x le — i^ re ^ ie — ' 1^ f^ l^~ c* ^— r7 '~r to X ot — re ^'** to x c^ — ■ re -"^ to cc * -^ 7-1 -^ to t- Ot C; o O O o o o — — — — '— — 71 ej ei 7-1 ej 7<) re re re '?*'?'i*''?i'^'?*'?^i**^y''?'"?'":**'?''?'T'T*'?''? -* T ^ 'T ,• o >— f ei re -^ Le to I— X o: o "^^ >i re -J >e to i;~ co qj o — ei re -f i _- ■^ 71 71 ei 71 71 ei 7-"i 71 7*1 7*1 ei ri tI tI 7! t") ei tm ei ?i ei ei ei 7i ei ^ 398 LOGARITHMS OF NUMBERS. ^^ l^ ^■" >o r^ _», y: CO l— C-l «i Ol CC _ JO ^ »o - lO --. I^M »o """* 10 "^ ""*"* d l^ I- ^ '■^ lO "-^ ^ CO CO -M >1 i—i *— 1 '^ ^ * X X X 1- l^ •^ ^ to *-~ en 2 i5 lO lO 1— 1 lO IC lO lO lO lO »o 1— ( — to to to to ^ ^ ^ ^ -Tl "X I— < ^^ I— 1 ^^ " t— (^ lO "o^ "cT T^ , — "^ -f 7-1 To T^ to "tT T^ I— ( •^ Ic X O C-1 iC — i to r— ( CO CO CO C^l ,~-« >0 s o OJ -f -^ .— ' l- CO CO -j- c» d -t^ LO o o to CC »c (M CO -^ (^ »o c; * Ci ' CO -T^ -* -* •r}1 CO ^-4 cr. l- rt« I— I'- ^^ t- CO CO ^ lO O t^ *— V^ M i^ 71 l^ 7<1 t^ 71 l^ cC CC <;^ to c^ Ci ^rf O I- t^ CV ■M O O) c% o l^ X fl to -^ ^^ !-• X CO -^ ■^ ^^ f ■-■. -J^ c* ■r*l I- •^ ■^ ^ l^ * c^ c^ c» X t- 'O 71 Ci CC 71 X CO S ^« t-^ ?1 oc CO Ci -f ^ to o iO ^ »o :3 to o to o lO Ci -f * X QO o (M CO to CO ~ — CO '*■ -^ o -M CO to CC X Ci o 7-1 CO iO X f* -^ -* ^^ ^-u ■«-^ Tt< -* iO i-O to to to to ^^ ^^ CC CC CC CC CC l- t- l^ l^ t^ 1** 'f' "f "^ -4< T "T '^l ^ "^ ^+1 'T '^ -* -^ T -+( _^ -:*l -* -+I _^ ■ '' -*! "^ ~ CO (M c^ O lO 'rf CO 00 "co to 7-1 t- Ci CO »-^ o -rH 71 "/^ ^ CO to X CC Tsl CC CO O |;~« t*** r^l ^^ L~* ■tx. t-^ JO Ci '*! CO ->5 UO GO r-< CO rf CO 71 o X to X I- 7-i CO t>. •^ o IS cr: ?1 t^ CO l^* t- O (M CO •»*i -^ -* CO •^ >o C-1 lO -+I to 1.0 >o Xr- to X to -3 CC ^ -* ^ I- •S r^ 71 I- i-^ to l^ r ^!J4 ^^ ^^ *^ ^^ ^P ■^ -f ^ r« -* -ri •TtH f z. -# ^ "^ ■5* -* "?* -* "7* -^ '^ (M o CO — . t^ '^ I— ( Ci X 1 Ci !M a ^^ t^ t^ -M Ol -^ CO tC X t^ X to 1 — ■^ CC 71 — H -^ 7-1 Ci »— ' t^ >^ CO X l^ o CO 'O I- X c* o^ ^i X t^ to CO 'O l^ CO X -t" to 01 CO .^4. lo (^ ^ I- 71 I- 71 I^ 'M t' 71 l^ 71 L- 7-1 1 ^ ^^ CC c to CC o CO ^ CO Ci ^u O^! -* to t- X o r— i CO -+< CC I- C^ o G-1 CO to CC •rM -** "^l ^ ^ lO O to to to to •tc CC CC CC CC CC CC I- I^ l— I- l^ -^ ■^ 'f' •^ -* ^ -+ •^ ^ •^ ^ ^ '^l '^ '^ "^ _^ •^ -^ -# T ■rH -^ "T** •^ ~ «c ^_^ o "M CO CTi -Tf lO ^^ CO '- '-C X CC •-0 ^ ^ ,^. — CO to t- ^ CO X lO Ol :» CO CO C-1 ->c — ^ l^ »::: X r^ 3D to X to I— •^ I- ■^ CC c^l ;* "f-i CO t^ lO CO O 71 CO -:** -* -+ r^ 7-1 o X lO 7-1 X '^ CI ic - — ■M cc ^ -:f< o iO ^-t tc F-^ tc ^i^ ^ ^- CC o to o — r * CO t.O >-v ^^ ^ l^ Ci ■M •? 5 l- X :^ »— « r^ ^ cc I- c* o 71 CO '^ CC ■^ -f< «jii -^ ■^ ^^ -+< iO »o i-O to LO to ^ o f_^ z^ CC CC CC i^ l- L~ t^ I- -* '?' f ■rf -^ ^ -^ '?'?'?' "* ■^ 'f 1** '?' ^ 'f -* -* -* 1*^ '^ ^ "? ^ ^ lO 7^1 "# "^ -^ t^ X -o o '^ CO CO to ^ to -* — H X to (71 O X X ^ 35 35 •*44 * -^ CO to X tc * l~ ^ l^ ^ CO ^-• X ^^ •rj^ '^. «.;; Z^ ^ ?1 I- S ^ ?: ri tn '^ X X g -p 71 X tlo ^ Ir' -^ I- X -rf T-l to 71 rj» I— 1 ■rT< 71 CO ^ •^ '^ •^M •»M -*l LO *o to to to lO to cC CC c^ CC t^ CC l- t- l^ t- t^ '^ ■^ "T** ■rf •^ -i* -# ^ -JH "T ~T -Jl r^l -*l -+I "^ "?* T' "? -t+ T -^ f -*i "7" ^1 'ZT "^ :m "^ ~ZJ t^ "-T ".T C-. CO To" TZ" IF '^l "^ >,0 X _, lo "^ t^ "cT ^ 1^ ^ — ; -*^ CO ;^ ^ ^ — - ^M lO -:h X 1~ ^ X 71 1^ CO ^^ ^ CO CC '"• :^ *^ ^— !_- 7-1 *^ _ -^i t"* ^ 71 cO -# ■^ -+• -t- 71 — c^ CC CO d to o -^ ■X. /-» lO ^^ !_- ■M Lr %; CO ■"0 X r^ X CO X CO I- 71 I— — CC r— CO — ?J :^ l^ C- M X c^ —. 71 ~^ t.O I- :c O >— 1 CO -f CC rc -^ -.^ v-ri — . •-— < — ^ i *■ J-0 to lO I'*' to to ^^ CC CC CC CC ''•^ i^ t^ l^ l- t^ '^ '*' ^ '^ ■^ "T -+ "? ^ "^ -+I -;>< -* ^ "T* -^ -^ -* ■* rh 'f -* "^ -^ t^ (M — 1 «* I— I CO o 0<1 r-(liOO'#OCO^-* 71 O t- '# ,-, X CC CO X C^ i^ »o CO o I- CO >* Ci C5 •^ -^ X t^ r— ^^ -H o» CC to X t- •— • -^ c^i * T.^ ^- lO Ci 71 ■* ^ CC Ci Ci CI c% X CC •^ f— < X *** o CC ?^1 Cl r-^ ^ * -^ o UO .^ ^C ^^ w ^-< CC — « CC — ^ CC >— • i-O o t.0 c^ 7-1 * •M ■^M ih I- o ■M CO lO ^ X * r-H 7-1 '^ uO l~ X O f— " CO *^ to CO -f -+■ ■^ -^ -* •^ "O to to to to >o tc CC tc cC CC CC l~ l^ I- l- I- -:t< "f •f '+' 'T 'f -* -? '^ -t< -^ •i* ^ ■^ "* -+ ■^ -f '^ -* •^ ^ "?* ~T '?' "^ 'fl lO "^ CO , CO , ^ 71 71 ^ 71 CO _^ t- 71 CC X C: 7-1 -f 1^ o -* o 1^5 tc .— . CO CO c^ r- CO X I- 71 ^— lO t.O C^ X 71 71 CC CO ^\ *-0 ^.^ *<*■ '^ cO l^_ —^ "A HH -^ — 4> — *- CO — ^ ^ CC CO o CC •-*' -^ ^ o tc ■M 1-^ CO CO ■^ ^ — H ci 'O o ^ 1.0 3 to o -* c^ •r^ * CO X >— ' ^ ^1 -f o I- CO ,'•"■ «_4 -^ »*( w" X w^ f— < 71 -f lO 1— X Ci ^— >1 ^"t* 10 CO -t* -*■ •-** '+< -* »0 lO lO lO lO lO >o tc ^ CC CC cC CC CC 1^ I'- 1- l^ -^ '^ ^ 4- -t< ^ -f ■^ -t4< -^ -t' "7^ -:*< '?' -+< ■^ "j^ •if -r -:f -:t< T "?* -* •>*1 1^ _^ CO CO -M O To" 1— 1 -H ^ -^ "^ c^ lO X O "^ 'cT *^ "co" I— "^ CO CN ■M Ci -M CO • ^ .-** CO ^ * ^H 01 t— X c^ 7-1 1^ l^ 7-1 '— < CC CC ^^ o r^ --M — ^ >o ^ Cw X — -f lO X ^ C5 c^ X CC -^ •M wi to I— 1 I— ■^ ^ --* .* 1 — Ol 1 — CO rf: ^ X CO X r^ -y: CO X CO X 71 1^ o CC X * -^ 71 'f' uo C^ — f -t" ii •-*< -H -t< *^ 1^ I'o to 4 1- CO CO X 7) CO 7» X X 71 71 X 71 X 7-1 X 71 X 71 71 71 71 71 71 71 71 71 71 71 d LOGARITHMS OF NUMBERS. 399 • 1~ ^ .- — -^ •M I- ta X cc X -+I ^ .- — cc -^1 I- -f -n it — 1 I- CO X ! . • i ~ n^ ■^ re cc :m :^^ — i ^- o o ^ Ci ^ en X I- t- cc cc it it it -r^ -:f( ~: 1 c: 1 ~ z :!; -f -* -+' -H ■^ "*! *^ ;»^ ^^ ,*.-. cc cc cc cc /N-« r^> CO CO cc cc cc M -^ -^ -' '^ '-' — C^J — CO t^ -* o ~. 1- — cc t^ -* cc cc I- cc t-j It it CM l^ O O Ci t^ C^l cr. O O t^ ci cc ^ t 1 o it it 1— ( CM en c^.| o ^ -* C^ — 1 X O C5 7q «r w^ CO It t'^ c^ f^' — ' ■>3 ri _ ri o c^ I- -* CM c; i-t 1— H cc C^ cc — -:*< r*» "^ CO J^J t-- »^ It c^ -*■ X CM cc o -^ t^ — It C: CM cc ^ cc r^ O -^ t^ _ -• , CO ^ ^« ^1 -*■ It tc X * — - tj -tl it cc X Ci cr; CM cc i'*^ cc t^ Ci cr> — 1 : '-'* I- t^ CO CO X X X X X n c^ Ci Ci c^ Ci Ci o o (^ o o o o — < ^ I 1 ^ ^ ^ ~i^ -^ -t< ^. T "f T^ 1** '+' '* "T '*':*< iJt i-t i-t it ip it it >p Ip 1 1 CO Ol ~ CO o O -f cc cc ^ O .— ( t— Ci 1- — 'M ~ CO it -* O iO X o 1 r- o o I- o I- It X t— .—4 ^^ CC cc -M -^ CM it CC r/j X -* cc CO CO cc 1 t^ It CO — cc *^ cc t- X X 1^ ^^ it CC — X it —I 1^ cc cc CO t^ — < 1 -M 1^ ^.< O O '-r' X CM cc O ■r+^ X >» CC O ~^ t^ — * *-c X ■CM it Ci (>■» CO ., 1 T) . — t '^^ -*! It tc X Ci — ! M CC it cc X ~ O "M cc cc 1- X O — I p iit ip »p lO -+ ~. •rM CO >t -* X tt cc ^M CC X — it I- CC 1— ' C4 ^ cc O -I o t- cc 1 1^ fM rj< CM It cc r^ CC — ' 1— 1 cc t-- ■Tt" CC -:>* X I- -N C>J Ci — • Ci ■>i .CM *v-» r- -^i I- c^ •-^ a-i cc -*! -+I cc ->< — H Ci I- -^f — or. Ci It Ci -+" 00 ' t>. ^^ l'^ O -*' CO CM tr- '-^ it! C-. cc l~ ^-* it X CM CC O cc 1^ O -*< t ^ '^i ; t^ T"^ '^- ^— . t-l Ct tt CC X ^* ^^ CM cc o CC l^ C. O CM ''^ •rrl cc 1- X c -- ' 1^ 1^ CO zc CO X X X X CI Ci c; c^ CI C. C5 o o o o r "? -r ^ *!:*" -^ -f ^ -*• <* '^^ T" -r -*l ti '^ it O it it ip "t it iO lO ^ cc c-^ CO 0-. cc -M cc Ci o lO -^ o — -1 t— O O it X X it O '+' it iO — _, ^^ -T) •—'■ Ci CC :M t^ Ut 1— 1 ^^ r- •■» cc X 1 - o o it cc cc cc >:*< X X ^ X -1- 7 Ct 1^ IC l- r^ It X cc X o M it a^ t- it cc r-( X 1- .— 1-t X •>! CZ^ cc — I cc O "^ CS cc 1^ ;^ -M Ct >t cc I— W- ■-—' C>4 cc "^ cc 1- Ci O — 1 cc >* it i~ X O ^ r— 1^ CO CO CO X X X X c: ^i Wi c; Ci Ci Ci o o o f^ O O O — 1 r-l 1** •^ ^ -tH ^ ^ -* y 'r' '^^ "^l ■^ "? -^ ^+1 -* jp »o 'P iO ip ip ip ip lO — 1^ ^ -* t- ct >1 >t ^ -M h- CC , <— » it cc ^ «^ t— -p Ci o o t^ cc t~ J -■ CO l~ .-H O ,^^ X r**. 1— O Ci cc cc ! >t X — cc X It CC cc it CO c^i t^l C5 t^ -t- CC CI cc l^ o t- -«+. ■>i X ;^^ 1^ 1 X — It X C-~l U5 CM CC o 1^ ^ ■>i cc It cc 1^ c; O * "T "T -*l -+' -* "^ it ir; it it it O it It it -^ (M on ;r; CO o X — '^ cc t— f- o o it I— ».t C5 _ Ci lO c; o o GO Ci Cit -H It --ri C-. X cc -* o ^-* X — ( cr. ?i — cc cc CO -* ■M >t it O O — ^ '+' CO — 1 M- cc X o — ' 5 ^~i i^ O ^ X CM 1^ -H It Ci cc t- C -+I X CM cc Ci cc cc CD cc t- O -^ 1^ C5 o CM CC "*! cc 1^ C-.' o t—t cc -# cc I- X O -^ CM '^ It 1-- X CI .^ i^ t- 00 X CO X X ■X X CT. Ci CiOiO CiCiOOOOOOOO'-i rtH -rh ■* ^^^^^^^^^^ T" '* ■* »o »o »o »n »o o »t «o «c r^ 1—1 ^ 'f' O X o cc O Ci X _4 o -^ -*■ O M '- CO CJ Ci t- CO t-- C3 1 1." '•' ^o CM 7\ ri -+■ ■M I- bl X r^ CM c ~*< -^ ~^ C: -+' C>J C-. t- ;:X X cc o c it 0 1- X O — ' ■M ■^ It cc xs cri o 1^ 1- CT/ X X X X X X r; Ci * Ci Ci Ci C-. o o o O O O O O — ' 1 1 T 'i' -* -* -^ -t- "* ■^ '^. -* ■* '+' '?' ■^ -* 'St* it »t JO it »t iO It it >rt 1 t^ O O (M (M >0 -H :^'; Ol —^ CO cc — cc ■_■ ■>i —' it it —I CO ' >c c^i »t X CD' n ,«^ *^ "-^^ cc cc -M CD X cc cc ^ cc Ol t- CM J^ 1 CN rfi 'T, -M V^ —^■ >^ o -rfl X -M » c^ -*< X -M it Ci cc t- o -+-»-. ^ X 'CM 1^ (T) -~^ ct '^ cc 1^ X o — cc "^ it t- X Ci r-l CM •*< it cc X Ci CD 1 t- t^ CO X X X X X X — * c; c^ Ci Ci Ci Ci O o ^ o o o o — ^ •<** 'iH -* ■* tH -~t< '+I -* -* rt* -^ -^ rH -* -* tH jp iC "P ip »p ip Jt »p 1 ^ or) »^ c. -* ■M cc CO t- O X ^ X _ C^ -^ it 'M b- X l— cc t^ o o ^ <^ O It tc n cc — - o cc CC J— X '♦I «c CO cc it O CD 1- CTi '-^ -^ It X — 1-^ Ci c^ ■',^ Cj: * X cc -^ 3^1 Ci cc -M y: -*' c. CO b- 1— ( M r- »^ It o -+ X ■M cc O *C w^ C>J cc C> ^ CC <-^ It Ci CM cc Ci cc cc .—1 t^ 00 ^ — "^ -+ Jt r^ X O •—1 ^\ -+< it t- X cr> — 1 CM cc it cc l^ Ci O l^ t- CO X X X X X X c~. cr- OS C: c^ (Tj Ci Ci O c c o cr o — ^ 'f -5H "* "* "i^ 1*" "T "f 'f "* ^ ^ rf r^t '^ -* it it iC it iJt it It it ^ O -. cc cc X '=*' •4^ «>• tt h- "^ cc CC CC CC — 1 CC ^ t^ cr O Ci it o , — 1 to ^ •M cc X O X ^^ ^ ■-*« -+ cr. o t- Ci t— o o it It o^ ic ■>l ^ '•^ ■^ I- a: 'M CC IC It CC cc it -M — X it 'M Ci it CD it O -^ ^ It ,:;*; -+ X c> 1 r^ ^^ It C-. cc »- ^- it ■Ci cc cc o •r*- t^ — it X O-l it o t~- '/-/ ^ — CM -# >c 1— X a . — 1 CM -^ it cc X Ci — CM cc it cc I- Ci O 1— l^ CT; X X X X X X X cr. c^ C: w- Ci Ci Ci O o o O O O O -H -*i -+ •rti ^ -+I "7^ "* '?' "?* ^ ■^ ^ '+ 'T -* '^ '^ IP it it JO »0 JO iO lO ^ J tq ~* ,4.- It cc r— X c; ^ _, (M cc '*' it cc t~ X Ci O — 1 oq cc ■* 1 _• >^ ~^ 5: ^ ? S ?c cc cc cc cc ^. S cc cc cc cc cc cc cc CM -M CM -M -M 1 ^ cc CO cc cc cc ^ 400 LOGARITHMS OF NUMBERS. , o C — cc X -f O — 7-j Ci i~ — ' -o cc ri ic -M X -^ O t^ — ^' ~ tr fe I r^ cc -M n — — — ' O O r: C-. r: cc -X r^ t- r- -^ tc -^ l- i~ ir: — re ic cc rt r^ re rt re cc c^i c^:/:'0^?ire>e--cr^5:c: — e^fc ~ r^ — — ^ — -^ n -M c^i 'M M -M "M re ?e ce ce re ce :e re -+ -* -+ -rt< o o o in le o »e >e o o lO ic JO o »o lO iO o ic le le o lo le ir; i :^oc^oct^o-Krere-^t^— '^-^-rftcr— cor;reo— ^tciecc ! t— ei CO — ' o >e " re tc >e o ei r; re re ~ "M o >e 1— le ~. ~ tr c. lO c/j o re le w t^ 00 cC' CO 00 L^ i.e -:f 'm c; t — ?< o w ei i— c-i i ■ a; 'M o n M >e CO — ' -^ r-- o re o Ci n ^ i— o re >e co c re o v c^ -*i le "-r^ CO c: o "M re -+' :c i^ Oj c. — ' 'M re o o t-- CO o — r^i re I— i^Hr-i— (— I— i.T^e^eie^>s>iiM'M!Mrerererecerere-+<-+-:f-:t< |0»OU3»0»0»ei»OlGllO»0»QlOOlS»QiO»QiO»QO»aOie»CO Ico— '-+'i^c;e^i»ncoe-it^re — —''Mie — ^o— 'vere-fcot^Citsi 1 t^ c: ^r cj ** "*; "^ '■-'' ^ '-"^ ■;:; -^ "" *-" !i? "^^ ' - "* ^ "- i: ^ 'i; ""' i5 I t>. ' OC' — -^ T-^ -H ^ t- O re tS Ci ei o CO o re -i c^ -H -+i -^ ri ri -*i "3 I ^ -M -# Le to CO c^ o ^1 re -+i o t- oo' ci — ei re -^ --c t- co — — ri re I ^ -- — — I ^^ e^i .->a "! e^j re re re re re re re re -^ -*• -^ I I in le 'e le o o o »o o »o »o o »c »e le >-e i.e >-e ie 'e le le >e >c 'e ! -^i ts re t- r^ e<) -* ei ^ o 'M -* ■>! t~ t — t' r^ t^ re le ce co c. i^ — ' CO ■ f tr 00 o — ei ei ei ei -H o GO '^ -H — I CO le 1— I i^ -m i^ ei i- t^ o re --C r: re tc cr. e^i le co ^ -f — cr. ■M >e i^ o re le co o re le •M-fieot-ciO— 're-Hiei^coc^O'Mre-^:ci-cor: — ■Mre ^^ ^^ I— I — < 1— ( -^ (M e-i -M c-1 ea e-i c>i e^ re ce ce re ce re re re •* -^ -r o»ciooio»oio»o»c»oiom»c»cic>e»ooo»oio»n»cic>o o "M re -*^ -^ »e »e to co -^ m — ' cC' i- co — ' i^ " t- ei --j ei k cc ei ! — i re — ' "•"• »e >— I re — I >e ^c ej le re CO C". t^ C O — ~ CO re -* •M t - I i.e CO — I ce m t-- CO Ci c; ci Ci cc L~ 'e re — C". — ej CO — o 'e o -*• ic i »e ao ei le CO ^^ -t' t~ O re cc c: ei le CO — re •-C r; ^-< -r i- ~ ei -:r I e^> re >e to t— c^ o — ' re -* »n to CO c; o e-i re -r' le t^ cO' r: c; ei re ! — . — — ^- — 1 — I e>i iM ei iM ei •>! M ei re re re re re re re re -^ -*• — ' o o Le o o ue lue le 'e 'e le le ic rj t- — ^ O m o w re — o — ■* CO tD le CO ce — re c: CO ^ c; t- O O) re re O "M — ' >e o re tr o O — ^ Ci re re o re -M t- c. CO ei — I i~ t^ o "M -^ le 'o tc o ?c >e -^1 re — I CO !C re o «c ei L- ei t- e>i -f 1^ O -+< t^ o re ^ ~ (M >n cc — ' -^ t~ c; ei le CO o re le CO' c re e^i re m tc i^ c". o ^^ c-i -+i le ^c CO ~ o 1— ' re -+• i.e i~ X' c^ o ei re _ _ ^^ ^- ,_i ,_! s-i ei cieioto»oio»c»o«!eioioiCK;oiciomu; 1 _ — o n t- cr -# re re -^ CO cr. re o c: c: ce Ci CO o «c »-e CO »e '-c ! -Tt- t~ ^ O -- X -^ O »e --r re ^ :o ei re — ' ?o -o re r- --0 ei -f re X I CO — -fj 1^ c~- o ei re re re re ei — o X CO re O t^ re n le ~ »e C". ei CO r: ei le c. ei le X — -*■ i^ o re »e CO — ' -f to c; ^ i~ c~. — eice-+-^t~xo— 'ej-^ie'-oxo^o— 'ce-*'>etoxc:0-^re — ^- _i _i ,— i ,— I rM ei e-T e'l -M ei ei ei re re re re re re re re -*' -i- -^ I i~ ,,~ ,^ ,~ ,^ if^ ,-; ,r; ,^ 1^ ,-^ 1^ le >e 'e te in le in »e «-e "n >e «e >e '-^ -^ ^ '^1 — ; t-~ r^ :ri le CO -f' t^ t^ X re c;: o re o X -^ t- X 'M *,. _^ TT. :^^ :y^ ^- , , ^„ ■^.p, — T- ^^^ ».— ^-^ v^ ^_ - • '^^ -^ -•» ^- >-»- • ' ■- ^ ^^^ -• « /^ _+, r<- ^ ^. 1^ (^ ^. )'; '^ -jj t^ i^ ce »e -* ~ o t-- o o t^ c: X -^ le X — ' -^ >^ I- "■- :r. o o c ~. CO t- le re o X -+I 1-H ^- ei i^ e-i t— ' -M , ^ :r — -^ I- o re I- C re >e X -^ -ti i^ O CM le X o re >e X c; n ei re -+ CO i~ X o — ei -^i )e CO t^ o: o — re -*> >e cr X C-. C — re ^ _j ,— 1 ^- -J eq e'T ei ei e-i e-i ei c>T re re re re ce re re re -^ -^ -r o le >e lo lO »o »o in m m >n m in o m m »n >n >n »n in »n m m »n I C X in "M ci in r: r/j 'Z: ei -^ m co t- t-- i^ co m -+< ■T'I o x m ei x -^f o m c -+• >-< 1 O re CO C: re CO o; "M m X — -r< r^ C re CO X -^ '*' CO cr. CI -^ I- ~. — ; e-i re -+' CO I- X 0-. — ' ■>! re m CO I- c^ o --^ c^i -t' m CO t^ c^ c: — ei — _< — — .-J ^-^ ^^ oi (M c>i ea e^ ■?! (M ce ce re re re re re re -:^ -+• -f in in »n »n in »n m m »n m m m «n in m in >n in m »n in in in in le -"CO XX ~o;o—''Mi^corec:t^t-cr. -*'--'— ^-+'—''^ in el-*- ! re I- I- re m re CO xj -+! CO ^ c. c; CO c: X ^ CO ■* X o; CO r. — m , x- — -f I- o; — I (M re ^+1 -* -f re ei — 1 Ci t^ le cm c: m — i- ei i-~ ox O X -M m X ^ m X — ^ -+' «^ cT: re CO 5J t^ -+ t^ C m m X c re le X o — re -+ m i~ X Oi — ' c^i re le CO t- X o — •>! -^ m CO I- r. ~ — ei , ^ — I _. ^ -H ^ CM ni ?» n -M ei ■>! re re re re re re re re -r — -*• in m in m m in lo >n lO lo »n in in lO in m in in m in in m m m le , 1^ CO t- X ^. O — ' CM re -+■ m CO t^ X Ci o ^ c-ci ce -** m CO t— X c; | _• ^1 ^ I ' -1 "* 1 ei re re re re re re re re re re — " —H -f — ^ -f -^i '+' -* -f -* ; ,," re re re re re re ce re :' re re re re re re re re re re re re re re re re . '^ LOGARITHMS OF NOIEERS. 401 . — L."^ •M ~ Le r^i ^ , X te 7! X J- n X le ei X to ei rt to re c ^^" rt ^ r: >-i CM -^i . — — — c^ — c. X X X L^ t^ I— to to to to ^ ^ T^J T^i » ^1 7-i T'l ri ?i r-i C-) ?j M — - - - — - - -* r-t t~ iC >.e t- 10 X — t^ — ■— 1^ X t^ re X e^T -<* Le -# yt ^ — ^ tZ X tc — i:; X I- M re ^1 1^ .;,-: X -r- to to c-1 Le Le yj ^— iC X C ?1 r^ tr i^ X X X X Of. i^ ^ •r^ ei c X le ei c. Le — „,^ ^^ '^ ■-^ X — Jr^ ue t^ r: — "^^ LT*' i^- * w-^ r^ le i^ 0^ ei ■"■*• le i^- 0^ "" i^ t^ t- X ^™ :^^ re -^ to t- cc cr. ^ z-i rt -r- Le to X ot '^ e ! re s,.» -^ ■"-*• "^ "^ ue »e u: m u: u: le le tC to tc to to to to to l^ I- I- I- • u: le le ue le ue i-e ip le ip le »p Le Le AC Le Le Le Le Le Le Le Le Le iC X ■~z X — Ci X re "*■ ij le I— c; c^i •^ tc X ei "■*" to X '^ :m '•— to I- 0-. — re 'e to i^ o t^ !>• X c^ ^^ ■>\ re "^ to t- X cr. •.;_^'. ei cc •"?— le to It- c: — -M re -!*' ■— • ^<* -^ 'i- le le le le ir: le i^ w ^ tZ: ^ to to to tc t- t^ l^ t^ O tc *? f *? *? ip jp »p jp jp *r Le iC »p »p >p ip ip ip ip IC iC — oo ^ ue — :<] — ,_ Ci ■* -r — ue /M. X ^ — — — Le to to ^ r-j b- «-"; C^4 le ue — ^ re X — c: --r X 1- c^ le »c '"' ^* re —. CM ei c. re re re r^ ^ iC: X ^ "TT" *^ "^ -^ ■r- re ri ..— * 1^ — ^ ei c: Le ei x r^ * ^— •^ vc X c; re i-e t- Ci ^- re le I— * .^ — CC -T*« to X ^ re Le -^ t- ■-^ t^ t- X * •— ' <>> re '^ le t^ X cr. :._; _« -s^ T:r le to i^ —. — ei re — ^ -— - ■^ -r" '^ ir; le le le te Le le 1:; t^ tc ^ w t^ to to tc I^ t>- l~- I— l3 u'; *r '.'- ip ip •f le »p ip ip ip ip ir: le ic IC Le Le ip Le Lc »p le Le ~ C: yi re t^ cc re re r^ t- ■<:^ X ^ i~- ^1 IC Le re '^ t^ c: c: oo — • X ^1 :m cc — *-^ t- c: le X t- rt >^ to CI Le Le ei Le ic •>! ^ 1^ — -— ■ X — re t^ X ^ ^— — ei c^i e^i :^) -.— — 1^ t^ X '^ M u: t^ c^ — re tr X >T — ■^ y :;^ . — Le t- ot CM -*■ Le * — "^ ^ t- X ■J^- '-.^- '>\ re -* te I- X r; * _ ■-^ ■«-4a »e to i^ X CC; — ei re ■T*- 15* ^ '^ -^ ue ts: le ic u; te te te ^ ^ ^ C^ to to to to l^ t~ l~. l^ o jp ip ip »p 'r ip 10 10 la »e ip ir: Le IC le ip iG la u: iQ »o »c ^ -^ —^ ^ M --C le -M -- ^ -^ -- .-^ 1^ _u ^ ^ Le -rt- ei X re w X cc c^ c^ w Ci ^ X le X X .« r-- 1^ *« r- oc .-1 •ex — ce le t- X * * — :^ * X r^ re — X Le CM cr: Le *a ^ J; —^ ce ■^ X c ?^i -* to wi — re *^ ^ X :;^ ei ia ■•^ t- X >^- '^^' :>) re "^ »e to X * 3 — ■^»» Le to t- X — p- ei re "^ ■^ TT -:*• -* te te 10 le «e le ue ■t^ tc :C ^ to to to to to; t- t- t- u; u." ue u; *r f *f ic ip »p ip ip ip i-t i^ Le »C ie ip ip Le jp Le Le te — oo c. ^ t^ -^i r^ 10 — X — e-i -^ -^ Le iC ei tr ce Ci t^ re X ■^ ^- >^ tiT re i^^ r^ -* X X ^ X X ■r*- or X *c * ^^ ■^^ re * c. — ^ re le — t- X X X -r 1^ ^ 1^ *^ ^— Ct I^- ^*" ""^ l~- "* Tf ir: X c: ea -^ r^ ^ — — 7^ ue t— c^ — u* 1^ * _-. re ~^ to X ^^ — ce •^ "T^ L-^ t- X * ^— re ■^*- »e to t- — :^ — . 7-J cc ie to t- X cr: — CM ce ••?*■ •rr "^ "^ ■# le »i; le le le »e le le w t^ tc ^ to to to to to t- 1- tr- ^ ue ip la ip ip »p le le *r iC Le tc ip ip »p ip ip ip »p ip ^ _ t^ eq -M X X -M re re ^ ->^ »^ ^ — t^ re t^ c: 00 ;^ X ^1 :e — i:e re t^ t- -:^ X X i-'^ * ^ tc ^ — ct re "^ re X' 0% ■^ [^ — ■'r^ t^ c: *— ■ re "^^ le to to to ^ i-': — ^ -^ ei cr r- Le -M c: Le — cn "^ w * — ce 'e X C^l ■^ to X Ti — i- tc rr ei re iC i^ X e-^ r^ --r 4:: •^ X ^- re -^ LC to I- r: ^ .« ei cc Le to r- X c; c^i re "^ T -^ -^ ^ 10 ir: le le »e le le tc ^ tc ^ to to -^ to to t- t- I- — AC *r tp "r *? to ip ip jp >p ip »p ic *r Le iC ip ip Le ip ip ip ip ip — IC -^ h^ .- ^ ^ le ->r :-: t— t^ -^ rr -^ X ^ r-~ X t- ^ cr ^ :0: X tc ■— c^ X ce re — te to re i- X »o * ^ rr ei — >^ X -M "-*■ i^ c\ — e^ re -^ -^ -r -^ r^ re ^- :;*^ X to re cr t- -^ cr N r^ ^: 1-^ ^ ^1 -* cr * — re le t^ n M« ^* »e r^ * ei ^ to i^ c. — rq -* u: •-r X * — ei -e le to t^ X :;^ — ei — i- to t- X c: cr — re ■^ •rr ■^ "^ ^ lO i^ le le le ue tc w to ^ to to to to to l^ t- t- — »C le ip If to ip ip jp jp iC ^ le iC Le le le Le Le »e Le Le ^ X t^ c; t- — ^ ^ -~ 10 to ^1 r— X 1^ _!, X cr — t^ e^cj t- «^( '^_ =c t- le :m r: '^f »e re t^ X :^ ^-*- '^» rc ^'-B - C. :m i-e t^ X — ^ CM ei -M 7J ^^ * X to -r — c; Le ei X 1"^ "^ '^ --C X — re le t- c; CM -e- to X •*■ ^:i '— 1— r: — re "T^- to- X Ci ^^ tr t^ c^ — ^ C^l -TT Le to I- X ^ —^ "M -^ Le I- X r; r- ep »p le ir: iC Le iC Le Le ip ie ip le ip ip o — t- r- re^ -Tt- 'M '^ te :m to .-. -«i — 1 _ •y:: -^ t^ a a oo io i^ re X c^ X re -::« ri t^ X t^ ^m -M _. 'r «>- to — re -M X -- ^^ -?- t^ -M te tr X ct ^ * rr; tc ^ M cr i^ -:^ t- '^^ *^. re 7^ '^ sr X re le r^ * *~ ■ej to X cr ei re le i^ X "^ 'e tr t^ r; ^-* ?5 re ue to t^ X * — ei -^ •*. Le t~ X c^ C — 1 CM "^ '^ ■rt* "r^ le le ue Le te ie is: tC to tc tc to to tc to t— t- t— f-' *:• jp 10 i-e Le le Le 'e Le i^ i^ 1^ »c Le Le le Le Le le io 10 ^ o ^ ri ce '^ t- X c: — ' c-1 ^ ■rff ».e to i^QOCiO-^er!'«* ■ ^z m ir; le Le le le te le »e to to to ^ tc t* ^ te to to I-- h- b- t— t^ c U. • • • - ^^^ ^ ^ ^ re Z D D 402 LOGAEITHMS OF NUMBERS. . 1 t^ re ^ CO IC r-l - ^ -^ — w ^ — t^ J": ■M * w -*• C X uo 01 t^ a 1 o ».-: o ■^ -:h -^ cc cc cc ^1 M CM -M i-^ ^- ^- ■3^ 00^ C^ Ci ^ Ci X C^ ■^ \'il,'^ '^ ^ z: ;:: ^ ^ ^ t-^ - Z "iZ ~ ~ ^ "^ ;:;:::: - M 1-H 3i iC CC O t- ^ I— i a X t^ X Cl 3^1 t- CO O O i-H ^ '-' a (O -^ cq tc tr c: ^ c» IC OL: t- -^ X c; t- CO IC ij:: ;m ?c t^ to ^ CO -- F— ^ I- '^^ t^ ri ^ ^^ ■^ CO r— < —s t^ Ci r-^ CO IC ^ i^ XXX X X t- "-C UO ^ 1 O :m c: l.O ■s CO en o M ce -+' U-: i^ X ~ :3 ^^ ■M CO -f >o tir 1- X ~ Ci o -o l^ a; ^i s,^' »— ■ cc ■— *- u:^ C t^ X C- '-O c^^ -^ ■^ LO s^ t^- X '^ "^ — t- i~ t- i^ I— CO CO a; CO a: X X X X c; * c^ Z^ Ci Cm o o o «^ m o o ip n^ ic o "r ^ ip tp o ic ip ip >P »p tp ip -p; -p; 1 CO Ci O ^ o — . — Cl .■^ ,_, C<1 O X CO X tr •^ iO t- — X t- X — X '-D O C-1 o o ^ lt: i—i Le 1* 'M ^^ i^- tC r— ' -+> ^ r- UO t- iO r-J -;f LO -M ^ >x-5 ^ -^ T— 1 »~ c; cc t- ^ 7t :^ X C C^l -^ Le tz l^ t- t^ 00 Ci -^ -M -Tf »e o CO cr^ . — . 7-} ce -Tt — L^ X C ^ ^ 01 ro '^ 1-0 -^ t- X X -:H t: t- CO C5 O 1—1 :m ■-i- >C: w r- X C^ O r—. cc -3". «0 tr t- X c: ^ t- t- L- t— t~ CO CO :/. CO CO X X X X c. C» wi Ci wi C^ Ci C* wi ^^ '"^ ) »C O O o IC o to O ip. ip o O iC Ip ip iC »p 10 »o IC ip ip ip :c :c 5^1 O O ^ •o cr. rM o oc -M w r^i X »-■: -r -H cc — oi tc Ol . 1— 1 lO t-~ o O M ■>\ CO ^— ^) d ■^ ic -rr O Cft CO ^ t^- w oi 10 tc -* •rf a -Tt* c^ ^ CO -M o cr. 7>J ■^-H t"^ c^ ^^ cc — *. LO cc tc ^^ ^^ w io — ^ ro t>. 00 Ci — H S-l -*" lO r^ CO f— , 3^1 cc ^ w C^ X Cl ^ 01 CO "^ iO * t--- t^ -* iO h- r/-; r; o — ^ C^l ce O ;^ t^ X C^ ^ ^-^ c>l -f 10 tr t^ CO Ci ^^ t- t- t^ l^ l^ CO CO CO CO CO X X X X — . * ^ * C^ wi Ci c^ c^ »0 ITi O ip le ue o le iC u': 1,"; o o ui ip o uo 10 10 >-0 ip ip ip tr t2 — «0 CO o .-. M CO _». o tc «-, o X X X Gt C<1 t^ CO — i r-^ CO t^ '^i CO 10 >c o oi -^ --C CO on l-C cc Ci r- ^^ ce CM X C^l CM b- •0 OT tC t^ iO iM oc re CO '>l tc ^O ^^ t^ ;^ ce ^ X -^ 1—4 CO -t> UO »c 10 10 iO -^ CO Ol -^ t- cc O ce "^ t^ l^ CO ^ -^ cp lO >p ip o jp JP JP jp ip >p »p ip ip JP ip ip ip «p cp Ci o o a CO t~ L'^ '^l -+I cc -fl O b- O le crs X t- t^ iO 01 — CO X Ci le t- i:: -^ -+■ '-*J » Le ^ -T- C^ !— ! 1— ' t^ c "^ l^ tC CO 1^ X tc O --= r-^ tj: — 'C c^ ce tc c^ :^^ — H l-^ Ci O OJ CO CO -+■ -^ ■^ ""^ CO Ol — ia\iz> t^ CI ^ -M cr. -^ ^ I- X ^ -M cc u": :^ l^ X Ci ! -*i >e w c/^ ' "i •— ^ p— C^l ce -t- t- X C- Oi- »^ rj CO "*• t^ t^ X c^ ■"* t^ t^ t-- r- t^ CO' CO c/,. CO X X X CO X — . wi C^ Ci C^ ^ * * :3 ^ »o o o o ip ip *c IC l_t »-": o Ul JP le lt: te 10 LO 10 10 iO io 10 ;c ^^ ~5 -^ ^ _ cc le t-- — ,_, -^' (^ — w n c C/1 X C: -i- -C. t- :r ci CO — -* CT. — 1 r— 1 l^- o '^ I-- :M ce t— c: C. i^ a~. -ri- -^ ^ CO t- c: X Ci -^f o - ^ ^ ■M •M CO CO CO CO 01 I— c;; "^ rt< CC ac C^ O e'l c^ o tc «^ c. '^ ^- e-T TC ist tC I- X — . C: — M CO -r .^4J -* lO CC l^ - C 1—1 -M ce ■^T** la l- X * O f— 1 01 t— tx. Ci ^^ t- I- I- 1^ t- CO a;. a CO X X X X X r: Ci * Ci * c^ ■* c^ c^ 1 ip IC ip IC o ip »p ip ip »p »c. ip JP »p »p »p ip iO IC i-0 ip ip ic :p tp ?0 O'l CX) CO CO cc X ce CO '^l O t~ lO ^ 'Tf iC X CO r; X CO ^ CO cc cc CO -r^i -^ c^ -M ;r '» ■-T^ c/-) .^^ ^\ -* t^ t- -r" ■X c^ X CO — •^ -^ cc r^- t- CC GO ce CO M ^ ^ ce r^ ^ r^ -^ tZ X :^ ^ M 01 ■^1 ^i ^- — * CO ce »0 siC 00 * ^- 'M -:+• >c tc r^ — C<) CO »^ --r I - X C^ — :M -M cc -^ iO --I r^ cc O r-^ -M ce T*" *e C3 X CI o 1—1 ■>! CO -^ 10 w cC' Ci •™' t- t- I- l- t- CO CO X CO X X X X X — . Ci C* C • 'Si w • w'i W* Cj» t,_^ 0^' 1 O »0 lO IC o o IC iri ic le »r: »C IC O iO ip »P JP ip ip 10 i^ IP tp tp 1 xxt^ioco'— c:t--io-!t"CococoiO)ooccxtrio;rc;iOOicoio C^Xi— ii— (X'>JO:jOiOt-tD01iO»OO^t^Xt^COwCC':h~ — C; •-o r-i b- 01 --T — ^ io r; 01 10 X — ' CO »o i^ X CT. o — -— o o c: (N -*i iC l^ CO O — H 01 -^ 10 O Oj r: O i— ' OJ CO iO w t^ X c. c — ■— -* iO o 1^ X o ^M oi CO -^' io vr I— cr: o — O) CO -^ iO t2 i^ — C — t^ i^ t^ I- I- X X X X X X X X X Ci CTi C-. r: Ci ~. r: c. c: O C «OiOlOiOtOlOiO»0'OiOiOiOiOiO»0»C>CiCiCiOl010lOww !i— lOQiOt^XCTJCiO'MCO'OXCIiZOlOXXO-^OXCiC-Tl^j I t^ CO o tc CO i-~ X I— c^ -Tt" CO Ci CO CO "-^ w t— ^r CO w t^ — ^ c: 01 — i Tt* o o o 'O r: CO I- ^ Tf i- ~ M -*< -^ i-^ X r; CD o o O c; c: X >— I ro -*i "-s I- X CD •— I CO -+ 'O tr X ~ o '— oi CO "o w 1^ X X c; o I ,_( '^ lO ^ i^ X C5 .-( oi CO -* io -~r t^ X o ^ oi CO -f 10 cr i^ X c: —i I t--t--i^c^t^t^xxxxxxxxcjCicrjCiCi05C^c:5;c;0 OiO»OiO»oiaioicioioioooo»OiOioioio»ooo»Oioto 1 CO X -*< X 01 ;:2 O -+ X 01 t- CO CD t^ ^ •~r X -H CD ri ^ oi iO — 5: I _^ t^ ^ rt c^ :o o tr I— X ~ o j I-. I, 1^ i^ 1^ 1^ •x X X X X X X X X ci r; c; ri r: Ci r: r: ~ C2 O »0 iO 'O iO 10 O 'O Vt >0 >0 >0 iO IC iO iO 'C5 'C^ 10 »0 >~ iO iO iO o 10 CO t- CO CS O -^ -N CO -* 1^ t^ »-- t-- t-^ X) X X X X X 00 CO CO CO CO CO CO CO C" o?r!t^x3i0>-<0'icc-*»oecb-xcs I -• K'Xoooox>cni1 re -^ Le — t.^ X; ci o ?j CO -^^ te tr t^ CO ; O O O O O O O — — ' — ^ — -^ 0-1 e-l >1 >l -M >« ri 3<1 I re X — t~ M o "M ^^ ^ c c— r} le le c^ o 1^ le c o ue tr M ue re ^ C: X le H o i^ -^ C: 2 ^J 30 rt oc re X Vl --i c Te r^ o fe le 30 c- Ore?t — ^lei^i- — r-xr-xienoc-icciociir;. c^oc5tDO; ~ r: t~ ^ JJ ^ '-^ re r: ie -— •-- r^ X p^ 1-- — -^ ri re --^ o 32 ^ "'^ :m re -^ ie 1-- X c. c — 7i re -* ^e t£ i^ x :n o — >i re -r!^ Te tH ?- I C O O O O O C: — — -— '— 1 — T— ,— — T^ r— 3-1 TCI >! >) :>j :M S-l M le »^ ~ i-e le r: o tc -^ ~ c-1 X o j-e le c o ic >e o o w t^ -^ t^ ej le ce re x o o i^ m re re — . -r ue -r — o :r ic m --^ t^ tr re c- O X — -^ ^ ~. ;^ e^i cr. le — ;r: Ti t^ ?i t le ~ re — r; >i 'e t^ vr :r t— ct c; r; c; — — ■>! re re '^^ '^ ic i~ ^ •--: ic i^ t- i^ x x x ir^ ^^C:;; — t^t^Ct^X'^xt^ot>.iCt^'r«t----;ti w r; CM ue t^ :M -^ -* le •^ I- X ^ _ f>) re ~~- ue w I— X — — ; „ e! -^ -a. ».e ;^ r^ ^ ^ ^ ^ c^ ^ ^— f — »— ' — ^— ^ ^- — . *— — — e J rt >j C-) >i M ■M rj ■-f *-f ■T- '-f «p ?f '"T ^ '^ ^ ~r T ^ Cf •r •o :p ?£ ;c cp «C *-f «3 -i re -^ ie t5 t^ o c c c o c; c — ■— r^ rz rz r;z rz r:: ri rx ?!' r^ ^'' ^ ^J ^2 "^ ^^ trt xr:>io— i^iex»ec:e^:>^t^t^'M-Tt-»>-e-i'T'c:re X ei -^ ei X ■>! '•^ — tc ~ ~. t~ CM urt »e c-i 1-- z; C: t- >i le le re CO ^ le ce — X vr re o — -M X -T o le c te r: '^ X — ue x ^r- ;;; — ci re -:i^ »e tr t^ 3 o o o c; o o o --1 ^ -^ — ' '— ^ — — ' —^ ^-i -1 ^1 ^1 ^T '^ ^ ^1 ■^ ic tc -^ IC '^ -^ ^ ^ tc tC '~C^ ^ ^ ^ ^ ^ '^ '~p '^ ^ ^ '-p ~ '-p C: -^ T— o -* o o -^ ^1 re r: X ■>! — re 1— re >— re Cc re — ' lO ■* C5 O -:^ '-D >e — ui «: -:« O re ri- — t- o O x re ^r tr t Ci m e-i o o ■-^ -^ -M o X »e e-i c^ ^ ■>) X — ~ ic o -^ ~ re I' i- x r-^ -h tc r^ i o — CM re re -*' le ic w t^ i^ X X c". c o c: 1 "m c-i 'm re re re :^^ re -r^ le •-c i^ X r: o — cm re -=- iC t^ x ~. cr; — cm ce -^ te ^ t~- O O cr O cc; c;: O O T-^ — — -^ ^ — ' — Tl ri cm 3^1 CM CM cm (M vc::r:^2w'Oo;cicoi05£^0'iC«Otp:p:p:po^tCt£w53':o CD — 3-"ire-rictoi-~xc". c::;i-^ ■■™' « ■M -4 — « ^^ •— I ^~' 'Z^ '^ '^ ^ w- Oi n r^ X X X X r^ t^ I- i^ t^ to to >- — - — - ^ tc ri ?c o — , le ■M to t- -** Ci oitoo^oi^ — '^la-^ — too 1^ t^ "M t^ t^ tc re r- r: r: to -M -^tc-fO-^ieieeit^O — ojto _^ ^"j ...*. )-^ ^ t- X X X X X X t-toic-^^eioxtore — x-r — -- rt r^ r^ r^ re re re re re re re re ?e re re re re re 71 71 71 71 Oi C". -^ — '^^ re -rr •e to t— X O". — 71 re — ue tc I- X — — 71 re :m re re re re ce re re re re re Ti -*"'^-r-:^-r-f-:^-=t'-#Leieieiie 7' '~r *-? ^ to T' t^ ■r-. X X -4, r^ to re le ue ei to X to 71 to t^ to re X — 71 — X i-e Lt — o ■■^ to te ei t^ T. ~ I- ei te tc le — ' le t- t^ re ->• 71 r; ^ TT ""-^ i-fZ to 1- X X X X X X t?— toie'"r"7ioxto~*"^— XL"'^— X 2J 21? ^1 - 1 "xX M ■>( ri ei 71 71 71 — 71 re -^ ue to t^ X — . — 71 re X -M r^ M r^ re re re re re re re "^ -r-^'^-r'-r-:^-:*' — -f*o>cueLe 'r ■■z. -o ;p to to to to to to to to to to to to --0 to to to to to to to iO r^ •-C re \Ci ^*< — r. -^ to to 71 u^t^^e — -"ic-^ — --oo —. ^ "M ~ re ie ■^^ 7) to r; r; t^ re to t- to re I- c. — t^ 7) t^ b- -.o 71 ^^ ^ •^^ re le to t^ X X X X X X t- ,, ^ — "^T re '^ ^ to t^ X Oi — 71 re "^ ie to i^ X oi — ^ 71 Ti '4 "^'4 •f 'r V^vfif^o^ptp -r "^ -r -r- -r ^- — — — 'e Le Le le to to to to %p to --0 to to to to to to 5C c. t- rj -^ -> '^ IS re t~ Oi t- ce to t- iG — 'Tf' ic ue M t ^ re re — O t^ ■>! 'Tf* -rr* — to Oi * t'^ re t- X t^ -*i c: — — — . ue X Oi -- ,^ o ■M re le to I- X X X X X X t"- to )e -r 71 — C to -:+• — ~. le 71 — ^ C ^ C — -. _. 0-. X X X *" *. "3 — ^1 re — ie to i^ X ot "• rj re re re re re re re re re re -^ -T"~f-^-7'-r'-r--?"-r'^-^Leie»-e i2 t-^ to t^ to to to to to to to totototototototototctototo .- O — X ej ei X X 71 71 — to — . - t- re -.0 X I- c -^ I- c^ C- '— o re re "O to C- Oi X "^ 5c X — re ie ^ I- X X X X X X ^ ^ ii: Z" ^ — r- '^ TT ''' ~ '■— '' o 5; S — ei ^ — ^ ■— t^ ^ ■—' Ul c^ >a o O ^ r^ t^ ':*^ r; 'M ■M ue X 0; X ^ xoOt-'Mtaie^O-^^tOLere c^ — ^ re ""^ to r^ X X X X X X i~^ t^ to '^ re — * t^ le 71 01 to re t- r^ XXX a. -/. X X X X X X X X X X X X t^ t^ t- I- tr to to X CO — ej re -»" >e to 1^ X c — 71 re -r le to i^ X 0; — 7i - ?j ei re re re re — -TT-^-^-T-^-^-^f-ri-eiei-e ^ tC i2 O to to to to to to iO CO totoiotocototototototototo '^ re r. — - ~. le 71 b- X t^ 71 Le le 71 t- ~. ~ 1-- re t^ r; c t^ '^ *^ ue ei X ■3 ^^ ■r. le X r: X ue c^^^ — xretot^to7itoxxto r". ^- re ""^ to r^ I- X X X X X i^ t~ to -r re — r: r- 17 71 ~ to re ^s<» ^ t^ t^ t^ t^ t^ t^ t^ t^ 1^ J- r- t^ t^ t^ 1^ t~ 1— "o to to to 17 te Le »^ X * O ^" ei -H- ie to t^* X T'. ™- 71 re "T" le to t^ X oi — ■ ei .M ri re re to to to to to to to to to to to --o to to ~ir '^ ei t- i~- _ij X X -:?' X rr. t~ 7-1 -f re "^ t^ X to 71 to X 0: t^ r^ ^ X -^ X 0; X ue C 71 71 Le X r; X le r-. > r. Ct — re -*• >e 1- t- X X X X X X i^ to Le re — r: t^ Le 71 t- r7 ^ tc to to to to to to to le le Le »e »e "^ ■"■ X c% o "" ei re -" ue to t^ X r; — " 71 re ""T" 1 e to t^» x * — 7 1 ■M ^1 re re re re re re re re re re — -^-'• — — -r'-r-^-r*'-*iLeLe»e *-w ■-C tc to to to to to to to to to to to to to to to to _ ue tc ei '.e _M re 71 X — — xre-:^-r> — lexxtont-oj- — . "-* r^ le X * orere— t^o— "O b- 71 — — " re ~. — 71 -" le to 1^ X X X X X X t^ to >e re ei x Le re o t- — X *- "w ^^ Tl ^^ -^ le to t^ X 0; '—' 71 re ""*■ *e to »^ x * ^^ 7i Tl ei re re -3 re to to to to to to ;t^;t;t^;t^2^^i§i2ij "" _" --or. X «/i^ re t^ ri t^ je le -* — Le t^ to re h- r: >e c re • X C^ l^» ^^ t^ X 1- re r^ r; r: to f -" 71 X 71 re re c -*• 1^ X to X -^ -^ J •« t- X X X X X X t^ to le re 71 X to re c t f .^m. -^ «W ^-M •«* *M *-M — -f- -f re re re re 71 71 1 •*. "^ X rt o — 7 1 :■; -*" 17 "O I-- X ~ "" T\ tototototototototototototo 'w \Z ^Z '^ to to to to to to to * * 1^ •-5 1- X — r^ — 71 re -f >e to I- X — . — 71 re -*- le to 1- X r. v; x: ^,^ ^^ LOGARITHMS OF NUMBERS. 405 , -M n re ?-T Ci w ■>! tr . ^ l^ C •>! -^ i^ l^ C". O ^1 -^ b- O C^ ~ n L- X C "M IC w CC O J- O O O Cri' C^ CC' CO l^ t— O 'O »C -+ -^ re 'M CM r— o O ~ CC I— t; tc — -*'»0'~c:^c^acciO'-^'Mre-+iCcci^ccc:C — -Itvire-Tfi^tc*' ' re --c^ -^ -* O '^i tc lo cc cr. M re re o w — c ~ I- re vc I- I- le o X -^ C — t^i t^ ei i^ ei w r— o c. re w ct re ts X — re le t- c: — i ry- ] r: r; r; X X t^ t^ w cr »e ue -r re re ei c; c: c; x i- --r te >e j re -^ ic ?c 1-- X r; o ^ e^5 re -^*< i-e ir i^ X c. c; ~ — ei re -* »c to ' le le lO le 'e le »e w w cr CO tc w tc tr tr w i^ i^ r^ i^ i^ t~ t^ r^ lec^ — OGO'*xc;Cii^^wt^t--ico-*'ieierec: •M t- re X e^ t^ -^ le cr. re t^ o re ?c n -M -*- ^o X c: — ' k^ fr- t"-- »^ !'*■ •-IJ *^ r^ ^•\ rvi .m ..^ r"^ -^ '/" -/~ i^ --• i-^ i^ .^ :r; r-i ue X O i^ o o CO le . X >e -- to ei t- re X e^ t^ -. -- . . -^ _ - . . , . ■ ^ X X X t^ t^ ic to »e le -+ -^ re C-) 'M — — O cr. X X i^ vD le le re -?■ ic to t^ X (T. O — ■>! re -* le CO i^ X ~ r; o — ei i:e te »o lo ic »c le to to to w to CO CO io to to CO I- i^ i^ L^ I— i-^ i^ totctocototocototctototototototototctotccototoccco ^ t-e 75 t^ ceto-+-oce»eier':~re-:f-t'eix— rere — t^ o: ic — i^ c^i X re X re X ei to C -^ i^ r- i^ C "m >-e i- ~. — ei fo 1^ t^ t^ to to le »e -f -r^- re re ■M c~j — c o c. co x i^ to • i^ i-~ i— i— i^ t-~ xxcocec.'^cc — ceuetob-t-t-t-t-tocotototot^ X O C^T ■^ L"^ xi>"rexOT— 'c^iectr— ^— otiecir— — oi*e *— • ^- o to o;>er-Ji^rex-^c;rexeit-' ^x — lexOreuex "O ei re »c to to to o ic ^ -:f re re ei ei — ^ o r: r. X I- I- to i.e -r- "-f* -* ^i iC re -* le to L^ X en c — M re ":?• »e to tr t- X o: O — c^i re — !*^ U; to icicieietciieictotototctocototocototot^t^t^t^ t^ t- t^ — totococctctocccotocDcocctocococctococotococc to CO CO '+'tcto»cccoto>iMcoorecoxO'M^*'»ef^r:^--cetc c: ce t^ X — c^i — X re Le to —' T— le t— t:^ to n to X X to re t^ r: ~ X -f c to :m X re r; -+• c: -^ r. re I le r: ei le x — '^ to x c: ei -:^ '^ le le »e -* -^ re re e-i ei — — c O r: X X t^ to to >e -* re re 71 — .. re -T le to «^ X c; o — M re — >e le tr t- X c: c — ->! re "■^ i.^ '^ le >e io »e »e ue »e to CO CO CO CO to to to to to to i^ t^ i^ i^ t^ t^ t- totototototocccocctococotrtotctototococccoto to CO to o; re to t^ I- t^ t^ ei c; "^i o »e Ci ce i>- — -:^ X eMt^o>-^r:co — rereeixreicie-fOiet^ t^ to ce ~ to ■>! X -^ c; le o "^ c. -^ X ei to c: re to o: ei lo 1— c: t-^ le -^ -*■ re re ei ei ei — c C r; ~ x i^ i- to ue »e — re m -M — O 1"^ re "-^ *e to i^ x c^ o ' — ^ ei re re ~~" >e to i^ x ci o t— ej re -*< le CO * * »e le le 'C le »e le to to to tr to to to to tr to to i^ i-- i^ i-- i^ t- t^ tototototccctotototocccotocotctocccocococotc CO CO to ' xo^tocii-iceceCT — (T.tocecscc^t-rex'ifOtore o i^ ue »e ~ o ot i^ ei le CO io — to c. r: X le o: 71 ei — X ei le O to re X -^ o ic c: »e o -?- X ej to c re I- o re )e X c •M — to 0^ -f re re 'M ei ei 1 c o o: x x i- r- to le le -^ re e» ei 71 re -f le to t- X o: O — ei e-1 re -^ ie to i^ X o: c: --^ ei re ■"*■ ie *e ic le le »e le >e le to to to to to to to to to to to i^ i^ r^ t^ t- i^ I-- to to CO to to to to to to to to to to to to to to to to CO CO to * C X >e ^ i-e X -^ ei re re (M " o X to re — ~ to -* re — ' ^ « -^ ~ -M -*• -* 1— to o — o (^ "M le to -tj- — to o: r: X le o re ^ 7) o 1 o t^ re o; »c o to ^ CO c; »e c; re 1- — -*■ I- o re to —. -- re ue t- 1 T— ^ re M n ^ —1 ^ C o o: cr. X I- i- tr to ue -+ -^ re ei — — O ~ X : , re -# ic to t- X o: O C ^ e-i re -" le to r- X cr. o — ei re -« -* Ke 1 ie ic »e le le le o CO to to to CO to to to to to to i^ t^ i— i^ i^ -^ J:: 1 1 oie'^^Ci-<*'xcq»ot-co<:r:ooooc:c:c^xc:ct o — ' re ei to X X m -J -+! to ic e-1 t^ o :m — ' X re le to le e^i b- o 71 — X 1 — L^ re o: jc — 1 to — to — le o -r- X — le X -^ — i^ c: -M -e to t- ^— ^ ' >i , O O O o: 0-. X X i- t- to le JO -* re re ei — o o — X »-- O UL. 1 re ^ le -r t- X X Oi c — ) re rc "^ » "^ 1 Le le le le lO lo o ui CO CO to to to CO CO to CO to I- i-^ t- t^ L^ l^ l^ to to to to CO to to CO to CO CO to to CO to to CO CO to to CO to CO to to _■ ■ O — ' ei re -^ »e to 1^ X ~ c: — •M re -t- )~ to i^ X o: o ^ (M re '*! j ►r le le ic le le le »e o te le to to to '^ to to to to to to t^ t^ i>- t- t- 1 iz; '!+< '^ -*l 406 LOGAEITHMS OF yUMBEEl! ~. ciCT-. 5^5iCiCic;o6X5icc5Dx:cxxx>coc/:ccx-X/XX I— tcc;xl■-oc3ir^^'^^cr;ooQOO'+lOcocoo^o — o — >j:^ ir: i-~ I- — r c; -T- w t— »c c^i ic Ci o O «>- c^ ^ cc Ci t — r ~ "M r^ J^ — ■>! ^^ -^ i^ Lt O t:^: iC »C ^r 7C CC "M O ~ l^ O M — ~. "-^ -:" — X ir: -t- r?^ :m — • O r; cc L- ID m '^ jc "M — r; rX' i~ w i- r: im — ox I— X ~. O — >i M i^ -f o iS t^ x; r; o o — >i r^ -Tt tc: -^ c- X X L^t^t^xxxxxxxxxxxc. ~. CiSiCicncjdcic.Ci -f -M — -- ^ ir: X re ~ '— ic v£ X ^j X ts o t:^ — 1 t^ 1^ ?£ cr. -+; c^ -*! re :^^ — C; C. X i^ ts ir; -^ :c '>i — O c; i^ --C i.e -h ~t — , ::^ ^. ^-, t- X ~. O ■ :m ^: -^ ir: w t^ X c: o O — :m ce -ri ue v:; i^ c~ -.'. t- i^ i^ X X X X X X X X X X X c. ~- ci ~. r; c; ~. ~ ~ c. ~ ce o o ce en -f o t^ ^ ce a; f-i (M c-"i O ^ o >< re I— X -e t^ CI c; re-t-ut^^ci^t— t— t-i— otc»c-rJer-ox«c^'-c::crec; re c^ — o r: CO t^ tc o -^i te c^i — o ~. X I— ir: -*| CO i-i o ~. X I- i^ X c c o — 'M ce -* '^ w t~ X c; r: c; — ' "M re '^ Le w vr 1^ X I— i^ t~ X X X X X X X X X X X X cv c; Ci ci ci c c; c; ~ ~ ^ 1 X o re t- -M X >c re re , . f ^ -M ~. re ^ t- •>:: -Ht—— .t^i^-firici-^jvii— ret^Tfre-i- X ■?;) re re — t^ ^ ^ 'e -*i — i:c o c^t <>i "* o w I- 1-- X X X X X i^ i^ tr le -t- >T ■— CI 1-- >-e re o X »r; -m r- ■>! — c; c: X i^ ;c o -?• re -M — O c; X I- ^ -^ re "M — c X i^ tc 1 i^ X r; — . o ^ ^1 re -f »r; — t^ X X ~ o r— >i re -^ >^ ic — t- X t— i^ I- t^ X X X X X X X X X X X ci ~. c; Ci Ci c — . ~. r; c: l»i::c;^~. ir-+irereir:x(MXO:ot-oocM5vte I o 3)Oxo-+— c; eire 1 — o ~. X i^ ^ !3 '^i re 'M -^ o C". X t^ w i-e -^ "M — o r; I— --r ic t- X X ~. 3 — ' M re '^ o :c t^ t~ X r-. o ^ c^i re -T- Le i* tr t^ x t^ i^ I— I- X X X X CO X X X X X X ~. r; ~. r; c; r; cr. ci r: r; ^ix-T'^iOO — re^' — x^iet^O'e'MO'— '»eoxxoo ~. — ~! — X re — L^ w -+- C". re ir; 1^ -H O le X r; X — r— ic X X i^ ^ yi ~. c^ ^ iz c^ p <0 ^ ^ y:. t^ '^ i^ ^ — c. t^ >^ ■yt ^ t>- Tf t^ T-^ X r; o ^ c^'i re -^ 'e 2 2 t~ X ci o — c^i re "^ ^ >-e cr i-^ X ■t- I— t^t^XXX XXXXXXXXCiClCiClClC: ~. CiCiC^ icccOwOO:CO:Dtcoo:c;occe£!«oeCwOO":cocc^ |x--c-:r^-*«ociccxicrero-:ht* »-- -'"* ,■'"*^ »^^ ^^n l-^N K ^^\ -.^ /""^ _JJ Pj^ •!— V ce I 5 — _ ^ , ^ ^ ^ — ... ^^ — ..,,.. .. ^- ^ , Ci X CO — CO ro re lO c: «o t^Oi— oi-^-Miat-co-^o-H^toortiooi^ox-^xo- cc x~. oc:—'^'— *— 1 — — 'OCixt^co-^'M — -)aixxorex^fro-eor-e; -Tf 'M C X f^ re o «>• M X i^ 1- tc Trere-:*! 1^ ^.^ — r-». ..^ ^^ ^ 1-- w ^ ir; -f^ T^Z :m — O ~ X »- -.c lo re » — O Ci X 'T i~' -^ -^ 'to l^ X w- '--.^ r— ■ rj re -** -fi ».e tc i^ X c; o — tM 0\ rr •*^ >^ 1^ t^ 1^ l^ r- X X X X X X X X X X X ci r-v Ci c^ — * — ^ — ^ ^ -r r- — ^ — r T ''p ^ ''p '-f '•? r r r '•f 9 -r re t- X i^ lo -^ o t-- t~ o ^^ tc c^ c- oo 'io ^^ o cc tr le — • ^s ^ o ' r; C: — "M re -f -f -* -M -+| -f re -^! — o — . -o !X! -x M c X >e -M o — ■ 1 :c w le -+ re ■>! — ^ w ~. X i^ u: »o -:^ re — o ci x i— -.r -* re >j — -^ ] cc i~ X n o >— ^1 re re -ri ic cc t~ X C". o — ^ >i re -t" >e -cr t— X " t- I- I-- I— X X X X x> CO X X oo OD X cr. 3- Ci o cr. Ci ci Oi Oi -Ti i Ow"-Cw;c^tr«c«c«ocooto«D«c>o;occ>tc>«o«occ>eo:c«o m '-T t^ X ~. c — ' ">! re -^ !e cr t^ X r: c — I, 1^ I- (.- I-- X XXX X X X X X X r; rt LOGAEITHMS OF NUMBERS. 407 fc ?s CC '^l cc— ^Cib-CC-HSq^CiCO cc •rh cc ^ c; X cc — - cc r-l ^. X iJ ccj S cc cc CCCC»CiCOiCiCUC-rH-* Tt< -* ^:M ~^ cc cc cc CC CC cc cc P CO CO CO GOcoGOccx-C rt* CM CO CM O »C IC C<) X Oi IC cc iC cc c -+ I- X x^ -r^ O ^ CO -t-< cc o rn •^ O O CO Ci CO cc CM l-^ C O O CO -r*i X . — 1 C3 CM cc , — 1 ''tc iC IC ?c cc Ci IC — ' l^ cc Ci -H Ci '-C O -^ Ci cc X CI cc cc t- O cc cc Ci ao o i-o cc CM-— 'C5COCCiOCCC<5r^Cr. I- cc -Th cc r— 1 o X ^ iC cc t— ( Ci CO ^^ O r^ CM cc cc -?• iC cc l^ CO Ci Ci ^^ f— 1 CM cc -^ o i-C cc t- X Ci Ci C5 oooooooooooo r— t I-^ »— ! r- i ^^ r- 1 I— ( >— 1 r— 1 1—1 I— 1 f..^ — "t "r t;- «J- b- b- b- b- I- L;- b- b- l- b- b- b- b- l;- b- l;- I- b- t^ b- b- o I— 1 c; OCCOCisNb-CCcTi'^CC rr* cci ,_, »c CT CI >- -^ X IC cc ^ , t- -* CO CM CC CC O (- r— '^i O O CC Ci '^ I- X X ^ CM t- O CM c^ f_, 1— -^ o I- CC Ci i-C O CC -^ CC — CC o IC Ci ■cc 1- ^^ UC X cq UC CO 1— ( L'; •^ cc — O X I- CC -f^ CC -— ' O CO l^ »c cr: C^) o Ci l^ »c -^ CM Ci b^ o^ .—I c^"^ cc cc -f o cc t- CO Ci cr. o V— • CM cc -:^ 'Tf 'C ^ L- X Ci ^^ o o oooooooooo ^^ ^^ 1~^ 1 — 1 I— 1 ,—1 1— , >— ( r^ r— > 1— ( r-( ^ t- l- l^ b- I- l^ l^ l;- l^.^ t_- b- I;- b- b- t^ b- b- b- b- b- t- b- b~ Ij- GO »c »c b~CMrH(M5b-c^)COCCt- CI cc 1 — i >o Ci cc CC o CC cc CTi CI •.-K -^ cc 1 cc ^ti i.C cc I— CO OO Ci o I— 1 >l cc cc ^ i-C cc l^ X X r: c; oooooooooooo t— i rH ^-^ 1— 1 r-< r— ; t-^ r— 1 rH .—1 cp I;- I;- 1-- t^ l~, t- ^ t- i^ t- I;- l_- b- b- l- b- b- b- b- b- b- b- b- t^ --■ _ IC CI 'M ^ O O (M b^ CC CTi >+ •4^ 1^ cc _t X cc 1^ CC cc t^ IC '-i O cc -Ci — — Ci CC r— 1 -* iO tC CC o IC X ^ o X i-C o •4' cc CC »c ^ t^ :c Ci tr CI 1^ CC Ci '^ Ci '^i c:i -Tf oc; CM I^ '^ X; C^l c fH^ 1^ — cc CCI .-^ CO -^ — 1 t^ cc Ci iC O >C -— 1 cc o LC o -^ CO, CI cc o ^/■s CC ^ cc . — i o Ci 1- cc -* cc CM O Ci b^ cc — V r-. Ci rr, O IC cc >■" o X CC ■* Oi •^ y— -H CI cc -^ ^ c-i I- CC •c; Ci Ci I- ^ o cc >c o -^ . — ■ tz :^ CI CI .-H X CC l^ o O cc Ci cc CM X -T^ O cc CM b- (M b~ CM t- ,-H cc r^ ^..^ X r-« IC X c-ci i^ f^ cc d O Ci X --cr ic — CI --I Ci CO cc o cc C) :^ C: 1— IC ."i- CI O Ci »^ iC C^ ^^ 'O — ' CI cc -H LC cc cc I— CO Ci o ^-< C4 CI cc ^.4^ IC cc 1^ l^ Xl Ci c; o o oooooooooo .— . ,— ; ■— t _ ^ I- l_- '.^ t- t;- t- l^ t;- b- t;- t- l- I- l- l^ l~ l^ b~ t- l- l^ l- l^ l_^ b~ I— i l:~. CCXCCi— iClt^-^lOOCCC cc T— 1 Ci o IC ^ t^ cc -* Ci b- <— J CC r~t CC O C-1 cc C-1 Ci -H CO O O C: fc Cv) 'C CO CO l~- -H ':^ t^ 1^ -T^J — - t^ '+' o --C CI 1-- cc cc 'T^ Ci cc X cc 1- y—^ i^C CT. CC b~ o cc cc Ci CM CC iC O CC c; O C' — .c? cc ^ ic --,c cc r- CC Ci o ,_, _l CI cc •rt^ CC X ^•^ C<1 ct o o oooooooooo ,— 1 ^.^ ,— , ■— 1 1—1 r-H ^H o b- l^ t- t, t-- l^ b- b- I - b- b- Ij- t-- l- t^ t^ b- b~ I;- b- b- b- b- l;- a "^i C5 CCt-^CCCCCvI— ICCCSCOO cc o cc. o cc O Ci rH t^ »^ cq o ■"^ o "^ccb-cccccr. cci^ic-^ CJ I- f— i CC tH CC —' CC -^ -^ LC C^-J Ci lO —^ t^ cc Ci 'i- O >-C O >C o — M ,^ cc 1- ,_, IC X C5 IC rr. -^ j^ o Ci »^ CC' i-c cc c") o Ci X cc 1-c cc CM o X 1^ uc -+< CM o Ci b- JO -^ , Ci Ci o — ' CT cc -^ >c; Lc cc i^ CO Ci o •— 1 _4 CM cc ~f »c cc cc b- X Ci Ci O oooooooooo -H —H -H I— i 7—1 1—1 T-^ 1—1 o cc b- t-- t^ t- t^ t, t-- t>. t-, b_ l~_ b- I— b- b- l- b- b- b- l^ b- b~ b- -^ l^ l- O >C ^ lO O t^ X (M Ci o •rf , : C:j t^ »c ro _, r^ t^ iC 1- CC :^ 1- cc X O — O X cc t- o o o b- cc 1— o 1^ cc b- O y-^ I- CC o cc cc Ci »0 O cc — 1 1- CI l- ^^ cc o ^T*< C^l CC o cc t^ <^,' CC /-^ ^'m 'JZ 1 — lC -:f CI — 1 C X I- i-C -H Cq w^ Ci re CC .^.JJ CC 1 — ' f-: X tc JC ^c O -X Ci "^ — C^i cc '^ lO iC cc 1— cc Ci c~. ^""^ CI /-^^ *»4 IC CC cc l^ X CT5 ~ Ci'O OOOOOOOOOO r— _ _l ,— , ,— 1 ,— 1 1—1 "T" ^ '^r L-_ Ij- l^ l~- t^ t^ i^ I- t_ i^ l^ L^ I;- l^ l^ b~ b- l- I;- b- b- b- r- (M cc -T^ ic cc t- CO cri O i-i c;i ^ 'f i-C cc l^ X ^ O ^ CM CC -+| i o o c_; C^ 'O O Clj O^ '-— ^ ^~^ ^^ ■"" .— .^ ^^ ,_, — f -— I y^ 1— ' CI CI CJ CM CM 1 i-C iC ic 'C ic Lc o uc uc j-C i-c .'c; -C »c uc iC iC tCt IC »c IC lO »C iC ! 408 LOGArjTHMS OF XL'MBERS. <5 t^ le re rj -^ ~. t- te -^ -J — ~. X i^ le re m — r; x ^ o -r ^i o <□ 'M'M'M-MT-l— 1 — — ^rt— ^OOOOOOOCiClCiCi CiCiCi 5 XXXXXXOOXOOXXXQOXXXXODl^t^l^t-l^t^t- O ' iM ri I- ?c — . i^ c I- r; vr cc -+• u: rT re o Ti tc — tc ^ - r^ -m t- : j eft 'Ji '"^ -* ^c '^1 * i^ * "1 re rt ""^ VI. re 1 ~" ^ c". * '^ cc I'- "^ re ce 0'>i'ei~c". — 'te— ^'ei-xcoo — ■ — ' — — ^— < — ■ — ^ i^ '-^. zc ! rtj ' a; I- ue re — o 'x -,::-+ :m c x i- -.e re — - i~ i.e re r-i - i- -^t- -m i O -^ fM re -^ L,e o tc t- X r; ~. o ~ "M re re -r le ic I- i^ X r: c .■^ ■*" -M -M -M :m 'M -m -m -m r\ tM 7-1 -M re re re re re re re re re re re re -r [ t- I- I;- L_- I- l_- L;- I- I- I- l;~ l- L~ I;- L- I- I- I- 1- I;- I- l~ I- L- t- 1 tc X ^ '^ o ~ re ii tc -t- I— i-e X o r; i^ O CO 'M .— 1 lo lo o ~ i-^ 1 O »^ ~. 1— I e^a o X -t< CO -^ -M >! O I- "M ^ c^ r: Ci t- re X M -+ -ri' ■ ?^ 'f ^ c; — re -H " I- r: C: — 1 :m c-i re re re re re re re n ei — o X X :i -r" ei — ' C". L- lO re — ' O X vr -T' "M C; X "^ -^ "M ':^ X " -^ ei X C — ei re -+^ -T' Le --r: i- x r-. ~ o — ' "m :e re -^ le --r i- i~ x ~ c ei ei e^i -m ei ei ei •m ei "m ri "m re re re re re re re re re re re re -t- i i_- I- 1- I- t.- i^ 1- i.^ t- t;^ t- t- t~- i;- I;- l;- V" 'ii' 'T '^T t;- *;- i^ i^ t^ o re — re o — 1 I- t- e-i -M o o o O -f -:*• X X re -* o T-i i-^ O t^ X re I- r; o r; vr CI t- o — ' — o i- "M '^ xi t. as i- — ^ ci ei le ic re c^ X o re ~ --r X —. — ei re ~ -7- le Le Le Le Le Le le -+ -*- re :m r^» ■ 1-— le re ei o x ^ *""f ei ^^ c^ i^ i^ re ^— c^ i^ »^e re r— * i~~- ie re — t- o — eice-+^-H0tc:t^xxc;0'^ei£jre^iewwt^xr-. o r~ i'^ r- t^ L— t- I- I— t- t- t^ t- i~^ i-^ t-^ i- i- i- i-^ i-^ i-^ i-^ i-^ t^ t^ -*• ~. X ei c ei c: ei X c •-:: i- re re ~. O i- x -f cc -h vr -m x t- le o -^ I- X i^ Le — Le r". o O c^ "~r — vr X c; ~- 1— "* r; re le '-c i.e X o ei -^ -^ X o — ei -T Le Le ir i^ t- i- i- i~ i^ i^ -.c tr Le -f «o tc *■" re — ^ ^ t-- »*^ — *■ ei o X tc -^ ei o x t^ *■•■' e*! o x ^ --+ '^i O •>c o r-j ei re re -+• »e ^ I- X X ~. c — Ti -,-1 re -* Le -^ -^ i- x r; c; C'l -M •M ei e^i ei ei -M >i c^i CI -M re re re re re re re re re re re re -*i L~- r~ tr- t-- C-- t- I- b- ^- b- 1-- t- t- 1-- I- b- t- 1-- 1-- 1-- 1-- i^ i~- t- b- L^-^iooc;-+'re:c-r't^iet~iei^-rt) — T^ v^ X o ^" re "^ le tc i^ x*c*c^c^*c^c^xi^w m le re ei o X if le re — ~ i-- o re — ~ i^ Le re i— t. i^ Le re — r; o O ^ :m re re -ti le tr; i^ t- X r; o ■ ei re 't- ie »e tr i- x c". c ^ ■M c;i "M e-i d d e-1 ei ei "M ei ei re ce re re re re re re re re re re re l^-, t- l^ b- b- b- t- t- t- I;- I;- I- I;- t-- I;- «;- t- 1^ I;- t;- t~ I- I;- t- I;- — n— 'XCiLeooo-^-t^xt^T— cererei^^.— i.— 'b-x-^t- O lo O ei re re — X re tc X X I- Le o le X r: Ci X »c o -^^ t- X ;-. — -H o X o ei re >e — I- X cr. O — ^ -^ — .— i r-i -^ -^ C cr. x •^ -»■ re — c: 1-- c^ -*' CI o X --T: -f ei — c. 1- Le re T— r: i^ le re o x •^ O — ei CI re -r- Le ■~ x~~ i— x c: c — — ri re ~^ Le >e " i— x —. r^ c) ej ei ei ei ci ei ei ei ei ei ei re re re re ?: re re re re re re re re I, I-- b- t^ i^ i-, |-_ t;, t- 1-, 1-- l^ lj~ u- t- l_- I- t_- L- l^ b- I- t- l- I-- -*■ -T« X t:D ?: --C •' »e ->c CI eq X X -f -+i o O ^ b- re i-e c-i '^ (M ic i^ re i~ o ^ — i c". ^ — ' le t^ t— tr -*" o >e X ri ~ X Le — >e X n o re le X o ci re le t^ X ~ c: — c) re re re re re re re re ci — o cc — C) c; X I- Le re — c. 1^ le -r ci o X -c -* ci o x^ tc -« ci o X re o — ci ci ce -Tt< le tc ~ I- X r: o — ' '-' ci re -f le Le t:; t^ X rv r~. CI ci ci rj c-1 c-j ci ci ci ci ci ci re re re re re re re re re re re re re 1^ t^ (^ t- b- 1^ t;- t- t^ b- t;- t^ t^ t;- t^ I- t- t^ b- t- b- l_- t;- t- t- t^ X -+' -" X t— — r: '— r; — X c t- r; ^ X le X le X i^ ^ c 'e -^ O o X c". ~ X -f o re --T --T " re r; -^ t- ~. c; x le — ^ c: O o^ Le t~ c^ — re »e 1- r; ~ ^ — ci re — -f le le o Le le Le Le -r*- re re c^ re 1—* c^ r^ ^ -^ c I o X t ^ » e re — ci t^ * e re ■ — ■ Ci t^ ^-e re — c i^^ d O •— 1 — 1 ci re -+• 'e vr tf t^ X r; o C — ci re -f -^ le — i- x x t. (CI ci CI CI CI ci ci CI CI CI CI CI re re re re re re re re re re re re re b- t^ t-- 1- I-- t- t- b- I- t^ t- t^ t- i^ t~ t_- t^ b- t- t;- t^ I;- I;- "^ •;- o re o CI X X re re ir- io O X (M o re CI Le -:!• X t- eo»erec:^hit^cr:r;x'^cicr:c;^^ — tc Ci — 1 re >e r^ r; o ci re -f "e •-= --r t- t- r~ i- i- r~ i- ir le le — ci o X 1- le re — r-. X "-C -f ci c X vr -" ci o X :c -ri CI o X — O -^ — ' CI re -" 'e >e --C r— X r: o o — 1 CI re -^ -:h le '^ i~ x x t. CI (Ci CI c^ ci c-i CT ci cm ce re re re re re re re re re re re re 1— 1 — l^lr-t-b-t-b-l^t-b-t-t-t-lr-t-f-t-t-t-t-l>-b-l;-i;-t>«^ reb-"cc5t^c;i-etocirexxrerexxrerexcr. letcre^re r: le o re le le — — r > re -^ -^ ci x re i- z. ~. x tr ci i— c ci le X — re "O i- r; — ci — - >e -^ i^ x x r. — — . r: cs r^ r. x x t- o — ci X to -t< CI o r; 1- ie re — n i— le re — ■ — i^ >e re — c; i^ o c o -^ -^ CI re -^' 'e >e "O r- X CI r; o — ci re re -f le vr t^ i~ x ~. ci c^i ci ci CI CI CI CI CI CI CI CI CI re re re re re re re re re re re re 'r '^ ^r 'r 'r "^^ "^r '^ ^^ *:" 'r 'i' ^" ^^ *;" "i" 'r 'r* "^ 'r* ^ 'r T* ^ T* 6 ! >e to t^ X c» o — ' CT re -" le --:: t- X r. o — CI re -« le tc 1-- X o ,. c 1 ci 'CI CI CI ^^ ^^ re re ^e re *e re ^e re *■*• "*• ""^ ""** ***■ ■•■*' ^^^ "** ^^ "^ X. '^- ' le le le \ct ^.e le i6 o ^e o >o o >!e lo >ra ic o jo le le »o >e le "e >e LOGAEITHMS OF NUMBEES. 409 >J - t- -^ ic re M — ~. 3C -^ ic cc c^) O r: ac t- w -^ >i — O X I- u; - 1 55 X ^ aoxocaoi-^t^i— i-^i— i^L^w:c«C=^5Cc^«£wicoiO i - 1 ^ l^ L^ t^ t^ l^ t^ l^ l^ t- I- I- I- t- I- I- I- t- L- t- t- t- t- t- t- t- 1 W 1 cc -^ tr -f. t^ t- CI -*■ — ic -^ O :m — -O t^ lO r; c: t- O — X c^ CC — * .-4a ^ 1^.^ ^ .-^ * i^ ^ — ^"J — ^ ^^ — iC X O O X ^ ^1 t~ "3 -^ CC— rit^LCcc — xicccc:t~ccot~ccr;«rMi^ccc; ' r> O 3C w -^ M r^ L^ ic cc — X •-:: -Ti- c>j — . i^ ic :m o t- ic cc o X »c ! — ^ C^ — ^ —t- LC ^ l^ X' X C* ^ ~~ ~" "^1 cc ""T* 1-C *-C ^ L^- X X C^ i '^ "7" '^ ~i!t^':^-^-?"-*--r--^-^'-CLCiCiCLC'-Ci-CiC»COlCiCU: , t^ tl. t^ t^Ct— (>.l>.l.^t-~t-l--t^t^t.--t^t>"t-t^t^t^t^t^t—t~t>' : c: c^ O OiO»0'?^»C-^^Ot^^^— it^C^aO-^tC'^O^IOiOOC ^ —' yz ccccxr: x^c1 CC -rr -;?■ o :r I- X X Ci •«^ r^ -rtl ^'^'^-T'-*iT<'*-r-Tit::»oic»coi2ia»cic»c»o»c»o "^ "^ '^r t^t_t^^>.t:^t^^-^-^-t-t^-^^-b-^-^-^-^-l^-^-^-^-^- ' I— 1 O *2 ic >T -^ c^i --r t- cc ^ -?• C". C X c tc ^t C: -^ 3C o — c; X uc o — -~r X I- --C cc X cc :C t- X t^ r — t^ lC -r- -M w t- UC cc C t^ -^ — X iC r- aC -;?■ C: VT ^ X -^J- t-- ~ t^ -^ -^T^ — ^:X^'^"^*l^*2'^l^X^CC— ^XtC"^ — C^"* "^ t^ -^ .-— *— tS2 ^ t~^ t^- X C^ O '"" "* -^1 CC '^ ■T' ^ ^ t^- t^~ X C^ H* "■^ ''^ ^-i.-i.-:i'-#-^-^-+-Ti-u-:LciCLCiciCiC»cicici-cuco t;- t- l5* t— t^ l^ l^ t^ t^ t^ l^ t- t^ L^ t^ tr- t^ «^- t^ t- t- l^ tj- b- t-- ■M rc Ci — C: C^T C-l OO C: t^ — — ' t~ O Ci -f tC 'rw C-. O X S^l CC — . «C -^ — . c-1 cc c^i o I- cc t- r; — o C5 -o oioiciB»cicicic -V- l-^v -v- :i-^_-MC:c^T-^^x»cc:c:^rr;Ci»ct^tcc^iiC':^0 t~^ irf ri 1 ^ , .^mt jl^ .^ cCO*Cd'^1CCCCC'lC^^COCCiCtC*CCCw t^ -* Tt •^ ^^ yz ^ "Tt- ^loi^'^cic^ccc^^ccdic — t"^ cc r^ 12 1^ ir: c^ .1— — • ■* -^ '*-"i ''^ c/~ i ^ cc ^~" X ^ *^ •""" ^ t~~* "^ C"! O t^- IC 71 LC cc cc ^ ic ic t^ t^ X ~. O i — c-'i cc cc ^ UC tc l^ t~ X Ci •*4^ ~^ .-U« _i..^_w^_+i-ti-*.-Tf<-f-icLCicoo»cc:uci-c«c»cir:iL-c — l^ t^ l>- t^ L-_ t~- t-- C^ t^ t^ t^ t~- t- t- t^ t^ t- t— t- t- t^ b- b- t- t>» -" cc t^ C^A-MCi — 0-:^>2— '-*'^^XC;X!MCCr-lC»-C'MCCt^-?< X w cc Ci cc ic I— t-- uc 7-1 cc - t^t^tv-t^t—t^t^r^b-t-t—t^t-t^t-b-t^t-t-t^t— t- L':oob-citr-— 'OosotrO'-oo — 05^cot-c-i'*^it-CiGO r: cc L- o — t- -: — 1 w o LC X r: o c: t::; 7^ t^ — cc -* cc ^ oc ~ cc i^ cc -:*< 71 O X --C <# 71 C. tC cc — t ^ — l^ -rf- O cc 71 X CC ''/S IC c^ — r; t^ LC cc c; X "-C -Tf* — r. i- lc 7i o x lc cc — x cc cc r-. ^k^ t- t- t^ -;-^ _«_i--^_H_j^_j^-jj-t.-«-*.LCL7LCLCLCLCl-CLCLClCLCtC t- t^ l>. t^l>.t^b_l^t~t^l^t^t-b-l--b-t-t-b-b-t~-b-b-b-t>- j ?o ^ I-- t--7171C:7-10»CCCCCCCLC— 'CC71tC00tCO71C:'^»C cc X — 71 C7 71 r; LC c; cc ic cc ic cc Ci -*! X — ' 71 — ■ O t^ t ^■^ cc — " c X i^ lc cc — X cc -T- — X LC 71 C-. LC CI X LC — t- cc X -^ ' •?< ^ -^ X ic cc — r: t^ — e^ 71 c x ic cc — x cc -:*■ — c: t^ -^ 7i Ci 71 cc -^ IC LC cc I- X r-. C: O — 7-1 71 cc ':?' LC LC cc t- X X ~~ "^ t^ t- t_t^l>.t-.t^t-t^t-t— t^l— t— t~t— b^t— b-b-t^-t-t-t- r— tr — -^xox7i7ixc:r;cc-^---*-^ — ccccci-rOccci ^^ ^- ^ ic r; cc -# LC -^ — X 7) cc X r; X cc cc X 71 '^f cc cc -t' — t::: LC cc 71C:~. t^LCCC — XCCCCOt^-^-^X-T- — t^CCCllCi— 1 o 1 ^^ ^.. * I- LC 71 X cc -^ — r: t- ic 71 X LC cc — X cc cc — c. O ' — • ^-■ 71 cc -" LC LC cc I- X X C: — 71 71 cc -^ LC LC cc t- X X *~* *^ *^ -T*--T^^^-^-r^ — — -^-^lcolcl::lclclclclclc»clc t- t^ I- t-- t^ t~ O- t- t- t- 1-- I;- t^ i^^ t;- t- b- t- t- t- t- t^ t_^ t~ t- 1 1 O — r^T cc -* iC cc l- X C: — ' 71 cc -f 'C cc t- X —. — CI cc -t- _. o 1 o ic ir. LC >c LC ic LC ic ic cc cc ec cc cc cc cc cc cc cc 1- 1- i^ i- »~- yi PC 1 >.C LC .c 'C LC LC LC IC IC >C LC LC IC C LC LC LC 410 LOGAPJTn:iS OF XUMFEES. ;_• O r: :>i •—k - oc b- •-r -j. ^ ?:« - -~ CC 1^ »e ^"* -^ -M "^ _ Ti r^ ■■^ O =: u; o o ^~i -f* -:*' -^ •-^ ■^ "^ — r *r- rc CC ce ce ^e re *^ ■M 7J ei "M e-i - t~ i^ i^ i^ t- t— t^ t^ l^ l^ I- l^ I— l^ l^ l^ i^ i^ L^ t^ t~ l^ t^ l^ i^ ^ ^ ic c~ 1^ _, ■^ -f _ O t^ - t;- t- l— l^ 1— I;- l^ r^ i— t^ l^ ^:- l^ ZTi O >^2 ■M :^ i^ ti , -^ ^ ,^ .-s CC 30 »e — . -M -M — ,-. on ^, T. »o •— »C l^ 1 1. ^~ r^ * — 1^ * ^^ "*^ l~. vi— •'"* 1'^ T „ -~~ 1- r^l l^ •M y— Tl 1- — •^ -*■ ~ Z 1 ^ — I^ — 1- — , 71 Le rr. . — . T ■M o L"* u: ■>\ :,^ I— o -M '^ 1^ T" c^t c^ I- "TT — ^ "^^ ^ ce — . -r o e-1 ^^ X _^ « .— ■ :r4 -C — T^ -^ •-C t^ I- i -» 3U ^. ^ ■^^ r—m :^^ ri *^ -r o »e i'- rri ^ tC w t^ ^ O w w o w tc ^ t^ t^ «^ v— t^ t^ 1^ r- 1^ t~- r^ 1^ t» t;- t- t- i- t^ t- t- I;- l- t~ t- t- L^ l^ I— t- t^ t;- I;- 1-- 1^ L;- t-- 1-- t- ?1 — -I. »c ~> 3C _^ o t^ __ C^T _ r^ -^ __ — .^ — — — .» ^ ~. ^ ^ O -M ^ rt —^ ^ *:; * re Le t^ Le '— ^^ t- *_ — 1^ r- i~ • — -* -^ *^ Wi »^ o iC ■:^ iC c^ -T" a. ce t^ — ic ^ ce tc :3. r't ^ ^ ^j l'^ T. -* ^•^ ^^ C^ I'— -r" Ti ^^ vc ■"T7" —« *, ^ -T— *^ V. •^ re w« '/: i.e -M -"^ I-- ■*^* "M /— ^ t* o r^ _ ■>\ r^ ri — o "^ :^ 1^ -J. C- Ci o — e^i -:\ re o — t^ r^ c. *» '-^ \^ c; ,^ ^^ ,^ ■^— ' ^v ^_. ,^ ^«, -^ i^ 1— I— 1- l^- J- »^ 1^ 1^ t^ l^ r^ t- t~ 1- L;- i^ t;- L- I;- I- l;- l;- l^ l- i~ i^ y— c^ l^ t- i;- t^ l^ i^ X vr -M — — o -* 13 r^ 00 — — 00 re lo -^ •>! --C c; c: t^ -M tr t^ -^ lO-^i^xxt- — r-O'^x — CI — ct^rex — re-^-rceO"-c — evj i^ c^i i^ - J i^ ■>! t i.e c: — x -i le r; ■m --^ ~ ?i le x — ^e tr 'o — 2 j£ if^ ~ S ^ V ~ ?^ — r^, 5 i '^ '"^ 5^ Lr ~ 32 £: '-€ rr "~ =^ ir I— t^ ?^ t^ t^ t^ t^ t^ t^ C^ t^ t^ t^ l~ t— L— t^ l^ t^ t^ t^ t^ I— t~ t^ -:?C"I»icieo-r^c:-^XTjv£o-— X— oxc-iJiex— revrx t^ t~ t^ t~ t^ t-~ t^ t^ L— t^ l^ 1-- t- t^ t— t^ t~ t-~ C^ t^ i^ i^ t^ (^ t^ CiOooro»o-*0'^'=**ocoooo<^^cc^-~ Ci-r• 7] t^ -M '-£ — i-e n re I- x ^ r^ O re tr: c: — Ho — e'lJice-^ootct^-xxr^o ^?jre — -^oirtDt^ — ' tc t:; "M ^ w ?e CC o ~. ie c} o X — X c; r^ re i~ ~. x Le o ce c; 5 — M ri re -r o ii: -3 i-^ X X c^ o — — 71 re — — ue S C:; r^ " u; vr " vr ts tc — — tr; " w CC ir ■^C i^ r^ i^ I- I- t^ (^ i^ 1^ t^ i^ l^ I— I— i^ i^ I- i^ t— l^ L^ t~ I- t~ i^ i^ l— L- I- t^ t^ i^ i^ t^ i-^ t^ ir- r. rere— i:ct-"e-ioocexiOC:t^'M-^-^.-'?oc:ooo-*oo X re cp X Xi i^ o ei i^ — - -^ Le ^ '^- ^J £;• It S ::: 2 ^J ~ "• ^ "" -v-i x> i.e re c: X ie re c X le re c: 1^ 'e fi r: T-l — ri H t5 re — x le -m ci o — C"! fi re -:• »e ».e — I— X X r; c O — ei re re -r le cr cc »>• ' ; O tr tC tc t: — ts tr vr ^ :C tc -.r "-C I- l^ l^ I- t^ l^ t^ l^. l^ l^ 1^ I t~ t^ 1:^ t— t— i^ t- t- r^ t^ t^ t^ l~ t- i^ t- l^ i^ t^ t~- t^ r^ i^ t^ 1^ I I oe i^ — ' re re ei c t— re 1^ O — "M — ~ i^ — i.e i^ cr. o ~ i~ re t. ■^ ~. o c le c; 'e ~ -^ X re I— — le x e-i tc ~. ?i i.e ~ — — i- n t^ -*• ei o t^ 'e r>i ~ i- " "M r; i- — • — ~ :c re — x - I- I- i^ t— t^ t^ t^ i~ i^ t~ t^ t^ t^ t^ t^ »^ i^ t-_ i^ t~ t— t^ t^ i^ 00 o CC 00 o o — ' o o X Ci ^ — re ce c >e I- 1^ — ^ o re re e-j x i r^ e-i lO i^ cc oc CC ce X "M >e t^ oc i^ i.e ri 1^ ^ cr i- u: -tm — tr I cr e-i t— -M i^ e^i i— 'M — — >c r: re <— — »e x rj le x — -r i^ o "M i --> iCC -^ — ~ — — — c; •— — — X — re — X »e re o t^ >e ■?! r; r^ "*< <-; ■-' ; c: c — — "M re -*• — I i^ t^ t— t^ t^ 1-- t-^ t— t— i^ t>- t^ i^ t- i^ 1-- i^ t^ t— i^ i~ t-- 1-- t-- i^ • i «e CC t~ X Ci O -- e-i ce -^ 'e CC t^ X n o — c^j re -^ le CC t-- X r; - . i^ (, i_ i~ I, :c X X X X X X X X x'~. r. r-. c. ~ r: r: r: r: T. LOGARITHMS OF NUMBERS. 411 ^ -M ^ - CO t^ .- »o ^ Ol ^"^ ^^ "^^ CO l~- ■""■" lO ^■^ OJ "■" ^ _ r I- CO ' ~ M -M >i !M — r— ^- — < ^- — -^ ' — .— ~ o ~ :^ C: o ~ 3 ~ ~ C". cr. .~ P, I— t- c- t- l^ t- t- V— t- tr- t- t^- t- l- L- 1^- I— l- L— L- L- — ~ O lO '"' 'M CO co 3 -^ o .— 1 _^ O O „ O to to O CM X CM lO O 1^ CO TO r« ■«r^ »r: IC CO i^ ■>\ ee ^ (^ C^ t^ O 'r^ X o (T-. rr> lO o o :>j ■^ tr CO '^ *— 00 'TTl o 1- cr^ GT) 05 C^ ,— ^ ^ CM Ol CM CM a: C/J o c^i d v^ ~T* — « CO UO Ol ■C^ to CO ■^ X lO Ol o^ CO :^ r- •^ , r y^ cc ■J- c^: O' — ^ ■>i CO rc -^ to lO w 1 — CJ -* r- CO CI o y-A C^l CO CO ■^ lO lO lO >o — ^ -i^ -i' ^ ^ -* T— I CO c^ CO o 1^ ~^ I— 1 C/J to CO v_^ t- '*. •— < X lO o^ Ci • * ^^ O tr- X * ^ ^ ^n C^J CO CO — ^ >o »o CO t- COi X ^-» ^ o 01 on ^^-^ -r+j 5 lo I- l^ CO OO CO CO CO CO CO CO CO CO CO CO X X Ci Ci cr. CTi CTi Ci Ci i^ l^ t- tr- t- tr- l^ b- t;- l;- I;- tr- I— t— I— t— I— t- t- tr- t-- t>- t- b- t_- •^ — r-1 r— 1 CO lO (M w* tr: ■r:+^ *— 4 crj iO CM CI to CO O' r- '^ 1—1 X lO 01 t* 5C ^ o o ^« ?M C^4 00 -T+^ o lO CO I- I— X O; C^l Ol CO -r* ^ lO l^ t- oo oo O) CO CO CO CO CO CO GO CO GC^ CO X c; * cr. c:: -^ Ci Ci t^ t- t-- t- t~ b- b- b- l^ Ij- t;- t— t- tr- t- I;- t* tr- t- Ir- t- tr- t- tr- l- 1 CO GO ^^ (M „H C5 -*l CO c: O GO lO O 't- -tc -^ ic ee -~ -* X o 1 — O 1 '^ t^ Oi C5 CO O CM r^ ^— »o ?w 1- t- io C^l zr, cc 1- c^ 1— 1 ^^ ^^ Ci to Ol CO C-i ^ tr^ CO o ^— CO' — O to t^ CO o^ 0^ c*^ <^ ■C^ *^ ^■^ ..^ o o o o ^ CO :.^ t^ -# ^™ ^-. tc CO ■:^ r- ■•-H ^— 00 lO C-l ■'"' 1 — T ,—1 ■X. 1^ 01 — ., -.-K -^ ^ CO Ci o o ^^ 04 C^l CO -* lO lO to 1- I-- X * ^ o C^J CM CO "* -:^ lO >^ t- t~ GO aj QO CO C*^ GO CO CC CO ■CO oc CO X X Ci Ci Ci Ci C7i (75 Ci Ci t~ I;- l^ t>- t- tr- b- t^ b- t;- "T" *r I;- I- t^- t- t- b- b- tr- b- l^ tr- b- tr- o '^ ^ r^ CO ^_ cc CO w f— 1-- to „j to — ^_ _^ O b- oc rr 1—1 _« io -ri* CO O l^ t- ^ Tl '^ w ^ CO lO to to -T* ^-. X cc t^ r-H j^ •— ^ CO <7■^ ^- CO 1^ l^ wi . — 1 CO ■rt> o t— CO Ot o -^ CM CM cc CO co *^ •^ '^ -/.- CO CO o iii ;m Ci t^ CO f— 1 CO to C^J Cj o CO ^- oc lO CM c^ CO CO o t^ ^^ X lO o CO w ;^ o »— 1 ,^^ C-) CO ■■^ ■'T^ »o c^ 1- r- x; C 'C^ o ^^ Ol ot CO ■v^ -h lO l^ t^ l^ CO CO OO o^ CO GO GO CO CO CO 00 X X CO c^ Ci CT. sTi t-- l^ t^ i^ t- t- t- l^ b- t^- b- t- t- t^ tr- I- tr- b- ^r tr- '^ t- tr— l^ t- l^ "* c^ cc lO '^t* CM oq «n CO Ci o CC lO »o -* CM Ci T' C^ OJ •»-< »o »o CO ^^ t- C^l rr^ to C-T ■r^ w CO o ->4 "^ l^ t- CO c_^ ^-^ C-1 CO -rt* lO lO to w b- b- t- to CO to -* 1 '^ ^- CO w CO o l- -:r — * w ^'-' c_^ t^ — H ^« X lO CJ to tr— ■>:*< '^l ' CO ■J- C^» r,^ ^^ CM C^J CO -Tt" — ^ o to t^ t^ X c^ CTi o f-—! f— • c^n r-^ '^t lO t- i^ l^ CO CO CO GO CO GO c/j CO OO CO 00 X X X cr: ■TTi m cr. t- t^ t>- t;- ^- t;- t- t- b- t- t- t;- t- t-- t^- b- b- b- 1^ t^ t;- i^ t- I;- tr- Ci^ !M — CO to t- tc CO CO CM -*H '<*' f^ o o — -CM ^ -M r— cr -M X — CO CO ^^ C-1 CO (M O I- CO t^ T~^ CO ■•■* *^ ^- to Ot to ,-_i ^. tr— CO o CI T— 1 CO »-0 l^ CO c^ ^^ 00 -u- »o to t- X' X C^ C^ r^. c. <-^, '^ CTi Ci '^^ t cc ■^^ CO l:^ '>\ Ci tc -r*" ^— cr. *:: C-l :,; to CO c_; t- —tw 1-^ C^ to CC r^ 00 "• cc- Ci Z1 '^' ^^ ^-' C^l CO "r^ ■r*H o to to I— X Oi C^ o ^- r^ on r^. CO -Tf- lO t^ t^ l^- ZIj CO^ C/J C/^ CO CO —* 1 CO o; c; ',^ ^H ^ Ol CO CO ■r*-. t-O to to t^ X X c^ r-. 01 y-«.« ># o L- t^ 00 aj CO CO CO CO CO CO' c/. 00 CO X X X Ci c^ CI r^ .--^ t- t^ I;- I;- I;- t^ t- t^ 1^ I— I- I— I- I- t— i- tr- I— l- t- t— T- b^ CO lO l-O CO CT. -t^ CC i^ ^ CO 00 CM -r^ IC ■rtH , i r- CM io CO t^ co -* o to »o 1^ 3=:' t- CO LO C^l I— GO %: t— c>i ^ o O CO to to lO b- O lO X O X T. c: CTi OX'*' Ci X CO x X ■^ »— C/_' uo CO . t- ■— ' r— w- ^e 'Z1 •,_^ r— "r^ ■—1 X lO O) rr. t^ 00 f-: t— ■^ 1—1 o V '^^ ^— <— ', c^^ CO co "^ LO to to t^ X X Cw o O ,_ Ol ■-o '^ lO l^ l^ t~ oc ■J^ CO X' QfJ orj CO -Jj GO' GO C/^ X X X Ci ■■T; --- Ci Ci ! '-r t- t;- L^ t- b- t- t- b- t;- t^ l;- «r- t^- t- b- t- b- tr- ^- t^ t- tr- b- t- _• 1 o ^ C^-J CO '^t O ^ r^ CO ^ ,— - -^ CM cc -+( lO t-r t— ro ,-^ r^ . 'M /vv ,^ zi^' £ i — Tt _£ Z. ^ _£ _^ — 1 ^ ^ """^ CM CM CM S§ o 412 LOGAEITHMS OF ^•^MBEES. T CC C^l ^ o ^'"' X i^ CC i-c -f ^1 , --^ ,^ X l~ UB^ ^"^ ^*" ^1 __ , ! * C^ Ci X X X X X X X X X l^ 1 — t~ ( ^ l~ 1^ i^ 1^ ,^ I, — ! ^ iC CC CC CC CC CC CC CC CC CC CC CC CC -' ^ ^ CC CC CC CC CC CC CC 1 " O CC IfJ CC CC o '^ ^ X CC x <^4 tc er. — r-^ ^ -4. •T+i ■r« CC C^l O X o l^ 00 ..■^ <— ' 1—1 :": X iir: o ic X ^ CM CI M f,^. 1^ CC X CM ic 1- X I- CC Ci oc i^ CC -^ CC C^l O X 1 — 1 C CC 1 — *■; CC "— — Ci C5 o — 00 i^c cq Ci CC CC O t -• CC o t^ ""^ ' — I- ^-*- » — X -;^ ^ X IC r- X ^^ •c t^ t^ CO Ci Ci X !^ — 23 i^ ~ ~~ ^Jl ':£ X 1^ X X ;i^ O O — M (M t- t- t- t- t- t- X X X CO X) ex 00 ex ex. ex X ex ex X CO X CO X X o o IC »o t^ t^ CC IC C^J Ci iC — »C Ci cq lO t^ cr. c o Cr O Ci X t- »:; c; 1— J ) O I- CC X CM '^ CC CC CC — - M X CC i^ C C( CC CC M tC i^ IC »-— ^^ C) — Ci X CC »c CC !— Ci t^ LC CC '^^ X «c r't O I "— """^ '"" l> CC C 1^ •"^ ^— X. IC — X IC CI Ci CC M Ci tc CC :^ t^ CC O I ^ CC w t^^ t^ to t- 1^ CO c; Ci O ^ ^" C-1 CC CC — *■ *c »c ::^ I- X X Ci O O — ^J C"! t^ t- t- t- t;- ^- X' X CO CX) X X ex ex cz; X ?- ?" X CC :c CC (X- oo CC b- -*■ Ci -t- t- O r-l (M C1 -C C^l ^ CC ^_ t^ CI CT O '^ CO 'M O t- t- r- »C CC Ci -f X ,_ ^ .— , — - CC r~ CC '/ ' — 1 "^*" iC tC iC CO CC «^ CC UC ■^ CC CM O Ci 1- t^ "T^ ">! O -r CC CC _ X CO C t- ■-*i — CC' IC CM c. CC '^; c: '-T: CC O I' CC ^ t^ T— D l^ '— ' , — . t^ — ^ ^ iO ^ t^ CC CC ^^ ^^ w """^ CI C i CC ■**" iC LC CC r- r^ v c d ^^ »— ' ^^ C'l o c:^ o o o c t>. b- t- t- t-- t- X (X c^ CC X ex ex ex- ex X X :/j C C^l Ci CC CC ^ \z CC ^ CC CC CD l^- CC Cej «o ;:: 1^ 00 CC Ci c:; cr -^ CM c J CC *^ -^ KC CC CC t— cy. Ci C5 Ci Ci Ci Ci Ci o c c:; o c o c o o o o o o o O — ^- '— ■— *i~ "^ ^ ^ t^ b- ex X OD CO CO ex X ex CO ^ exj °p ?^ ex X X CO e» c» »o (>• i"n CO t— tr- ^ -^ CD b- -^ O CC <^ oc Ci X >C IM l^ 'M UC I- X X 1- IC CI X -^ -*i -*< •M — o Ci X t^ ic: ^t" CI O X CC -^ M ^ I- >c ■M c; t * — Ci W CC O t- o I- -^ — X IC CT X IC C) Ci CC M Ci CC CI Ci CC CC , »o iC 1- 00 CC c^ O C^ — CI CM CC ■"*' '■*' IC tj^ CC l^ V, r/. Ci CD C — CM Ci Ci c; o o o cr; t-- t;- t~- t^ I;- t;- X X X X CC X ex CC X X CX- c/. CC X ex ex CC ex ex O 7C IC tC' CC IC ■r** ^ b- CC t— — Th r— c: ^ ^ o O ^. t- IC CC O t^ O -:*' t- Ci C ^ c; I- CC ~ CC i^ Ci o o c X >~ *.- IC Ci CI -f »c -^ CC l^ CC »-C >c -♦1 c^i -^ o X I- IC CC ■ 1 o X >c CC y—m X »c CC o t — ■*' o 00 »c C^l Ci CC CC ^^ X IC CI X IC M Ci »C CI c »o '^ 1^ t— CO <-^ O O ^ CM CI CC '^ "^^ IC CC CC t^ X X Ci O O — CM Ci Ci Ci Ci .— CD O O O er o o o o o o o o o O '— — ' ^- — »>. t^ l_- t- t-- b- ex CO OO CO 00 op X CO c» X X X au (X) X X X 00 00 ■ kct -s: 1^ CC Ci O — CI CC -:*< "C CC t~- X Ci --. __ CI CC _H IC CC t^ X Ci y. X 1 — — — — - - — — ~ — ~ ~ ^^ ^^ ~ LOGAEITHMS OF NDMBEES. 413 rt5 (M c\i COQOa)OOCC>C»OOGOOOCOCOX)GOOOGOCX;QOGOCOC»COOOCX)CX)cyO CO -:*< O t- cc ~. iO rt t^ !M cc' re ~ ■■ " JC "# -* Kl 1— I C<) 0cX)aC)GOC»cX)00 ^-i ' — ! -'• -*» •'• jn ^>— ^ ^^-^ s»N i,^* ^-^ —i 1^. t^». '^L,^ t;; i^.:; ;^ t;^^ r-H icj O t- CC ~. iO rt t^ CM CC' n Ci -^ Ci -ri- C5 '^ Ci -f CO Cit L^ ^i^-^ot-'+iOt^cftOoiccito^iaiiocvicoo.— cot^coioroiMOcot^iOcoT— iOGOt^o»c-*icccf;cc:c-ico-Tt*r-i j!^-:t<'+IOO«Ct— CX-COCiOO-— '— i!M:CCC-*»C10?Cl^t-(X)C: ^ -H ^ -H ri 1— I ^ r^ r-J ^ C^^ -M !M C^l !M CM 5>1 (M !M (M CM CM C^ CM (M C»C»CCaOCX)C)OOOCOC»COCXiCX'COaO(X)OOC»GOC)OCCCOiX)OOCX)00 1— (rHr-ioocic;oot-ooiO-^'^rcccrccO'*'*ot-c^.— (cc •^r-(l>-C<4«r>(300'-^i-HOCOlOi— liOOCitOCO^OCOO^OCM^ r-C0'#^t-CCOO0ClC0C0C?JOOtC^«0'-HC0^5C>— (100-* f^C5icrcc^ocC'3;cD(MCJiC(MCiioc0QOODa0COCX)COQO0OC»C»O0C»00O0O0 C(^':tH»OtDt^t^t^COQOCX)COCX)C^CiO^M!Mf^'*COOOOCCtOO t- -^ o o cr: M "-r* 10 tr: '^i c-COC5C50 — ^-^GvirOCOTriiCOCCl^t^csOOO ■-^ — ^ r-l — I " -— 1 ^ -— I '— ! ^ 3^1 C<) 3-1 iM M Ol C^l CM -M CM (M (M 2^1 Sq COXCOCOX'XXXXXOOCOXCOCOXQOCKCOCOaOaOCX)<»00 ic; X o >— 1 re -f w' -'• .-* '-J^ '-. •-. ^^ «.^ *.— -f-i ^. ^— ICOCMt^CMtCF— OOlC rr r^t-'#>— (t^'^i— t-^MOt^CC50t^?COOCOcr:OCMaiO(MCO • CC«-<*^O^CCC:l:-t-C0C5ClO--1>--CMr0I0-^-*iC«r^^^^~X^0 ^ — i-H 1— I ^ rH ,-^ r— r-H r- ^ CM CM CM C^l CM » CM Cq Cq C<1 C<1 (M C<1 (M aDCO00COGO(3OCO00COCX>COO0COCOCOXQ0COCO0000CX)COaDC» O-TlCOC-liOCr. CMOCiCMiOXCqoCiCMtOOOCi'^ClW.-it- t^-^OCOOCCOt^t^t-lC'MCi-^X'CM-^OCOiOTHi— lOO'^'CX) Oi^ccot-ccc:orc:o^roc^toc^ic^ocM^-ooqa)'*rHt- co cc ^ o ic ^ I- I- X cr: cr. O' o -^ CM CM n? -*H ^ o tcco t- 00 00 ,., r-^ ^ ,_, ,H r-^ r— — . r— ^ -^ -M C^^ (M C^CI CM !M CM CM (M (M (M CM ?1 -M COOOCOCOCOC»COOOC)OCOC»000000<»COOOOOOOCX)X(X'aJX X O »0 Ci o ^ — OCJ t^ Cq t^ — ' CO O 1-- -+I Ci -*i t- X -* — ( t-- ^ CD "Occc^iCi?T~trcq^ c-lcO'*i'*r: ^ X o CO -— cr. ---::;'i^t^~^'-'^ic>ir;co-^QC'COX'— -+io«ccC'*cM XT ;5 32 2 "5 ^1 ~- >-'^ CM X ic --1 CO o r::; X ^ ;.; ^:: ^ o ?- CO o -3 CMCC-^-^OOCOt-XCOCT. OO-^:MC?)C0-+-^i0 5cCt-0000 '~*'~^'~*'~''^'~''~^^— •'^^^CqcqC^IC^lCqc^JC^CICqCqcMcMCMCMSq COC»COC»OOC30a)cpcOCOCOCOOCCOCOCCXC»CCXOO(Xl(X)C»C» z, ' ^ 414 LOGAEITHMS OF XL'MBEFvS. ^ re 71 — o r: X t^ tr le -r re 71 71 — o ~ X t^ --c le ~ — re 71 — ^ - -^ ^ — :■: r^ r^ r^ re r^ :- :^ ?t re ^^ M -M c— ~— ^ >— >— — >^^ ^~,-,^,i^>_.^>^>^-_._. — . — .^ ^^ — >»^_ — '^ L-- vr X o re t ->r: :m oo L.e ce ^3 — -M re vr — re X Le :m o O ?j -* le ce ie re c: i~- M t- 3 re o tc ^s o re o --r r-i -c; c; re i~ -o 00 :m — C: "^ X >! o r; ri ^r r; M le X r t^ — 7^ — t- — — re -^ ; X 'e — X -?" C: t^ re ~ ^ :m :c le — t~ -^ c; — >i n le t^ — ^ — _ ^* xxxxxxxxxxccxxxxxxxxxxaoxxx • ^i--+*^oc7iieOi2— txiere7i — 7-ireicr:rex-^ — oc:^ 3CO— • ^c. b-reciret-0 7i^:re7ior — r — -^r-x — 7ire — — C: --• X — i-e r; m tc r. re -.r r. 7i ue x c; re ue x o re o t- X ■JO *^ •"■ k^* ^ '^^^ » T-J Ci i-T^ ^— l/^ ■"* ^ c^» CC '^- » C'l cc —^ ^™ l>- *^ "^ X CiO-"— 'Tir^CC'^ "^ L^ vc tc t~^ V" >I C^ '^ o — 7 i Z-Z ^'^ ^ — ■tsiccrcrcc^r^c^c^rcccc^ccccr^r^cccc "^ ""^ ""^ ""^ """ ■•^ ■'^ "^ ajoccoxocxxxxiaococcc^ Ci -M le X re X re — tc re — o C: — M le X ri i^ re c: X r- i^ c; re ^ t^ t"^ t>* te re * *e O- -^^ ("^ c» o o cv i^ i-e — t^ t-i i-e x '^ ^ 1^ * 7^ f. — L!^ * ^"J %^ ^ ^^ '^ ^ ;* -^ y^ _ ■— j- f^^_ ^ ■^ ^ »— ^ * ^_ t>. 1 t>- re o — re n Le c^i X le — t ~ re o "-C 71 c^ i-e — P- — o — 7 ) ~ t- 1 ^* '^-^ ^" "~* ^i ^ i re ■"?■ *■" i^rz » '-^ L"^ ~j X ~ 3^ ^ — "— T"! re *^ -^ -™ ;^r:c^:':rcr:r:r^r^r':^t:':r^rtr^-^'^ — . — — iCOXXXXXXCCXXCCCCO^aCXCCXX-JCXCCCCXXOC coo-rci'*citrre— iCixxc:c:retc:C;»c — xccueiev^x ci :m re re re — r^ tD 7-1 :c o re le t- t- tr »c ei c; ~i — . re -~r X n =c rt i^ ue n c-i ^ O re t- c re u; c; ei le X c re ue X o ei ^r •■* ' w re c^ » ei X le — X *^ o i^ re Ci *-e ei x *"• — t^ re ^ * 7i x .-. cTi o O ■" ej ej re *™ "^ le tc *» t^- t^- c/^ ir. c^ o ^^ "™ 7j ei *^ -^" *?" •^ 71 re re re re re re re re re re re re re re re re ""^ -^ -rr -rr -^ -^ ■*— •^- !Xxxxxxcx><»xxacaDXxccxxccccc^ 1 -5f X re r: Le — — . t^ »e ue le Le r^ — re t^ ei X le re 7i -e t>- lO t~^ c^ c^ c X 1-^ 1M X r^ t^ O C'T r^ ~^ rt C'l c^ vi; ^^ t-^ — '■^ tc c^ C^l^O'T'XTlt^ Or^t^O T l"^ O -* ^ C^ — "^ l^ C^ CI *— tC X sC "^1 c^ iC ^- X — ^ ^- t^ re c^ tc !M * tc ^ i>- "^^ o » ■?'i c^ *-t — t"^ iC C» O" ^ ^~ 'i'j T^j r^ ""^ ■"■" i-T; ;^ t^ t"^- t"^ "yz in c^ "^ — ^— d ci re "^^ •■« c^ rc re re ce re re re ce re "^e re re re r^ re *^ ■•™' -^ "^ ■*"■ ■™ ■■• —*• ^^^ ooxxxxxxxxxxxxxxxxxxxxxxxx — '-C 71 X »-e re — o O o — re ue — re X -^i- -T X ir- t-- X r; 71 ic ^ce»^»eu:-r7^rii-eo-#t-r:0 — oc-. i^rer-. -r'x — -?-i-e :c o -f X 71 tr c; re t *■ t- c -^ t- c 7i le x o re i-e x c 7i le e*! X '"■^" — " I"- —^ o * ^ * '-e e^] x ■*• ^~ i~-- re c^ ^-^ e'^ x "^^ *^ t>* ■^ c^ O o — " e 1 ei re ~" '^ *-e o ^ i>- t^ t/^r^c^.^o — et'cice ■■*• "^^ e*^ ce ce ^ re re re re re re re -^ ce ** re re re "■"" —^ —^ ■"■ ■*• """^ •• — ^ aOXXXXXXXXOOOOOOOGXXXCCCCCCXX t^-^ — xtCLe-*^-^'<^ir:t^Orexrer:^reci'M'M-^t^'^io ^o C- •■" -^ ^" ^ X '-^ ^^ » * ""^ '-^ t"^ y v^ » "^ — t"^ "^i "'*' '"' '*'j "^^ * re X "CI * O re r^* — -^ x -^ "T" t^^ o re w c^ 71 — r^ "^ ^- ""^ v^ /»^ ■^ — " t"~ T t"~ ^ ce ^ ei X — I"- "T" ^ ei "■. le — • t^ "•*• '"^ t^ <»^ • • e^^ecer:rere:erecerererererer:rece^ — -3--^ — ^-r — cpxxxxxxxxxxxxxxxxxxxxxxxx -^-Mrixt-trtri^^ — -tPt^Mt-reot^tctctrxo — X — T^ Le ^ i^ t^ ^r •^ — t- ce r^ -e -T^ ce — 1 '-/' ^!- ce --r t- ce — cc I- Le c: re t r* x — i-c x ^ i-- o ce ^e x c; re le 1^ C M "^ '-^..^ t~^ ^ C- "-^ -"J c^ *^ — vj ""^ ^ t^* cec^tcT'i'v'-'— — ^t^'"e^^ ' c^^ w* '-^ -^ *"" "^ Ci c* re 'T' *rt i "z '-^ r^— I-— V" '>" ce "^ "" e^i '^t '"■^ ^^ ■^** I ^1 ce ce ce ce ce ce ce ce ce ce 'e *^ *^ ** ^^ "'^ "^ ""~ ""^ •■" *^** ■•*• "^^ ■•*■ 1 - WOCXXXCXlXXXCCCOGCXCCOOCCCOXOCOOCOXCC I • • • t — X -^ c: -^ 1- c: — 71 71 c X ue 71 i^ o t^ c: tc •— ' *-e c^ ce t '^ ™" "^ X ^™ ie X' -^ ^ e t/" — '^ vc ce t^i '■^ t^ -^^ •— ■ "'^ ^^ ceo^eic^'eeiX"^ — t^ce*^e'ic^ie — i^ ~^ ■ — ■'ic'^i'^ir' c;oc eicece'^-'ejet^t-i-xxceoc: — c^ie^jcece^-* "CI ce ce ce ce ce ce ce ce ce ce ce ce ce ce ce ce "^*' "^ ■■■*^ ■'^ "^r* '■^ ■•*" '■^ xccxxxxxxxxccxccooccxaoxxxobccxxx , QO t- t^ t- X n t^ — --r — r — - ei — o — 71 le X 71 X -rr M 1 1 O -f X 71 tc o -f x 71 i-e — 71 »e cc r t~ c re o x c re '.e r^ 1 Q re — 'e 71 X ie — I — ?> c -— re ~ le ei x -^ — t- re r. -^ 7i x -r „,^ 1 Ci Ci O ""^ '~~* Ti r^ r^ *■" '* 4^ '-^ t^ I'— VI "Ui. c^ o O ^~ ^^ T^j "^ ■""* ■■"*• "■^ j ^'i '^-t ^-* *»^ *»^ *-^ *■* *>» *^ -^ *^ «^ -Y* *^ -^ *>v- — • »». .^^ „^ ..^ ,,^, .^ ..^ ^^ aoxxxxx)xxxxxx>aDxxxxxxxxxxxoo 'etsr^acc. o-— e-ire-#ictcr^aoci0^^7-icc->*4»occh-ooc: ,' ■ '.r ?:: rr r:: Ir ?S ?2 ;5 ?5 ?5 '^ X ?5 ?S '* '^ r!r ^ ^ ^ ^- ^ ^ ^ ^• >^ LOGARITHMS OF A'UilBERS. 415 rnTo •^ "/' r^ CC iC IC 'tH ■^ ■M ^"" ^"" Ci 00 t- t— cc IC -rl- cc cc Cl _ ^ S 'fc ■~ 1 >i ,_^ —i — H ^^ ^-, ^-^ —^ ,— ^ O CT) O O O O 'O o o ■^ 'C_; o o cr: cc cc CC cc cc cc CC cc cc cc cc cc cc cc cc cc cc cc c:: ~ ^ — ^ -^ trl lO r- o CM ^ ^ ^ -M IC o cc -tH cc -* cc O iO CM O ■^ CM cc _ ;r> o M .— ^ t^ CC x; 1—^ lO le- 00 QO t- ic C^l GO -r CO' CM lO t^ GO 00 00 iO t-- rrt , — i CM -*! 1-C t-~ cc ers o ^-4 CM cc -*i UC IC cc cc I- l^ t— t— l^ t^ ^•^ ?r; 'M T) o r— • 1^ cc ^ lO CO-:f T' 'CO ,—> ^~i >4 cc cc -^ -^' o cc cc t- t— cc Ci Cl ■:'__;; « -^ ^+1 •^ -+i -f -* ^ io iC iO lO O O iC IC >o ^ IC lO IC iC IC »c cc CO CO QO CO CO CC CX) cc CXI CO CX GO CO CO GO 00 CO 00 oo CO CX) CX) CO CX) cc ^ -^ CO cc 05 r~- in cc t^ o ■rH ot-ioiot^oo— 1 00 '^ C>) * *C ^— • 1^ cc c; uc — t- cc o cc CM CO -rt- 1^ w r^i CO o O tc t~- OT) CT; c^ Ci .^ r-H ^H CM c>:i cc -* -^ ic cc cc t- t— oc. Cl en o 'TtH ~^ '^ tH •^ -* ^+1 "^^ ICT IC iC o ic o ic lo o ir; io iC »c id IC IC cc c» CO oo CO CO CX) COOOC>0 o UC t^ L^ -H t^ JO o cc CM GO -rf o cc CM c; o — c— cc CTi ic ^ t- cc ^ IC 1— 1 t- lO «o w l^ CO GO a a O .— ^ r—t '^^ CM cc -* -^ LC ic: cc t- tr- CO CO n ^ ■'T+j ■<-H -* -«— 1 "rr* -H -** •^ O o tC o IC O O O O iO lO ie lO o o l-C CD GO CO CO x> cc Oj CX cc CXI C/J c^ CX) CCCCCCCCCCCCCCCCOO CO CO CO CX) 1 —1 o : — 1 cc cc cc cc GO t— 1 cc CM — 00 CO O cc GO -^ Cl CM cc iC O cc cc O C5 CO O 0<1 CO cc l^ f— 4 cc: o iC IC -"*! cc O cc CM t- — ^ -rtH cc cc oo CO t- CO O C IC O IC IC >C IC IC l-C IC cc CO CO GO CO CO CO GO CO cc GO CX) CO CX) CO oo CX C» r: s^ t:; t^ t^ oo (Ji w. o o — ■ CM •M c-: -* -^ ic lo cc t- tr- rr. err ^^ <-) "* 'jW -:t< -^^ .**^ ■■* -r*- -* o tC tc O IC lO o tc o tc o ie IC IC IC IC cc ao 30 CO CO CX5 CXI CXi GO CO Ct) cc y^ »: CO cxi CX) C30 CO CO ex CO GO op CO CO ^ CO IQ ^ CC »::: l^ _ cc CM C: 0-. C: -^ Tti c: cc 'T- cc '^ t- __ tr- IC 't* o o '^ CM Ci o c *C GO r— ■ cc CC cc cc r-H CO' IC — cc err cc cc t— oo CO '^ CD o. o t— 1 cc IC CC l~ w^ o ^- CM C- -f' -rtH O cc cc t- tr- I-- 1^ 1^ t— T^ CO o o J^J cx) '^l o cc CM C/J o 1— ■ 1— cc <3i IC — 1 i^ cc 1- i (M cc cc '^^ iC >^ cc ee t- rr, -r rr. ^ -^ -T*- -r*^ 'TfH -+ 'J+H — ^ iC iC^ o o IC LC »C lO IC »C IC IC iC 1^ IC IC IC CO ao CO CO CO cc CX; CX CX) CDO ^J CO CXi C» CO CC' QO CO O) CX CX GO CO oo CO ,_ CO 1^ cc r^ a CM t^ CM — ; CX) 00 Ci CM t- CM O oo Ci -^ -*< c-fi cc l-v JC ^ CO 3^1 ^ 1^ cc CT. cc l^ o 1 — I CM CM Cl O CO »C O LC O cc IC t— re re CO o Ol ■^ lO r^ 00 o 1 — t cc -^ »c cc t- oo CO Ci O O re 3^4 Ci »o 1—1 »^ cc C5 cc ^1 GO -Th o cc CM 00 '^l O t- cc ■Ci IC ^^ f— r^ -— '^ ^v^ lO o o r^ 1^ GO C/^' a o o .—i CM CM cc cc -* O iC cc cc i- m rr Cl J x.- -T- O I- CC- >c — ' t- cc Ci »o — 1 I- cc »"> 1 1^ S o cc I— 1- CO CO C5 o o ^—t Cl cc cc -# '^ »ci cc cc l- r— CX) '"J r-" -* -^ '^h '+• ^rf ■^ 'tK '^ IC o iC >-C »c »o O >C tC IC »c O iC IC iC 'C iC GO CO CO' CO CO »^ CO GO CX) CX) C/J GO CO C» QO 00 CO CX) cc CX cc CO op <» (^ o ^H cc t^ ^ 1-- -^H r^ Cd ^-. cc O >C Cl O O CM "* Ci >c CiO 7<1 c^c cc CO aj t^ IC CM (3^ ■^^ w» CC c^ CO OS c:: Ci GO cc cc Ci -* 00 CI i.O 1 — CO OT' w** — ' c^^ »c 1^ CO o T~- cc -^ »c cc 00 c/: c; o — 1 — CM Cl cc cc fy^ X cc ,^ '^^' i^ cc a: iO 1— ( CO ■tH o o CM QO ^5^ 'O cc cc Ci o — t- cc Cl o r^ 1^ ,_^ »c iC cc c^ 1^ CO C/J cTi cO :^ ^^ I— < CI cc cc -« -rfi O cc cc l- 1— re <-r. ■^^ ■^ ■^ T^ Tf^ ^*' ■Tt- '^i -^ o IC i^ »c O JC O O iC IC iC iC c CO CtJ CfJ CO »J cx CX! CO cao CX) CX) CO CX) CO CX) CX) CO c» CX' GO cc CX CX X) op -• 1 o 1—1 Oi cc -* IC CC i^ 00 ^, ,^ ^ CM cc -^ O cc t- CO Ci O r— 1 Cl ■^ •*< i -: ^ig o o o o o o o o o ^^ -H <— 1 — ( '— 1 1-H r-t 1 — 11-^ — 1 CM Cl Cl Cl Cl I S t^ I- L^ L- L- l^ t^ t- I- i^ t- l^ t- t- l^ t- t- l- tr- l- l- I— J- .- ■< 416 LOGARITHMS OF NUMBERS. -^ c^ oo t- CO lO ^ CO CM (M 1— ( 1 — ' o c. X X t^ CO iO -* cc,- OI o:i O Oi ^ • -• ^- w* wJ CI w^ o C- Ci Ci Ci c; Ci X X) CO X CO GO m on on rr m GO t- — [ ^ t:: o iO lO lO CD IO 1^ X -^ -o '*! I—I CO CO oo 1—1 -*< CO X X CO t^ to CO CJi to O -^ GO O OI CO' CO CO 1^ l^ CO CD lO -* CO !M 1"-1 o Ci OO t~- )0 -t< CO f^—i Ci cr> co '^ OI o -*s •JO '+I '.^ w >l GO ^ o CO ;^^ X ^ w^ \r^ 1—1 l^ CO Ci »o (~. CO Ol Xi ■+ o Oi o l-H C^l (M CO CO -+I iC iO cc CO r^ 1^ cci Ci Ci CT) O OI OI CO 'n ■^ to ^^ ^ CO CO CO CO CO o CO CO CO CO CO CO cc CD l^ l^ 1^ r^ l^ h- r^ l^ t^ aj CO Oj CX) (X) CX) CX) CX) CO CO CX) CX) CX) \ l^ CO •J^* to 1—1 1- CO Ci to —J l^ CM X -+I C-- CO OI r^ CO Ci 00 '^ f— ( (>) C^J CO CO -rH -^ lO CO CO 1- 1^ y.. Ci c: c c OI OI rr rr ■»*' -^ <^ «i; CO CO CO CO CO CO CO CO cc CO cc CO cc CO I- 1^ 1^ 1^ h- h- h-. t- l>- GO OO CO CO CO CX) CO tX) CX) GO CO CX CX CX CX X CO ex GO GO CX CX GO GO CO ^ (^^ ^+1 O) CO r-i — ^ CM »o o CO I- X , , CO CO CO '^ CO 1* 1 — 1 OI ■^ 00 to i^ o 7-i CO -f Ci cc X o ■'^ '■^ CTi rr to c:i X CO 1^ -H CO >^ t^ t^ «s u; to '^ "^ CO CO O) l-H 1-^ c^ Ci »^ CO 10 -r OI 1— CO CO -+< OI C X t^ Er C5 *o 1—1 l^ CO Ji to T-~ l^ c^J X -+ O cc OI X CO CTi l~ t^ CO X b- 1—1 1—1 (>< CO CO ^ -* to CO CO I- l~ X Ci ^i <^ ^""^ ,_^ OI CO r- •^ -rf -o '^ ci; C!,- CO ClJ CO cc CO CO CO CD CO CO CO t^ l^ (^ t^ h- l^ t— t- t^ 1 C/J 1 • CX) OJ CO CO CX) cyj CXJ cc CX C/J CX) i aj -f C' CL.' O^l X ■r;^' CJ, CO CM X "* cr. to —1 t- CO 10 1 — CO 05 X cc c f— 1 — 1 CM CO -+I -* >o CO CO l- 1- X) CO Ci o cr y—t r^ OI r^ '-f -n «c ^.^ CO S^ CO CO CO CO cc CO CO co CO cc cc CO l^ t- 1^ t~~ »^ «^ 1^ t~ t^ OD a; CO oo CX) CO CX) CX) CX) CO 4 1-^ o cr. t^ CO to CO o^ C-. GO t-' »o CO 1—1 ^ 1 i2 I-H h- CO C5 >o 1—1 l^ CO Ci to 1—1 l^ OI CO '^ O CO CM CTi CO CTi JO l-H t^ ■^ o I— 1 T—t (>4 0>1 CO -^ ^ to to CO t^ t^ XJ OO Ci o o 1—1 1 — 1 OI OI CO -^ -^l 1 <^ CO CO CO CO cc CO CO CO CO CO CO ?^ CO CO CO t^ t^ r^ t^ r— t^ h~ t- t- '^ CO CO 00 CJO o Ci 1— I CO ^ 'tH ~¥ o cc r- t^ x X Ci Ci o 1— t OJ OI CO -Tf -+* o ^ t^ t- t- X a: Oj U.' C/J CX) CfJ CO CX' GO CX) CX) cx> CX 00 X CX CO CO CX) GO CX) 00 cc cc — , i~- t^ CO cq r- lO '^l lO CO CO O O r- -H O (^ t^ X OI ^ t^ X t^ CO CO CO T— 1 cc (3i OA "^l to iO to -+I Oq Ci to 1—1 to Ci OI to cc b- l- u; lO lO lO -+I -^ CO Cvj C-J f~^ O CTi CO r^ CO -* CO O^) o X •^ to CO y-^ Ci >4 -* c^ CO (M CO -+ o CO rj X —w Ci »o 1—1 t- CO Ci to CO C^ '^^ CO o CO to cc CO cc to CO o CO OI I— y~^ ~r 1- X Ci Ci Ci c» C5 Ci oo aj t~ 1- CO to 'f CO fM ,— « O c: i^ cc ■Tfl CO Ci r- to CO ^^^ cc OS o ^— 1^ co Ci JO — ^ b- cc: to 1—1 t- 03 oo -V r^ CO OI 1- CO Ci »o . o o 1-^ CM '>i CO 00 -f- to to CO CO r^ CO CO Ci Ci O 1—1 r— ■ OI OI CO CO -^ c^ CO CO CC CO cc CO CO cc CO CO CD CO CO CO CO cc t^ h- r— l~- r^ t^ l^ b- ?- CO 00 CO CX) CX) CX) CO CO CO CX CX CX) CX CX) QO oc CO co X GO CX CX CX CX (^ CO -f -« lO Ci •rH ,_ o _ CO X to -tH -Tt' l^ (M Ci or, -- CO X cc cc cc oc CO '+I r-* l^ o CO cc t~- X oo cr. Ci' O o .— ^ OI OI CO ^^ CC CO CO CO z^ CO cc to ?^ CO cc CO CO v^ CO CO cc i^ 1^ »- t- 1^ 1^ b- t- op CXj C/J ?-" CfJ CX) CX) "X) GO c» CO CX) J Cv) O) CO CO CO CO CO CO CO CO CO CO -f ^ -* — *^ ■^ ~f -r -+ -n t< '^ '^ l~ 1^ l- (~ I-- h- I— I- I-- 1- l~. 1- r^ l^ 1^ l^ 1^ t— l^ 1^ 1- l~ LOGARITHMS OF NUMBERS. 417 -^^ O 1-- cc cr.' -*i CO ^ -+' CO i^ cc i^ lis -^ CM CO -f C-. -rt- X' T— M -+ o coccrc-— icows^^-ocinc-iciwccot— ^O«occc:i^'><^'^ i": iC 5C t-- 1-- X cr; c; O O r— — :m re cc "*< -^ lO o tr cr t- CO GC' Ci I- t^ l^ t^ l^ l^ t^ t^ OD DO OC' CO CO GO CO OO CC CO GO CO aC CO CC GO CO GOCOGOCCGOCOCOCOCOOOCOGOCCCOCOOOCOCOCO X COCOCOGOCO -H c^i -: o — w ^ -^ i^ Ci O -— O cr. I- »~ CM GC JC i^ — -^ i^ CO r; C^l C: I- '-': ro C X> O CM r: I — ?- — l^ r — x -* — i^ +• c^ •— i GO -^^ c- >-T — ^ c^i CO re r; »c o ^ — ^ i~ ;r w i^ L^ CO 00 ci O O — ,-i c^ c-i cf: ~ri- -f lo i:t ^ zs: i-^ CO CO a l^ I, t^ L^ t-- l^ l^ I- CO CO CO GO CO GO CO GC CO CO CO- CO CO' CO CO CO CO CXCOCO'CCCOCOCCCCCCCCGOCOCOOOCOCCCOOCCOOCCOCOCOCOCO -^ -t, ^ oD -rt« r; re i.^ O CM re '^ CO -M .— oo ic ^ w ^ »-e CO O 'M re o -^ CM c: 1-- — " ei c: i^ — 1< — go >e (M c. le c^i c; »e ei co -^O^Cr-^i^rec^-^ocicMt-r' t^re^'+icO'iS'— 'l^ceGO'*Ole — ^ CM c>ci re -:*^ -+i »^ >e w tc I- CO- GC- c: I- (^ l^ t- t^ L^ l^ t- CO GO GO QC X X CO QO CO CO CO GO CO CO CO CO ^ XXXXXXXX'XCOXXXXCOXXGOXXXXXXOD ■-cx(Mc;x5:ceoxorec:xc:rec;coo-*<^^ocMt-ieo xcTJ^Owr-it^OCMiewvrwie-*' — cooo»e~. cM-?-wt^ OXw-*i'--Ci^'^'— x»ocMc^oceotcreo^'>4cri»e — J-- ^ c. ic ^^ 1-- CM CO -^ o >c r-H t^ CM t» ■* O lO — i^ CM X re c; '.e o ue ue tc t^ t^ X CO c; o o ^ — ' i o X o »c c^i CM -* cr. tr ^ cr; le ce -+ o r; cc re ~. -" CO ei le i^ Ci cr. ~. c^ i— »c (M X re x cm i-e x ._. i— le CM o X >e re o X le oi C". 1^ CO o I— -+■ -— I' — »' o i- ce c: co 53 rec:oo^iMcorec7:>eo«Mxceci»c04CcMi^cccO'«tiO ue>etr;i--i^xxc^cr;o— ^i-<(M(Mcece'3'ieiCw':it~t-xc; I- l^ t^ 1-- l^ 1-- t^ t-- I- X X X CO X X CO X GO X X X X X X X xxxxxxxxxxxcoxxxxxxxxxcoxxx 00 T-j X <£^ t- o ic ce re cc — Ci Ci (M t^ le ue X 'rf o>i re 3£ re 22 « CM ^ X le 1— I t- i-i le X O n — i^ _f. -^, (— ; ^ i-j ^1 C-. i^ ■^ '-^ X rec; t2recr-iec>4X-+'0 •M X re r: »e — w (M 00 -^ a^ >ie >— < o ej X' re r". »e o ^ ■>) i^ re -.. ue »e --^ "-C i-^ X X c: cr. o o r-1 iM CM re re -* -*' r-iereo^'M!COre~t-xooxi-'^e '— t-rer:-*'0^e ijc; ^ -u: t- X X ~ cr. o O •— • ^M CM re re -:f -^i le — "v:: I— t^ X X l^t^l— t^t--l^t-t^l^XCOCOXXXXXXCOXXXXXGO XXXCOXXXCOCOX^OOCOGOXXCOXXCOXXCOCOXX t^ r- ( CM X ?c «o c^ lie CM c>:i le O t^ t^ o le CM e-T »e o X X — t— ti a: i^ »e iM X -« c. ce «? a; O ^-t CM ^ O X le (M t- ce I— --c cm i— he o tr cc i-^ X CO cr. ::; o c — ' c^ ei « re -t< '^ le >e tc t— I— CO X il-t^lr-t-t-t^l-t^l~-XCOXXXXXXXCOXXX XXX jXXXXXXOOXXXXXCOXXXXCOXXXX XXX re r: X o CO O X o (M 00 CO O >e -^ -^ X -^i CM re r- -* re >e c j.e -^ re r— X le — cc o -* l>- en ■— _ ... ». t- -^ sM c; to -^ ' — X »e (M ct CO ei r; to C'l c; »e — t^ -+' C --o -M t- re c. ue o CO ei x re c le o co c-i r^ re cr. -*■ o co — ■ t- »e »e CO CO t^ i^ X c: C'. o o — ' — "'o<] re re -:f -^ le »e cc t^ i~- X CO l^ 1-- t^ t~- t~ l~- t^ l^ t— X CO X 00 CO CO 00 X CO X X X X X X X XOOXXCOXXXXOCCOCOOOXCOXXCOXCOCOOOXXCC •lo — iecire-+'iecoi--xcr:0'— icMce-^ieeoi-^cociO— 'e co co co co co co co co co co t^ r- t— t- t^ ^ I i^ i^ t- i^ i~ i^ i^ I— I- i^ t^ I— i^ i^ i^ i^ i^ i^ i^ t^ i^ t^ t^ t^ t-. E E 418 LOGARITHMS OF XUMBERS. C; c. :/: X i~ i~ ti i-'i 'C -- c<: >i '-' — C ~. ci GC 1^ i^ i^ «^ i~ -*- r^ I ;_ o >f5 o '": i-t '."; i": o o u: o I.-: 1^ ir: o -;f -+i -*i -+i -:r -*i -Tt" -t> -+i -r .t: »c 't '-"^ »c o »o lO o »o o »^ 'C o o »n lo ir; 1:3 o o »:r: o »c o in \ P;, • T ir; cf: — I 00 la •— — C -*• I- ~- — 71 -M — O O) o ■>j y:: r~ :c :>» k-^ O '-r -M X r: cr. 10 c: --r^ — ' — — t~ :m »- 7a i- — ■-:: — »~ o -f- ~. cc aoocXXoo55a6XXoo3o5DccXajaDajX'a/^c^5ii'. c^r^ 3^ ~ I- o cc c: o O >2 -- '^' "* ^ '-^ '^ ^ "^ ^r '^ '^ ^ ^ 3* ^ — ci 1^ — ' i^ 7^1 CO -r< r; -* o o o — " "-^ — vc o »c o 'f c; ?^ cc M O -M c» c^ r-. -^t' O 'O — ' I- -M OD JQ Ji -^ C: l 'Xj TT. z:i CiOO — — •>ir:rt-f'^ij;»20'-£i--x^c}r^c;0;^ — 7i7i aj5iaDxa536xxxXx55X3£5D3£55xx~. ?. 3;r;c^r; X)-+i?c-+'X»»a;cX'*ccu:c:i^co^» t^ tc — I t^ rs X -*« c: '-": o ^ 7i i- r: x -t- ~. 1-"; c w — ■ " 7i i^ ?;; i — o r; ^ ^7 o o o -*- 00 I X -T" o '-'I — t^ 71 X r^; Ci -r n o ^ ir: o iS ~. -*i r: rf« X ;o t^ — i i-^ c; — I- •M X re c; -* O i"^ — ;c 71 X cc ~. -^ Ci <~ C: — — i^ 7-1 X Ixxxxxxxxxxxxxxxxxxx5;5;ri5i5ic^ X) i:; »o X -* cc -*• X ;r: ts c. o -*< tr — ~ o i.-^ ce — - I- 71 X 77 Ci -*- o »e — I w 71 I- re X -- r. 17 o w — i^ — . c c ;^ — ' 71 7^1 ce ^ 3*i le i3 ^ ~ i;^ 1^ X X ~ c; 2 ~ '"' ^1 ^^ xxxxx5oxxxxxxxixxxxxxxic;ai?iriC5 X --c X 71 r; ~. 7 1 -X t^ X ce o "-1 't' —1 o re X »^ ^ -+ 7^1 ce t- ic ^7 re 71 — H X i.e 7i i- ei ^o c: re ue to i— i- :r -t^ t-i c; «c Tvi i— 1— 1 le — I- re ~ -f o w -^ i^ 71 X ce X re X re X re X 71 i^ 71 :o — le -ri -" ~ le c vZ 71 i^ re X -*• r: »e o "o — ' i- 71 x re ri -f o »e ^ :r * I — . r; 2 ■^ ^ 7^1 7j 2? -3 Tl* ;i' !2 ":r '5 tr '_r ^ ^ ~ ir ^ "• ""' '^' 2? ixxxxxxxxxxxxxxx56xxxx5:c:;cici5: t'-t^r:-+<7^ice':coecnox~ceO'— -fOC. TqoooGOoe'— I i^ t— ti le re o w ei i^ ^ i.e i- ai — • 7c — >e o re r: -+ o 'e — ' I- 71 X re ~. -^ o '7 — vr 71 1- re X -K r; le o to ~- ~- 5 ~ '"' ^ 2.^ ^ ^ I? "^ '-"^ ^ — '"~ Lr -'^ -'^ -• -• --^ —' '"' ''' '^' X X ^ ^ . C^ C^ w • C- v^ C^ C^ wi w- C^ ^ * wi v^ Ci C^ O O O o o XXXOOXXXXXXXOOGOXXXXXXXCiCiOiUS^ j i~ I- o o le •o o X X •— I t^ >e t~ 7>i o — '7 71 71 17 — — ; re c^ X I c: — 7^1 I- re ~ >e c: vr ^ i 7i i~ 7>i 1- T-i i- 7'i i~ ti -.c — '.7 C: -^ o I re X -t< cr; le o "-C 7! 1^ re X -+• ~ "e c — — 1- 71 X r7 j; -f C: 17 I o -• ^' 2 5 z? 2." 2? ^ ^ ^ ■^ i2 '"^ '^ It '-T ^ i i: £: ■-- 5 ~ ii,' 2 I ixxxoDxXxSDXxxX'XaDxxa^^ ^' I le o t— CO ci o — ' 7>i ce -^ le to t^ X Oi o -^ 7^1 ce -* »e o i^ X ct: I ^ - I 1^ I- 1- I - I- X X xi X X 00 X X x> X ji c: C-. r; c^ r: ~ ~ — - c: ■ S. ^ I 1- I- I- I- t- 1- t- i~ I- I- I- i^ i~ i^ I- 1- I- I- 1- I- I- I- I- I- i~ ^' LOGARITHMS OF NUMBERS. 419 i*^ \ '>i 1>^ ^■"' O CS O ao Ir- tr- O er; in -+i -H ""■ re 7^1 1—1 ^^ O O c; 00 1-- !C S3 re re 71 71 (7i 7i ^ TZl "^ "-*^ -+< -*. cc ^5 « cc If; cc re re re re re re re re re ;:z i ir: o to ir; o o o o »o ic o o in o in in in in m >s: i:: la y::> ^c> ^ a -X o l^ O c: -M t- «i> 00 "*< 7) i-H — 1 Ci I- -^ 1— i~^ m 7) -H o C— 1— < .^ ^ „ Ci re o 1— 1 in ce '^i -^ X o o O O Ci t^ •*! ^^ 1- c^i «o O -* X 1-^ to cv cc ;^ o re t- ^ re CC i2 Ci — -*' t- O ''■^ '^ "-H o C^l t- 71 X « 00 -* ^^ o o • rC "^ -*< »0 lO - 1— > -+1 1^ ^ -*< I- — -+< Ci "* o »n — 1 ic — t^ 71 X re X -f 'i' o *n "^ T^ — — t^ — l-^ T'l •--' ~ *c r^ -+( m m — w t^ i^ oo X * C-, 71 7^1 re ^ ,^ — ,■ i^ j'-^ :^ ~ o o ^ o o C:' c o o o ^ ^ o ^^ ^^ -• ":• ":• ":• x' 9 Ci 9 Ci Ci Ci Ci Ci Ci Ci Ci cc o ^ cr. o m -Tf m o cr. o m -*i o .^ ,^ <71 X t- j_. .^ .^ _ .__ j^ 1 -— ^ O »n <^ m -t< 71 o i^ re X re i^ »^ -+< sC I- X Ci X <- m re c; -o o -*■ 00 ^1 --C O -+I X r-^ m X- 71 in c^ m m X — ' ~^ I- o re --c Ci -* c. m o -o — 1 io 71 i^ CM X re X ^ Ci -rt- O le c; w — ?c: ^ ,-, r7 -f -* n c: -*- t- o en 'T^ m tc m 'Tt^ re o X t o -f ^ m ir- --1 m X 71 :c ■^'* re «C O re 1^ ,^ CC w^ Ci 71 m cc — re ' ■*-*l r': CO cc cr. -* o »o O 'O — 1 ^ 71 t^ re X ro r- , '^ Ci T^ C^ — H -H m tC' cc i^ 1^ X ortD C^ c. o r"^^ 71 71 o o ooooooooo O O -1 ^-, 1— ( -;' -• —J ci Ci c:i Ci ci ci Ci ";• X" ^• -:• X' Ci Ci Ci Ci Ci Ci Ci Ci X — X » -H t^ t^ o w m X IC m X ># -+i X; m tc re 71 — c X I'e >j o tc I- X t^ tr m 71 — m . — . --i; O -+ r-- — < 71 *^ Ci TZ i^ — 1 m c: '': 1- O "rt- GC -^ m X r-^ '^ X — 1 -rH t^ o ce m GO -r •M t- ct Xi -+i c; -+i c m -^ t^ — I- 71 I-- re X' re Ci -^i ~ le o m o /^ rc *^ ■^ 'Tf m :i o — 00 C-. re — ' 71 C^ '+' m o =c -. _*, 71 -* Ci X ^ «C «C Ci — *< ^ X 3; o cr. X I- m 7i X '+' ~ -+' 1^ ^ re tr tC -O -H 71 Ci' 'T^ X c^l O — ^ X 71 -^ O i^ O -*' t-- —t -* t^ ^^ I— 1 — vi^ '->\ I- « X. re c^ ^ O m o ^ ^-^ tn 71 ^>- 7i X re X re Ci -*i Ci r^ r^ -+• -+' un in :r --n t- x x; c^ »• '-^ <^ —4 T-^ 71 71 re re -rf '+' m »n O O' o o o o o o o c o o — ^— ^ .— ^^ - * . * ": n . ^ n -; "r* ^" ":• ":• o »o -*( m o ci o m '<*i m o — o in -:^ — 71 — re Ci Ci 71 X X 71 ' O •>! "* in ?^ in en -:f r^ in o tC *^ tC 7( l- 71 X re X re c. 1 ^ T^ CC — +■ -r^ in m -^ --c t- i^ -/. * -J: :'_ _■ -^ ^^ ^.^ 71 71 ?7 re -^ -+ m m j ^ o o o o o o o o o o ^ :^ o ^-* ^-^ ^^ , — ( .• 1 • ": .» • ^ "^ "t* -• X* i o -^ 71 re -r+< m 1^ t^ 00 s: O _i 71 re '^i in .-> t- X Ci O — '^1 ->~ -n ' • { £. O — o o o o o o o o ^— -^ — —• 71 71 71 71 71 1 ^ 1 X X x> X X X X X X X GC y. X X X X X X X X X X X X X •< 1 E E 2 420 LOGARITHMS OF NUMBERS. ^ to «o >o -*' ^^ re 'M ei """" t—i c; c; CO t- t-~ CO le le '^l '+' ^^ 71 c-1 ^ Sq TZ >1 M o 10 »e »e •c 10 le «e »e «c »c le le »o »e io le -H »0 O CS r-l 00 CO ^ c^ le re CO (M ei CO -^ 10 —1 ^ ^^ -- »e 00 i^ r: t- -M le CO ^ (M -*i -ti -*i re Oi t- '^t^ «0 F-H »o 71 -# CO t- -M «o t- o -M -)H t^ C5 r— 1 re 10 t- ■^ . .—1 C^l -* CO 1-- cs I— 1 'f* CO -^, Cl "*! C5 lO >o »o f-^ CO y-> CO ^- t- iM t- CM t~ 71 CD ce CO CO ce cs to t^ t^ CO 'O^ Ci ^^ 1—1 CI C-1 ■re ■re. ^ -M 10 le CO CO r- 1- T CO cs r-^ -H 1— 1 P-l 1—1 ^ M (M » CM -M c^j CM 0 CS t^ '^ f t^ "+1 '+( cs l^ cs 10 CS t- CO -+i ■* I-- •r^ '^ la CT: C^ CO CO 1—4 et 71 —1 CO c-1 00 -* cs re t~- c~. r^ >e CO t^ O — • -f ^ x. ei -f >e t- cs 71 re >e CO CTS CO CO -* o -^ :o t~ t^ CO c: rs le ^ ^ -M ?i re CO ^ CO C71 t- (7» -^ le CO CO c-1 1^ l^ I— 71 ro 00 ce CO crs 00 a; .— 1 »— 1 1— t f-H ^^ —1 C^l C^J 'M •M 7T C^l c^^ CM (M — CO >-H CO — 1^ C-l 1— 7^1 t- c-1 t- ?o t- I- cc :/j C » w - ^ 1—! «— r CM !>J re re -*< 't^ le ue CO W 1- r— CO CO cs 1— t ^H 1—. ei \ CM (M (M CM 71 7^1 7^ 7-1 71 ei c-1 c-1 C-. ci C5 c; 'P ^- *• ■:• C5 ^' CI c; ^.' CS CS W. -. w. _. ^.* ^•' CS cs crs cs t^ ^ >{; c: t- C5 i-e -f t H -:f< C5 t- CS -+l -ti t— »0 CO 1— « ce cs »e c; lo o -+" CO ^ ^ «^ on 00 ir- •^ ^ oq cs i^e ^— CO 1— 1 CO ce -^ --r; c. c^"i -tH cc Ci -— I re •e r^ es ^ re le i^ 00 CM re 1^ CO i~ cs — ' -— l^ M CO cc cr. c^ -+i Oi ^+1 d * le I— ^ CO f— ' z^ f— I CO —J t- 71 CO to t^ t^ CO -tr: c^ c^ ^^ ^-1 — CM re re -:*< — H le CO t^ 1— 1^. X) CO' cr. •— 1 F— 1 1— 1— 1 r—i 1-^ 1~~^ ei ei 'T^ CM ei ei ei ei 71 CM CM 7-1 7-1 71 71 C^l 71 C-1 c-. Ci Ci q~. 9 q-. 05 C5 ^• ^- CS cr. — 1 ci o »o -f • -^ .-^ ei — CO le ~ CO '-< t- t- — —. -^ t- CO h- CO -+I t- C>q 00 (M C •>! -+■ le CO le -*' re t 1- cr: C^ re l- ^ ^ '^ CO Ci r— re CO cr ^ ->) H^ CO 00 e-i re CO CO C^ r— 71 ce »e CO iTi l^ (M t^ (M CO re X' re ^ •^ cr. ^ c^ le 'e en: le *— CO T~* CO r-< »o •^ t- t^ CO >co 'C^ * ■ ■- '^ y—U r- -M \ ei ">» CM ei CM 74 n r- ' re ^ 10 ^ >ie c re >e -* CO CM 71 t- le ir- ce -t' o >o O -f t~^ ^ ej ^^ -f ^ -^ re — cs CO 71 CO re CO 7*1 le C^) 7-1 '^ Cr. r-l -f CO X -H re le I— re le to CO — re — ^ CO t- XI ■^ to -^ l^ CI c te CO ^- •^ CO l^ t-- 00 00 Ci' Ci (^^ ^. — ei ei re re -^ le >e CO c*; r— t^ X CO cs .— I ^- ,— ,— 1 T— 1 ^.< 1—1 ei ej ei ei ei ei 71 7-1 7^1 71 C-1 7-1 7-1 7-1 7-1 7-1 7-1 71 C: -. o: Ci ^" Oi Oi Ci CS CS crs crs cs cs T' CS ^• ■^1 CS CS X CO o i- l^ CO -, -H ei 1— ' 1— 1 •e 71 -f cr: CO --J -ri -- ce 7-1 »e —1 •^ 1^ re t- ^^ le t^ cTi ei CM ei 1-^ cr: »^ -H CO 71 CO -^^ t^ 'C: — -^ re CO CO ^^ re *e t^ -^ ei -+< CO -r cs r-^ re le CO CO * c-1 ce -^ CO ^i-- CO ^^ CO '— ' t- ei t^ (M -r re CO re 00 re cs -:+< C -+> cs -t< »e ue c^ re CO 1- t^ CO co c; c: ^ ^^ —1 -M -M re re — f^ --* le >e CO r^ r- cyj X c ^w _^ _4 ^^ ^— ^^ T-^ ei e^i e^ CM CM ei 7^1 CM 71 71 71 71 71 CM 7-1 oi 71 71 Ci r: o: o T- 9 Cs CTS CS CS CS CS CS ^• c^ cs C3S CS cr. "M :M le "M re t^ >e t^ rv^ ei le CM e-i h- »e t— re 71 CO re la cr. ce cr 0:1001^0 C- "M »o r^ Ci i^ >e ei c: le le Ct le X' c CO ^ CO »e CO "M -+< 1^ c; 1 — ' ei -H CO CO en — re -:H iO t- CO ■M 'e CO r-i CO r— « 1"^ ei l~ ei t^- re T re CO r^ r/^ — tf ^ -H C^ -r C^ "■*' '^ 71 CO l^ l^ CO CO Oi -Oi ^■^ ^^ —• ei -M re re -t' -:*' le >e CO CO t^ 1- X cr. ^^ — ^ — ^ , — i ^-t -H — ^ e-i -M c^ CM CM ■M 7^1 71 71 71 71 71 C-1 c-1 7-1 c-1 7-1 71 Oi c; Ci c: 01 Ci Oi ^- CS C^ C5 CS CS CS CS CS OS C?5 CS cs CS cs cs crs CO CO CO' — -tw r' le ^_ 1 le ei ^ X I- r- CO t^ — • ■crs _^ t- t- ^ X CO c CO — CO — CO n t- CM (^ 71 X re -r ce CO re C^ -* rs -" cs ,_ CO I- r^ CO X w. Oi '"' ^ •— « —1 -M ri ^0 *C -f -+ le "O CO * r^ r— CO CO ^-i — i-H *— • r-H ■— H ^^ ei ei 7) ei -M ei 71 71 71 71 71 71 71 71 71 7-1 C-l CM -^^ -•v ^.v -^^ -•^ r: 05 cs • • • • • • • • • • • • • 0: c: >e cc ue _• ^ re _4 ic re >e r—. ^— . re — CO -h 1^ -f _u c: t- re 10 -+! CO — «^ le t^ *^ *^ le -^ 71 /— ^ --V ^-J |.,^ ei CO 1 — ■ ce le t- le CO re ie 1^ ej ■^ ^ CO CD ei — u *^ (- C". — 71 •rU '7 1^ CO cr. --*v -H - le »e CO — ^ CO *-^ CO "M t— 71 t- 71 t- re CO re -x; re CO ce cs CO CO (- 00 «) c^ 0^ r> ~H — ' "M ei re re ■T^ -+ le •^ ^i* r— r— 00 CO r—l 1 .— 1 1— ' r^ ' — t —4 ei -M ei -M ej ei 71 CO 71 7-1 71 71 71 71 71 7-1 c-l 71 C5 rj ^TJ Cfi C5 •c: 35 9 OS C5 C5 OS OS CS cs CS 'CS CS CJS CS CS ^* V CS CjS _• ! o '^ 1- CO cr. — ei re _44 >e CO r— X c: r-l ei re -H »e CO t— CO cs 1 ^ .- C-l C>l -M -M ei re re ^y^ r*^ 'V*' re ^^ *^ cc ^^ -fci -H -H -f — ^ -h — ^ —4- —4- -i^ ] ^'^ '^ v r. x X X CO' -JO X CO CO X> X X CO CO CO X CO X CO X X X X X i ^ LOGARITHMS OF NUMBERS. 421 t- t^ C- -o -th cq ci o -■>! I- c-T tc O cc o I- oo cs CI c; CO -o -* :m cc coco:y0TCooeOGOcO3Ococi>*iC5-fir-. -#c:i-^c^'i;C5'iHC;;^Ci c: r-I ci oi c^ Oi ci c: c^ ci cr- o 9. c^ c: c^ c^ c^ c^ o c:i o Oi cs o -7HO«rct-ocO'#oc:co.-iooc5»C':tHco-£cO>£:tc:pocOr-; ^ O CO O CO r-^ CC U^ -^ i;C .-' ^ O 0^ 10 !>; CO Ci C^ - . CC L- 10 C^ Ci L-7^ir--ccaocccoco:cccoo«corCGOfCoor?icorc^22aococo ^ o o rH rH -M (M cc rt -+i -# lo io «c CO t- I- CO 00 c; C; O O " ^. ri ci OT c^ o o 05 ci CI ci ci Ci Ci C5 c; Ci o cr. CI c; CI Ci ci ci ci _t^ i^ ~5 _+< CJ CO r-^ 00 C. -H CC l-^ IQ CC ?1 (M t- O CO >0 t^ 'M C<1 ID 10 »r;o>c^'MOoo-+-otO'-'>oc:!M>Qt~oocic:Cicoi^i t-" (M l~- C:i ■•— O --^ --^ <>1 !M ^t re -^ -+l >0 O CO CO t^ »>• CO CO Ci C5 O O '— ' T— I ^-^^^^^r^^r'^^^r^-r^r^ irf\ cf;^rcreccccfefCrerCfC-t<'+'-+i— H c^ ci c~. c^ c^ o c^ ci c; c; Ci 'C; Ci c: C5 Ci C". c: o Ci ci c: C5 Ci C5 CC CO -H CO C^l 01 1-- 'O t^ CO -+i Ci l^ O CO Ci "^ -+l 00 CO Ci CO t^ C<1 -— I cecere^i-HCico7eci'>-eo-"cO(M-^cocociCiCicot-»ocoo ■MCC-i-OCOCOt^COCOCiOOO — T— I'— -— (^^--^^Hr— 11— l-Hr--i— I 5^ _ CO r— CO >— I CO — CO f— l^ C^l I- "M l^ Cq t^ *>! t^ 3^1 t^ 01 t^ (M t^ CiOO"— i^-^'M'^icecc-^'+i'eiecocot^t—'XicociCiOO^^^^ 'M?ececococeccceicccccccccMroceicceccjorO'*'+i-+^-* CiCiciCiCiCiCiCiCiCiClCiCiOCiCiCiCiCiCiCiCiCiCiCi (MCOOt HiOCi!:»^^COCi-+i-Ht>-iCt^CO?COOt^Ot^Ci»OlO (Me-T'M^OCOlfI(MCi-^Ci-*CO— i-rtOreO t-COCiOi— l"^lCCCe'+l'+l«0>QCOCOCOCOCOCOtOCOCOCOCOCO ueOlOT— iCOr-HCO^HCO^^COr— ICO— (CO.-^COr^COr— ICO-HCOr— ICO CiOO^-— i!Meiccocoi^»^cOQOCiCiOO'— 't— I c-iccccccrccccocecccecoiccccereccccrefCcocc-+i'*-+'-*i CiCiCriCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiOCiCiCiCi -H Ci >— I t^ CO Ci t- CO ce o i-HcoiOcocot^c^i(Mioce'tioO'*co>oe"ircQOt^ ^„^OCit^lO(MCO'-HCi-HCX)r^-*cOCOCiCiCi CqCC-HlC-in'COt— CZJCOCiCiOOi— I^^^Hr-J^-,-Hr-i'-^i— Ir— Ir-^ ICOOO'OOlCOieO'OT-JCOr-JCO— JCOi— ICO^-JCOr--COi-^ CiOO'-< — -0 0'00>000 0'»00»OOi-CO'OC'0 C-CiOO'— iC^(MCC?0-*l-+'»C>0C0C0l-~l^G0C0CiC:iOO'— 1^ c.r-cii— it-corocviioc'i'^ COCiCiC5t:^COCCOt:^CCCOcet>-OCOCOt>-Ci05CiCiOOCO-TfQOiOO»0 CiCiOO'— I"— i'?q'MrecC'+i>c»ococoi--t^oocociCiOOr-^'— I (M(McccccococcceccfCcccciccccececcrcicrere-rfH-H-+i-+i CiCiCiCiCiOiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCiCi O'-HCC|CC'*l»0C0t~-C0CiO^^C<>CC-+ll0C0t^C0CiOr-He^lCC-+l LO to »0 10 IC >0 >C ».0 IC 10 CO CO CO CO CO «0 CO CO CO CO t^ l^ l^ >— 1-- coaDcooocoooaooococooDcooocooooococoaoaococoQO(Xco 422 LOGARITHMS OF NUMBERS. -^ . -^ L- i^ ^ -r re 7q :m „ ^^ - ^ - - X X (^ I- — — lO o -* ce ce fc ■"t^ 1 c^ Ci * C^ * * Ci C^ * * * * X X X X X X X X X X XXX, p\ 1 "^^ ~r "^ ■rf ■^ "^ ■^ "^ ~?" ~^ "^ -r -^ ^ -+< ■^ ■^ 'Tf -* -f< ~^ -tl -:f -^ , ^ o ^_ ^ l**^ ?t us ce ic ,_, CM r^ t^ ^_ -- '>^ -~ CM X o; IC te --_ o ^ ce •r^ (^ O * re u ci ^^ ce T ^ 't^ ^ M ^^ c^ t^ CM X -* ^ -Tf X ^ -# -r t- r^ -* -4. * -* c; ^ X X -^ X X ce X c^^ -M e^i t^ I^ ,^ "' c^) r^j r^ re -Tf -*■ >c i-e w ;^ I- I- X X ^ ^ O ""^ Jl CM s,. ;;+ -* T- ^ 5 '^ 3* 3+ ^ ^ ^ 3j ^ ^ -^l ^ll »e O iC c; 9 IC i£ le i£ — o w o O' ?e — ce ^ _ t^ t-- ^_ o _M -M ue (M ^ ^ ^^ .- to — c -^ c^ 1^ ^ — t^ C: :^i -*l te le le le CM t-- -:t< O t^ ,—1 ue ex ce to * c_; Ct c^ CO t- l^ cc IC -r ce CM ^ (^ * X to to '^ ri ^^ Ci l^ to ""^ X -t- Ci CC VJ re CO ce 00 ce X ce X ce X CM l^ CM t- CM r^ CM to ■ — ' to ^- X :m :^^ rc cc -* r4< le le \^ w 1- l^ X x; C5 Ci ^ p— ( CM t^l ■T*" ■— T* *^ -4- -hi ^ -+' -+■ -f ^^ •^ -rt" -f -^ -f -f »e ic le »e le ue ue le ue T- -" t' f -^• f ^' f ^• -.' ■r- f '^^ -:• r" r- 9 9 9 'w -- 9 9 9 -^ — _ -^ •-C O X — -- 3 tc t- CM evi to i^ X »e X -* — ei CM b- t^ )-"; ~- t^ CJ; 't" CO o ce -f t^ w CO to le ce T-H X IC -- «- ei t^ ^ -+1 t- i-~ ir* -f -T** le -M ?i — ■^ Ci X t^ to o '+ ce f— " ^D Ci 1^ to -T" t^ r^ CO re OO ce Gc ce X ce I- M t- M t^ CM j^ C>) l^ r-F to I--I *c — to o l> :m ^1 re ce -^^ -* ue le w sC l^ t^ X X * * ^ O "— ' T— 1 :m c>) C^ C<1 -rH v^ -^ -+ -* "-^ -v-^ -f -* -Tf -+ -^ '^ -Tf -rf -^ '^ te le le le »e »e le ue »o ^• Ci 9 -- -r- 9 C5 9 ^• •r- 9 9 -:• ex 9 9 w« 9 9 ex ex ex X iC tc eq eq i^ w c« r— c; to t^ -^ ^ X t^ „ c; CM — ,__ t^ X ^ -* »c r--i ^ ^^ 1^ CO I— 1 ce o ^ t^ t- l^ :^ -* 'M ^ X -t- X ^ o wi c^ CO 1- t^ u le -* ce CT) ^« ^ * X I- *e "^ -M X to *** • * cc XI :m r^ rj «- ei t- -M t^ CM I— c^l 1- r— ^ to f— 1 to r— ' to 1—^ le O ue o .« "" M ^i ce ce ^ -r te Le ^ 1^ f- X- X 05 * ^ o ^-^ ei CM ce ce -^i -^ -+• -* -* -rr -t -+i -** -^ -* 'Tf •^ -* -+ — *« -+ -Tf le »e lo »e le »e le j t^ ^^ t^ ^^ le c; C-l '^ w j^ X X X l^ to ce ^M X -* o le o -* X ^ u; L': ^ -+ ce ^ ^1 ^ ^^ w5 X r^ to le -It" ce -M ^^ -cx X to ue ce r- o c^i t^ CM t^ :m t^ (M t- ei w .— i ^o ^^ to r^ to to o le o iC o ue o ue to r^ 1^ X X. ^ c^ ^ ^^ (M '>i ce ce -+■ 'T^ ■rt- — JH -+ -* -*< '^ -* ^ -f -+' ^Tf* "* '-^ -+• -* le ic le »e >e »e »e ie »c t- '.• -^• ^■ ^. ^. 9 9 w* 9 0* ^• ^• -■ -• ^• w. w. 9 9 ex ex crx IC ^ ;- ^ ue o O t2 -H r^ o t- ^ lO _ e^i .- X ^„ X -^ i^ ex tC 1>} t^ CM u c ce '^ l^ X ^ X t^ te CM ex le r^ to ex -M o o Ci C- CO CO l^ w 'e — *- ce -M ^M ^ ^ X r^ le -r ce *-^ ^ X to ue _u C^l t^ — cr F— ( W <— ' w ^— t^ ^^ sO ^^ vo :^ le c Le o >e ^ ue -+< rM M re re -f -t^ le o r^ io 1^ t^ X X. w5 * ;^ ^O •"* ^^ -M e) CM ce ce ■-;t^ -+ •r« -+ 'f' -f -+i -+^ -f '+' -h -4- -+ -T' -* -*i »o ue le le \ 7-i 7e re -* 't" le ic ^^ t^ 1^ r- X X ex o o o ..i^ ^^ ->) (M ce ce ' • -+ ■^ -Tf — *< -^ -* -f -+ -+ -^ -* -* -tl -* -^ ^-^ >e >e le >'^ f^ 1'^ je »e ue 1 -• ^- -:• -:• 9 cr. 9 -:• T- 9 n- T- T- 9 9 Js ■— ' >»v .— s — ^ >-». .^^^. — ' ,— ^ /— ^ • • • • • '• t t 'T cc ^1 ■^ •^ t^ -* lO ^ ,^ »o >* t^ la X le CO e>i fM b- r^ ^ -. ce — ce t- C^ X; ce t- ^ -:H t- Ci o ^^ — ^ ^^ o Ci r^ •e -M X -+■ O -t- ex ce to o '>— »- 35 ^^ CO 00 t- U X i^ »e n ^^ t^ o ^ ce ri '— ' O T. r— ir -^ CM O i-^ '^ l-*^ o le O o CD >e ^ >-e C* ^ Ci 'T^' C^ -*< c: "^ Ci ce X ^t^ X ce X r-i ->! re re -+ -# >-e le u w to t^ 1- cc 00 wi c^ o o ^^ ei CM ce ce -f- -r tH "+ -f -* -h -+■ -f *4rf -H ^ -f v4rf -^ -f •4* le le »e 9 ex t* f X* ^' w- t' ^' X* t* '^* "^* *•* ^* '^' ^* "i' "i' ~" "' __, I— 1 •^ U5 Ci t>- 05 U »^ cc ce h- to O X C r^ T-. »0 »Q o o -f ce b- X) — h C5 -+ OO G^l m X ^ 3^J ee ce ce ce 1-^ 't^ t^ -+i ^ 1-- ce X. -M to ex ■,,^ ■^ wv o; C» 00 t^ IC r^ m -* ce -M v^ c * 1^ to le ce ei '*^ ex 1-- ue .->. ^ o Ci '^ cn ^ C5 -:^ ^ -^ * '^ 05 -f ce X ce X ce X ^-■ t^ -M l^ /-^ ^^ M M ei oe ce '^f ^ le le ?S u I-- t- X x; * ^ ^^ ■M '>'\ Tfi :^ -+ -« -:f -Tt< ^ -Tf ^ -+. -f -H -+< ^4^ -f "^ ~f ~f -+ >e »e i^ 1^ 1'^ ue »e >e t' Ci ";• 5i w. w. f ":• ^.' C?5 9 ^' ^' f w« ^- _. _. ".■ ■J- 9 9 9 9 ■ 1 o .- t^ CO -^ o — ei ^ -+i »e to t^ X -~ o ,^ ■M ce '+< >e — t^ X ex - I ^ ( ^ 1^ r^ 1- X X X 'jj X X X v: x; X * C^ — ^ — x X X X X X X X X X x: X X X X X X X X X X X XXX LOGAEITHMS OF NUMLERS. 423 ■~ M CM c O Ci C". oo GO t^ t^ tr o ui -# -+■ rc ^t cr:-#0-^t^t^'MC^ii:^w-^0-*rc:ri~x:r ex •oxc^OOOCiX•-occOl-r^oOl:^xc^^l::xo:^^^^-^''~-H i^ io fc c>4 O X 1.-: c^ r-i Ci t- '^i c^^ c^ I — r- cm ct ^d -:t^ -- x ir: cm r; -•^ t^ •■^ t^ ^^ t^ ^ lT^ .;^ lO wi ■'^ C^ ■'^ CC CC OC CC l'^ C^l t^- ^1 ^^ '"^ "^ -^ <«^ " -^" *^ *-^ t^ * l^ l^ CO GO CO Ci C^ O O ^- ' — ' ^4 7^1 CO CC "^ -^ *^ »^ ^ w» IC 1^^ 1^ It i^ i*^ i*^ iC O O lO lOCCs^^tC^CCtCww'^wt^^^ ' -. c-. z~. r; 9 ci c^ cr. 9 cr. 9 9 9 9 9 9 9 9 9 9 99999 j -f M "-^ re --T: re ic; M re r; c; w :r M CM --r^ cr — cr: -* r^ I- tr X ; X — c>j M CM — X i.e re 05 i-e — to cr -*! X — re i.e to i^ x i^ M — 0; I- re — 0; to -t- CM r. i^ ue CM cr 1^ — *■ cm n --o re cr i^ — ^^ GO to — iecrLec;'CjOi-#c:-+'xrexrexcMi^->)to — t:; — LCiO CO -* le to to L- i-- i— X X ci oc: — — c^i CM re re -r -Ti 1:1 to «e ic e en -:*^ cr- -# X re X re X CM I- ej I vr — ue t- ^ ' -:+. ue -^ --C vr I- J- X X r: r: C cr cm c:-i re re -^ -t- i.e le -o g|gg||||^gggS|5SSS5==S|5S ~. ~ ?e CM tr -+' r^ i-e t^ --c re ue — > -m x r: -* -^ cr le le r: r: ; — re 10 tr vr :r .-e -:?^ CM I- -^ cr -o — ic -. re c: en r- CM re re re re — r: t- »e re -^ — i- »-e cm r; x le re r; i- c i-e te le ■ ue »c 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 to ' to le ot X — i r: -M re cT' •>! Ci — 1 t^ 0; ue to CM re c; d ue 1- Oi cr cr 0; X t^ '^ -M cr lo — to >c X »- to x a: cr en re — rcxtr-:?-— ni^oreccxtore — xuereci-'*'- xi.e '^ -* r. re X re X re I— -M t^ CM i^ to — to cr 'e cr 'e n -* r: re x -*--t.iciotctot--t--xxo;ciCC ■M:Mrerere-f^*'>eice LoicieicieieicieiCieLei-etotototototototot^totototo '^ re -+' cr >e i.e cr — re CM i-e -** t^ le t^ i^ t- -* vr re -:f ^ e- t^ X X c^ c^ ^"^ ^^ ^^ e^i e^T ^1 "v* '■^^ -rw -^ »e * ic ic >o le >c le ic ue ic >e ue -^ tr: to -^ tr tr: -o -^ -^ cc <£: -~c -^ : 9 =r- r- 9 9 =r- T- 9 9 9 -- 9 9 9 9 9 9 9 9 9 9 9 9 9 9 I'crcMxr. ue>o--oia^cot ^ricqcrecrrecrMcs — xcr cr — CM re -^ -rf -# re — cc: ir re c i-e — -^^ cr -^ t- -M re la ic ->r , re -M X -^ '^ -M cr X ue re — n tr -* -- Ci cr: re — X ue >i en tc -.toc^^-oxoxoxo o:e-*iiotototo>.ere— 'C5to?cixrex(MtoOM>etoxocc: cri^»iere-^c;i>-i^re — xto-*- — crto-r»<— «c:toreot — ^ — - CM i^ -M t^ j-1 to — to — to le »e as -^ Ci -+> X re X re i^ CM t- 1 -rf '*' ic ue --o to r- t- X X c: c. c ->ci ->i re re -* — »e i le ue le ue u^ ue le <-e 10 --o --o -^ --o to to to to to to to to to Oi Ci n. c: Ci ci c^ c; cr c; Ci cr cr. c: c^ ~. . X X X t- m re r-4 X -:f cr to — le cr CM le i^ c: cr — CM -t*- CM cr X to -*i M X to -* — cr i^ -*! !>> C5 to -Tt- — X re t- !M«>-->cito— 'to — tco»oOiec^-*c::-^xcexrei-cMt-cMto cr -^i-THueuetotot^t^xxciCicrcro cM-Mrere-^-^-oic te »-e lO t^ i?: »o le »e to to to to to to to to to to to to a Oi en a a a a a oi a a c: c. a c^ ci Ci c^ cr-. a en c a-, a ^ «:0— 'CMre-^ictot^xcio — cMre-^uetri^xr: c: — cMre-*! z 424 L0C4AEITHMS OF ^XMBEE3. ^ 5 S X X t— i^ tr tr i-": L.-: to to to to to CO ^ to cc n 71 71 — CO to 3 Ci Ci X X t- UC <-C lO iO i-C id •- ' "^ "^ ■^ - '^ ~- ■ ^ ~ '^ ~~ ~^ "^ -^ "^ — -T*' -e -n '^ -e rf ^ — ~ ■M c :m C: cc r— -^ T) — ■M »~ -^ ;_^ O X to O X ^, _l J- _l^ ^ T ^1 ;^ to c^ o S -? 5 t§ fi i^ >-C ^ X i-C ^ 71 O I- t2 71 X ■^ Lrt O L-^ CJ; 7J to — to >0 O X cc 1 - 71 to ^ -^ lr~ L^ t^ X X C; Ci :^ O -— ' — 71 71 CC cc -^ -ri Tf< to LC tc to I- l^ -• ^• ^.' ~' ":• ^• T* Ci Ci Ci ^1 C: •^ ?5 LC ^r L^ LC Ci t^ ^ Ci 7^1 _^ ,J< cc t- tC O Ci cc 71 t- t- 71 L^ ^ "^ ^1 J"- to >i X 7C '.X. cc to -^ CC JO l^ X Ci X to -rr 71 — cc X i-c n X i-c — cc X cc I- :^^ i- X ^r^l — ' l^ cc — X cc Ci uc — t- cc X; '- — X t— l^ -f- jC -C :^« '...,^ -^ •— I ..— 7J 71 A^ -^ ^^ le "* "~ t— I— t^ v^ to to ^ * tC t^ !>, i^^ t^ t^ 1— t— r^ 1-- I— t^ t^ l^ t^ I— [>• t— t— ""- C^ * o^ ». w. C- C» '""; '"; '"^ '^' -^^ • • -O: ^ X t^ o 'e ci -- »-C 71 --1 tc ,-. -* -* — -N -i, i~ rc i_^ M Ci tr CI l^ CI -r> 3 I- -?- C: t^ :::: 5C CC 71 X t ^* cc ^ cc CC 71 O X to LC — ' t^ 71 X t>. ~^ -- •^ X CC X M t^ CM to — to ^ iC *" Ci cc X 71 I- 71 to —1 l-C t^ ^ t^ t^ t-- X X Ci Ci o o — — 7) 71 7; cc cc ^v ,^, ^.^ «— ' ^— ^_ ^.^ ^.^ I-^ I^ t^ f^* I- i^ I- I- L- t~ l^ t- i^ I- r- r— ":' T' -r- -• ":• ~- ".' ".• '• c;t CO t— l^ -^ O i-C 'T X X 7^1 7-1 to 1!C O O 'T^ <:>< Ci Oi iO IC ^^ -^ !>. w-MOaoy3r:c:u:ou; ^ *7^ t-^ ""- t^ r- t~- ■M C^ tO' M Ci to CT Ci tC 71 T. »C — X —r — to 71 X 'ei ^ to 71 X -^ ?c -:r X ^i X 75 t~- >i tc — . tc Cj: ».c :3 --- O; — • X cc I- 71 I— ■ — to ' — 1'^ ,^ t», ,^ t^ t^ X X Ci Ci c o ^— «-« 74 71 7^1 rc cc 1'^ to "* 1— l^- -"•^ :c -^ to to to to to to L^ t^ r- l-~ ;^ l^ h- t- t— r^ t^ t- l^ I- t^ 1^ t^ T" -.■ Oi Oi Oi Ci Ci Ci Ci Ci ^- Ci Ci Oi f Oi 05 — -5f -e T-. Ci -^ ^ -; cr -^ -- cn X -- CC X. X cc '* Ci o i-^ to 71 cc — «r u-; rc — ~. to >T X -r- Ci T^ l^ . — — i- to X o — ^ — 71 — o c; I- LC t- -:r — ^ X ■"T" — * "y T* — i~^ ■^^ ^^ r^ cc Oi »0 7^1 X "^^ '"' U5 re X Ct^ t^ CT t tr — i-C ^ IC Ci — - X CC X 71 t^ 71 to ^— i*- ^^ -— ' 15 w tC t^ l^ X X Ci C: O C r^ 1^ r- 71 71 l~ t-^ t^ l^ tr- iC to to l~- L- l^ t^ t-- t^ I--. to to ^i '"^ ^•^ •^ — — " — ' -— ^ '-^ 1 >-- u- _, ^ t~ t^ M cc X X ^ ^ -, — -^ to 7-1 ee Ci O tC I- -:- LC 7-1 1 C. 3C I-- LC tM Ci tr -M l^ -M t^ ^— '— ih X ^ 7-1 -^ iC l-C tc "T-l * to cc ^ to cc * to cc O; to CI X i-C ^~ L~^ cc Ci 1^ tH 1 « t- --) c^ 71 to — to O l^ ^ — i- cc X CC t- 71 to ^^ to O iC- Ci "•*■ T^ ^ --; t^ t^ X X Ci Ci C: O r~ t^ 1^ 71 71 l^ ^ I— t^ t- 1= to to to t- t^ 1-- t^ t^ I— ^.^ to to to to to t^- t^» Oi Ci • 1 tc t- -^ -- — — •O: t^ ^ -j. _ 7i X C t- X uc X lO 3^1 — ■,,_^ X UC cc ""i L^ ^- to * »o -J ^ -T^ to X Ci Ci "C Ci X t- lO cc X ir: ■>\ X UC 71 X iC 71 X i~ — r^ — ^ tC 71 X ■■?** ^— to 71 X -r O CC ?q t- M to — to * i-C c^ — Oi -^ X re X 71 l^ ^— to — cc tc tc t^ t^ X X Ci Ci 3 O o — — 71 71 CC cc -^" -r^ IC iO to to to l~— 2r — o^ :r !£ ':£ '■— '■£. 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Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci ci Ci Ci Cicncncicr. CiCiCiCiCiCiCi rc -M O iS r-J C-l X O t^ Ci X r-i f-j )~ to c-i re o re -^ le o c:d O t- ^ T— 1 o Ci r^ tc X ---•>) re -t- le ue -^ re ei c: X le -M X -^ o *— «0 O -+I Ci re i^ ■M tc o -^i X ei to o -*' X ^ ue Ci CM to c re to t>. n 1^- C-J tD O le Ci -+' X re t^ ^^tooiecirexcMto-^iec; •^ X t- Ci Ci C C — ^ — 23 2? £2 re '^ '^i ec::;':fxrei-'Mtocr:»eci rt X • A ^ Ci o o — ^ r-t c-i c-i re re -f -+ c t^ I- to le -* CM cc; i^ '^^ 1— t^ CM X CM ■M l^r-t^O-^CiC^t- ' le Cire 1- ^leciretoo^t- — X iS "M tC ^ lO O' ^ X re i^ CM tc o»eci-^xc>cii-— (too^Ci re t- ue Ci Ci o o ^^ ^ ^ ej rj re re '^^'Tf-fieictotoi^t^xxx Ci Ci VD CO Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci W* Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci cOi Ci Oj CO re -^i .— re r-. -+i -M o o to re 00 l^ CT'-^OxtO'*i— It— reci ~¥ Ci O -f X re t^ I— 1 le Ci re t— ^H -^ X ei to Ci re 'C ^^ rc^ "* t— tc o o Ci ■^ X M 1- — tc O -^ Ci re X -M to ^ le Ci -^ X re t-I rf Ci o o o — ^ >i ^1 re ..^ ..U4 <^ 1^ i^ *^ -^ j^,_ 1^^^ 1^^^ f-J^ ^ cr. C-. Ci Ci X CO Ci Ci Ci Ci Ci ci Ci ci ci Ci Ci Ci Ci Ci Ci ci — w* Ci Ci Ci Ci ci Ci ci ci ci 9 CiCiCiCiCiCiOiCiCiCiCiCiCiCl rl cc Ci ^ CC' o X -^ o le le CD^XOX — 0-*-rt<0 — X , ^— CO « I- J-l o C^ !— • ""^ C^ l^* X Ci Ci X X to le re o t— -*' c^ >e — 'e cc CO S-1 l- — »e o -i- X ei tc o-^xeitoo-^x — lecTiei ^!* ^^ »--5 O -^ Ci re X M «c ^ le o -* X re I- ei to o 'e cr. re x ei S *^ wi Ci o c o :r !r' 2? £.' " re -r — " -^ »e te to to I- t^ I— X X -■. ^ CO Ci Ci Ci Ci *t* Ci Ci Ci Ci "T" cr. Ci cr. Ci Ci Ci ci ci ci ci ci ci t^ CO l-O t^ '^t^ t^ieciCi-^'+iOi— (X— icireeit^x-#iere to -* r^ CO CC t- .-! -^ r^ Ci I— 1 re -*i Le »e -T" -^ ei — Ci t:; re o to OI I— ■>! Ci re CO C-1 t- ^ i-e Ci ':*' X ej t:: O -^ X CM t= Ci re I- r-i ^ X — le 3-1 5 O Ci -rM X c: Ci C C re t^ ^ tc c ^ ^ >i 2^ f- »e Ci -^ X CM 1- — 1 le o -^ Ci re t- re -^ ■* -' — o cr Ci t- i.e re o tr CM X -^ Ci "^ Ci re X (M 1^ >— 1 >.e Ci re X CM to o -f t- ^ «e Ci re to crc' re t-» ''"*■ <—( o -* Ci re ! t:; c^ 1— CO L't c5 >e ■^ -^ Ci re X 'T' 1 ^^ *— H i:;> ^, re 1- ^^ >e Ci re t^ -J -^ X ei le Ci o 5 ^ X re 1^ •^ ^ "M rl ?1 ;^ X £2 t;^ -;; tc C ^2 H: ^ ^ ?^ — ^ 5 -z. o£ X X Ci Ci C^ C^ wl 'C^ 'C^ ci C^ C^ Ct C^ Ci —..—.,—* r— » ,— ^ -— Z ^ .— ' .^ 7' f Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci Ci ci ci ci ci ci ci ci Ci Ci o ;c t^ oo Ci OC Ci i o I— 1 e to !>• CO Ci o — < ■M re -^ »e to t- 1- ij- »^ r^ (^ X X CO X x CO x X X X c7i Ci cr. c. Ci — cr: c: ?. -• w. Ci Ci STEENGTH OF MATERIALS. 427 STRENGTH OF MATERIALS, Definitions. 1. Direct exferisiMlittj or comjyresgiUlity is the amount of direct strain produced by each pound on the square inch of direct stress. 2. Direct jyliahility includes both of the terms 'direct extensi- bility ■ and 'direct compressibility.' 3. Elasticity is that property of a body by which it tends to occupy a determinate bulk and figure at' a given pressure and temperature. 4. Elastic strength is the utmost amount of stress which a body can bear without set. 5. Factors of safety are of three kinds — I. The ratio in which the breaking load exceeds the proof load. n. The ratio in which the breaking load exceeds the work- ing load, in. The ratio in which the proof load exceeds the working load. 6. Modidns of elasticity is the reciprocal of the direct plia- bility when the stress does not exceed the proof strength. 7. PUahility is the inverse of stifEness, and is measured by the quantity of strain produced by a certain hxed stress. 8. Proof strain is the utmost strain which a body can bear without injury. 9. Proof strength or proof stress is the utmost stress which a body can bear without suffering any diminution of stiffness or strength. 10. Proof load is the load which produces the proof stress. 11. Set is the permanent strain or alteration of shape which remains in an imperfectly elastic body after a stress has been removed. 12. Sinking or resilience is the quantity of work which is re- quired to produce the proof strain, and is one-half of the product of the proof strength of the body by its proof strain. 13. Stiffness is measured by the intensity of the stress re- quired to produce a certain tixed quantity of strain. 14. Strain is the measure of the alteration of the shape of a body, corresponding to a given stress. 15. StrenfftJi or idtimate strength is the utmost amount of stress which a body can bear without breaking. 16. Stress is the intensity of a load which tends to alter the •42'8 STRENGTH OF MATERIALS. shape of a body, and it is also the equal and opposite resistance olfered by the body to its load. 17. UUimate strain is the utmost strain or alteration of shape which a body can bear without breaking, 18. Wurki/iff strength is the utmost strength to which it is considered safe to subject a body during its ordinary use as part of a structure. 19. Working load is the load which produces the working stress. Notes on the Strength, kc, op Materials. {Prom ' Ski^y-buildi/ig, Theoretical and Pi'acticaV) 1. The tenacity of wrought iron and puddled steel is greater in the direction in which they are rolled than in the direction of their breadth, but in cast steel it is the reverse. Table of the Coxditioxs in which Iron is Found in its Ores. Oxides of Iron JJy Chemical Equi- valents* By Weight Per- centage of Iron Native ii-on is nearly pure, or combined with one-fourth to one-hundredth j^art of its weight of nickel Protoxide or black oxide of iron . -' ^^^' [ oxygen Peroxide or red oxide of iron . ■! i™?„ * ( oxygen ^Magnetic oxide of iron . . .I ''■''^^ ' ( oxygen Hydrate /"peroxide of iron, ) (iron . of per- 2 equivalents ) * ( oxygen oxide water, - ) , f oxygen of iron I 3 equivalents J '• ( lijdrog. Carbo- f Protoxide of iron, ) (iron . nate • ^ equivalent ) " 1 oxygen of iron carbonic acid, \ f oxygen \ 1 equivalent j ' i carbon 2 1 4 3 3 2 8 6) 3; 6 2 w 2 5* 1 116 o2 ) 224 \ 144 -374 56 'l 48 rll6 12 j 80 to 100 78 70 72-4 CO 48-3 2. Brimn iron 07'e is hj'drate of peroxide of iron nearly pure or mixed. When nearly pure and compact it is called Irowti hematite ; when earthy and mixed with clay, yellow ochre. 3. Carbonate of iron, when pure and crystallised, is called sparry ??v/// ore, or sj?athose iron ore ; when mixed with clay and sand, clay ironstone. When clay ironstone is coloured black by carbonaceous matter it is called black-band ironstone. * The chemical equivalents adopted in the above table are as follows : — oxygen, 16 ; carbon, 6 ; hydrogen, 1 ; iron, 28. STRENGTH OF MATERIALS. 429 4. ^lagnetic iron ore consists of magnetic oxide of iron, and contains about 72 per cent, of iron. 5. Red iron ore is peroxide of iron pure or mixed. When pure and crystalline it is called apecular iron ore, or iron glance ; when pure or nearly so, and in kidney-shaped masses showing a iibrous structure, it is called red hematite ; when mixed with more or less clay and sand it is called red ii'onstone and red oclire. 6. The strength of iron depends mainly upon the absence of impurities, such as sulphur, calcium, and magnesium, w^hich make it brittle at high temperatures, while silicon and phos- phorus make it brittle at low temperatures. 7. Cold-bla»t iron is stronger than hot-blast. 8. Annealing cast iron diminishes its tensile strength. 9. The strength of cast iron to resist crushing or cross- breaking is increased by repeated meltings, but after the twelfth melting the resistance to cross-breaking begins to diminish. 10. Good cast iron should show a good, clear skin, with regular faces and sharp angles, and when broken the siu'face of fracture should be of a light bluish-grey colour and close-grained texture with a uniform metallic lustre. 11. Cast iron becomes more compact and soitnd when cast under pressure. 12. Strength and toughness of bar iron are indicated by a fine, close, and uniform tibrous structure, free from all appearance of crystallisation, witli a clear bluish-grey colour and silky lustre on a torn surface where tlie fibres are shown. 13. "Wrought iron has its longitudinal tenacity increased by rolling. 14. The tenacity of ordinary boiler plate is not appreciably diminished at a temperature of 395° Fahrenheit, but at a dull red heat it is diminished to about three-fourths, and the tenacity of good rivet iron increases wnth elevation of temperature up to about 320° Fahrenheit, at which point it is one-third greater than at ordinary atmospheric temperature. 15. Wrought iron should not be used in ship-building which will not bear a tensile strain of 20 tons per square inch. 16. The tensile strain for wrought iron should not exceed §• or \ of the breaking weight. 17. Steel is made by adding carbon to malleable iron or by abstracting carbon from cast iron. 18. The hardness and toughness of steel is increased by being hardened in oil, but its strength is reduced by being hardened in water. 430 STRENGTH OF MATERIALS. 19. The shearing strain of steel rivets is about one-fourth less than their tensile strength. 20. Case-hardening bohs weakens them. 21. Bessemer steel is made by blowing jets of air into mol- ten pig iron and stopping tiie process at the instant when the proper amount of carbon remains in tlie iron, or else the blast is continued until all the carbon is removed, and then the proper amount of carbon along with manganese and sihcon is added, the usual way of adding the carbon being by running into the molten pig iron a sufficient quantity of highly carbonised cast iron. The steel thus produced is run into ingots, which are hammered and rolled like blooms of wrought iron. 22. Blister steel is made by embedding bars of pure wrought iron in a layer of charcoal and subjecting tliem to a high tem- ]^.erature, or by exposing the sm-face of the iron to a cm-rent of carburetted hydrogen gas at a high temperature. 23. Cast steel is made by melting bars of blister steel in a crucible along wdth a small quantity of carbon and some manga- nese, or it may be made by melting bars of pure malleable iron with manganese and the whole quantity of carbon required to make steel. 24. Granulated steel is made by running melted pig iron over a wheel into a cistern of water, the lumps being then taken out and emliedded in pulverised hematite or in sparry u'on ore, and exposed to a sufficient temperature to cause part of the oxygen of the ore to combine with and extract the carbon from the superficial layer of each of the lumps of iron, each of which is reduced to the condition of malleable iron at the sm-face while its heart continues in the state of cast iron. A small quantity of malleable iron is produced by the reduction of the ore. These ingredients being melted, produce steel. 25. Puddled steel \s made by puddling pig iron and stopping the process at the instant when the proper quantity of carbon remains ; the bloom is then shingled and rolled like bar iron. 2B. SJirar steel is made by 1u-eaking bars of blister steel into lengths, and making them up into bundles or fagots, and rolling them out at a welding heat, repeating the process until a near approach to uniformity of texture and composition is obtained. 27. The ultimate elongation of really good and tough specimens of iron and steel may be taken as follows : — In r>ar iron . . . from •l.'s to '30. Plate iron, lengthwise ,, -0-1 ,, -17. „ crosswise ,, '015 „ -11. Steel bars . . „ '05 „ "19. „ plates . . „ '03 „ '19. I STRENGTH OF CYLINDERS, TUBES, ETC. 431 Eesistaxce of Thix Hollow Cylinders and Spherical Shells to Bursting. p ^ bursting pressui'e in lbs. per square inch. T = tensile strength of material in lbs. per square inch (see tables, pp. 269, 270). f = thickness of material in inches. r = radius in inches. For Thin Holloiv Cijlinders. Tt . P?" p = — t = — r T Fw Thin Spherical Shells. 2Tt . pr p = - — - r = — r 2T Resistance of Thick Hollow Cylinders and Spherical Shells to Bursting. p = bursting pressure in lbs. per square inch. T = tensile strength of materials in lbs. per sqtlare inch. R = external radius in inches. r = internal radius in inches. Fo?' Thick Hollon' Cylinder's. p ^ T(R._ __,.2)_ ^^^ /T^'V) ,.^K /(T - P) ^^ / (T + P) ,. ^E /(T- V (T - P) V (T + R- + r-' V (T - P) V (T + P) For Thick Spherical Shells. 2t(r3 - r^) P = jj3 + 27-3 ^,. y2(T + P) ^^E y(2T - P) V (2t - p) V 2(T + P) Tenacity of "Wrought-iron Eiveted Joints in Lbs. per Square Inch of Entire Plate. Double-riveted. Diameter of each hole = ^ of distance from centre to centre of holes ^ 35,700 lbs. Single-riveted = 28,600 lbs. Kesistaxce of Wrought-iron Tubes to Collapsing. p "= collapsing pressure in lbs. per square inch. l = length of tube in inches. d = diameter in inches. t = thickness of metal in inches. P _ 9672000^^ lud 432 FACTORS OF SAFETY, ETC, Ebsilience of Tie Baes. s = proof stress. E = modulus of elasticity (see table, p. 269). A = sectional area. L = length. B = resilience of bar. m = modulus of resilience. B = S'AL s* Table op Examples op Moduli of Resilience. 3*£at? *^ ' rivet. 436 steexgth of riyeted joints, kixds of stress, etc Kelative Texacity of Eiveted Joixts. Rivet Holes Deducted. Rivet Holes Included. Continuous plate . . . 100 100 Double-riveted joint . . . 100 70 ' 8ingle-riveted „ ... 79 56 Eesistaxce to Shearing. In plate and rivet iron the resistance is nearly equal to the tensile strength. In metals such as cast iron it is somewhat greater than the tensile strengtli. In timber it is nearly equal to the tenacity across the srrain. Modulus of Elasticity. To Determine the Modulus of Elasticity from Extension o^^- Comjn'ession. E = modulus of elasticity (for values see pp. 269, 270). a = area of section in square inches. L = length of materials in feet. W = load applied in lbs. I = elongation or compression in feet. eaZ E = WL ~Tl w = 1= WL EA Table of Different Kinds of Load and Stress. {JRankine.) Load axd Stress Strain STniTTESs Plia- I Way of BUITY i BilEAKIXG Strength v Pull or tension Stretch Resistance or I to extension ' extension Direct exten- ' Tearing sibiUty i Tenacity or resistance to tearing Thrust or pressure Squeezing Resistance or com- to com- pression pressiou Racking or , Rigidity distortion ■ Direct i compres- ; Crushing sibility ' Resistance to crushing Resistance to shearing a. S j ^1 Shearing or racking stress Twisting stress Lateral Shearing, pliability 1 slid"ig or ^ •' 1 detrusion Torsion Resistance or 1 to twisting 1 twisting — [Wrenching 1 Resistance to Wrenching Resistance to breaking across Trans- verse stress Indirect thrift Trans- Bending verse stiffness "-"^'J-, \"?ot^ Bending with com- — pres.-ion 1 Breaking 1 across 1 Resistance to indirect crushing EACKING, MODULUS OF RIGIDITY, ETC. 437 Kacking or Distortion. In the diagram (fig. 154) abcd represents the original form of figure before distortion, and ahcd represents the distorted form of ABCD. T,. . ,. 2(AC — ac') 2(Jr7— BD) ., , Distortion = —^^^ ■' = -^^ ^ sensibly. AC BD Sacking or Shearing Stress Is that kind of stress that produces distortion, and the racking stress at the two pairs of faces of a distorted particle is of equal intensity ; also every racking stress on a particle is equivalent to the combination of a tension and thrust of the same intensity acting diagonally or at an angle of 45° as regards the stress. Example (see fig. 154), R = racking stress in ri lbs. on the square inch of surface represented by the arrows r. T = tensile strength in n lbs. on the square inch acting parallel to the diagonal bd. s = compressive strength in 7i lbs. on the square inch acting parallel to the diagonal AC. R = T + s. To Determine the Modulus of Elasticity from a Rect- angular Beam supported at both Ends. L = length of beam or distance between supports in feet. w = weight in lbs. D = depth of beam in inches. b = breadth of beam in inches. d = deflection of beam in inches. E = modulus of elasticity. ^_ WL^ .^^ WL' ^BD^d 4bd3e Modulus op Rigidity. M = modulus of rigidity. R = racking stress. D = distortion. M = 5 R = MD D = - I> M 438 BEX;>IXG MOMENTS AND SHEARING FORCES OF BEAMS- Table of Moduli of Rigidity for Yaeious Substances. Metals ;Modii]us of pLigMity. Lbs. on Sq. In. i Timber Mo. lulus nl Ri^iduy. Lbs. on Sq. In. Brass wire . 5,330,000 I Ash Copper . 6,200,000 Oak Cast iron . 2,850,000 Elm -rv ^ w ' from 8,500,000 -r, . . ^\roughtiron|^^ 10,800,000 ' ^^^ ^^^ pme 76,000 82,000 I 76,000 ( !from 62,000 ' to 116.000 Bending Mo3IENts and Shearing Forces of Beams. W = total load. ■w' = additional load. ■'// Between b and c Between b and a _ W(L— £) - _^ BENDING MOMENTS AND SHEARING FORCES OF BEAMS. 439 Between B and c r M = -i ^ I p ^iZl^w Fig. 159. loaded at oxe side of MIDDLE. Between b and A < _ (t}x(\j — a?) M-- ^ I <- — X'- ■jr—A W F = — wy /m. -G-^) Fig. 160. t'n'iformly loaded, <: ^ ^ ^ ^ (a) ^ ^ Fig. 161. partially uniformly loaded. Between j ~^ W^ 2l/ 2^/ B and c j V- X LF = «(2/-|^-a?) Between! 2l B and A ^ cwy- ^ ~~2i7 To Find the Bending Moments and Shearing Forces AT ANY Cross Section of a Beam Unequally Loaded. s = supporting force at A. s' = supporting force at b. 03, Wj, a>2 = loads at the respect- ive points C, D, E. R = magnitude of resultant load. Fig. 162. fW'/ ^ R = a> + Wj + a»o {oj X AC) + (q>j X ad) + (gjg X AE) R Rx a? x = s=^^^ x + y S' = : x + y The shearing stress on any cross section (fig. 162) between AC=S, between CD = s — «, between de = s — w — Wj. To Determine the Bending 3Ioment at any given Loaded Point. EuLE. — Multiply each shearing force by the length of the division on which it acts ; the algebraical sum of the products corresponding to the divisions which lie between that point and either end of the beam will be the bending moment at the given loaded point. 440 MOMENT OF RESISTANCE OF BEAMS TO CROSS-BREAKING. Examjyle. Bendiyig moment at A = 0, at c = SxAC, at d = Sxac + (S — w)CD, at E = SX AC+(S — w)CD + (S — £0 — a>,)DE. Note. — The maximum bending moment is at R, the shearing force being zero at that point. Modulus of Kupture. In a beam the modulus of rupture is the intensity of the stress which is just sufficient to cause breaking to commence, and in skeleton beams is simply equal to the tenacity or crushing stress of a separate bar of the material. When the section of the beam is symmetrical above and below, so that the neutral axis lies at the middle of the depth, then the beam will give way according as the tenacity or the resistance to crushing is the less ; but in beams of the more ordinary form of cross section (see p. 441) it is generally different from either the direct tenacity or its re- sistance to crushing, and is generally taken at eighteen times the load that is required to break a bar of an inch square sup- ported at two points one foot apart and loaded in the middle. Values of Moduli of Rupture. Cast-iron open-work beams .... 17,000 „ solid rectangular bars . . . 40,000 Wrought-iron plate beams . - . . 42,000 „ bars and axles .... 54,000 Puddled steel 62,500 Steely iron 52,500 Moment of Resistance of Beams to Crobs-breaking. Note. — The moment of resistance at a given cross section should be at least equal to the greatest bending moment. Skeleton Beam. M = moment of resistance at given point. A = sectional area of stringer at given point, s = greatest safe intensity of stress. D = perpendicular distance of centre line of stringer from joint. M = ASD. Beams of Various Sections. M = moment of resistance at given section. I = geometrical moment of inertia of section (see p. 441) relatively to neutral axis. A = area of cross section. D = depth of beam. B = breadth of beam. d = distance of most severely stretched layer from neutral axis. VALUES OF COEFFICIENTS OF FORM 441 d' = distance of most severely compressed layer from neutral axis or centre of gravity. s = greatest tensile stress on stretched layer. s' = greatest compressive stress on compressed layer. gO g g/ — = lesser of the two quotients - and — . d° '' "" ~ d d' Ji and (? = coefScients of form (see annexed table). d° The same unit of measurement to be taken all through. Table of the Values of Coefficients of Form. Solid Rectangle. D 12 Hollom Rectangle. «&■■&"•» 1= c= 12 6BD^ BD^-J»d^ Solid Circle. ThickRollow Circle. 1= 64 6D*(BD— 6rf) TJiicJi Hollow Square. j_B*-6* 12 6B* 6b^ Thin Hollow Sqiuire. 12 ^^2(B-6) 3b yi 320* Thin Hollow Circle. ^1^^ c=i Solid Elli_pse. 64 /l- 7. =-0982 32 TAtcA Hollow Ellipse. TTfBD^-M^) : U^ y 32bd» ~Sd-(bd — 6(f) jT/ii/^ Hollow Ellipse. BD'' — 26a^ ..^^ yt= 12 BD^-26rf^ 6bd= BD^-26(f^ '6BD*(BD-26d) 55 Weights axd Test Loads of Anchors. W = weight in cwts. (exclusive of stock). T = test load in tons. L = length of anchor in feet. A = area of augmented surface in square feet. w = weight of stock in cwts. ^"800 '^-50 ^^"^T L = ^SOW Weight and Test Loads of Chains and Kopes.. w = weight in tons per hundred fathoms. T = test load in tons. D = diameter of chain in inches. C = circumference of rope in inches. T = -loC > hemp rope shroud-laid. W = -009602 1 , . 1 , . -, T = -l^cr ^ hemp rope cable-laid. w = -03902 1 . . T= -7502 J> iron-ware rope. w = -04c2 1 , , . T = 1-12502 f steel wire rope. w = 2-9D2 T = 2-9D2 1 . . = 12d2 J> ngging chain. = 2-43D2 -| = 18d'^ D = /I ' V 18 J TV = 2-43D2 -I T = 18D''^ ,— "^ chain cable THICKNESS OF IROX SKIN, ETC. 447 Proportions of Chain-cable Links. Length outside . . . = 6 diameters of bolt. „ inside . . . = 4 „ ,, Breadth outside . . . = 36 „ „ Thickness of stud at end . = 1 „ ., „ „ middle = 2 „ „ " ' " Description of Cables. Hemp is laid up HgM-lmnded into yarns. Yarns are laid up left-handed into strands. Three stj'ands laid up 7'ight-handed make a hawser. Three harvse?'s laid up left-handed make a cable. Shroud-laid rope has a eore surrounded by four strands. Thickness of Iron Skin in Ships. W = displacement in tons. L = length of vessel in feet. B = breadth of vessel in feet. D = depth of vessel in feet. T = thickness of skin in inches. WL ~ 800BD Eesistance of Iron Skin to Buckling. E = ultimate resistance to buckling in tons on the square inch. T = thickness of plating in inches, s = space between frames in inches. ^ 400T s Eesistance of "Wrought-iron Armour Plates. iFairhairn.) Tensile Strain. M = coefficient of dynamic resistance in foot lbs. S = breaking strain in lbs. per square inch. L = elongation of material per unit of length. M = ^ 2 Punching Iron Plates. P = pressure in tons to punch a plate. w = work in foot lbs. to punch a plate. r = radius of punch in inches. t = thickness of plate in inches. P = lU;f W = 10640?T(2 448 PENETRATION OF SHOT, IMPACT OF SHOT, ETC. Penetration op Shot into Ibon Aemoue. d = distance of penetration in inches. Tt' = weight of shot in lbs. V = velocity of shot in feet per second at time of impact. r = radius of shot in inches. r_ / rvv 2 ^^^^ for round-ended cast-iron service shot. 494:0r for fiat- ended steel shot. \/ 1571360r Velocity op Shot. W = weight of shot in lbs. TV = weight of charge in lbs, V = initial velocity of shot in feet per second. V = velocity of shot at n feet per second. r = radius of shot in inches. 2800 -y/w ._ V ^^"^ 1 + ^000063 -Vw Impact of Shot. w = weight of shot in lbs. V = velocity of shot at time of impact in ft. per second. I = force of impact in foot lbs. per second. i = ---=^=01553wv2 2ff 644 To Determine the Size of the Rim of a Fly-wheel. V = velocity in feet per second at the periphery. 71 = number of revalutions per minute. d = diameter of wheel in feet, w = weight per foot of rim. a = sectional area of rim in square inches, c = centrifugal force for one foot of rim, s = strain on any section of rim. n= '''^'' cd _ TVV^ 161 = height water is to be lifted. G = -0340lNLD- C = -OO-DioeNLD- ^'-y 29-4G XL H NLD-A Ch ' 97020 ^^ 15557 Hydraulic Press. p = pressure in tons, D = diameter of ram in inches. L = distance between fulcrum and axis of small pump. d = diameter of small pump in inches. I = length of pump handle from the fulcrum to paint application of power. f = force applied to pump handle in lbs. D-/7 of p = 2240^'=L Table of the Pressure of Water at Different Head^ H = head in feet. P=pres.sure in lbs. per s^i. foot. />= pressure in lbs. per sq. inch. 1 1-25 1-5 1-75 62-4 78-0 93-6 109-2 124-8 187-2 249-6 p H 5 •4333 -.5416 - 6 •6500 7 •7583 8 •8666 9 1 1-3000 10 1 •7333 20 312-0 374-4 436-8 499-2 561-6 624-0 1248-0 2-1666 2-6000 30333 3-4666 3-9000 4-3.3.33 8-6666 30 40 50 60 70 80 90 1872-0 2496-0 3120-0 3744-0 4368-0 4992-0 5616-0 13-0000 17-3333 21-6666 26-0000 30-3333 34-6666 39-0000 Discharge of Water from Sluices and Orifices. V = theoretical velocity due to head of water in feet second. H = head of water in feet. A = area of aperture or outlet in square feet. Q = quantity discharged in cubic feet per second, g = force of gravity = 32-2. per Mlling cisterns, etc. r = velocity of real discharge in feet per second. k = coefficient for different diameters of sluices. 453 V = H = V2^ = 8-065 a/h = •01o53v2 Q = A^ ^/2^H = 8-025A* ^H ^^ Q Q k^2ffii 8•025A'^/H v=k'^2ffH =8-025A;a/h Table of the Values of Coefficient k. For Short Square Tubes For Short Cylindrical Tubes. Lgth. Dia. 2 10 •617 •814 •75 Lpth. Dia. •69 •65 •62 Lffth. k •59 •56 •48 Lgrth. Dia. 1 2 4 k Lgth. ^. Dia. Leth. ^. Dia. 49 •eo 60 -or. 100 -48 Dia. 50 60 100 20 30 40 •62 •82 •77 13 ^73 25 -68 37 -63 Discharge of Water from a Cistern. T =time of discharge in seconds. Q =rate of discharge (found by above formula). W = volume of water in cistern in cubic feet. T = 2w for vertical-sided cistern. 4w T - — for wedge-shaped cistern. T = — for pyramidal-shaped cistern. Time of Filling a Cistern when Supply and Discharge ARE GOING ON AT THE SAilE TiME. f = cubic feet of water going in per minute. / = cubic feet of water going out per minute. T = time required to fill cistern in minutes. t — time required to empty cistern in minutes. e = contents of cistern in cubic feet. T = F-/ t = /-F 454 rORCE OF WATER, ETC. Pressure of Water on Dock Gates. D = depth of water in feet. L = length of one gate in feet. T = thrust on ribs in lbs. N=normal pressure on the surface of the gates in lbs. <^ = distance from point where gates meet to a right line joining their hinges. T = 312d^l^ n = 32ld2 Force of Water in Motion. F = force of water against surface in lbs. A =area of surface in square feet. V = velocity of water in feet per second. Vj = velocity of water in miles per hour. y2 = velocity of water in knots per hour. =sine of angle of incidence with opposing surface. F = 0AV2 = 2-1510AVi2 = 2-8o20AV2% Table of the Force OF Water in Motion. 1 \'elocity iu Feet per Second 1 Pressure in Lbs. per Square Foot Velocity in Miles per Hour Pressure in Lbs. per Square Foot Velocity in Knots per Hour Pressure in Lbs. per Square Foot 1 1 2-1511 1 2-8524 2 4 2 8-6044 2 11 4004 3 9 3 19-3600 3 25 6711 4 16 4 34-4177 4 45 6375 5 25 . 5 53-7777 5 71 3087 6 36 6 77-4400 6 102 6844 7 49 7 105-4044 7 130 7640 8 64 8 137-6711 8 182 5501 81 9 174-2400 9 231 04(X) 10 100 10 215-1111 10 285 2346 11 121 11 260-2844 11 345 1330 12 144 12 300-7600 12 410 7.378 13 169 13 363-5377 13 482 0465 14 196 14 421-6177 14 559 0508 15 225 15 484-0000 15 641 7170 16 256 16 .-50-6844 16 730 200f*) 17 289 17 621-6711 17 824 3280 18 324 18 696-9600 18 924 1601 10 301 19 776-5511 19 102!) 6060 20 400 20 86()-4444 20 1140-0384 1 FRICTION OF SURFACES, ETC. Vol Flow of Water through Pipes. H = loss head of water in feet. L = length of pipe in feet. D = diameter of pipe in feet. Q = quantity discharged in cubic feet per second. V = velocity of discharge in cubic feet per second. k = coefficient of friction =0258 for rough approximation. H = ^LV2 64-4D = -02C 12D/64-4D = 8-025y55 *=02(l.^^) Q = .78o4VD= Table of Coefficients of Friction and Angles OF Eepose. r = resistance of friction to the sliding of two surfaces. p = pressure over the surfaces. k = coefficient of friction. R = P^' X-, ^f -\T„i.^^„-i„ Coefficient of Name of Aiatenals -r._. ». » , Friction = tan (}> Angle of Repose =:/a;(A-t- x) x = y = A(a± \/a- + Ai/-J a\/x(A + x) Catenary. Fig. 179. If a uniform chain be freely suspended from two points, A and B, the curve in which it will hang is termed a common catenary ; the parameter oc is equal to the length of a piece of the chain whose weight is equal to the tension at the lowest point c in the curve. The directrix ox is a horizontal line drawn through the extremity O of the parameter. The tension at any point P in the curve is equal to the length of a piece of the chain whose weight is equal to the tension at the point, and is thus equal to the ordinate pm. JEJquatio/is to the Catenary (see fig. 179). a? = abscissa. ' y = ordinate. c = tension at c. s = length cp of chain. CcM'tesian. Approximate Equation. x^ = 2ciy-c)-^iy-cy CATENARY. 459 FormulcB for the Catena/)'y when the points of support a/i'e in tlie same horizontal plane (see fig. 180). S =span. h = height or dip. J) = parameter. I = length of chain. w = weight of unit of chain. t = tension at A or B. c = tension at c. y = ordinate at A or B. c w t , S^ S^ y = - = p + h = p + — + + + &:c. 46080/^^ t=-ym Fig. 180. ^ W- =0- -^ B 7i 1 -1^ 1 \^ 1 \ \\ 1 1 1 \\ ^^J / / 1 1 X'' 1 tJ> \^. ' / y V^ 1 y y 1 i^^^^^ ! c 1 1 \ X A M }0 c=pw ^ = S + + s= 24/?- 1920/?* + (SCC. 7. S2 S* , Sp 384i?3 46080/? . + &C. Approximate Formulae. ^-[^^v^^^^^-^^Gyj^s-^e^^^^^^- ^ = ^[%-v/(3y)'^-2l(|y] =g nearly. Sh- l=yCs^.^I>--) = s^'^n.^lj. ., S2 7h s- , Catenaries that naake equal angles at the points of suspen- sion with their ordinates or horizontal dimensions are similar figures. 460 CATENARIAN CURVES. Table of Relations of Catenarian Curves, the Parameter being TAKEN AS Unity. Anffle of h s / 1 -^^ Suspension 1 "^ 1 ! 2 i ^ i 1°" 0' •00015 •01745 1 ♦01745 i 1-0001 114-586 2°" 0' •00061 •03491 •03492 1^0006 57-279 3°" 0' •00137 •05238 •05241 : 1^0014 38-171 4°" 0' •00244 •06987 •06993 1^0024 28^6 13 o°" 0' •00382 •08738 •08749 1-0038 •22-874 6^" 0' •00551 •10491 •10510 1-0055 19-046 7°" 0' •00751 •12248 •12278 1-0075 16-309 8°" 0' •00983 •14008 •14054 1-0098 14-254 9**" 0' •01247 •15773 •16838 1-0125 12^654 10°" 0' •01543 •17542 •17633 1-0154 lb372 U°" 0' •01872 •19318 •19438 1-0187 10^820 12°" 0' •02234 •21099 •21256 1-0223 9-444 13°" 0' •02630 •22887 •23087 1-0263 8^701 14°" 0' •03061 •24681 •24933 1-0306 ^•062 15°" 0' •03528 •26484 •26795 1-0353 7-508 16°" 0' •04030 •28296 •28675 1-0403 7-021 17°" 0' •04569 •§0116 •30573 1-0457 6-591 18°" 0' •05146 •31946 •32492 1-0515 6-208 19^" 0' •05762 •33786 •34433 1-0576 5^863 20°" 0' •06418 •35637 •36397 1-0642 5-553 21°" 0' •07114 •37502 •38386 1-0711 5-271 22°" 0' •07853 •39376 •40403 1-0786 5-0 14 23°" 0' •08636 •41267 •42447 1-0864 4-778 24°" 0' •09484 -43169 •44523 1-0946 4-562 25°" 0' •10338 •45087 •46631 1-1034 4-361 26°" 0' •11260 •47021 •48773 M126 4-176 28°" 0' •13257 •50940 -53171 1^13-26 3-843 30'-" 0' •15470 •54930 •57735 1-1547 3-551 32=" 4' •18004 ' -5912 •62649 1-1800 3-284 34°" 16' •21003 -6371 •68:30 1-2100 3-034 36°" 52' •24995 •6932 -74991 1-2499 2-773 39°" 11' •29011 •7443 •81510 1^2901 2-567 4l°"44' •34004 •8029 •89-201 1-3400 2-362 4i°" 0' •39016 •8566 •96569 1-3902 2-196 46°" 1' •43999 •9066 r0361 1-4400 2-060 48°" 11' •49981 •9623 1-1178 1-4998 1-9-25 50=" 8' •56005 1-0142 1-1974 1-5800 1-^11 52°" 9' ' •62973 1-0706 1-2869 , 1-6297 l-(;99 54°" 13' 1 •71021 1-1304 1-3874 1-7102 1-592 56°" 28' , •81021 1-1995 1 -5089 1 1-8102 1-481 5H°" 3' ! •88972 1-2510 1-6034 1-8897 1-416 60°" 0' 1-0000 1-3169 1-7321 2-0000 1-317 64°" 6' 1-2894 1-4702 2-0594 2-2894 1-140 67°" 2'^' ! 1-6095 1^6 135 2-4102 2-6095 1-002 67^" 32' ' 1-6168 1 1-6164 2-4182 2-6168 0-9998 WEIGHTED ROPE. 461 To Construct a Catenary GEOMETRicALiiY. Fig. 181. M « P Let E be the lowest point in the curve, OE its parameter, and ox its directrix. Make ab equal to OE ; then with A as centre and AE as radius dessribe the small arc EF. Join FA and pro- duce it to M and to b, making bf equal to fm ; then with B as centre and BF as radiiis describe the small arc FG. Join BG and produce it to N and to C, making CG equal to GX ; then with C as centre and CG as radius describe the small arc GH. Proceed in a similar manner till the curve is of the required length. Weighted Rope. To determine the position a weight will take when hun-g on a rope suspended from two points not in the same horizoivtal plane. Fig. 182, Let A and b be the two points of suspension ; make bC equal to the length of the rope ; bisect AC in D : the point E where the perpendicular DE cuts BC will be the point at which the weight will hang. I = length of rope. d = distance between points of suspen- sion. h = height of one sup- port above the other. X and y — co-ordinates of the point. f. -^„c^- y= _^{l'-d')-h yd 462 MECHANICAL POWERS. MECHANICAL POWERS. The power applied and the weight lifted are directly propor- tional to the distances moved through by each body in a given time. w = weight to be raised. p = power applied. D = distance of power from fulcrum. d — distance of weight from fulcrum. n = number of movable pulleys. L = length of inclined plane and wedge. H = height of inclined plane. c = circumference described by P. t = thickness of wedge. s = distance moved through by P. s = distance moved through by W. E = resistance to wedge. jp = pitch of screw. General Formula for all the Powers. W: SP P = W,9 p SP w The Lever and Wheel and Axle. w = PD p = w^ D D = Vid d = PD W Fig. 183. Fig. 184. I (i) Pig. 18-5. P A r o ■^^ Fir;. 186. inclined plane, wedge, screw. The Pulley. 463 w = 2P71 Fig. 187. one movable pullky. P = w 2/1 Pig. 188. two movable plt-lets. 2^ote.—¥or revolutions of wheels see p. 451. The Inclined Plane. WH w = PL H p = H = £^ W l = L WH The Wedge. R = ^ p = ?^ l PL t = The Screw. w = ^ PC w _wp p = c = :^ Fig. 189. Fig. 191, 4_ cm i\Y>f<9.— One-third more power than is obtained by the fore- going formulae is generally allowed, in order to overcome the resistance due to friction, &.C., weight and power being in equilibrium. 464 FORMULA FOR MECHANICAL WORK. Force, Power, and Work. (^John W. Nyatrom.) s = space in feet passed through by the force F in the time T. F = force or pressure in lbs. V = velocity in feet per second. T =tinie of operation in seconds. p = power in foot lbs. of one pound raised one foot per second. H —horse power of 550 lbs. raised one foot per second. w = physical work expressed in workman days of 1.980,000 foot lbs. M = weight in lbs. of moving mass, or the weight of a mas> acted upon by a mechanical force. G = acceleration of the combined gravity and mechanical force. g = accelerating force of gravity =32-166 feet per second. L = number of labourers employed (not workman days). ■ D = number of days of eleven working hours. N = number of horses (not horse power). n = number of blows of steam hammer or pile-driver. Note. — By a workman day is meant a man's day's work of 11 hours in the day when the work done is supposed to be equal to the work accomplished by one horse-power in the time of one hour. Formula for Mechanical Work. ^ FT _ 550th ^ 550 X 3600W F F F _ P _ 550HT _ . 550 X 3600W _ 550 x 3600w V S VT S V _ S _ P _ 550H ^ 550 X 3600W T F F FT T=-= — = -^^ ■_ 550x3600w V P 550h~ fv P=FV = ^^ = 550H=^^Oii3600w T T H = ^ ^^^ -r ^^ :^ 3600W 550''550~550t'^ T 550 X 3600 ~ 550 x 3600 ~ J50 x 3600 ~ 3600 WORK UNDER THE ACTION OF GRAVITY. 465 L W FV FS N = -- = 11 llD 550 11x550x3600d \ _. w _ jw;_ ^ 50w ^ FS ^~L~llN~FV 550x3600l FVD F^VS W = DL=llDN 50 50 X 550 X 3600L Formula for Work under the Action of Gravity. gj,2 VT _ PT _ 4 X 550'^H- ^ 550 X 3600W ^ 2 2 2m 2gM'^ M 2x550 X 3600W 550 x 3600W ^ 2xj50H ^ 550 x 3600wg' x 2 ^" gT-r S ■ 2s 2 X 550H ,-- /i 2s_2x550h_ /_-_ /550 x 3600> 'S'^'S ■i - ^ >>^-s Nature of Labour ^ ^ ? •& o = 6 o i^ O O ai •5«i:>-=o '3 "" "A ■A ^ ^ 1 p. 1. liaising Weights Vertically. A man mounting a gentle in-\ cline or ladder without bur- 1 8-0 203,200 4,230 145 29 den — i.e. raising his own weight '' Labourer raising weights with \ rope and pulley, the roj)e re- • 6-0 563,000 1,560 40 39 turning without load ) Labourer lifting weights by [ hand i 6-0 631,000 1,480 44 34 Laboui"er carrying weights on\ his back up a gentle incline i or up a ladder, and returning [ unladen -' 6-0 406,000 1,130 145 8 Labourer wheeling materials \ in a barrow up an incline of 1 in 12, and returning with j 100 313,000 520 130 4 empty barrow ■' Labourer lifting earth with a'* spade to a mean height of 5^ \ feet ) 10-0 281,000 470 6 78 2, Action on Machines. Labourer walking and pushing [ or pulling horizontally j 8-0 150,000 3,130 27 116 1 Labourer turning a winch . 8-0 1,250,000 2,600 18 144 Labourer pushing and pulling \ alternately in a vertical di- I rection J 8-0 1,146,000 2,390 11 216 1 Horse yoked to a cart and i walking j 100 ; 15,688.000 26,150 150 175 Horse yoked to a whim gin 8-0 8.440.000! 17,600 100 175 Do. do,, trotting 4-5 ! 7.036,000126,060 66| 391 One man can lift with both hands 236 lbs. „ „ „ support on his shoulders 330 Ib.-s. A man's strength is greatest in raising a weight when his weight is to that of his load as 4 is to 3. jVote. — In the above table the unit of work is taken at a presMire of 1 lb. exerted through 1 foot. BOARD OF TRADE REGULATIONS FOR MARINE BOILERS. 467 Table giving the Useful Effect of Agents employed IN the Horizontal Transport of Burdens. {From \ Ttvisdeiis ^Fractical MeoJianics.'^ "^-^ o 'O o Agekt O u seful ffect Minut eight sporte Lbs. S e8 pw« fifi 1 » u. Man -walking on a horizontals ; road without burden — thatis^ I 10-0 25,398,000; 42,330 145 292 transporting his own weight ] Labourer transporting material in a truck on two wheels,! .^.J lof^TOOo' 21710 returning with it empty for a '/^ "j l*^."^o,OOU| J1,/I0 220 99 new load '' 1 Do. do., with a wheel-barrow . 10-0; 7,815,000 13,030 130 160 Labourer walking with a weight on liis back 7-0 .0.470.000' 13.030 90 145 J } Labourer transporting mate- rials on his back, and return- ing unburdened for a new 6-0 5,087,000 14,100 145 97 load ' Do. do., on a band-barrow . 10-0 4,298,000 7,16a 110 65 Horse transporting material in \ a cart, walking, always laden i 10-0 200,582,000 334,300 1,500 223 Do. do., trotting 4-5 90,262,000 334,.300 750 44 Do. do., transporting materials \ in a cart, returning with the [ 1 10-0 10,940,800182,350 1,500 121 cart empty for a new load ) Horse walking with a weight ) on his back / 10-0 34,385,000 57,310 270 212 Do. do., trotting 7-0 .32,072,000 76,410 180 424 Note. — The usefiil effect in the above table is the product of the weight in lbs. and the distance in feet. BOARD OF TRADE REGULATIONS FOR MARINE: BOILERS, ETC. Boilers and Superheaters^ Pressures on Flat Swfaces. On flat surfaces the pressure should not exceed 5,000 lbs. to each effective square inch of sectional area of stay ; but if in any case a greater pressure is required, where the flat surfaces are stiffened by X or L irons, tlie mode of stiffening must be sub- mitted to the Board of Trade for approval. H H 2 458 BOARD OF TRADE REGULATIONS FOR MARINE BOILERS. To find the area of any diagonal stay. Rule. — Find the area of a direct stay needed to support the surface ; multiply this area by the length of the diagonal stay, and divide the product by the length of a line drawn at right angles to the surface supported at the end of the diagonal stay. jVbte. — When gusset stays are used their area should be in excess of that found by the above rule. Girders for Flat Surfaces. When the tops of combustion boxes, or other parts of a boiler, are supported by solid rectangular girders, the following formula may be used for finding the working pressure to be allowed on the girders, assuming that they are not subjected to a greater temperature than the ordinary heat of steam, and in the case of combustion chambers that the ends are fitted to the edges of the tube plate and the back plate of the combustion box : — FORMULA. p = working pressure. L = length of girder in feet. D = depth of girder in inches. T = thickness of girder in inches. w = width of combustion box in inches. p = pitch of supporting bolts in inches, d = distance between the girders from centre to centre in inches. k = 500 when the girder is fitted with one supporting bolt, = 750 when fitted with two or three supporting bolts, = 850 when fitted with four supporting bolts. ^ X D- X T (yi—]))d X L Plates for Flat Surfaces. The pressure on plates forming flat surfaces may be found by the following formula : — FORMULA. w = working pressure. T = thickness of plate in sixteenths of an inch. s = surface supported in square inches. k = constant according to the following circumstances : - BOARD OP TRADE REGULATIONS FOR MARINE BOILERS. 469 k - 100 when the plates are not exposed to the impact of heat or flame and the stays are fitted with nuts and washers, the latter being at least three times the diameter of the stay and two-thirds the thickness of the plate they cover. )fe = 90 when the plates are not exposed to the impact of heat or flame and the stays are fitted with nuts only. & = 60 when the plates are exposed to the impact of heat or flame and steam in contact with the plates, and the stays fitted with nuts and washers, the latter being at least three times the diameter of the stay and two- thirds the thickness of the plates they cover. A = 64 when the plates are exposed to the impact of heat or flame and steam in contact with the plate, and the stays fitted with nuts only. ^ = 80 when the plates are exposed to the impact of heat or flame with water in contact with the plates, and the stays screwed into the plate and fitted with nuts. A = 60 when the plates are exposed to the impact of heat or flame, with water in contact with the plate, and the stays screwed into the plate having the ends riveted over to form a substantial head. ^ = 36 when the plates are exposed to the impact of heat or flame and steam in contact with the plates, with the stays screwed into the plate and having the ends riveted over to form a substantial head. s-6 Cylindrical Boilers. When cylindrical boilers are made of the best material, with all the rivet holes drilled in place and all the seams fitted with double butt-straps, each of at least | the thickness of the plates they cover, and all the seams at least double-riveted with rivets having an allowance of not more than 50 per cent, over the single shear, and provided that the boilers have been open to inspection during the whole period of construction, then 6 may be used as the factor of safety ; but the boilers must be tested by hydraulic pressure to twice the working pressure. But when the above conditions are not complied with the additions in the following table must be added to the factor 6, according to the circumstances of the case. 470 BOARD OF TRADE REGULATIONS FOR MARINE BOILERS. Table ghing the Constants to be Added to the Factor of Safety for Cylindrical Boilers. Mark' A D E* F G H I J* K M N P Con- stants •15 •o •o •15 •15 •15 Circumstances in which the Constants have to be added When the holes are fair and good in the longi- tudinal seams, but drilled out of place after bending. When the holes are fair and good in the longi- tudinal seams, but drilled out of place before bending. When the holes are fair and good in the longi- tudinal seams, but punched after bending instead of drilled. When the holes are fair and good in the longi- tudinal seams, but punched before bending. When the holes are not fair and good in the longitudinal seams. When the holes are fair and good in the circum- ferential seams, but drilled out of place after bending. When the holes are fair and good in the circum- ferential seams, but drilled before bending. When the holes are fair and good in the circum- ferential seams, but punched after bending. When the holes are fair and good in the circum- ferential seams, but punched before bending. When the holes are not fair and good in the cir- cumferential seams. When double butt-straps are not fitted to the longitudinal seams, and said seams are lap and double-riveted. When double butt-straps are not fitted to the longitudinal seams and the said seams are lap and treble-riveted. When only single butt-straps are fitted to the longitudinal seams and the said seams are double-riveted. When only single butt-straps are fitted to the longitudinal seams and the said seams are treble- riveted. When any description of joint in the longitudinal seams is single-riveted. When the circumferential seams are fitted with single butt-straps and are double-riveted. • The allowance may be increased still further if the workmanship or material is very doubtful or very unsatisfactory. BOARD OF TRADE REGULATIONS FOR MARINE BOILERS. 471 Table givln^g the Constants to be Added to the Factor OF Safety for Cylindrical Boilers (concluded). Mark W'' Con- stants •2 •1 •1 •2 •25 •3 1-65 Circumstances in which the Constants have to be added When the circumferential seams are fitted with single butt-straps and are single-riveted. When the circumferential seams are fitted with double butt-straps and are single-riveted. When the circumferential seams are lap joints and are double-riveted. When the circumferential seams are lap joints and are single-rivet-ed. When the circumferential seams are lap and the strakes or plates are not entirely under or over. When the boiler is of such a length as to fire from both ends, or is of unusual length, such as flue boilers, and the circumferential seams are fitted as described opposite p, r, and s ; but when the circumferential seams are as described opposite Q and T, v -S will become v '4. When the seams are not properly crossed. When the iron is in any way doubtful and the surveyor is not satisfied that it is of the best quality. When the boiler is not open to inspection during the whole period of its construction. Strength of Joints in Cylindrical Boilers. FORMULA. P = percentage of strength of plate at joint as compared with the solid plate. p' = percentage of strength of rivets as compared with the solid plate.f p = pitch of rivets. 3-26 „ 3-75 3"75 ,, 4'75 JS^ote. — The above weights are for expansive engines of good make ; compound engines average from 10 to 20 per cent, heavier. Consumption of Coal per Day, Hour, &c. I.H.P. X -06429 = tons per 24 hours at the rate of & lbs, per hour. •0.5893 = •05357 = •04821 = •04286 = •037.50 = •03214 = •02679 = •02143 = •01071 = » 4^ 4 3^ 3 ■'a 2 1 4i H Stowage of Coal, &c. The Admiraltj' allowance for coal = 48 cubic feet per ton of 2,700 lbs. = 40 cubic feet per ton of 2,240 lbs., which is the average generally allowed for coal-bunker space. The bulk of wood is about 6 times as much as an equivalent of coal. A cord of wood = 4 feet x 4 feet x 8 feet = 1 28 cubic feet. A cubic foot of tallow weighs about 59 lbs. ,, ,, waste „ 9, II M oil .1 )« 56 ,, 480 WEIGHT OF MARINE ENGINES AND BOILERS. 02 m Z I— I o I— I _ 02 —J << 05 ^ S3 c 2 ^ >>-3. g S '-^ r? ^ o ■•5 . A '2 H ;5 ^ £-3 a: O C5 ;5^-K j3 00^—99 «o •* o ' - 00 us 00 o;g ' ^ 'HI e» ■* -^ d-- o ® ® — ^ 6 c 1.-5 O O I .0 ^ <=«'*99gg^^^o«^o -, 0^2 ■= tf J I •= * 5 -^ >« 2 - ^5 - c 9 9 2 a -.3 X X g 5 » « ;s ^ a ~ -95 «t-i o a ~ s .3 • c 2 — C p*. u >-* = d '^2 = 2 — ._ K ir — - , aS *= s ^ •»- £ = i 4) ~ 3 >> -s — 2. ""- = 2 a l^^i C :»: S > t y. G C :- 0- - 1^ ^. rS ^ C. . a: = ~ c « >, .H tx ® Z-kJZ 3 ii!z: ^ 6 r-Z a o I « - _ WEIGHT OF MARINE ENGINES AND BOILERS. 481 ;5 w c ^ K W K iTI' O ^" Oi c <1 h- 1 ^ pi r/: -< ^ g O cc rr. C f^ t) O <^ < 2 PS < E-t O -< : (M ~^ c "M ^ ,- C^ — r-l oa HH ^ r- ^ 'C '^ ^^ lO -^^ p. .-j^.s o c= - 2? S ^, > o c- t^ o 9 S • " — s ^ °° . — ^ O ^ g^ 9 goo:o«_ co»-^ 00 r? r? O © c: S 5 „ M r: t't [S22S^"0®??^w to • ^ ;; — is :r sc f- r* j„ -•„ ='^ ^ "^ s « s .-- c 2 F <; ~ ^ ■ jo > o;30oo9 5'='5<'SqD_'> = c::;^ ca o cc o' > o J: '^ Tj< o ,- t, o o « S CO ^ c _ _ ^, :-> 1- iM r^ ;; t- S f=H s^ .-1 .-I 2 © <=> L-t tr CO ^^ -V ?t _i •■ o - -, -^ ^ 5rc5 e^ is I ff^^i ®00' SS'-dC'S't- ^A.ciiao.2 _"^fe S'-' S c o2«5 - ,©©.c«ui9goo©. C CO '^ o «^ o oc ;? ■* ^- = . . . .^ . . .^ . I 'V % ^ 03 .^ B. T^ fe fa. 1 1 "111 .| = ^11 .|i|.i I I 482 WEIGHT OF MAKINE ENGINES AND BOILERS. o 2: 5 ""^ ^\ -> a; — >i s^liir I ;^s~^ 1 5 '^ 5- =i la y o V 5 _?^_ — r~ " -,M'ac-^~:Cr-t _ lift-' — ~-i -"^ r =_' • - -r' "^ « =5 = S M ^ ■-" 'i" ~ c ""■ - - rTi r- CI OC f; ^ _ X J2 "^^ 2 . P S -S la S C: r- ? S S — — '* ~ -^ ~ '^ = = i t^^„'«-EH--^:::^i~:i,5£- _ ;:^ c = i i 5 .1 X S - i -i C^ g S W « ,_ _ ■ JO CO I (?a "^r^ >*^ r- I 1 s— ' Cj ^^ •*:' ^ e I fi - _ t- TT o 'E^ ;x x;x ' ' c»= ■T) — C-1 o . -r 2 »1 :S aa. M r-i S o ,2 « - 2 " o 5 — r-< «" > — 'a .2 ic ^ ^ £ _!: -^ --^ ? o _ 5 S g - O O = _ =! " ~ ^ - " .^ « u ~ . -=, 5 -S £: S , -c ; 3 'rH "- = a 5?=^ a **^ — *^- f^? - -< =<"'»- +^-^ ci a ttrt : = c WEIGHT OF MAKIXE ENGINES AND BOILERS. 483 02 xn. "A (—4 ft o SD c 02 ®^ 32 . r5 o ^ - c: W ro _ j; - ' I -~ ~- .V -V is * ~ r~ I w — ^, ^ ?* rg c^ S^ o a ^ OS ■?^ "to !3 E tc --. '■^ i-'H C o =; — i ^^^u-T"?!: 2'='^®~-,o So®, « .H S S = cs ^^ o p-*^ I I "^^ ^t ^ Av ~ irx -"^ o . ^ o 1-1 I- ?J °^ :_ ^ ^ ; -^ , - 2 ~ ^ :r S ~Vx'z-^^':^^^^o O 53 ;5 O -a O 4^ b"" S a -• *-t i-t ^ ';r r; *^ :j^ r^ U C ^ O -1 S - ~^ -« -<^ ■•'COfA— -.- V® c o ^ to >< — OCD»0 S c» =-S ;£ p. ^ n = -~5 c. ^ • = i^ •£? r~ "r-S — < TO a "^ :l = X g c !£■■ ^ ~ .3 •^ .2 ^ P '-»3 XT. 7; S s >, •^^r-'- >■-= c •= o CO . '^ c 5 -e ji; o -5 5"^ a o - u J3 ^-a-S £ O -J ■■ o v: : - > CJ t : o ^ ( tjc o so 5c' .^ , a o s • ; r ^ ::, o ; -If c: i?;^ = =- 484 PROCESSES FOP. SEASOXi:^'G TIMBER. SEASONING TIMBEE. JSatural Seasoning. This is performed by exposing the timber freely to the air in a dry place sheltered from the wind and sun, and so stacked as to admit of the air passing freely over all the surfaces of the pieces. Timber for carpenter's work will require about two years to season it properly ; for joiner's work, about four years, or even longer. Seasoning hy a Vacuum. The timber is placed in a chamber from which the air is exhausted, heat being at the same time emploj'ed so as to vaporise the exuded juices, the vapour being conveyed away by means of pipes surrounded by cold vrater. Seasoning hy Hot Air {Davidson^. The timber is placed in a chamber and exposed to a current of hot air impelled by a fan at the rate of about 100 feet per second, the air passages, fan, and chamber being so arranged that one-third of the volume of air in the chamber is blown through it per minute. The temperature of the hot air varies for different kinds of timber as follows : — Oak of any dimensions . 105° F. Bay mahogany 1" boards . 280°-300° Leaf woods in logs . . 90^-100° Pine woods in thick pieces 120° Water Seasoning. Tliis is done by immersing the timber in water — if shallow and salt it is better than fresh — and letting it remain there for periods averaging from 10 to 20 years, but it is sometimes only allowed to remain 14 days, when it is taken out and stood upright in some sheltered place where the air can get at it thoroughly, so as to render it quite dry. Sometimes it is thoroughly boiled or steamed for a day or two instead of being immersed in cold water for longer periods. All these processed tend rather to injure the strength of the wood, making it softer, althougii it tends to prevent cracking, war^nng, and shrinking. Note. — Slowly seasoned timber is tougher and more elastic than when it is rapidly dried. Seasoning by heat alone is very injurious to timber, as it produces a hard crust on the surface and prevents the moisture from evaporating. For joiner's work and carpentry natural seasoning should have the preference. PROCESSES FOE PEESEEYING TDIBER. 485 PRESERVING TIMBER Ceeosoting- (Bethell.) The timber is first well dried, either by being freely exposed to the thorough circulation of the air or dried in an oven at a temperature varying from 90° to 100° Fahr., depending on the kind of timber. One process is then to place the timber in a strong iron cylinder, and subject it to a vacuum of 6 to 12 lbs. per square inch for 30 or 40 minutes. The creosote is then allowed to flow in, and a pressure put upon it, varying from 100 to 150 lbs. per square inch, for about 1 to 2^ hours. The other process consists in simply immersing the timber in an open tank containing hot creosote, the temperature beingkept up to about 120°to 150" Fahr., and left for some time to the natiu'al process of absorption. JVate. — Ordinary fir timber absorbs from 8 to 10 lbs. of creosote per cubic foot of timber : red pine, from 15 to 16 lbs. ; memel, from 10 to 12 lbs. ; oak, from 4 to 5 lbs. This method of preserving timber is the most generally used : it is a sure pre- ventive against the attack of the teredo and other marine worms. Impeegnatiox with Metallic Salts. Kyan's Process. This consists in immersing the timber in a solution of bichlo- ride of mercury diluted with about 100 to 150 j^arts of water, or about 1 to | of a lb. of the salt to 10 gallons of water. Twenty-four hours are usually allowed for each inch in thickness for boards, kc. Jlargai'ij's Process. Margary employed sulphate of copper diluted with about 40 to 50 parts of water, applied with pressure varying from 15 to 30 lbs. per square inch for 6 or 8 hours. Burnett's Process. A solution of about 1 lb, of chloride of zinc to 4 or 5 gallons of w^er is injected and applied with a pressure varying from 100 to 120 lbs. per square inch for about 15 minutes. The timber is then taken out and allowed to dry for about 14 days. The timber should remain immersed for about 2 days for every inch in thickness. Payne's Process'. Payne's process consists in impregnating the timber with a strong solution of sulphate of iron, and afterwards forcing in a solution of anv of the carbonate alkalies. 486 3IEASUEEMENT OF TIMBER. TIMBEE MEASUEZ. In estimating quantities of timber duodecimals are usually employed — that is, the foot, inch, seconds, >xc., are each di%'ided into twelve parts instead of ten, as in common decimal fractions ; so that by this means feet, inches, and seconds can be directly multiplied by feet, inches, and seconds. Thus : — 12 inches make 1 foot. 12 seconds make 1 inch. 12 thirds make 1 second. 12 fourths make 1 third. And— Feet multiplied by feet give feet. Feet multiplied by inches give inches. Feet multiplied by seconds give seconds. Inches multiplied by inches give seconds. Inches multiplied by seconds give thirds. Seconds multiplied by seconds give fourths, Sec. ■ To Multiply by Duodecimals. SuLE. — Place the several denominations of the multi|)lier directly under the corresponding denominations of the multi- plicand. Then multiply each denomination in the multiplicand by the number of feet in the multiplier, and place each product under its corresponding denomination in the multiplicand, always carrying one for every twelve. In the same manner multiply by the number of inches, and set each product one place farther to the right hand. Then multiply by the number of seconds, and set each pro- duct another place farther to the right hand. Thus proceed with all the other denominations, and the sum of all the products will' be the whole product required. Examjyle 1. Example 2. Multiply 3 ft. 6i ins. by Multiply 2 ft. 7 ins. 4. sees. 2 ft. 5^ ins. 8 thirds by 1 ft. 2 ins 3 sees. ft. ins. sees. 3 thirds. 3 6 6 ft. ins. ?6C5. thrds. 2 o 3 2 7 4 8 7 1 1 2 3 3 15 8 6 2 7 4 8 10 7 6 . 5 2 9 4 Ans. 8 7 7 16 7 10 2 7 10 2 An$. H 1 2 11 4 2 MEASUREMENT OF TIMBER. 487 To Find the Solid Contents of Eound or Unsquared Timber. EULE 1. — Multiply the square of the quarter-girt by the length, and the product will be the solid contents. Rule 2. — Find the area in the following table which cor- responds to the quarter-girt .in inches, and multiply it by the length of the timber in feet ; the product will be the solid con- tents in cubic feet and decimals of a cubic foot. What is the solid contents of a tree whose girt is 60 inches and whose length is 18 feet ? By Rule 1. By Rule 2. 4)60 fb. ius. ins. 15 = 1 3 1 r{ ft. 16 9 ft. ins. sees. 18 1 6 1) 18 9 1 1 6 Ans. 28 1 6 4)60_ 15 ins. Corresponding to 15 ins. in the table is 1-562 feet, and sq. f t. 1-562 IS 12196 1562 Ans. 28-112 Table of Constants for Measuring Timber. ' Girt Area. Girt Area. Girt Area. Girt Area. Girt Area. \ 4 Sq. Ft. 4 Sq. Ft. 4 Sq. Ft. 4 Sq. Ft. 4 Sq. Ft. Ins. Ins. Ins. Ins. Ins. 6 •250 9f •660 13i 1-266 m 2-066 24 4-000 6i •271 10 •694 13f 1-313 m 2-127 24i 4-168 4 •293 10\ •730 14 1-361 nf 2-188 25 4 -.340 6f •316 m •766 14} 1-410 18 2-250 2H iol6 7 •340 lOf •803 14i 1^460 18i 2-377 26 4-694 7i •365 11 -840 14f 1-511 19 2-.507 26i 4-877 n •391 11t •879 15 1-562 191 2-641 27 5-063 7f •417 lU •918 15i 1-615 20 2-778 274 5-252 8 •444 111 •959 15i 1-668 20i 2-918 28 5-444 8t •473 12 1-000 15f 1-723 21 3-063 28i 5-641 8i •502 12i 1-042 16 1-778 2H 3-210 29 5-840 8f •532 12i 1-085 16i 1-834 22 3-361 2H 6^043 9 •563 12| 1-129 16i 1-891 22i 3-516 30 6-250 9| •594 13 1-174 16f 1-948 23 3-674 31 6-674 n •626 13| 1-219 17 2-007 23i 3-835 32 7-111 488 MEASUREMENT OF TIMBER, AND BRICKLAYING. Timber Measures. 40 cubic feet of unhewn timber 50 „ „ squared ,, 600 superficial feet of 1-inch planks or deals 400 300 240 200 170 150 100 120 deals n [ =1 load. 3 ^, „ 5> '■'■2 5> 55 >J ^ J3 55 J „ make 1 square of boarding, flooring, Sec. 1 hundred. Battens are 7 ins. wide, deals 9 ins., and planks 11 ins. Waste on Converting Timber. English oak = 200 per cent „ „ plank = 50 55 Greenheart = 25 J5 Mahogany = 30 55 Quebec oak = 10 55 Teak 15 ;5 African oak =100 per cent. American elm = 15 Dantzic fir plank = 25 ,, oak = 50 „ „ plank = 40 English elm = 200 Dantzic fir, when cut from planks . . = 10 per cent. Yellow pine, when cut for head and stern work = 200 „ „ „ „ decks . . . = 10 „ Plastering. 1 In. TMck. I In. Thick. i In. Thick. 1 bushel of cement will cover 1^ sup. yd., 1^ sup. yd., 2 J sup, yds. 1 do. and 1 of sand , „ 2^ sup. yds., 3 sup. yds., 4^ „ 1 2 31 4i fii -■■ 55 ■^ 55 55 "3 55 ^2 5> "4. 55 -"- 55 ^55 55 *o 55 '-' 55 «^ 55 1 cubic yd. of lime, 2 yds. of sand, and J ^!^ ^' ^ ^' ^ -. ' 3 bushels of hair will cover . • 1 1^ " " -, ^-u ' (60 ,, ,, laths. Bricklaying. Size in In 5. Weight in Lbs London stock bricks 8f x 4^ x 2-| 6-81 Red kiln . . ditto. 7-00 Welsh fire . . 9 x 4^ x 2| 7-84 Paving . . . 9 X 4^ X 1^ 5-00 Square tiles . . 9| x 9| x 1 5-70 ., . . G X G X 1 2-lG BRICKLAYING AND PAINTING. 489 A rod of brick-work = I6i ft. x 16^ ft. x l^ brick thick. = 306 cubic ft. = 11| cubic yards. = 272 sup. ft. li brick thick. = 4,.3o2 stock bricks i courses 1 ft. high. = 4,533 „ if „ measure 11^ ins. = 5,371 „ laid dry. Bricks absorb about ^ their weight of water. A rod of brick- work requires about 3 cu. yds. of mortar, or 1^ cu. yd. of chalk lime and 3 loads of sand, or 1 cu. yd. of stone lime and 3^ loads of sand, or 36 bushels of cement and an equal quantity of sand. A load of mortar or sand = 1 cubic yard. A bag of cement = 3 bushels, a sack = 5 bushels. A load of mortar requires about 9 bushels of lime and 1 cu. yard or load of sand. 500 bricks = 1 load, 830 stock bricks weigh 1 ton. 1,000 bricks loosely stacked occupy about 72 cu. ft. „ closely ,, „ 56 „ MoHar is composed of 1 of lime to 3 or 3^ ©f sharp sand. Concrete „ „ „ 4 of gravel and 2 of sand. Cement „ „ Portland cement to 3 of sand, or the cement may be used alone. Painting. As an average ^ lb. of paint should be allowed per sq. yd. for the first coat, and about ^ lb. for each additional coat. 1 lb. of stopping should be allowed for every 20 sq. yds. A gallon of tar and 1 lb. of pitch wall cover about 12 sq. yds. the first coat, and 17 yds. each additional coat. Priming consists of white lead and linseed oil. Knotting „ red lead and size. Putty „ Spanish whiting and linseed oil. White Paint. 28 lbs. white lead, 6 pints linseed oil, 2 pints turpentine, and 1 lb. litharge will cover about 100 sq. yds. Black Paint. 28 lbs. black paint, 10 pints linseed oil, 2 pints turpentine, and 1 lb. litharge will cover about 160 sq. yds. JDistenqjer. 112 lbs. whiting, 28 lbs. dry white lead, and 7 lbs. glue, mixed with boiling water. 4^0 ALLOTS, SOLDERS, AND VARNISHES. Table of Alloys. (Component Paxrs Allot i Copper , Tin Zinc Brass Soft gun-metal Metal for toothed wheels . » J> 55 Hard bearings for machinerj Gtm metal for mathematical in strtiments Speculum metal , Sound copper castings Tombac, or red brass Ked sheet brass , Brass that solders well Ordinary brass Muntz metal Extremely tenacious metal . Bearings to stand great strains Extremely hard metal Government standard metal Articles for turning Bearings, nuts, &zc. Bell metal .... Statuary bronze . 16 lOf 16 12 91, 1" 8 H n 2 U 16' 16 16 Ui 16 90 1 1 2i 1" 1 1 32 1 1 1 — . 1 — 1 , u 1 2 91 1 -'■2 2 2^ 91 8 -2 14^ 2 2| 5 2 5 12 u 1* Table of Solders. Solders Component Parts j 1 1 JrlUi 1 Copper Tin Lead Zinc Bismuth! 1 Coarse solder for plumbers . 1 3 — Resin Fine solder for plumbers . — 1 2 — — Solder for tin — H 1 — — „ or chloride of zinc „ pewter . — 3 4 — 9 «. •» „ bismuth — 2 2 — 1 Brazing, soft 4 1 3 ~ ' f Sal ammoniac or 1 „ hard ,, hardest . 1 - 3 — — , ' r chloride of zinc | Varnishes. Bhick Japan for Metals. — Burnt umber 4 ozs.,asphalttim 1^ oz., boiled oil 2 quarts. Mix by heat and thin with turpentine. Another Recipe. — Amber 12 ozs., asphaltum 2 ozs. Fuse by VAENISHES. 491 heat ; add boiled oil half a pint, resin 2 ozs. ; when cooling add IG ozs. of oil of turpentine. Black Japan VarnisJi. — Bitumen 2 ozs., lamp black 1 oz., Turkey umber ^oz., acetate of lead ^ oz., Venice turpentine ^ oz., boiled oil 12 ozs. Melt the turpentine and oil together, carefully stirring in the rest of the ingredients, previously powdered. Simmer all together for ten minutes. Cahinetmaher's Varfiish.—'Pale shellac 700 parts, mastic 65 parts, strongest alcohol 1,000 parts by measure. Dissolve and dilute with alcohol. Cabinet Varnish. — Fused copal 14 lbs., hot linseed oil 1 gallon, hot turpentine 3 gallons. Properly boiled, dries very quickly. CkeajJ Oak Varnish. — Dissolve 3Hbs. of paJe resin in 1 gallon of oil of turpentine. Common Varnish. — Dissolve 1 part of shellac in 7 or 8 of alcohol. Co2)al Vai-nish. — Copal 300 parts, drying linseed oil 125 to 250 parts, spirit of turpentine 500 parts. Fuse the copal as quickly as possible ; then add the oil, previously heated to nearly boiling point ; mix well ; then cool a little and add the spirit of turpen- tine ; again mix well, and cover up till it has cooled down to 130° Fahrenheit ; then strain. Co2)al Varnish for Metals, Chains, ^'e. — Copal melted and dropped into water 3 ozs., gum sandarach 6 ozs., mastic 2^ ozs., powdered glass 4 ozs., Chio turpentine 2^ ozs., alcohol of 85 per cent. 1 quart. Dissolve by gentle heat. Gold Varnish. — Turmeric 1 drachm, gamboge 1 drachm, oil of turpentine 2 pints, shellac 5 ozs., sandarach 5 ozs., dragon's blood 7 drachms, thin mastic varnish 8 ozs. Digest with occa- sional shaking for 14 days in a warm place ; then set it aside to fine and pour off the clear. Elastic Varnish. — Gum mastic 5 lbs., spirits of ttirpentine 2 gallons. Mix with gentle heat in a close vessel ; then add pale ttirpentine varnish 3 pints. Table Varnish. — Dammar resin 1 lb., spirits of turpentine 2 lbs., camphor 200 grains. Digest the mixture for 24 hours. The decanted portion is fit for immediate use. Another Recipe. — Oil of turpentine 1 lb., bee's wax 2 ozs., colophony 1 drachm. Turpentine Varnish. — Eesin 1 part, boiled oil 1 part. Melt and then add turpentine 2 parts. Varnish for Iron-rcoi'Ti. — Dissolve 10 parts of clear grains of mastic, 5 parts of camphor, 15 parts of sandarach, and 5 parts of elemi in a sufficient quantity of alcohol, and applj' cold. 492 LACQUERS. Another Recipe. — Dissolve in about 2 lbs. of tar oil \ lb. of asphaltum, \ lb. of powdered resin. Mix hoi in an iron kettle and apply cold. Varnish for Metals. — Dissolve 1 part of bruised copal in 2 parts of strongest alcohol. It dries very quickly. Another Recipe. — Copal 1 part, oil of rosemary 1 part, strongest alcohol 2 or 3 j^arts. This should be Applied hot. White Copal Varnish. — Copal 16 parts ; melt, and add hot linseed oil 8 parts, spirits of turpentine 15 parts. Colour with the finest white lead. WJiite Priming for Japanning. — Parchment size |, isin- glass |. White Varnish. — Tender copal 1\ ozs., camphor 1 oz,, alcohol of 95 per cent. 1 quart ; dissolve, then add 2 ozs. of mastic, 1 oz, of Venice turpentine ; again dissolve, and strain. White Spirit Varnish. — Sandarach 25 parts, mastic in tears 6 parts, strongest alcohol 100 parts, elemi 3 parts, Venice turpen- tine 6 parts. Dissolve in closely corked vessel. Lacquers. To niaTte Lacquer. — Mix the ingredients and let them stand in a warm place for 2 or 3 days, shaking them freely till the gum is dissolved, after which let them settle for 48 hours, when the clear liquor may be poured off ready for use. Pulverised glass is sometimes used to carry off impurities. Gold Lacqxier. — Ground turmeric 1 lb., gamboge 1^ oz., powdered gum sandarach 3^ lbs., shellac | lb., spirits of wine 2 gallons. Shake till dissolved, then strain and add 1 pint of turpentine varnish. Gold Lacquer fm' Brass not Dipped. — Alcohol 4 gallons, turmeric 3 lbs., gaml^oge 3 ozs., gum sandarach 7 lbs., shellac \\ lb., turpentine varnish 1 pint. Gold Lacquer for Dijrped Brass. — Alcohol 36 ozs., seed-lac 6 ozs., amber 2 ozs., gum gutta 2 ozs., red sandal-wood 24 grains, dragon's blood 60 grains, Oriental saffron 36 grains, pulverised glass 4 ozs. Good Lacquer, — Alcohol 8 ozs., gamboge 1 oz., shellac 3 ozs., annotto 1 oz., solution of 3 ozs. of seed-lac in 1 pint of alcohol; when dissolved, add Venice turpentine \ oz.j dragon's blood \ oz. Keep in a warm place 4 or 5 days. Good Lacquer f 01' Brass. — Seed-lac 6 ozs., amber or copal 2 ozs., best alcohol 4 gallons, pulverised glass 4 ozs., dragon's blood 40 grains, extract of red sandal-wood obtained by water 30 grains. Lacquer for Dipped Brass. — Alcohol of 95 per cent. 2 gal- DIPPING ACIDS. 49S Ions, seed-lac 1 lb., gum coiDal 1 oz., English sa&on 1 oz., an- notto 1 oz. Another Recipe. — Alcohol 12 gallons, seed-lac 9 lbs., tur- meric 1 lb. to a gallon of the above mixture, Spanish saffron 4 ozs. The saffron is only to be added for bronze work. Lacquer VarnisJi. — Add so much turmeric and annotto to lac varnish as will give the proper colour, and squeeze through a cloth. Pale Lacquer for Brass. — Alcohol 8 gallons, dragon's blood 4 lbs., Spanish annotto 12 lbs., gum sandarach 13 lbs., turpen- tine 1 gallon. Dipping Acids. Aquafortis Bronze Dii?. — Nitric acid 8 ozs., muriatic acid 1 quart, sal ammoniac 2 ozs., alum 1 oz., salt 2 ozs., water 2 gallons. Add the salt after boiling the other ingredients, and use it hot. Brown Bronze Dip. — Iron scales 1 lb., arsenic 1 oz., muriatic acid 1 lb. ; a piece of solid zinc, 1 oz. in weight, to be kept in while using. Brown. Bronze Paint for Copper Vessels. — Tincture of steel 4 ozs., spirits of nitre 4 ozs., essence of dendi 4 ozs., blue vitriol 1 oz.,water ^ pint. Mix in a bottle. Apply it with a fine brush, the vessel being full of boiling water. Varnish after the appli- cation of the bronze. Bronze for all kinds of Metals. — Muriate of ammoniac (sal ammoniac) 4 drachms, oxalic acid 1 drachm, vinegar 1 pint. Dissolve the oxalic acid first. Dipping Acid. — Sulphuric acid 12 lbs., nitric acid 1 pint, nitre 4 lbs,, soot 2 handfuls, brimstone 2 ozs. Pulverise the brimstone and soak it in water 1 hour ; add the nitric acid last. A?i other Becipe. — Sulphuric acid 4 gallons, nitric acid 2 gal- lons, saturated solution of sulphate of iron (copperas) 1 pint, solution of sulphate of copper 1 quart. Good Bippintji Acid for Cast Bi'ass. — Equal quantities of sul- phuric acid, nitre, and water. A little muriatic acid may be added. Green Bronze Bip. — "Wine vinegar 2 quarts, verditer green 2 ozs., sal ammoniac 1 oz., salt 2 ozs., alum \ oz., French berries 8 ozs. Boil the ingredients together. Ormolu Dipping Acid far Sheet Brass. — Sulphuric acid 2 gallons, nitric acid 1 pint, muriatic acid 1 pint, water 1 pint, nitre 12 lbs. Put in the mm^iatic acid last, adding a little at a time, and stir with a stick. Another ^^^f^f?.— Sulphuric acid 1 gallon, sal ammoniac 1 oz.. 494 CEMENTS AND GLUES. flowers of sulphur 1 oz., blue vitriol 1 oz., saturated solution of zinc in nitric acid mixed with equal quantity of sulphuric acid 1 gallon. Vinegar Bronze for Brass. — Vinegar 10 gallons, blue vitriol 3 lbs., muriatic acid 3 lbs., corrosive sublimate 4 grains, sal ammoniac 2 lbs., alum 8 ozs. Cemexts astd Glues. ' Cement f 01' EaHlien and Glass Ware. — Isinglass dissolved in proof spirit and soaked in water 2 ozs. (thick) ; dissolve in this 10 grains of very pale gum ammoniac (in tears) by rubbing them together, then add 6 large tears of gum mastic dissolved in the least possible quantity of rectified spirit. Cement for Iron Tiiltes, S'c. — Finely powdered iron 60 parts, sal ammoniac 1 pint, sufficient water to form into a paste. Cement for Plumbers. — Black resin 1 part, brick dust 2 parts. Melt together. Cement for Leahy Boilers. — Powdered litharge 2 parts, fine sand 2 parts, slaked lime 1 part. Cement for Joining Jletals and Wood. — Stir calcined plaster into melted resin until reduced to a paste ; add boiled oil till brought to the consistency of honey. Apply warm. Cast-iron Cemetit. — Clean iron borings or turnings pounded and sifted 50 to 100 parts, sal ammoniac 1 part. When it is to be applied moisten it with water. Turner's Cement. — Bee's was 1 oz., resin ^ oz., pitch ^ oz. Melt and stir in fine brick dust. Coppersmith's Cement. — Powdered quick lime mixed with bullock's blood and applied immediately. Engineer's Cement. — Equal weights of red and white lead mixed with drying oil. Spread on tow or canvas. Cement for Joining Metal and Glass. — Copal varnish 15 parts, drying oil 5 parts, turpentine 3 parts, oil of turpentine 2 parts, liquid glue 5 parts. Melt in a bath and add 10 parts of slaked lime. Gasfitters Cement. — Resin 4| parts, wax 1 part, Venetian red 1 part. Cement for Fastening Blades into Handles. — Shellac 2 parts, prepared chalk 1 part, powdered and mixed. Cement for Pots and Pans. — Partially melt 2 parts of sulphur and add 1 part of fine blacklead. Mix well. Pour on stone to cool, and then break it in pieces. Use like solder with an iron. Cement foi' CracJis in Stoves. — Finely pulverised iron made into a thick paste with water glass. WOOD-STAINING AND ENAMELS. 495 Very Strong Glue. — Mix a small quantity of powdered chalk with melted common glue. Glue to Resist Moisture. — Boil 1 lb. of common glue in 2 quarts of skimmed milk. Marine Glue. — Cut caoutchouc 4 parts into small pieces and dissolve it by heat and agitation in 34 parts of coal naphtha, add to this solution 64 parts of powdered shellac, and hea't the whole with constant stirring until combination takes place, then pour while hot on to metal jDlates to form sheets. When used must be heated to 280^ Fahr. Liquid Glue. — Dissolve 1 part of powdered alum in 120 parts of water ; add 120 parts of glue, 10 parts of acetic acid, and 40 parts of alcohol. Digest. Another Becipe. — Dissolve 2 lbs. of good glue in 2| pints of hot water, add gradually 7 ozs. nitric acid, and mix well. Parckment Glue. — Parchment sha\dngs 1 lb., water 6 quarts ; boil until dissolved, then strain and evaporate slowly until of proper consistency. Drauglitsmaw s or Mouth Glue. — Glue 5 parts, sugar 2 parts, water 8 parts. Melt in water bath and cast in moulds. For use dissolve in warm water or moisten in the mouth. Wood-staining. MaJiogany Colour {DarTi). — Boil together in a gallon of water ^ lb. of madder and 2 ozs. of logwood. "V\Tien the wood is dry, after having been washed over with the hot liquid, go over again with a solution of 2 drachms of pearl ash in a quart of water. Maliogany Colour {LigM). — Wash the stirface with diluted nitrous acid, and when dry use the following : — dragon's blood 4 ozs., common soda 1 oz., spirits of wine 3 pints. When well dissolved, strain. Rose Wood. — Boil 8 ozs. of logwood in 3 pints of water until it is reduced to half. Apply boiling hot two or three times. The stain for the streaks is made from a solution of copperas and verdigris in a decoction of logwood. E'bony. — Wash the wood with a solution of sulphate of iron ; when dry, apply a mixture of logwood and nut galls ; when dry, wipe with a sponge and polish with linseed oil. Enamels. White Enamel. — Potash 2.5 parts, arsenic 14 parts, glass 13 parts, saltpetre 12 parts, flint 5 parts, and litharge 3 parts. 490 KECIPES, CASK- GAUGING, ETC. JBlacTi Enamel— QlSij 2 parts, protoxide of iron 1 part. Blue Unm^ieL— Fine paste 10 parts, nitre 3 parts : colour with cobalt. Green Eyiamel.—Yiii 1 lb., oxide of copper \ oz., red oxide of iron 12 grs. Yellon- Enamel. — "WTiite lead 2 parts ; alum, white oxide of antimony, and sal ammoniac, each 1 part. Tracing Paper. Nut oil 4 parts, turpentine 5 parts ; mix and apply to the paper, then rub dry with flour and brush it over with ox gall. Indians' Ink. Finest lamp black made into a thick paste with thin isin- glass or gum water, and moulded into shape. It may be scented with essence of musk. Copying Ink. Add 1 oz. of moist sugar or gum to every pint of common ink. Staircases or Companion Ladders. The ordinary tread of a stair or step is 8 ins., and rise 7^ ins. ; above or below that |- in. rise must be subtracted or added for every inch added to or taken from the width of tread, as the case may be. Cask- gauging. c = contents of cask in gallons. D = middle or bung diameter in ins. L = length in ins. d = end or head diameter in ins. c = •0009442l(2d- + d-) considerably curved. c = •0009442l(2d2 ^ ^p^ _ |(u _ ^y. moderately curved. C = -0014162L(D2 + c = -OOOOSISlCSGd^ + 2od' + 26Dd) any form. Variations of Tides. The difference in time between high water and high water averages about 49 minutes. MISCELLANEOUS RECIPES, ETC. Compositions of Gunpowder. 497 America 75 saltpetre 12-5 charcoal 12-5 sulphur Austria 72 17 „ 16 England 75 15 „ 10 France 75 12-5 „ 12-5 Germany 75 „ 13-5 „ 11-5 Russia 73-78 „ ]3-59 „ 12-63 Spain 76-47 „ 10-78 „ 12-75 Sweden 76 15 „ 9 Average wei ght per cubic foot = 56-42 lbs. • Cubical cont ents of 100 lbs. = 1-773 cu. ft. »> >J V = 3063-7 cu. ins. Tempering Steel. Colour. Temperature. Purpose. Light straw . . 430°-410° . . turning tools for metals. Dark „ . 470°-480° . . tools for wood, screw taps, and dies. Dark yellow . 500<'\ 530°/ hatchets, chipping chisels, Light purple . saws, kc. Dark „ . 550° springs, &c. Conducting Powers of Various Substances. Soft woods are not such good conductors of sound as the harder kinds. The comparison between metals is as follows : — Gold =1,000. Copper = 898. Zinc =363. Silver = 973. Iron =374. Lead =180. Sizes for Lightning Conductors. Copper rod, f in. diam. „ pipe, 1^ in. diam., ^ in. thick. Iron rod galvanised. 1| in. diam. „ pipe, 2^ ins. diam., ^ in. thick. Flat copper bar, 3 ins. wide by ^ in. thick. Preservative for Steel. Caoutchouc 1 part, turpentine 16 parts, and boiled oil 8 parts, well mixed and boiled together. The caoutchouc should first be dissolved in the turpentine by a gentle heat, and the boiled oil then added. It should be applied with a brush, and it may be removed by turpentine. Specific Gravity. "W = weight of body in air. w = weight of body in water, L = weight of lead and body in water. I = weight of lead in w-ater. (1) Bodies heavier than water. (2) Bodies lighter than water. w r, w Sp. gr. = Sp. gr. =. W-7V ' " (W-rZ)-L Kbte. — Tn the second example the body is sunk by attaching to it a heavy substance such as lead. KK 498 ADMIRALTY REGULATIONS FOR THE TRANSPORT SERVICE. Admiralty Regulations for the Transport Servich. Transports must have a height of 6 feet from deck to beam ; in ships conveying horses, 7 feet, and 12 in hold from ceiling to beam. Measurement stores are usually rated at 40 cu. ft. to the ton ; heavy stores, at 20 cwt. In freighting siorc skips the Government stipulates for the conveyance of one passenger to eveiy 25 tons of stores (if requii-ed), at the rate of 6 tons freight fot every first-class passenger, 4 for every second, and 3 for every third. Ships conveying over 50 troops are to have a free-board of not less than 4 ins. for every registered foot of depth of hold. The dimensions of a cabin for one officer, 30 superficial feet ; for two, 42 ; 10 additional for every ofiicer in addition — all independent of the bed-places, which are to be 6 feet long and 27 inches wide. The standing bed-places for one woman and two children under ten years of age, or for two women, are to be ,6 feet long and 3 feet wide. x\ll standing bed-places to be kept ii ins. from the ship's side. Hospital accommodation, 2 or 3 per cent, of the passengers. The hammocks are to be 6 feet long ; each is to have a space 9 feet long by 16 ins. wide. The crews of transports are to be four men to every 100 tons register, with two boys in addition in every ship : paddle- wheel steamers, five men to every 200 tons gross register ; screws, three to every 100 tons gross register; engineers, &c. (in addi- tion), one man to every 15-horse power. Horses should be allowed daily 6 lbs. oats, 10 lbs. hay, half -peck or 2^ lbs. bran, 8 gallons of water, and such quantities of vinegar and nitre as may be required. Their stalls should be about 8 feet long, 3^ to 4 feet broad, 5 to 6 feet high, rising at head to 7^ and 8 feet. Table giving the Total Weight and Measurement allowed for officers. ^^aval Officers. Coramander-in-chief Admiral, vice-admiral, rear admiral Captain of fleet, commodore, inspector-general of hospitala ) and fleet f Caprain, chaplain . Staff captain, df^puty inspector-peneral of lidspitai? and fleet. ^ secretary to c )nimander-in-ctiief or flasr officer, inspector of ( machinerj' atl'">ut, communder, staff commander, staff sur- j ?eon, lientenant. master, snrpreon, paymaster, chief engineer ' Bocsretary to commodore, naval instructor, as.«i.sfcant surfreon Sub lieutenant. c)iief warrant officer, .'^eiond master, as.'^i-tant pavmaster, eufrineer. ns^ilstant engineer, warrant oflioer and all subordiiut-fi officons ) n Cwt. Cu. Ft 40 I 200 36 j 180 30 I 150 26 I 130 18 i SO 12 ; 60 i 6 i 30 PROVISIONS ALLOWED IN THE KOYAL NAVY. 499 Table of THE Weight of Provisions as allowed in the j KOYAL Xavy for One Man for Fourteen Days and FOB 1,000 Men for F OUR Months. Kind of Provision For 1 Man for 14 Days For 1 ,000 Men for 4 Months 1 Net I Gross Net Tare of Casks Gross lAllowance Weight Allowance and Packages T.Cwt.Qr.Lb. Weight ' Lbs. Lbs. T.C^vt.Qr.Lb. T.Cwt.Qr.Lb. Bread 14 14-25 53 19 3 22 19 1 2 54 19 "24 Spirits ; 4-016 5-0 15 9 3 14 3 16 1 14 19 6 1 Salt beef . 5"25 8-78 20 4 3 24 13 12 -24 33 17 20 Salt pork . b'2b 8-48 20 4 3 24 12 8 3 26 32 13 3 22 Flour 5-2.J 6T5 20 5 9 3 9 26 •23 14 1 7 Peas . 3-5 4-125 13 10 9 2 8 15 18 8 Oatmeal . •75 -88 2 17 3 10 9 3 1 3 7 2 11 Sugar 1-31 1-601 5 3 23 1 2 1 12 6 3 17 (Jocoa •875 1-105 3 7 1 21 17 2 13 4 5 6 lea . •218 •295 16 3 6 6 9 1 2 3 16 \'inegar 1-3 1-59 5 22 12 13 6 2 1 26 Tobacco — — 3 11 1 -20 18 2 8 5 Soap . — — 1 15 2 24 7 3 4 2 3 2 Total . — — 166 5 1 4 42 8 2 2 208 13 3 € Table of the Weight of Provisions as allowed in the Royal Navy foe each Man per Diem for Fourteen Days. Daj'S Sunday . Monday . Tuesday Wednesday Thursday Friday . Saturday Sunday . Monday. Tuesday Wednesday Thursday Friday Saturday Total for 14 days n Lbs. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 Pts. X Lbs. Lbs Lbs. I 3. I — I -S I 4 f 3. 4 — i 5i Pts. — 5 3. 1 2 2. i H H ■*- d Id C3 s bD .p ^ rt 02 O Pts. Ozs. — H — U — u i H — u — n — H — H — 1* — H 1 2 H 14 H li 1 21 Ozs. 14 Ozs. i. 4 i 4 i Pts. 4 — 3i ' » Bread takes 6 cu. ft. of stowage ior a b-3.g of 112 lbs. = 124 cu. ft. rer ton. t One gallon of spirits=9-18 lbs. One gallon of vinegar =l'i-4 lbs' „ peas =8-0 ,, oatmeal -&0 ^ Kk2 WEIGHT, SIZES, ETC., OF PROVISION PACKAGES. Z a: Q < «■;■-::; -r .-t ~ r. o -fi o -^ — cc "M — < o 1^ --^ O lO 'w ^1 — v= iC T 1 L^ cro %D T- u: — X t^ ?-> o (_; :o "M -H -i-i T-H I— I ,-1 i^ o cc G^» 1—' ^ _ (?j ^ x o ci o cri !M 5^ lo -n* o C5 o Ci cc »c c^i I - t^ ci i>. ci o (TJ --r -o t^ 1-3 T— ( 1— I ,— t 25 i os-orO(M'M'-Mt>.OX!OC5-*53^1^«3w!rJCiO'MOOaO SI- 1^ 1 -- ,— 1 HW -"in-*!) G HC« ^ X ir: -^ X 'M ec ^ ^ ro c^ ija t-il CO (N rH ,-t I 1 w ^ O O O -f o -< 00 t^ CO eo ' "^ ^ TJ a: 5: c o) ' X3 ^ • ' OS aQ 2 "73 2 C5 c cS X o o cy ^ jj &jc~ u ;/. ^ br,:^ S cS "1^ — « _U,-7^ -1 . *"' "^^ L.-T r'. 'T -^ ' •• -o J. -t; -(CsX rJ- --^ — -«, 1^ lO y^ ^ O * — ^ rj C^ — " ^ CO <-M ■— ^^ o -^ T— * tO t-- ,^ y. ^ -^. X. C^ 1—1 ^ I- to ■^ 5^1 ■M ^ ^ CO 3-4 ^ '^ o CO o CO CO' — ' CO 'N CO '>! ooo-t-5^c. I- — f<3^c;r^ CO o^ 34 3^4 th 1—1 !^^ 3^ 1—1 1— I -^1 r^ H;< "^ HC' ^>T ic 4 — ^^ if ^ r/ 1 I -** -^ -*^ -«i-ts -tji -w -m c 1 I — r- -^ x i-o 3o «M 'TJ t~ o r-' o i^ ^- uo -^ >.o oo lO 04 -+1 CO CO ■>4 3-1 3^4 CO "M TJ C<1 » 1— ' CO ?^ CO 04 CO 3<1 CO T'l I^ 3^4 34 a:C;0»-<"!*IO»0003<4 3^JOX50ir5C;OX-TrX 0■t^XX'^4-f< .2oOt>-»0Kl(M'-(r-i05X--0r-iXS03<»OC!O-J*34C0-O;--0 — X t_;.— I C0 3^1i— It— ' a33MC034 CO^-r- 1— « - — "5 -^ S3 "3 ci 3 a: C =? fl V3 rP ? z:. ^ -^ 72 EMIGKANTS riETART SCALE. ^01 J o O o < > > < jact^AV J Q If X : c ' c ;: (•Bipni m) in 1 III 1 N rji '^l 00 O ' III 1 c o N o 1 o a S" o u "3 o 3h o s o o «< d ?! = c 6 '^ "^ «> »o c-l C^l -M N -N C-1 'iJb ^TUDSig; '4 ?^ ?1 C^ J1 1^1 •M C-l --- > p C ^ ^ -^ re ^1 L^ --£ C^ «0 — ^^ 6 '"' "3 ■J. % 33 S! o a 2 C >■ o o a l-H > a r^ jauna |i- 1- 1 I-- c 3 ^ai^S 'i^ \ 1 ^1 1 1 ^""'^ ,-K -u 1 Complement of m^^n and officers Officers' men and ell'ecis . Officers' stores and slops . Water Tare of tanks to do. . paAiasajj oil 1 1 ^' ^•lOJ ^ 1 1 <» 1 1 1 ^-^^ Q 1 1 III r-. i '^ 1 00 1 1 1 CO 1 tc s \ III \ ^ 1I r- — o> 1 ^ a S9 Sunday . Monday > Tuesday Wednesday . Thursday Friday . Sat urday Wc(>kly*tolals ^02 STOWAGE OF VARIOUS SUBSTANCES. TA.BLE SHOWING THE Number of Cubic Feet requihed TO Stow One Ton Weight of Various Substances.. Substances Cu. Ft. to a Ton Substances .^"•,J''^- to a Ton Ashes, pot and pearl 40 Indigo, in cases 66 Ballast, Thames 22 Linseed . 56 Rirley .... 47 Marl 28 Bread, in bulk 124 Molasses . 60 Goal, Admiralty 48 Oats, in bulk . 61 „ Newcastle 45 Kice, in bags . 45 „ Welsh . 40 Rum, in casks . 60 Coffee, in bags 61 Saltpetre . 36 Cotton, compressed . 50 Sand, pit . 22 Earth mould . 33 „ river 21 Firewood 288 Sandstone 14 Flax .... 88 Shingle, clean . 24 Flour, in barrels 50 Slate 13 Freestones 16 Sugar, in bags 39 Ginger .... 80 Tares, in bulk . 48 Granite stone . 14 Tea, in boxes . 111 Gravel, coarse . . ,. 23 Timber, hard . 40 Hay, compressed . , 105 „ soft 50 „ uncompressed 140 Turmeric . 66 Hemp 64 Silk, in bales . 128 Hides, well packed . 64 „ pieces, in cases 110 „ loosely packed 84 Wheat, in bulk 45 Table giving the Various Substances which in India | are reckoned AT 50 Cubic Feet TO THE Ton Mea- surement. Apparel Elephants' teeth Roping, in coils Arrowroot, in cases Ginger, in bags Sago, in cases Bee's wax .Gums, in cases Sal ammoniac Blackwood Gunnv bags Sarsaparilla Books Hemp, in bales Senna, in hales or bags Borax, in cases Hides and skins. in Shellac, in cases Camphor, in cases bales Silk piece goods Cassia, all kinds Indigo, in cases Skins Cigars, in boxes Mace, in cases Soap, in bars Cinnamon, in bales Mother-of-pearl, in Stick lac, in cae-es Cloves, in chests cases Tallow Coffee, in cases or bags Musk, in cases Tea, in chosts Coir fibre, in liales Nutmegs, in cases or Timber. hcAvn Colocvnth. in cases casks Tobacco, in bales Cotton, in bales Nux vomica, in bags Tortoise shells Cowries, in bags Haw silk, in bales Wines, in casks Cummin seed Khubarb, in cases Wool, in bales Note. — In England 40 cubic feet is generally taken as a ton measure- ment (see Tonnaye, p. 83). WEIGET AND SCANTLINGS OF SBIPS' BOATS. 503 Table givi^^g the W 'eight of Ships Boats as USED IN Her Majesty's Navy. Boats Lgth. Breadth D. pth \\ ei^nl of ench cwt. qrs.lbs. U.itiu. (1 It. ft. ins. Ii-. iUS. cwt.qrs. Its. Launch not sheathed . 42 11 3 9 89 2 74 3 ,, ., 40 10 6 3 9 82 3 14 67 1 14 Pinnace 32 8 9 3 2 48 2 14 43 1 21 ,, ... 30 8 7 3 1 45 3 14 40 ;^ 21 ., ... 28 8 4 2 10 43 2 21 38 2 14 ,t ... 26 8 n 2 8 36 1 8 34 12 Cutter 34 — - — 26 1, ... 32 — _ 22 3 14 ,. ... 30 8 1 2 8i 17 3 14 16 7 ,. ... 28 7 lOJ 2 8' 15 3 10 14 1 14 ,, ... 26 7 5 2 7 12 2 6 11 1 ,. ... 25 7 3 2 6i 12 1 11 u i> ... 23 6 11 2 5i 10 1 21 9 3 Cutter or Jolly boat . 18 6 2 2" 7 2 14 7 Jolly boat . 16 5 7 2 2 4 3 21 4 1 14 Gig .... 32 6 6 2 2 9 3 14 ,.•••• 30 6 6 2 2 9 7 8 2 14 ,.•••• 28 5 6 2 2 8 10 7 2 7 , 26 5 6 2 2 7 14 ■ 6 2 14 „ . . . . 24 5 6 2 2 6 2 21 6 7 ,.•••• 22 5 6 2 2 5 3 21 6 1 .1 • -. . 20 5 6 2 2 6 14 5 2 14 Cutter gig . 20 6 6 2 6 7 Dingy 14 5 2 2 1 4 7 3 2 7 ,,..,. 12 5 2 3 7 2 3 Whale boat 27 5 3 2 6 8 2 ,. ,, , . 25 5 3 2 4 7 3 Troop boat 38 — 26 3 Lifeboat ('White's';. 32 — — 10 1 19 26 — — 7 3 11 - Table giving the Scantlings of Ships' -Boats as built \ IN Her Majesty's Navy. 1 It. ins. ft. ins. ft.iup ft.ins. ft. ins. ft. ins. ft. ins. ft. ins. Length 34 32 80 25 23 20 18 16 Hreadch 8 10 8 6 8 1 7 3 6 11 6 4 6 5 7 Depth . 2 11 2 10 2 8i 2 6i 2 5A 2 3 2 2 2 1 Keel \ ^'^^"^ • 31 3 3 22 2| n 21 2i 1 moulded 5 42 4.^ 4^ U 0' 4A 4 4 Stem and Post sided 3 22 22 2i 2i 2i 2i 2i ( sided If u li lit 1^ li 1 1 Floors -, moulded li u 1^ H H 1 0| 04 inimiber 22 21 20 17 16 14 13 12 ' j sided . u Is' 1| U 11 1 0| 0^ Futtocks \ moulded i number 1^ 1 1 Oi 0| Of Oi c^ 68 64 60 50 46 4a 36 32 ' Gunwales -! '^^^P ' 2i 2 2 1ft 1| If 15 If Muuwdies -, ^jj.^j^ 2i 2^ ^ 2 2 15 If If Number of knees 20 20 20 18 16 16 14 12 Numi .er of breast hook s 2 2 2 2 2 2 2 2 Extra fixed thwarts fo'c 1 — — — — 1 1 ,, No. double-kneec 3 4 4 4 3 3 3 2 2 „ loose . 3 3 2 2 1 1 1 1 Mast thvrarts, f thick oak t broad If If 1? U n U U n 9 9 9 9 9 8i 8 s- Other thwarts j*^i^^ n 7 u 7~ OH on 7" 7 H 7 H 7' H 7 u 7 Blank of bottom thick Oi Oi Oi 0,^ 0^^, 0^ 0^ 0,^ 504 VENTILATION, ANCHORS, CHAIN CABLES, ETC. "U'EiaHT OF Men and Animals. A crowd of people closely packed — 85 lbs. per sup. ft. The average weight of a man = 140 lbs. 6 ozs., or about 15 men to a ton. A strong cart-horse = 14 cwt. and a cavalry horse = 11 cwt. An ox = 7 to 8 cwt. and cow = 6| to 8 cwt. A pig = l to 1^ cwt. and a sheep = f to l^- cwt. Space Allotted to Animals. A horse = 120 sup. ft. A bullock = 40 to 60 sup. ft. A cow =90 to 100 sup. ft. Sheep and pig«s = 10 „ 12 „ Ventilation, &c. Each pei-son should be allowed at least from 2h to 4i cu. ft. of fresh air per minute. Tlie following are average velocities of air in feet per minute in different positions : — At outlets where foul air escapes from cabins = 1.50 to 198. At inlets where fresh air enters cabins = 78 to 96. In tubes, trunks, chimneys, &c., for fresh or foal air = 720 ; or — Y = velocity in feet per minute in chimneys, &;c. H = height of shaft, trunk, ko., in feet. T =mean temperature in trunk in degs. Fahr. t = temperature of external air in degs. Fahr. k ■- coefficient of dilatation of air for 1° Fahr. = '002 V = 8-025 VB.Jc{T-t) Inclination op Ships' Sliding Ways. For small vessels = 1 in 12 to 1 in 14. For average „ = 1 „ 16 „ 1 „ 18. For largest „ =1 „ 20 „ 1 „ 24. Test Loads of Anchoes and Chain Cables. To find the test load for a qiven chain cable in tons. KuLE. — Square the diameter of the bolt of the cable in ins., and multiply the result by 18. To find thfi diameter of a cahh in ins. to suit a gireti test load. Rule. — Divide the test load in tons by 18, and sefetract the .square root of the quotient. ^ /^ To find the worMng load of chain ')'itjging. Rule. — Square the diameter of the bolt in ins., and mul- tiply the result by 8, To determine the diameter of holt for a chniyi cable in ins. Rule. — Extract the cube root of the load displacement in tons, and multiply the result by -125. ADMIRALTY SIZES OF ANCHOES AND CHAIN CABLES. -505 ■^^ "^^ -S o ^ ^ t:i O o """*" •^ C v-H ^ •rO (^COr-l O rH 1 1 1 1 1 1 i 1 f==< P c< T)< i» OrHO O •— i o c ^^ (M '-' c< c^ IM — 1 ,d ". 1-1 C^ O O l-H c^ o IM CO NOOO rH aj c rt\ ^o "S -* c^ ■* 00 ?^ (^^r-l iCi r^ t~ ■^ 00 Ci t- CO «-1 __ c^ c; — j: Ci 00 1^ t- CO ■^ -< :2 B. O '-"-~ — — ' -^ < S OQ ^ 1 Z "* -"l* -rt" ^ rjl Tj< ■o i-i S -^ Tl< r)i CO CO -^s _^^ .i;_ _^^ ■^"" ^^^"" _,^ ^i_ 05 :^ 'i. -ci'.ix, -« .. ^Xr<4»-i:itH» r«N •-H■K^ *-»^-Hi S 73 i «S •- " '^ rH ri I- r-^ rH rH rH r- rH rH r-l '^ ^ - 1 ^ C^ -p r-^ •— -H .g IM C-< ^ cte ■■Jx-<-> rC« -» «-* U^X © S a> o fi| s^ ss c^ (^^ o LI tH ^ CO C-: !M 1 1 S g ^g -rH -^ rH ^ '"' " " < o J-J 1 5 43 ^. cc o r,- r- rH rH r-H 1-1 rH rH rH 1 1 t3 c 1 C-. OG -CI t^ 1 f~\ lisi •' '-'~ — r- rH '"' '-' "" " ^ c d . 1 , H "^ [ r-,-^ r^ r-rH 1-^ rnrH 1 1 S *""" r-- 1^ -^ f^ — ' t^ ^ ■* Ol- r- _t O Z suox C^ CO c> 'T O CO 'a- 1—1 =C LO -i- u I ^.na o":^ of to sc" t- , CO > ^ ,1 - - ^ o S Si CO c3 O O g 1^ o -e "J;. - ^ 2Q K +; •.- 1:2 o c'd 1 S a cc 'o. «: CHAIN CABLES. ■S -J ^H^ ^^^ i-r< a Bo 1 1 1 1 1 MM 1 ! «* -2 a OQ A ^ 7^ 1 1 1 1 ' MM 1 1 ^ •laasi 1 _• '^ f. -S o 1 — » o o o «~- T ? = -" e 5 "o •§2 & O O (M c» •M 0-1 ~, — _ ~; X. ^ .[_• fe = & L- ti5 IS t^ •■O T»< t-- n T> C< d i.r,^. ^ ~ — «, *, _ i 3 CJ3 2 c-> ^■ ^ ~ o o o -* _« K C X ,~ C) ^ ::^ £ : i&o ^ c OOO CI o ! O O — _ 1 ^ 1 o •*" — 1 : ^ ° -»i IS IT-- 'M c; C-l 1- -x --c *-t X 5 5^ C*» 5-1 »-l ■"^ '"^ a: i 1 ° ^^^ ^ ,-,^_-, F-l 1 ■«1 3 ; ^ 1 ■S o — c: O O O O ^^ .^ , rr -^ „ 'Z '[- ^ 1<5 J-< O •— ' T-H CC CC _ 1 ,_ ~ _ — «•< c< S ►- c ^o 5 ^ — u^ _ — CO r^ Tt< ^ c; X r~ < ^ p c: O t-^ CO M -<' Wrt .-1 ^§ — 1 -H o 2; ■*^ -*tl^ •* f o cc M M ■M W ^ r; -# ^^^^ ^^^^ 1 •_ r. ^ s 1 i^ « r»c-4:i-r-> -^ -ec I t.ixt-?; ■ 1 ' 1 suoy. """■' -t< -1 o ■•* ■^ I^ C to o ^^^^^ ^ C-l CC C'J C5 1 t- l~- ?3 CO >a ^ u I ^notnaryBidsTQ W5 'O -** Co" Ci 1 00 MS -* cc •>« ^"T ^ a-, 1.^ S^ . * » c Lw D • • • •A ^ tJ O C-2 = . ti "i ' s . ^ t- = S fe |81 % 5 o c (Z3 — '_ . '/~* "Tt -^ Ui 5 o a; H ^ P^^-i ci W zr.y,'^< ^^^ O bending moments, horse power, etc. 507 Bending Moments and Sizes of Paddle and Spring Beams. w = load in Ions. D = diameter of shaft in ins. L = lentfth of outboard part of shaft in ins. l' = length of projecting part of paddle beam in ins. M = approximate bending moment of paddle beam. m' = approximate bending moment of spring beam at middle. E =etfecrive sectional area of iron paddle-beam in sq. ins. B = depth of iron paddle beam in ins. b' = depth of square wood in paddle beam in ins. 8 = span from centre to centre of paddle beams in ins. ^ D^ X L „ „. w »,' ^' ^ S W=- M = WXL M= 4000 2 3m Note. — The breadth of the spring beam should not be less than I of the depth of the paddle beams. Nominal Hoese- power. {Loyv-preamre Engines^) Y = assumed velocity of piston = 128 feet per minute x cube root of length of stroke. v' =real velocity of piston in feet per minute. D = diameter of piston in ins. N = nominal horse-power for Admiralty ^a<^- 480-483.) I.H.P. . . 8,000 7.000 6,000 5,000 4,000 3,000 2,000 1,000 100. Diamr. in feet 24-0 23-5 23-0 22-0 21-0 19-0 15-0 100 6-0. Table giving the Breaking Strain of Tiller Ropes. Hide Ropes Cir Breaking f^train Ins. T. Cwt. Qr. 2^ 1 5 2 3 1 16 3 3i 2 10 4 3 5 Cir. Breaking Strain Ins. 5 5i T. Lwt. Qr. 4 2 2 5 2 6 3 2 White Ropes Cir. Ins. 2i 3'^ 3i 4" Breaking i^train T Cwt. Qr. 2 6 3 6 4 10 5 17 2 Cir. Breaking Strain Ins. 4| 5 5 T. Cwt. Qr. 7 8 2 9 3 2 11 2 1 Number of Shot in Piles. • In a triangular pile = i | n.^n + 1 ) x (;i + 2) } = num"ber ; when 71 = number in each side of base. In a square pile =i^/i(%+ 1) x (2% + 1)} = number; when n = number in each side of base. In a rectangular pile = 1}3n-(w-1)x(w + 1)xw}= number ; when N = number in longest side of base and 71 = number in shortest side of base. Diameter of Iron for Shackles to Chains. {Admiralty Utile.) From \io\ inch chain, the iron in shackles to be ^th of an inch more in diameter than the chain. Above i to ^ incli cliain, the iron in shackles to be \ of an inch more In diameter than the chain. Above ^ to 1 inch chain, the iron in sl.ackles to be |th of an | inch more in diameler than the chain. INDEX, ACC i CCELERATED rotation, US I\. Accelerating force, 147 Accommodation, passenger, 111 Acids, dipping, to make, 493 Air, quantity allowed in cabins, o04 Ale measure, 117 Alloys, component parts of, 490 Alphabet, Greek. 3 Anchors as supplied to H.M. ships, oOo test loads of, rule for, 504 weight and test loads of. 44G. Angle bars, weight of, rule for, 289 iron, weight of, 279 rule for, 273 of heel for vessels, 202 steel, weight of, 287 Angles, circular measure of, 29 • measurement of, 21 of heel for vessels, 203 Angular measure, 118 Animals, space allotted to, 504 ■ weight of, 504 work done by. 466 Annuities at interest, 45G Arc. circular, centre of, 70 eTlititic, to describe, 9 of circle, length of, 34 geometrically. to describe, 8 parabola, to desci'ibe, 16 Arcs of circles, length of, 29 Ardency, 197 Area of circle. 34 cycloid. 36 ellipse, 36 parabola, common, 36 ])iirallelogra:u, 3"2 polygon, regular, 33 rin^, cii'cular, 36 sp.ils fitted in vessels, 203 ■ — — suitable for vessels, 202 se-:tor of circle, 35 seg-ment of circle, 35 trapezium, 33 — — —trapezoid, 32 triangle. 32 — 70ue. circular, 36 Areas by polar co-ortliuates, 40 BED Areas by Simpson's rules, 36 multiplier.* for, 37 mensuration of, 32 of circles advancing by 8ths. 4S lOth.5. 5S 12ths, 65 polygons, regular. 33 — formulae for, 27 segments of circles. 67 triangles, expressions for, 25 rules for, 26 water planes, 168 Arithmetical progression, 456 Armour bolts, sizes of, 508 penetration of shot, 448 plates, strength of, 447 AvoirduiX)is weight, 115 Axis, instantaneous. 150 of level motion, 171 Axle and wheel, power of, 463 BALLS, cast-iron, weight of, 294 Bar iron, weight of, 276 Bar iron, round and square, weight of. 282 - steel, weight of, 285 ■ round and souare, weight of, 290 Baraue-rigged vessels, masts and spare of. 336 Barque-rigged vessels, masts and spars of, fact ore to determine, 344 Bars, angle, rale for weight of, 289 Beam iron, bulb, weight of. 283 Butterly. weight of, 283 Wallace's, weight of, 284 of strongest section, 11 Beams, bending moments of, 438 paddle. 507 spring, r07 ;- deflection of, 444 '- load? on. 438 — of equal strength, 442 shearing forces of, 438 strength of, 440 Bed berths, sises of, 498 L L ol4 INDEX. BEE Beer measnre, 117 Bending moments of beams, 438 paddle,507 spring,507 Berths, sizes of, 498 Bessemer steel, 430 Birmingham wire gauge, 307 Blocks, iron, sizes of, 302 wood, sizes of, 356 Boats, scantlings of, 503 to be carried by ships, 108 weights of, 503 Bodies, falUng, 148 Body plan, expansion from, 510 Boilers, Board of Trade regulations, 467 cylindrical, factors of saiety for, 470 for, 472 valves, 472 471 diagonal stays for, 468 — gauges for, 474 — girders for, 468 — plates for flat surfaces, 468 — pressures on flat surfaces, 467 — safety valves for, 474 — testing, 474 Bolt heads, weight of, 300 Whitworth's sizes, 309 Bolts, armour, sizes of, 508 Brass, sheet, weight of, 278 weight of cubic foot, 293 inch, 293 ■\vii-e, weight of, 300 Bread tanks, sizes and weight of, 310 Bricklaying, 488 Bricks, sizes and weight of, 488 stowage of, 489 Brickwork, rod of, 489 Brigs, masts and spars of, factors for, 344 Bristol Channel, distances down, 213 Builder's tonnage, rule for, 84 ■- • tonnagr-s, ts blt-s of, 85 Bulb iron, weiglit of, 283 Buoyancy, centre of, method of com- puting.', 155 ■ to determine, 152 BuOVP, Board of Trade regulations, 110 Butterly beam iron, weight of, 283 Wallace's, weight of, 284 Butt straps, percont.nges for, 274 .strength of, 435 CABINS, size=? in transports, 498 Cahl" chain and rope, equivalent strengths o!'", "0") •'able, cliiiiij. jir'jportions of links, 447 stowage of, 303 — plates, rivets, &c., — pressure on siifety — strength of joints, CEN Cable, chain, weight and strength of, 304 Cables as fitted in H.M. ships, 505 description of. 447 hempen, weight of, 350 relative proportions of, 350 rule for test loads of. 504 Canvas for sails. 351 sail, weight and strength of ,303 Capacity, scale of, 153 Cargo, dead-weiglit factor, 83 measurement factor, 83 Cask-gauging, 496 Casks, sizes of, 122 Castings, shrinkage of, 295 weight from patterns, 302 Cast-ii'on balls, weight of, 294 ■ pipes, weight of, 292 Catenary, formulae for, 458 to construct. 461 Caulking, weight of, 283 Cement, a bag of, 489 composition of, 489 Cements, surface a bushel will cover, 488 to make, 494 Centre from two known centres, 73 of arc, circular, 70 buoyancy, 152 method of com- puting , 155 circle by lines, 7 circular arc, 70 segment, 71 cone, 72 curved line, 70 irregular, 77 198 — solid, cylinder. 72 effort, 196 relative position of. ment, 161 elliptic segment, 71 figure, part shifted, 77 gravity by experiment, 173 of bodies. 146 — .ship and equip- 159 --ship's huU. 159 plating. water planes, 1C8 hemisphere, 72 irregular figure, 73 lateral resistance, 196 relative] 0- sition of, 1 98 parabola, 71 paraboloid, 72 plane area by ordinate?, 7" polar coordi- nates, 75 prism. 72 — pyrauiiil. INDEX. olo CEN Centre of sector of circle, 71 plane ring, 71 segment of circle, 71 ellipse, 71 . parabola, 71 semicircle, 71 _ by polar co-ordi- nates, 77 trapesium, 70 trapezoid, 70 triangle, 70 wedge-shaped solid, 77 Centrifugal force, 151 Certificates, passenger, 103 Chain cable and rope, equivalent strengths of, 805 links, proportions of, 447 stowage of. 303 weight and strength of, 304 505 cables as fitted in H.M. ships, — rule for test loads, 504 504 rigging, rule for working load, — shackles, proportions of, 512 weight and strength of. 804 Chains, weight tod test loads of, 440 Change wheels for screw-cutting, 508 Circle, arc of, length of, 34 geometrically. area of, 34 centre of, by lines, 7 circumference of, 33 diameter of, 33 from chord and sine, 34 properties of, 34 sector of. area of 35 centre of. 71 segment of. area of. 35 ship's, to measure. '221 useful numbers for, 311 Circles, areas of, adyancing by 8tbs. 4S lOths, .58 12ths, 65 segments of, areas of, 67 Circular arc. centre of, 70 arcs, lengths of, 29 furnaces, working pressure of, 473 measure, 21 of angles, 29 •segment, centre of, 71 ■ speed for saws, 512 ■ zone, area of. 36 Circnmferenc« of circle. 33 Circumferences of circles bv 8ths, 48 1 lOths. 53 12ths, 63 Cissoid. to describe. 19 Cisterns, urt5 of the foot, 1 39 '. pence. 3S3 siiillinn-3. 383 the yard, 139 fractions, 4 Decimals, fractions to, 140 redu(ti'ev inch immersion, 153 thrust, to constnict, 194 Cycloid, area of, 36 ~ curtate, to construct. 15 prolate, to construct, 15 to construct, 14 Cylinder, centre of, 72 — ^^ fi-ustum of, development of, 12 solidity of, 41 Cylinders, strength of, 431 Cylindrical boilers, Board of Trade rules for, 469 factors of safety plates, &c., for. ELA Development of segment of parabo- loid, 13 sphere, 13 Deviating force, 151 Diameter of circle, 33 from choril and versed sine, 34 Diagonal stays for boilers, 468 Dies, Wiiitworth's. 309 Dietary scale, Emigration Board's, 501 ■ royal navy, 499 Dipping acids, to make, 493 period of, 1S4 Discharge of water from cisterns, 453 sluices, ic, 452 Discount table, 384 Displacement, 151 ciurve of. 153 sheets, 155 explanation of, 158 Distances down the courses of rivers, &.C., 210 measurement of, 10 by trigono- meny, 26 Distemper, composition of, 489 Distortion, 437 Distress signals, 111 Dock erates, pressure on. 454 Draught of water, difference in salt and fresh water. 509 Drawing papers, sizes of, 122 Drawings, working, colours for, 511 Driving piles, 510 Dry measure. 117 — metrical, 123 lUiodecimals, 48 G Dynamical .stability, 163 approxiuiate, 165 curve of. 166 . exact calculation of, 166 geometrically.! ' siu-face of, 164 EFFECTnT. liorse power, 192 Eflfoit, centre of. 196 Effort, centre of, relative position <>f, 198 Elasticity, definition of, 427 niodulusof , definition of, 427 from bendintr, 437 extension. 436 &c..2n hquMs, 271 metiils. 26:» stones, eiirth, timber. 270 INDEX. 517 ELL Ellipse, area of, 36 formulas for, 457 perpendicular to, 9 tangent to, 9 to describe, 8 Ellipsoid, frustum of, solidity of, 48 s^ment of, solidity of, 43 solidity of, 43 Elliptical bars, rule for weight of, 272 Elliptic arc, to describe, 9 segment, centre of, 71 Emptying cisterns, time of, 453 Enamels, to make, 495 Engineer's stores, stowage of, 479 Engines, marine, consumption of, 479 friction of, 193 thrust curve, 194 weight of, rule for, 479 weights of. &c., 480 English and French vocabulary, 361 coins, 142 Entablature plate, development of , 14 Epicycloid, to describe, 15 Equal parts, to divide a line into, 7 Evolution, 267 Expansion of body plan, 510 Extensibility, definition of, 427 r EXPRESSING- a function, 3 ? Factors for rigging of ships, 354 Factors, miscellaneous, 273 of safety, formulae for, 432 kinds of, 427 Falling bodies, 148 Feet, inches to, 139 per second to knots per hour, 273 miles per hour, 273 to metres, 133 yards, 139 Filling cisterns, time of, 453 Fire hose, Board of Trade rules for, 111 Flat iron, malleable, weight of, 276 steel, malleable, weight of, 285 Flotation, surface of, 171 Flow of water through pipes, 455 Fluids, weight of, 272 Fluxes for solders, 490 Fly wheel, arms of, 449 strength of, 448 Force, accelerating, 147 centrifugal, 151 of gravity, 147 water in motion, 454 power and work, 464 • retarding, 147 Forces, parallel, 144 resultant and resolution of, 143 Foreign measures, useful numbers for, 311 money, weights, and measiires, 141 Fractions, decimal, 4 HEE Fractions to decimals, 140 Free board, rules tor, 50i) French and English vocabulary, 368 measures, 123 weights, 122 Friction, coefficients of. 455 of marine enguii^s. 193 Frustum of cone, development of, 11 ■ solidity of, 41 surface of. 47 cylinder, development of. 12 ■ — soliditj- of, 41 — hj-perboloid, solidity of, 44 — paraboloid, solidity of, 44 — pyramid, solidity of, 41 surface of, 47 Furnaces, circular, working pressure of, 473 (^ SIGN of force of gravity, 2 T? Gauges for boilers, 474 Gauges, values of, 3U7 Gauge, wire, Whitworth's, 308 Gauging casks, 496 Geometrical moments of figures, 72 progression, 456 Geometry, practical, 7 Girder iron, \\'eight of, 284 Girders for flat surfaces of boilers, 468 Girth, mean curved, 47 Globe, segment of, solidity of, 42 solidity of. 42 Glues, to make, 494 Gravity, 147 centre of, bodies, 146 by experiment, 173 ship's hull, 159 and e- quipment, 161 159 plating, water planes, 168 specific, 497 of metals, 269 work under action of, 465 Greek alphabet. 3 Guns, ammunition for, 298 dimensions of, &c., 297 Gun slides, sizes of, 297 weight of, &c., 298 Gyration, least radius of, for special figures, 433 radii of, for special figures, 81 ~ radius of, geometrical, 78 of weight, 149 HAilMOOKS, sizes of, 498 Hand cranes, 451 Harmonic curve, to construct. 18 Heel, angles of, for vestels, 203 118 INDEX. HEE Heeling moment of sails, 202 H"eit^ht3, measurement of, 25 Hemi-ellipsoid. centre of. 7'2 Hemisphere, centre of, I'i Hemp and iron cables, relative pro- portions of, a50 wire rope, equivalent strensTths of, 351 roi)e, fiat, weight and strength of. 3<);j weight and strength of, 304 — of, 350 Horizon, distances of, 3<>6 Horse-power, effective, 192 indicated, 507 nominal, 507 to drive flat-fronted vessels, 189 Horse, weight of, 504 Horses, stores allowed for, 498 Hose, fire. Board of Trade rules for, HI Humtjcr, river, distances down, 216 Hundredweights to kilograms, 134 tons, 138 Hyiraulic presa, 452 Hyperbola, formulae for, 457 to describe, 17 Hy{>erbolic logarithms, 377 H y!)er bo loid, frustum of, solidity of, 44 solidity of, 44 Hyiwoycloid, to construct, 16 Hypotenuse of triangle, to find, 32 IMPACT of shot, 448 Impregnation or oimber, 485 Impulse, 147 — of wind, 2ol in lbs. per square foot, 200 on sails, 199 on soUd body, 149 Inacces-sible objects, distances between, lu Inch, divisions of, to decimals, 140 millimetres, 132 laches, metres to, 131 millimetres to, 124 to feet, 139 millimetres, 133 Inclined plane. 463 Income table, 383 Indian ink, to make. 496 Indicated horse-power, 507 ratio of effec- tive to, 192 thrust, 192 Inertia, moment of, geometrical, 78 • plane area, 79 sjxjcial figures, 441 weight, 149 Ini*:ial friction of engines, 193 luii, copying, to make, 496 KNO Ink, Indian, to make, 496 Instantaneous axis. 150 Interest, simj)le and compound, 455 annu- ities at, 456 Involute of a curve. 18 Iron and hemp cables, relative pro- pf;rtions of, 350 angle, weight of, 279 rule for, 27vt balls, cast, weight, 294 blocks, sizes and weights of, 30* bulb plate, weight of, 283 Butterly beam, weight of, 283 Wallace's, weight of, 284 corrosion of, 509 corrugated, weight of, 296 cubic foot of, weight of, 293 inch of, weight of, 293 dead-eyes, weight of, 301 H. weieht of, 284 hoop, weight of, 300 kinds of, 428 malleable flat, weight of, 276 pipes, east, weight of, 292 malleable, weight of, 291 round bar, weight of, 282 sheet, weight of, 278 skin, thickness of, 447 square bar. weight of, 282 wire, \\eight of, 3i>0 - rope, and chain cables, equi- valent strengths of, 305 — weight and strength of, 304 Isochronous rolling, 182 form of section for, 184 JOINTS of cylindrical boilers, strength of, 471 riveted, strength of, 431 formulae for, 434 435 notes on, K ILOGRAMS, hundredweights to, 134 ounces to, 134 pounds to, 134 quarters to, 134 tons to, 134 — — - to pounds, 135 tons, 135 BZilometres, knots to, 209 to knots, 209 Knot, Admiralty, 208 table, time and, 205 Knots, kilometres to, 209 miles to, 208 INDEX. 519 KNO Knots per hoiir to feet per second, 273 to kilometres, 209 miles, 208 Knotting, composition of, 489 LACQUERS, to make, 492 Ladders, companion, tread for, 496 Lancashire wire gauge, 307 Land measure, 115 Lanterns, Board of Trade rules for, 114 Laps, &c., percentages for, 274 Launching ways, inclination of, 504 Laws of motion, 147 Lead lines. 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