iiiil ~ ~, ~i,- ~I;~,,,, -,-..... _$,d,,.-..= =. - -, cr-i i i- i i ii! K a A- - )L = -d- q - - - - - _A! =__. L ~~~~~~-,~ L~..- rJ _-_ <.~~~~~~-.. k / Locolno've buaft by.4o/7's Bother~ fo 4 e Sy cuvste~ & tzic~a Ra Road. 183/. with Septt' u.ts iVorri~Patet/ Diirelet~cton Varuzesb' s A a a/zi a/t Vltl. NORRIS'S HAND-BOOK FOR LOCOMOTIVE ENGINEERS AND MACHINISTS: COMPRISING TIE PROPORTIONS AND CALCULATIONS FOR CONSTRUCTING LOCO MOTIVES, MANNER OF SETTING VALVES, TABLES OF SQUARES, CUBES, AREAS, &c. &c. BY SEPTIMUS NORRIS, CIVIL AND MECHANICAL ENGINEER. "Knowledge is power." PHILADELPHIA: HENRY CAREY BAIRD, SUCCESSOR TO E. L. CAREY. 1854. Entered according to Act of Congress, in the year 1852, by SEPTIMUS NORRIS, in the Clerk's Office of the District Court of the United States in and for the Eastern District of Pennsylvania. STEREOTYPED BY L. JOHNSON AND CO. PHILADELPHIA. PRINTED BY T. K. AND P. 0. COLLINS. TO WILLIAM NORRIS, ESQ. AS A TESTIMONY OF ESTEEM AND ADMIRATION OF HIS GREAT INTELLECTUAL ATTAINMENTS, AND OF HIS SCIENTIFIC AND CLASSICAL KNOWLEDGE; DESCRIBING THE GREATEST WORK OF MAN, ~1? JLoohnotlb& $feih %gin4 IS AFFECTIONATELY INSCRIBED BY HIS BROTHER, THE AUTHOR. PREFACE. IN presenting this work to the Engineers of the United States, I beg they will study it with attention, as it is the result of many days' close application and research. I have taken care to present all formulas and rules in the most simple manner, so that there will be no danger of the young student being discouraged by unnecessary display of algebraical formulas, which the sight of frightens the timid. All may understand who are familiar with the simple rules of Arithmetic,,-Addition, Subtraction, Multiplication, and Division. I give here the result of my experience after a study of twenty years, and for the last twelve years engaged with my senior brother, William Norris, to whom I am entirely indebted for all the infor5 6 PREFACE. mation I have received relating to locomotives. He built the first locomotive in this country, and was the first engineer that ever attempted to surmount the Inclined Plane across the Schuylkill, where there is a rise of I in 14, for 1 mile equal to 377 feet ascent. This wonderful performance was made amid the shouts of thousands: no one has ever attempted such a feat since. In connection with my brothers, I have constructed and built some five hundred and thirty locomotives; one hundred and seventy of which are now successfully running on roads in England and the Continent, seventeen of which are running on the Birmingham and Gloucester Railway, England. Some builders, or perhaps foremen of the locomotive-shops of this country, may think it unwise in my giving to all mechanics the secrets (which they consider) of the business. My belief is, that all I can teach a man or apprentice, so much the better will be the success of my business; and the million should be learned in all things, as well as the few illiberal-minded. I give here every thing PREFACE. 7 relating to the construction of locomotives; and I hope my feeble efforts may prove of value to many who seek after this great science, Mechanics. It is the greatest of all sciences, teaches the mind to think correctly, and produces that intellectual enjoyment which no other study can approach. TABLE OF CONTENTS. Page AREAS of Circles, Diameters, and Circumferences.. 40 Areas of Segments and Zones of a Circle............ 46 Boiler of Locomotive..................................... 238 Cast Iron and Wrought, Elasticity of.................. 206 Conic Sections........................................... 102 Connecting Rods of Locomotives......................... 271 Cylinders and Valves of Locomotives................ 261 Dimensions of Parts of Locomotives................... 218 Eccentric, and Mode of Setting in Shaft............... 174 Eccentrics and Rods................................... 273 Feed Pumps of Locomotives.......................... 265 Fractional Parts of an Inch................. 167 9 10 TABLE OF CONTENTS. Page Framing of Locomotives................................. 252 Friction...................................................... 198 Grades, Resistance per Ton.............................. 203 Lever............................................... 168 Mechanical Powers........................................ 192 Mensuration of Solids.................................... 117 Mensuration of Surfaces............................. 104 Miscellaneous Remarks................................... 205 Proportions of Length of Circular Arcs.............. 55 Proportions of Length of Semi-Elliptic Arcs........ 58 Radius of Curves.......................................... 190 Railways.................................................... 151 Reciprocals of Numbers.................................. 31 Resistance against Piston............................... 207 Revolutions of Driving Wheels.................... 191 Safety Valve and Lever............................... 178 Safety Valves.............................................. 257 Setting Valves............................................. 280 Shrinkage of Tire Bars.................................. 188 Spring Steel........................................... 189 TABLE OF CONTENTS. 11 Page Squares, Cubes, Square Roots, Cube Roots........... 69 Steam Engine.................................. 143 Time of Running One Mile............................ 204 Tractive Power of Locomotives......................... 173 Tubes of Locomotives................................. 249 Valve Motions........................................ 275 Weight of Materials....................................... 130 Weights and Measures.................................... 13 WYheels of Locomotives............................ 267 built brthe AMOSKEAG MANUFACTURING Co...Jt......'STER.A, _.f 1851. _I!! I )I,,..ca:e -'~ ~ z -Foo' NORRIS'S HAND-BOOK. TROY WEIGHT. grains 24=- 1 dwt 480= 20 -= 1 oz 5760 = 240 = 12 = 1 lb AVOIRDUPOIS WEIGHT. drachms 16=- 1 oz 256 —= 16- 1 lb 7168 = 448 = 28 = 1 quarter 286782 = 1792= 112 = 4 = 1 cwt 573440 = 35840 - 2240 = 80 = 20 = 1 ton 1 lb. = 14 oz., 11 dwt., 151 gr. troy. 1 oz. = 18 dwt., 5~ gr. troy. N. B.-7000 troy grains make 1 pound avoirdupois; hence 175 pounds troy are equal to 144 pounds avoirdupois. APOTHECARIES' WEIGHT. grains 20 = 1 scruple 60=- 3= 1 drachm 480=- 24 = 8 = — 1 oz 5760 = 288 - 96 - 12 =- lb 2 13 14 WEIGIHTS AND MEASURES. LONG MEASURE. inches 12 = 1 foot 36 = 3 =1 yard 72 = 6-= 2 =1 fathom 198= 16 —= 5~= 2- = 1pole 7920= 660= 220 =110= 40=lfurlong 63360 = 5280 = 1760 = 880 = 320 = 8 = 1 mile A mile contains 80 chains, land measure; and a chain contains 100 links, or 22 yards. An inch contains 12 lines. MEASURE OF THE CIRCLE. seconds (") 60 = 1 minute (') 360 = 60 = 1 degree (0) 32400 = 5400 = 90 = 1 quadrant 129600 = 21600 = 360 = 4 = 1 circumference WINE MEASURE. pints 2-= 1 quart 8- 4 = 1 gallon 336-= 168 = 42=1 tierce 504 = 257= 63 = 11 =1 hogshead 672= 336= 84=2 = —l1 luncheon 1008= 504 = 126=3 =2 =l =l lpipe 2016 1008 — 252 — 6 = 4 - 3:2=1 tun WEIGHTS AND MEASURES. 15 ALE AND BEER MEASURE. pints 2= 1 quart 8= 4= 1 gallon 72= 36= 2= 1 firkin 144= 72= 18= 2 = 1 kilderkin 288=144 = 36 = 4 -= 2 = barrel 432 = 216 = 54 = 6 = 3 = 1 = 1 hogsead 576=288 72= 8=4=2 =l1 =lpunch'n 864 =432=108 = 12= 6 = 3 =2 ==1butt N. B.-The pint, quart, and gallon, for wine, ale, and beer, and grain or'corn, measure the same with regard to their magnitude; 8 of these gallons make one bushel; and 1 gallon contains 277'274 cubic inches, or 10 lbs. of distilled water, at 62 degrees Fahrenheit. DRY MEASURE. pints 8 = 1 gallon 16 = 2 = 1 peck 64 = 8= 4 = 1 bushel 256 = 32 = 16= 4= coom 512 = 64- 32= 8= 2= 1 quarter 2560=320=160=40=10= 5 = 1 wey 5120 = 640 = 320 = 80 = 20 = 10 = 2 = I last 16 WEIGHTS AND MEASURES. CLOTH MEASURE. inches 2= 1 nail 9 - 4 = I1 quarter 36 = 16 = 4=-1yard 27 =- 12 = 3 - 1 lemish ell 45 =- 20 = 5 = 1 English ell 54 = 24 = 6 = 1 French ell SQUARE MEASURE. inches 144 = 1 foot 1206 = 9 1 yard 39204 = 2 12 = 30- 1 pole 1568160 = 10890 = 1210 = 40 = 1 rood 6272640 -— 43560 = 4840 = 160 = 4 =1 acre 10Osquare chains make 1 acre; 640 acres make 1 square mile; 30 acres 1 yard of land; and 100 acres 1 hide of land. SOLID MEASURE. inches 1728 = 1 foot 46656 = 27 =- 1 yard 1 cubic foot - 2200 cylindrical inches = 3300 spherical inches = 6600 conical inches. WEIGITS AND MEASURES. 17 MISCELLANEOUS. 1 Acre, Scotch, 1'271 acres English, or.............................. 6084 sq. yards. 1 Acre Irish, 1'638 acres English, or 7840 sq. yards. 1 Barrel, imperial measure......... 9981'86 cub. in. " soap........................ 256 lbs. 1 Bushel, imperial measure......... 2218'19 cub. in. " Winchester measure..... 2150'42 cub. in. barley...................... 50 lbs. coal...................... 88 lbs. flour or salt............... 56 lbs. " oats........................ 40 lbs. " wheat...................... 60 lbs. 1 Chaldron coals, Newcastle....... 53 cwt. 1 Chain..................... 100 links. 1 Clove of wool................. 7 lbs. 1 Fodder of lead, Stockton......... 22 cwt. "~'~at Newcastle.... 21 cwt. at London....... 191 cwt. 1 Gallon, imperial measure......... 27727 cub. in. " distilled water, 62....... 10 lbs. proof spirit or oil......... 9'3 lbs. " former wine measure.... 231 cubic in. " former ale measure....... 282 cubic in. " Irish measure.......... 217'6 cubic in. 1 Geographical mile................. 1'15 Eng. miles. cc degree............... 6912 Eng. miles. 1 Gross................................. 12 dozen. 1 Great gross.......................... 12 gross. 1 Hand...................... 4... 4 inches. 1 Hundred of deals.' 120 in number. nails. 120 in number.'." salt..................... 7 lasts. 1 Last of salt................... 18 barrels. 2* 18 WEIGHTS AND MEASURES. 1 Last of gunpowder................ 24 barrels. 6" potash, soap, pitch, or tar 12 barrels. " flax or feathers............ 1i7 cwt. C" wool........................ 4368 lbs. 1 Link................................. 7'92 inches. 1 Load of boards, inch............... 600 square ft. c" bricks....................... 500 in number. " hay or straw.............. 36 trusses. 4" lime....................... 32 bushels. "6 planks, two-inch.......... 300 square ft. (" sand........................ 36 bushels. timber, squared........... 50 cubic feet. " timber, unhewed......... 40 cubic feet. 1 Mile................................... 80 chains. 1 Pack of wool...................... 240 lbs. 1 Palm.................................. 3 inches. 1 PQle, Woodland..................... 18 feet. " Plantation................ 21 feet. " Cheshire............... 24 feet. 1 Sack of coals...................... 224 lbs. " wool....................... 364 lbs. 1 Seam of glass..................... 124 lbs. 1 Span................................. 9 inches. 1 Stone of meat or fish.............. 8 lbs.' horseman's weight......... 14 lbs. c glass.......................... 5 lbs. " wool.......................... 14 lbs. 1 Thousand of nails.................. 1200. 1 Truss of new hay................ 60 lbs. old hay................... 56 lbs. c" straw...................... 36 lbs. 1 Tun of vegetable oil............... 236 gallons. " animal oil.................. 252 gallons. i Tod of wool......................... 28 lbs. 1 Wey of wool........................ 182 lbs. WEIGHTS AND MEASURES. 19 Relative Value of the Imperial and Old English Mleasures. Wine Gallon. Imp. gallon.. 1 2 3 4 5 6 7 8 9 Old gallon... 1200 2-401 3-601 4801 6002 7202 9-402 9-602 10-803 Old gallon.. 1 2 3 4 5 6 7 8 9 Imp. gallon.. 0-833 1'666 2-499 3-332 4166 4-999 5832 6665 498 Ale Gallon. Imp. gallon.. 1 2 3 4 5 6 7 8 9 Old gallon... 0 983 1-966 2-950 3-933 4-916 5-899 6-883 7-866 8-849 Old gallon... 1 2 3 4 5 6 7 8 9 Imp. gallon.. 1017 2 034 3-051 4068 5-085 6-102 7119 8-136 9-153 Coran Bushel. Imp. bushel.. 1 2 3 4 5 6 7 8 9 Old bushel.. 1-031 2-063 3095 4-126 5-158 6-189 7-221 8-252 9284 Old bushel... 1 2 -3 4 5 6 8 - 9 Imp. bushel. 0-969 1-939 2908 3-878 4-8471 5817 6-786 7-755 8-725 N. B.-The foregoing relative values are computed in whole numbers and decimal parts; they exhibit the value of units only, but the value for tens, hundreds, thousands, &c. may be found by changing as many of the decimals into integers as there are ciphers in the numbers sought. EXAMPLE.-Required the number of imperial gallons in 4743 old wine gallons:4000 - 3332 700- 583'2 40= 33'32 3= 2'490 3951'019 imperial gallons. 20 WEIGHTS AND VALUES IN DECIMALS. TROY WEIGHT. AVOIRDUPOIS AVOIRDUPOIS ENGLISH MONEY. WEIGHT. WEIGHT. Dec. parts of a lb. Dec. parts of a cwt: Dec. parts of a lb. Decimal parts of ~1. Ozs. Decimals. Qrs. Decimals. Ozs. Decinsals. Sh. Dec. Sh. Dec. 11'916666 3'75 15 *9375 19'95 9'45 10'833333 2' 5 14'875 18'9 8'4 9'75 1 *25 13'8125 17 *85 7'35 8 *666666 __ _ 12 *75 16 *8 6 *3 7 *583333 11 *6875 15 7 5 5 *25 6'5 Lbs. Decimals. 10 G625 14'7 4 *2 5'416666 27'241071 9'5625 13'65 3'15 4'333333 26 232142 8'5 12 *6 2'1 3'25 25'223214 7' 4375 11'55 1'05 2'166666 24 214286 6 3 75 10 5 1'083333 23'205357 5'3125 22 *196428 4' 25 Pence. Decintals. 21 *187500 3 *1875 11 *045833 Dwts. Decimals. 20 178572 2 125 10 041666 19'079166 19'169643 1 9 *0375 18'075 18 *160714 8'033333 17'070833 17'151785 7 *029166 16 *066666 16 *142856 Dr. Decimals. 6'025 15'0625 15'133928 15'058593 5 *020833 14'058333 14'125 14'054686 4' 016666 13 -054166 13'116071 13'050780 3' 0125 12'05 12'107143 12 6046874 2' 008333 11 -045833 11'098214 11'042968 1' 004166 10 -041666 10.089286 10'039062 9'0375 9'080357 9 *035156 Farthing. Decinals. 8'033333 8 -071428 8'03125 7'029166 7'0625 7'027343 3 003083 6 025 6'053571 6'023437 1 002083 5'020833 5'044643 5'019531 4 *016666 4'035714 4'015625 3'0125 3'026786 3'011718 ENGLISH MONEY. 2'008333 2'017857 2 0007812 1'004166 1'008928 1'003906 Decimal parts of Is. G o1& D Pecimals. Os. Decimals. LONG Pence. Decimals. zs.Decials. IASURE. 11' 916666 15'002604 15'008370 10 1 833333 14 6002430 14'007812 Dec. parts of a foot. 9'75 13'002257 13 6007254 8 6;6'666 12 |002083 12'006696 les. Dciisals. 7' 583333 11'001910 11'006138 11'91(;6; 6ij 6j 6.5 10.001736 10.009580 10.833333 5. 416666 9.001562 9.005022 9.75 4.*53333 8'001389 S 0044G4 8 2G66636 3 25 7 001215 7 803306.7 | 58 I 33 2 152663 6.001042 i6 6.0-55:248 6.5 1 9083333 5.000868 5.602790 5.416666. 4 1000694 1 4 | 002282 4 33| 333 Forthilig. Dccibolos. 3'000521 I 3'001674 3.25 3'0625 2 0000347 2 001116 2 166266 2'041666 1'000173 1 1 000558 1'08333:3 1 020833 WEIGHTS AND MEASURES. 21.Relative Value of British and French Weights and Measures. FRENCH DECIMAL, OR MODERN SYSTEM. WEIGHTS. MEASURES-OF LENGTH. French. British. French. British. QRAMME........ 15'434 grains. METRE...... 39'371 inches. Decigramme.. 1.5434 " Decimetre.. 3-9371 " Centigramme.. 01543 " Centimetre. 0.3937 inch. Milligramme.. 00154 " Millimetre. 0-0393 " Decagramme.. 154-34 "' Decametre. 32.809 feet. Hectogramme. 3-2154 oz. troy. Hectometre 328.09 " or 3-527 oz. avoir. Kilometre... 1093'6 yards. Kilogramme.. 2-6795 lb. troy. Myriametre 6-2138 miles. or 2.2048 lb. avoir. Myriagramme 26-795 lb. troy. OF SUPERFICIES. or 22-048 lb. avoir. ARE...... 119-60 sq. yards. Quintal......... 1 cwt. 3 qrs. 24y Deciare...... 11-960 " lb. Centiare.... 10-764 sq. feet. Millier or Bar 9 tons, 16 cwt. 3 Milliare.....1155-00 sq. inch. qrs. 12 lb. Decare..... 1196-0 sq. yards. Hectare 2-4712 acres. MEASURES —OF CAPACITY. LITRE*......61'028 cubic in. OF SOLIDITY. or 1-761 imp. pint. STERE...... 35-317 cub. feet. Decilitre........ 6-1028 cubic in. Decistere... 3-5317 " Centilitre...... 06103 " Centistere.. 61.0-28 cubic in. Millilitre....... 0-0610 " Millistere... 61.028 " Decalitre....... 61028 " Decastere... 13'080 cubic yds. or 2-2 imp. gallons. Hectostere.130-80 Hectolitre...... 3-5317 cubic feet or 2-75 imp. bush. * The Litre - a cubic decimetre. Kilolitre....... 35 317 cubic feet f The Are a square decimetre. Myrialitre...... 353-17 "; The Stere - a cubic metre. NoTE.-The decimetre, centimetre, and millimetre are respectively formed by dividing the metre by 10, 100, and 1000; and the decametre, hectometre, kilometre, and myriametre, by multiplying the metre by 10, 100, 1000, and 10,000: the other measures and weights of the decimal system are formed in a like manner from their respective units. 22 WEIGHTS AND MEASURES. THE OLD SYSTEM, OR SYSTEME USUEL. WEIGHTS. MBEASURES OF LENGTII. French. British. French. British. Grain............. 0837 grain. Ligne............... 0091 inch. Gros (72 grains) 60o285 grains. Pouce (12 lignes) 1-093 " Once (8 gros)... 1-1024 oz. avoir. Pied (12 pouces) 13-12 " Livre (16 onces) 1'1024 lb. avoir. Aune (3 pieds).. 3 937 feet. Toise (6 pieds)... 6-562 " MEASURES OF CAPACITY. Litron.........1376 imp. pints. The Livre is -500 grammes. The Boisseau is = 12'5 litres. Boisseau..... 2-7515 imp. gal. The Toise is = 2 metres. British and Foreign Weights and feasures. 1ELATIVE VALUE OF BRITISH AND FOREIGN COMMIERCIAL WEIGHTS. Coultry or Place. Weights. No. equal t 1I cwt. British. Aleppo...................... Oke......................... 40-10 Alexandria............... Rottolo For................ 119-84 Amsterdam............... Pound Flem............... 5079 Algiers.........o...... Rottolo..................... 94-12 Barcelona.................. Pound...................... 126.97 Berlin...................... Pound...................... 108.42 Bremen..................... Pound..................... 101.95 Cairo........................ Rottolo...................... 117.89 China........................ Catty........................ 84-00 Cologne..................... Pound...................... 108'64 Constantinople............ Oke........................... 39.53 Copenhagen............... Pound...................... 101.55 Cyprus..................... Rottolo...................... 21.35 Damascus.................. Rottolo...................... 28-44 Dantzic..................... Pound...................... 108-42 Florence.................... Pound..................... 149'61 France........... Livre usuel................ 101-59 Frankfort....... Pound...................... 10873 Geneva..................... Pound, heavy........ 92-25 Genoa................. Pound, heary............. 145-69 Hamburgh................. Pound.. 104-86..~~~~~~~~~~~148, WEIGHTS AND MEASURES. 23 Relative Value of British and Foreign Commercial Weights —continued. Country or Place. Weight. No. Britisho wt. British. Hanover.................. Pound................... 104'37 Konigsberg............ Pound...................... 108.42 Japan........................ Catty........................ 86'15 Leghorn................. Pound...................... 149-61 Leipsic...................... Pound...................... 108'79 Lubec....................... Pound...................... 104-82 Madeira................. Pound..................... 110'79 Malta........................ Rottolo..................... 64.17 Milan....................... Pound, new.............. 5079 Naples....................... Cantaro gro................ 5699 Nuremberg............... Pound...................... 99-61 Persia...................... Batman..................... 8831 Poland..................... Pound...................... 125-72 Portugal.................. Pound...................... 110'68 Prussia............ Pound...................... 108-60 Riga............... Pound...................... 121.51 Rome........................ Pound...................... 149.79 Rostock.................... Pound...................... 99.84 Rotterdam.................. Pound...................... 102-82 Russia....................... Pound..................... 124-08 Sardinia.................... Pound...................... 128- 00 Sicily........................ Pound..................... 160.00 Smyrna..................... Oke.......................... 39-53 Spain....................... Pound.................. 110-40 Sweden..................... Pound...................... 149-33 Trieste...................... Pound...................... 90.75 Tripoli...................... Rottolo.................... 100.00 Tunis.............. R....... Rotul........................ 100-85 Venice..................... Pound, new............... 50.79 Vienna.......... Pound...................... 90-68 Zurich....................... Pound, heavy............ 96-33 24 CORN OR DRY MEASURES. Relative Value of British and -Foreign Corn or Dry Measures. No. equal to 1 Country or Place. Name of Measure. quarter, or 8 bush. English. Alexandria.......... rebebe................. 1.85 Algiers.................... zarrie................ 14'54 Amsterdam............. mudde...................... 2.61 Ancona..................... rubbio............... 1'01 Antwerp.................... mudde....................... 291 Barcelona.................. quartera................... 425 Berlin............... scheffel..................... 58 Bruges...................... hoed................ 1-74 Cologne..................... malter...................... 1.79 Constantinople............ killow........... 8'77 Copenhagen............... toende....................... 209 Cyprus............... medimno....... 3'87 Dantzic..................... scheffel..................... 5-32 Dunkirk..................... rasiere................. 2.18 Embden.................... tonne........................ 1'51 Florence.................... staja......................... 11'95 France................... boisseau...... 23.32 Frankfort................... malter.................... 2-69 Genoa...................... mina............... 2-40 Groningen.................. mudde...................... 3.19 Hague..................... sack.......................... 271 Hamburgh............... scheffel...................... 276 Hanover.................... himten...................... 935 Leghorn.................... sacco...................... 4.00 Leipsic...s........ scheffel.................... 2 09 Lisbon...................... alquiere..................... 21 50 Lubec....................... scheffel.................... 8'70 Majorca.................... quartera................... 4.13 Malaga............. fanega...................... 516 Malta........................ salma................... 1.00 Munich..................... scheffel..................... 0.80 Naples....................... zomolo.................... 5.69 Netherlands............... mudde................2..... 2.91 Persia....................... artaba....................... 4-42 Poland...................... korzec.............. 5'69 Prague...................... strick....................... 2.72 Prussia..................... scheffel,.............. 5.29 Riga....... loop........................ 4.26 Rome.................... rubbio..................... 0.98 1 ~~~~~~~~~~~i BRITISH AND FOREIGN MEASURES. 25 Relative Value of British and Foreign Corn or Dry Mleasures-continued. No. equal to 1 Country or Place. Name of Measure. quarter or 8 bush. English. Rostock..................... scheffel...................... 748 Rotterdam................. sack......................... 2'80 Russia....................... chetwert................... 138 Sardinia.................... starello..................... 5'94 Sicily........................ salma........................ 105 Smyrna..................... killow....................... 5-67 Spain.................... fanega...................... 516 Sweden.................... tunna........................ 198 Texel........................ loop.................. 4-64 Trieste..................... metzen...................... 4'79 Tunis....................... 055 Venice...................... stajo......................... 3 (,; Vienna...................... metze........................ 4'73 Utrecht..................... sack........................ 3.22 Wirtemberg................ scheffel..................... 1 61 Zante........................ misura...................... 13-84 Zealand.sack.......................... 3.89 Zurich....................... mutt......................... 3-51 Relative Value of British and Foreign JVine or Liquid Measures. Content in Country or Plaoe. Name of Measure. British Imp. Gallons. Amsterdam............... wine stekan............... 427 Antwerp............ stoop........................ 060 Barcelona................. carga........................ 27.24 Berlin....................... anker........................ 8-24 Bordeaux.................. velte......................... 1 58 Burgundy................. quartant................... 22-63 Canaries................... arroba...................... 353 Champagne............... quartant................... 19.82 Cognac.................... brandy velte............... 1.60 Cologne................ viertel. 131 Constantinople............ almud...................... 115 8 26 BRITISH AND FOREIGN MEASURES. Relatitu Value of British and Foreign Wine or Liquid Measures-continued. Content in Country or Place. Name of Measnre. British Imp. Gallons. Copenhagen.............. viertel..................... 1'70 Cyprus...................... cass.......................... 1 04 Dantzic.... ohm......................... 32-96 Dresden.................. eimer....................... 14-88 Florence................... wine barile................ 10-03 France...................... setier........................ 1-63 Frankfort.................. viertel...................... 1 60 Geneva...................... setier........................ 9.95 Genoa....................... wine barile............. 16-33 Hamburgh...... ahm.........................' 3186 Hanover................... ahm.......................... 34-23 Heidelberg................ aas........................ 051 Leghorn..................... wine barile................ 1003 Leipsic................ eimer.................. 16-74 Lisbon...................... almude...................... 364 Lucca........................ oil coppo................ 21-97 Malaga................. arroba...................... 2'49 M alta........................ oil caffiso.................. 458 Marseilles.................. millerolle.................. 1415 Naples............. wine barile.......... 9'17 Netherlands....... at........................... 2201 Oporto...................... almude..................... 5.61 Poland.................... garniec.................. 0'35 Prussia.................... eimer....................... 15-12 Riga......................... anker........................ 8-61 Rome....................... wine barile................ 12-84 Rostock..................... anker........................ 7'96 Rotterdam............... ahm......................... 33'32 Rouen...................... barrique.................. 43-06 Russia...................... vedro........................ 2-09 Spain........................ wine arroba............. 3.53 Strasburg.................. ohm......................... 10.14 Sweden..................... kann......................... 0-57 Trieste...................... eimer........................ 12-45 Venice..................... secchio..................... 2-37 Vienna..................... eimer........................ 12-42 Zante........................ barile........................ 14'68 Zurich...................... maas........................ 0-36 BRITISH AND FOREIGN MEASURES. 27 Relative Value of British and -Foreign Measures of Length. I No. of each Country or Place. Name of Measure. equal to 100 English Feet. Amsterdam................... Foot.................... 107.71 Antwerp...................... Foot....................... 106'76 Augsburg..................... Foot....................... 103'00 Berlin..................... Foot...................... 98.44 Berne........................... Foot...................... 103-98 Bremen........................ Foot.................. 105'44 Brunswick..................... Foot...................... 10685 Carrara............... Palmo.................... 12513 China....................... Foot...................... 94'41 Cologne........................ Foot...................... 11080 Copenhagen................ Foot...................... 97.16 Cracow........................ Foot...................... 85-53 Dantzic...................... Foot.................. 106.19 Dresden.................... Foot..................... 107-71 France........................ Foot................... 91.46 Frankfort..................... Foot.................... 106.38 Geneva........................ Foot..........62.46 Genoa.......................... Palo................ 123-45 Gottingen................. Foot...................... 104-80 Hamburg..................... Foot...................... 106.38 Hanover................ Foot........1....... 04.80 Leipsic................ Foot................... 108.01 Leyden........................ Foot...................... 9724 Liege........................... Foot...................... 10600 Lisbon...................... Foot..................... 92-73 Malta........................... Foot................... 107.52 Mecklenburg............... Foot...................... 104-80 Milan........................... Foot................. 76.82 Moscow........................ Foot...................... 9111 Munich................. Foot...................... 105'54 Naples...................... Palmo.................. 115-60 Neufchatel.................... Foot...................... 101.60 Nuremberg.................. Foot...................... 100.33 Padua.......................... Foot...................... 86.14 Pisa............................ Palmo................. 102.21 Prague........................ Foot..................... 101-52 Prussia......................... Foot...................... 97'16 Riga............................ Foot...................... 111.21 28 BRITISH AND FOREIGN MEASURES. Relative Value of British and Foreign Measures of Length-continued. No. of each Country or Place. Name of Measure. equal to 190 English Feet. Rome................... Foot...................... 102-38 Rostock........... Foot...................... 105-44 Russia... Foot..................... 87-27 Sardinia....................... Palmo.................... 12269 Sicily........................... Palmo................ 125.91 Spain........................... Foot................ 10791 Sweden................. Foot...................... 102-73 Venice.................. Foot...................... 87-71 Vienna.............. Foot................. 96-38 Ulhn................... Foot...................... 105-35 Wirtemberg.............. Foot...................... 106-57 Zurich......................... Foot...................... 101-60 Relative Value of British and Foreign Square and Cubic Measures. Country or Place. Square Foot in Cubic Foot in EngEnglish Sq. Inches. lish Cubic Inches. Amsterdam.......................... 124-255 1385-070 Antwerp.............................. 126'337 1420'027 Augsberg.........7............... ] 35722 1581-161 Berlin................................. 148'693 1813-162 Berne.................................. 133-287 1538'798 Bremen............................... 129504 1473'755 Cologne........................ 117.288 1270'229 Dantzic............................... 127-690 1442-897 Dresden.............................. 124.099 1382'463 France........................ 163-558 2091-743 Geneva............................... 369-024 7088-951 Hamburg............................ 127441 1138-684 Hanover.............................. 131.194 15!)2 696 Leipsic............................... 123.432 1371 329 Liege................................. 128-142 1450-577 Lisbon................................. 167 547 2168-728 Milan................................. 243984 3811-030 BRITISHI AND FOREIGN MEASURES. 29 Relative Value of British and Foreign Square and Cubic 2lfeasures-continued. Country or Place. Squ-ire Foot in Cubic Foot in EngCountry or Plce. English Sq. Inchlos. lish Cubic Inches. Munich............................... 129-390 1471.811 Nuremberg........................... 143 041 1710-770 Prussia.............................. 152.670 1886-390 Rhineland........................... 152-670 1886'390 Riga............................... 116.424 1256-215 Rome.................................. 137-358 1609.835 Spain.................................. 123 832 1378 -002 Sweden............................... 136-515 1595-041 Venice................................ 187'142 2560.102 Vienna................................ 155-002 1929-774 Zurich................................. 139-476 1647.211 Relative Value of British and Foreign Land Mleasures. English No. of each Country or Place. Name of Measure: Square equal to 10 Yards. English Acres. Amsterdam........... Morgen............. 9722 4.978 Berlin................. Great 1lorgen...... 6786 7-132 Dantzic................. Morgen............... 6650 7-278 France................. Hectare............... 11960 4.046 Geneva................. Arpent................ 6179 7.833 Hamburg.............. Morgen............... 11545 4-192 Hanover............... Morgen............... 3100 15-613 Naples.............. Moggia............... 3998 12-106 Portugal............... Geira.................. 6970 6.944 Prussia................. Morgen............... 3053 15'853 Rhineland............ Morgen............... 10185 4-752 Rome................... Pezza................. 3158 15-196 Russia.................. Dessetina............ 13066 3-704 Saxony................. Acre................... 6590 7-344 Spain.................. Fanegada............ 5500 8 800 Sweden................. Tunneland............ 5900 8-203 Switzerland........... Faux.................. 7855 6-161 Vienna................. Jock................... 6889 7 025 Zurich.................. Common Acre....... 3875 12'488 3* 30 BRITISH AND FOREIGN MEASURES. Relative Value of British and Foreign Road Measures. No. of each Country or Place. Name of Measure. English. equal to 100 English Miles. Arabia........... Mile....................... 2148 81-936 Brabant.......... League................ 6077 28-966 China............. Li......................... 632 278.481 Denmark......... Mile...................... 8244 21 348 Flanders......... League.................. 6864 25-641 France........... League of 2000 Toises 4263 41-285 Germany......... Mile, long............... 10126 17-381 Hamburg......... Mile....................... 8244 21 348 Hanover.......... Mile..................... 11559 15-226 Holland.......... Mile....................... 8103 21-725 Hungary......... Mile....................... 9113 19-313 Netherlands...... Mile, metrical.......... 1093 161-024 Persia............ Parasang.............. 6086 28-918 Poland............ Mile, long............... 8103 21-725 Portugal.......... League.................. 6760 26.035 Prussia........... Mile...................... 8237 21-367 Rome............. Mile...................... 1628 108.108 Russia............. Werst..................... 1067 150-814 Spain............ League, common...... 7416 23-732 Sweden........... Mile..................... 11700 15-012 Switzerland...... Mile................. 9153 19 228 Turkey............ Berri...... 1826 96-385 31 A TABLE Of the Reciprocals of Numbers; or the Decimal.Fractions corresponding to Vulgar -Fractions of which the Numerator is unity or 1. [In the following tables, the decimal fractions are reciprocals of the denominators of those opposite to them; and their product is = unity. To find the decimal corresponding to a fraction having a higher numerator than 1, multiply the decimal opposite to the given denominator, by the given numerator. Thus, the decimal corresponding to -1 being'015625, the decimal to ~ will be *015625 x 15 ='234375.] Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/2'5 1/20 *05 1/38'026315789 1/3'333333333 1/21 047619048 1/39'025641026 1/4'25 1/22 -045454545 1/40'025 1/5 *2 1/23'043478261 1/41'024390244 1/6 -166666667 1/24 041666667 1/42'023809524 1/7'142857143 1/25'04 - 1/43'023255814 1/8 *125 1/26 -038461538 1/44 *022727273 1/9'111111111 1/27 -037037037 1/45 *022222222 1/10 *1 1/28'035714286 1/46'02173913 1/11'090909091 1/29 -034482759 1/77'0212766 1/12'083333333 1/30'033333333 1/48'020833333 1/13'076923077 1/31'032258065 1/49 02'0408163 1/14'071428571 1/32'03125 1/50 *02 1/15'066666667 1/33 -030303030 1/51'019607843 1/16'0625 1/34 -029411765 1/52'019230769 1/17 *058823529 1/35 028571429 1/53'018867925 1/18 *055555556 1/36 027777778 1/54 *018518519 1/19'052631579 1/37 027027027 1/55'018181818 82 RECIPROCALS OF NUMBERS. Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/56'017857143 1/98'010204082 1/140'007142857 1/57 01754386 1/99 01010101 1/141'007092199 1/58.017241379 1/100 01 - 1/142.007042254 1/59 *016949153 1/101.00990099 1/143.006993007 1/60.016666667 1/102 009803922 1/144 *006944444 1/61 *016393443 1/103 009708738 1/145.006896552 1/62 *016129032 1/104 009615385 1/146 006849315 1/63'015873016 1/105 *00952381 1/147 006802721 1/64.015625 1/106 009433962 1/148 006756757 1/65.015384615 1/107.009345794 1/149 006711409 1/66.015151515 1/108 009259259 1/150 *006666667 1/67.014925373 1/109 009174312 1/151 -006622517 1/68.014705882 1/110 *009090909 1/152 006578947 1/69.014492754 1/111 *009009009 1/153.006535948 1/70.014285714 1/112 *008928571 1/154.006493506 1/71.014084517 1/113 *008849558 1/155.006451613 1/72.013888889 1/114 00877193 1/156.006410256 1/73.01369863 1/115 008695652 1/157.006369427 1/74 *013513514 1/116 00802069 1/158 -006329114 1/75 *013333333 1/117 008547009 1/159 *006289308 1/76 *013157895 1/118 *008474576 1/160.00625 1/77 *012987013 1/119 *008403361 1/161 *00621118 1/78.012820513 1/120.008333333 1/162 *00617284 1/79 -012658228 1/121 *008264463 1/163 *006134969 1/80.0125 1/122 *008196721 1/164 *006097561 1/81 *012345679 1/123.008130081 1/165 006060606 1/82.012195122 1/124 *008064516 1/166.006024096 1183.012048193 1/125 *008 1/167 005988024 1784.011904762 1/126 007936508 1/168.005952381 1/85.011764706 1/127 007874016 1/169 *00591716 1/86 -011627907 1/128 *0078125 1/170 *005882353 1/87.011494253 1/129.007751938 1/171 *005847953 1/88 *011363636 1/130.007692308 1/172.005813953 1/89.011235955 1/131.007633588 1/173 *005780347 1/90.011111111 1/132.007575758 1/174 *005747126 1/91.010989011 1/133.007518797 1/175 *005714286 1/92.010869565 1/134'007462687 1/176.005681818 1/93'010752688 1/135'007407407 1/177.005649718 1/94.010638298 1/136 -007352941 1/178 *005617978 1/95.010526316 1/137 -00729927 1/179 *005586592 1/96.010416667 1/138 *007246377 1/180.005555556 1/97.010309278 1/139.007194245 1/181 [005524862 RECIPROCALS OF NUMBERS. 33 Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/182.005494505 1/224 *004464286 1/266 -003759398 1/183.005464481 1/225 -004444444 1/267 -003745318 1/184 005434783 1/226 *004424779 1/268 -003731343 1/185.005405405 1/227 *004405286 1/269.003717472 1/186 *005376344 1/228 -004385965 1/270 -003703704 1/187'005347694 1/229 *004366812 1/271'003690037 1/188 *005319149 1/230 -004347826 1/272 -003676471 1/189 *005291005 1/231 *004329004 1/273 003663004 1/190 *005263158 1/232 *004310345 1/274'003649635 1/191.005235602 1/233 -004291845 1/275 -003636364 1/192.005208333 1/234 *004273504 1/276 -003623188 1/193 *005181347 1/235 *004255319 1/277'003610108 1/194 *005154639 1/236 -00423f72 88 1/278 -003597122 1/195'005128205 1/237 -004219409 1/279 -003584229 1/196 *005102041 1/238 -004201681 1/280'003571429 1/197 -005076142 1/239 -0041841 1/281 -003558719 1/198'005050505 1/240 -004166667 1/282 -003546099 1/199'005025126 1/241 -004149378 1/283 -003533569 1/200 -005 1/242.00413-2231 1/284'003522127 1/201 -004975124 1/243 *004115226 1/285 -003508772 1/202.004950495 1/244 004098361 1/286'003496503 1/203 *004926108 1/245.004081633 1/287 -003484321 1/204 -004901961 1/246 -004065041 1/288'003472222 1/205 |004878049 1/247.004048553 1/289'003460208 1/206 *004854369 1/248.004032258 1/290 -003448276 1/207 *004830918 1/249.004016064 1/291 -003436426 1/208 -004807692 1/250.004 1/292 -003424658 1/209 *004784689 1/251 -003984064 1/293 -003412969 1/210.004761905 1/252.003968254 1/294.003401261 1/211.004789336 1/253 -003952569 1/,295 -003389831 1/212 -004716981 1/254.003937008 1/296.003378378 1/213 *004694836 1/255.003921569 1/297'003367003 1/214.004672897 1/256.00390625 1/298 -003355705 1/215 *004651163 1/257'003891051 1/299'003344482 1/216'00462963 1/258 *003875969 1/300'003333333 1/217 [004608295 1/259.003861004 1/301.003322259 1/218.004587156 1/260.003846154 1/302 -003311258 1/219.00456621 1/261 [003831418 1/303 -00330133 1/220 -00-1545455 1/262.0038167941 1/304 -003289474 1/221.004524887 1/263.0038022811 1/305 003278689 1/222 -004504505 1/264'003787879 1/306 -003267974 1/223'004484305 1/265'0037735851 1/307'003257329 34 RECIPROCALS OF NUMBERS. Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/308.003246753 1/350.002857143 1/39.2.00255102 1/309.003236246 1/351.002849003 1/393 *002544529 1/'310.003225806 1/352.002840909 1/394.002538071 1/311.003215434 1/353.002832861 1/39.5 002531646 1/312.003205128 1/354.002824859 1/396 *002525253 1/313 -003194888 1/355.002816901 1/397.002518892 1/314.003184713 1/356 -002808989 1/398.002512563 1/315 *003174603 1/357.00280112 1/399.002506266 1/316.003164557 1/358.002793296 1/400.0025 1/317.003154574 1/359.002785515 1/401.002493766 1/318.003144654 1/360.002777778 1/402.002487562 1/319.003134796 1/361 -002770083 1/403 *00248139 1/320 003125 1/362.002762431 1/404 -002475248 1/321'003115265 1/363.002754821 1/405 *002469136 1/322.00310559 1/364.002747253 1/406.002463054 1/323.003095975 1/365 *002739726 1/407.002457002 1/324'00308642 1/366.00273224 1/408'00245098 1/325'003076923 1/367 -002724796 1/409. 002444988 1/326'003067485 1/368 -002717391 1/410'002439024 1/327'003058104 1/369 *002710027 1/411'00243309 1/328.00304878 1/370 -002702703 1/412.002427184 1/329 -003039514 1/371 *002695418 1/413.002421308 1/330 008030303 1/372.002688172 1/414.002415459 1/331.003021148 1/373.002680965 1/415 -002409639 1/332.003012048 1/374.002673797 1/416.002406846 1/333.003003003 1/375 -002666667 1/417.002398082 1/334.002994012 1/376.002659574 1/418.002392344 1/335.002985075 1/377.00265252 1/419.002386635 1/336.00297619 1/378 -002645503 1/420.002380952 1/337.002967359 1/379 *002638521 1/421 002375297 1/338 00295858 1/380 -002631579 1/422 -002369668 1/339.002949853 1/381 *002624672 1/423.002364066 1/340.002941176 1/382.002617801 1/424.002358491 1/341.002932551 1/383 *002610966 1/425.002352941 1/342.002923977 1/384.002604167 1/426'002347418 1/343.002915452 1/385 *002597403 1/427.00234192 1/344.002906977 1/386.002590674 1/428.002336449 1/345.002898551 1/387.002583979 1 1/429'002331002 1/346'002890173 1/388.00257732 1/430'002325581 1/347.002881844 1/389 *002570694 1/431'002320186 1/348 *002873563 1/390 *002564103 1/432'002314815 1/349 00286533 1/391'002557545, 1/433 *002309469 RECIPROCALS OF NUMBERS. 35 Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal or Numb. Reciprocal 1/434 u002304147 1/476.00210084 1/518'001930502 1/435'002298851 1/477 002096486 1/519 *001926782 1/436'002293578 1/478.00209205 1/520.001923077 1/437'00228833 1/479.002087683 1/521'001919386 1/438 -002283105 1/480 *002083333 1/522 *001915709 1/439 -002277904 1/481'002079002 1/523 *001912046 1/440 *002272727 1/482'002074689 1/524 *001908397 1/441 *002267574 1/483'002070393 1/525 *001904762 1/442 *002262443 1/484'002066116 1/526 *001901141 1/443 -002257336 1/485.002061856 1/527 *001897533 1/444.002252252 1/486 -002057613 1/528.001893939 1/445 *002247191 1/487.002053388 1/529 *001890359 1/446 *002242152 1/488.00204918 1/530 *001886792 1/447 *002237136 1/489.00204499 1/531 *001883239 1/448 *002232143 1/490.002040816 1/532'001879699 1/449.002227171 1/491.00203666 1/533 *001876173 1/450 *002222222 1/492'00203252 1/534'001872659 1/451'002217295 1/493'002028398 1/535 -001869159 1/452'002212389 1/494 -002024291 1/536 *001865672 1/453'002207506 1/495 -002020202 1/537'001862197 1/454'002202643 1/496'002016129 1/538'001858736 1/455'002197802 1/497'002012072 1/539 -001855288 1/456'002192982 1/498'002008032 1/540'001851852 1/457'002188184 1/499'002004008 1/541'001848429 1/458'002183406 1/500'002 1/542'001845018 1/459'002178649 1/501'001996008 1/543'001841621 1/460'002173913 1/502 -001992032 1/544 -001838235 1/461'002169197 1/503 -001988072 1/545 -001834862 1/462'002164502 1/504 001984127 1/546'001831502 1/463'002159827 1/505'001980198 1/547 -001828154 1/464'002155172 1/506'001976285 1/548'001824818 1/465'002150538 1/507'001972387 1/549'001821494 1/466'002145923 1/508'001968504 1/550'001818182 1/467'002141328 1/509 -001964637 1/551'001814882 1/468'002136752 1/510'001960784 1/552'001811594 1/469'002132196 1/511'001956947 1/553'001808318 1/470'00212766 1/512'001953125 1/554'001805054 1/471'002123142 1/513'001949318 1/555'001801802 1/472 -002118644 1/514'001945525 1/556'001798561 1/473'002114165 1/515'001941748 1/557'001795332 1/474'002109705 1/516'001937984 1/558'001792115 1/475'002105263 1/517'001934236 1/559'001788909 36 RECIPROCALS OF NUMBERS. Fraetion Decimal or Fraction Decimal or Fraction Decimal or or Numo. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/560'001785714 1/602.00166113 1/644'001552795 1/561 *001782531 1/603.001658375 1/645'001550388 1/562.001779359 1/604.001655629 1/646 *001547988 1/563.001776199 1/605.001652893 1/647.001545595 1/564'00177305 1/606'001650165 1/648'00154321 1/565'001769912 1/4(07 *001647446 1/649 -001540832 1/566'001766784 1/608'001644737 1/650'001538462 1/567'00176.3668 1/609'001642036 1/651'001536098 1/568'001760563 1/610'001639344 1/652'001533742 1/569 *001757469 1/611'001636661 1/653'001531394 1/570'001754386 1/612'001633987 1/654 001529052 1/571'001751313 1/613'001631321 1/655'001526718 1/572'001748252 1/614'001628664 1/656 00152439 1/573'001745201 1/615'001626016 1/657'00152207 1/574'00174216 1/616'001623377 1/658'001519751 1/575 *00173913 1/617'001620746 1/659'001517451 1/576'001736111 1/618'001618123 1/660'001515152 1/577'001733102 1/619'001615509 1/661'001512859 1/578'001730104 1/620'001612903 1/662'001510574 1/579'001727116 1/621'001610306 1/663 i 001508296 1/580'001724138 1/62 2'001607717 1/664 I-001506024 1/581'00172117 1/623. 001605136 1/665 -001503759 1/582 -001718213 1/624'001602564 1/666 001501502 1/583.001715266 1/625'0016 1/667.00149925 1/584.001712329 1/626'001697444 1/668'001497006 1/585.001709402 1/627'001594896 1/669 I001494768 1/586.001706485 1/628'001592357 1/670 i001492537 1/587 *001703578 1/629'001589825 1/671'001490313 1/588 -00170068 1/630'001587302 1/672.001488095 1/589'001697793 1/631.001584786 1/673 -001485884 1/590'001694915 1/632'001582278 1/674'00148368 1/591.001692047 1/633'001579779 1/67.5'001481481 1/592 -001689189 1/634'001577287 1/676 -00147929 1/593 -001686341 1/635'001574803 1/677.001477105 1/594 -001683502 1/636'001572327 1/678'001474926 1/595'001680672 1/637'001569859 1/679'001472754 1/596'001677852 1/638 -001567398 1/680'001470-88 1/597'001675042 1/639'001564945 1/681 -001468429 1/598'001672241 1/640 -0015625 1/682'001466276 1/599'001669449 1/641'001560062 1/683'001464129 1/600'001666667 1/642'001557632 1/684'001461988 1/601'001663894 1/64&'001615521' 1/685.001459854 RECIPROCALS OF NUMBERS. 37 Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/686 *001457726 1/728.001373626 1/770.0012981701 1/687 *001455604 1/729.001371742 1/771 *001297017 1/688 *001453488 1/730.001369863 1/772 *001295337 1/689.001451379 1/731.001367989 1/773 *001293661 1/690 *001449275 1/732.00136612 1/774 *00129199 1/691 *001447178 1/733 -001364256 1/775 *001290323 1/692 *001445087 1/734.001362398 1/776 *00128866 1/693.001443001 1/735.001360544 1/777 *001287001 1/694 *001440922 1/736 -001358696 1/778 *001285347 1/695.001438849 1/737.001356852 1/779.001283697 1/696 *001436782 1/738.001355014 1/780:001282051 1/697 *00143472 1/739.00135318 1/781 *00128041 1/698 *001432665 1/740.001351351 1/782 *001278772 1/699 *001430615 1/741.001349528 1/783 *001277139 1/700 *001428571 1/742.001347709 1/784 -00127551 1/701 *001426534 1/743.001345895 1/785 *001273885 1/702.001424501 1/744.001344086 1/786 *001272265 1/703 [001422475 1/745 -001342282 1/787 *001270648 1/704.001420455 1/746.001340483 1/788 *001269036 1/705 *00141844 1/747.001338688 1/j789 *001267427 1/706.001416431 1/748.001336898 1/790 *001265823 1/707 *001414427 1/749.001335113 1/791.001264223 1/708.001412429 1/750'001333333 1/792 -001262626 1/709.001410437 1/751'001331558 1/793.001261034 1/710 *001408451 1/752'001329787 1/794.001259446 1/711 *00140647 1/753.001328021 1/795.001257862 1/712.001404494 1/754.00132626 1/796 -001256281 1/713.001402525 1/755.001324503 1/797 *001254705 1/714.00140056 1/756.001322751 1/798.001253133 1/715.001398601 1/757 -001321004 1/799 -001251364 1/716 *001396648 1/758 -001319261 1/800 *00125 1/717.0013947 1/759.001317523 1/801.001248439 1/718.001392758 1/760.001315789 1/802'001246883 1/719 *001390821 1/761 -00131406 1/803 *00124533 1/720 *001388889 1/762.001312336 1/804 *001243781 1/721.001386963 1/763.001310616 1/805.001242236 1/722 *001385042 1/764.001308901 1/806.001240695 1/723 *001383126 1/765.00130719 1/807 *001239157 1/724 *001381215 1/766.001305483 1/808.001237624 1/725.00137931 1/767.001303781 1/809.001236094 1/726.00137741 [l1/768.001302083 1/810.001234568 1/727 *001375516fj 1/769.00130039 1 1/811 -001233046 4 38 RECIPROCALS OF NUMBERS. Fraction Decimal or Fraction Decimal or Fraction Decimal or or Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/812 *001231527 1/854.00117096 1/896'001116071 1/813 *001230012 1/855.001169591 1/897.001114827 1/814'001228501 1/856 001168224 1/898.001113586 1/815 *001226994 1/857.001166861 1/899.001112347 11/816.001225499 1/858.001165501 1/900 *001111111 1/817 -00122399 1/859.001164144 1/901.001109878 1/818 *001222494 1/860.001162791 1/902 *001108647 1/819.001221001 1/861 O00116144 1/903.00110742 1/820.001219512 1/862.001160093 1/904 *001106195 1/821 *001218027 1/863.001158749 1/905.001104972 1/822 *001216545 1/864.001157407 1/906 *001103753 1/823 *001215067 1/865.001156069 1/907'001102536 1/824.001213592 1/866.001154734 1/908 *001101322 1/825'001212121 1/867.001153403 1/909 *00110011 1/826 *001210654 1/868.001152074 1/910 *001098901 1/827 *00120919 1/869.001150748 1/911'001091695 1/828 -001207729 1/870.001149425 1/912 *001096491 1/829 *001206273 1/871 *001148106 1/913 *00109529 1/830 o001204819 1/872 *001146789 1/914 *001094092 1/831 s001203369 1/873.001145475 1/915.001092896 1/832 /001201923 1/874'001144165 1/916'001091703 1/833 -00120048 1/875 *001142857 1/917 *001090513 1/834'001199041 1/876.001141553 1/918 *001089325 1/8I5 /001197605 1/877 *001140251 1/919 *001088139 1/836 *001196172 1/878.001138952 1/920 *001086957 1/837.001194743 1/879 -001137656 1/921 *001085776 1/838 -001193317 1/880.001136364 1/922 *001084599 1/839 *001191895 1/881.001135074 1/923 *001083423 1/840'001190476 1/882.001133787 1/924 *001082251 1/841 *001189061 1/883.001132503 1/925 001081081 1/842 *001187648 1/884 -001131222 1/926 *001079914 1/843 /00118624 1/885.001129944 1/927 *001078749 1/844 *001184834 1/886.001128668 1/928 *001077586 1/845'001183432 1/887.001127396 1/929.001076426 1/846.001182033 1/888.001126126 1/930 *001075269 1/847 *001180638 1/889.001124859 1/931 *001074114 1/848 *001179245 1/890'001123596 1/932.001072961 1/849 *001177856 1/891.001122334 11/933 *001071811 l/850'001176471 1/892.001121076 s 1/924'001070664 1/851'001175088 1/893'001119821 1/93)5 *001069519 1/852'001173709 1/894'001118568 1/936 -001068376 1/853'001172333 1/895'001117818 1/937 1-001067236 RECIPROCALS OF NUMBERS. 39 Fraction Decimal or Fraction Decimal or Fraction Decimal or Jr Numb. Reciprocal. or Numb. Reciprocal. or Numb. Reciprocal. 1/938'001066098 1/959'001042753 1/980'001020408 1/939 *001064963 1/960'001041667 1/981'001019168 1/940'00106383 1/961'001040583 1/982 -00101833 1/941 o001O62699 1/962'001039501 1/983 001017294 1/942'001061571 1/963 *001038422 1/984 -00101626 1/943 -001060445 1/964'001037344 1/985'001015228 1/944'001059322 1/965'001036269 1/986'001014199 1/945 -001058201 1/966'001035197 1/987'001013171 1/946'001057082 1/967'001034126 1/988'001012146 1/947'001055966 1/968'001033058 1/989'001011122 1/948'001054852 1/969'001031992 1/990'001010101 1/949'001053741 1/970'001030928 1/991'001009082 1/950'001052632 1/971'001029866 1/992'001008065 1/951 *001051525 1/972'001028807 1/993 -001007049 1/952'00105042 1/973'001027749 1/994'001006036 1/953'001049318 1/974'001026694 1/995'001005025 1/954'001048218 1/975'001025641 1/996 -001004016 1/955'00104712 1/976'00102459 1/997'001003009 1/956 -001046025 1/977 -001023541 1/998'001002004 1/957'001044932 1/978'001022495 1/999'001001001 1/958'001043841 1/979'00102145 1/1000'001 TABLE OJf the Diameters, Cirvcumferences, and Areas of Circles, in Inches. Circum- Area in Circum- Area in z Circum- Area in ferenc in Square a 8 ferencein Square ferencein Square Inches. Inches. Inches. Inches. Inches. Inehes. T *1963 000306 4 12.566 12-566 9 28.274 63.617 8 3927 -01227 4 12'959 13-364 28-667 65396 sT3 *5890 *0271 i 13-351 I-186 4 29-059 67-200~ & *~7854 o04909 4o 13'744 15-033 { 29'452 69-029 9 14-137 15-904 29-845 70-882 T 06 4 14-529 16-800 4 30-237 72 759 1.1781'110441 4 14-f922 17'720 4 30.630 74.662 T 1.3744.15033 4 15 315 18 665 4 31023 76-58S - 1~5708 -19635 5 15-708 19.635 10 31-416 78.540 19 7 67671 *21850 4 16 100 20-629 4 31'808 80-515 8 1-9635 *30 68) 16-493 21-647 4 32 201 82-516 r 2'1598 *37122 16'886 22 690 4 32594 845-0 I 2-3562'44172 17'278 23-758 32-98 86590 a 2'5525'51849 i 17'671 24'850 33-379 88'664 1 6 27489 60132 18-064 25'967i 33772 90-762 t 29452 69030 B 18-457 27-108 34-164 92-885 1 3-141 *785 6 18'849 28-274 11 34-557 95-033 * 3534'994 I 19-242 29-464 4 34950 97-205 3927 1-227 I 19-635 30-679 4 35-343 99-402 4.319 1-484 4 20027 31-919 4 35-735 101.623 4 4-712 1-767 4 20-420 33-183 4 36-128 103-869 5'105 2-073 4 20-813 34-471 4 36-521 106-139 5-497 2-405 4 21-205 35-784 4 36-913 108-434 4 5890 2-761 4 21-598 37-122 4 37-306 110-753 2 6-283 3-141 7 21'991 38-484 12 37-699 113-097 6675 3-546 4 22-383 39-871 38091 115-466 7-068 3-976 4 22-776 41'282 38.484 117-859 7461 4'430 4 23'169 42-718 38877 120-276 7854 4-908 4 23-562 44-178 4 39-270 122-718 8246 5-411 23954 45-663 i 39-662 125-184 8-639 5-939 4 24-347 47-173 4 40'055 127-676 9-032 6-491 4 24-740 48-707 40-448 130-192 3 9-424 7-068 8 25-132 50-265 13 40'840 132-732 9-817 7'669 25-525 51-848 4 41.233 135-297 4 10 210 8-295 4 25-918 53456 41'626 1378586 10-602 8 946 4 26-310 55-0881 i- 42'018 140'500 10-995 9'621 4 26-703 56-745 i 42-411 143-139 4 11-388 10-320 i 27096 58-426 42-804 145-802 11-781 11-044 27'489 60-132 i 43-197 148-489 * 12-173 11-793 4 27-881 61-862 4 43-589 151-201 40 CIRCUMFERENCES AND AREAS OF CIRCLES. 41 Diam. Circum. Ar a. Diam. Circum. Area. Diam. Circum. Area. 14 43-98 13-93 20 62-83 314-16 26 81-68 530-93 I 44-97 156-69 I j 63-22 318-09 t 8207 53f604 [ 44-76 159-48 ] 63'61 322-06 82-46 541-18 45-16 16229 [ 64'01 326-05 82-85 546-30 [ 45-55 165-13 64-40 330-06 83-25 551-54 4594 167-98 64-79 334-10 83-64 556-76 I 46-33 170-87 0 65-18 338-16 81403 562-00 46-73 173-78 65-58 342-25 84-43 567-26 15 47'12 176'71 21 65-97 346-36 27 84-82 572-55 [ 47-51 179-67 [ 66-36 350-49 85-21 57787 [ 47-90 182-65 66B75 354-65 [ 85-60 583'20 48-30 185-66 67-15 358-84 86-00 588-57 I 48'69 188-69 I 67-54 363-05 3 86-39 593-95 I 49-08 19174 I 67-93 367-28 I 8678 599-37' 49-48 194-82 H 68-32 3.71354 87-17 604-80 49-87 197,93 X 68-72 375-82 87-57 610-26 16 50-26 201-06 22 69-11 380-13 28 87'96 615-75 [ 50-65 204-21 ] 69-50 384-46 [ 88-35 621-26 51-05 207-39 ~ 69-90 388-82 8875 626-79 51-44 210-59 i 70-29 393-20 89-14 632-35 i 51-83 213-82 70-68 397-60 W 89-53 637-94 52-22 217-07 I 71-07 402-03 i 89-92 643-54 [ 52-62 220-35 71-47 406-49 [ 90-32 649-18 I 53-01 223-65 7186 410-97 90'71 6-54-83 17 53 40 226-98 23 72-25 415-47 29 91'10 660'52 k 53-79 230-33 * 72'64 420-00 k 91'49 666-22 [ 54-19 233-70 1 73-04 424-55 91-89 671-95 Q 54-58 237-10 { 73-43 429-13 92-28 677-71 a54-97 240-52 3 73-S2 433-73 92-67 683-49 55-37 243-97 g 7421 43836 93-06 689-29 55-76 247-45 74-G1 443 01 93 46 695-12 Q 56-15 250 94 75300 447-69 9385 700-98 18 56354 254-46 24 75-39 452-39 30 94-24 706 86 356S-94 258 01 * 75379 457 11 94-61 712-76 ~ 57-33 261-58 i 7618 461'86 l 95-03 718'69 I 53772 265-18 I 76.57 466.63 I 95-42 724-64 } 58-11 268.80 [ 76-96 471.43 [ 95.81 730-61 58'51 272-44 1 77-36 476.25 [ 96-21 736-61! 58I90 276-11 I 77-75 48110 96.60 742.64 0 59-29 279-81 * 78-14 485-97 96-99 748-69 19 59-69 283-52 25 78-53 490-87 31 97-38 754-76 60-08 287-27 * 78-93 495-79 97-78 760-86 * 60-47 291-03 79-32 500-74 9817 766-99 1 60-86 294-83 s 79-71 505.71 1 98-56 773.14. 61.26 298-64 I 80-10 510-70 | 98*96 779-31 * 61-65 302-48 I 80'50 515-72 ] 99-35 785-51. 62-04 306-35 [ 80-89 520.76 99-74 791'73 * 62.43 310-24 l 81-28 525-83 } 100-13 797'97 4q; 42 CIRCUMFERENCES AND AREAS OF CIRCLES. Diam. Circum. Area. Diam. Circum. Area. Diam. Crcum. Area. 32 100-5 804-21 38 119-3 113411 44 132 1520-53 / 100-9 810.541 1197 114159 * 136 1529-18 / 101'3 816-86 l 120-1 1149'08 I 139' 0 1537-86 I 101"7 823-21 { 120-5 1156-61 I 139-4 1546-55 / 102-1 829.57 1 120-9 1164-15 ~ 139-8 1555'28 [ 102-4 835-97 i 121-3 1171-73 I 140-1 1564-03 [ 102'8 842-39: 121-7 1179-32 1405 1572-81 / 103-2 84883 122-1 1186-94 t 140-9 1581-61 33 103-6 855 30 39 122-5 1194-59 45 141-3 1590-43 b 104-0 861-79 * 122-9 1202-26 * 141-7 1599-28 i 1044 868-30 4 123-3 1209-95 I 142-1 1608-15 i 104-8 874-84 i 123-7 1217-67 - 142-5 1617-04 a 105-2 881-41 a 124-0 1225-42 2 142-9 1625'97 i 105-6 888-00 124-4i 23318 143-3 1634-92 a 106l0 894-61 a 124-8 1240'98 I 143-7 1643-89 * 106-4 901.25 I 125-2 1248-79 144-1 1652-88 34 106.8 907-92 40 125-6 1256-64 46 144-5 1661-90 107-2 914-61 i* 126-0 1264-50 * 144-9 1670-95 107151 921-32 I 126-4 1272-39 ~ 1145'2 1680-01 107-9 928-06 I 126-8 1280-31 i 145-6 1689-10 108-3 93-82 127-2 128825 146-0 1698-23 108-7 941-60 [ 127-6 1296-21 I 146-4 1707-37 q 109'1 948-41 t 128-0 1304 20 146-8 1716-54 g 109 5 955 25 * 128 4 1312.21 | 147.2 1725'73 35 109-9 962-11i 41 128-8 1320'25 47 147-6 1734-94 110-3 96899 129-1 1328-32 2 148S0 1744'18 i 0ll-7 975'900 ~ 129-5 1336-40 I 148-4 1753-45 - 111-1 982'84i9 - 129-9 1344-51 I 1488 1762'73 115S 98980 jI 130'3 135265 B 149'2 1772'05. 111'9 996-78i 130-7 1360-81' 149.6 11781-39. 112-3 1003'71 131-1 1369-00 i 150-0 1790-76 L 112-7 101OOS1 II 131-5 1377-21' 150-4 1800-14 36 113-0 1017'87 42 131-9 1385'44 48 150.7 1809'56 113-4 1024-95!1 132' 3 1393a70 * 151'1 1818-99 / 113-8 1032-06il 132-7 1401-98 151-5 182846 114-2 1039-191 133-1 1410-29 i 151-9 1837-93 3 114-6 1046-391 1335 14186lS 2 I 152.3 1847-45 115-0 105352 - 133-9 1426-98 if 152-7 1856-99 f 115-4 1060.73i il 134-3 1435-36 t 153-1 1866'55 t 115-8 1067-95 i 134-6 1443-77 ft 1535 1876'13 37 116-2 1075-21 4t 135-0 1452.20 49 153-9 1885-74 * 116-6 1082-48 * 135-4 14:60-65 * 154:3 1895-37 117-0 1089-79 ~t 135-8 1146913 t, 154-7 1905-03 117-4 1097-11 ft 136-2 1477-63 t 155-1 1914-70 t 117'8 1104-46. 136-6 1486-17 ft 155-5 1924-42 f 118-2 1111-84l t 137-0 1494-72 155-9 1934-15 f 118-5 1119-24 137-4 1503'0 i 156-2 1943-91 118-9 1126-66 f 137-8 1511-99 156-6 1953-69 CIRCUMFERENCES AND AREAS OF CIRCLES. 13 Dm. Cirom. Area in Ar in Diam. Circum. Area in Area in Inchts. Inches. Square nc Sqare Feet. Inc.les. Inches. Square Inches. Square Feet. 50 157.0 1963.5 13-63 61 191-6 2922-4 20-29 1 157.8 1983-1 13-77 1 192-4 2946.4 20.46 - 158-6 2002-9 13-90 2 193-2 2970.5 20-62 15941 2022-8 1404 i 1939 2994-7 20-79 51 160-2 2042-8 14-18 62 194-7 3019-0 20-96 ~ 161-0 2062'9 14-32 1 195'5 3043'4 21-13 161-7 2083-0 14-46 196-3 3067-9 21-20 a 162-5 2103-3 14-60 197'1 3092-5 21-47 52 163'3 21.23'7 14-74 63 197-9 3117'2 21-64 a 164'1 2144'1 14-89 I 198-7 3142-0 21-81 164-9 2164-7 15-03 l 199-4 3166-9 21 9S; 165-7 2185-4 15-17 [ 200-2 3191-9 22'16 53 160-5 2206-1 15.32 6-1 201-0 3216-9 22-34 167-2 2227-0 15-46 [ 201-8 3242-1 22-51 168-0 22-18-0 15-61 / 202'6 3267'4 22-68 16-'8 2269'0 15-75 I 203-4 3292-8 22-86 54 169.- 2290-2 15-90 65 204-2 3318-3 23-04 170i)4 2311-4 16-05 H 204-9 3343-8 23-22 17i12 2332'8 16-20 H 205-7 3369-5 23-39 1 1720 2354-2 16-3-1 H 206-5 3395-3 23-57 55 172 7 2`375-8 16-4-9 66 207-3 3421-2 23-75 175 23974 16-64 4 208'1 3447-1 23-93 - 17, 24,19-2 16-80 208-9 3473-2 24-11 175-1 24410 1695o 209-7 34993 24-30 56 175-9 24530 17-10 67 210-4 3525-6 24 48 1 17; -7 2485-0 17-25 2112 3552-0 2' 66 W 17-5 250o7'1 17-41 - 212-0 3578-1 24-84 178'2 2529 4 17-56 212-8 3605-0 25-03 57 179'0 2551-7 17-72 68 213'6 3631-6 25-22 179' 8 2574-1 17-87 2144 3658-4 25-40 186S6 259367 18-03 - 215-1 3685-2 25 59 I 81-I4 26(19-3 18 19 a 215'9 37122 2577 58 182 2 2612-0 18 34 69 216'7 3739 2 25-96 4 182'9 2664-9 1850 1 217-5 3766-4 2615 183-7 2687-8 18-68 218-3 3793-6 26.34 1845 2710-8 18-82 2191 3821o0 26-53 59 185-3 2733-9 18-98 70 219-9 3848s4 26-72 I 186-1 2757-1 19-14 220-6 3875-9 26-91 186-9 2780-5 19 30 221-4 390306 27-10 187-7 2803-9 1947 1 -?2-2 3931-3 27,30 60 1884 28274 19-63 71 2230 3959-2 27-49 189-2 2851-0 1979 223-8 3987 27-68 190'0 2874-7 19296 4 224-6 4015.1 27-87 190'8 2898-5 20-12 225'4 4043-2 28-07 44 CIRCUMFERENCES AND AREAS OF CIRCLES. Diam[ Circum. Area in Area in i Diam. Circum. Area in Area in \lache~s. Inches. Square Inches.f Square Feet. Inches. Inches. Square Inches. Square Feet 72 226-1 4071-5 28-27 83 260-7 5410-6 37.57 [ 226-9 4099-8 28-47 1 261.5 5443-2 37-79 ~ 227-7 4128-2 28066 262-3 54-76-0 38-02 2285 4156-7 28-86 2631 5508-8 38 25 73 229-3 4185-3 29-06 84 263-8 5541-7 23848 ~ 230-1 4214-1 29-26 j 264-6 5574-8 38-71 ~ 230-9 4242-9 29-46 H 265-4 5607-9 38-94 ~ 231-6 4271-8 29-66 4 266-2 5641-1 39-07 74 232-4 4300-8 29-86 85 267-0 5674-5 39-40 ~ 233-2 4329-9 30-06 ~ 267-8 5707-9 39-63 ~ 234-0 4359-1 30'26 I 268-6 5741-4 39-87 [ 234-8 4388-4 30-47 I 269-3 5775-0 40-10 75 235'6 4417-8 30-67 86 270-1 5808-8 40-33 ~ 236-4 4447'3 30-88 l 270-9 5842-6 40-57 [ 237'1 4476-9 31-09 ~ 271-7 5876-5 40-80 [ 237-9 4506-6 31-30 H 272-5 5910-5 41-04 76 238-7 4536-4 31-50 87 27:3-3 5944-6 41-28 ~ 239-5 4566-3 31-71 ~ 274-1 5978-9 41-52 ~ 240-3 4596-3 31-91 ~ 274-8 6013-2 41.75 l 241-1 4626-4 32-12 1 275-6 6047-6 41-99 77 241-9 4656-6 32-33 88 276-4 6082-1 42-23 ~ 242-6 4686-9 32-54 l 277-2 6116-7 42 47 ~ 243-4 4717-3 32-75 [ 278-0 6151-4 42-71 z 244'2 474'77 32-96 278-8 6186-2 42-95 78 245-0 4778-3 33-18 89 279-6 622141 43-20 ~ 245-8 4809-0 33-39 I 280-3 6256-1 43-44 ~ 246-6 4839-8 33-60 [ 281-1 6291-2 43-68 247-4 4870-7 33-81 281-i9 6326-4 43-92 79 248-1 4901-6 34-03 90 282-7 6361-7 44-17 ~ 248-9 4932-7 34-24 [ 283-5 6397-1 44-42 ~ 249-7 4963-9 34-46 I 284-3 6432-6 44-66 / 250-5 4995-1 34-68 H 285-1 6468-2 44-81 80 251-3 5026-5 34-90 91 285-8 6503-8 45-16 ~ 252-1 5058-0 35-12 I 286.6 6539-6 45-41 ~ 252-8 5089-5 35-34 I 287-4 6575-5 45.66 t 253-6 5121-2 35-56 288-2 6611-5 45-91 81 254-4 5153-0 35-78 92 289-0 6647-6 46-16 1 255-2 5184-8 36-00 i 289-8 6683-8 46-41 4 256-0 5216-8 36-22 1 290-5 6720-0 46-66 256-8 5248-8 36-44 a 291-3 6756-4 46-91 82 257.6 5281-0 36-67 93 292-1 6792-9 47-17 ~ 258-3 5313-2 36-90 k 292-9 6829-4 47-43 ~ 259-1 5345-6 37-12 I 293-7 6866-1 47-68 [ 259-9 5378-0 37-34' 294-5 6902-9 47-93 CIRCUMFERENCES AND AREAS OF CIRCLES. 45 Diam. Cirnum. Area in Area in Diam. Circum. Area in,rea in inches. Inlines. Squ.re Inches. Square Feet. Inches. Inch.s. Squa.e Inches. square Feet 94 295 3 6939 7 48-19 121 380'1 11499-0 79-85 29;50 6976-7 48'45 122 383'2 11689'9 81-18 29-3 7013-8 48'70 123 386'4 118823 8251 j 297'6 7050'9 48'96 124 389'5 12076-3 83'86 95 2984 70882 4922 125 392-7 122718 8522 Q29'2 71.25'5 49'48 126 395'8 12469'0 86.59 1- 3000 7163'0 49-64 127 398-9 126677 87-91 3 3C 08 7200-5 50'00 128 402'1 12867'9 89-36 96 301'5 7238' 2 5026 129 405-2 13069'8 90'76 1 3023 72759 5052 130 408-4 13273-2 92'17 ~ 303'1 7313'8 50'78 131 411'5 13478-2 9359 303'9 7351-7 51'05 132 414'6 13684-8 95'03 97 3047 7389 8 5131 133 417-8 13892-9 96'47 + 05- 7127'9 51'57 134 420'9 14102.6 97-93 I 306.f3 7,166'2 5184 135 424-1 14313'9 99'40 le 307.U0 7504 5 52'11 136 427,2 14526'7 10088 98 307.8 7542-9 52'38 137 430'3 14741-1 10236 308'6 7581'5 5265 138 433-5 14957'1 10387 ~ 309'4 7620-1 52'91 139 436'6 15174'7 105'37; 310G2 7658'S 53'18 140 439-8 15393'S 106'90 99 311-0 7697'7 53-45 141 442'9 15614-5 10843 41 311-8 7736-6 53-72 142 4461 158368 10997 ~ 312'5 7775'6 53'99 143 449'2 16060'6 11153 ] 3133 7814'7 54-26 144 452-3 16286'0 113-09 100 314'1 7854'0 54'54 145 455-5 16513'0 114-67 101 317'3 8011' 7 55-63 146 458-6 16741-5 11626 102 320'4 8091-2 56'74 147 461-8 16971-7 117-86 103 323'5 8332'3 57-86 148 464'9 17203-4 11946 104 326'7 8494'9 58-99 149 468-0 17436-6 12108 105 329'8 8659'0 60-13 150 4712 17671-5 122-71 106 333'0 8824' 7 61'28 151 474'3 17907'9 12436 107 330i1 8992-0 62'44 152 477'5 181459 126-01 108 339 2 9160'9 63 61 153 4806 18385'4 12767 109 34.2'4 9331-1 64-80 154 483-8 186265 129-35 110 345-5 9503' 3 65-99 155 486-9 18869-2 13103 111 348-7 9676-9 67-20 156 490'0 19113-5 13273 112 351-8 9852'0 6S-41 157 493'2 19359-3 134-44 113 355-0 10028-7 69-64 158 496-3 19606-7 13615 114 3531 10207-0 70-88 159 499-5 19855-7 137-88 115 361.2 10386' 9 72-13 160 502-6 20106-2 13962 116 36-q4 19538-3 73-39 161 505-7 203583 14137 1li7 36;. 5 1051'3 74'66 162 508'9 20612-0 14313 118 370' 10935'9 75-94 163 512-0 20867-3 144'91 119 373. 8 11122-0 77-23 ]64 515-2 211241 146'69 120 3766 11309'7 78'54 165 518-3 21382-5 148-4 46 AREAS of the Segments and Zones of a Circle, of which the DIAMETER is ldity, and supposed to be divided into 1000 equal parts. Heigh. Area of Area of Area of Area of Heigt. Segment. Zone. H Segment. Zone. *001 -000042 *001000'036 -009008'03.969 *002 *000119'002000'037'009383'036967 ~003'000219'003000 *038'009763'037965 ~001'000337 *004000'039'010148 *038962 ~005'000470'005000'040'010537'039958 *006'000618'006000'041'010931'040954'007'000779'007000 -042'011330 -041951 *008'000951'008000'043.011734'042947,009'001135'009000'044.012142.043944'010'001329'010000.045'012554'044940 ~011.001533,011000'046'012971.045935 ~012 -001746'011999'047 -013392'046931 ~013'001968,012999'048.013818 -047927 ~014'002199'013998'049.014247'048922 ~015'002438'014998'050'014681 -049917 ~016'002685'015997'051.015119'050912'017 -002940'016997'052'015561'051906 ~018'003202'017996'053'016007.052901 ~019.00347L'018996'054'016457'053895'020'003748'019995'055'016911'054890 ~021'004031.020994'056'017369.055883'022'004322'021993'057'017831'056877'023'004618'022992'058'018296 -057870'024'004921'023991'059'018766'058863 ~025'005290'024990'060 — 019239.059856'026'005546'025989'061'019716'060849'027'005867'026987'062'020196'061841'028'006194'027986'063 -020680'062833 *029'006527.028984'064'021168'063825'030'006865'029982'065'021659'064817'031'007209'030980'066.022154 -065807 ~032'007558'031978'067'022652'066799 ~033'007913'032976'068'023154'067790 ~034'008273'033974'069.023659 -068782 ~035'008638'034972'070'024168'069771 AREAS OF THE SEGMENTS OF A CIRCLE. 47 Area of Area of eight. Area of Area of Segment. Zone. Segment. Zone. *071 *024680'070761 -111 -047632 110082 ~072'025195'071751'112'048262 111057 ~073'025714'072740'113'048894'112031 -074'026236 073729'114'049528'113005'075'026761'074718'115'050167'113978 ~076 *027289'075707'116'050804'114951 ~077'027821'076695'117'051446'115924 ~078'028356'077683 118'052090'116896 ~079.028894'078670'119.052736'117867 ~080.029435'079658'120'053385'118838 ~081'029979.080645'121.054036'119809'082'030526'081631 ~122'054689'120779'083'031076'082618'123'055345'121748'084'031629'083604'124'056003 -122717'085'032186'084589'125'056663'123686'086'032745.085574'126'057326'124654'087'033307'086559'127'057991'125621'088'033872'087544'128'058658'126588'089'034441.088528'129 -059327 -127555 ~090'035011 -089512'130'059999.128521'091'035585.090496 -131'060672.129486'092'036162'091479'132.061348'130451'093'036741'092461'133'062026 -131415'094 -037323.093444'134'062707'132379'095 -037909'094426'135'063389'133342'096'038496'095407'136'064074'134304'097'039087'096388'137'064760'135266 ~098'039680'097369'138'065449'136228'099'040276'098350 -139'066140'137189'100'040875'099330'140'066833'138149 ~101'041476'100309'141'067528'139109'102'042080'101988'142'068225'140068'103'042687'102267'143'068924'141026 ~104'043296'103246'144'069625'141984'105'043908'104223'145'070328'142942'106'044522'105201'146'071033'143898'107'045139'106178'147'071741'144854'108'045759'107155'148'072450'145810'109'046381'108131 -149'073161 146765'110'047005'109107'150'073874'147719 48 AREAS OF THE SEGMENTS OF A CIRCLE. Area of Area of Area of Area of Height. Segment. Zone. Height. Segment. Zone. *151'074589 *148673 -191'10468.5 186248 ~152'075306 *149625.192.105472 *1 87172 ~153.076026'150578 *193 *106261 *188094 *154'076747'151530'194 -107051.189016 ~155 *077469 *152481 4195'107842'189938 *156'078194'153431'196.108636 *190858 ~157 -078921'154381 197' 109430'1913 777 ~168'079649 *155330'198'110226'192696 *159'080380 *156278 *199 *111024'193614.160 *081112'157226 *200'111823'194531'161 *081846 *158173 *201'1126-4'195447'162 *082582'159119 *202 *113426 196362'163'083320'160065 *203 *114230'197277 ~164'084059 *161010 *204 *115035'198190 ~165 *084801 161954 o205 *115842 *199103 *166'085544'162898 *206'116550'200015 *167 *086289 *163841'207 *117460'200924'168'087036'164784'208 -118'271 201835'169'087785'165725'209.119083'202744'170'088535'166666'210'119897'203652'171'089287'167606'211'120712'204559'172'090011'168549'212'121629'205465 ~173 *090797'160484 *213'122387'206370'174'091554'170422'214'123167'207274 ~175'092313 *'171359 *215'123988'208178 ~176'093074'172295'216'124810'209080 ~177'093836 *173231'217'125631'209981 ~178 *094601'174166 *218'1264593 210882'179'095366'175100 *219: 127285'211782'180'096134'176033'220'128113'212680 ~181'096903'176966'221 -128942'213577 ~182.097674'177897 -222'129773.214474'183'098447'178828'223'130605'215369 *184'099221'179759'224'131438'216264'185'099997'180688'225.13227 2'217157'186'100774'181617'226.133108'218050 ~187.101.553'1825415'227.133')14 218941 ~*18 102334 -183-472.228 -13"8l S 2198 32'189'103116'184398'229'135624 -22072l'190'103900'185323'230'136465'221610 AREAS OF THE SEGMENTS OF A CIRCLE. 49 Area of Area of H Area of Area of Height. Segment. Zone. Height. Segment. Zone. *231 -137307 -222497 -271 -171978.257075 *232'138150'223354'272'172867'2;57915 *233'138995 *224269'273 *1'73758'258754 *234'139841'225153 *274'174649 *259591 *235'140688 *226036'275'175542'260427 *236'141537 *226919 *276 *176435'261261 237'142387'227800 *277 *177330 *262094 *238 4143238'228680'278'178225'262926 *239'144091'229559 *279 *179122'263757 *240 *144944,230439 *280 *180019 o264586 *241'145799 *231313 *281 *180918'265414 *242.146655 *232189 *282 *181817.266240 *243.147512.233063 *283 5182718'267065 *244.148371'233937 *284 5183619.267889 -245 *149230 1234809 *285 *184521 *268711 *246 *150091 -235680 *286 5185425 *269532 *247 *150953 236550 *287 *186329 *270252 *248 *151816 *237419 -288 *187234 *271170 *249 *152680 *238287 *289 5188140 -271987 *250 -153546'239153 *290 *189047 -272802'251.154412'240019 -291 1859955 *273616'252'155280'240883 *292 [190864'274428'253 *156149'241746'293 [191775'275239 *254 *157019'242608 *294 *192684'276049'255 4157890 *243469 *295 4193596'276857 *256 *158762 *244328 *296 *194509 *277664 *267'159636'245187 *297 *195422 *278469 *258 *160510 *246044 *298 *196337 *279273 *259.161386 *246900 *299 *197252 *280075 *260 *162263 *247755 -300 *198168 *280876 *261 *163140 *248608 *301 4199085 *281675 *262 *164019 *249461 *302 200003 *282473 *263.164899'250212'303 200922 *283269 *264 *165780' 251162'304 201841 *284063 *265 *166663 *252011 305 *202761 *284857 *266 4167546 *252851 *306 *203683 *285648 *267 *168430 *253704 *301 *204605 *286438 *268 *169315 *254549 *308 *205527 287227 *269 4170202 *255392 309 *206451 1288014 270 171089 266236 810'207876'288799 5 50 AREAS OF THE SEGMENTS OF A CIRCLE. e Af Area of Area of ight Area of Area of Height. Segment. Zone. Segment. Zone..311 ~208301.289583 -351 *245934.319538 *312 *209227 *290365 352 *246889 *320249'313'210154'291146'353'247845 *320958 -314'211082 *291925'354 *248801 *321666 ~315.212011 *292702 *355.249757 *322371 *316.212940.293478.356 *250715 *323075 *317'213871 *294252.357 *251673 *323775 ~318.214802.295025 *358.252631.324474 *319.215733.295796.359.253590 *325171 *320.216666 *296565.360 *254550.325866 *321.217599.297333.361 *255510 326559 *322 *218533 *298098.362.256471 *327250 *323 *219468 *298863.363.257433 -327939 *324 *220404.299625 *364 *258395 *328625 *325.221340.300386.365 *259357.329310 *326 *222277.301145 -366.260320.329992 *327.223215 -301902.367.261284.330673 *328.224154 *302658.368 *262248.331351 *329 *225093 *303412 *369 *263213.332027 ~330 *226033.304164 -370 *264178 *332700 ~331 *226974 *304914.371 *265144 *333372 *332 *227915.305663.372.266111.334041 ~333.228858 *306410.373.267078.334708 *334.229801.307155.374.268045.335373 ~335. 230745 *307898.375.269013 *336036 ~336.231689.308640.376.269982 *336696 ~337.232634.309379.377.270951.337354 ~338.233580.310117.378.271920.338010 ~339 *234526 *310853.379 -272890 *338663 ~ 40 *235473 -311588.380 -273861.339314 ~341'236421 *312319.381 *274832 *339963 *342 *237369 *313050'382'275803'340609 ~343 -238318 *313778'383.276775.341253 ~3414 239268.314505 *384.277748'341895 ~345.240218 *315230 *385'278721 *342534 ~346 *241169.315952 *386 *279694 -343171 ~347.242121'316673 *387 *280668'343805 *348'243074'317393 *388 -281642'344437 ~349 *244026 *318110 *389'282617 *345067'850 *244980 318825!390 1'283592'345694 AREAS OF THE SEGMENTS OF A CIRCLE. 51 Area of Area of Area of Area of [eight. Segment. Zone. Height. Segment. Zone.'391 *284568 *346318 -431 *323918'369040 ~392 *285544'346940'432'324909'369545 ~393 *286521 *347560'433 *325900'370047'394'287498'348177 -434'326892'370545'395 *288476 *34S791.435 *327882.371040 *396 *289453.349403 *436.328874.371531.897 *290432.350012'437 -329866 *372019'398 *291411.350619'438.330858 *372503'399,292390.351223'439'331850'372983'400'293369.351824'440'332843'373460'401'294349'352423'441'333836'373933'402'295330'353019'442.334829'374403'403'296311'353612'443.335822'374868'404'297292'354202'444.336816'375330'405'298273'354790'445'337810'375788'406'299255'355376'446'338804'376242'407.300238'355958'447'339798'376692'408.301220'356537'448'340793 -377138'409'302203.357114.449.341787'377580'410'303187''357688'450'342782'378018'411 -304171.358258'451 -343777'378452'412'305155'858827'452'344772'378881'413 ~306140'359392.453.345768'379307 *414'307125'359954.454'346764'379728'415'308110'360513'455'347759'380145'416'309095'361070'456'348755'380557'417'310081'361623'457'349752'380965'418'311068'362173.458 -350748'381369'419'312054 -362720.459'351745'381768'420'313041'363264'460'352742'382162'421 ~314029'363805'461'353739'382551'422'315016'364343'462'354736'382936'423'316004'364878'463'355732 -383316 *424'316992 -365410'464'356730'383691'425'317981'365939.465'357727'384061'426 ~318970.366463'466'358725'384426'427'319959'366985'467'359723'384786'428'320948'367504'468'360721;385144'429'321938'368019.469'361719'385490 *430'322928'368531'470'362717'385834 52 AREA OF THE SEGMENT OF A CIRCLE, ETC........ Area of Area of Area of Area of HIeight. Segilmet. Zone. Height. Segment. Zone. *471'363715'386172 *486.378701.3905I:0 4472'364713'386505 *487'379700.390730 ^473'365712'386832 *488 -380700.390953 [474'366710 *387153 -489 *381699 *391166 *475 -367709 -387469.490'382699 *391370 476 -368708 *387778 *491 -383699.391564 *477.369707.388081 *492 -384699.391748 *478 -370706.388377 *493'385699 *391920 479.371704.388669'494'386699.392081 480 372704 *388951 -495'387699 -3922229 *481.373703.389228 1 496 388699.392362.482 *374702.389497.497.389699.392480 -483 *375702'389759 *498'390699'392580 *484 *376702.390014 1499 *391699.392657.485.377701.390261 500.392699'392699 To find the Area of the Segment of a Circle. RULE.-Divide the height, or versed sine, by the diameter of the circle, and find the quotient in the column of heights. Then take out the corresponding area, in the column of areas, and multiply- it by the square of the diameter; this will give the area of the segment. EXAMPLE.-Required the area of a segment of a circle, whose height is 3-1 feet, and the diameter of the circle 50 feet. 31 = 3'25; and 325 -. 50 = 065. ~065, as per Table, ='021659; and'021659 x 502 = 54'147500, the area required. AREA OF A CIRCULAR ZONE. 53 To find the Area of a Circular Zone. RULE 1. -When the zone is less than a semicircle, divide the height by the longest chord, and seek the quotient in the column of heights. Take out the corresponding area, in the next column on the right hand, and multiply it by the square of the longest chord; the product will be the area of the zone. EXAMPLE.-Required the area of a zone whose longest chord is 50, and height 15. 15. 50 = -300; and'300, as per Table, = -280876. Hence'280876 x 502 = 702'19, the area of the zone. RULE 2.-When the zone is greater than a semicircle, take the height on each side of the diameter of the circle, and find, by Rule 1, their respective areas; the areas of these two portions, added together, will be the area of the zone. EXAMPLE.-Required the area of a zone, the diameter of the circle being 50, and the height of the zone on each side of the line which passes through the diameter of the circle 20 and 15 respectively, 20 - 50 = -400;'400, as per Table, = 351824; and'351824 x 502 = 879'56. 15. 50 -='300;'300, as per Table, -= 280876; and'280876 x 502. — 70219. Hence 879'56 + 702'19 - 1581'75. 5* 54 AREA OF THE SEGMENT OF A CIRCLE. Approximating Rule to find the Area of a Segment of a Circle. RULE.-Multiply the chord of the segment by the versed sine, divide the product by 3, and multiply the remainder by 2. Cube the height, or versed sine, find how often twice the length of the chord is contained in it, and add the quotient to the former product: this will give the area of the segment very nearly. EXAMPLE.-Required the area of the segment of a circle, the chord being 12, and the versed sine 2. 12 x 2 = 24; -=8; and 8 x 2 =16. 23 *. 24 (12 x 2) — = 3333. Hence 16 +'3333 = 163333, the area of the segment very nearly. 55 TABLE V. Proportions of the Lengths of Circular Arcs. Height Length Height Length Height Length Height Length of Arc. of Arc. of Are. of Arc. of Arc. of Are. of Arc. of Are. *100 1.02645'135 1'04792'170 1'07537 *205 1.10855 ~101 1'02698'136 1 104862 *171 1 *07624 *206 1.10958 *102 1 02752 *137 1'04932'172 1 *07711 *207 1.11062 ~103 1 02806 -138 1 05003'173 1 *07799 *208 1.11165 ~104 102 860 *139 1'05075 -174 1'07888'209 1*11269 ~105 1'02914'140 1'05147 -175 1'07977 *210 1'11374 ~106 1'02970'141 1 05220'176 1 *08066 -211 1'11479 ~107 1-03026'142 1-05293 *177 1-08156 -212 1 11584 *108 1.03082 *143 1 105367 *178 1 108246 -213 1 -11692 ~109 1'03139'144 1105441 *179 1'08337 *214 1.11796 ~110 1'03196'145 1.05516'180 1.08428 *215 1 11904 ~111 1 -03254'146 1.05691 *181 1'08519 -216 1'12011 -112 11038312 *147 11.051,671 182 1'08t611 *217 1'12118 *113 1'03371'148 105743 183 1-'08701 -218 11'225 *114 1'03430i'140 1.05819 i'184 1-(087971 *.219 1-12334 ~115 1'03490 1650 1 08966'185 1-08890'220 1'12445 ~116 1-03551'151 1-05973'186 1.08984 1 221 1'12556'117 1 03611'152 1 *06051'187 1,09079 1 222 1 *12663 ~118 1'03672'153 1.06130'188 1.09174 *223 1.12774 ~119 1'03734'154 1.06209'189 1.09269'224 1.12885 ~120 1!03797'155 1 06288'190 1.09365 -225 1.12997 ~121 1'03860'156 1*06368'191 1.09461'226 113108 ~122 1'039823'157 1.06449'192 1 09557 -227 1 13219 ~123 1.03987 158 1.06530'193 1.09654'228 1 13331 ~124 1'04051'159 1*06611'194 1'09752i'229 1 13144 ~125 1-04116'160 1-06693 -195 1-09850 I'230 1-13557'126 1-04181'161 1-06 775'196 1 09949' 231 1 -13671 ~127 1-04247'162 1-06858 1 197 1-10:048i -232 1 13786 ~128 1'04313'163 11'06941!'198 110147 {'233 1-13903'129 1.0480 -164 1-070251'199 1!10247 1234 1-14020 ~130 1-04447.165 1-07109 1' 200 1-10348 j 235 1 14136 ~131 1'04515'166 1-07194 i 201 1'10447'236 1-14247 ~132 1-04584'167 1'07279'202 1'10548 237 1-14363''133 1.04652'168 1.07365'203 1-10650 -238 1-14480 ~104 1'04722!. -69 1.07451'204 1.10752'239 1.14597 56 PROPORTIONS OF CIRCULAR ARCS. Height Length Height Length Height Length Height Length of Are. of Arc. of Are. of Arc. of Are. of Arc. of Arc. of Arc.'240 1-14714 *283 1-20146 *326 1-26286 *369 1-33069 ~241 1-14831 *284 1-20282.327 1-26437 I370 1-33234 *242 1-14949 *285 1 -20419 *328 1-26588 *371 1 33399 *243 1 15067 *286 1 20558 *329 1-26740 *372 1.33564 *244 1-15186 *287 1 20696.330 1-26892 -373 1-33730 *245 1-15308 *288 1 -20828 *331 1 27044 *374 1.33896 ~246 1-15429 *289 1 20967 *332 1,27196 t375 1-34063 *247 1115549 *290 1-21202 *333 1-27349 *376 1-34229 *248 1.15670 *291 1 -21239.334 1-27502 *377 1-34396 ~249 1-15791 *292 1-21381.335 1 27656 -378 1-34563 *250 1.15912 *293 1-21520 *336 1-27810 *379 1-34731 *251 1'16033 *294 1.21658 -337 1 -27864 *380 1.34899 *252 1.16157 *295 1'21794 *338 1-28118 *381 1-35068 *253 1.16279 *296 1 21926 *339 1 -28273.382 1-35237 *254 1-16402.297 1 22061 I340 1-28428 *383 1 35406 *255 1-16526 *298 1-22203 *341 1 28583.384 1135575 *256 1,16649 *299 1-22347 *342 1-28739 9 385 1 35744 *257 1.16774.300 1.22495 *343 1-28895 *386 1135914 *258 1-16899 -301 1-22635 -344 1-29052 *387 1-36084 *259 1.17024.302 1 22776.345 1-29209 *388 1-36254 *260 1.17150.303 1-22918.346 1-29366.389 1136425 *261 1-17275.304 1-23061.347 1-29523.390 1136596 *262 1.17401.305 1 23205 *348 1.29681.391 1 -36767 *263 1.17527.'06 1.23349 *349 1 -29839 *392 1-36939 *264 1-17655.307 1-23494.350 1.29997.393 1-37111 *265 1.17784.308 1-23636 *351 1-30156 *394 1-37283 *266 1.17912 -309 1 -23780 *352 1 30315 *395 1-37455 *267 1-18040.310 1 -23925 *353 1 -30474 /396 1 37628 *268 1-18162 *311 1-24070 *354 1-30634.397 1.37801 *269 1-18294 8312 1 24216.355 1 30794 *398 1-37974 *270 1-18428.313 1-24360 *356 1-30954 *399 1-38148 -271 1-18557'314 1.24506 *357 1831115 *400 1 38322 *272 1-18688 *315 1 24654 *358 131276 *401 1 38496 *273 1-18819 *316 1.24801 *359 1-31437 *402 1-38671 *274 1-18969 *317 1 -24946 *360 131599 -403 1 38846 *275 1-19082.318 1250951 *361 1-31761 404 1-39021 *276 1.19214.319 1 252431 *362 1-31923 405 1*39196 *277 1.19345.320 1-25391 363 1-32086 *406 1-39372 *278 1-19477'321 1'25539 *364 1-32249 407 1-39548 *279 1-19610.322 1.25686 *365 1-32413 *408 1-39724 *280 1-19743 *323 1-2-5836 *366 1.32577 409 1-39900 *281 1.19887.324 1-25987 *367 1-32741 *410 1 I40077'282 1120011'325 1'26137 *368 1.32905 *411 1 40254 PROPORTIONS OF CIRCULAR ARCS. 57 Height Length Height Length Height Length Height Length of Are. of Arc. of Are. of Are. of Arc. of Are. of Are. of Are. -412 1-40432 *435 1-44589 -457 1-48699 *479 1 -52931 *413 1 40610 -436 1-44773 *458 1-48899 -480 1-53126 ~414 1-40788.437 1 44957 *459 1-49079 *481 1-53322 -415 1-40966 *438 1-45142 *460 1-49269 *482 1-53518 -416 1-41145 *439 1-45327 *461 1-49460 -483 1-53714 ~417 1-41324 -440 1-45512 *462 1-49651 -484 1-53910 ~418 1-41503 *441 1-45697 -463 1 -49842 *485 -54106 -419 1-41682 *442 1-45883 -464 1-50033 *486 1-54302 *420 1-41861 443 1-46069 -465 1 -50224 -487 1-54499 -421 1- 42041 -444 1-46255 -466 1-50416 488 1-54696 -422 1-42222 -445 1-46441 -467 1-50608 -489 1-54893 -423 1-42402 -446 1-46628 -468 1 -50300 -490 1-55090 -424 1-42583 -447 1-46815 -469 1-50992 -491 1-55288 -425 1 42764 -448 1-47002 -470 1-51185 -492 1 -55486 -426 1-42945 -449 1-47189'471 1-51378 -493 1-55685 -427 1 -4127 -450 1-473771 -472 1-51571 -494 1-55854 -428 1-43309 -451 1-475651 473 1-51764 -495 1-56083 -429 1-43491 -452 1-47753 -474 1-51958 -496 1-56282 -430 1-43673 -453 1-47942 -475 1 52152 -497 1-56481 -431 1-43856 -454 1-48131 -476 1-52346 -498 1-56680 -432 1-44039 -455 1-48320 -477 1-52541.499 1-56879 -433 1-44222 -456 1-48509 -478 1-52736 -500 1-57079 -434 1-44405 58 TABLE VI. Proportions of the Lengths of Semi-Elliptic Arcs. Height Length Height Length Height Length Height Length of Are. f ilrc. of Arc. of Are. of Are. of Are. of Are. of Are..100 1-04162 *135 1-07726'170 1-11569 *205 1-15602 *101 1-04262 *136 1-07831 *171 1-11682.206 1-15720 *102 1-04362 /137 1-07937 *172 1-11795 *207 1-15838 *103 1-04462 *138 1-08043 *173 1-11908 *208 1-15957 *104 1-04562'139 1-08149 *174 1-12021 *209 1'16076 *105 1'04662 *140 108255 *175 1-12134 *210 1'16196 ~106 1-04762'141 1 08362.176 1-12247 *211 1-16316 ~107 1-04862 *142 1-08469 177 1-12360.212 1-16436 ~108 1-04962 *143 1 08.576 *178 1-12473 *213 1-16557 *109 1-05063 *144 1 08684 *179 1-12586 *214 1-16678 ~110 1-05164.145 1-08792 *180 1-12699.215 1-16799 *111 1-05265 *146 1-08901 *181 1'12813.216 1116920 *112 1 05366 *147 1-09010 *182 1 12927.217 1 17041 *113 1'05467 *148 1-09119 *183 1-13041 *218 1-17163'114 1-05568'149 1-09228 *184 1-13155 *219 1-17285 *115 1 05669'150 1-09330 *185 1-13269.220 1-17407 ~116 1-05770'151 1 -09448 *186 1-13383 *221 1-17529 *117 1 05872 *152 1-09558 -187 1-13497 *222 1-17651 *118 1-05974 ~153 1 -09669 *188 1-13611 *223 1-17774 *119 1-06076 *154 1.09780 *189 1-13726 *224 1*17897 *120 1-06178.155 1109891 *190 1*13841 *225 1-18020 *121 1 06280.156 1-10002 *191 1-13956 *226 1118143 ~122 1-06382 *157 1-10113 *192 1-14071.227 1*18266 *123 1-06484 *1,8 1-10224 *193 1*14186!1228 1118390 4124 1'06586 *159 1-10335 *194 1-14301 *229 1-18514 *125 1-06689.160 1-10447 *195 1.14416 *230 1.18638,126 1-06792 *161 1-10560'196 1.14531 *231 1.18762 ~127 1'06895 4162 1-10672'197 1-14646 *232 1118886 ~128 1-06998 -163 1310784 *198 1-14762 *233 1-19010,129 1-07001 *164 1.10896 -199 1-14888 *234 1-19134 ~130 1-07204 *-165 1-110081.200 1-15014 *235 1119258 *131 1-07308'166 1-11120 201 1-15131 *236 1-19382 ~132 1P07412'167 1'11232 *202 1-15248 *237 1-19506 *133 1P07516 *168 1-11344 *203 1-15366 *238 1.19630.134 1.07621.169 1-11456 *204 1-15484 *239 1-19755 PROPORTIONS OF SEMI-ELLIPTIC ARCS, 59 Height Length Height Length Height Length Height Length of Are. of Arc. of Arc. of Are. of Are. of Arc. of Are. of Are. *240 1i19880 *283 1'25406'326 1-31198 1369 1-37268 *241 1 ~20005 *284 1 2.5538 *327 1-31335 *370 1-37414 *242 1-20130 *285 1'25670 *328 1-31472'371 1 37662 ~243 1 20255.286 1225803 *329 1-31610 *372 1-37708 ~244 1-20380'287 1-25936.330 1-31748 *373 1-37854 *245 1'20506 *'288 1'-26069 -331 1 -31886 *374 1-38000 *246 1 -20632 *289 1 26202 *332 1 -32024 *375 1-38146 *247 1-20758 -290 1-.26335 *333 1.32162 *376 1.38292 *248 1.20884 -291 1-26468.334 1-32300 -377 1-38439 -249 1-21010 *292 1 26601 -33.5 1-32438.378 1-38585 -250 1-21136 -293 1 26734 *336 1-32576 *379 1-38732 *251 1 -21263 *294 1-26867 *337 1-32715 *380 1-38879 *252 1-21390 -295 1-27000 -338 1-32854 *381 1-39024 *253 1-21517 -296 1-27133'339 1-32993 *382 1-39169 *254 1-21644 -297 1-27267'340 1-33132 *383 1-39314 *255 1-21772 -298 1-27401 -341 1 33272 *384 1-39459 *256 1-21900 *299 1 27535 *342 1-33412 *385 1-39605 -257 1-22028 /300 1-27669 /343 1-33552 -386 1-39751 -258 1-22156 -301 1-27803 -344 1-33692 H387 1.39897 ~259 1-22284 -302 1-27937.345 1-33833 I388 1-40043 ~260 1-22412 303 1-28071 *346 1-33974 l389 1-40189 ~261 1 22541.304 1-28205'347 1 -34115 -390 1-40335 *262 1-22670 -305 1-28339.348 1-34256 -391 1-40481 ~263 1-22799 -306 1-28474.349 1-34397 -392 1-40627 264 1-22928 -307 1-28609 -350 1-34539.393 1-40773 ~265 1-23057 -308 1-28744 -351 1-34681 /394 1-40919 *266 1-23186 -309 1-28879 -352 1 -34823 395 1 -41065 ~267 1-23315 -310 1 -29014.353 1 -34965 -396 1-41211 ~268 1-23445 -311 1-29149 *354 1-35108 I397 1-41357 *269 1-23575 -312 1-29285.355 1 -35251 398 1-41504 ~270 1-23705 |313 1-29421 *356 1 -35394 [399 1-41651 ~271 1-23835 -314 1-29557.357 1-35537 -400 1-41798 -272 1-23966 -315 1-29603 -358 1-35680 -401 1-41945 ~273 1-24097 -316 1-29829.359 1-35823 -402 1-42092 *274 1-24228 *317 1-29965 -360 1-35967 -403 1-42239 ~275 1-24359 -318 1-30102 *361 1-36111 404 1-42386 ~276 1-24480 -319 1-30239 -362 1-36255 -405 1-42533 *277 1-24612 -320 1-30376 -363 1-36399 -406 1 42681 278 1-24744 -321 1-30513 -364 1-36543 -407 1-42829 *279 1-24876 -322 1-30650 -365 1-36688 -408 1-42977 ~280 1-25010 -323 1 30787 -366 1-36833 -409 1-43125 *281 1 -25142 -324 1-30924 -367 1-36978 -410 1-43273'282 1-25274 -325 1-31061 -368 1-37123 -411 1-48421 60 PROPORTIONS OF SEMI-ELLIPTIC ARCS. Height Length Heigiht Length IHeight! Length -eight Length of Are. of Are. of Are. f Are. ef Are. of Are. of Aro. If _r_. of rc. I _of _ r. ~412 1143569.455 1150077 *498 1 56763.541 1.63465 ~413 1143718.456 1 50230 -499 1 56921 *542 1 63623 ~414 1'43867 i457 1 50383'500 1 57089'543 1163780 ~415 1'44016 *458 1150536 6501 1 57234.544 1163937 ~416 1,44165,459 1,50689,502 1.57389'545 1164094 ~417 1 44314 -460 1'50842 *503 1 57544.546 1'64251 ~418 1,44463.461 1.50996.504 1 57699.547 1.64408'419 1'44613 *462 1l51150'50.5 1157854 *548 1164565 ~420 1.44763 *463 1.51304.506 1.58009.549 1164722.421 1 44913.464 1.51458'507 1.58164.550 1'64879 ~422 1'45064'465 1.51612'508 1'58319'551 1.65036 ~423 1 45214'466 1.51766'509 1'58474'552 1'65193 ~424 1'45364'467 1'51920'510 1'58629'553 1'65350 ~425 1.45515.468 1.52074'511 1 58784'554 11655()7 ~426 1'45665'469 1.52229'512 1.58940.555 1 65665 ~427 1.45815 -470 1.52384.513 1 59096 -556 1 65823 ~428 1.45966 -471 1.52539'514 1 59252.557 1165981 ~429 1-46167'472 1 52691'515 1.59408 6558 1166139'430 1.46268'473 1.52849'516 1'59564'559 1166297'431 1.46419. 474 1.53004'517 1.59720'560 1'66455 ~432 1.46570'475 1-53159'518 1.59876'561 1 66613 ~433 1.46721'476 1.53314t.519 1.60032'562 1 66771 ~434 1-46872'477 1 53469'520 1-60188'563 1'66929 ~435 1 47023 -478 1.53625'521 1.60344'564 1 67087 ~436 1.47174'479 1.53781 ~522 1.60500'565 1 67245 ~437 1.47326'480 1.53937 ~523 1.60656.566 1167403'438 1-47478'481 1'54093'524 1.60812 567 1 67561 ~439 1.47630'482 1.54249 ~525 1 60968'568 167719 ~440 1.47782'483 1 54405 ~526 161124 -569 1.67877 441 147934 -481 1.54561'527 1.61280'570 1168036'442 1-48086'485 1-54718 1528 1161436'571 1-68195'443 1'48238'486 1.54875 529 161592 *572 1-68354'444 1'48391'487 1-55032'530 1-61748'573 1'68513'445 1.48544'488 1-55189 1531 1'61904'574 1-68672'446 1.48697'489 1-55346 i532 1-62060 *575 1'68831'447 1-48850'490 1.555031 533 1-62216' 576 1168990'448 1~49003'491 1.55660'534 1.62372 577 1-69149'449 1.49157 -492 1-558i7 535 1-62528 578 1'69308 -450 1-49311'493 1 55974'536 11-62684 *579 1.69467'451 1.49465'494 11561311 *537 11*62840'580 1.69626 4152 149i.61$ i495 1.562i9 i8 162996'581 1 69785'453 1-49771'496 l 1 564i47 Ii 59'59l63152 i 52 1-69945 1'46 1'49924'497 1156605 i 540 1163309 I'583 1'70106 PROPORTIONS OF SEMI-ELLIPTIC ARCS. 61 Height Length Height Length |Height Length Height Length of Are. of Arc. of Arc. of Are. of Aru. of Are. of Arc. of Arc. __- _ —--— j I ___ _ —-- *584 1-70264 *627 1-7197' 670 1 84226.713 1 91355 -585 1-70424.628 1-77359'671 1.84391 *714 1-91523 *586 1-70584 *629 1'77521 *672 1-84556 -715 1'91691 6587 1-70745'630 1-77684i *673 1 84720'716 1-91859 *588 1 70905 *631 1-77847 *674 1584885.717 1-92027 *589 1.71065 *632 1.78009 *675 1-85050.718 1-92195 *590 1.71225 633 1 78172 *676 1'85215 *719 1'92363 *591 1-71286'634 1178335 *677 1 853791 720 1192531 *592 1-71546 *635 1-78498 *678 1.85544 *721 1'92700'593 1 71707'636 1'78660 *679 1 85709 *722 1'92868 *594 1-71868 *637 1 78823 *680 1'85874 *723 1193036 *595 1'72029 *638 1-78986 *681 1'86039 *724 1-93204 *596 1-72190 *639 1'79149 *682 1-86205'725 1-93373.597 1-72350'640 1-79312 *683 1 86370 *726 1-93541 *598 1-72511.641 1-79475 *684 1-86535 *727 1-93710 *599 1-72672 *642 1-79638 *685 1-86700.728 1 93878.600 1-72833 -643 1 79801 *686 1-86866 *729 1.94046 ~601 1172994 *644 1.79964 *687 1.87031 -730 1'94215 ~602 1.73155'645 1580127 688 1.87196 *731 1 94383 ~603 1.73316'646 1-80290 689 1587362 *732 1194552 ~604 1.73477'647 1.80454 690 1.87527.733 1-94721'605 1-73638.648 1 80617.691 1.87693.734 1.94890 606 1'73799'649 1'80780 *692 1-87859'735 1-95059'607 1 73960 *650 1-80943 -693 1-88024 *736 1'95228 ~608 1-74121'651 1-81107 *694 1.88190 *737 1-95397 ~609 1-74283'652 1181271'695 1'88356'738 1-95566 ~610 1'74444'653 1-81435 -696 1-88522'739 1-95735 ~611 1'74605'654 1'81599 1'697 1-88688 *740 1-95994 ~612 1174767'655 1-817631.698 1-88854'741 1.96074'613 1'74929'656 1 -81928.699 1 89020'742 1-96244'614 1-75091'657 1 -82091 i700 1-89186 *743 1 -96414'615 1'75252'658 1-82255'701 1-89352'744 1-96583 ~616 1-75414'659 1.82419'702 1.89519.745 1 -96753 ~617 1.75576'660 1-82583 703 1-89685.746 1-96923 ~618 1.75738'661 1-82747'704 1-89851 *747 1 197093 ~619 1.75900'662 1 82911.705 1 90017 *748 1 -97262 ~620 1.76062'663 183075'706 1.90184 *749 1'97432 ~621 1.76224'664 1-83240'707 1-90350'750 1-97602 ~622 1.76386'665 1-83404'708 1-90517'751 1-97772 ~623 1.76548'666 1-83568.709 1-90684'752 1-97943 ~624 1-76710'667 1-837331 710 1-90852 -753 1-98113 ~625 1'76872'668 1.83897)'711 1.91019 *754 1-98283'626 1'77034 669 1-84061.1 712 1'91187 7655 1'98468 6 62 PROPORTIONS OF SEMI-ELLIPTIC ARCS. Height Length Height Length Height Length Height Length of Are. of Are. of Are. of Are. of Are. of Arc. of Are. of Arc. *756 1-98623 *799 2-06027 *842 2.13618 -885 2.21388 *757 1.98794 *800 2-06202 *843 2213797 *886 2-21571'758 1 98964 *801 2-06377 *844 2.13976 *887 2'21754 *759 1-99134 *802 2.06552 *845 2-14155 *888 2-21937 *760 1.99305.803 2-06727 *846 2.14384 *889 2.22120'761 1 99476 *804 2-06901 *847 2-14513 *890 2-22303 *762 1 99647 *805 2-07076 *848 2-14692 *891 2-22486 *763 1-99818 *806 2.07251'849 2-14871 *892 2-22670 *764 1 99989.807 2-07427'850 2-15050 *893 2-22854 *765 2-00160 *808 2-07602 *851 2'15229'894 2-23038 -766 2-00331 *809 2.07777'852 2.15409'895 2.23222 *767 2-00502 *810 2.07.953'853 2-15589 *896 2.23406.768 2.00673.811 2.08128.854 2-15770 *897 2-23590 *769 2-00844 *812 2-08304.855 2.15950 *898 2-23774'770 2-01016 -813 2.08480 -856 2-16130'899 2-23958 *771 2-01187'814 2.08656'857 2.16309 *900 2-24142'772 2-01359 *815 2-08832.858 2-16489.901 2-24325 *773 2-01531 *816 2-09008'859 2-16668 *902 2-24508 ~774 2.01702 -817 2-09198'860 2-16848 *903 2.24691 ~775 2-01874 *818 2-09360.861 2-17028.904 2-24874 *776 2-02045'819 2.09536 *862 2-17209 *905 2-25057 ~777 2.02217 *820 2-09712 *863 2-17389 *906 2.25240 ~778 2-02389'821 2-09888 *864 2.17570.907 2'25423 ~779 2-02561 *822 2-10065'865 2.17751 *908 2'25606 ~780 2'02733 *823 2-10242'866 2'17932 *909 2.25789 ~781 2-02907 *824 2'10419 *867 2-18113 *910 2.25972 -782 2-03080 *825 2-10596 *868 2.18294.911 2.26155 ~783 2-03252 *826 2-10773 *869 2'18475 *912 2-26338 ~784 2-03425'827 2.10950 *870 2.18656 *913 2-26521 ~785 2-03598 *828 2'11127'871 2-18837'914 2-26704 ~786 2-03771 *829 2-11304'872 2-19018'915 2.26888 *787 2.03944 *830 2-11481 *873 2-19200 *916 2-27071 ~788 2-04117'831 2.11659 *874 2.19382'917 2-27254 ~789 2.04290 *832 2-11837 *875 2.19564.918 2'27437 ~790 2.04462'833 2.12015'876 2.19746'919 2-27620 ~791 2 04635'834 2-12193'877 2-19928 *920 2-27803 *792 2'04809'835 2'12371'878 2-20110'921 2.27987 ~793 2.04983'836 2-12549 *879 2'20292'922 2-28170 ~794 2-05157'837 2-12727.880 2-20474'923 2-28354'795 2-05331'838, 2-12905'881 2-20656'924 2-28537 ~796 2-05505'839 2.13083'882 2-20839'925 2'28720 ~797 2-05679'840 2-13261 *883 2-21022 *926 2-28903 ~798 2-05853;841 2-13439'884 1 221205'927 2-29086 RULES FOR FINDING THE LENGTHS OF ARCS. 63 Height Length I Height Length Height Length Height Length of Arc. of Are. of Arc. of Arc. of Arc. of Arc. of Are. of Arc.'928 2-29270.947 2o32785'965 2o36191 *983 2.39631 ~929 2'29453'948 2.32972 o966 2.36381 *984 2.39823 ~930 2'29636 -949 2'33160 *967 2o36571'985 2'40016 ~931 2'29820'950 2.33348 *968 2.36762'986 2.40208 932 2'30004'951 2833537 *969 2-36952 *987 2 40400 *933 2'30188'952 2'33726'970 2387143'988 2'40592'934 2 30373'953 2 33915'971 2 37334'989 2'40784 *935 2'30557'954 2 34104'972 2 37525'990 2'40976 *936 2 30741'955 2 34293 *973 2 37716 *991 2 41169.937 2'30926'956 2'34483'974 2 37908 *992 2'41362 *938 2'31111'957 2 34673'975 2 38100'993 2 41556 *939 2'31295 *958 2 34862 *976 2 38291 *994 2 41749 940 2'31479'959 2 35051'977 2 38482'995 2 41943'941 2'31666'960 2'35241 *978 2 38673'996 2-42136 *942 2 31852'961 2'35431'979 2 38864'997 2'42329'943 2'32038'962 2'35621 *980 2'39055'998 2'42522'944 2'32224'963 2'35810'981 2'39247'999 2'42715.945 2'32411'964 2'36000'982 2'39439 1000 2'42908'946 2-32598 To find the Length of an Are of a Circle, or the Curve of a Right Semi-_Ellipse. RULE. —Divide the height by the base, and the quotient will be the height of an arc of which the base is unity. Seek, in the Table of Circular or Semi-elliptical arcs, as the case may be, for a number corresponding to this quotient, and take the length of the arc from the next right-hand column. Multiply the number thus taken out by the base of the arc, and the product will be the length of the arc or curve required. EXAMPLE 1. —In Southwark Bridge, London, the profiles of the arches are the arcs of circles; the 64 RULES FOR FINDING THE LENGTHS OF ARCS. span of the middle arch is 240 feet, and the height 24 feet; required the length of the arc. 24 -- 240 ='100; and'100, as per Table V., is 1'02645. Herice 1'02645 x 240 = 246'34800 feet, the length required. EXAMPLE 2.-The profiles of the arches of Waterloo Bridge are all equal and similar semi-ellipses; the span of each is 120 feet, and the rise 28 feet; required the length of the curve. 28 -- 120 = -233; and'233, as per Table VI., is 1'19040. Hence 1'19010 x 120 = 142'81200 feet, the length required. In this example there is, in the division of 28 by 120, a remainder of 40, or one-third part of the divisor; consequently the answer, 142'81200, is rather less than the truth. But this difference, in even so large an arch, is little more than half an inch; therefore, except where extreme accuracy is required, it is not worth computing. These Tables are equally useful in estimating works which may be carried into practice, and the quantity of work to be executed from drawings to a scale. As the Tables do not afford the means of finding the lengths of the curves of elliptic arcs which are RULES FOR FINDING THE LENGTHS OF ARCS. 65 less than half of the entire figure, the following geometrical method is given to supply the defect. Let the curve, of which the length is required to be found, be A B C. k Produce the height line, B d, to meet the centre Produce the height line, B d, to meet the centre of the curve, in g. Draw the right line, A g, and from the centre, g, with the distance, g B, describe an arc, B h, meeting A g in h. Bisect A h in i, and from the centre, g, with the radius, g i, describe the arc i k, meeting d B, produced to k; then i k is half the arc AB C. *, 66 The following TABLES of Areas and Solidities will be found considerably to diminish the labour of calculation. The numbers in the first column after the names represent the areas or solidities, when the length of the side or edge is 1, or unity; and the numbers in the other columns are multiples of those in the first, by the unit over each. To find, therefore, the area of a Polygon, take the square of the length of the side, and seek in the proper columns the multipliers, which are to be ranged under each other as in common multiplication. The sum of these will be the area required. EXAMPLE.-To find the area of a Pentagon, whose side is 18 inches, the multiplier being the square of 18 = 324. The number in the Table under 4 is 6'8 8 1 9 " " " "'" 2 "' 34 4 0 9 " " " " " 3 " 5'1614 Answer, the last four figures being - - decimals - - - -(the Area,) 5 5 7'4 3 0 9. The same RULE is applicable to Table X., of the Solidities of Regular Bodies, using the cube instead of the square. To find the Areas of Regular Polygons. RuLE.-Square the length of side of the Polygon, and take the products from the subjoined Table, as directed in page 66. 1 2 3 4 5 6 7 8 9 Trigon............. -4330'8660 1'2990 1.7320 2-1651 2.5981 3.0311 3.4641 3.8971 Pentagon......... 1-7204 3 4409 5.1614 6-8819 8.6024 10.3229 12.0433 13.7638 1564843 Hexagon.......... 2.5981 5.1961 7.7942 10-3923 12.9903 15.5884 18.1865 20-7846 23.3826 Heptago......... 3-6339 7'2678 10.9017 14.5356 18.1695 21-8035 25-4374 29 0713 32-7052 Octagon........... 4-8284 9'6568 14.4852 19-3137 24-1421 28-9706 33.7990 38-6274 43-4558 Nonagon... 6-1818 12.3636 18.5455 24.7273 30.9091 37.0909 43-2728 49-4546 55.6364 Decagon........... 7-6942 15-3884 23.0826 30.7768 38-4710 46-1652 53 8595 61.5537 69-2479 Undecagon. 93656 18-7313 28-0969 37-4626 46-8282 56-1938 65-5595 74-9251 84-2908 _ Dodecagon.... 1-11961 22-3923 33-5884 44'7846 55-9808 67-1769, 78-3731 89.5692 100-7654 To Jind the Areas of Circles and Spheres. RULE. —Square the diameter, and take the products from the subjoined Table, as directed in page 66. To find the circumference of a circle, take the products of the diameter only. 1 2 3 4 5 6 7 8 9 Area of Circle... 7854 1 5708 2*3562 3-1416 339270 447124 5 4978 622832 7-0686 Area of Sphere.. 3-1416 6-2832 9-4248 12-5664 15*7080 18-8496 21 9912 25 1328 28 2744 Circumference I 3.1416 6-2832 9 4248 12.5664 15*7080 18'8496 21'9912 251328 i 28'2744 _of Circle..... To.find the Areas of the Regular Bodies. RuLE — Square the length of one of the edges, and take the products from the subjoined Table, as directed in page 66. 1 2 3 4 5 6 7 8 9 Tetrahedron..... 1.7320 3 4641 541962 6-9282 8.6603 10.3923 12.1244 13.8564 15.5885 Hexahedron...... 6'0000 12.0000 18-0000 24.0000 30.0000 36'0000 42.0000 48 0000 54'0000 Octahedron...... 3.4641 6-9282 10-3923 13-8564 17-3205 20'7846 24-2487\ 27-7128 31-1769 Dodecahedron... 20.6457 41.2915 61-9372 82-5829 103.2287 123.8744 144.5201 165*1658 185-8116 Icosahedron...... 8.6603 17.3205 25-9808 34-6410 43-3013 51'9615 60.6218 69.2820 77.9423 To find the Solidities of the Regular Bodies. RIuLE.-Cube the length of one of the edges, and take the products from the subjoined Table, as directed in page 66. 1 2 3 4 5 6 7 8 9 Tetrahedron..... 1178.2357.3536.4714 *5893.7071.8250 *9428 1 0607 Hexahedron...... 1 0000 2 0000 3.0000 4.0000 5 0000 6 0000 7 0000 8.0000 9.0000 Octahedron....... 4714.9428 1-4142 1-8856 2 3570 2.8284 3 2993 3 7712 4-2426 Dodecahedron... 76631 15.3262 22-9894 30.6525 38-3156 45-9787 53 6418 61.3050 68'9681 Icosahedron......! 2.1817 4.3634 6.5451 8'7268 10.9085 13-0901 15-2718 17-4535 19'6352 TABLE of Squares, Cubes, Square Roots, and Cube Roots. Number. Square. Cube. Square Root. Cube Root. Number. 1 1 1 1.0 1.0 1 2 4 8 1*4142136 1-2599210 2 3 9 27 1-7320508 1.4422496 3 4 16 64 2.0 1-5874011 4 5 25 125 2-2360680 1*7099759 5 6 36 216 2-4494897 1-8171206 6 7 49 343 2-6457513 1-9129312 7 8 64 512 2-8284271 2-0 8 9 81 729 3 0 2-0800837 9 10 100 1000 3-1622777 2-1544347 10 11 121 1331 3-3166248 2-2239801 11 12 144 1728 3-4641016 2 2894286 12 13 169 2197 8-6055513 2-3513347 13 14 1]96 2744 3.7416574 2-4101422 14 15 225 3375 3-8729833 2-4662121 15 16 256 4096 4'0 2-5198421 16 17 289 4913 4-1231056 2'5712816 17 18 324 5832 4-2426407 2-6207414 18 19 361 6859 4-3588989 2-6684016 19 20 400 8000 4-4721360 2-7144177 20 21 441 9261 4-5825757 2-7589243 21 22 484 10648 4-6904158 2'8020393 22 23 529 12167 4-7958315 2-8438670 23 24 576 13824 4-8989795 2-8844991 24 25 625 15625 5 0 2 9240177 25 26 676 17576 5-0990195 2-9624960 26 27 729 19683 5-1961524 3 0 27 28 784 21952 5'2915026 3-0365889 28 29 841 24389 5-3851648 3-0723168 29 30 900 27000 554772256 3-1072325 30 31 961 29791 5-5677644 3'1413806 31 32 1024 32768 5'6568542 3'1748021 32 33 1089 35937 5-7445626 3'2075343 33 34 1156 39304 5'8309519 3-2396118 34 35 1225 42875 5'9160798 3'2710663 35 36 1296 46656 6'0 3-3019272 36 37 1369 50653 6-0827625 3.3322218 37 38 1444 54872 6-1644140 3-3619754 38 39 1521 59319 6-2449980 3'3912114 39 40 1600 64000 6-3245553 3'4199519 40 69 TO SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 41 1681 68921 6-4031242 3-4482172 41 42 1764 74088 6.4807407 3.4760266 42 43 1849 79507 6-5574385 3.5033981 43 44 1936 85184 6-6332496 3-5303483 44 45 2025 91125 6-7082039 3-5568933 45 46 2116 97336 6-7823300 3.5830479 46 47 2209 103823 6-8556546 3-6088261 47 48 2304 110592 6-9282032 3'6342411 48 49 2401 117649 7-0 3-6593057 49 50 2500 125000 7.0710678 3.6840314 50 51 2601 132651 7-1414284 3-7084298 51 52 2704 140608 7-2111026 3-7325111 52 53 2809 148877 7-2801099 3-7562858 53 54 2916 157464 7-3484692 3-7797631 54 55 3025 166375 7-4161985 3'8029525 55 56 3136 175616 7'4833148 3-8258624 56 57 3249 185193 7-5498344 3-8485011 57 58 3364 195112 7'6157731 3-8708766 58 59 3481 205379 7-6811457 3-8929965 59 60 3600 216000 7-7459667 3'9148676 60 61 3721 226981 7-8102497 3'9364972 61 62 3844 238328 7 8740079 3-9578915 62 63 3969 250047 7.9372539 3-9790571 63 64 4096 262144 8-0 4*0 64 65 4225 274625 8-0622577 4-0207256 65 66 4356 287496 8*1240384 4 0412401 66 67 4489 300763 8 1853528 4-0615480 67 68 4624 314432 8-2462113 4-0816551 68 69 4761 328509 8-3066239 4'1015661 69 70 4900 343000 8-3666003 4-1212853 70 71 5041 357911 8-4261498 4*1408178 71 72 5184 373248 8'4852814 4'1601676 72 73 5329 389017 8-5440037 4-1793392 73 74 5476 405224 8 6023253 4-1983364 74 75 5625 421875 8-6602540 4-2171633 75 76 5776 438976 8-7177979 4.2358236 76 77 5929 456533 8-7749644 4-2543210 77 78 6084 474552 8-8317609 4-2726586 78 79 6241 493039 8 8881944 4.2908404 79 80 6400 512000 89442719 4'3088695 80 SQUARE ROOTS, AND CUBE ROOTS. 71 Number. Square. Cube. Square Root. Cube Root. Number. 81 6561 531441 9 0 4-3267487 81 82 6724 551368 9'0553851 4-3444815 82 83 6889 571787 9-1104336 4-3620707 83 84 7056 592704 9-1651514 4'3795191 84 85 7225 614125 9-2195445 4 3968296 85 86 7396 636056 9-2736185 4'4140049 86 87 7569 658503 9-3273791 4*4310476 87 88 7744 681472 9-3808315 4-4479602 88 89 7921 704969 9'4339811 4-4647451 89 90 8100 729000 9'4868330 4-4814047 90 91 8281 753571 9'5393920 4-4979414 91 92 8464 778688 9-5916630 4-5143574 92 93 8649 804357 9-6436508 4-5306549 93 94 8836 830584 9-6953597 4-5468359 94 95 9025 857375 9'7467943 4-5629026 95 96 9216 884736 9'7979590 4-5788570 96 97 9409 912673 9-8488578 4'5947009 97 98 9604 941192 9-8994949 4-6104363 98 99 9801 970299 9*9498744 4-6260650 99 100 10000 1000000 10'0 4'6415888 100 101 10201 1030301 10'0498756 4'6570095 101 102 10404 1061208 10-0995049 4-6723287 102 103 10609 1092727 10-1488916 4'6875482 103 104 10816 1124864 10'1980390 4-7026694 104 105 11025 1157625 10-2469508 4'7176940 105 106 11236 1191016 10-2956301 4-7326235 106 107 11449 1225043 10'3440804 4-747'4594 107 108 11664 1259712 10-3923048 4-7622032 108 109 11881 1295029 10-4403065 4-7768562 109 110 12100 1331000 10*4889885 4-7914199 110 111 12321 1367631 10'5356538 4-8058955 111 112 12544 1404928 10-5830052 4'8202845 112 113 12769 1442897 10-6301458 4-8345881 113 114 12996 1481544 10-6770783 4-8488076 114 115 13225 1520875 10'7238053 4-8629442 115 116 13456 1560896 10-7703296 4*8769990 116 117 13689 1601613 10-8166538 4-8909732 117! 118 13924 1643032 10-8627805 4-9048681 118 119 14161 1685159 10'9087121 4.9186847 119 120 14400 1728000 10.9544512 4-9324242 120 72 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 121 14641 1771561 11'0 4-9t60874 121 122 14884 1815848 11'0453610 4-95967157 122 123 15129 1860867 11'0905365 4'9731898 123 124 15376 1906624 11'1355287 4-9866310 124 125 15625 1953125 11-1803399 5*0 125 126 15876 2000376 11'2249722 5-0132979 126 127 16129 2048383 11'2694277 5 0265257 127 128 16384 2097152 11'3137085 5-0396842 128 129 16641 2146689 11-3578167 5'0527743 129 130 16900 2197000 11'4017543 5'0657970 130 131 17161 2248091 11*4455231 5'0787531 131 132 17424 2299968 11 4891253 5-0916484 132 133 17689 2352637 11'5325626 5*1044687 133 134 17956 2406104 11'5758369 5-1172299 134 135 18225 2460375 11'6189500 5'1299278 135 136 18496 2515456 11'6619038 5'1425632 136 137 18769 2571353 11'7046999 5'1551367 137 138 19044 2628072 11'7473401 5-1676493 138 139 19321 2685619 11'7898261 5-1801015 139 140 19600 2744000 11-8321596 5-1924941 140 141 19881 2803221 11'8743422 5'2048279 141 142 20164 2863288 11'9163753 5'2171034 142 143 20449 2924207 11'9582607 5*2293215 143 144 20736 2985984 12'0 5 2414828 144 145 21025 3048625 12-0415946 5'2535879 145 146 21316 3112136 12-0830460 5-2656374 146 147 21609 3176523 12-1243557 5-2776321 147 148 21904 3241792 12-1655251 5*2895725 148 149 22201 8307949 12-2065556 5-3014592 149 150 22500 3375000 12-2474487 5'3132928 150 151 22801 3442951 12-2882057 5-3250740 151 152 23104 3511808 12'3288280 5'3368033 152 153 23409 3581577 12-3693169 5-3484812 153 154 23716 3652264 12'4096736 5*3601084 154 155 24025 3723875 12-4498996 5-3716854 155 156 24336 3796416 12*4899960 5328321264 156 157 24649 3869893 12'5299641 5'3946907 157 158 24964 3944312 12-5698051 54'4061 202 158 159 25281 4019679 12-6095202 5 4175015 159 160 25600 4096000 1.2'6491106 5'4288352 160 SQUARE ROOTS, AND CUBE ROOTS. 73 Number. Square. Cube. Square Root. Cube Root. Number. 161 26921 4173281 12-6885775 5*4401218 161 162 26244 4251528 12-7279221 5-4513618 162 163 26569 4330747 12-7671453 5-4625556 163 164 26896 4410944 12'8062485 5-4737037 164 165 27225 4492125 12-8452326 5 4848066 165 166 27556 4574296 12-8840987 5-4958647 166 167 27889 4657463 12.9228480 5-5068784 167 168 28224 4741632 12-9614814 5-5178484 168 169 28561 4826809 13-0 5-5287748 169 170 28900 4913000 13-0384048 5-5396583 170 171 29241 5000211 13-0766968 5-5504991 171 172 29584 5088448 13-1148770 5-5612978 172 173 29929 5177717 13-1529464 5.5720546 173 174 30276 5268024 13-1909060 5-5827702 174 175 30625 5359375 1382287566 5-5934447 175 176 30976 5451776 13.2664992 5-6040787 176 177 31329 5545233 13.3041347 5-6146724 177 178 31684 6639752 1933416641 5-6252263 178 179 32041 5735339 13-3790882 5-6357408 179 180 32400 5832000 13 4164079 5-6462162 180 181 32761 5929741 13-4536240 5-6566528 181 182 33124 6028568 13-4907376 5-6670511 182 183 33489 6128487 13-5277493 5-6774114 183 184 33856 6229504 13-5646600 5-6877340 184 185 34225 6331625 13-6014705 5'6980192 185 186 34596 6434856 13-6381817 5-7082675 186 187 34969 6539203 13-6747943 5-7184791 187 188. 35344 6644672 13-7113092 5-7286543 188 189 35721 6751269 13-7477271 5-7387936 189 190 36100 6859000 13-7840488 5-7488971 190 191 36481 6967871 13-8202750 5-7589652 191 192 36864 7077888 13-8564065 567689982 192 193 37249 7189097 13-8924440 5-7789966 193 194 37636 7301384 13-9283883 5-7889604 194 195 38025 7414875 13-9642400 5-7988900 195 196 38416 7529536 14-0 5-8087857 196 197 38809 7645373 14-0356688 5-8186479 197 198 39204 7762392 14-0712473 5-8284767 198 199 39601 7880599 14-1067360 5-8382725 199 200 40000 8000000 14-1421356 5-8480355 200 74 SQUARES, CUBES, Number.| Square. Cube. Square Root. Cube Root. Number. 201 40401 8120601 14*1774469 5 8577660 201 202 40804 8242408 14-2126701 5.8674643 202 203 41209 8365427 14-2478068 5-8771307 203 204 ]41616 8489664 14-2828569 5.8867653 204 205 42025 8615125 14-3178211 5.8963685 205 206 42436 8741816 14-3527001 5-9059406 206 207 42849 8869743 14-3874946 5.9154817 207 208 43264 8998912 14.4222051 5.9249921 208 209 43681 9129329 14-4568323 5-9344721 209 210 44100 9261000 14*4913767 5-9439220 210 211 44521 9393931 14-5258390 5-9533418 211 212 44944 9528128 14-5602198 5-9627320 212 213 45369 9663597 14-5945195 5-9720926 213 214 45796 9800344 14.6287388 5.9814240 214;-.215 46225 9938375 14-6628783 5-9907264 215 216 46656 10077696 14-6969385 6-0 216 217 47089 10218313 14-7309199 6G0092450 217 218 47524 10360232 I14.7648231 6-0184617 218 219 47961 10503459 14'7986486 6-0276502 219 220 48400 10648000 14-8323970 6-0368107 220 221 48841 10793861 14.8660687 6-0459435 221 222 49284 10941048 14.8996644 6-0550489 222 223 49729 11089567 14-9331845 6-0641270 223 224 50176 11239424 14-9666295 6-0731779 224 225 50625 11390625 15-0 6-0822020 225 226 51076 11543176 15-0332964 6-0911994 226 227 51529 11697083 15-0665192 6-1001702 227 228 51984 11852352 15-0996689 6-1091147 228 229 52441 12008989 15-1327460 6-1180332 229 230 -52900 12167000 15-1657509 6-1269257 230 231 53361 12326391 15.1986842 6-1357924 231 232 53824 12487168 15-2315462 6.1446337 232 233 54289 12649337 15.2643375 6.1534495 233 234 54756 12812904 15-2970585 6-1622401 234 235 55225 12977875 15-3297097 6-1710058 235 236 55696 13144256 15.3622915 6.1797466 236 237 56169 13312053 15-3948043 6-1884628 237 238 56644 13481272 15.4272486 6-1971544 238 239 57121 13651919 15-4596248 6-2058218 239 240 57400O 13824000 15.4919334 6.2144650 240 SQUARE ROOTS, AND CUBE ROOTS. 75 Number. Square. Cube. Square Root. Cube Root. Number. 241 58081 13997521 1555241747 6-2230843 241 242 58564 14172488 15-5563492 6-2316797 242 243 59049 14348907 15-5884573 6-2402515 243 244 59536 14526784 15-6204994 6 2487998 244 245 60025 14706125 1.5-6524758 6-2573248 245 246 60516 14886936 15-6843871 6-2658266 246 247 61009 15069223 15-7162336 6.2743054 247 248 61504 15252992 15-7480157 6-2827613 248 249 62001 15438249 15-7797338 6-2911946 249 250 62500 15625000 15.8113883 6.2996053 250 251 63001 15813251 15-8429795 6-3079935 251 252 63504 16003008 15-8745079 6.3163596 252 253 64009 16194277 15.9059737 6.3247035 253 254 64516 16387064 15'9373775 6 3330256 254 255 65025 16581375 15 9687194 6-3413257 255 256 65536 16777216 16-0 6-3496042 256 257 66049 16974593 16-0312195 6-3578611 257 258 66564 17173512 16-0623784 6 3660968 258 259 67081 17373979 16-0934769 6-3743111 259 260 67600 17576000 16-1245155 6-3825043 260 261 68121 17779581 16-1554944 6-3906765 -261 262 68644 17984728 16-1864141 6-3988279 262 263 69169 18191447 16-2172747 6-4069585 263 264 69696 18399744 16-2480768 6-4150687 264 265 70225 18609625 16 2788206 6*4231583 265 266 70756 18821096 16-3095064 6.4312276 266 267 71289 19034163 16-3401346 6.4392767 267 268 71824 19248832 16'3707055 6-4473057 268 269 72361 19465109 16-4012195 6 4553148 269 270 72900 19683000 16-4316767 6-4633041 270 271 73441 19902511 16-4620776 6'4712736 271 272 73984 20123648 16-4924225 6'4792236 272 273 74529 20346417 16-5227116 6'4871541 273 274 75076 20570824 16-5529454 6.4950653 274 275 75625 20796875 16-5831240 6-5029572 275 276 76176 21024576 166132477 6-5108300 276 277 76729 21253933 16-6433170 6-5186839 277 278 77284 21484952 16'6733320 6-5265189 278 279 77841 21717639 16-7032931 6'5343351 279 280 78400 21952000 16-7332005 6-5421326 280 76 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 281 78961 22188041 16 7630546 6 5499116 281 282 79524 2 2125768 1'*7928556 6-5576722 28 2 283 80089 22665187 16.8226038 6-5654144 283 284 806566 22906304 16.8522995 6'5731385 284 285 81225 23149125 16-8819430 6'5808443 285 286 81796 23393656 16-9115345 6-5885323 286 287 82369 23639903 16-9410743 6-5962023 287 288 82944 23887872 1.6-9705627 6'6038,545 288 289 83521 24137569 17-0 6-6114890 289 290 84100 24389000 17-0293864 6-6191060 290 291 84681 24642171 17.0587221 6'6267054 291 292 85264 24897088 17-0880075 6-6312874 292 293 85849 251.53757 17-1172428 6-64185'22 293 294 86436 25412184 17.1464282 6'6193998 294 295 87025 25672375 17.1755640 6-6569302 295 296 87616 25934336 17.204650.5 6-6644437 296 297 88209 26198073 17.2336879 6.6719403 297 298 88804 26463592 17-2626765 6-6794200 298 299 89401 26730899 17-2916165 6-6868831 299 300 90000 27000000 17-320o081 66943295 300 301 90601 27270901 17.3493516 6'7017593 301 302 91204 27543608 17.3781472 6-7091729 302 303 91809 27818127 17-4068952 6-7165700 303 304 92416 28094464 17'1355958 6-7239,508 304 305 93025 28372625 17-4642492 6-7313155 305 306 93636 28652616 17.4928557 6.7386641 306 307 94249 28934413 17-5214155 6-7459967 307 308 94864 29218112 17.5199288 6-7533134 308 309 95481 29503629 17-5783958 6'7606143 309 310 96100 29791000 17'6068169 6-7678995 310 311 96721 30080231 17.6351921 6-7751690 311 312 97344 30371328 17'6635217 6'7824229 312 313 97969 30664297 17'6918060 6-7896613 313 314 98596 30959144 17*7200451 6-7968844 314 315 99225 31255875 17-7482393 6-8040921 315 316 99856 31554496 17-7763888 6 8112847 316 317 100i89 31855013 17'8044938 6-8181620 317 318 101124 32157432 17.8325515 6 8256242 318 319 101761 32461759 17.8605711 6'8327714 319 320 102400 32768000 17 888-5438 6-8399037 320 SQUARE ROOTS, AND CUBE ROOTS. 77 Number. Square. Cube. Square Root. Cube Root. Number. 321 103041 33076161 17'9161729 6-8470213 321 322 103684 33386248 17'9443584, 6'8541240 322 323 104329 33698267 17'9722008 6'8612120 323 324 104976 34012224 18,0 6'8682855 324 325 105625 34328125 18o0277564 6 8753443 325 326 106276 34645976 18-0554701 6 88 3888 326 327 106929 34965783 18.0831413 6.8894188 327 328 107584 35287552 18-1107703 6-8964345 328 329 108241 35611289 18-1383571 6.9034359 329 330 108900 35937000 18-1659021 6-9104232 330 331 109561 36264691 18.1934054 6'9173964 331 332 110224 36594368 18.2208672 6'9243556 332 333 110889 36926037 18'2482876 6'9313008 333 334 111556 37259704 18'2756669 6'9382321 334 335 112225 37595375 18'3030052 6-9451496 335 336 112896 37933056 18-3303028 6.9520533 336 337 113569 38272753 18-3575598 6-9589434 337 338 114244 38614472 18-3847763 6-96581-98 338 339 114921 38958219 18-4119526 6.9726826 339 340 115600 39304000 18-4390889 6.9795321 340 341 116281 39651821 18-4661853 6-9863681 341 342 116964 40001688 18-4932420 6-9931906 342 343 117649 40353607 18'5202592 7'0 343 344 118336 40707584 18-5472370 7-0067962 344 345 119025 41063625 18'5741756 7-0135791 345 346 119716 41421736 18-6010752 7-0203490 346 347 120409 41781923 18-6279360 7-0271058 347 348 121104 42144192 18-6547581 7-0338497 348 349 121801 42508549 18-6815417 7-0405806 349 350 122500 42875000 18-7082869 7 0472987 350 351 123201 43243551. 18'7349940 7-0540041 351 352 123904 43614208 18'7616630 7-0606967 352 353 124609 43986977 18.7882942 7'0673767 353T 354 125316 44361864 18.8148877 7 0740440 354 355 126025 44738875 18'8414437 7-0806988 355 356 126736 45118016 18.8679623 7-0873411 356 357 127449 45499293 18.8944436 7.0939709 357 358 128164 45882712 18-9208879 7-1005885 358 359 128881 46268279 18-9472953 7-1071937 359 360 129600 46656000 18-9736660 7-1137866 360 7188 78 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 361 130321 47045881 19-0 7-1203674 361 362 131044 47437928 19-0262976 7'1269360 362 363 131769 47832147 19-0525589 7*1334925 363 364 132496 48228544 19-0787840 7-1400370 364 365 133225 48627125 19-1049732 7'1465695 365 366 133956 49027896 19-1311265 7-1530901 366 367 134689 49430863 19-1572441 7-1595988 367 368 135424 49836032 19-1833261 7-1660957 368 369 136161 50243409 19-2093727 7-1725809 369 370 136900 50653000 19-2353841 7-1790544 370 371 137641 51064811 19'2613603 7-1855162 371 372 138384 51478848 19-2873015 7.1919663 372 373 139129 51895117 19.3132079 7.1984050 373 374 139876 52313624 19.3390796 7-2048322 374 375 140625 52734375 19-3649167 7-2112479 375 376 141376 53157376 19-3907194 7-2176522 376 377 142129 53582633 19-4164878 7 2240450 377 378 142884 54010152 19.4422221 7-2304268 378 379 143641 54439939 19-4679223 7-2367972 379 380 144400 54872000 19-4935887 7-2431565 380 381 145161 55306311 19.5192213 7'2495045 381 382 ]145924 55742968 1965448203 7 2558415 382 3883 146689 56181887 19-5703858 7.2621675 383 384 147456 56623104 19.5959179 7'2684824 384 385 148225 57066625 19.6214169 7-2747864 385 386 148996 57512456 19'6468827 7-2810794 386 387 149769 57960603 19-6723156 7 2873617 387 388 150544 58411072 19-6977156 7'2936330 388 389 151321 58863869 19 7230829 7.2998936 389 390 152100 59319000 19.7484177 7 3061436 390 391 152881 59776471 19'7737199 7-3123828 391 392 153664 60236288 19.7989899 7.3186114 392 393 154449 60698457 19.8242276 7.3248295 393 394 155236 61162984 19-8494332 7-3310369 394 395 156025 61629875 19.8746069 7-3372339 395 396 156816 62099136 19'8997 187 7 3434205 396 397 157609 62570773 19-9248583 7.3495966 397 398 158404 63045792 19-9499373 7'3557624 398 399 159201 6352 1199 19-9749844 7.3619178 399 400 160000 64000000 20-0 1 73680630 400 SQUARE ROOTS, AND CUBE ROOTS. 79 Number. Square. Cube. Square Root. Cube Root. Number. 401 160801 64481201 20.0219844 7-3721979 401 402 161604 64964808 20-0499377 7.3803227 402 403 162409 65450827 20-0748599 7-3864373 403 404 163216 65939264 20-0997512 7-3925419 404 405 164025 66430125 20-1246118 7 3986363 405 406 164836 66923416 20-1494417 7 4047206 406 407 165649 67419143 20-1742410 7-4107950 407 408 166464 67917312 20-1990099 7-4168595 408 409 167281 68417929 20-2237484 7-4229142 409 410 168100 68921000 20-2484567 7-4289589 410 411 168921 69426531 20-2731349 7-4349938 411 412 169744 69934528 20-2977831 7-4410189 412 413 170569 70444997 20-3224014 7-4470342 413 414 171396 70957944 20-3469899 7 4530399 414 415 172225 71473375 20'3715488 7 4590359 415 416 173056 71991296 20'3960781 7*4650223 416 417 173889 72511713 20-4205779 7-4709991 417 418 174724 73034632 20'4450483 7'4769664 418 419 175561 73560059 20'4694895 7'4829242 419 420 176400 74088000 20-4939015 7 4888724 420 421 177241 74618461 20-5182845 7'4948113 421 422 178084 75151448 20-5426486 7 5007406 422 423 178929 75686967 20-5669638 7-5066607 423 424 179776 76225024 20-5912603 7-5125715 424 425 180625 76765625 20-6155281 75184730 425 426 181476 77308776 20-6397674 7'5243652 426 427 182329 77854483 20-6639783 7-5302482 427 428 183184 78402752 20-6881609 7-5361221 428 429 184041 78953589 20'7123152 7'5419867 429 430 184900 79507000 20-7364414 7 5478423 430 431 185761 80062991 20'7605395 7*5536888 431 432 186624 80621568 20-7846097 7'5595263 432 433 187489 81182737 20-8086520 7 5653548 433 434 188356 81746504 20'8326667 7-5711743 434 435 189225 82312875 20-8566336 7-5769849 435 436 190096 82881856 20-8806130 7-5827865 436 437 190969 83453453 20-9045450 7-5885793 437 438 191844 84027672 20-9284495 7-5943633 438 439 192721 84604519 20-9523268 7-6001385 439 440 193600 85184000 20-9761770 7'6059049 440 80 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 441 194481 85766121 21.0 7.6116626 441 442 195364 86350888 21.0237960 7.6174116 442 443 196249 86938307 21-0475652 7.6231519 443 444 197136 87528384 21.0713075 7.6288837 444 445 198025 88121125 21.0950231 7 6346067 445 446 198916 88716536 21-1187121 7-6403213 446 447 199809 89314623 21-1423745 7-6460272 447 448 200704 89915392 21-1660105 7-6517247 448 449 201601 90518849 21-1896201 7.6574138 449 450 202500 91125000 21-2132034 7.6630943 450 451 203401 91733851 21.2367606 7-6687665 451 452 204304 92345408 21.2602916 7-6744303 452 453 205209 92959677 21'2837967 7'6800857 453 454 206116 93576664 21-3072758 7'6857328 454 455 207025 94196375 21-3307290 7-6913717 455 456 207936 94818816 21'3541565 7-6970023 456 457 208849 95443993 21-3775583 7-7026246 457 458 209764 96071912 21-4009346 7-7082388 458 459 210681 96702579 21-4242853 7-7138448 459 460 211600 97336000 21-4476106 7'7194426 460 461 212521 97972181 21-4709106 7*7250325 461 462 213444 98611128 21-4941853 7-7306141 462 463 214369 99252847 21'5174348 7-7361877 463 464 215296 99897344 21 -406592 7-7417532 464 465 216225 100544625 21.5638587 7-7473109 465 466 217156 101194696 21-5870331 7-7528606 466 467 218089 101847563 21-6191828 7-7584023 467 468 219024 102503232 21-6333077 7-7639361 468 469 219961 103161709 21'6564078 7'7694620 469 470 220900 103823000 21-6794834 7-7749801 470 471 221841 104487111 21-7025344 7'7804904 471 472 222784 105154048 21'7255610 T7859928 472 473 223729 105823817 21-7485632 7-7914875 473 474 224676 106496424 21-7715411 7-7969745 474 475 225625 107171875 21-7944947 78024538 475 476 226576 107850176 21-8174242 7.8079244 476 477 227529 108531333 21-8403297 7.8133892 477 478 228484 109215352 21-8632111 7 8188156 478 479 229441 109902239 21-8860686 7-8242942 479 480 230400 110592000 21-9089023 7-8297353 480 SQUARE ROOTS, AND CUBE ROOTS. 81. Number. Square. Cube. Square Root. Cube Root. Number. 481 231361 111284641 21.9317122 7-8351688 481 482 232324 111980168 21-9544984 7-8405949 482 483 233289 112678587 21-9772610 7-8460134 483 484 234256 113379904 22.0 7-8514244 484 485 235225 114084125 22-0227155 7-8568281 485 486 236186 114791256 22-0454077 v7-8622242 486 487 237169 115501303 22-0680765 7 8676130 487 488 238144 116214272 22-0907220 7-8729944 488 489 239131 116930169 22-1133444 7-8783684 489 490 240120 117649000 22-1359436 7-8837352 490 491 241081 118370771 22-1585198 7-8890946 491 492 242064 119095488 22-1810730 7'8944468 492 493 243049 119823157 22-2036033 7-8997917 493 494 244036 120553784 22-2261108 7-9051294 494 495 245025 121287375 22-2485955 7-9104599 495 496 246016 122023936 22-2710575 7'9157832 496 497 247009 122763473 22-2934968 7'9210994 497 498 248004 123505992 22-3159136 7-9264085 498 499 249001 124251499 22-3383079 7-9317104 499 500 250000 125000000 22'3606798 7-9370053 500 501 251001 125751501 22-3830293 7-9422932 501 502 252004 126406008 22-4053565 7.9475739 502 503 253009 127263507 22-4276615 7-9528477 503 504 254016 128024064 22-4499443 7-9581144 504 505 255025 128787625 22'4722051 7-9633743 505 506 256036 129554216 22'4944438 7-9686271 506 507 257049 130323843 22-5166605 7-9738731 507 508 258064 131096512 22 5388553 7-9791122 508 509 259081 131872229 22-5610283 7-9843444 509 510 260100 132651000 22-5831796 7-9895697 510 511 261121 133432831 22-6053091 7'9947883 511 512 262144 134217728 22-6274170 8-0 512 513 263169 135005697 22.6495033 8-0052049 513 514 264196 135796744 22*6715681 8-0104032 514 515 265225 136590875 22-6936114 8-0155946 515 516 266256 137388096 22-7156334 8.0207794 516 517 267289 138188413 22-7376340 8-0259574 517 518 268324 138991832 22-7596134 8-0311287 518 519 269361 139798359 22-7815715 8-0362935 519 520 270400 140608000 22-8035085 8-0414-515 520 82 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 521 271441 141420761 22-8254244 8-0466030 501 522 272484 142236648 22'8473193 8'0517479 522 523 273529 143055667 22'8691933 8'0568862 523 524 274576 143877824 22-8910463 8-0620180 524 525 275625 144703125 22-9128785 8-0671432 525 526 276676 145531576 22-9346899 8'0722620 526 527 277729 146363183 22-9564806 8'0773743 527 528 278784 147197952 22-9782506 8-0824800 528 529 279841 148035889 23-0 8'0875794 529 530 280900 148877000 23-0217289 8-0926723 530 531 281961 149721291 23-0434372 8-0977589 531 532 283024 150568768 23-0651252 8-1028390 532 533 284089 151419437 23-0867928 8-1079128 533 534 285156 152273304 23 1084400 8'1129803 534 535 286225 153130375 23-1300670 8-1180414 535 536 287296 153990656 23'1516738 8-1230962 536 537 288369 154854153 23-1732605 8-1281447 537 538 289444 155720872 23 1948270 8-1331870 538 539 290521 156590819 23'2163735 8'1382230 539 540 291600 157464000 23-2379001 8'1432529 540 541 292681 158340421 23-2594067 8'1482765 541 542 293764 159220088 23-2808935 8-1532939 542 543 294849 160103007 23-3023604 8'1583051 543 544 295936 160989184 23-3238076 8-1633102 544 545 297025 161878625 23-3452351 8-1683092 545 546 298116 162771336 23-3666429 8-1733020 546 647 299209- 163667323 23-3880311 8'1782888 547 548 300304 164566592 23-4093998 8'1832695 548 549 301401 165469149 23'4307490 8-1882441 549 550 302500 166375000 23-4520788 8-1932127 550 551 303601 167284151 23 4733892 8:1981753 551 552 304704 168196608 23.4946802 8-2031319 552 553 305809 169112377 23 5159520 8.2080825 553 554 306916 170031464 23.5372046 8-2130271 554 555 308025 170953875 23 5584380 8-2179657 555 556 309136 171879616 23.5796-522 8.2228985 556 557 310249 172808693 23-6008474 8-2278254 557 558 311364 173741112 23-6220236 8-2327463 558 559 312481 174676879 23-6431808 8-2376614 559 560 313600 175616000 23-6643191 8-2425706 560 SQUARE ROOTS, AND CUBE ROOTS. 83 Number.I Square. Cube. Square Root. Cube RMot. Number. 561 314721 176558481 23-6854386 8 2474740 561 562 315844 177504328 23-7065392 8'2523715 562 563 316969 178453547 23-7276210 8.2572633 563 564 318096 179406144 23-7486842 8.2621492 564 565 319225 180362125 23.7697286 8'2670294 565 566 320356 181321496 23'7907545 8'2719039 566 567 321489 182284263 23-8117618 8-2767726 567 568 322624 183250432 23-8327506 8-2816355 568 569 323761 184220009 23-8537209 8 2864928 569 570 324900 185193000 23-8746728 8-2913444 570 571 326041 186169411 23-8956063 8-2961903 571 572 327184 187149248 23'9165215 8'3010304 572 573 328329 188132517 23-9374184 8-3058651 573 574 329476 189119224 2389582971 8'3106941 574 575 330625 190109375 23'9791576 8-3155175 575 576 331776 181102976 24'0 8-3203353 576 577 332929 182100033 24'0208243 8-3251475 577 578 334084 183100552 24-0416306 8-3299542 578 579 335241 184104539 24-0624188 8-3347553 579 580 336400 195112000 24 0831891 8-3895509 580 581 337561 196122941 24-1039416 8-3443410 581 582 338724 197.137368 24-1246762 8-3491256 582 583 339889 198155287 24-1453929 8-3539047 583 584 341056 199176704 24.1660919 8-3586784 584 585 342225 200201625 24-1867732 8-3634466 585 586 343396 201230056 24-2074369 8-3682095 586 587 344569 202262003 24-2280829 8-3729668 587 588 345744 203297472 24-2487113 8.3777188 588 589 346921 204336469 24-2693222 8-3824653 589 690 348100 205379000 24-2899156 8-3872065 590 591 349281 206425071 24'3104916 8-3919423 591 592 350464 207474688 24-3310501 8-3966729 592 593 351649 208527857 24-3515913 8-4013981 593 594 352836 209584584 24-3721152 8-4061180 594 595 354025 210644875 24,3926218 84108326 695 596 355216 211708736 24.4131112 854155419 596 597 356409 212776173 24.4335834 8-4202460 597 598 357604 213847192 24.4540385 8-4249448 598 599 358801 214921799 24-4744765 8-4296383- 599 600 360000 216000000 24-4948974 8'4343267 600 84 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. I- 36120 —— I 271 - 601 361201 217081801 24-5153013 8.4390098 601 602 362404 218167208 24-5356883 8 4436877 602 603 363609 219256227 24-5560583 8 4483605 603 604 364816 220348864 24-5764115 8-4530281 604 605 366025 221445125 24-5967478 8-4576906 605 606 367236 222545016 24-6170673 8-4623479 606 607 368449 223648543 24-6373700 8 4670000 607 608 369664 224755712 24-6576560 8-4716471 608 609 370881 225866529 24-6779254 8-4762892 609 610 372100 226981000 24 6981781 8-4809261 610 611 373321 228099131 24.7184142 8.4855579 611 612 374544 229220928 24.7386338 8.4901848 612 613 375769 230346397 24.7588368 8'4948065 613 614 376996 231475544 24.7790234 8.4994233 614 615 378225 232608375 24 7991935 8 5040350 615 616 379456 233744896 24-8193473 8-5086417 616 617 380689 234885113 24-8394847 8-5132435 617 618 381924 236029032 24-8596058 8.5178403 618 619 383161 237176659 24-8797106 8-5224321 619 620 384400 238328000 24-8997992 8-5270189 620 621 385641 239483061 24-9198716 8-5316009 621 622 386884 240641848 24-9399278 8-5361780 622 623 388129 241804367 24-9599679 8'5407501 623 624 389376 242970624 24-9799920 8-5453173 624 625 390625 244140625 250 85498797 625 626 391876 245314376 25-0199920 8-5544372 626 627 393129 246491883 25-0399681 8.5589899 627 628 394384 247673152 25-0599282 8.5635377 628 629 395641 248858189 25.0798724 8'5680807 629 630 396900 250047000 25-0998008 8'5726189 630 631 398161 251239591 25.1197134 8.5771523 631 632 399424 252435968 25-1396102 8.5816809 632 633 400689 253636137 25.1594913 8'5862047 633 634 401956 254840104 25-1793566 8 5907238 634 635 403225 256047875 25 1992063 8.5952380 635 636 404496 257259456 25-2190404 8-5997476 636 637 405769 258474853 25'2388589 8-6042525 637 638 407044 259694072 25-2586619 8.6087526 638 639 408321 260917119 25-2784493 8-61.32480 639 640 409600 1262144000 2-52982213 8-6177388 640 SQUARE ROOTS, AND CUBE ROOTS. 85 Number. Square. - Cube. Square Root. Cube Root. Number. 641 410881 263374721 25*3179778 8 6222248 641 642 412164 264609288 25-3377189 8'6267063 642 643 413449 265847707 25-3574447 8-6311830 643 644 414736 267089984 25-3771551 8-6356551 644 645 416025 268336125 25-3968502 8-6401226 645 646 417316 269586136 25-4165301 8.6445855 646 647 418609 270840023 25'4361947 8-6490437 647 648 419904 272097792 25-4558441 8-6534974 648 649 421201 273359449 25-4754784 8'6579465 649 650 422500 274625000 25'4950976 8-6623911 650 651 423801 275894451 25'5147016 8-6668310 651 652 425104 277167808 25.5342907 8-6712665 652 653 426400 278445077 25-5538647 8.6756974 653 654 427716 279726264 25-5734237 8-6801237 654 655 429025 281011375 25-5929678 8 6845456 655 656 430336 282300416 25-6124969 8-6889630 656 657 431649 283593393 25'6320112 8-6933759 657 658 432964 284890312 25-6515107 8-6977843 658 659 434281 286191179 25-6709953 8-7021882 659 660 435600 287496000 25.6904652 8-7065877 660 661 436921 288804781 25*7099203 8-7109827 661 662 438244 290117528 25'7293607 8-7153734 662 663 439569 291434247 25'7487864 8-7197596 663 664 440896 292754944 25'7681975 8-7241414 664 665 442225 294079625 25-7875939 8-7285187 665 666 443556 295408296 25'8069758 8'7328918 666 667 444889 296740963 25-8263431 8-7372604 667 668 446224 298077632 25-8456960 8-7416246 668 669 447561 299418309 25-8650343 8-7459846 669 670 448900 300763000 25 8843582 8 7503401 670 671 450241 302111711 25-9036677 8-7546913 671 672 451584 303464448 25-9229628 8-7590383 672 673 452929 304821217 25*9422435 8-7-633809 673 674 454276 306182024 25'9615100 8'7677192 674 675 455625 307546875 25-9807621 8-7720532 675 676 456976 308915776 26-0 8'7763830 676 677 458329 310288733 26-0192237 8'7807084 677 678 459684 311665752 26'0384331 8-7850296 678 679 461041 313046839 26-0576284 8-7893466 679 680 462400 314432000 26-0768096 8-7936593 680 8 86 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 681 463761 315821241 26-0959767 8-7979679 681 682 465124 317214568 26-1151297 8-8022721 682 683 466489 318611987 26.1342687 8-8065722 683 684 467856 820013504 26-1533937 8-8108681 684 685 469225 321419125 26 1725047 8-8151598 685 686 470596 322828856 26-1916017 8-8194474 686 687 471969 324242703 26-2106848 8-8237307 687 688 473344'325660672 26-2297541 8-8280099 688 689 474721 327082769 26-2488095 8-8322850 689 690 476100 328509000 26-2678511 8-8365559 690 691 477481 329939371 26-2868789 8-8408227 691 692 478864 331373887 26.3058929 8-8450854 692 693 480249 332812557 26.3248932 8-8493440 693 694 481636 334255384 26.3438797 8-8535985 694 695 483025 335702375 26.3628527 8-8578489 695 696 484416 337153536 26.3818119 8.8620952 696 697 485809 338608873 26-4007576 8-8663375 697 698 487204 340068392 26.4196896 8.8705757 698 699 488601 341532099 26.4386081 8.8748099 699 700 490000 343000000 26-4575131 8-8790400 700 701 491401 344472101 26 4764046 8.8832661 701 702 492804 345948408 26.4952826 8-8874882 702 703 494209 347428927 26.5141472 8-8917063 703 704 495616 348913664 26.5329983 8.8959204 704 705 497025 350402625 26 5518361 8'9001304 705 706 498436 351895816 26'5706605 8.9043366 706 707 499849 353393243 26.5894716 8'9085387 707 708 501264 354894912 26-6082694 8.9127369 708 709 502681 356400829 26 6270539 8*9169311 709 710 504100 357911000 26'6458252 8-9211214 710 711 505521 359425431 26-6645833 8.9253078 711 712 506944 360944128 26-6833281 8 9294902 712 713 508369 362467097 26-7020598 8.9336687 713 714 509796 363994344 26'7207784 8.9378433 714 715 511225 365525875 26o7394839 8 9420140 715 716 512656 367061696 26-7581763 8-9461809 71 6 717 514089 368601813 26'7768557 8-9503438 717 718 515524 370146232 26-7955220 8-9545029 718 719 516961 371694959 26-8141754 8-9586581 719 720 618400 373248000 26-8328157 8-9628095 720 SQUARE ROOTS, AND CUBE ROOTS. 87 Number. Square. Cube. Square Root. Cube Root. Number. 721 519841 374805361 26-8514432 8*9669570 721 722 521284 376367048 26-8700577 8-9711007 722 723 522729 377933067 26*8886593 8-9752406 723 724 524176 379503424 26-9072481 8'9793766 724 725 525625 381078125 26-9258240 8-9835089 725 726 527076 382657176 26-9443872 8-9876373 726 727 528529 384240583 26-9629375 8-9917620 727 728 629984 385828352 26.9814751 8-9958829 728 729 531441 387420489 27-0 9.0 729 730 532900 389017000 27-0185122 9-0041134 730 731 534361 390617891 27-0370117 9-0082229 731 732 535824 392223168 27-0554985 9-0123288 732 733 537289 393832837 27-0739727 9-0164309 733 734 538756 395446904 27-0924344 9-0205293 734 735 540225 397065375 27-1108834 9 0246239 735 736 541696 398688256 27-1293199 9-0287149 736 737 543169 400315553 27.1477439 9 0328021 737 738 544644 401947272 27-1661554 9 0368857 738 739 546121 403583419 27-1845544 9 0409655 739 740 547600 405224000 27-2029410 9-0450417 740 741 549081 406869021 27'2213152 9'0491142 741 742 550564 408518488 27-2396769 9 0531831 742 743 552049 410172407 27-2580263 9 0572482 743 744 553536 411830784 27-2763634 9-0613098 744 745 655025 413493625 27-2946881 9-0653677 745 746 556516 415160936 27-3130006 9-0694220 746 747 558009 416832723 27-3313007 9-0734726 747 748 559504 418508992 27*3495887 9'0775197 748 749 561001 420189749 27-3678644 9-0815631 749 750 562500 421875000 27-3861279 9-0856030 750 751 564001 423564751 27'4043792 9-0896392 751 752 565504 425259008 27 4226184 9 0936719 752 753 567009 426957777 27.4408455 9.0977010 753 754 568516 428661064 27-4590604 9-1017265 754 755 570025 430368875 27-4772633 9-1057485 755 756 571536 432081216 27 4954542 9 1097669 756 767 573049 433798093 -27*5136330 9-1137818 757 758 574564 435519512 27-5317998 9-1177931 758 759 576081 437245479 2765499546 9.1218010 759 760 577600 438976000 27-5680976 9-1258053 760 88 SQUARhS, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 761 579121 440711081 27.5862284 9-1298061 761 762 580644 442450728 27.6043475 9-1338034 762 763 582169 444194947 27.6224546 9-1377971 763 764 583696 445943744 27-6405499 9-1417874 764 765 585225 447697125 27 6586334 9-1457742 765 766 586756 449455096 27-6767050 9-1497576 766 767 588289 451217663 27-6947648 9 1537375 767 768 589824 452984832 27-7128129 9-1577139 768 769 591361 454756609 27.7308492 9-1616869 769 770 592900 456533000 27'7488730 9-1656565 770 771 594441 458314011 27*7668868 9-1696225 771 772 595984 460099648 27'7848880 9'1735852 772 773 597529 461889917 27'8028775 9'1775445 773 774 599076 463684824 27'8208555 9.1815003 774 775 600625 465484375 27-8388218 9-1854527 775 776 602176 467288576 27'8567766 9'1894018 776 777 603729 469097433 27-8747197 931933474 777 778 605284 470910952 27.8926514 9'1972897 778 779.606841 472729139 27.9105715 9-2012286 779 780 608400 474552000 27'9284801 9'2051641 780 781 609961 476379541 27-9463772 9*2090962 781 782 611524 478211768 27-9642629 9*2130250 782 783 613089 480048687 27'9821372 9'2169505 783 784 614656 481890304 28-0 9'2208726 784 785 616225 483736625 28'0178515 9'2247914 785 786 617796 485587656 28'0356915 9'2287068 786 787 619369 487443403 1 28-0535203 9-2326189 787 788 620944 489303872 28-0713377 9-2365277 788 789 622521 491169069 28-0891438 9-2404333 789 790 624100 493039000 28-1069386 9-2443355 790 791 625681 494913671 28*1247222 9 2482344 791 792 627264/ 496793088 28-1424946 9-2521300 792 793 628849 498677257 28'1602557 9.2560224 793 794 630436 500566184 28'1780056 9-2599114 794 795 632025 502459875 28*1957444 9'2637973 795 796 633616 504358336 28'2134720 9-2676798 796 797 635209 506261573 28-2311884 9-2715592 797 798 636804 508169592 28'2488938 9'2754352 798 799 638401 510082399 28-2665881 9-2793081 799 800 640000 512000000 28-2842712 9-2831777 800 SQUARE ROOTS, AND CUBE ROOTS. 89 Number. Square. Cube. Square Root. Cube Root. Number. 801 641601 513922401 28-3019434 9-2870440 801 802 643204 515849608 28'3196045 9-2909072 802 803 644809 517781627 28'3372546 9-2947671 803 ) 804 646416 519718464 28-3548938 9-2986239 804 805 648025 521660125 28-3725219 9 3024775 805 806 649636 523606616 28*3901391 9 3063278 806 807 651249 525557943 28-4077454 9-3101750 807 808 652864 527514112 28-4253408 9-3140190 808 809 654481 529475129 28-4429253 9'3178599 809 810 656100 531441000 28-4604989 9-3216975 810 811 657721 533411731 28-4780617 9-3255320 811 812 6.59944 535387328 28-4956137 9-3293634 812 813 660969 537367797 28-5131549 9-3331916 813 814 662596 539353144 28-5306852 9-3370167 814 815 664225 541343375 28 5482048 9-3408386 815 816 665856 543338496 28-5657137 9-3446575 816 817 667489 545338513 28-5832119 9-3484731 817 818 669124 547343432 28-6006993 9'3522857 818 819 670761 549353259 28-6181760 9-3560952 819 820 672400 551368000 28-6356421 9-3599016 820 821 674041 553387661 28'6530976 9'3637049 821 822 675684 555412248 28-6705424 9-3675051 822 823 677329 557441767 28 6879766 9-3713022 823 824 678976 559476224 28-7054002 9-3750963 824 825 680625 561515625 28-7228132 9-3788873 825 826 682276 563559976 28'7402157 9 3826752 826 827 683929 565609283 28-7576077 9-3864600 827 828 685584 567663552 28-7749891 9-3902419 828 829 687241 569722789 28-7923601 9-3940206 829 830 688900 571787000 28-8097206 9-3977964 830 831 690561 573856191 28'8270706 9-4G15691 831 832 692224 575930368 28-8444102 9'4053387 832 833 693889 578009537 28-8617394 9 4091054 833 834 695556 580093704 28-8790582 9'4128690 834 835 697225 582182875 28s8963666 904166297 835 836 698896 584277056 28'9136646 9-4203873 836 837 700569 586376253 28-9309523 9.4241420 837 838 702244 588480472 28-9482297 9-4278936 838 839 703921 590589719 28.9654967 9.4316423 839 840 705600 592704000 28 9827535 9 4353880 840 s~o~o~oo~~o~oo 90 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 841 707281 594823321 29-0 9 4391307 841 842 708964 596947688 29'0172363 9 4428704 842 843 710649 599077107 29-0344623 9-4466072 843 844 712336 601211584 29-0516781 9-4503410 844 845 714025 603351125 29-0688837 9'4540719 845 846 715716 605495736 29-0860791 9'4577999 846 847 717409 607645423 29-1032644 9 4615249 847 848 719104 609800192 29-1204396 9-4652470 848 849 720801 611960049 29-1376046 9-4689661 849 850 722500 614125000 29'1547595 9-4726824 8-50 851 724201 616295051 29-1719043 9-4763957 851 852 725904 618470208 29-1890390 9-4801061 852 853 727609 620650477 29-2061637 9-4838136 853 854 729316 622835864 29'2232784 9-4875182 854 855 731025 625026375 29-2403830 9'4912200 855 856 732736 627222016 29-2574777 9-4949188 856 857 734449 629422793 29-2745623 9-4986147 857 858 736164 631628712 29-2916370 9-5023078 858 859 737881 633839779 29'3087018 9-5059980 859 860 739600 636056000 29-3257566 9-5096854 860 861 741321 638277381 29'3428015 9'5133699 861 862 743044 640503928 29-3598365 9-5170515 862 863 744769 642735647 29'3768616 9-5207303 863 864 746496 644972544 29-3938769 9-5244063 864 865 748225 647214625 29-4108823 9-5280794 865 866 749956 649461896 29-4278779 9'5317497 866 867 751689 651714363 29-4448637 9-5354172 867 868 753424 653972032 29-4618397 9-5390818 868 869 755161 656234909 29'4788059 9-5427437 869 870 756900 658503000 29-4957624 9 5464027 870 871 758641 660776311 29-5127091 9 5500589 871 872 760384 663054848 29-5296461 9-5537123 872 873 762129 665338617 29'5465734 9-5573630 873 874 763876 667627624 29-5634910 9-5610108 874 875 765625 669921875 29'5803989 9'5646559 875 876 767376 672221376 29-5972972 9-5682982 876 877 769129 674526133 29-6141858 9-5719377 877 878 770884 676836152 29-6310648 9-5755745 878 879 772641 679151439 29-6479342 9-5792085 879 880 774400 681472000 29-6647939 9-5828397 880 SQUARE ROOTS, AND CUBE ROOTS. 91 Number. Square. Cube. Square Root. Cube Root. Number. 881 776161 683797841 29'6816442 9-5864682 881 882 777924 686128968 29'6984848 9 5900939 882 883 779689 688465387 29'7153159 9'5937169 883 884 781456 690807104 29'7321375 9'5973373 884 885 783225 693154125 29'7489498 9'6009548 885 886 784996 695506456 29'7657521 9'6045696 886 887 786769 697864103 29'7825452 9'6081817 887 888 788544 700227072 29'7993289 9'6117911 888 889 790321 702595369 29'8161030 9'6153977 889 890 792100 704969000 29'8328676 9'6190017 890 891 793881 707347971 29'8496231 9-6226030 891 892 795664 709732288 29-8663690 9-6262016 892 893 797449 712121957 29-8831056 9'6297975 893 894 799236 714516984 29-8998328 9-6333907 894 895 801025 716917375 29 9165506 9-6369812 895 896 802816 719323136 29'9332591 9-6405690 896 897 804609 721734273 29-9499583 9-6441542 897 898 806404 724150792 29-9666481 9-6477367 898 899 808201 726572699 29-9833287 9-6513166 899 900 810000 729000000 30-0 9-6548938 900 901 811801 731432701 30-0166620 9-6584684 901 902 813604 733870808 30-0333148 9-6620403 902 903 815409 736314327 30.0499584 9-6656096 903 904 817216 738763264 30-0665928 9-6691762 904 905 819025 741217625 30.0832179 9-6727403 905 906 820836 743677416 30.0998339 9-6763017 906 907 822649 746142643 30-1164407 9*6798604 907 908 824464 748613312 30-1330383 9-6834166 908 909 826281 751089429 30-1496269 9-6869701 909 910 828100 753571000 30-1662063 9-6905211 910 911 829921 756058031 30'1827765 9-6940694 911 912 831744 758550528 30-1993377 9-6976151 912 913 833569 761048497 30-2158899 9.7011583 913 914 835396 763551944 30.2324329 9-7046989 914 915 837225 766060875 30-2489669 9-7082369 915 916 839056 768575296 30.2654919 9-7117723 916 917 840889 771095213 30-2820079 9-7153051 917 918 842724 773620632 30-2985148 9-7188354 918 919 844561 776151559 30-3150128 9-7223631 919 920 846400 778688000 30-3315018 9'7258883 920 92 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 921 848241 781229961 30-3479818 9-7294109 921 922 850084 783777448 30-3644529 9'7329309 922 923 851929 786330467 30-3809151 9-7364484 923 924 853776 788889024 30-3973683 9-7399634 924 925 855625 791453125 30-4138127 9 7434758 925 926 857476 794022776 30*4302481 9-7469857 926 927 859329 796597983 30.4466747 9-7504930 927 928 861184 799178752 30-4630924 9.7539979 928 929 863041 801765089 30-4795013 9-7575002 929 930 864900 804357000 30 4959014 9 7610001 930 931 866761 806954491 30'5122926 9-7644974 931 932 868624 809557568 30-5286750 9-7679922 932 933 870489 812166237 30-5450487 9'7714845 933 934 872356 814780504 30-5614136 9 7749743 934 935 874225 817400375 30-5777697 9'7784616 935 936 876096 820025856 30'5941171 9'7819466 936 937 877969 822656953 30-6104557 9-7854288 937 938 879844 825293672 30-6267857 9-7889087 938 939 881721 827936019 30-6431069 9'7923861 939 940 883600 830584000 30-6594194 9-7958611 940 941 885481 833237621 30-6757233 9'7993336 941 942 887364 835896888 30-6920185 9'8028036 942 943 889249 838561807 30-7083051 9-8062711 943 944 891136 841232384 30-7245830 9'8097362 944 945 893025 843908625 30-7408523 9-8131989 945 946 894916 846590536 30-7571130 9-8166591 946 947 896809 849278123 30-7733651 9-8201169 947 948 898704 851971392 30-7896086 9*8235723 948 949 900601 854670349 30-8058436 9*8270252 949 950 902500 857375000 30-8220700 9-8304757 950 951 904401 860085351 30'8382879 9'8339238 951 952 906304 862801408 30-8544972 9-8373695 952 953 908209 865523177 30-8706981 9-8408127 953 954 910116 868250664 30'8868904 9-8442536 954 955 912025 870983875 30'9030743 9'8476920 955 956 913936 873722816 30'9192497 9'8511280 956 957 915849 876467493 30'9354166 9'8545617 957 958 317764 879217912 30'9515751 9-8579929 958 959 919681 881974079 30-9677251 9-8614218 959 960 921600 884736000 30 9838668 9-8648483 960 SQUARE ROOTS, AND CUBE ROOTS. 93 Number. Square. Cube. Square Root. Cube Root. Number. 961 923521 887503681 31-0 9 8682724 961 962 925444 890277128 31'0161248 9-8716941 962 963 927369 893056347 31-0322413 9-8751135 963 964 929296 895841344 31-0483494 9-8785305 964 965 931226 898632125 31-0644491 9-8819451 965 966 933156 901428696 31-0805405 9-8853574 966 967 935089 904231063 31-0966236 9 8887673 967 968 937024 907039232 31-1126984 9-8921749 968 969 938961 909853209 31-1287648 9'8955801 969 970 940900 912673000 31-1448230 9-8989830 970 971 942841 915498611 31'1608729 9'9023835 971 972 944784 918330048 31-1769145 9-9057817 972 973 946729 921167317 31"1929479 9-9091776 973 974 948676 924010424 31'2089731 9-9125712 974 975 950v25 926859375 31.2249900 9-9159624 975 976 952576 929714176 31-2409987 9.9193513 976 977 954529 932574833 31-2569992 9.9227379 977 978 956484 935441352 31-2729915 9-9261222 978 979 958441 938313739 31.2889757 9.9295042 979 980 960400 941192000 31-3049517 9.9328839 980 981 962361 944076141 31'3209195 9-9362613 981 982 964324 946966168 31-3368792 9-9396363 982 983 966289 949862087 31'3528308 9'9430092 983 984 968256 952763904 31-3687743 9-9463797 984 985 970225 955671625 31-3847097 9-9497479 985 986 972196 958585256 31'4006369 9-9531138 986 987 974169 961504803 31-4165561 9-9564775 987 988 976144 964430272 31-4324673 9-9598389 988 989 978121 967361669 31-4483704 9-9631981 989 990 980100 970299000 31-4642654 9-9665549 990 991 982081 973242271 31'4801525 9*9699095 991 992 984064 976191488 31'4960315 9-9732619 992 993 986049 979146657 31-5119025 9-9766130 993 994 988036 982107784 31-5277655 9-9799599 994 995 990025 985074875 31-5436206 9 9833055 995 996 992016 988047936 31-5594677 9*9866488 996 997 994009 991026971 31-5753068 9'9899900 997 998 996004 994011992 31-5911380 9-9933289 998 999 1998001 997002999 31-6069613 9-9966656 999 1000 1000000 1000000000 31-6227766 10'0 1000 94 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 1001 1002001 1003003001 31-6385840 10*0033322 1001 1002 1004004 1006012008 31.6543836 100066622 1002 1003 1006009 1009027027 31-6701752 10-0099899 1003 1004 1008016 101.2048064 31.6859590 10-0133155 1004 1005 1010025 1015075125 31-7017349 10-0166389 1005! 1006 1012036 1018108216 31-7175030 10'0199601 1006 1007 1014049 1021147343 31-7332633 10-0232791 1007 1008 1016064 1024192512 31.7490157 10-0265958 1008 1009 1018081 1027243729 31-7647603 10-0299104 1009 1010 1020100 1030301000 31-7804972 10-0332228 1010 1011 1022121 1033364331 31.7962262 10.0365330 1011 1012 1024144 1036433728 31-8119474 10-0398410 1012 1013 1026169 1039509197 31-8276609 10-0431469 1013 1014 1028196 1042590744 31-8433666 10-0464506 1014 1015 1030225 1045678375 31-8590646 10-04975'1 1015 1016 1032256 1048772096 31'8747549 10'0530514 1016 1017 1034289 1051871913 31.8904374 10-0563485 1017 1018 1036324 1054977832 31.9061123 10'0596435 1018 1019 1038361 1058089859 31-9217794 10.0629364 1019 1020 1040400 1061208000 31-9374388 100662271 1020 1021 1042441 1064332261 31'9530906 10'0695156 1021 1022 1044484 1067462648 31'9687347 10'0728030 1022 1023 1046529 1070599167 31'9843712 10'0760863 1023 1024 1048576 1073741824 32'0000000 10'0793684 1024 1025 1050625 1076890625 32-0156212 10'0826484 1025 1026 1052676 1080045576 32'0312348 10.0859262 1026 1027 1054729 1083206683 32 0468407 10*0892019 1027 1028 1056784 1086373952 32 0624391 10 0924755 1028 1029 1058841 1089547389 32'0780298 10'0957469 1029 1030 1060900 1092727000 32 0936131 10 0990163 1030 1031 1062961 1095912791 32-1091887 10'1022835 1031 1032 1065024 1099104768 32'1247568 10-1055487 1032 1033 1067089 1102302937 32-1403173 10'1088117 1033 1034 1069156 1105507304 32'1558704 10.1120726 1034 1035 1071225 1108717575 32 1714159 10'1153314 1035 1036 1073296 1111934656 32.1869539 10.1185882 1036 1037 1075369 1115157653 32'2024844 10'1218428 1037 1038 1077444 1118386872 32 2180074 10'1250953 1038 1039 1079521 1121622319 32.2335229 10-1283457 1039 1040 1081600 1124864000 32'2490310 10'1315941 1040 SQUARE ROOTS, AND CUBE ROOTS. 95 Number. Square. Cube. Square Root. Cube Root. Number. 1041 1083681 1128111921 32-2645316 10.1348403 1041 1042 1085764 1131366088 32.2800248 10-1380845 1042 1043 1087849 1134626507 32-2955105 10.1413266 1043 1044 1089936 1137893184 32-3109888 10-1445667 1044 1045 1092025 1141166125 32.3264598 10.1478047 1045 1046 1094116 1144445336 32.3419233 10-1510406 1046 1047 1096209 1147730823 32-3573794 10-1542744 1047 1048 1098304 1151022592 32-3728281 10-1575062 1048 1049 1100401 1154320649 32-3882695 10.1607359 1049 1050 1102500 1157625000 32.4037035 10.1639636 1050 1051 1104601 1160935651 32.4191301 10.1671893 1051 1052 1106704 1164252608 32-4345495 10-1704129 1052 1053 1108809 1167575877 32-4499615 10-1736344 1053 1054 1110916 1170905464 32.4653662 10.1768539 1054 1050 1113025 1174241375 32-4807635 10-1800714 1055 1056 1115136 1177583616 32-4961536 10-1832868 1056 1057 1117249 1180932193 32.5115364 10.1865002 1057 1058 1119364 1184287112 32-5269119 10.1897116 1058 1059 1121481 1187648379 32.5422802 10-1929209 1059 1060 1123600 1191016000 32-5576412 10.1961283 1060 1061 1125721 1194389981 32-2729949 10-1993336 1061 1062 1127844 1197770328 32-5883415 10-2025369 1062 1063 1129969 1201159047 32-6036807 10.2057382 1063 1064 1132096 1204550144 32.6190129 10-2089375 1064 1065 1134225 1207949625 32-6343377 10-2121347 1065 1066 1136356 1211355496 32-6496554 10-2153300 1066 1067 1138489 1214767763 32-6649659 10.2185233 1067 1068 1140624 1218186432 32.6803693 10.2217146 1068 1069 1142761 1221611509 32-6955654 10-2249039 1069 1070 1144900 1225043000 32-7108544 10-2280912 1070 1071 1147041 1228480911 32.7261363 10-2312766 1071 1072 1149184 1231925248 32.7414111 10-2344599 1072 1073 1151329 1235376017 32.7566787 10-2376413 1073 1074 1153476 1238833224 32*7719392 10.2408207 1074 1075 1155625 1242296875 32-7871926 10-2439981 1075 1076 1157776 1245766976 32.8024389 10.2471735 1076 1077 1159929 1249243533 32-8176782 10-2503470 1077 1078 1162084 1252726552 32-8329103 10-2535186 1078 1079 1164241 1256216039 32-8481354 10-2566881 1079 1080 1166400 1259712000 32-8633535 10-2598557 1080 96 SQUARES, CUBES, Number. Square. Cube. Square Root. Cube Root. Number. 1081 1168561 1263214441 32-8785644 10.2630213 1081 1082 1170724 1266723368 32-8937684 10-2661850 1082 1083 1172889 1270238787 32-9089653 10.2693467 1083 1084 1175056 1273760704 32'9241553 10'2725065 1084 1085 1177225 1277289125 32-9393382 10-2756644 1085 1086 1179396 1280824056 32-9545141 10'2788203 1086 1087 1181569 1284365503 32-9696830 10'2819743 1087 1088 1183744 1287913472 32-9848450 10'2851264 1088 1089 1185921 1291467969 33-0000000 10'2882765 1089 1090 1188100 1295029000 33-0151480 10l2914247 1090 1091 1190281 1298596571 33-0302891 10'2945709 1091 1092 1192464 1302170688 33'0454233 10'2977153 1092 1093 1194649 1305751357 33-0605505 10-3008577 1093 1094 1196836 1309338584 33-0756708 10-3039982 1094 1095 1199025 1312932375 33-0907842 10-3071368 1095 1096 1201216 1316532736 33'1058907 10.3102735 1096 1097 1203409 1320139673 33-1209903 10-3134083 1097 1098 1205604 1323753192 33-1360830 10-3165411 1098 1099 1207801 1327373299 33'1511689 10-3196721 1099 1100 1210000 1331000000 33-1662479 10-3228012 1100 1101 1212201 1334633301 33-1813200 10.3259284 1101 1102 1214404 1338273208 33'1963853 10-3290537 1102 1103 1216609 1341919727 33'2114438 10-3321770 1103 1104 1218816 1345572864 33'2264955 10-3352985 1104 1105 1221025 1349232625 33 2415403 10-3384181 1105 1106 1223236' 1352899016 33.2565783 10-3415358 1106 1107 1225449 1356572043 33-2716095 10-3446517 1107 1108 1227664 1360251712 33.2866339 10-3477657 1108 1109 1229881 1363938029 33-3016516 10-3508778 1109 1110 1232100 1367631000 33-3166625 10-3539890 1110 1111 1234321 1371330631 33.3316666 10-3570964 1111 1112 1236544 1375036928 33-3466640 10-3602029 1112 1113 1238769 1378749897 33.3616546 10-3633076 1113 1114 1240996 1382469544 33.3766385 10-3664103 1114 1115 1243225 1386195875 33-3916157 10-3695113 1115 1116 1245456 1389928896 33-4065862 10-3726103 1116 1117 1247689 1393668613 33-4215499 10-3757076 1117 1118 1249924 1397415032 33.4365070 10-3788030 1118 1119 1252161 1401168159 33'4514573 10-3818965 1119 1120 1254400 1404928000 33.4664011 10-3849882 1120 SQUARE ROOTS, AND CUBE ROOTS. 97 Number. Square. Cube. Square Root. Cube Root. Number. 1121 1256641 1408694561 33'4813381 10'3880781 1121 1122 1258884 1412467848 33-4962684 10-3911661 1122 1123 1261129 1416247867 33'5111921 10*3942523 1123 1124 1263376 1420034624 33-5261092 10-3973366 1124 1125 1265625 1423828125 33'5410196 10'4004192 1125 1126 1267876 1427628376 33*5559234 10-4034999 1126 1127 1270129 1431435383 33'5708206 10'4065787 1127 1128 1272384 1435249152 33-5857112 10-4096557 1128 1129 1274641 1439069689 33'6005959 10'4127310 1129 1130 1276900 1442897000 33'6154626 104158044 1130 1131 1279161 1446731091 33'6303434 10'4188760 1131 1132 1281424 1450571968 33'6452077 10'4219458 1132 1133 1283689 1454419637 33'6600653 10'4250138 1133 1134 1285956 1458274104 33'6749165 10'4280800 1134 1135 1288225 1462135375 33'6897610 10'4311448 1135 1136 1290496 1466003456 33'7045991 10'4342069 1136 1137 1292769 1469878353 33'7194306 10'4372677 1137 1138 1295044 1]478760072 33-7342556 10-4403267 1138 1139 1297321 1477648619 33'7490741 10'4433839 1139 1140 1299600 1481544000 33-7638860 10-4464393 1140 1141 1301881 1485446221 33-7786915 10-4494929 1141 1142 1304164 1489355288 33'7934905 10-4525448 1142 1143 1306449 1493271207 33-8082830 10-4555948 1143 1144 1308736 1497193984 33'8230691 10-4586431 1144 1145 1311025 1501123625 33-8378486 10-4616896 1145 1146 1313316 1505060136 33-8526218 10-4647343 1146 1147 1315609 1509003523 33-8673884 10-4677773 1147 1148 1317904 1512953792 33-8821487 10-4708185 1148 1149 1320201 1516910949 33-8969025 10-4738579 1149 1150 1322500 1520875000 33-9116499 10-4768955 1150 1151 1324801 1524845951 33'9263909 10'4799314 1151 1152 1327104 1528823808 33-9411255 10'4829656 1152 1153 1329409 1532808577 33-9558537 10'4859980 1153 1154 1331716 1536800264 33-9705755 10-4890286 1154 1155 1334025 1540798875 33-9852910 10-4920575 1155 1156 1336336 1544804416 34-0000000 10-4950847 1156 1157 1338649 1548816893 34-0147027 10-4981101 1157 1158 1340964 1552836312 34-0293990 10-5011331 1158 1159 1343281 1556862679 34-0440890 10-5041556 1159 1160 1345600 1560896000 34'0587727 10'5071757 1160 1 9 98 SQUARES, CUBES, SQUARE ROOTS, ETC. Number. Square. Cube. Square Root. Cube Root. Number. 1161 1347921 1564936281 34-0734501 10.5101942 1161 1162 1350244 1568983528 34-0881211 10.5132109 1162 1163 1352569 1573037747 34-1027858 10-5162259 1163 1164 1354896 1577098944 34-1174442 10-5192391 1164 1165 1357225 1581167125 34 1320963 10-5222506 1165 1166 1359556 1585242296 34-1467422 10-5252604 1166 1167 1361889 1589324463 34-1613817 10.5282685 1167 1168 1364224 1593413632 34.1760150 10-5312749 1168 1169 1366561 1597509809 34.1906420 10-5342795 1169 1170 1368900 1601613000 34-2052627 10-5372825 1170 1171 1371241 1605723211 34-2198773 10-5402837 1171 1172 1373584 1609840448 34-2344855 10'5432832 1172 1173 1375929 1613964717 34'2490875 10'5462810 1173 1174 1378276 1618096024 34'2636834 10*5492771 1174 1175 1380625 1622234375 34-2782730 10-5522715 1175 1176 1382976 1626379776 34-2928564 10'5552642 1176 1177 1385329 1630532233 34'3074336 10-5582552 1177 1178 1387684 1634691752 34-3220046 10-5612445 1178 1179 1390041 1638858339 34-3365694 10-5642322 1179 1180 1392400 1643032000 34-3511281 10-5672181 1180 1181 1394761 1647212741 34.3656805 10-5702024 1181 1182 1397124 1651400568 34-3802268 10-5731849 1182 1183 1399489 1655595487 34-3947670 10.5761658 1183 1184 1401856 1659797504 34-4093011 10-5791449 1184 1185 1404225 1664006625 34-4238289 10-5821225 1185 1186 1406596 1668222856 34.4383507 10-5850983 1186 1187 1408969 1672446203 34-4528663 10-5880725 1187 1188 1411344 1676676672 34-4673759 10-5910450 1188 [ 189 1413721 1680914269 34-4818793 10-5940158 1189 1190 1416100 1685159000 344963766 10 5999850 1190 1191 1418481 1689410871 34'5108678 10-5999525 1191 1192 1420864 1693669888 34-5253530 10-6029184 1192 1193 1423249 1697936057 34-5398321 10-6058826 1193 1194 1425636 1702209384 34.5543051 10.6088451 1194 1195 1428025 1706489875 34-5687720 10-6118060 1195 1196 1430416 1710777536 34-5832329 10-6147952 1196 1197 1432809 1715072373 34-5976879 10'6177228 1197 1198 1435204 1717374392 34-6121366 10'6206788 1198 1199 1437601 1723683599 34-6265794 10-6236331 1199 1200 1440000 1728000000 34-6410162 10-6265857 1200._ _ _ _ _. _ _ _ _ - _ _ _ _ _.1 RULES FOR SQUARES, CUBES, ETC. 99 To find the square of a greater number than is contained in the table. RULE 1. —If the number required to be squared exceed, by 2, 3, 4, or any other number of times, any number contained in the table, let the square affixed to the number in the table be multiplied by the square of 2, 3, or 4, &c., and the product will be the answer sought. EXAMPLE.-Required the square of 2595. 2595 is three times greater than 865; and the square of 865, as per table, is 748225. Then, 748225 x 32= 6734025, Ans. RULE 2.-If the number required to be squared be an odd number, and do not exceed twice the amount of any number contained in the table, find the two numbers nearest to each other, which, added together, make that sum; then, the sum of the squares of these two numbers, as per table, multiplied by 2, will exceed the square required by 1. EXAMPLE.-Requir'ed the square of 1865. The two nearest numbers (932 + 933) = 1865. Then, per table (9322=868624)+(9332=870489) = 1739113 x 2 = 3478226 - 1= 3478225, Ans. To find the cube of a greater number th'an is contained in the table. RULE.-Proceed, as in squares, to find how many times the number required to be cubed exceeds the 100 RULES FOR SQUARES, CUBES, number contained in the table. Multiply the cube of that number by the cube of as many times as the number sought exceeds the number in the table, and the product will be the answer required. EXAMPLE.-Required the cube of 3984. 3984 is 4 times greater than 996; and the cube of 996, as per table, is 988047936. Then 988047936 x 43= 63235067904, Ans. To find the square or cube root of a higher number than is in the table. RULE.-Refer to the table, and seek in the column of squares or cubes, the number nearest to that number whose root is sought, and the number from which that square or cube is derived will be the answer required, when decimals are not of importance. EXAMPLE.-Required the square root of 542869. In the table of squares, the nearest number is 543169; and the number from which that square has been obtained is 737. Therefore, V 542869 = 737 nearly, Ans. To find more nearly the cube root of a higher number than is in the table. RULE.-Ascertain, by the table, the nearest cube number to the number given, and call it the assumed cube. Multiply the assumed cube and the given number, SQUARE ROOTS, AND CUBE ROOTS. 101 respectively, by 2; to the product of the assumed cube add the given number, and to the product of the given number add the assumed cube. Then, by proportion, as the sum of the assumed cube is to the sum of the given number, so is the root of the assumed cube to the root of the given number. ExAMPLE.-Required the cube root of 412568555. Per table, the nearest number is 411830784; and its cube root is 744. Therefore, 411830784 x 2 + 412568555 = 1236230123. And, 412568555 x2+ 411830784 = 1236967894. Hence, as 1236230123: 1236967894:: 744: 744'369, very nearly, Ans. To find the square or cube root of a number containing decimals. Subtract the square root or cube root of the integer of the given number from the root of the next higher number, and multiply the difference by the decimal part. The product, added to the root of the integer of the given number, will be the answer required. EXAMPLE.-Required the square root of 321'62, V 321 = 17'9164729, and V 322 = 17'9443584; the difference ('0278855) x'62 + 17'9164729 - 17'9337619, Ans. 9* 102 THE CONIC SECTIONS. THE plane figures formed by the cutting of a cone by a plane, are five in number, viz: The Triangle, the Circle, the Ellipse, the Hyperbola, and the Parabola. The methods of finding their linear and superficial admeasurement have been already described; the several directions in which the section of the cone is to be made, in order to produce them, are as follows:The Triangle is formed by cutting the cone through the vertex and any part of the base. The Circle, by cutting the cone through the sides, parallel to the base. The Ellipse, by a cut passing obliquely, or at an angle with the base, through both sides of the cone. The Hyperbola, by cutting through one side and the base parallel to the axis, or at a greater angle with the base than that made by the opposite side. The opposite Hyperbola is formed by continuing the cutting plane through an opposite and equal cone, produced by continuing the sides of the first cone through its vertex. The Parabola, by cutting through one side and the base of the cone in a direction parallel to the opposite side, or making an equal angle with the base. THE CONIC SECTIONS. 103 The Ellipse has two vertices, being the points in the curve at the extremities of the longest diameter; the Hyperbola has one vertex,: or, rather, the opposite Hyperbolas one each; the Parabola has one only. The Transverse Axis is the line uniting the two vertices. The Conjugate.Axis is a line drawn through the centre of the transverse axis, and at right angles to it. A Diameter is a right line drawn through the centre, in any direction, and terminated at each end by the curve. A Conjugate Diameter is a line drawn through the centre of any diameter, parallel to the tangent of the curve at the extremity of such diameter. An Ordinate to a Diameter is a line between the diameter and the curve, parallel to its conjugate. The part of the diameter cut off by an ordinate and terminated by its vertex, is called the Abscissa. The Parameter, or latus rectumn, is a line drawn through the focus, at right angles to the transverse axis, and terminated by the curve. The parameter of a diameter, in the ellipse and hyperbola, is a third proportional to the diameter and its conjugate; in the parabola, it is a third proportional to one abscissa and its ordinate. The Focus is that point in the transverse axis where the ordinate is equal to half the parameter. 104 MENSURATION OF SURFACES. By the foregoing proportions, therefore, the focus of either curve may be found. The Ellipse has two foci; as have likewise the opposite Hyperbolas; but the Parabola has one only. The Ellipse has its several parts lying within the circumference of the curve; the axis and centre of the Hyperbola lie on the outside, in consequence of the axis being drawn between the vertices of the two opposite Hyperbolas. The axis of the Parabola is of infinite length, because the axis can only touch one point or vertex in the curve. MENSURATION OF SURFACES. OF FOUR-SIDED FIGUREWS. 2 3 4 c~ld id To find the area of a four-sided figure, whether it be a square, fig. 1, aparallelogram, fig. 2, a rhombus, fig. 3, or a rhomboid, fig. 4. RULE.-Multiply the length, a b, or c d, by the breadth or perpendicular height; the product will be the area. MENSURATION OF SURFACES.' 105 OF TRIANGLES. Fig. 5. Fig. 6. Fig. 7. C C C a Z~\b a/~ b a b d d To find the area of a triangle, whether it be isosceles, fig. 5, scalene, Jig. 6, or right-angled, fig. 7. / RULE.-Multiply the length, a b, of one of the sides, by the perpendicular, e d, falling upon it; half the product will be the area. To find the length of one side of a right-angled triangle, when the lengths of the other two sides are given. RULE 1.-To find the hypothenuse, a e, Jig. 7, add together the squares of the two legs, a b and b c, and extract the square root of that sum. RULE 2.-To find one of the legs, subtract the square of the leg, of which the length is known, from the square of the hypothenuse, and the square root of the difference will be the answer. OF REGULAR POLYGONS. To find the Area of a regular Polygon. RULE.-Multiply the length of a perpendicular, drawn from the centre to one of the sides (or the 106 MENSURATION OF SURFACES. radius of its inscribed circle) by the length of one side, and this product again by the number of sides; and half the product will be the area of the polygon. [For a table of the areas of regular polygons, see pages 67, 68.] OF TRAPEZIUMS AND TRAPEZOIDS. Fig. 8. Fig. 9. b a b n a *3c I, d dp c To find the Area of a Trapezium, fig. 8. RULE 1. —Draw a diagonal line, a c, to divide the trapezium into two triangles; find the areas of these triangles separately, and add them together. RULE 2.-Divide the trapezium into two triangles, by the diagonal a c, and let two perpendiculars, b f, and d e, fall on the diagonal from the opposite angles; then, the sum of these perpendiculars multiplied by the diagonal, and divided by 2, will be the area of the trapezium. To find the Area of a Trapezoid, fig. 9. RULE. —Multiply the sum of the two parallel sides, a h, d c, by a p, the perpendicular distance MENSURATION OF SURFACES. 107 between them, and half the product will be the area. RULE 2.-Draw a diagonal, a c, to divide the trapezoid into two triangles; find the areas of those triangles separately, and add them together. OF IRREGULAR FIGURES. Fig. 10. Fig. 11. b o find te Area of an Irregu To find the Area of an Irregular Polygon, ab ed efg, fig. 10. RULE.-Draw diagonals to divide the figure into trapeziums and triangles; find the area of each separately, by either of the rules before given for that purpose; and the sum of the whole will be the area of the figure. To find the Area of a Long Irregular Figure, d c a b, fig. 11. RULE. —Take the breadth in several places, and at equal distances from each other; add all the breadths together, and divide the sum by this number, for the mean breadth; then multiply the mean 108 MENSURATION OF SURFACES. breadth by the length of the figure, and the product will be the area. OF CIRCLES. Fig. 12 Fig. 13. Fig. 14. a 6a a/ Id /a f To find the Circumference of a Circle when the Diameter is given; or the Diameter when the Circumference is given. RULE 1.-Multiply the diameter by 3'1416, and the product will be the circumference; or divide the circumference by 3'1416, and the quotient will be the diameter. RULE 2.-As 7 is to 22, so is the diameter to the circumference. As 22 is to 7, so is the circumference to the diameter. RULE 3 — As 113 is to 355, so is the diameter to the circumference. As 355 is to 113, so is the circumference to the diameter. To find the Area of a Circle. RULE 1. —Multiply the square of the diameter.by'7854; or the square of the circumference by MENSURATION OF SURFACES. 109 *07958; the product, in either case, will be the area. RULE 2.-Multiply the circumference by the diameter, and divide the product by 4. RULE 3.-As 14 is to 11, so is the square of the diameter to the area. Or, as 88 is to 7, so is the square of the circumference to the area. To find the length of any Are of a Circle. RULE 1.-From 8 times the chord of half the arc, a c, fig. 12, subtract the chord, a b, of the whole arc; one-third of the remainder will be the length of the arc, nearly. RULE 2.As 180 is to the number of degrees in the arc; So is 3'1416 times the radius to its length. Or, as 3 is to the number of degrees in the arc; So is'05236 times the radius its length. To find the Area of a Sector of a Circle, fig. 13. RULE 1.-Multiply the length of the are, a d b, by half the length of the radius, a c; the product will be the area. RULE 2.-As 360 degrees is to the number of degrees in the arc of the sector; so is the area of the circle to the area of the sector. To find the Area of a Segment of a Circle, fig. 12. RULE 1. —To the chord, a b, of the whole arc 10 110 MENSURATION OF SURFACES. add the chord, a c, of half the arc and one-third of it more. Then multiply the sum by the versed sine, or height of the segment c d, and four-tenths of the product will be the area of the segment. RULE 2. —Divide the height, or versed sine, by the diameter of the circle, and find the quotient in the column of versed sines, at the end of Mensuration of Solids. Then take out the corresponding area in the next column on the right-hand, and multiply it by the square of the diameter, for the answer. To find the Area of a Circular Zone, fig. 14. RULE 1.- -When the Zone is less than a Semicircle, to the area of the trapezoid, a b c d, add the area of the circular segments, a c and b d; the sum is the area of the zone. RULE 2.- When the Zone is greater than a Semicircle, to the area of the parallelogram, e f g h, add the area of the circular segments, e k g and f 1 A; the sum is the area of the zone. To find the Area of a Circular Ring, or Space, included between two Concentric Circles. RULE.-Find the areas of the two circles separately; then the difference between them will be the area of the ring. MENSURATION OF SURFACES. 111 OF ELLIPSES. Fig. 15. Fig. 16. Fig. 17. a a a I \ I, "d' b - b b To find the Circumference of an -Ellipse, fig. 15. RULE.-Square the two axes, a 6 and c d, and multiply the square root of half that sum by 3'1416; the product will be the circumference nearly. To find the Area of an Ellipse, fig. 15. RULE. —Multiply the transverse diameter, a b, by the conjugate c d, and the product by'7854. To find the Area of an Elliptic Segment, fig. 16. RULE.-Divide the height of the segment, a p, by the axis a b, of which it is a part, and find, in the table of circular segments at the end of Mensuration of Solids, a circular segment having the same versed sine as this quotient. Then, multiply the segment thus found and the two axes of the ellipse continually together, and the product will give the area requir&d. 112 MENSURATION OF SURFACES. When the transverse, a b, the conjugate, c d, and the abscissw, a p and p b, are given, to find the ordinate, p q, fig. 17. RULE.-Multiply the abscissae into each other, and extract the square root of the product; this will give the mean between them. Then, as the transverse diameter is to the conjugate diameter, so is the mean.to the ordinate required. When the transverse, a b, the conjugate, c d, and the ordinate, p q, are given, to find the abscissa, fig. 17. RULE.-From the square of half the conjugate, take the square of the ordinate, and extract the square root of the remainder. Then, as the conjugate diameter is to the transverse, so is that square root to half the difference of the two abscisse. Add this half difference to half the transverse, for the greater abscissa; and subtract it for the less. When the transverse, a b, the ordinate; p q, and the two abscissx, a p and p b, are given, to find the conjugate, c d. RULE.-As the square root of the product of the two abscissae is to the ordinate, so is the transverse diameter to the conjugate. Note.-In the same manner the transverse diame MENSURATION OF SURFACES. 113 ter may be found from the conjugate, using the two abscissa of the conjugate, and their ordinate perpendicular to the conjugate. When the conjugate, c d, ordinate, p q, and abscisse, a p and b p, are given, to find the transverse diameter. RULE.-From the square of half the conjugate subtract the square of the ordinate, and extract the root of the remainder., Add this root to the half conjugate if the less abscissa be given; but subtract it when the greater abscissa is given. Then, as the square of the ordinate is to the rectangle of the abscissa and conjugate, so is the reserved sum, or difference, to the transverse diameter. OF PARABOLAS. Fig. 18. Fig. 19. Fig. 20. V V V b b b i d e f d f d e f To find the Area of a Parabola. RULE.-Multiply the base by the height, and two-thirds of the product will be the area. 10* 114 MENSURATION OF SURFACES. To find the Area of a Frustum of a Parabola, fig. 19. RULE.-Multiply the difference of the cubes of the two ends of the frustum, a c d f, by twice its altitude, b e, and divide the product by thrice the difference of their squares. To find the Abscissa or Ordinate of a Parabola, fig. 18. RULE. — The abscissae, v b and b e, are to each other as the squares of their ordinates, a b and d e, that is, as any abscissa is to the square of its ordinate, so is any other abscissa to the square of -its ordinate. Or, as the square root of any abscissa is to its ordinate, so is the square root of another abscissa to its ordinate. To find the Length of a Parabolic Curve, cut off by a Double Ordinate, fig. 20. RULE.-To the square root of the ordinate, a b, add of the square of the abscissa, v b; the square root of that sum, multiplied by 2, will give the length of the curve nearly. MENSURATION OF SURFACES. 115 OF HYPERBOLAS. Fig. 21. Fig. 22. V V d e f d e f To find the Area of a Hyperbola, fig. 21. RULE.-To five-sevenths of the abscissa, v b, add the transverse diameter, v e; multiply the sum by the abscissa, and extract the square root of the product. Then multiply the transverse diameter by the abscissa, and extract the square root of that product. Then, to 21 times the first root add 4 times the second root; multiply the sum by double the product of the conjugate and abscissa, and divide by 75 times the transverse; this will give the area nearly. To find the Length of a Hyperbolic Curve, fig. 22. RULE.-TO 21 times the square of the conjugate, a b, add 9 times the square of the transverse; also, to 21 times the square of the conjugate add 19 times the square of the transverse, and multiply each of these sums by the abscissa, v b. To each of the two products add 15 times the product of the transverse and square of the conjugate. 116 MENSURATION OF SURFACES. Then, as the less sum is to the greater, so is the ordinate to the length of the curve nearly. When the transverse, v e, the conjugate, d f, and the abscissae, v b and b e, are given, to find the ordinate, a b, fig. 21. RULE.-As the transverse diameter is to the conjugate, so is the square root of the product of the two abscissae to the ordinate required. Note.-In'the hyperbola, the less abscissae added to the axis gives the greater; and the greater abscissa subtracted from the axis gives the less. When the transverse and conjugate diameters, and the ordinate, are given, to find the abscissa. RULE.-To the square of half the conjugate add the square of the ordinate, and extract the square root of that sum. Then, as the conjugate diameter is to the transverse, so is the square root to half the sum of the abscissae. To this half sum add half the transverse diameter for the greater abscissa, and subtract it for the less. When the transverse diameter, ordinate, and abscissa, are given, to find the conjugate. RULE.-AS the square root of the product of the two abscissae is to the ordinate, so is the transverse diameter to the conjugate. MENSURATION OF SURFACES. 117 When the conjugate diameter, the ordinate, and the two abscisse, are given, to find the transverse diameter. RULE.-TO the square of half the conjugate add the square of the ordinate, and extract the square root of that sum. To this root add the half conjugate when the less abscissa is used; and subtract it when the greater abscissa is used; reserving the sum or difference. Then, as the square of the ordinate is to the product of the abscissa and conjugate, so is the reserved sum or difference to the transverse. MENSURATION OF SOLIDS. OF CUBES AND PARALLELOPIPEDONS. Fig. 23. Fig. 24. b To find the Solidity of a Cube, fig. 23. RULE.-Multiply the side of the cube by itself, and that product again by the side; the last product will be the solidity of the given cube. 118 MENSURATION OF SOLIDS. To find the Solidity of a Parallelopipedon, fig. 24. RULE. —Multiply the length, breadth, and depth or altitude, continually together, or, in other words, multiply the length, a b, by the breadth, a c, and that product by the depth or altitude, c d; this will give the required solidity. OF CYLINDERS AND PRISMS. Fig. 25. Fig. 26. To find the Solidity of Cylinders and Prisms. RULE.-Multiply the area of the base by the height of the cylinder or prism, and the product will give the solid content. To find the Convex Surface of a Cylinder. RULE.-Multiply the circumference by the length of the cylinder; the product will be the convex surface required. MENSURATION OF SOLIDS. 119 OF CONES AND PYRAMIDS. Fig. 27. Fig. 28. Fig. 29. Fig. 30. C c To find the Convex Surface of a Right Cone, or Pyramid, fig. 27. RULE.-Multiply the perimeter, or circumference of the base, by the slant height, or length of the side of the cone, and half the product will be the surface. To find the Convex Surface of a Frustum of a Right Cone, or Pyramid, fig. 28. RULE.-Multiply the sum of the perimenters of the two ends by the slant height or side of the frustum, and half the product will be the surface required. To find the solidity of a Cone, or Pyramid, figs. 27 and 29. RULE. —Multiply the area of the base by the height, c d, and one-third of the product will be the content. 120 MENSURATION OF SOLIDS. To find the Solidity of the Frucstum of a Cone, fig. 28. RULE.-Divide the difference of the cubes of the diameters of the two ends by the difference of the diameters; this quotient, multiplied by'7854 and again by one-third for the height, will give the solidity. To find the Solidity of the Frustum of a Pyramid, fig. 30. RULE.-Add to the areas of the two ends of the frustum the square root of their product, and this sum, multiplied by one-third of the height, will give the solidity. OF WEDGES AND PRISMOIDS. Fig. 31. Fig. 32. I I To find the Solidity of a Wedge, fig. 31. RULE.-TO the length of the edge of the wedge add twice the length of the back; multiply this sum by the height of the wedge, and then by the breadth of the back; one-sixth of the product will be the solid content. MENSURATION OF SOLIDS. 121 To find the Solidity of a Prismoid, fig. 32. RULE.-Add into one sum the areas of the two ends and four times the middle section, parallel to them; then, this sum multiplied by one-sixth of the height, will give the content. Note.-The length of the middle section is equal to half the sum of the lengths of the two ends; and its breadth is equal to half the sum of the breadths of the two ends. OF SPHERES. Fig. 33. Fig. 34. Fig. 35. I / To find the Convex Surface of a Sphere, or Globe, fig. 33. RULE.-Multiply the diameter of the sphere by its circumference. Or, multiply 341416 by the square of the diameter; the product will be the convex surface ret quired. Note.-The convex surface of any zone or segment may be found, in like manner, by multiplying its height by the whole circumference of the sphere. 11 122 MENSURATION OF SOLIDS. To find the Solidity of a Sphere, or Globe, fig. 33. RULE.-Multiply the cube of the axis by'5236; the product will be the solidity. To find the Solidity of a Spherical Segment, fig. 34. RULE. —To three times the square of the radius of its base add —the square of its height; then, multiply the sum by the height, and the product by'5236. To find the Solidity of a Spherical Zone or Frustum, fig. 35. RULE.-TO the sum of the squares of the radius of each end add one-third of the square of the height of the zone; this sum, multiplied by the said height, and the product by 1'5708, will give the solidity. OF SPHEROIDS. To find the Solidity of a Spheroid, fig. 36. RULE.-Multiply the square of the revolving axis, c d, by the fixed axis, a b; the product, multiplied by'5236, will give the content. To find the Solidity of the Segment of a Spheroid, figs. 37 and 38. RULE.- When the base, e f, is circular or parallel to the revolving axis, c d, fig. 37, multiply the MENSURATION OF SOLIDS. 128 Fig. 36. Fig. 37. C f d Fig. 38. 3 dg9. e'Fig. 40. d e Fi3 e:zfIi~x:> f / aIb'' d dk fixed axis, a b, by 3, the height of the segment, a g, by 2, and subtract the one product from the other; then multiply the remainder by the square of the height of the segment, and the product by'5236. Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment. RULE. — When the base, e f, is perpendicular to the revolving axis, c d, fig. 38, multiply the revolving axis by 3, and the height of the segment, c g, by 2, and subtract the one from the other; then, multiply the remainder by the square of the height of the segment, and the product by'5236. Then, as the revolving axis is to the fixed axis. so is the last product to the content. To find the Solidity of the Middle Frustum of a Spheroid, figs. 39 and 40. RULE. — When the ends, e f and g h, are circu 124 MENSURATION OF SOLIDS. lar, or parallel to the revolving axis, c d, fig. 39, to twice the square of the revolving axis, c d, add the square of the diameter of either end, e f, or g A; then multiply this sum by the length, a b, of the frustum, and the product again by -2618; this will give the solidity. RULE.- When the ends, e f and g h, are elliptical, or perpendicular to the revolving axis, 1 k, fig. 40, to twice the product of the transverse and conjugate diameters of the middle section, a b, add the product of the transverse and conjugate of either end; multiply this sum by the length, I k, of the frustum, and the product by'2618; this will give the solidity. OF CIRCULAR SPINDLES. Fig. 41. Fig. 42. d h g I " b To find the Surface of a Circular Spindle, fig. 41. RULE. —Multiply the length, a b, of the spindle by the radius, o c, of the revolving arc. Multiply also the said arc, a c b, by the central distance, o e, or distance between the centre of the spindle and MENSURATION OF SOLIDS. 1.25 centre of the revolving arc. Subtract this last product from the former; double the remainder; multiply it by 3'1416, and the product will give the surface of the spindle. Note. —The same rule will serve for any segment, or zone, cut off perpendicularly'to the chord of the revolving are; but, in this case, the particular length of the part, and the part of the are which describes it, must be used, instead of the whole length and whole arc. To find the Solidity of a Circular Spindle, fig. 41. RULE.-Multiply the central distance, o e, by half the area of the revolving segment, a c b e a. Subtract the product from one-third of the cube, a e, of half the length of the spindle. Then, mul. tiply the remainder by 12'5664, or 4 times 3'1416, and the product will be the solidity required. To find the Solidity of the Frustum, or Zone, of a Circular Spindle, fig. 42. RULE.-From the square of half the length, h i, of the whole spindle, take one-third of the square of half the length, n i, of the frustum, and multiply the remainder by the said half-length of the frustum. Multiply the central distance, o i, by the revolving area, which generates the frustum. Subtract the last product from the former; and the re11* 126 MENSURATION OF SOLIDS. mainder, multiplied by 6-2832, or twice 3-1416, will give the content. OF ELLIPTIC SPINDLES. Fig. 43. Fig. 44. Tofn.hoi i.... To find the Solidity of an Elliptic Spindle, fig. 43. RULE.-To the square of the greatest diameter add the square of twice the diameter at one-fourth of its length; multiply the sum by the length, and the product by -1309, and it will give the solidity very nearly. To find the Solidity of a Frustum or Segment of an Elliptic Spindle, fig. 44. RULE.-Proceed, as in the last rule, for this, or any other solid, formed by the revolution of a conic section about an axis; namely, Add together the squares of the greatest and least diameters, and the square of double the diameter in the middle between the two; multiply the sum by the length, and the product by'1309, and it will give the solidity. Note.-For all such solids, this rule is exact when the body is formed by the conic section, or a MENSURATION OF SOLIDS. 127 part of it revolving about the axis of the section; and will always be very near the truth when the figure revolves about another line. OF PARABOLIC CONOIDS AND SPINDLES. Fig. 45. Fig. 46. Fig. 47. —. —. —- Cf C To find the Solidity of a Parabolic Conoid. RULE. —Multiply the square of the diameter of the base by the altitude, and the product by'3927. To find the Solidity of a Frustum of a Paroboloid, fig. 45. RuLE. — Multiply the sum of the squares of the diameters of the two ends, a b and d c, by the height of the frustum, e f, and the product by *3927. To find the Solidity of a Parabolic Spindle, fig. 46. RULE.-Multiply the square of the middle diameter by the length of the spindle, and the product by ~41888, (which is eight-fifteenths of'7854,) and it will give the content. 128 MENSURATION OF SOLIDS. To find the Solidity of the Middle Frustum of a Parabolic Spindle, fig. 47. RULE.-Add together 8 times the square of the greatest diameter, c d, 3 times the square of the least diameter, g h, and 4 times the product of these two diameters; multiply the sum by the length, a 6, and the product by'05236, (which is I of 3'1416;) this will give the solidity. OF CYLINDRICAL RINGS. Fig. 48. a' To find the Convex Surface of a Cylindrical Ring. RULE.-To the thickness of the ring, a b, add the inner diameter, b c; multiply this sum by the thickness, and tlie product by 9'8696, (which is the square of 3414159,) and it will give the superficies required. To find the Solidity of a Cylindrical Ring. RULE.-To the thickness of the ring add -the inner diameter; then multiply the sum by the MENSURATION OF SOLIDS. 129 square of the thickness, and the product by 2.4674, (which is one-fourth of the square of 3'1416,) and it will give the solidity. To find the superficies or solidity of any regular body. RULE 1. —Multiply the tabular area by the square of the linear edge, and the product will be the superficies. RULE 2.-Multiply the tabular solidity by the cube of the linear edge, and the product will be the solidity. Surfaces and Solidities of the Regular Bodies, when the linear edge is 1. No. of Sides. Names. Surfaces. Solidities. 4 Tetrahedron 1-73205 0-11785 6 Hexahedron 6.00000 1.00000 8 Octahedron 3-46410 0-47140 12 Dodecahedron 20-64573 7-66312 20 Icosahedron 8-66025 2-18169 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. TABLE of the Weight of Fat and Rolled Iron per foot, in length..a c. BREADTH IN INCHES AND PARTS OF AN INCH. cl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.4 3 a 8 3a 2 21 2 2 1{ 1 ~ ~ 1} 1 - 1 1.68 1.57 1.47 1.36 1.26 1.15 1.05 0.94 0.84 0.73 0.63 0.57 0.52 0.42 0.31 0.21 252 2386 2.20 2.04 1.89 1.73 1.57 1.41 1.26 1.10 0.94 0.86 0.78 0.63 0.47 0.31 836 3815 2.94 2.73 2.52 2.31 2.10 1.89 1.68 1.47 1.26 1.18 1.05 0.84 0.63 042 5604 4*72 4.41 4.09 3878 3.46 8.15 2.83 2.52 2.20 1.89 1.73 157 1.26 094 0.63 6.72 6.30 5.88 5.46 5.04 4.62 4.20 3878 3836 2.94 2.52 2.31 2.10 1.68 1.26 8.40 7*87 7.35 6.82 6.30 5.77 5.25 4-72 4.20 367 3.15 288 2.62 2.10 157 T 10.08 9.45 8.82 8.19 7.56 6.93 6.30 5.66 5.04 4.41 38.78 3.46 3815 2.52 11.76 11.02 1029 9.55 8.82 8.08 7.35 6.61 5.88 5.14 4.41 4.04 3867 2.94 1 13844 12.60 11.76 10.92 10.08 9.24 8.40 7.56 6.72 5.87 5.04 4.62 4.20 M 1} 15.12 14.16 13820 12.28 1134 10.39 9.55 8.50 7.56 6.60 5.67 5.19 4.72 11 16.80 15.75 14.70 13865 12.60 11.55 10.50 9.45 8.40 7.35 6.30 5.77 19 18.46 17.32 16.16 15.01 13886 12.70 11.55 10.39 9.24 8.07 11 20.18 18.90 17.64 16.38 15.12 13.86 12.60 11.34 10.08 8.80 1{ 23854 2205 20.58 19.11 17.64 16.17 14.70 13822 2 26.88 25.20 23852 21.84 20.16 18.48 16.80 15.12 28 3365 31.50 29.40 27.39 25.20 23810 3 40.32 37.80 35.28 32.76 8j 47,04 131 TABLE of the Weight of Cast-Iron Pipes, in Lengths. Weight. Weight. Weight.. 9 Weight. _ In. In. Ft. C. qr. lb. In. In. I Ft. C. qr. lb. Inch. In. Ft. C. qr. lb. 1 I 31 12 61 9 2 0 16 11 f 9 5 0 7 f 3~ 21 3 9 2 3 20 - 9 6 112 1 ~ 41 21 i 9 3 2 21 } 9 7 2 8 it 4~ 1 4 9 4 121 1 9 10 1 2 2 i 6 1 8 4 12 9 5 024 f 6 2 0 7 f 9 30 7 9 62 8 2t f 6 116 9 93 320 f 9 7320 i 6 210 9 49 3 5 1 9 103 0 f 6 3 10 1 9 6 2 4 121 9 5 1 16 3 i 9 2 20 931 6 f 9 63 9 ft 9 100 6 9 4022 f 9 81 0 t 9 11 21 12 9 5'0 10 1 9 11 0 21 f 9 13 6 13 f 9 5 220 9 21 0 8 9 32 4 f 9 7 014 31 9 9 3 0 9 4125 9 82 7 f 9 1 0 21 ft 9 5 118 1 9 11 2 12 f 9 12 14 1 9 7 116 131'I 9 5 3 7 9 20 8 8~ i 9 33 2 1 9 7112 9 22 0 f 9 4226 ti 9 8316 54 f 9 1 11 f0 t 9 5 2 22 1 9 11 3 24 4 9 2 3 17 10 1 9 73 8 9 1 312 14 14 9 6 0 4 f 9 2 112 9 f 9 40 0 ft 9 7 216 9 2 3 21 9 5 0 4 1 9 9 1 0 41 9 1 2 2 9 60 2 1 9 12 114 f 9 2 0 4 1 9 8 026 14f 9 6 0.24 f 9 2 2 14 91 f 9 4 0 18 f 9 7 314 9 3 021 t 9 51 0 t1 9 9 2 2 5 f 9 4 12 22 9 6 1 6 1 9 12 3 6 9 2 1 19 8220 15 9 6 121 9 231 17 10 f 9 4 1 10 9 93 7 9 3124 ft 9 5 1 26 1 9 13 026 5~ftf 91310 ft 9 6214 1~ 9 163 5 9 22 0 1 9 90 8 15f 3 9 6214 ft 9 3 0 18 10 9 4 2 14 91 0 10 9 39 3 7 f 9 53 7 1 9 13 217 1 9 5012 f 9 7 00 1 9 171 6 6 f9 2 0 0 1 9 92 0 16 f 9 7022 9 2 2 21 11 f 9 4 314 f 9 10 120 9 3 1 17 1f 9 6 011 1 9 14 0 8 9 4 0 16 9 717 1 7 1 9 17 314 1.9 5 2 20 1 9 9 320 1f 9 1213 4 132 WEIGHT OF MATERIALS. Table of the Weight of one foot length of Malleable Iron. SQUARE IRON. ROUND IRON. Scantling. Weight. Diameter. Weight. Circum. Weight. Inches. Pounds. Inches. Pounds. Inches. Pounds. 021 016 1 0.26 0-47 037 1i 0.41 0-84 4 0 66 14 0.59 4 1.34 1.03 13 082; 1.89 3 1-48 2 1.05 257 7 2.02 24 1-34 1 3'36 1 2.63 21 1.65 1* 4.25 14 3.33 2t 2-01 1i 5-25 1i 4.12 3 2.37 1i 6-35 1i 4.98 34 2-79 14 7.56 14 5-93 31 3-24 1I 8-87 14 6-96 34 3.69 14 10.29 14 8-08 4 4.23 1~- 11-81 14 9.27 44 5.35 2 13.44 2 10-55 5 6.61 24 17.01 24 13.35 54 7-99 24 21-00 24 16'48 6 9'51 24 25-41 2t 19'95 64 11'18 3 30.24 3 23.73 7 12-96 34 41-16 34 27.85 74 14.78 4 63.76 34 32-32 8 16.92 41 68-04 3t 37-09 8S 19.21 5 84-00 4 42.21 9 21.53 6 120.96 44 53.41 10 26.43 7 164.64 6 65.93 12 31.99 Weight of Cast-Iron Plates, per superficial foot, from oneeighth of an inch to one inch thick. 3 inch.,4 inch. Y inch. Y inch. % inch. 4 inch. % inch. 1 inch. lbs. oz. lbs. oz. lbs. oz. lbs. oz. lbs. oz. lbs. oz. lbs. oz. lbs. oz. 4 131 9 104 14 8 19 53 24 24 29 0 33 134 38 101 WEIGHT OF MATERIALS. 133 Table of the Bore and Weight of Cocks. Content of Bore of Weight of Content of Bore of Weight of Copper. Cock. Cock. Copper. Cock. Cock. Gallons. Inches. Pounds. Gallons. Inches. Pounds. 30 1 7 200 23 30 50 1Q 8 260 3 34 80 2 12 340 34 44 120 21 19 420 3~ 56 150 2i 26 430 and 33 70 upwards. _ Three-fourths of the diameter of the bore, taken at the hinder part, will give the diameter of the cock at the mouth. Table of the Dimensions and Weight of Coppers, from 1 to 208 galls. The Dimensions taken from lag to brim. Inches, W eight Inches, lagWto Gallons. Gllht Weight Inches, Weight, hlag to Gallons. Weight la.in bs. lag to Gallons. in lbs. in lbs. brim. in lbs. brim. brim. 93 1 1~ 24 15 221 29j 29 43~ 124 2 3 241 16 24 30 30 45 14 3 41 25 17 25~ 32 36 54 151 4 6 251 18 27 34 43 641 161 6 71 26 19 281 35 48 72 17J 6 9 261 20 30 36 53 791 18~ 7 101 26f 21 311 37 58 87 19~ 8 12' 27 22 33 38 63 94~ 204 9 13j 274 23 34~ 39 67 100J 21 10 15 271 24 36 40 71 106j 211 11 16j 274 25 37~ 45 104 156 22 12 18 28 26 39 50 146 219 22J 13 19~ 281 27 40~ 55 208 312 234 14 21 29 28 42 1 12 134 WEIGHT OF MATERIALS. Table of the Weight of Lead, per superficial foot, from onesixteenth of an inch to one inch thick. Thick Thick- Thick ThicknessT Weight. ness. Weight. Thick- Weight. Thick- Weight. inch. lbs. inch. lbs. inch. lbs. inch. lbs. 1-16th 3a 1-8th 71 1-4th 143 3-4ths 44} 1-12th 5 1-6th 10 1-3d 19 1 inch 59 l-lOth 6 1-5th 12 1-half 291 Weight of Lead Pipe of the usual thicknesses, per foot in length. i-inch bore..... 1 lb. 1 oz. i...... lb. 8oz. -— llb. 12 oz. - 2lbs. 1 "..... 2 lbs. - 2 lbs. 11 oz. - 2 lbs. 14 oz. it "..... 3 lbs. — 3lbs. 11 oz. - 4lbs. 7 oz. "..... 4 lbs. - 4 lbs. 11 oz. - 5 lbs. 9 oz. 2 "..... 5 lbs. 9 oz.- 7 lbs. - 8lbs. 5 oz. 2~ "..... 7 lbs. - 8 lbs. 9 oz. - 10 lbs. Weight of Copper Tubing of the usual thickness. When the inside diameter is i of an inch, 3 ounces; i of an inch, 5 ounces; ~ of an inch, 6 ounces; 4 of an inch, 8 ounces; and 4 of an inch, 10 ounces per foot. TABLE of the Weight of Metals, Woods, Stones, Earths, etc. Weight of Weight of Weight of METALS. a cubic foot, a cubic foot, a cubic inch, in Ounces. in rounds, in Pounds. Antimony in a metallic state, 6 4 4 fused.............. 6,624 4140 0238 " glass of............ 4,946 309.1 0.182 " sulphur of........... 4,064 254-0 0.147 " ore, gray and foliated 4,368 273-0 04159 " " radiated........ 4,440 277-5 0.161 Bismuth, cast...................... 9,823 614-0 0-355 native.............. 9,822 614-0 0-355 " ore, in plumes........ 4,371 273-2 0.159 Brass, common cast........ 7,824 489-0 0-283 " cast, not hammered..... 8,396 524-8 0-303 " wire-drawn............ 8,544 534-0 0-308 Copper, cast........................ 8,788 549-2 0-317 " wire-drawn............ 8,878 554.9 0.320 " pyrites.......... 4,080 255-5 0-148 " ore, Cornish............. 5,452 340-8 0-202 49" " white................ 4,500 281-2 0.163 " " gray............ 4,500 281-2 0'163,c " yellow........... 4,300 266-8 0.156 " " blue............. 3,400 212-5 0.123'C" " prismatic.......... 4,200 262'5 0-152 " " foliated, florid-red 3,950 247'0 0-143 " " radiated, azure... 3,231 202-0 0-117 " " emerald............. 3,300 206-2 0'120 Gold, pure, cast................... 19,258 1203-6 0-697 " the same, hammered..... 19,362 1210-1 0-701 " 22 carats, fine, standard 17,486 1093-0 0'633 " the same, hammered..... 15,589 974'4 0.564 " 20 carats, fine, trinket... 15,709 982-0 0-568 " the same, hammered..... 15,775 986-0 0-570 Iron, cast............................ 7,207 450-5 0-260 " bars.................. 7,788 486-8 0-281 " pyrites, cubic............ 4,702 294-0 0-170 " " radiated,.........4,775 298-5 0-173 " " dodecahedral... 4,830 3020 0-175 it" " from Freyburg.. 4,682 292.6 0-170 " " from Cornwall.. 4,789 299-4 0'173 " ore, specular............... 5,218 326-1 0-189 " " micaceous...........5,070 317-0 0-184 prismatic......... 7,355 4598 0266 135 136 WEIGHT OF MATERIALS. Weight of Weight of Weight of METALS. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Lead, cast........................... 11,3852 7095 0410 " litharge..................... 6,300 393-8 0-228 " ore, cubic................... 7,587 474-2 0.274 " 6" horned............... 6,072 379,5 0 220 " " corneous............. 6,065 379'1 0'220 46" " reniform.............. 3,920 345'0 0-142 " " blue.................... 5,461 341.4 0.198 " black................. 5,670 360-6 0'210 " " brown.............. 6,974 436-0 0.252 96" " white.................. 7,236 452.2 0.261.".. red or red-lead spar 6,027 376'8 0'219 ". yellow, molybde-} 11,352 natted........., ~ 11,352 709.5 0.410 Mercury, solid, 40 deg. below 15,632 9 0 0566 00 Fahr........... 15,632 977 " at 32 deg. of heat... 13,619 851-2 0 493 " at 60 deg............... 13,580 848-8 0.491 " at 212 deg............ 13,375 836-8 0-484 Nickel, cast......................... 7,807 488'0 00282 " ore, called Kupper- 6648 4155 0240 nickel, of Saxe....,648 415 t".".. of Bohemia.. 6,207 38880 0 225 Platina, crude, in grains........ 15,602 975-1 0.570 " purified.................. 19,500 1218-8 0-706 " the same, hammered.. 20,337 1271-1 0-736 69 " " rolled........ 22,069 1379.4 0-799 it " v wire-drawn 21,042 1315-1 0.773 Silver, cast, pure.................. 10,474 654-6 08379 c" ast, pure, hammered.. 10,511 657-6 0-381 " Parisian standard.......-10,175 636-0 0-368 " the same, hammered... 10,376 648.5 0-375 " French coin............... 10,048 628.0 0-364 " the same, hammered... 10,408 650-5 0.376 " shilling, George II.....10,534 658-4 0-380 Steel, soft................... 7,833 489-6 0-283 " hardened, not tempered' 7,840 - 490-0 0-283 " tempered, not hardened 7,816 488-5 0-283 " tempered and hardened. 7,818 488-6 0-283 Tin, pure Cornish............. 7,291 455-6 0-263 " the same, hardened........ 7,299 456-0 0.263 " Molacca, fused............ 7,296 456-0 0-263 " " hardened.........I 7,307 456-8 i 0-263 WEIGHT OF MATERIALS. 137 Weight of Weight of Weight of METALS AND WOODS. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Tin ore, red........................ 6,935 433,5 0.250 black..................... 6,901 431.4 0250 " white.................... 6,008 375-5 0.221 Tungsten........................... 6,066 879 1 0 220 Uranium.......................... 6,440 402 5 0'232 Wolfram........................... 7,119 4450 0257 Zinc, in its usual state......... 6,862 429 0 0-248 4 pure and compressed 7,191 4495 0-259 " formed by sublimation, 5,918 and full of cavities.. 2699 0156 sulphate of.................. 1,900 118-8 0 069 "I saturated solution, the00 temp. 42 deg. 1,386 866 0.050 WooDs. Alder........................ 800 50.0 0-029 Apple-tree........................... 793 49.6 0*029 Ash.................................... 845 529 0-031 Bay-tree............................. 822 614 0029 Beech............................... 852 53-2 0-031 Box, Dutch..................... 912 57'0 0[ 033 " French...................... 1,328 830 0. 048 " Brazilian, red............... 1,31 64'5 0[ 037 Campechy........................... 913 67 1 0-033 Cedar, American.................. 561 351 0I 020 " Indian............... 1,315 82-2 0'047 " Palestine.................. 613 38-4 0-022 " Wild..................... 596 37-2 0-021 Cherry-tree.............. 715 44I 8 0.026 Citron................... 726 45*4 0.026 Cocoa................. 1,040 65'0 0[ 037 Cork..................... 240 150 0.009 Cypress..................... 644 40.2 0-023 Ebony, Indian................ 1,209 75 6 0 044 i" American................. 1,331 83 2 0-048 Elder.............................. 95 43-5 0025 Elm...................... 671 42 0 0 024 Filbert............................... 600 37*5 0-021 Fir, yellow..................... 657 41-1 0-023 A" white....................... 569 35'6 0021 male........................... 550 344 0019 12 138 WEIGHT OF MATERIALS. Weight of Weight of Weight of WOODS, STONES, EARTHS, ETC. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Fir, female......................... 498 11 0018 Hazel.......................... 600 37 5 0-021 Jasmin, Spanish................... 770 48-1 0-028 Juniper.............................. 556 348 0020 Lemon-tree......................... 703 44.0 0'025 Lignum-vitse........................ 1,333 83.4 0-048 Lime-tree............................ 604 37-8 0-022 Logwood............................. 913 57-1 0.033 Mahogany.......................... 1,063 66-5 0-038 Maple................................ 750 47-0 0-027 Mastic-tree......................... 849 53-1 0.031 Medlar............................. 941 59.0 0'034 Mulberry........................... 897 56-1 0-032 Oak, heart of, 60 years old.... 1,170 73-1 0'043 " dry............................. 932 58-2 0.033 Olive-tree......................... 927 58-0 0.033 Orange-tree......................... 705 44-1 0 025 Pear-tree............................ 661 41-4 0.024 Pomegranate-tree................. 1,354 84-6 0.049 Poplar................................ 383 24-0 0-014 " white, Spanish........... 529 33-1 0.019 Plum-tree.......................... 755 47-2 0.027 Quince-tree........................ 705 44-1 0.025 Sassafras............................ 482 30-1 0-017 Vine................................. 1,327 83-0 0-048 Walnut..............................71 42.0 0.024 Willow............................. 585 36-6 0-021 Yew, Spanish....................... 807 50'5 0-029 "Dutch................... 788 49-2 0-028 STONES, EARTHS, ETC. Alabaster, yellow.............. 2,699 168-8 0.098 " stained brown.. 2,744 171-5 0.099 " veined,.2......... 2,691 168.2 0-098 " Dallias................. 2,611 1632 0.095 " Malaga................ 2,876 179-8 0-104 " Malta.................. 2,699 168-8 0-098 " Oriental, white...... 2,730 170-6 0.099 cc " semitransparent } 2,762 172'6 0'100 " Piedmont.............. 2,693 168-4 0-098 WEIGHT OF MATERIALS. 139 Weight of Weight of Weight of STONES, EARTHS, ETC. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Alabaster, Spanish Saline...... 2,713 169.6 0.099 " Valencia.............. 2,638 164-9 0 096, Ambergris......................... 926 58-0 0033 Amianthus, long.................. 909 57 0 0 033 " short.................. 2,313 144-6 0084 Asbestos, ripe...................... 2,578 161-1 0 094 " starry...................3,073 192 1 0 111 Borax................................. 1,714 1071 0062 Brick, earth........2,00...... 2,000 125-0 0.073 Chalk, British......8.......... 2,78......4 1740 0100 " Biangon, coarse.......... 2,727 170-5 0-098 " Spanish.................... 2,790 174'4 0.100'Coal, Cannel..............1......... 1,270 79'4 0-046'" Newcastle.................... 1,270 79.4 0-046 " Staffordshire................ 1,240 1 77'5 0'045," Scotch....................... 1,300 812 0 047 Cutler's-stone................2...... 2,111 132-0 0-076 Emery................................ 4,000 250-0 0'144 Flint, black......................... 2,582 162-0 0'094 " veined........................ 2,612 163-2 0'095 " white........................ 2,594 162-1 0*094 " Egyptian.................... 2,565 160q4 0.093 Glass, flint........................... 2,933 170-9 0099 it white.................. 2,892 168.2 0.098 " ottle........................ 2,732 170-8 0'099 " green........................ 2,642 165'1 0-096 " St. Gobin................... 2,488 155.5 01090 " Leith, crystal.............. 3,189 199-4 0-116 " fluid.......................... 3,329 208'1 0-120 Granite, Aberdeen, blue kind.. 2,625 164-1 0.095 " Cornish.................. 2,662 166.4 0.096,, Egyptian, red.......... 2,654 165!9 0-096 14 " " gray......... 2,728 170-5 0 099 " beautiful red...........2,761 172'6 0.100 " Girardmor............... 2,716 169-8 0-098 " violet, of Gyromagny 2,685 168'0 0'098 " Dauphiny, red.......... 2,643 165-2 0-096 99 " " green..... 2,684 167-8 0'098 St', radiated... 2,668 166'8 0'097 "a Semur, red..............'2,638 164-9 0-096 " Bretagne, gray......... 2,738 171'1 0-098,,', yellowish... 2,619 162'8 0.095 140 WEIGHT OF MATERIALS. Weight of Weight of Weight of STONES, EARTHS, ETC. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Granite, Carinthia, blue........ 2,956 184-8 0-107 Grindstone........................ 2,143 134-0 0'077 Gypsum, opaque................... 2,168 135-5 0'077'' semi-transparent.... 2,306 144-1 0 083 - fine ditto................ 2,274 142-1 0-082 " cuneiform, cryst....... 2,306 144.1 0-083' rhomboidal.... 2,311 144-5 0'083 " ditto, ten faces........ 2,312 144-5 0-083 Hone, white, razor................ 2,876 179'8 0-104 Jet, a bituminous substance... 1,259 78-8 0-046 Lime-stone, green.......... 3,182 199.0 0-116'L" ~arenaceous.......... 2,742 171.4 0.098 " white fluor.......... 3,156 197'2 1-014 r" compact............. 2,720 170'0 0.098 " foliated............. 2,837 177-4 0-102 9" granular............. 2,800 175-0 0.101 Manganese....................... 7,000 437.5 0.252 " gray ore, striated.. 4,756 297-2 0.172 " gray, foliated....... 3,742 234-0 0-135 " red, from Kapnick 3,233 202-1 0-117 " black................ 3,000 187'5 0'108 " scaly............ 4,116 257-2 0.149 " sulphuret of......... 3,950 246-9 0.142 " phosphate of....... 2,600 162-5 0-094 Marble, African............ 2,708 169-2 0.098 " Biscayan, black........ 2,695 168'5 0.098 " Brocatelle............... 2,650 165-6 0.097 " Campanian, green..... 2,742 171-4 0.099 " Carrara, white.......... 2,717 169-9 0.098 " Castilian................. 2,700 168'8 0.098 " Egyptian, green........ 2,668 166.8 0.097 " French.................... 2,649 165'6 0'099 " Grenada, white......... 2,705 169-1 0 098 " Italian, violet......... 2,858 166.1 0.097 " Norwegian............... 2,728 170.5 0.099 " Parian, white........... 2,838 164.9 0.096 " Pyrenean................ 2,726 170-4 0-098 Red........................ 2,724 170'2 0-098 " Roman violet............ 2,755 172-2 0.099 " Siberian................. 2,718 169-9 0'098 " Siennian.............. 2,678 167-4 0-097 I" Switzerland.............. 2,714 169-6 0-098 WEIGHT OF MATERIALS. 141 Weight of Weight of Weight of STONES, EARTHS, ETC. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Marble, Valencia................ 2,710 169-4 0.098 Mill-stone........................... 2,484 155-2 0'090 " phosphoric.......... 1,714 107-1 0-062 Porcelain, China.................. 2,385 149-1 0-086' Limoges.......... 2,341 146-4 0-084 " Sevres................. 2,146 134-1 0-077 British................. Portland-stone..................... 2,5 70 160-6 0-094 Pumice-stone....................... 915 57-2 0-033 Paving-stone....................... 2,416 151.0 0-088 Purbeck-stone..................... 2,601 162-6 0-094 Porphyry, red...................... 2,765 172-9 _ 0-099 " green................. 2,676 167-2 0.097 " red, from Cordone.. 2,754 172-1 0-099 " green, from ditto... 3,728 233-0 0.135 " red, from Dauphiny 2,793 174-6 0.101 Pyrites, copper.................... 4,954 309-6 0-180 " ferruginous, cubic..... 3,900 241-2 0-140 49 " " round..... 4,101 256-4 0149 Domingo) 3,440 215-0 0-125 Rotten-stone........................ 1,981 124-0 0-071 Salt................................... 2,130 133-1 0-077 Serpentine, opaque, green,a 2430 152-0 0-088 Italian "9 clack & oliveined, 2,594 162-1 0-094,I (' red and black 2,627 164-2 0-095 " semi-transpa- _ 2,586 161-6 0-094 rent, grained. j " fibrous............... 3,000.187-5 0.108 " from Dauphiny.... 2,669 167-0 0-097 Slate, common..................... 2,672 167-0 0.097 " new........................... 2,854 178-4 0-104 " black stone................. 2,186 136-6 0-079 " fresh polished............. 2,766 173-0 0-099 Stalactite, opaque................. 2,478 154-9 0-090 " transparent.......... 2,324 145-2 0-084 Stone, Bristol...................... 2,510 157-0 0.091 " Burford..................... 2,049 128-1 0-075 " common..................... 2,520 157-5 0.091 142 WEIGHT OF MATERIALS. Weight of Weight of Weight of STONES, EARTHS, ETC. a cubic foot, a cubic foot, a cubic inch, in Ounces. in Pounds. in Pounds. Stone, Clicard, from Brachet.. 2,357 147-4 0-085.. 4" from Ouchain.. 2,274 142-1 0-082 " Notre-Dame............... 2,378 148-6 0.085 " Oriental blue.............. 2,771 173-2 0.099 " paving...................... 2,416 151.0 0-088 " Portland................... 2,570 160-6 0'094 " pumice...................... 915 57-2 0.033 " Purbeck.................... 2,601 162-6 0.094 " prismatic basaltes....... 2,722 170-1 0.099 rag........................... 2,470 155-4 0.090 " rotten....................... 1,981 124-0 0-071 rock of Chatillon........ 2,122 132-6 0-076 " Siberian blue............. 2,945 184.1 0-107 " St. Cloud................... 2,201 137-6 0-079 " St. Maur................... 2,034 127'1 0.075 " touch.................... 2,415 151.0 0-088 Sulphur, native.................... 2,033 127-1 0.075 melted................... 1991 124-5 0-071 " melted. 1,991 124.5 0071 Talc, black....................... 2,900 181-2 0-105 " crayon....................... 2,089 130-6 0-075 " German...................... 2,246 140-4 0-081 " Muscovy................... 2,792 174-5 0.101 " yellow........................ 2,653 166-0 0097 APPLICATION. RsULE.-Find, by the rules in the "Mensuration of Solids," the solidity of the material of which the weight is required, and multiply that solidity by the factor, in the foregoing table. EXAMPLE 1.-Required the weight of a bar of iron 12 feet long and 1 inch square? Weight of 1 inch bar-iron, as per table, page 135, is 0.281. 12 feet = 144 inches. Then -281 X 144 inches = 40'464, or 40 —5-, or 401 lbs. nearly. EXAMPLE 2.-Required the weight of a plank of yellow fir, 11 inches wide, 3 inches thick, and 20 feet long? 20 feet - 240 inches. One cubic inch yellow fir, as per table, = 0-023; therefore *-023 X 240 X 11 X 3 = 182160 2, or 1821 lbs. nearly. 143 THE STEAM-ENGINE. THE power of the steam-engine is estimated by that exerted by the horse. A horse-power, as fixed by Watt, is equal to 33,000 lbs. avoirdupois, raised one foot high per minute; and one day's work of a horse, is this power, acting through eight hours. The pressure of steam is calculated in pounds avoirdupois on the square inch, in perpendicular inches of mercury, and in atmospheres; each atmosphere is estimated as equal to the average pressure of our atmosphere at the surface of the earth. The pressure of the atmosphere is reckoned as equal to that of 30 perpendicular inches of mercury; or 14'7 lbs. per square inch, or 11955 lbs. per circular inch. To find the Horses' power of an Engine, according to AMr. Watt's rule. From the diameter of the cylinder in inches, subtract 1, square the remainder, multiply the square by the velocity of the piston in feet per minute, and divide the product by 5640. The quotient will be the number required. GENERAL PROPORTIONS OF CONDENSING ENGINES. Cylinder. —The best proportion is when the length is twice the diameter; because the cooling 144 STEAM-ENGINE. surface is then least, in proportion to the content of steam. Air-Pump and Condenser.-In double condensing engines, these are made, by Boulton and Watt's rule, each to measure one-eighth the content of the cylinder. Velocity of the Piston to produce the best effect.In engines working the steam expansively, 100 times the square root of the length of the stroke in feet, is the best velocity in feet per minute. In engines not working expansively, 103 times the square root of the length of the stroke in feet, is the best velocity in feet per minute. To find the quantity of Water required for Steam and Injection.-Multiply the area of the cylinder in feet, by half the velocity in feet for single, and by the whole velocity in feet for double engines. Add - for cooling and wasting; and this, divided by 1497, (at the common pressure on the valve of 2 lbs. per circular inch,) will give the quantity of water required for steam per minute. The quantity of water for injection should be 24 times that required for steam. The diameter of the injection-pipe should be oath part of that of the cylinder. The valves should be as large as practicable. The boiler should be capable of evaporating about 12 gallons per hour for each horse-power. STEAM-ENGINE. 145 NON-CONDENSING, OR HIGH PRESSURE ENGINES. The length of the cylinder should be at least twice its diameter. The velocity of the piston, in feet per minute, should be 103 times the square root of the length of the stroke in feet; or 100 times, if the steam is worked expansively. The area of the cylinder should be, to the area of the steam-passages, as 4800 is to the velocity of the piston, found as above. Form and Direction of Steam-pipes. —Enlargements in steam-pipes succeeded by contractions, always retard the velocity of the steam-more or less according to the nature of the contraction-and the like effect is produced by bends and angles in the pipes. These should, therefore, be made as straight, and their internal surface as uniform and free from inequalities as may be practicable. The following proportions of velocity, from Mr. Tredgold, will exemplify this:The velocity of motion that would result from the direct unretarded action of the column of fluid which produces it, being unity.............. 1000 or 8 The velocity through an aperture in a thin plate by the same pressure is...............................625 or 5 Through a tube from two to three diameters in length, projecting outwards......................... -813 or 6-5 Through a tube of the same length, projecting inwards.................................................... 681 or 5-45 Through a conical tube, or mouth-piece, of the form of the contracted vein......I................... 983 or 7-9 13 146 STEAM-ENGINE..Friction of the parts of Engines.-The amount of pressure upon the piston, expended in overcoming friction, appears, on an average, to be not more than 1 lb. to the square inch, in well-constructed engines. The difference in loss of power from this cause, between beam and direct action engines, is found by experiment to be inconsiderable. Mr. Tredgold's Estimate of the Distribution and Expenditure of the Steam in an Engine. IN A NON-CONDENSING ENGINE. Let the pressure on the boiler be.................. 10.000 Force required to produce motion of the steam in the cylinder will be.............................. 0-069 Loss by cooling in the cylinder and pipes........ 0.160 Loss by friction of piston and waste.............. 2.000 Force required to expel the steam into the atmosphere....................................... 0069 Force expended in opening the valves, and friction of the various parts..................... 0.622 Loss by the steam being cut off before the end of the stroke........................................... 1.000 Amount of deductions 3.920 Effective pressure................ 6-080 IN' A CONDENSING ENGINE. Let the pressure on the boiler be................... 10'000 Force required to produce motion of the steam in the cylinder.................................... 0'070 Loss by cooling in the cylinder and pipes........ 0.160 Loss by friction of the piston and waste.......... 1 250 STEAM-ENGINE. 147 Force required to expel the steam through the passages................................................ 0.70 Force required to open and close the valves, raise the injection water, and overcome the friction of the axes.................................. 0.630 Loss by the steam being cut off before the end of the stroke...................................... 1 000 Power required to work the air-pump............ 0.500 Amount of deductions 3'680 Effective pressure............... 6-320 Steam-rower required to drive various kinds of Machinery. A series of experiments instituted by Mr. Davison, at Messrs. Truman and Co.'s Brewery, gave the following results: 1st. That an engine which indicated 50 horses' power when fully loaded, showed, after the load and the whole of the machinery were thrown off, 5 horses', or one-tenth of the whole power. 2d. 190 feet of horizontal, and 180 feet of upright shafting, with 34 bearings, whose superficial area was 3300 square inches, together with 11 pair of spur and bevel wheels, varying from 2 feet to 9 feet in diameter, required a power equal to 7'65 horses. 3d. A set of three-throw pumps, 6 inches in diameter, pumping 120 barrels per hour, to a height of 165 feet, = 4'7 horses. 4th. A similar set of three-throw pumps, 6 inches 148 STEAM-ENGINE. in diameter, pumping 160 barrels per hour, to a height of 140 feet, = 6'2 horses. 5th. A set of three-throw pumps, 5 inches in diameter, raising 80 barrels per hour, to a height of 54 feet, = 1 horse. 6th. A set of three-throw " starting" pumps, pumping 250 barrels of beer per hour, to a height of 48 feet, = 4'87 horses. 7th. Two pair of iron rollers and an elevator, grinding and raising 40 quarters of malt per hour, 85 horses. 8th. An ale-mashing machine, mashing at the time-100 quarters of malt, = 5'68 horses. 9th. Two porter-mashing machines, mashing at the time, 250 quarters of malt, = 10'8 horses. 10th. 95 feet of horizontal Archimedes screw, 15 inches diameter, and an elevator, conveying 40 quarters of malt per hour, to a height of 65 feet,3'13 horses. Steam-engines for Cotton-mills.-With a mean pressure on the piston, with low pressure steam, of 5 lbs. per circular inch, each circular inch will drive three spindles of cotton yarn twist with the machinery. For mule yarn, add 15 to the number of the yarn, and multiply the sum by'26, for the number of spindles for each circular inch of piston. Or, one-horse power will drive 100 spindles with cotton yarn, and machinery. For mule yarn, add STEAM-ENGINE. 149 15 to the number of the yarn, and multiply by 8, for the number of spindles for each horse-power..Economy of Steam-jackets.-The following Table presents the results of three experiments made in France to ascertain the economy of steam-jackets to the cylinders of engines, in the consumption' of fuel. In the 1st, the steam first entered the jacket round the cylinder, and passed from thence into the cylinder. In the 2d, the steam entered the cylinder directly, without passing into the jacket. In the 3d, the steam entered both the cylinder and jacket directly, by means of separate communications between them and the boiler. The result shows an increase in the consumption of fuel of nearly five-sevenths, in the 2d experiment, over that in the 1st. Total Consumption in Mean Pressure in Consumption Water d Dura- pounds avoirdupois. Atmospheres. per hour in lbs. evapog tion of rated Experi- by 1 lb. meats. Cylin Con- of Coals. Water. Boil'r. Cylin- Con- Coals. Water. Coal. der. dens'r. h. m. 1 43 165 1482-7 8387'1 3'82 2'57 0-26 34-28 193'9 5'66 2 33 30,1982-12 11111 59 3-5 2'55 0-28 58'16 331-7 5'61 3 32 3041469-5 7822'23 3-5 2'73 0'24 45-22 240'7 5632 13* 150 STEAM-ENGINE. MARINE ENGINES. The following Dimensions are given by Mr. Russell, for the Cylinders of Marine Engines of various power: Horse Inches Horse Inches Power. Diameter. Stroke. Power. Diameter. Stroke. 10 20 2 ft. 0 in. 125 59 6 ft. 0 in. 20 27 2 ft. 6 in. 150 62 6 ft. 3 in. 30 32 3 ft. 2 in. 175 66 6 ft. 6 in. 40 36 3 ft. 6 in. 200 70 7 ft. 0 in. 50 40 4 ft. 0 in. 250 76 7 ft. 6 in. 60 48 4 ft. 3 in. 300 82 8 ft. 0 in. 70 46 4 ft. 6 in. 350 87 8 ft. 6 in. 80 49 4 ft. 9 in. 400 92 9 ft. 2 in. 90 52 5 ft. 0 in. 500 100 10 ft. 0 in. 100 55 5 ft. 6 in. The improvements in Marine Engines have of late years been various and extensive. Those in oscillating and direct action engines have far exceeded previous calculation. In recently constructed war-steamers with screw-propellers, the whole machinery is placed seven or eight feet below the water-line, and the screw is driven by direct action at the rate of 45 revolutions per minute. 151 RAILWAYS. Summary of the average Items of the Construction of a Mile of Railway. By Mr. DEMiIPSEY. "The average quantities, per mile, of the several items which are involved in the formation of a double line of railway, of the 4 ft. 81 in. gauge, up to the completion of the permanent way, and exclusive of the stations and buildings, and locomotive and carrying stock, may be computed as follows: "The quantity of excavations in 342 miles of double line of railway, (comprised in ten railways,) amounted to 35,338,000 cubic yards, giving an average of about 103,330 yards per mile, or 58'71 cubic yards of earth-work for each yard forward of the line. Assuming the width of the formation level to be 10 yards, or 30 feet, (which is about the average,) with an additional width of 5 yards on each side, for ditches, hedges, &c., the slopes at 1l base to 1 of height,-and also assuming the whole line to be, either in cutting or embankment, of an average depth of height of 11 feet,-we shall require 56'73 cubic yards of earth-work per yard forward of the line. This is sufficiently near to the actual average of 58'71 yards to answer the pur 152~ RAILWAYS. pose of this general calculation. The average width of land required will thus be, Central width. Base of Slopes. Ditches, &c. 30 + 16.5 + 16-5 +- 15 + 15 =93ft. or31yds., which will give about 11] acres of land per mile. Allowing for severance, &c., this may be assumed at 12 acres. "The quantity of ballasting, 30 feet wide, and 18 inches thick, will equal 5 cubic yards per yard forward, or 8800 cubic yards per mile. " The sleepers, transverse, 8 feet long, and 10 by 5 inches, placed 2 feet 6 inches apart, will require 11'733 cubic feet, or 235 loads of timber; or 4224 sleepers per mile. "' The chairs required, supposing the rails to be rolled in lengths of 15 feet each, will be 1408 joint chairs, and 7040 intermediate; and their weight, reckoning each joint chair at 20 lbs., and each intermediate chair at 15 lbs., will be 12 tons 11 cwt. 1 qr. 20 lbs., and 47 tons 2 cwt. 3 qrs. 12 lb., respectively, or 59 tons 14 cwt. 1 qr. 4 lbs. altogether. "The rails, assuming the weight at 56 lbs. per yard, will weigh 176 tons,-1408 lengths being required. "If two oak trenails and two iron spikes be required for each chair, 16,896 of each will be wanted per mile, with 8448 wooden keys for fixing the rails in the chairs. RAILWAYS. 153 "If felt be interposed between the chairs'and sleepers, and the former be assumed at 10 x 5 inches bearing surface, 2933 square feet of felt will be required per mile. "The timber in the side fences, formed of posts 8 feet long, 6 x 4 inches, 9 feet apart, with four rails 5 x 2~ inches, and intermediate upright stay 3 x 2 inches, will consume as follows: 1174 posts = 1565 cubic feet; 4696 rails =-3666 cubic feet; 1174 stays = 269 cubic feet; or a total of 110 loads. "'Of the masonry, timber, iron, &c., &c., in bridges, viaducts, culverts, drains, retaining walls, &c., scarcely any estimate can be formed. Taking the average of a few cases, the masonry would appear to amount to about 110,000 cubic feet per mile; but in some cases from 30 to 50 per cent. of this quantity is substituted by timber and iron." Weight of Rails. On railways with much heavy traffic, the weight of the rails should be, to insure firmness and durability, as on the London and North-western Railway, about 75 lbs. per yard, and their bearing-surface about 21 inches broad. The best distances for the bearings being about 2 feet 9 inches to 3 feet asunder. 154 RAILWAYS. Atmospheric Railway. An experimental line of Atmospheric Railway on Hallette's principle, about 100 yards long, has been laid down, to exhibit its peculiar valve, and its working power, on a small scale. The valve is closed by longitudinal caoutchouc pipes, covered with cotton and leather, and filled with compressed air, at about 5 lbs. to the inch pressure. The wear and tear of these elastic lips, by the continual rubbing of the wedge which opens them as the train passes, can be satisfactorily ascertained only by experiments on a large practical scale. Menai Tubular Bridge. The tubular bridge designed by Mr. Stephenson for crossing the Menai Straits, or the line of the Chester and Holyhead Railway, is proposed to be rectangular, and of the following dimensions, viz. length 450 feet, width 15 feet, height 30 feet; made of iron plates one inch in thickness. Numerous experiments have been made on the strength of iron tubes, by.Messrs. Hodgkinson and Fairbairn, to determine the requisite strength, and the weight it would support. The estimated strength of this tube would be equal to 1100 tons applied in the centre, including its own weight; or 747 tons, dedlucting its own weight. But this being the full strain that the tube would bear without breaking, a RAILWAYS. 155 much less weight must be fixed upon as within the point of safety. The addition of chains is proposed to add to the support of the tube, and experiments are still in progress to determine a form that would sustain a more considerable weight. Its practicability has been established, in the opinion of the engineers, by the results of experiments on a tube 75 feet long, 21 feet wide, and 41 feet deep, weighing about 5 tons; which broke with a weight on the middle of 35 tons. Resistance to Railway Trains. The resistance of the air to railway trains is estimated, by Mr. Barlow, at not more than ten pounds on each ton weight, on the average. The loss of velocity estimated by comparison of the actual with the theoretical velocity, is caused by the consumption of power in overcoming the inertia of the train, and not from defect or loss of power in the action of the engine. A paper was read at the meeting of the British Association in September, by Mr. Scott Russell, on " The law which governs the resistance to the motion of railway trains at high velocities." His experiments have been undertaken " on a large scale, with railway trains of a great variety of size and weight, and at velocities as high as sixty-one miles an hour," and were combined with those formerly 156 RAILWAYS. made by the Association, and by Mr. W. Harding; and he presents the results in the following Table. Mr. Russell remarks that the experiments show not only a great amount of resistance at high velocities, but likewise a great variation and anomaly in the results. He describes the resistance as consisting of three elements. Firstly, the friction of the axles and wheels, as an ascertained quantity, equal in the best carriages to 6 lbs. per ton of the train. Secondly, the resistance of the air; which, acting on a solid body such as a railway train, he regards as much less in amount than that inferred by Smeaton, from experiments on thin plates. And, thirdly, a large amount of resistance, increasing with the velocity of the train, and amounting, at ten miles an hour, to about 3 lbs. per ton, at thirty miles to 10 lbs., and at sixty miles an hour to 20 lbs. per ton; and which Mr. Russell ascribes to the concussions, oscillations, frictions of various kinds, &c., which are produced at high velocities. Mr. Russell has constructed a formula compounded of'these resistances, the comparison of the results of which, with those of the actual experiments, is shown in the Table; but the great anomalies observable at the various velocities remain yet unexplained, and appear incapable of being accounted for on any theory. Much is probably due to the combined action of inaccuracy in construction, and variation in the quality of the materials RAILWAYS. 157 employed. A doubt may likewise be suggested whether, in practice with heavy trains, the quantity of the first-named element of resistance, the friction of the axles, &c., is constant at all velocities. TABLE referred to above. No. of Uniform velocity Resistance Resistance Experiment. maintained in in lbs., per ton, by in lbs,, per ton, by miles, per hour. Experiment. Formula. 1 10 8.40 9.30 2 14 12'60 13-90 3 14 12'60 13-90 4 29 16'50 15'70 5 31 23'30 25.40 6 31 18-20 16'30 7 32 22'50 27'20 8 33 22'50 22*70 9 83 15-68 16'90 10 33 15'96 17-00 11 34 16-60 17-30 12 34 16-95 17'30 13 34 17-70 17'30 14 34 23'30 27'20 15 34 25'00 23-10 16 35 22.50 26'10 17 36 22'50 22'40 18 36 22.40 21-50 19 37 17'50 18-20 20 37 25.00 28 40 21 39 30'00 31'00 22 41' 22-99 19-60 23 41 26.78 19'60 24 45 21-70 21'00 25 46 23'10 21'30 26 46 30-31 31'00 27 47 33.70 33.10 28 50 32'90 36'30 29 51 26.40 23'00 30 53 41-70 42.10 31 61 52.60 54.80 14 158 RAILWAYS. New Sand Cement. A metallic sand cement, of great hardness and tenacity, has been lately used on the London and South-western Railway and elsewhere, in forming mortar and concrete. The sand is brought from Swansea, and the proportions of the cement are,for mortar, the metallic sand, ordinary sand, and lime, in equal quantities; for concrete, metallic sand 1 part, lime 1 part, gravel 6 parts. The iron contained in the metallic sand becomes disseminated through the mass, and acts as a firm bond to the whole composition. GAUGE OF RAILWAYS. The experiments instituted by the advocates of the broad and narrow gauge, respectively, with the object of testing their respective merits as to speed and power, have not led to any satisfactory conclusion. The direction of the wind, the state of the rails, and the inclination of the road operated so variously during the trials, as to destroy all uniformity in the conditions of the several experiments. Mr. Bidder, in his report on the recent gauge experiments, gives the following results:NARROW GAUGE. Date of experiment............... Dec. 30. Dec. 31. Dec. 31. Draft in tons...................... 50 6.... 0...... 80 RAILWAYS. 159 Date of experiment............... Dec. 30. Dec. 31. Dec. 31. Distance travelled in miles..... 42...... 85...... 42 Time in minutes and seconds... 72 6......106 12...... 56 52 Water evaporated, lb............. 12215..... 19708...... 9900 Ditto do. per mile, lb.. 291...... 232..... 235-7 Ditto do. per hour, lb.. 10160...... 11150...... 10430 Cubic feet per hour............... 1621...... 178..... 167 Coke consumed, lb................ 1381............ 1176 Water evaporated per lb. of coke, lb.......................... 9. 3...... 9. 6...... 8.8 Coke consumed per mile, lb.... 31'2...... 24...... 26-6 Pressure............................. 71...... 60 Engine................ A Surface of fire-box, feet square...... 58 Areas of blast pipes in circular inches.............................. 33641 Contents of cylinders, do.............. 4725 BROAD GAUGE. Date of experiment............... Dec. 17. Dec. 16. Dec. 16. Draft in tons....................... 60...... 20...... 80 Distance travelled in miles..... 101...... 1001...... 1014 Time in minutes and seconds...112 42...... 117 4...... 121 3 Water evaporated, lb............ 22596..... 23489...... 24838 Ditto do. per mile, lb.. 232...... 2333 245 Ditto do. per hour, lb. 11820...... 12020...... 12300 Cubic feet per hour............... 198...... 192...... 1961 Coke consumed, lb. Water evaporated per lb. of coke, lb........................... 78...... 7-12...... 7'12 Coke consumed per mile, lb.... 29-6...... 33.6...... 33'6 Pressure. Engine............................... Ixion Surface of fire-box, feet square 97 Areas of blast pipes in circular inches......................... 37211 Contents of cylinders, do........ 4961 160 RAILWAYS. The friction of air through tubes, Mr. Bidder observes, is tolerably well ascertained: it appears that, with a pressure of'04 lbs. per inch, the velo. city of the air through the long tubes of the A engine used in the narrow gauge experiments was 16 miles per hour; and through the shorter tubes of the Ixion 18 miles per hour. Opinion of Mr. W. CUBITT on Uniformity of Gfauge, (Evidence before Select Committee of the House of Lords.) Proposes a 6-feet gauge, the alteration to which would cost between ~500 to ~1000 per mile. Says that the bridges and tunnels on the existing narrow gauge railways are wide enough to admit of rails 6 feet apart, the only alteration necessary in the carriages being that of bringing out the wheels 6 or 8 inches each side, as the carriages themselves are already wide enough, and the wheels would still be under the body of the carriage. The enlarging this gauge to 6 feet would make a better gauge, and enable them to bring the centre of gravity of the engine lower, and allow larger engines than can now be used wiLh safety upon the narrow gauge. Long engines (as long as 20 feet) may be made to run with very large driving wheels, and go safely round curves, by using what is called in America a "Bogy Carriage," viz. supporting the engine on RAILWAYS. 161 two trucks with four low wheels each, which trucks could turn independently at each end, while large driving wheels without flanges may be placed between the trucks. Estimates the altering of carriages to the 6 feet gauge at less than ~30 each, and the engine and tender at ~350 to ~400 each. Extract firom a Return to the House of Lords, of the WORKING STOCK of Existing Railways. Locomotive Passenger Luggage Engines, Carriages. Vans, &c. Arbroath and Forfar...................... 5......... 12...... 110 Birmingham and Gloucester.........40......... 46......... 586 Bristol and Gloucester.................... 11......... 20...... 213 Chester and Birkenhead.................. 10........ 60......... 36 Dublin and Drogheda................. 15......... 69...... 105 Dundee and Newtyle...................... 7......... 9......... 138 (Also 3 stationary engines.) Durham and Sunderland........................... 23...... 28 (13 stationary engines.) Dunfermline and Charlestown.................... 2......... 189 (Horses used on this line.) Eastern Counties........................... 66.........204.........1142 Edinburgh and Dalkeith............................ 28...... 104 (Horses used on this line.) Edinburgh, Leith and Granton................. 8......... (Horses used on this line.) Glasgow, Paisley, Kilmarnock, &c.... 31..........133...... 1334 Grand Junction-including Liverpool and Manchester and Bolton and Leigh....................................128.........343.........1978 Gravesend and Rochester................ 4:. 16......... 6 Great North of England................ 37..... 46......... 717 Great Western..............................127.........232....... 919 Hartlepool....................... 5......... 8...... 6 14* 162 RAILWAYS. Locomotive Passenger Luggage Engines. Carriages. Vans, &c. Hayle and Redruth........................ 7......... 6......... 119 (Also 2 stationary engines.) Hull and Selby............................. 17......... 45......... 238 Lancaster and Preston Junction....... 6......... 3'7....... 36 Leicester and Swannington.............. 8......... 4......... 13 Llanelly and Llandillo....4.......... 4......... 2........ 454 London and Blackwall.............................. 47......... 7 (Eight stationary engines.) London and Brighton..................... 44.........163......... 423 London and Croydon...................... 8......... 56......... 89 London and Southwestern............... 47.........212........ 508 Manchester and Birmingham.......... 27.........100......... 961 Manchester, Bolton, and Bury.......... 12......... 5......... 228 Maryport and Carlisle.................... 8......... 16......... 135 Midland.......................................109.........251......... 1842 Newcastle and Darlington......... 37. 81.........2515 Newcastle and North Shields............ 5......... 28......... 124 Newcastle and Carlisle.................... 26......... 67......... 653 (Also hire 470 coal-wagons.) Newtyle and Coupar....................... 1......... 2......... 48 Norfolk........................................ 18......... 50......... 497 North Union.................................. 19......... 49......... 54 Pontop and South Shields................ 13..................2649 (Newcastle and Darlington passenger-carriages.) Preston and Wyre.......................... 8......... 40......... 108 St. Helen's................................... 9.................. 20 Sheffield, Ashton, and Manchester..... 25......... 105......... 469 Stockton and Hartlepool, and Clarence....................................... 19......... 23. 67 South-eastern............................. 90.........409......... 881 Taff Vale...................................... 12......... 23......... 328 Ulster............................... 11......... 34......... 102 Wishaw and Coltness...................... 11......... 10.........1016 York and North Midland................. 48.........109.......1050 RAILWAYS. 163 COST OF EUROPEAN RAILWAYS PER MILE, extracted from a Report published by order of the House of Commons. Belgian Lines. Ghent to Courtray.................................... ~6.620 Ghent to Bruges....................................... 7,675 Landen to St. Trond................................. 8,990 Louvain to Tirlemont................................ 19,957 Liege to Prussian Frontiers........................ 40,797 Ans to Liege........................................... 62,325 Average of State Lines.............................. 17,132 French Lines. Paris and Orleans.................................... 24,390 Paris and Rouen.................................... 23,754 Strasbourg and Basle................................ 18,485 Amiens and Boulogne................................ 20,000 Rouen and Havre.................................... 28,300 Avignon and Marseilles............................. 28,600 Orleans and Bordeaux.............................. 20,830 The Centre.............................................. 18,050 The North with Calais Branch.................... 19,900 Paris and Lyons.................................... 24,840 Lyons and Avignon, with Branch to Grenoble 25,800 Austrian Lines. Olmiitz to Prague..................................... 11,657 Briinn to Bohmisch Tribau........................ 16,360 Prussian Lines. Berlin and Potsdam................................... 12,323 Magdeburg and Leipsic............................. 10,179 Cologne to Belgian Frontiers...................... 28,334 English Lines, with Scotch and Irish. Arbroath and Forfar................................. 9,214 Chester and Birkenhead............................ 34,198 164 RAILWAYS. Dublin and Drogheda................................ ~15,652 Dublin and Kingstown............................. 59,122 Dundee and Arbroath................................ 8,570 Durham and Sunderland.......................... 14,281 Edinburgh and Glasgow............................. 35,024 Eastern Counties and North-eastern............ 46,355 Glasgow, Kilmarnock, and Air................... 20,607 Glasgow and Greenock.............................. 35,451 Gravesend and Rochester........................... 13,333 Great Western......................................... 43,885 Hartlepool............................................. 26,660 London and Birmingham........................... 38,440 London and Blackwall............................ 287,678 London and Brighton................................ 56,981 London and Croydon................................. 80,400 London and South-western......................... 28,004 Manchester, Bolton, and Bury.................... 70,000 Manchester and Birmingham..................... 61,624 Manchester and Leeds.............................. 64,582 Midland.................................................. 30,949 Newcastle, Darlington, and Brandling.......... 22,992 Newcastle and Carlisle............................. 17,837 Newcastle and North Shields..................... 44,233 Norfolk.................................................. 13,150 North Union, and Bolton and Preston.......... 27,799 Preston and Wyre................................... 22,261 Sheffield and Manchester.......................... 48,543 South-eastern.......................................... 44,412 Taff Vale..................................... 21,610 Ulster'.............................................. 14,334 York and North Midland, &c..................... 25,924 RAILWAYS. 165 Problems, and one which is most generally used. For example, supposing we wish to find out the variation from a straight line on a curve of 400 feet radius with a locomotive whose extreme centre of wheels are 20 feet. We wish to know this to find the width of tire to be put on the forward driving wheels to prevent falling off the rails. We are supposing now an 8-wheel engine, 4 drivers, and 4 wheels in a truck. Fig. 49. 10 10 b Now we have the radius, a b, 400 feet, and the chord, a c, 20 feet, which is the distance the wheel centres are apart; from b, as a centre, we describe the arc, a d c, with b a, as radius. Bisect the chord, a c, at f; then a f will equal 10 feet, and f c equal 166 RAILWAYS. 10 feet. Now to find the distance,f d, which is the versed sine: RULE.-Find the square of a b, and from it subtract the square of a f; the remainder find the square root of, then the product; subtract from a 6, and the remainder will give the distance, df. 400 100 400 160000 100 3)159900(399'87 400 9 399'87 69) 699 0'13 Feet, Answer. 621 789)7800 7101 7988)69900 63904 79967)599600 559769 39831 RAILWAYS. 167 A TABLE of the Fractional Parts of an Inch, when divided into thirty-two parts; likewise a foot of twelve inches, reduced to decimals. Parts. Decimals. Parts. Decimals. Parts of Decimals. a Foot.'03125 i and A'53125 11'9166'0625 and T1'5625 10'8333'09375 i and A'59375 9.75 ~'125 -'625 8'6666 ijand3 ~'15625 and A'65625 7'5833 *and *'1875 4and i'*6875 6.5 iand z'*21875 and A'71875 5'4166 ~'25 i' 75 4'3333 and A'28125 4 and A'78125 3'25 T'3125 J and -'-8125 2'1666 ~375 i and A'84375 1'0833 s and'*40625 7'875 7'07291 iand'4375 - and A'90625 i'0625 and A'46875 7 and T''9375 -'0528 { -.5 - and A'96875 i'04166 A.03125''02083 _01031 The great utility of the above table is to facilitate the multiplication and division of parts of an inch; also in calculations. For example, suppose a sheet of iron to be 20 inches long, 121 and A inches broad, and i and A of an inch in thickness, what number of cubic inches does it contain? 168 RAILWAYS. 20'625 263'58 12'78' 84 165000 105432 144375 210864 41250 20625 221'4072 cubic inches of iron. 263'58150 THE LEVER. i. A LEVER iS an inflexible bar, either straight or bent, supposed capable of turning round a fixed point, called the fulcrum. According to the relative positions of the weight, power, and fulcrum, on the lever, it is said to be of three kinds, viz. when the fulcrum is somewhere betwixt the weight and power, it is of the first kind; when the weight is between the power and the fulcrum, it is of the second kind; and when the power is between the weight and the fulcrum, it is of the third kind, thus: 1st. P. w. F 2d. P. F. w. P. w THE LEVER. 169 2. In the first and second kinds there is an advantage of power, but a proportionate loss of velocity; and in the third kind there is an advantage in velocity, but a loss of power. 3. When the weight x its distance from the fulcrum = the power x its distance from the fulcrum, then the lever will be at rest, or in equilibrio; but if one of these products be greater than the other, the lever will turn round the fulcrum in the direction of that side whose product is the greater. 4. In all the three kinds of levers, any of these quantities, the weight, or its distance from the fulcrum, or the power or its distance from the fulcrum, may be found from the rest, such, that when applied to the lever, it will remain at rest, or the weight and power will balance each other. Weight X its distance from fulc. 5. =- power. Dist. of power from fulc. Power X its distance from fulc. 6. - - weight. Dist. of weight from fulc. Weight X dist. weight from fulc. 7. - dist. power from fulc. Power. Power X dist. power from fulc. 8. - dist. weight from fule. Weight. 9. In the first kind of lever, the pressure upon the fulcrum = sum of weight and power: in the second and third =- their difference. 15' 170 THE LEVER. 10. If there be several weights on both sides of the fulcrum, they may be reckoned powers on the one side of the fulcrum, and weights on the other. Then, if the sum of the product of all the weights x their distances from the fulcrum be = to the sum of the products of all the powers x their distances from the fulcrum, the lever will be at rest; if not, it will turn round the fulcrum in the direction of that side whose products are greatest. 11. In these calculations the weight of the lever is not taken into account; but if it is, it is just reckoned like any other weight or power acting at the centre of gravity. 12. When two, three, or more levers act upon each other in succession, then the entire mechanical advantage which they give, is found by taking the product of their separate advantages. 13. It is to be observed in general, before applying these observations to practice, that if a lever be bent, the distances from the fulcrum must be taken, as perpendiculars drawn from the lines of direction of the weight and power of the fulcrum. Example. —In a lever of the first kind, the weight.s 16, its distance from the fulcrum 12, and the power is 8; therefore by No. 7 of this chapter, 16 x 12 - 24 the distance of power from the fulcrum. THE LEVER. 171 In a lever of the second kind, a power of 3 acts at a distance of 12; what weight can be balanced at a distance of 4 from the fulcrum? Here, by No. 6, 3 x 12 9 weight. In a lever of the third kind the weight is 60, and its distance 12, and the power acts at a distance of 60 x 12 9 from the fulcrum; therefore, by No. 5, - 9 80 the power required. If there be a lever of the first kind, having three weights, 7, 8, and 9, at the respective distances of 6, 15, and 29, from the fulcrum on one side, and a power of 17 at the distance of 9 on the other side of the fulcrum, then a power is to be applied at the distance of 12 from the fulcrum, in the last-mentioned side; what must that power be to keep the lever in balance? Here (6 x 7) + (15 x 8) + (29 x 9) = 423 = the effect of the three weights on the one side of the fulcrum, and 17 x 9 = 153 = the effect *of the power on the other side. Now, it is clear that the effect of the weight is far greater than the effect of the power; and the difference, 423-153 = 273, requires to be balanced by a power applied at the distance of 12, which will evidently be found by dividing 270 by 12, which gives 22'5, the weight required. 172 THE LEVER. 14. The Roman steel-yard is a lever of the first kind, so contrived that only one movable weight is employed. TABLE showing the Effects of a Force of Traction of 100 pounds, at different Velocities, on Canals, Railroads, and Turnpike Roads.* VELOCITY OF LOAD MOVED BY A POWEIR OF 100 LBS. MOTION. On a Canal. On a level Railway. pike Road. Miles Feet per per _ Hour. Second. Tot. Mass Useful Tot. Mass Useful Tot. Mass Useful moved. effect. moved. effect. moved. effect. lbs. lbs. lbs. lbs. lbs. lbs. 2j 3-66 55,500 39,400 14,400 10,800 1,800 1;350 3 4.40 38,542 27,361 14,400 10,800 1,800 1,350 3 5-13 28,316 20,100 14,400 10,800 1,800 1,350 4 5-86 21,680 15,390 14,400 10,800 1,800 1,350 5 7-33 13,875 9,850 14,400 10,800 1,800 1,350 6 8-80 9,635 6,810 14,400 10,800 1,800 1,350 7 10*26 7,080 5,026 14,400 10,800 1,800 1,350 8 11-73 5,420 3,848 14,400 10,800 1,800 1,850 9 13'20 4,282 3,010 14,400 10,800 1,800 1,350 10 14'66 3,468 2,462 14,400 10,800 1,800 1,350 13'5 19 9 1,900 1,350 14,400 10,800 1,800 1,350 * The force of traction on a canal varies as the square of the velocity; but the mechanical power necessary to move the boat, is usually reckoned to increase as the cube of the velocity. On a railroad or turnpike, the force of traction is constant, but the mechanical power necessary to move the carriage increases as the velocity. ! TABLE of the Tractive Power of the Locomotive Engine, when the adhesion is from one-fifth to one-fifteenth that of the insistent weight of the Driving Wheels. Insistent TRACTION IN LBS. WHEN THE ADHESION IS IN THE FOLLOWING RATIOS. weight on driv. wheels, 1 | 1 1 1 5 2240 1866-6 1600 1400 1244-4 1120 1018-1 933-3 861-5 800 746-6 6 2688 2440 1920 1680 1493'3 1344 1221'8 1120 1033-8 960 896 7 3136 2613-3 2240 1960 1792 1568 1425'5 1306.6 1206'1 1120 1045-3 8 3584 2986-6 2560 2240 1991'1 1792 1629 1493'4 1378-4 1280 119446 9 4032 3360 2880 2520 2240 2016 1832-7 1680 1550-7 1440 1344 10 4480 3733.3 3200 2800 2489 2240 2036.3 1866-6 1723 1600 1493-3 11 4928 4106 6 3520 3080 2737'7 2464 2240 2053-3 1895'4 1760 1642-6 12 5376 4480 3840 3360 2986'6 2688 2443'6 2240 2067-7 1920 1792 13 5824 4853-3 4160 3640 3235'5 2912 2647'2 2426-6 2240 2080 1941-3 14 6272 5226-6 4480 3920 3484'4 3136 2851 2613-3 2412'3 2240 2090'6 15 6720 5600 4800 4200 3733'3 3360 3054.5 2800 2584-6 2400 2240 16 7168 5973-3 5120 4480 3982'2 3584 3280 2986-6 2757 2560 2389-3 17 7610 6346-7 5440 4760 4231'1 3808 3483-7 3173-3 2929-3 2720 2538'6 18 8064 6720 5760 5040 4480 4032 3687.5 3360-0 3101 6 2880 2688 19 8512 7093-3 6080 5320 4729 4256 3891 3546'7 3274-0 3040 2837 -3 20 8960 7466-7 6400 5600 4977'7 4480 4094-8 3733'3 3446-3 3200 298687 21 9408.7840 6720 5880 5226'6 4704 4298-3 3920 3618'6 3360 3136 22 9856 8213-3 7040 6160 5475-5 4928 4502 2 4106'7 3791'0 3520 3285'3 23 10304 8586-7 73,60 6440 5724'4 5152 4705-7 4293-3 3963-3 3680 3334.6 24 10725 8960 7680 6720 5973-3 5376 4909.5 4480 4135-6 3840 3484 25 11200 9333'3 8000 7000 6222 2 5600 5091 4666'7 4307'6 4000 3633'3 174 ECCENTRIC WHEEL. TO CONSTRUCT AN ECCENTRIC WHEEfL. From the centre of the shaft 0 take 0 P equal to half the length of the stroke which you intend the wheel to work; and from P as a centre, with any radius greater than P D, describe a circle, and this circle will represent the required wheel. For every circle, drawn from the centre P, will work the same length of stroke, whatever may be its radius; as, whatever you increase the distance of the circumference of the circle from the centre of motion on the one side, you will have a corresponding increase on the opposite side equal to it. Thus, suppose an eccentric wheel to work a stroke of 18 inches is required, the diameter of the shaft being 6 inches; and if 2 inches be the thickness of metal necessary for keying it on to the shaft, then set off, from 0 to P, 9 inches; and 9 + 5 14 inches, the radius of the wheel requlired. Fig. 50. D Pormulz. Let S represent the space the end A is moved through by the eccentric wheel, and s the space the slide moves. Then, A B x s = B C x S; and this equation, ECCENTRIC WHEEL. 175 solved for A B, B C, S, and s, gives the following: BCxS ABxs AB= (1.) S C (3.) s BC ABxs BCxS Bc= s (2.) s AB (4.) Mode of Setting the Eccentrics on the Main Shaft of -Driving Wheels of Locomotives. We suppose the use of an additional eccentric for working the valves half stroke. Fig. 51. Crank pin. Fig. 50 represents the true position of eccentrics on right-hand side of engine when the rock arm is used. Cylinders and eccentric rod supposed to be horizontal, the crank being on its forward centre, g. 176 ECCENTRIC WHEEL. B' G is the length of crank. B/ D is half radius of axle. B' B equal half the throw of valve. B' C equal half the throw of cut-off eccentric. B' A equal lap and lead. Draw the line, E F, perpendicular to the line of eccentric rod and tangent, to the lap and lead circle, and when it intersects the throw of valve at the points, F M and B M, is the centre of the eccentrics; the cut-off eccentric, c o, is set on the line of crank when the throw of eccentric intersects tha line. Crank.Drawn on the Centre. Fig. 52. Crank pin. Fig. 52 is the same construction as fig. 51, work ECCENTRIC WHEEL. 177 ing direct to the valve, without the intervention of rock arm, only the centres of eccentrics are on the left side instead of right, and the forward motion eccentric is below, and the backward motion eccentric above. E F line drawn at right angle with eccentric rod. A B' lap and lead. B B' 4 throw of valve. B' C 4 throw of cut-off eccentric. B' D radius of axle. For any other angle of cylinder or eccentric rod, the construction is precisely similar. The angle of connecting rod, being so trifling, is not taken into consideration in practice. Tire-Bars, Lengths required to make Inside Diameter. Thickness Inside Outside Length of Bar when finished. Diameter. Diameter. required. Inches. Inches. Inches. Feet. Inches. 1j 33 36 9 1t 381 42 10 8 21 39 44 10 9 1t 42J 46 11 7 2 44* 48 12 2 2 51 55 13 83 2 56 60 15 2] 2 68 72 18 3 2 80 84 21 3 2 92 96 24 2 178 BRASS. The alloy of 2 zinc and.1 copper may be crumbled in a mortar when cold. The ordinary range of good yellow brass, that files and turns well, is from about 4~1 to 9 oz. to the pound. Brazing solders-3 copper and 1 part zinc, (very hard;) 8 parts of brass and 1 zinc, (hard;) 6 parts brass and 1 tin, 1 zinc, (soft.) Solder for iron, copper, and brass, consists of nearly equal parts copper and zinc. Muntz's metal-40 parts zinc, 60 copper. Any proportions between the extremes of 50 zinc and 50 copper, and 37 zinc and 63 copper, will roll and break at the red-heat; but 40 zinc to 60 copper are the proportions preferred. Large bells-4~ oz. to 5 oz. of tin to 1 lb. of copper. Tough brass for engine work —1~ lb. tin, 11 zinc, and 10 lbs. copper. Brass for heavy bearings-2~ oz. tin, 1 oz. zinc, and 1 lb. copper. Babbit's metal —1 lb. copper, 5 lbs. regulus of antimony, and 50 lbs. tin. Melt copper first, add the antimony with a small portion of the tin, charcoal being strewed over the metal in the crucible to prevent oxidation. ON THE SAFETY VALVE AND LEVER. THE apertures for safety valves require no nice calculations. It is only necessary to.have the aper THE SAFETY VALVE AND LEVER. 179 ture sufficient to let the steam off from the boiler as fast as it is generated, when the engine is not at work. The safety valve is loaded sometimes by putting a heavy weight upon it, and sometimes by means of a lever with a weight to move along to suit the required pressure. When the whole weight is put on the valve, to find the pressure to each square inch: — Multiply the square of the diameter of the valve by 7854, and this product will give the area, or number of square inches in the valve. And if the whole weight upon the valve, in pounds, be divided by the number of square inches in the valves, the quotient will give the number of pounds pressure to each square inch in the valve. Ex.-If a weight of 40 lbs. be placed upon a valve, the diameter of which is 3 inches, what will be the pressure to each square inch? 32 x'7854 = 7 square inches; then, 40. 7 = 55 lbs. per square inch. ON THE SAFETY VALVE LEVER. This being a lever of the third order, it may be calculated as follows; and also the weight of the lever will be considered; for when the lever is large and the valve is small, the weight of the lever is such as to produce a very sensible pressure upon the valve to each square inch. But previous to the 180 THE SAFETY VALVE AND LEVER. calculation, it will be necessary to make the following remarks. Since the fulcrum is at one end, and the power or the action of the steam between that end and the movable weight, (see fig. 2, where F is the fulcrum, A is the point where the steam acts, and W the movable weight,) some have taken A for the fulcrum, and thereby have committed very great errors; for, according to this rule, a weight put on at twice the distance from A that A is from F, would, if the weight of the lever were not considered, be twice its own weight upon the valve; whereas, if it had been reckoned from F, its real fulcrum, it would be three times its own weight upon the valve. It has been shown, and indeed it is almost selfevident, that if we have two, three, or four times, &c. the leverage, we will have two, three, or four times &c. the effect produced respectively, the weight remaining the same. Therefore divide the length of the lever by the distance between the fulcrum and valve, and the quotient gives the leverage; and the leverage, multiplied by the weight, gives the whole weight upon the valve; and this product, divided by the number of square inches in the valve, gives the weight per square inch. Or, if the weight per inch be known, multiply the number of pounds per square inch by the number of square inches, and THE SAFETY VALVE AND LEVER. 181 this product gives the whole weight upon the valve, which, divided by the leverage, gives the weight. Or, if the weight be given, divide by it, and the quotient will give the leverage; and the leverage, multiplied by the distance between the fulcrum and the valve, gives the length of the lever. -Ex.-Given the whole length of the lever 24 inches, the distance between the fulcrum and valve 3 inches, the diameter of the valve 2y inches; required the weight put on at the end of the lever, so as to have 50 lbs. per square inch upon the valve; also to divide the lever so as to have 40, 30, 20 lbs. &c. upon the valve with the same weight. (2'5)2 x'7854 = 4'9 = area of the valve. 4'9 x 50 = 245 lbs. whole weight on the valve. 245 = -30'625 lbs. = the weight which must be put on at the end of the lever to give 50 lbs. per square inch. 4'9 x 40 And 30'625 = 6'4; then, 6'4 x 3 = 19'2 inches, the distance from the fulcrum the weight must be placed to have 40 lbs. 24 - 19'2 = 4'8; that is, the weight must be shifted in towards the fulcrum 4'8 inches to have 40 lbs. per inch; and for 30 lbs. per square inch, move it in 4'8 inches more, &c. 16 182 THE SAFETY VALVE AND LEVER. To find what weight must be put on at the end of a Lever to give any number of pounds pressure per square inch upon the Valve, the weight of the Lever being taken into consideration. RULE.-Find the area of the valve by multiplying the square of the diameter by'7854; then multiply this area by the number of pounds per square inch which you want upon the valve, and this product will give the whole weight upon the valve. Next divide the whole length of the lever by the distance between the fulcrum and, valve,* and the quotient will give the leverage which any weight will have when put on at the end of the lever. Multiply this leverage by half the weight of the lever, and the product will give the pressure on the whole valve from the action of the lever alone: add to this product the weight of the valve, &c. and subtract the sum from the whole weight on the valve above mentioned; the remainder will give the weight which will be pressing on the valve from the action of the weight alone; and this, divided by the leverage, gives the weight itself. Note. —If, instead of considering half the weight of the lever to act at the end, we conceive the whole weight to act at the centre of gravity, the result will be the same, the lever being uniform. * What is here meant by the distance between the fulcrum, and valve, is that part of the lever between the fulcrum and the point where the lever acts upon the valve. THE SAFETY VALVE AND LEVER. 183 Ex. 1.-Given the length of the lever 24 inches, the distance between the fulcrum and valve 3 inches, the weight of the valve 3 lbs. the weight of the lever 3 lbs.; it is required to determine what weight must be put upon the end of the lever that it may press 30 lbs. per square inch, the diameter of the valve being 3 inches. Now, 3 x 3 - 9, and 9 x'7854 -70686 area, or number of square inches in the valve; and 7 x 30 = 210 lbs. = whole weight upon the valve; and if we conceive the whole weight of the lever to be concentrated in its centre of gravity, and acting with the leverage of the centre of gravity, the lever being uniform throughout its length, we have, since 12 = the distance between the fulcrum and centre of gravity, 12 3 = 4, the leverage of the centre of gravity; and 3 lbs. the weight of the lever, multiplied by 4, gives 12 lbs. the weight that the lever will give upon the whole valve. 12 lbs. added to 3 lbs. the weight of the valve, gives 15 lbs. the weight from both lever and valve; and this, subtracted from 210 lbs. gives 195' lbs., the weight upon the valve from the action of the weight alone, independent of the weight of the lever; and this, divided by the leverage, gives the weight. Thus, 24. 3 = 8 = leverage of the end of the lever; and 195- 8 gives 24- = 24'375 lbs. the weight put on at the end of the lever to give 30 lbs. per inch, 184 THE SAFETY VALVE AND LEVER. when the weight of the lever is taken into consideration. Now, this being determined, we must mark the lever in those points, where the above found weight will give 20 lbs. per inch, and also 10 lbs. per inch. This is found by inverting the above operation; for you have the weight given, the valve, &c. to find the leverage; and the leverage, multiplied by the distance between the fulcrum and valve, gives the distance from the fulcrdm the given weight must be put. Thus 7 x 20 = 140 = whole weight upon the valve; from this subtract 15 lbs., the weight from the valve and lever, and the remainder gives what the weight must put on = 125 lbs.; and this weight, divided by the given weight, 24'375 lbs., gives 54128 = the leverage; and 5'128 x 3 = 15'384 inches from the fulcrum, for 20 lbs. per inch. Now, to determine the distance from the fulcrum when there are 10 lbs. per inch, proceed as above. Thus, 7 x 10 = 70 lbs. upon the whole valve; subtract from this again 15 lbs., the weight of the lever and valve, and 55 remains; and 55. 24'375 - 2'256 = leverage; and 2'256 x 3 = 6'768 inches from the fulcrum. Ex. 2. —Given the length of the lever 16 inches, and its weight 2 lbs.; the distance between the fulcrum and valve 2 inches, and the weight of the valve and spindle 1~ lb.; to find what weight must be put on at the end of the lever to press 40 lbs. THE SAFETY VALVE AND LEVER. 185 per square inch upon the valve, the diameter of which is 2 inches. The square of 2 is 4; hence, -7854 4 3'1416 = area of the valve, or number of square inches in it. 40 lbs. 125'6640 lbs. = weight on the whole valve. 16 -- 2 = 8 = leverage which the weight will have at the end. Now, to consider half the weight of the lever to act at the end, is the same as to consider the whole weight of the lever to act at its centre of gravity, the lever being uniform. 1 lb. = half the weight of the lever. 8 = leverage at the end of the lever. 8 lbs. = weight on the whole valve from the action of the lever. 1'5 lbs. = weight of valve, &c. 9'5 lbs. = weight on the valve from the action of both lever and valve. 125'664 9.5 116'164 16* 186 THE SAFETY VALVE AND LEVER. That is, the weight put on at the end of the lever must be such as to press 116'164 lbs. on the whole valve; but the leverage of the weight is 8, therefore one-eighth part of this weight will do. Thus, 116'164 -- 8 = 14'52 lbs. = weight which must be put on at the end to press 40 lbs. per inch upon the valve. Now, to mark the lever where we will have 35 lbs., 30 lbs., 25 lbs., 20 lbs., &c. per square inch, we must proceed thus:3'1416 = number of square inches in the area of the valve. 35 157080 94248 109.956 9.5 100'456 = weight on the valve from the action of the weight. And 100'456 -- 14'52 = 6'918 = the leverage which the weight must have; and if this leverage be multiplied by the distance between the fulcrum and valve, thus:6'918 = leverage. 2 = distance between the fulcrum and valve. 13'836 = the distance along the lever from the fulcrum. THE SAFETY VALVE AND LEVER. 187 16 -13836 = 2'164 inches = the distance which the weight must be moved in towards the valve. And if you want 30 lbs. per square inch, move it in 2'164 inches farther, and so on, as far as you please, making the division always 2'164 inches. TABLES OF SAFETY VALVE LEVERS. 1. Diameter of the valve 4 inches, weight of the valve, &c. 3 lbs., length of the lever 24 inches, the distance between the fulcrum and valve 4 inches, and the weight of the lever 8 lbs.; then the weight put on at the end of the lever, to give 30 lbs. per square inch upon the valve, will be 58'332 lbs., or 58 lbs. 54 oz. nearly. For 10 lbs. pressure, dist. from fulcrum 6'765 inches. 20 lbs.................................. 15382 inches. 30 lbs..................................24 inches. 2. Length of the lever 16 inches, distance between the fulcrum and valve 2 inches, diameter of valve 2 inches, the weight of the lever 4 lbs., and weight of valve 4 lb.; it will require a weight of 9'7185 lbs. to be put on at the end of the lever to give 30 lbs. per, square inch upon the valve; and the distances in inches which the weight must be from the fulcrum to give 10, 15, 20, 25, and 30 lbs. are respectively as follow: — 188 THE SAFETY.VALVE AND LEVER. 10 lbs. 15 lbs. 20 lbs. 25 lbs. 30 lbs. 3.068 6.301 9.534 12.767 16 If the weight be taken off from the lever, then the weight on the valve from the action of the lever alone will give 51 lbs. per square inch. Note.-9'7185 lbs.-is 9 lbs. 111 oz. 3. Weight of lever 4 lbs., and weight of valve, &c. 1 lb.; whole length of the lever 24 inches, distance between the fulcrum and valve 3 inches, diameter of valve 3 inches, weight put on at the end 42'05375 lbs. Distances from the fulcrum in inches:10 lbs. 20 lbs. 30 Ibs. 40 lbs. 50 lbs. 388292 8-8719 13 9146 189573 24 SHRINKAGE OF TIRE BARS. The general allowance for shrinkage is - of an inch to the foot of diameter of wheel centre. EXAMPLE.-Suppose we wish to turn a centre of driving wheel to fit a tire which is 5 feet the inside diameter, equal to 188k inches circumference; then we must turn the centre i larger in circumference, which is 1891 inches. SPRING STEEL. 189 Shrinkage of Castings. In making all patterns of work, we make an allowance of " inch larger per foot in cast iron; for brass allow I full. SPRING STEEL. I give the following result of experiment made by me with spring steel. The bar made use of was supported on both extremities, and the weight suspended in the middle. The following are the results:-Length between the fulcrum 2 feet; width of bar 1 inch. Number of Bending Deflection. Thickness Experiments. Weight. of Bar. 1 82 lbs. -f i s i 2 118 lbs. A| i4 The table shows that the deflection was equal in both cases. At the same time it appears that the squares of the thickness are to each other as the bending weights, -12 25 very near, thus conforming to the general theory. For calculating the Radius of a Curve when the Angle of Deflection and Chord are given. Railroad curves are always laid off with chords of 100 feet, and we often find, when speaking of 190 RADIUS OF A CURVE. curves, the angle of deflection is merely given. Now to find the radius: 5730 feet are a common radius, which is equal to a deflection of 1~. RULE. —Divide the number of degrees deflected into 5730; the product will be the radius of the curve. EXAMPLE.-We have a curve with a deflection of 60, and the chord 100 feet. 6)5730 955 feet radius of curve, Ans. REVOLUTIONS OF DRIVING WHEELS. 191 TABLE of Revolutions, per Mile, of Driving Wheels, and con. sumption of Steam, Water, and Fuel, for each sized Wheel; taking the Steam admitted to each Cylinder as exactly one cubic foot at a gross pressure of 114'7 lbs., or 100 lbs. on the spring balance. DRIVING WHEELS. CONSUMPTION PER MILE. Diame- Circum- Revolu- Anthrater. fere|ce. tions per Steam. Water. Wood. cite Coke. Mile. Coal. Feet. Feet. Number. Cubic feet. Pounds. Pounds. Pounds. Pounds. 3 9-4248 5602 2240 478 119-5 76-33 59 75 3j 11.0 480-1 1920-8 409 74 102 41 (;83 51-22 4 \ 12-566 420 3 1680-4 358.45 89*61 59-76 44*8 4f 14 137 373.4 1493-6 318-6 79 65 5381 39*8 5 15-708 33683 1345~2 286 94 71-73 47-82 35 87 55 17-278 305-6 1222-4 261835 65 34 43-56 32-67 6 18 849 280*5 1122 239 33 59 83 39189 29192 61 20O420 258 6 10344 220~65 55-14 3678 27-58 7 22-0 2400 960 204-78 51 2 34-13 25-6 71 23-562 224-0 906 193-26 48-31 32 21 24-16 8 25-132 210-1 840-4 179 27 44-82 2988 22-41 81 26-703 197-4 7896 168-43 42-11 2807 21-05 9 28-274 186-7 745 8 159-1 39.8 26-5 19-9 91 29-845 176-9 707-6 150-9 37-82 25-15 18-9 10 31-416 16880 672 143 -35 35-84 23-89 18 192 THE MECHANICAL POWERS, WEIGHTS REQUIRED TO CRUSH SOME OF THE MORE IMPORTANT MATERIALS. i. Metals. On the square inch. Cast iron.......................................115813-177776 lbs. Brass, fine...................................... 164864 Copper, molten............................... 117088 Capper, hammered.................... 103040 Tin, molten............................. 15456 Lead, molten.......................... 7728 2. Woods. Oak.............................................. 3860-5147 Pine.......................................... 1928 Pinus sylvestris.............................. 1606 Elm............................................ 1284 3. Stones. Gneiss......................................... 4970 Sandstone, Rothenburg.................... 2556 Brick, well baked........................... 1092 THE MECHANICAL POWERS. Power is compounded of the weight or expansive force of a moving body multiplied into its velocity. THE power of a body which weighs 40 lbs., and moves with the velocity of 50 feet in a second, is the same as that of another body which weighs 80 lbs., and moves with the velocity of 25 feet in a THE MECHANICAL POWERS. 193 second; for the products of the respective weights and velocities are the same. 40 x 50 = 2000; and 80 x 25 = 2000. Power cannot be increased by mechanical means. Power is applied to mechanical purposes by the lever, wheel and axle, pulley, inclined plane, wedge, and screw, which are the simple elements of all machines. The whole theory of these elements consists simply in causing the weight which is to be raised, to pass through a greater or a less space than the power which raises it; for, as power is compounded of the weight or mass of a moving body multiplied into its velocity, a weight passing through a certain space may be made to raise, through a less space, a weight heavier than itself. Power is gained at the expense of space, by the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. LEVER. Case 1.- When the fulcrum of the lever is bstween the power and the weight. RULE.-Divide the weight to be raised by the power to be applied; the quotient will give the difference of leverage necessary to support the weight in equilibrio. Hence, a small addition either of 17 194 THE MECHANICAL POWERS.'leverage or weight will cause the power to preponderate. EXAMPLE 1.-A ball weighing 3 tons, is to be raised by 4 men, who can exert a force of 12 cwt.: required the proportionate length of lever? 60 3 tons = 60 cwt.; and12 = 5. In this example, the proportionate lengths of the lever to maintain the weight in equilibrio, are as 5 to 1. If, therefore, an additional pound be added to the power, the power side of the lever will preponderate, and the weight will be raised. But, although the ball is raised by a force of only onefifth of its weight, no power is gained, for the weight passes through only one-fifth of the space. The products, therefore, arising from the multiplication of the respective weights and velocities are the same. EXAMPLE 2.-A weight of 1 ton is to be raised with a lever 8 feet in length, by a man who can exert, for a short time, a force of rather more than 4 cwt.: required at what part of the lever the fulcrum must be placed? 20 cwt. 4 cwt. 5; that is, the weight is to the power as 5 to 1; therefore, 5 x -1 foot and a third from the weight. EXAMPLE 3.-A weight of 40 lbs. is placed one THE MECHANICAL POWERS. 195 foot from the fulcrum of a lever; required the power to raise the same when the length of the lever on the other side of the fulcrum is five feet? 40 x 1 5 - 8 lbs., Ans. CASE 2.- When the fulcrum is at one extremity of the lever and the power at the other. RULE.-AS the distance between the power and the fulcrum is to the distance between the weight and the fulcrum, so is the effect to the power. EXAMPLE 1.-Required the power necessary to raise 120 lbs., when the weight is placed six feet from the power, and two feet from the fulcrum? As 8:2::120:30 lbs., Ans. EXAMPLE 2.-A beam, 20 feet in length, and supported at both ends, bears a weight of two tons at the distance of eight feet from one end; required the weight on each support? 40 cwt. x 8 feet 20 feet - 16 cwt. on the support that is furthest from the weight; and 20 feet = 24 cwt. on the support nearest to the weight. WHEEL AND AXLE. RULE.-As the radius of the wheel is to the radius of the axle, so is the effect to the power. EXA-MPLE.-A weight of 50 lbs. is exerted on the periphery of a wheel whose radius is 10 feet; re 196 THE MECHANICAL POWERS. quired the weight raised at the extremity of a cord wound round the axle, the radius being 20 inches. 50 lbs. x 10 feet x 12 inches 20 inches. --- 300 lbs., Ans. PULLEY. RULE.-Divide the weight to be raised by twice the number of pulleys in the lower block; the quotient will give the power necessary to raise the weight. EXAMPLE. —What power is required to raise 600 lbs., when the lower block contains six pulleys? 600 6 x2 50 lbs., Ans. INCLINED PLANE. IRULE. —AS the length of the plane is to its height, so is the weight to the power. EXAMPLE.-Required. the power necessary to raise 540 lbs. up an inclined plane, five feet long and two feet high. As 5: 2:: 540: 216 lbs., Ans. WEDGE. CASE 1. — Wen two bodies are forced from one another by means of a wedge, in a direction parallel to its back. RULE.-As the length of the wedge is to half its back or head, so is the resistance to the power. THE MECHANICAL POWERS. 197 EXAMPLE.-The breadth of the back or head of the wedge being three inches, and the length of either of its inclined sides 10 inches, required the power necessary to separate two substances with a force of 150 lbs. As 10 1:: 150: 22 lbs., Ans. CASE 2. —When only one of the bodies is movable. RULE.-As the length of the weight is to its back or head, so is the resistance to the power. EXAMPLE.-The breadth, length, and force, the same as in the last example. As 10:3:: 150:45 lbs., Ans. SCREW. The screw is an inclined plane, and we may suppose it to be generated by wrapping a triangle, or an inclined plane, round the circumference of a cylinder. The base of the triangle is the circumference of the cylinder; its height, the distance between two consecutive cords or threads; and the hypothenuse forms the spiral cord or inclined plane. RULE.-TO the square of the circumference of the screw, add the square of the distance between two threads, and extract the square root of sum. This will give the length of the inclined plane; its height is the distance between two consecutive cords or threads. 17-* 198 THE MECHANICAL POWERS. When a winch or lever is applied to turn the screw, the power of the screw is as the circle described by the handle of the winch, or lever, to the interval or distance between the spirals. Velocity is gained at the expense of power by the lever and the wheel and axle. LEVER. CASE. —When the weight to be raised is at one end of the lever, the fulcrum at the other, and the power is applied between them. RuLE.-As the distance between the power and the fulcrum is to the length of the lever, so is the weight to the power. EXAMPLE.-The length of the lever being eight feet, and the weight at its extremity 60 lbs., required the power to be applied six feet from the fulcrum to raise it? As 6:8:: 60:80 lbs., Ans. N. B. —Any other example may be computed by reversing any of the foregoing operations. FRICTIO N. WE have considered the effects of the first movers of machinery, and we must now direct our attention to the subject of Friction, which, as we have FRICTION. 199 frequently noticed, tends to diminish these effects. On this subject it is not our intention to dwell long, as all the researches that have been hitherto made in this branch of mechanical science are not of such a nature as to furnish means of deducing satisfactory laws. The resistance arising from one surface rubbing against another is denominated friction; and it is the only force in nature which is perfectly inert-its tendency always being to destroy motion. Friction' may thus be viewed as an obstruction to the power of man in the construction of machinery; but, like all the other forces in nature, it may, when properly understood, be turned to his advantage,-for friction is the chief cause of the stability of buildings or machinery, and without it animals could not exert their strength. The friction of planed woods and polished metals, without grease, on one another, is about one-fourth of the pressure. The friction does not increase on the increase of the rubbing surfaces. The friction of metals is nearly constant. The friction of woods seems to increase after they are some time in action. The friction of a cylinder rolling down a plane, is inversely ag the diameter of the cylinder. The friction of wheels is as the diameter of the axle directly, and as the diameter of the wheel in 200 FRICTION. versely. The following hints may be of use for the purpose of diminishing friction: The gudgeons of pivots and wheels should be made of polished iron, and the bushes or collars in which they move should be made of polished brass. In small and delicate machines, the pivots or knife edges should rest on garnet. Oily substances diminish friction-swine's grease and tallow should be used for wood, but oil for metal. Black lead powder has been used with effect for wooden gudgeons. The ropes of pulleys should be rubbed with tallow. As to the friction of the mechanic powers. The simple lever has no such resistance, unless the place of the fulcrum be moved during the operation. In the wheel and axle, the friction on the axis is nearly as the weight, the diameter of the axis, and the angular velocity —it is, however, very small. When the sheaves rub against the blocks, the friction of the pulley is very great. In most, if not in all screws, the friction of the screw is equal to the pressure —the square-threaded screw is the best. In the inclined plane, the friction of a rolling body is far less than that of a sliding one. To estimate the amount of the friction of a carriage upon a railway, we have, PxT P x T= friction, in which rule P signifies the power that will keep FRICTION. 201 the wagon on the plane, independent of friction; T the time of descent without friction,-both of which are to be determined by the laws of the inclined plane given before: and t is the time of actual descent of the wagon or carriage. There is a loaded carriage on a railroad 120 feet in length, having an inclination of one foot to the hundred. The carriage, together with its load, weighs 4500 lbs. Now, the height of the plane may be found by the principles of geometry, from the proportion of similar triangles. 100: 120:: 1: 1-2 = the height of the plane; and by the laws of falling bodies, and the properties of the inclined plane, x 120 — 2731 x 120 = 32'772 = the time in seconds in which the carriage would descend down the plane without friction —and by the properties of the inclined plane, 100: 1:: 4500: 45 = the force that sustains the carriage, without friction, from rolling down the plane; wherefore, by the rule, 45 x 32'772 45 - = 20'421 = the friction in pounds, which retards the carriage in rolling down the railway. TABLE Of Comparative Velocity of Driving Wheels to Pistons, the Circumference of the former taken as 1. LHalf [ DIAMETER OF WHEELS, IN FEET. travel of Piston, in. Inches. Inches. 3j 4 4j 5 5 6 6 7 7~8 24 0.3638 0'3182 0'2830 0'2541 0.2315 0'2122 0'1958 0'1819 0'1691 0-1592 22 0'3334 0'2918 0'2593 0'2334 0'2122 0'1945 0'1796 0'1667 0'1656 0-1417 20 0'3031 0'2652 0'2393 0'2122 0.1929 0'1768 0.1632 0'1516 0'1414 0-1326 L 19 0'2880 0'2519 0'2273 0'2016 0'1832 0'1679 0.1550 0'1440 0'1343 -01259 18 0'2728 0'2386 0'2151 0'1910 0'1736 0'1591 0.1468 0'1364 0'1272 0-1194 17 0-2577 0'2254 0'2034 0'1803 0'1640 0'1503 0.1387 0'1288 0'1202 0'1127 16 0.2425 0'2121 0'1914 0'1697 0-1543 0'1415 0'1305 0'1213 0'1131 0'1061 15 0'2273 0'1989 0'1795 0'1691 0.1447 0'1326 0.1224 0'1137 0'1060 0'0994 14 0'2122 0'1856 0'1675 0'1485 0.1350 0'1237 0.1141 0'1061 0'0990 0'0928 13 0'1971 0'1724 0'1555 0'1379 0'1254 0'1149 0.1061 0'0985 0'0919 0-0862 12 0'1819 0'1591 0'1436 0'1273 0'1157 0'1061 0'0979 0'0909 0'0848 0'0796 TABLE of Gradients, and Resistance per Ton for each Gradient. VERTICAL RISE. Gravity due VERTICAL RISE. Gravity due to incline, to incline, Ratio, Per Mile. per Ton. Ratio, Per Mile. per Ton. One in Feet. Lbs. One in Feet. Lbs. 100 52.80 22.40 60 88.00 37.333 99 53-33 22 626 59 89-49 37 966 98 53-88 22 858 58 91.03 38*620 97 54-43 23.092 57 9263 39-298 96 55o00 23-334 56 94-28 40.0 95 55.60 23,579 55 96.00 40 726 94 56'17 23'830 54 97'77 41'480 93 66.77 24.086 53 99.62 42-264 92 65752, 24-342 52 101-53 43-076 91 58-02 24-614 51 103-52 43 920 90 58-66 24-888 50 105-60 44-800 89 59 33 25'168 49 107-75 45-716 88 60-0 25.454 48 110-00 46'688 87 60.69 25.746 47 112-34 47.660 86 61-39 26.046 46 115-04 48-684 85~16 62,00 26,303 45 117.33 49-777 85 62.12 26-353 44 120.0 50.908 84 62.86 26,666 48 122-78 52.092 83 63-61 26-988 42 125.71 53.333 82 64-39 27.317 41 128.78 54.634 81 65.20 27.718 40 132.00 56.00 80 66.0 28.00 39 135-38 57.436 79 66.83 28-355 38 138.95 58.944 78 67-69 28-718 37 142-70 60.540 77 68.57 29.090 36 146-66 62*222 76 69*47 29-472 35 150-84 64.000 75 70-40 29-867 34 155-30 65-880 74 71-38 30 270 33 160-0 67 880 73 72.32 30 685 32 165-0 70.0 72 73*33 31-111 31 170-32 72-216 71 74*36 31-550 80 176-00 74.666 70 75.43 32.000 29 182-06 77.240 69 76-49 32.464 28 188.56 80.00 68 77-64 32.940 27 195-55 82.960 67 78,81 33*932 26 203.06 86.152 66 80.0 33.940 25 211.20 89.60 65 81'23 34.460 24 220'0 93'336 64 82'50 35.0 23 229'56 97'368 68 83'81 35'555 22 240 101-816 62 85-16 36-108 21 251 43 106*666 61 86-55 36-720 203 TABLE OF TIME Occupied in running One Mile, Speed in Feet per Minute, and Number of Revolutions of the Driving Wheels, or Double Strokes of the Piston, per Minute, at the following given speeds. REVOLUTION OF WHEELS, PER MININUTE-THE DIAMETER OF WHEELS BEING IN FEET. Speed Speed Time of per per running Hour. Minute. 1 mile. 3 feet. 33 ft. 4 feet. 4i feet. 5 feet. 5, feet. 6 feet. 6 feet. 7 feet. 7I feet. 8 feet. MIiles. Feet. Min. Sec. No. No. No. No. No. No. No. No. No. No. No. 10 880 6 00 93.37 80. 70- 62-25 56-02 50'93 46-68 43'09 40 37'35 35-01 11 968 5 27 102.71 88. 77- 68.47 61-62 56-02 51.35 47.40 44* 41-08 38-51 12 1056 5 00 112.04 96' 84' 74'70 67-23 61'12 56-02 51-71 48. 44'82 42-02 0 13 1144 4 37 121"37 104' 91' 80.92 72-73 66-21 60'69 56'02 52: 48-55 45-52 ] 14 1232 4 17 130-70 112' 98' 87-14 78'33 71'30 65-36 60-33 56- 62-29 49' 15 1320 4 00 140-06 120' 105' 93.37 84-03 76.40 70.03 64.64 60- 56-02 62-53 16 1408 3 45 149-39 128. 112. 99-69 89.63 81-50 77.70 68]95 64* 59-76 56 17 1496 3 32 158-72 136. 119' 105-82 95-23 86'59 79'37 73-26 68 63-49 59-5 18 1584 3 20 169.06 144. 126. 112-04 100-84 91-68 84-03 77'57 72- 67'23 63' 19 1672 3 09~ 178.39 152 133' 118'26 106'44 96-77 88-70 81-88 76' 7096 66-5 20 1760 3 00 187-72 160. 140- 124'48 112'04 101'86 93.37 86-19 80- 74'70 70. 21 1848 2 51~ 197-05 168. 147. 130-72 117-65 106-95 98-04 90o50 84- 78-43 73-5 22 1936 2 43] 206.38 176' 154' 136-94 123925 112-04 102'71 94-81 88- 82'17 77[ 23 2024 2 361 215.71 184- 161. 143'17 128.85 117-14 107-38 99.12 92' 85]50 80-5 24 2112 2 30 225.05 192- 168. 149-40 134-45 122.24 112-05 103-43 96' 89-63 84. 25 2200 2 24 233.48 200' 175 155.63 140.05 127.33 116.72 107-74 100. 93.37 87.5 26 2288 2 18j 242'77 208' 182 161.85 145-65 132.42 121.39 112-05 104' 97.14 9127 2376 2 13* 255 74 216' 189' 168-07 151-26 137.52 126.05 116.36 108. 100.87 94-5 28 2464 2 08~ 261.43 224' 196' 174.3 156,86 142.63 130.72 120.67 112' 104-57 98. 29 2552 2 04 270'78 232' 203' 180.52 162.47 147'74 135.38 124.98 116. 108.32 101.5 30 2640 2 00 280.11 240 2910 186.74 168.08 152.85 140.04 129.30 120. 112.02 105' 31 2728 1 56 289'52 248' 217' 192.97 173.78 157.98 144-71 133-61 124. 115-75 108-5 32 2816 1 52i 298.79 256' 224' 199-20 179-39 163' 149.38 137-62 128. 119-5 112' 33 2904 1 49 308012 264' 231' 205-42 185s 168.1 154-05 141.93 132- 123.23 115.5 34 2992 1 45i 317.46 272 238 211-64 190-60 172-99 158-72 146-24 136. 127.06 119' 35 3080 1 421 326-80 280 245| 217.13 196.21 177.29 163.33 150-85 140. 130-69 122.5 36 3168 1 40 336.13 288' 252 224-09 201.82 182.4 168' 155-16 144' 134.42 126. 37 3256 1 371 345-47 296- 259 230'32 207-42 187'5 172-67 159.41 148. 138.16 129-5 38 3344 1 34 354-81 304' 266' 286.54 213.03 192.6 177.34 163-72 152' 143 90 133' 39 3432 1 32i 364.15 312' 273' 24277 21863 197-95 182-05 168'03 156' 145-63 136.5' 40 3520 1 30 37357 320' 280' 252-72 224-24 203.76 186-72 172.40 160- 149.36 140' 42 3696 1 253 392.15 336' 294' 261-44 235.44 214' 196-05 181-02 168' 156-83 14745 3960 1 20 420-17 360- 315' 284-30 252-27 229-23 210-06 193-95 180. 168.03 157.5 48 4224 1 15 438.18 384' 336' 298.79 269.10 244.55 224.06 206.88 192. 179-23 118' 60 4400 1 12 446.85 400' 3508 315'90 280'30 254-75 233-70 215.50 200' 186.70 1853 55 4840 1 05~ 513-54 440- 385' 347-49 308.33 280-17 256-74 237.05 220. 205-37 192-5 60 5280 1 00 560'23 480' 420' 379-08 336'36 305.64 280-5 258.6 240- 224.04 210-1 65 5720 0 55 606.91 520' 455/ 416-67 364.39 331.11 303-42 280.15 260' 242.71 227.5 70 6160 0 51 1653'59 560' 490' 442'19 392'42 356-58 3267'6 301'70 280' 261-38 24575 6600 0 48 700o28 600- 525 473-85 420o50 372-05 350~10 323-25 300' 280o 05 262'5 80 7040 0 45 746.96 640' 560' 505-50 448'48 407.60 373.44 344.80 320' 298-72 280' 85 7480 0 42i 792'74 680' 59562 528.38 476.50 433-25 396'77 346-35 340' 317-42 297.6 90 7920 0 40 [40.34 720 630-27 560-23 504-62 45865 420-11 367'90 360' 336' 315-14 t:a._._.. _..~ 206 COMPARATIVE ELASTICITY OF WROUGItT AND CAST IRON. An able work on tubular bridges gives the following as'the comparative elasticity of wrought and cast iron: "The mean ultimate resistance of wrought iron to a force of compression, as useful in practice, is twelve tons per square inch, while the crushing weight of cast iron is forty-nine tons per square inch, but for a considerable range, under equal weights, the cast iron is twice as elastic, or compresses twice as much as the wrought iron. " A remarkable illustration of the effect of intense strain on cast iron was witnessed by the author, at the works of Messrs. Easton & Amos. The subject of the experiment was a cast-iron cylinder ten and five-eighths inches thick, fourteen and a half inches high, the external diameter being eighteen inches. If requisite for a specific purpose to reduce the internal diameter three and a half inches, this was effected by the insertion of a smaller cast-iron cylinder into the centre of the large one; and to insure some initial strain, the large cylinder was expanded by heating it, and the internal cylinder being first turned too large, was thus powerfully compressed. The inner cylinder was partly filled with pewter, and a steel piston being fitted to the bore, a pressure of 972 tons was LOSS FROM RESISTANCES. 207 put on the steel piston. The steel was'upset' by the pressure, and the internal diameter of the small cylinder was increased by full three-sixteenths of an inch; i. e. the diameter became 3}1 of an inch! A new piston was accordingly adapted to these dimensions-and in this state the cylinder continues to be used, and to resist the pressure; the external layer of the inner cylinder was thus permanently extended 3-1- of its length. In fact, it can only be regarded as loose packing, giving no additional strength to the cylinder. Under these high pressures, when confined mechanically, cast iron, as well as other metals, appears, like liquids, to exert an equal pressure in every direction in which its motion is opposed." LOSS FROM RESISTANCES AGAINST THE PISTON, Produced by imperfect action of the Valves. This is a branch of the subject deserving especial consideration. Its importance, as referring to the economical working of steam-engines, may be profitably illustrated by a brief historical account of the consecutive alterations and improvements in the valve arrangements and mechanism of the locomotive engine, and of the results produced in the saving of fuel. It may be premised that the same principles have 208 LOSS FROM RESISTANCES. their application not only to the engines of other railways, making due allowances for difference in the gradients, and difference in the loads and dimensions of engines, and difference of speed, but also to fixed engines in general. The history of the locomotive engine may, for the sake of convenient classification, be divided into two periods-the first a period of increasing, the second a period of decreasing consumption, as respects the article of fuel. Brief allusion may be made to the events of both periods, and a reference to the causes which retarded as well as to those which accelerated improvement. During the first few years after the opening of railroads, the class of improvements comprising the gradual enlargement of dimensions, as necessary for maintaining higher rates of speed and the transport of heavy bodies, the better disposition and proportionment of the component parts, and the selection of suitable materials, capable of resisting heavy strains and various other causes of derangement and decay, demanded, inY consequence of their direct influence upon the traffic of the companies, unremitting attention. The necessity of securing regularity in the transport of trains, whether of passengers or goods, was pressing and paramount, and afforded sufficient materials for thought and experiment. It is, therefore, a source of less surprise than regret, that little progress should have been LOSS FROM RESISTANCES. 209 made in diminishing the consumption of fuel. Trials of the consumption of different engines of similar size and power were made from time to time, and these agreeing pretty closely together, served to lull suspicion of unnecessary waste of fuel. As the engines increased in dimensions, the consumption of fuel increased also, which was considered a natural and inevitable consequence of the exertion of increased power. The adoption, in 1836, for the passenger traffic, of what were termed short-stroked engines, was attended with the establishment of a quicker rate of travelling than had before been known, but, unfortunately, also with an extravagant increase in consumption of coke. This was erroneously referred to the mechanical disadvantage of the short stroke, an explanation which for a time was deemed satisfactory. Attention was directed to schemes of smoke-burning, by which the use of coal, as a much cheaper fuel than coke or wood, might be rendered possible. Hitherto all engines were furnished with the slide valve ordinarily used in high pressure engines, the mode of operation of which is well known to every practical mechanic. It will be remembered that the operations of admitting the fresh steam and releasing the waste steam are alternately performed by the same valve and by the same motion. The valve, being made to slide backwards and forwards upon the face of the ports, opens and closes the 18* 210 LOSS FROM RESISTANCES. several passages in their turn. The two extreme ones, termed steam ports, communicate with either, Fig. 53.-Old Valve, I'd Lap. end of the cylinder. The middle one is termed the exhausting port, and its corresponding passage terminates in a pipe open to the atmosphere, and carried into the chimney. Steam is admitted freely into the steam-chest from the boiler. The valve is made of sufficient length to cover, when placed in the centre of the stroke, all the parts. In this position no steam can enter the cylinder; but as the valve moves on, one of the ports opens, and the arrangement of the valve-gearing is such, that when the piston is ready to begin its stroke, the steamport begins to open. During the forward progress of the piston, the valve not only travels to the end of its stroke, but returns to the point from whence it set out. Its continued motion in the same direction finally closes the valve, and prevents any further admission of steam. The steam has now done its work, and must be removed. In the middle of the valve a hollow chamber is formed, of sufficient LOSS FROM RESISTANCES. 211 length to span between the ports. As soon as the edge of this chamber passes the edge of the steamport, the pent-up steam finds vent, and rushing through the chamber into the exhausting passage, escapes into the chimney. It will be observed, by referring to fig. 53, that the exhausting port opens when the steam-port closes, and both events happen as nearly as may be at the end of the stroke. The perfection of a slide valve consists, other things being supposed equal, in the degree of nicety with which its motion is timed relatively to the motion of the piston. The functions of the piston are absolutely dependent Fig. 54.' Lap. upon the proper timing of the admission and release of the steam. A most slight and apparently trifling error in the adjustment produces a most serious effect upon the consumption of fuel. If from any cause the valve should open to admit steam for a fresh stroke before the preceding stroke is finished, it opens too soon, and an unnecessary resistance to the piston is produced. If, on the other hand, the valve should delay its 212 LOSS FROM RESISTANCES. opening until the piston has begun to return, it opens too late, because then the steam has uselessly to fill the space left vacant. Hence a waste of steam and loss of power. As far, then, as the admission of steam is concerned, it is a necessary condition that the steam-ports should open neither before nor after, but at the precise moment when the stroke commences. Some engineers, indeed, have recommended giving the valve "lead," as it is termed; that is to say, setting it so as to open a little before the completion of the foregoing stroke; but it seems very questionable whether the slightest advantage is gained by doing so to a greater extent than is necessary to compensate for any slackness in the parts of the valve-gearing, or for their expansion when hot; and about -1 of an inch may be considered sufficient -for this purpose in a well-constructed engine. The valve shown in fig. 53 satisfies the conditions required in the admission of steam. It opens exactly at the right time. The steam begins to enter as the piston begins to move, and follows it steadily and effectively throughout its course. Whatever time the piston takes for its journey, the steam is allowed as much time to follow it. At first the opening is small; but then the motion of the piston is comparatively slow, and therefore the supply keeps pace with the demand. As respects the release of the steam when the LOSS FROM RESISTANCES. 213 stroke has been completed, the performance of this valve is altogether unsatisfactory, and here lurks the cause of the difference in the performances of the old and the late engines. But it might be said the release does appear to take place at the right time, because it occurs just when the piston has finished the stroke; and if it were to occur before, a loss of power would ensue. This is a plausible view of the case, and one which undoubtedly delayed, for years, the saving of fuel which has since been effected. Sufficient attention was not bestowed upon the processes going on in the interior of the cylinder, or upon the facts which might have indicated them. Alternately to fill and empty the cylinder of its contents are operations requiring time. The time allowed for the first operation-that of filling the cylinder with steam-necessarily corresponds with the duration of the stroke, whatever its duration may be. But this cannot be the case as regards the second operation-the emptying of the cylinder. This ought to be performed in an instant, in the minutest fraction of the duration of the stroke; otherwise the steam continues pent up when it ought to be liberated-when it ought to assume its minimum pressure, the pressure of the atmosphereand exerts an injurious counter-pressure against the piston, tending to increase the resistance to be overcome. 214 LOSS FROM RESISTANCES. To effect the free and rapid discharge, it is necessary not merely to open the communication to the exhausting pipe, but to open a wide passage, and to have this down by the time the piston recommences its motion. The valve alluded to cannot accomplish this.. Its motion is gradual, not instantaneous. The passage only begins to open when the piston is in the turn, and it is not wide open until the piston has travelled through Ioth of its entire stroke. The steam in the cylinder is consequently restrained from escaping, being wire-drawn in the passage out, and consequently takes considerable time to assume the pressure of the atmosphere. In the mean while the new stroke has begun, and been partially completed; and so far the piston has had to contend with a resistance altogether illegitimate-a resistance which, in many cases, and especially at high speeds, has been nearly equal to all the other resistances put together. In the year 1838 the extent of the disease was first suspected, and a remedy attempted. It had before been observed that the giving of an engine "lead" tended to improve its speed when travelling already at a high speed, and with a light load. The Circumstance was attributed to the opening of the steam-port being wide at the time of commencing the stroke, thereby increasing the facility for LOSS FRO0M RESISTANCES. 215 the entrance of the steam in following up the piston. Its true explanation was found to be the earlier release of the waste steam, and consequent diminution of resistance. As sometimes 3ths of an inch, or even 1 an inch "lead" was given in passenger engines, it was' decided to try the effect of opening the exhausting passage earlier by the same amount, while the steam-port should still be made to open only at the turn of the stroke. An engine was chosen for the experiment. Its original valve resembled fig. 53. Placing the valve on the ports, so as to allow the exhausting passage to be -ths of an inch open, the steam-port would, at the same time, be ~ inch open. This space, therefore, was closed by adding to the length of the valve, at each end, 4 inch. The eccentric was, of course, shifted on the axle to correspond with the alteration, and the engine with the altered valve (Fig. 54) was again set to work. The amount by which the valve at each end overlaps the steam-ports, when placed exactly over them, is technically termed the lap. The lap of the valve being then 3ths of an inch, the exhausting passage was about 3ths of an inch open when the stroke was finished. The engine was made the subject of several experiments. With passenger trains the saving in fuel was very considerable. the consumption, while running, being only about 25 lbs. of coke per mile, with loads of 216 LOSS FROM RESISTANCES. five to eight cars, and the speed was considerably improved. It here becomes necessary to refer to the consumption of the Liverpool and Manchester engines before and at the time we speak of, in order to form a just conception of the position arrived at. In 1836 and 1837, larger engines were gradually introduced to replace the smaller class, which had become insufficient for maintaining the higher rate of speed then demanded; and their increased consumption of fuel was commensurate with their increase of size. For an idea of the general effect attendant upon their introduction, the following table, showing the coke consumed in several consecutive years, may be consulted. 11,561 trips in 1836, 7,907 tons (gross) coke. 12,063 trips in 1837, 9,876 tons (gross) coke. 12,953 trips in 1838, 10,816 tons (gross) coke. Thus, during three years, when the change went on, although the work done increased only in the proportion of 100 to 136, without material difference in the magnitude of the loads, in 1839 and 1840 the average consumption attained its maximum, being about 49 lbs. per mile, gross, with passenger trains averaging seven cars, and 54 lbs. per mile with freight trains averaging sixteen burthen cars. 40 lbs. net consumption with passenger trains, LOSS FROM RESISTANCES. 217 was moderate for such an engine used for the experiment; and the performance of the engine when altered, being under 30 lbs. net, was naturally considered favourable. This result was evidently obtained from the earlier exhaustion of the steam. Whereas previously the opening of the exhaustion passage was contemporaneous with the termination of the stroke; now it took place before, and was already 3ths of an inch open at the end of the stroke. A portion of the steam could by that time escape, and the back pressure was diminished. The valves of two engines, Nos. 10 and 12, were next altered to have 3ths of an inch lap-No. 10 in January, No. 12 in June, 1841. During the last quarter of the year 1841, the gross consumption of No. 10 was 36- lbs. per mile, and that of No. 12 40 lbs. per mile, and the net consumption about 30 and 33 lbs. No. 12 Valve, waith jt1s of anl iilch La)p. Coke. Week ending cwt. qrs. lbs. Jan. 4, 1841, 12 trips of 30 miles, 130 0 0 Jan. 11, 1841, 12 trips of 30 miles, 127 2 0 Per Trip. - - cwt. qrs. lbs. per mile. 24 257 2 0 = 10 2 25 =40'1 Valve with Iths of an inch Lap. Feb'ry 9, 1841, 10 trips of 30 miles, 88 3 0 March 7,1841, 8 trips of 30 miles, 71 1 0 March 21, 1841, 14 trips of 30 miles, 118 3 0 March 28, 1841, 16 trips of 30 miles, 137 2 0 48 trips. 416 1 0 - 8 2 19 = 32'4 19 Difference 7'7 218 DIMENSIONS OF PARTS OF LOCOMOTIVES. Diameter of Cylinder.-In locomotive engines the diameter of the cylinder varies less than in either the land or the marine engine. In few of the locomotive engines at present in use is the diameter of the cylinder greater than 18 inches, or less than 12 inches. The length of the stroke of nearly all the locomotive engines at present in use is 18 inches, and there are always two cylinders, which are generally connected to cranks upon the axle, standing at right angles with one another. Outside cylinders, operating upon pins in the driving wheels, have latterly been largely introduced. AREA OF INDUCTION PORTS. RULE. —To find the size of the steam-ports for the locomotive engine. —Multiply the square of the diameter of the cylinder by'068. The product is the proper size of the steam-ports in square inches. Example.-Required the proper size of the steam-ports of a locomotive engine whose diameter is 15 inches. Here, according to the rule, size of steam-ports = -068 x 15 x 15 = -068 x 225 = 15'3 square inches, or between 15k and 151 square inches. DIMENSIONS OF PARTS OF LOCOMOTIVES. 219 After having determined the area of the ports, we may easily find the depth when the length is given, or, conversely, the length when the depth is given. Then, suppose we know the length was 8 inches, then we find that the depth should be 15'3 -- 8 = 1'9125 inches, or nearly 2 inches; or suppose we knew the depth was 2 inches, then we would find that the length was 15'3 2 = 7165 inches, or nearly 71 inches. Area of eduction ports.-The proper area for the eduction ports may be found from the following rule. RULE.-1To find the area of the eductionports. — Multiply the square of the diameter of the cylinder in inches by'128. The product is the area of the eduction ports in square inches. Example.-Required the area of the eduction ports of a locomotive engine, when the diameter of the cylinders is 13 inches. In this example we have, according to the rule, area of eduction port = -128 x 132 ='128 x 169 = 21'632 inches, or between 21~ and 211 square inches. Breadth of bridge between ports.-The breadth of the bridges between the eduction port and the induction ports is usually between i inch and 1 inch. DIAMETER OF BOILER. RULE.- To find the inside diameter of the boiler.Multiply the diameter of the cylinder in inches by 220 DIMENSIONS OF PARTS OF LOCOMOTIVES. 3'11. The product is the inside diameter of the boiler in inches. -Example.-Required the inside diameter of the boiler for a locomotive engine, the diameter of the cylinders being 15 inches. In this example we have, according to the rule, inside diameter of boiler = 15 x 3'11 = 46'65 inches, or about 3 feet 10 inches. Length of boiler.-In the Northern and Eastern Counties Railway the length of the boiler is 8 feet; while in the North Midland Counties Railway, in the Great Western Railway, and in the Hartlepool Railway, the length of the boiler is 81 feet. In the Belgian railways the length of the boiler is 8 feet 2 inches. And in the Bordeaux and La Teste railway the length of the boiler is 8 feet 9 inches. In Stephenson's locomotive engines, the length of the boiler is between 11 and 12 feet. In this country the length is from 10 to 14 feet. Diameter of steam dome inside.-It is obvious that the diameter of the steam dome may be varied considerably, according to circumstances; but the first indication is to make it large enough. It is usual, however, in practice, to proportion the diameter of the steam dome to the diameter of the cylinder; and there appears to be no great objection to this. The following rule will be found to give the diameter of the dome usually adopted in practice. DIMENSIONS OF PARTS OF LOCOMOTIVES. 221 RULE.-To find the diameter of the steam dome.Multiply the diameter of the cylinder in inches by 1'43. The product is the diameter of the dome in inches. Height of steam dome.-The height of the steam dome may vary. Judging from practice, it appears that a uniform height of 2~1 feet would answer very well. Diameter of safety-valve.-In practice the diameter of the safety-valve varies considerably. The following rule gives the diameter of the safety-valve usually adopted in practice. RULE.- To find the diameter of the safety-valve. — Divide the diameter of the cylinder in inches by 4. The quotient is the diameter of the safety-valve in inches. Example. —Required the diameter of the safetyvalves for the boiler of a locomotive engine, the diameter of the cylinder being 13 inches. Here, according to the rule, diameter of safety-valve = 13 - 4 = 31 inches. A larger size, however, is preferable, as being less likely to stick. Diameter of valve spindle.-The following rule will be found to give the correct diameter of the valve spindle. It is entirely founded on practice. RULE. —TO find the diameter of the valve spindle.-Multiply the diameter of the cylinder in inches by'076. The product is the proper diameter of the valve spindle. 19* 222 DIMENSIONS OF PARTS OF LOCOMOTIVES. Example.-Required the diameter of the valve spindle for a locomotive engine whose cylinders' diameters are 13 inches. In this example we have, according to the rule, diameter of valve spindle ='13 x'076 ='988 inches, or very nearly 1 inch. Diameter of chimney. —It is usual in practice to make the diameter of the chimney equal to the diameter of the cylinder. Thus, a locomotive engine whose cylinders' diameters are 15 inches, would have the inside diameter of the chimney also 15 inches, or thereabouts. This rule has, at least, the merit of simplicity. Area of fire-grate.-The following rule determines the area of the fire-grate usually given in practice. We may remark, that the area of the fire-grate in practice follows a more certain rule than any other part of the engine appears to do; but it is in all cases much too small, and occasions a great loss of power by the urging of the blast it renders necessary, and a rapid deterioration of the furnace plates from excessive heat. There is no good reason why the furnace should not be nearly as long as the boiler: it would then resemble the furnace of a marine boiler, and be as manageable. RULE. —To find the area of fire-grate.-Multiply the diameter of the cylinder in inches by'77. The product is the area of the fire-grate in superficial feet. DIMENSIONS OF PARTS OF LOCOMOTIVES. 223 Example.-Required the area of the fire-grate of a locomotive engine, the diameters of the cylinders being 15 inches. In this example we have, according to the rule, area of fire-grate = -77 x 15 = 11'55 square feet, or about 111 square feet. Though this rule, however, represents the usual practice, the area of the fire-grate should not be contingent upon the size of the cylinder, but upon the quantity of steam to be generated. Area of heating surface. —In the construction of a locomotive engine, one great object is to obtain a boiler which will produce a sufficient quantity of steam with as little bulk and weight as possible. This object is admirably accomplished in the construction of the boiler of the locomotive engine. This little barrel of tubes generates more steam in an hour than was formerly raised from a boiler and fire occupying a considerable house. This favourable result is obtained simply by exposing the water to a greater amount of heating surface. In the usual construction of the locomotive boiler it is obvious that we can only consider four of the six faces of the inside fire-box as effective heating surface, viz. the crown of the box, and the three perpendicular sides. The circumferences of the tubes are also effective heating surface; so that the whole effective heating surface of a locomotive boiler may be considered to be the four faces of the inside 224 DIMENSIONS OF PARTS OF LOCOMOTIVES. fire-box, plus the sum of the surfaces of the tubes. Understanding this to be the effective heating surface, the following rule determines the average amount of heating surface usually given in practice. RULE.-TO find the effective heating surface.Multiply the square of the diameter of the cylinder in inches by 5; divide the product by 2. The quotient is the area of the effective heating surface in square feet. Example. —Required the effective heating surface of the boiler of a locomotive engine, the diameters of the cylinders being 15 inches. In this example we have, according to the rule, effective heating surface = 152 x 5 - 2 = 225 x 5 *- 2 = 1125 -- 2 = 5621 square feet. According to the rule which we have given for the fire-grate, the area of the fire-grate for this boiler would be about 111 square feet. We may suppose, therefore, the area of the crown of the box to be 12 square feet. The area of the three perpendicular sides of the inside fire-box is usually three times the area of the crown; so that the effective heating surface of the fire-box is 48 square feet. Hence the heating surface of the tubes = 526'5 - 48 = 478'5 square feet. The inside diameters of the tubes are generally about 1l inches; and therefore the circumference of a section of these tubes is 5'4978 inches. Hence, supposing the tube DIMENSIONS OF PARTS OF LOCOMOTIVES. 225 to be 8} feet long, the surface of one = 5'4978 x 82 -- 12 = -45815 x 81 = 3'8943 square feet. And, therefore, the number of tubes = 478'5 3.8943 = 123 nearly. Area of water-level.-This of course varies with the different circumstances of the boiler. The average area may be found from the following rule. RULE.-To find the area of the water-level.Multiply the diameter of the cylinder in inches by 2'08. The product is the area of the water-level in square feet. Example. —Required the area of the water-level for a locomotive engine, whose cylinders' diameters are 14 inches. In this case we have, according to the rule, area of water-level = 14 x 2'08 = 29'12 square feet. Cubical content of water in boiler.-This of course varies, not only in different boilers, but also in the same boiler at different times. The following rule is supposed to give the average quantity of water in the boiler. RULE.-To find the cubical content of the water in the boiler.-Multiply the square of the diameter of the cylinder in inches by 9; divide the product by 40. The quotient is the cubical content of the water in the boiler in cubic feet. Example.-IRequired the average cubical content of the water in the boiler of a locomotive engine, the diameters of the cylinders being 14 inches. 226 DIMENSIONS OF PARTS OF LOCOMOTIVES. In this example we have, according to the rule, cubical content of water -= 9 x 142 -- 40 4441 cubic feet. Content offeed-pump. —In the locomotive engine the feed-pump is generally attached to the crosshead, and consequently it has the same stroke as the piston. As we have mentioned before, the stroke of the locomotive engine is generally in practice 18 inches. Hence, assuming the stroke of the feed-pump to be constantly 18 inches, it only remains for us to determine the diameter of the ram. It may be found from the following rule. RULE.-To find the diameter of the feed-pump ram.-Multiply the square of the diameter of the cylinder in inches by'011. The product is the diameter of the ram in inches. Example.-Required the diameter of the ram for the feed-pump for a locomotive engine whose diameter of cylinder is 14 inches. In this example we have, according to the rule, diameter of ram = -011 x 142 = *011 x 196 - 2'156 inches, or between 2 and 2{ inches. Cubical content of steam room.-The quantity of steam in the boiler varies not only for different boilers, but even for the same boiler in different circumstances. But when the locomotive is in motion, there is usually a certain proportion of the boiler filled with the steam. Including the dome and the steam-pipe, the content of the steam room will be DIMENSIONS OF PARTS OF LOCOMOTIVES. 227 found usually to be somewhat less than the cubical content of the water. But as it is desirable, that it should be increased, we give the following rule. RULE.-To find the cubical content of the steam room.-Multiply the square of the diameter of the cylinder in inches by 9; divide the product by 40. The quotient is the cubical content of the steam room in cubic feet..Example. —Required the cubical content of the steam room in a locomotive boiler, the diameters of the cylinders being 12 inches. In this example we have, according to the rule, cubical content of steam room = 9 x 122 -- 4(p -'9 x 144 -- 40 = 32.4 cubic feet. Cubical content of inside fire-box above fire-bars. — The following rule determines the cubical content of fire-box usually given in practice. RULE. —To find the cubical content of inside firebox above fire-bars.-Divide the square of the diameter of the cylinder in inches by 4. The quotient is the content of the inside fire-box above fire-bars in cubic feet. Example. —Required the content of inside firebox above fire-bars in a locomotive engine, when the diameters of the cylinders are each 15 inches. In this example we have, according to the rule, content of inside fire-box above fire-bars = 152 - 4 = 225 -- 564 cubic feet. Thickness of the plates of boiler.-In general the 228 DIMENSIONS OF PARTS OF LOCOMOTIVES. thickness of the plates of the locomotive boiler is 9'32 inch, or No. 3 wire-gage. Inside diameter of steam-pipe.-The diameter usually given to the steam-pipe of the locomotive engine may be found from the following rule. RULE. —To find the diameter of the steam-pipe of the locomotive engine.-Multiply the square of the diameter of the cylinder in inches by'03. The product is the diameter of the steam-pipe in inches. Example.-Required the diameter of the steampipe of a locomotive engine, the diameter of the cylinder being 13 inches. Here, according to the rule, diameter of steam-pipe = —03 x 132 ='03 x 169 = 5'07 inches; or a very little more than 5 inches. The steam-pipe is usually made too small in engines intended for high speeds. Diameter of branch steam-pipes.-The following rule gives the usual diameter of the branch steampipe for locomotive engines. RULE. — To find the diameter of the branch steampipe for the locomotive engine.-Multiply the square of the diameter of the cylinder in inches by'021. The product is the diameter of the branch steampipe for the locomotive engine in inches. Example.-Required the diameter of the branch steam-pipes for a locomotive engine, when the cylinders' diameter is 15 inches. Here, according to the rule, diameter of branch pipe ='021 x 152- = *021 x 225' = 4'725 inches, or about 4i inches. DIMENSIONS OF PARTS OF LOCOMOTIVES. 229 Diameter of top of blast-pipe.-The diameter of the top of the blast-pipe may be found from the following rule. RULE.-To find the diameter of the top of the blast-pipe.-Multiply the square of the diameter of the cylinder in inches by'017. The product is the diameter of the top of the blast-pipe in inches. -Example.-The diameter of a locomotive engine is 13 inches; required the diameter of the blast-pipe at top. Here, according to the rule, diameter of blast-pipe at top ='017 x 132 = 017 x 169 2'873 inches, or between 23 and 3 inches; but the variable exhaust is now generally used. Diameter of feed-pipes.-There appears to be no theoretical considerations which would lead us to determine exactly the proper size of the feed-pipes. Judging from practice, however, the following rule will be found to give the proper dimensions. RULE.-TO find the diameter of the feed-pipes.Multiply the diameter of the cylinder in inches by:141. The product is the proper diameter of the feed-pipes. Example. —Required the diameter of the feedpipes for a locomotive engine, the diameter of the cylinder being 15 inches. In this example we have, according to the rule, diameter of feed-pipe = 15 x'141 = 2'115 inches, or between 2 and 21 inches. Diameter of piston-rod.-The diameter of the 20 230 DIMENSIONS OF PARTS OF LOCOMOTIVES. piston-rod for the locomotive engine is usually about one-seventh the diameter of the cylinder. Therefore, RULE.- To find the diameter of the piston-rod for the locomotive engine.-Divide the diameter of the cylinder in inches by 7. The quotient is the diameter of the piston-rod in inches. Example.-The diameter of the cylinder of a locomotive engine is 15 inches, required the diameter of the piston-rod. Here, according to the rule, diaimeter of piston-rod = 15 -. 7 = 21 inches. Thickness of piston. —The thickness of the piston in locomotive engines is usually about two-sevenths of the diameter of the cylinder. Therefore, RULE. — To find the thickness of the piston in the locomotive engine.-Multiply the diameter of the cylinder in inches by 2; divide the product by 7. The quotient is the thickness of the piston in inches. Example.-The diameter of the cylinder of a locomotive engine is 14 inches, required the thickness of the piston. Here, according to the rule, thickness of piston = 2 x 14 -- 7 = 4 inches. Diameter of connecting-rods at middle.-The following rule gives the diameter of the connectingrod at middle. The rule is entirely founded on practice. RULE. —To find the diameter of the connectingrod at middle of the locomotive engine.-Multiply the diameter of the cylinder in inches by'21. The DIMENSIONS OF PARTS OF LOCOMOTIVES. 231 product is the diameter of the connecting-rod at middle in inches. 3 Example.-Required the diameter of the connecting-rods at middle for a locomotive engine, the diameter of the cylinders being 12 inches. For this example we have, according to the rule, diameter of connecting-rods at middle = 12 x'21 - 2'52 inches, or 21 inches. Diameter of ball on cross-head spindle.-The diameter of the ball on the cross-head spindle may be found from the following rule. RULE. — To find the diameter of the ball on crosshead spindle of a locomotive engine.-Multiply the diameter of the cylinder in inches by'23. The product is the diameter of the ball on the crosshead spindle. Example.-Required the diameter of the ball on the cross-head spindle of a locomotive engine, when the diameter of the cylinder is 15 inches. Here, according to the rule, diameter of ball = -23 x 15 - 3'45 inches, or nearly 3~ inches. Diameter of the inside bearings of the crankaxle.-It is obvious that the inside bearings of the crank-axle of the locomotive engine correspond to the paddle-shaft journal of the marine engine, and to the fly-wheel shaft journal of the land engine. We may conclude, therefore, that the proper diameter of these bearings ought to depend jointly upon the length of the stroke and the diameter of the 232 DIMENSIONS OF PARTS OF LOCOMOTIVES. cylinder. In the locomotive engine the stroke is usually 18 inches, so that we may consider that the diameter of the bearing depends solely upon the diameter of the cylinder. The following rule will give the diameter of the inside bearing. RULE.-To find the diameter of the inside bearing for the locomotive engine.-Extract the cube root of the square of the diameter of the cylinder in inches; multiply the result by -96. The product is the proper diameter of the inside bearing of the crank-axle for the locomotive engine. Diameter of plain part of crank-axle. —It is usual to make the plain part of crank-axle of the same sectional area as the inside bearings. RULE.-TO determine the diameter of the plain part of crank-axle for the locomotive engine.-Extract the cube root of the square of the diameter of the cylinder in inches; multiply the result by'96. The product is the proper diameter of the plain part of the crank-axle of the locomotive engine in inches. Diameter of the outside bearings of the crankaxle.-The crank-axle, in addition to resting upon the inside bearings, is sometimes also made to rest partly upon outside bearings. These outside bearings are added only for the sake of steadiness, and they do not need to be so strong as the inside bearings. The proper size of the diameter of these bearings may be found from the following rule. DIMENSIONS OF PARTS OF LOCOMOTIVES. 233 RULE.-To find the diameter of outside bearings for the locomotive engine.-Multiply the square of the diameter of the cylinders in inches by'396; extract the cube root of the product. The result is the diameter of the outside bearings in inches. Diameter of crank-pin. —The following rule gives the proper diameter of the crank-pin. ft is obvious that the crank-pin of the locomotive engine is not altogether analogous to the crank-pin of the marine or land engine, and, like them, ought to depend upon the diameter of the cylinder, as it is usually formed out of the solid axle. RULE.- To find the diameter of the crank-pin for the locomotive engine.-Multiply the diameter of the cylinder in inches by'404. The product is the diameter of the crank-pin in inches. Example.-Required the diameter of the crankpin of a locomotive engine whose cylinders' diameters are 15 inches. In this example we have, according to the rule, diameter of crank-pin = 15 x'404 = 6.06 inches, or about 6 inches. Length of crank-pin. —The length of the crankpin usually given in practice may be found from the following rule. RULE.-To find the length of the crank-pin.Multiply the diameter of the cylinder in inches by ~233. The product is the length of the crank-pins in inches. 20" 234 DIMENSIONS OF PARTS OF LOCOMOTIVES. The part of the crank-axle answering to thcrank-pin is usually rounded very much at the corners, both to give additional strength and to pre.. vent side play. These, then, are the chief dimensions of locomotive engines, according to the practice most generally followed. The establishment of express trains and the general exigencies of steam locomotion are daily introducing innovations, the effect of which is to make the engines of greater size and power; but it cannot be said that a plan of locomotive engine has yet been contrived that is free from grave objections. The most material of these defects is the necessity that yet exists of expending a large proportion of the power in the production of a draft; and this evil is traceable to the inadequate area of the fire-grate, which makes an enormous rush of air through the fire necessary to accomplish the combustion of the fuel requisite for the production of the steam. To gain a sufficient area of fire-grate an entirely new arrangement of engine must be adopted; the furnace must be greatly lengthened, and perhaps it may be found that short upright tubes may be introduced with advantage. Upright tubes have been found to be more effectual in raising steam than horizontal tubes; but the tube-plate in the case of the upright tubes would be more liable to burn. We here give the preceding rules in formulae, in DIMENSIONS OF PARTS OF LOCOMOTIVES. 235 the belief that those well acquainted with algebraic symbols prefer to have a rule expressed as a formulae, as they can thus see at once the different operations to be performed. In the following formulse we denote the diameter of the cylinder in inches by D. Parts of the Cylinder. Area of induction ports, in square inches ='068 x D2. Area of eduction ports, in square inches ='128 x D2. Breadth of bridge between ports between i inch and 1 inch. Parts of Boiler. Diameter of boiler, in inches = 3.11 x D. Length of boiler between 8 feet and 12 feet. Diameter of steam dome inside, in inches = 1'43 x D. Height of steam dome = 21 feet. Diameter of safety-valve, in inches = D -- 4. Diameter of valve-spindle, in inches = -076 x I). Diameter of chimney, in inches = D. Area of fire-grate, in square feet = -77 x D. Area of heating-surface, in square feet = 5 x D2-. 2. Area of water-level, in square feet = 2.08 x D. Cubical content of water in boiler, in cubic feet = x D2. 40. Diameter of feed-pump ram, in inches ='011 x D2. 236 DIMENSIONS OF PARTS OF LOCOMOTIVES. Cubical content of steam room, in cubic feet = 9 x D2 -- 40. Cubical content of inside fire-box above fire-bars, in cubic feet = D2 - 4. Thickness of the plates of boiler =- inch. Dimensions of several Pipes. Inside diameter of steam-pipe, in inches ='03 x D2. Inside diameter of branch steam-pipe, in inches -021 x D2. Inside diameter of the top of blast-pipe = -017 x D2. Inside diameter of the feed-pipes = -141 x D. - Dimensions of several moving Parts. Diameter of piston-rod; in inches = D - 7. Thickness of piston, in inches = 2 D -- 7. Diameter of connecting-rods at middle, in inches =21 xD. Diameter of the ball on cross-head spindle, in inches = -23 x D. Diameter of the inside bearings of the crankaxle, in inches ='96 x v D2. Diameter of the plain part of crank-axle, in inches ='96 x /D2. Diameter of the outside bearings of the crankaxle, in inches = -Y396 x D2. Diameter of crank-pin, in inches'404 x D. Length of crank-pin, in inches ='233 x D. DIMENSIONS OF PARTS OF LOCOMOTIVES. 237 The following simple rules I have always made use of in the construction of locomotives, and the result of their performances proves most fully that my theory is correct, having never failed in one single instance in the performance of an engine, and having drawn the heaviest load ever drawn by a single locomotive in the world. The "Philaaelphia" hauled a load of 1268 tons from Pottsville to Richmond-94 miles-158 coal cars, containing 750 tons of coal. The train was 2020 feet in length. The load was started and drawn through curves of 750 feet radius, at a rate of 10 miles per hour, the engine only weiging 15x8 tons. The engine was upon 6 wheels, with my patent flexible vibrating truck, coupled, 42 inch diameter, cylinder 14-1 x 22. To find the capacity of boiler.-For every cubic inch of cylinder, make 13 square inches of fire surface,'th of which must be in fire-box, Oths in tubes. For area of steam-ports on valve face. —Divide the area of cylinder by 12, which will be the required area, making the ports never less than 1 inch, nor more than 14. For area of exhaust on valve face.-Multiply the area of steam-ports by 2'06. For steam pipe make it equal to area of steam-port. For large steam pipe, 11 the area of steam-port. For exhaust, make it equal to area of exhaust opening on face of cylinder. For high speeds, lap one inch, lead I6th. Slow speed freight engine, lap 2 inch, lead 3%ths. 238 LOCOMOTIVE ENGINES. General Features of the Boiler. The boiler is the most important part of a locomotive engine, and the useful effect of the machine depends in a great degree on the boiler being capable of generating the requisite quantity of pure steam, without requiring the draught of air and flame through the fire and tubes to be accelerated or forced excessively. The fire-box is that part of the boiler in which the heat is generated and partially absorbed, the remaining absorption taking place in the flue tubes, which convey the products of combustion from the fire, through the water, to the smoke-box, whence they are dissipated in the atmosphere. Of course, the more nearly these products of combustion, at their entrance into the chimney, are found to have been cooled down to the temperature of the water in the boiler, the more economical in fuel the boiler will, ceteris paribus, be. To obtain the utmost economy in this way, the superficial surface of the tubes has been increased to the utmost extent, by enlarging the diameter and increasing the number and size of the tubes. The boiler of Bury's 14-inch* engine contains 92 tubes of 21 inches external diameter, and 10 feet 6 * This dimension is the diameter of cylinder, by which dimension locomotives are distinguished. THE BOILER. 239 inches long; the boiler of Stephenson's 15-inch engine contains 150 tubes of 1% inch external diameter, and 13 feet 6 inches long. It will therefore be seen that the superficial surface in Bury's tubes is, comparatively, rather small, but yet the production of steam is found to be sufficiently copious, with a blast-pipe of rather more than the average diameter; on the other hand, notwithstanding its great surface, Stephenson's boiler is found to require a smaller blast-pipe than usual. It seems highly probable that the extra intensity of blast requisite in the latter case consumes so much power to produce it, as completely to countervail the economy of fuel consequent on the very complete abstraction of the heat, by the great length of tubes in proportion to their diameter. In an experiment tried by Mr. Stephenson, the heating surface of the fire-box, where the heat is received by radiation, was found to be more effectual than the tube surface, where the heat is received by conduction, in the ratio of 1 to 3, and hence the heating surface of a locomotive is sometimes estimated as the surface of the furnace plus one-third that of the tubes. The shell, which is cylindrical, is attached to the smoke-box and fire-box by angle-iron; the end of the shell next the smoke-box is closed entirely by the tube-plate, but at the smoke-box end the water has free access quite round the internal fire-box, 240 LOCOMOTIVE ENGINES. one side of which forms the tube-plate. The shell, external fire-box, aid the smoke-box are always of iron, the thickness of plate being 5 th in. in ordinary boilers of 3 feet 4 in. in diameter, though in some cases it is 3 in.; the pitch of rivets is 1i in., and the diameter of rivets 11th in. The shell is sometimes made with flush joints, a band of iron covering the joint attached by two rows of rivets. The boiler plates should have their fibres running round the boiler instead of in the direction of its length, as the plate is somewhat stronger in that direction. The boiler is secured endwise by longitudinal stays, which are fastened by cutters to jaws attached to the end plates. The blast-pipe is the eduction pipe diminished in area at the mouth to such a degree as to cause the steam to issue with a great velocity, whereby a powerful draught through the fire is maintained by the steam rushing up the chimney. The area of the mouth of the blast-pipe varies in different engines, but an area of Bad of the area of the cylinder is a common proportion. A variable blast-pipe, the orifice of which may be increased or diminished in area, is now much used. One arrangement for this purpose consists of the application of a regular plate at the top of the blast-pipe, with a hole through the centre of the plate, through which the nozzle of the blast-pipe passes. When this regulator plate is closed, the whole of the steam has to ascend THE BOILER. 241 through the central nozzle; but when the regulator is open, or partly open, a part of the steam escapes through the holes in it. Another plan consists in the application of a movable plug within the blastpipe, which may narrow the escape orifice to an annular space of small area, the plug being raised or lowered by a lever and rod. Stephenson's method of contracting the blast consists in making the nozzle of the pipe conical, and forming it to slide within the upright pipe, whereby an annular space is left for the escape of the steam around the nozzle when the nozzle is lowered. The man-hole, or entrance into the boiler, consists of a circular or oval aperture of about 15 in. diameter, placed by Bury at the summit of his dome, and by Stephenson in the front part, a few inches above the cylindrical part of the boiler. The cover that closes this aperture in Bury's engine also contains the safety-valve seats, thus simplifying the construction by preventing the necessity of an independent aperture and cover for the safety-valves, as in Stephenson's engine, where the safety-valves are placed independently on the top of the dome. The steam-tight joint of the man-hole cover is made in Bury's engine by a single thickness of canvas, smeared with red-lead; and the joint is not liable to become defective or leaky, because the surfaces are turned true and smooth, both on the cover and its seat. When these surfaces have not been made 21 242 LOCOMOTIVE ENGINES. true in this manner, it becomes requisite to use a number of thicknesses of canvas, or other material, to form the joint-; and the action of the steam soon rotting away the soft substance, a leakage is caused through the joint, which makes repair indispensable. The small domes are of the same form as those used on the Grand Junction Railway, which are cylindrical vessels of about 20 in. diameter, and 2 feet in height, with a semi-globular top, are generally made of plate-iron, about 3 in. thick, welded at the seam, and with the flange at the bottom turned out of the same piece. In some cases, domes of this form have been constructed of cast iron, about 3 in. thick, but they have been found objectionable from their top weight, and they cannot be considered as altogether safe from explosion. Fire-box.-Iron fire-boxes have been extensively tried by Bury and others, and in cases where the plate-iron of which they were formed has been of a peculiarly perfect texture, and not liable to laminate or crack under the action of the heat, they have been found to answer exceedingly well, and not only to be much cheaper than copper, but also to last at least twice as long before requiring renewal. If the materials be very carefully selected, the use of iron fire-boxes will be found productive of economy, if only uWed in situations where pure water is obtainable. The duration of ordinary copper fire-boxes depends in a great measure upon the original tex FIRE-BOX. 243 ture of the copper, which ought to be rather coarsegrained than rich and soft, and also particularly free from irregularity of structure and lamination. Considerable advantages have been found to arise from increasing the capacity of the fire-box, more especially its depth, which ought to be such as to allow of the requisite quantity of coke being placed within it without reaching above the mouths of the lower tubes, a fault which would cause the smaller pieces of' coke to enter and block up the tubes, to the manifest deterioration of the draught, and diminution of the efficacy of the engine. The heating surface in the fire-box being of an extremely valuable and efficient nature, and the extensive area of fire-bar surface being very conducive to freedom of draught, we are induced to question whether the large square fire-box is not pro tanto preferable to the round one, which must necessarily be very small, except on the 7-feet gage, in which case the round fire-box offers decided advantages. The square firebox is generally made of iron, - in. to - in. thick in every part except the tube-plate, which has been from X in. to y in.; but experience has shown considerable advantage in making the tube-plate 8 in. thick, as this great strength prevents the spaces between the tubes from being compressed, and the tube holes rendered oval; in the processes of drifting and feruling the tubes, however, this evil will be found to exist, even with a 8 in. tube-plate, if 244 LOCOMOTIVE ENGINES. the tube holes be placed in too close contiguity, as has been found the case in several of Stephenson's engines; and, from practical observation, we find that i in. should be the minimum distance between any two tubes. The sides, back, and front below the tubes, of the square fire-box, are stayed at intervals of 4' in. to 5 in. with either copper or iron stays, screwed through the outer case into the firebox, and securely riveted; but, as the riveting within the fire-box is found to decay rapidly, from the action of the heat, Mr. Dewrance, of the Liverpool and Manchester Railway, has adopted, with good results, stays formed with a large square head, and screwed from within the fire-box outwards, the square head projecting 2 in. into the flame. Iron stay-bolts for the fire-box are found to last nearly as long as copper, and, from their superior tenacity, are often considered preferable. Until lately, it had been supposed that round fire-boxes possessed such advantages in point of strength over square ones, owing to their arched form, that they were capable of resisting the pressure of the steam without the use of stays; but experience has shown that, whatever be the shape, a fire-box must be stayed more or less to render it safe, for the shell of the fire-box is liable to be wasted so much by the heat, that it is not safe to depend altogether upon the strength its form confers, especially as the form will be changed if the boiler be suffered to become short FIRE-BOX. 245 of water. In round fire-boxes, the sides near the crown part generally suffer most from waste: these portions are now provided with stays by Messrs. Bury, Curtis, and Kennedy, who are the main supporters of round fire-boxes; and with this provision the round fire-boxes are necessarily the stronger. The roofs of all fire-boxes require to be stayed by cross-bars; but the bars are required to be both stronger and more numerous for the square fireboxes, and should always be carefully made of wrought-iron, and very carefully fitted before being bolted on. Stay-bars of cast-iron have been employed, on account of their cheapness; but, having led several times to accidents from explosion, they are now discarded. These bars are only in contact with the fire-box at the part around the rivets, and in all the other parts they permit the access of the water below them. It is advisable to bring these bars to an edge on the under side, so as to facilitate the escape of the steam. In Sharp and Robert's engines, the fire-box is made of three plates; the tube-plate and front plate have their edges bent over, and to these are attached a single plate which forms the crown and two sides of the furnace. The interior fire-box is joined at foot to the exterior by a Z-shaped iron, which forms the bottom of the waterspace, and is preferred, inasmuch as it leaves a wide water-space, and is easily cleaned. The outer and inner fire-boxes are joined round the fur21* 246 LOCOMOTIVE ENGINES. nace door, which is double, to prevent inconvenient radiation. The external fire-box has sometimes a semi-cylindrical top, joined by turning over the sides like an arch, and sometimes a dome-shaped top. The fire-bars have always been a source of much expense in the locomotive engine, as they burn out very rapidly, and have to be often renewed: from the rapid combustion going on over their upper surfaces, they become heated intensely throughout, causing them to throw off scale, and to bend under the weight of the fuel. The best remedy has been found to consist in making the bars very thin and deep, so as to keep their lower edges exposed to a cooling draught of air, and to diminish the area of metal conducting heat downwards from their heated upper edges. Thin fire-bars admit of being placed nearer together than thick ones, thus offering no increased impediment to free draught, while preventing the loss of small pieces of unburnt coke, which might otherwise drop through into the ashbox and be wasted. Fire-bars have given much satisfaction when made 4 inches deep, (parallel,) and full 5 inch thick on the upper edge and 3 inch on the lower edge. The frame carrying the fire-bars has often been made capable of being dropped on the instant, with its fire-bars and fire, into the ashbox, or upon the road, by means of catches drawn back by levers; but though the fire-bar frame is FIRE-BOX. 247 thus left unsupported, very often it will not drop, and even cannot be forced down out of its place; owing to the clinkers and tarry products of combustion forming an adhesive binding between its edge and the fire-box: it has accordingly been found best to support the fire-box frame permanently, and when any cause requires the sudden withdrawal of the fire, to lift the fire-bars singly out of place, by means of the ordinary dart. It is necessary to place the fire-bars with their upper surface about 3 inches higher than the bottom of the water-spaces, which, by this means, will be allowed to contain quiescent water, ready to retain without injury any deposite that subsides from the water; and the water-spaces should be periodically cleansed, by means of the mud-holes placed opposite the edge of each waterspace in the lower part of the outer fire-box shell. These mud-holes are made water-tight by means of either a brass plug simply screwed in and with a slight taper, or by a door applied with a soft packing on its face, and screwed up with a bridge-piece and bolt, making the joint on the internal surface of the outer shell, the hole and door being made sufficiently oval to enable the door to be introduced into the water-space. The latter plan often gives rise to inconvenience, from the joint being found leaky when the steam is raised, rendering it necessary to drop the fire, and empty the boiler, before it can be renewed. In some very large square fire 248 LOCOMOTIVE ENGINES. boxes, such as those used on the Great Western Railway, a diaphragm, or divisional 4-inch waterspace, has been placed across the middle of the firebox, with the view of obtaining increased heating surface. This diaphragm has its lower edge (in which deposite takes place) made straight, and about 2 inches below the general surface of the fire-bars, but its upper edge is of the form of an inverted arch, in order to promote the free delivery of the steam generated within it into the steam-dome. The sides of the fire-box, where the diaphragm is attached, are not cut away to form passages for the water and steam, but are pierced with a series of circular holes, 3 inches in diameter, to permit a due circulation without uselessly weakening the fire-box; but the uppermost hole of the series must be placed at the highest point of the diaphragm, otherwise an accumulation of steam, and consequent injury at that point, will ensue. The use of a diaphragm is found to be beneficial in the case of a very powerful engine, provided its upper edge be made sufficiently low to admit of the tubes being conveniently drifted over it, and to allow the dart to be used with facility in dropping the front set of fire-bars. The ash-box consists of a plate-iron tray, placed below the fire-box, to receive the burning ashes that drop from between the fire-bars. In the earlier locomotives, no ash-boxes being used, the red-hot ashes were dispersed to a considerable distance by TUBES. 249 coming in contact with the wheels, and conflagrations were often thereby originated. The ash-box should be as large as convenient, and not less than 10 inches deep, otherwise it will materially impede the draught; but if of ample dimensions, and closed at the sides and back, it will increase the draught, particularly when running against a head wind, at which time a strong draught is required. A hanging shutter to open or close the front of the ash-box forms a good damper. The bottom of the ash-box is placed about 9 inches above the level of the rails, and should on no account be nearer than 6 inches, otherwise the engine cannot pass safely over stones or similar objects lying accidentally between the rails. Tubes. —The tubes are generally formed of brass; the ferules by which they are secured are for the most part made of steel at the furnace end, and of malleable iron at the smoke-box end, and the holes in the tube-plates are tapered, so that the tubes bind them together. Great care should be taken in securing the tubes, as any neglect will be productive of much inconvenience. The ferules are found to be very injurious to freedom of draught, particularly in very small tubes; and to overcome this objection the methods we have mentioned, and many others, have been tried for fastening the tubes in by riveting over or screwing into the tube-plates; but hitherto no method, except that of internal tube 250 LOCOMOTIVE ENGINES. rings, has been found to answer in the case of brass tubes; but we think it likely that, with wrought-iron tubes, internal tube-rings will be ultimately abandoned. Stephenson has frequently adopted iron tubes of late, in preference to brass, on the score of their greater cheapness and durability; and in some cases, where unusual attention has been paid to them, and pure water used, they have been found to answer very well. A common internal diameter of tubes is 1- in. If made very small, the tubes are liable to be choked by pieces of coke, and the sectional area will be inconveniently contracted, while, if made much larger, the heating surface will be unduly diminished. The number of tubes varies considerably in different boilers; in one species of locomotive in extensive use the number is 134, and the pitch 21 in. Sufficient space is left below the tubes for deposite, that it may not be in contact with the tubes and cause them to be burned: the extreme tube of the widest row is about the diameter of a tube from the boiler shell. In the long-boiler engines of Stephenson, from the volume of water contained in them, considerable time is required to get up the steam, even so much as three and a half hours where the ordinary engines take two hours, and they require great care in firing and feeding to prevent the steam running low. Smoke-box and chimney.-The smoke-box door of many engines is hinged at the bottom, and is kept SMOKE-BOX AND CHIMNEY. 251 shut by means of handles and catches; but the position of the door when open is in that case inconvenient, as it prevents ready access to the tubes. In some of Stephenson's engines, the smoke-box door is in two leaves, which open like the doors of a house, overlapping at the centre, where they are closed by a bar, and at the top and bottom by handles and catches. This door admits of the easy examination of the cylinders and valves. A small door is usually left near the bottom of the smokebox, by which the accumulated cinders may be removed. The bottom of the smoke-box should not be below the ash-pan, or be much nearer the level of the rails than 18 inches, else the waste-water cocks of the cylinder projecting through it, would be liable to injury from objects lying on the line. The smoke-box is lower in freight engines than in passenger engines, on account of the driving wheels being smaller; and, being coupled with the other wheels, the cylinder has frequently to be inclined to let the moving parts work clear of the front axle. The chimney must not stand more than 14 feet high above the rails. The sectional area of the chimney is about 1-10th of the area of fire-grate. The chimney is usually provided with a damper, similar to the disk throttle-valve of an ordinary engine; this is generally hung off the centre, and a hole is nade in it for'the top of the blast-pipe, which projects through it when it is closed. Another 252 LOCOMI-OTIVE ENGINES. damper has been applied by Messrs. Rennie at the smoke-box end of the tubes, consisting of a slidingplate perforated with holes, which, when opposite the ends of the tubes, will give a free current, and may be made to close them completely if required. Another kind of damper consists of an arrangement of thin bars similarly disposed to the laths of a Venetian blind; the plates being so hinged, that when placed with their edges to the tube-plate, they leave the flow of air through the tubes unimpeded, and when hanging down they close up the tubes, or they partially close the tubes in any intermediate position. By either of these arrangements, the hot air is retained for a longer period in contact with the tubes than if a simple damper were used, as each tube is virtually furnished with a hanging bridge which keeps in the hottest air and lets only the coldest flow out. An inconvenient degree of heat in the smoke-box is also prevented. The smoke-box is usually made of ~th plate; the chimney of 8th plate; the blast-pipe of 8th copper, and the steam-pipe of ]%6th copper. Framinig. —In some engines the side-frames consist of oak, with iron plates riveted on each side. The guard-plates are in these cases of equal length, the frames being curved upwards to pass over the driving-axle. Hard cast-iron blocks are riveted between the guard-plates, to serve as guides for the axle-bushes. The side-frames are connected across FRAMING. 258 at the ends, and cross-stays are introduced beneath the boiler to stiffen the frames sidewise, and prevent the ends of the connecting or eccentric rods from falling down, if they should be broken. The springs are of the ordinary carriage kind, with plates, connected at the centre, and allowed to slide on each other at their ends. The upper plate terminates in two eyes, through each of which passes a pin, which also passes through the jaws of a bridle, connected by a double-threaded screw to another bridle, which is jointed to the framing: the centre of the spring rests on the axle-box. Sometimes the springs are placed between the guardplates and below the framing, which rests upon their extremities. One species of spring which has gained a considerable introduction consists of a number of flat steel plates, with a piece of metal or other substance interposed between them at the centre, leaving the ends standing apart. A common mode of connecting the engine and tender, is by means of a rigid bar with an eye at each end, through which pins are passed. Between the engine and tender, however, buffers should al. ways be interposed, as their presence contributei greatly to prevent oscillation and other irregular motions of the engine. A bar is strongly attached to the front of the carriage on each side, and projects perpendicularly downwards to within a short distance of the rail, to clear away stones or other 22 254 LOCOMOTIVE ENGINES. obstructions that might occasion accidents if the engine ran over them. The axles bear only against the tops of the axle boxes, which are generally of brass; but a plate extends beneath the bearing, to prevent sand from being thrown upon it. The upper part of the box in most engines has a reservoir of oil, which is supplied to the journal by two tubes and siphon wicks. Stephenson uses cast-iron axle boxes with brasses, and grease instead of oil, which is fed by the heat of the bearing melting the grease, and causing it to flow down through a hole in the brass. All the engines with outside bearings have inside bearings also; they are supported. by longitudinal bars, which serve also in some cases to support the piston guides: these bearings are sometimes made so as not to touch the shaft, unless in the event of its breaking. Steam-dome pipes and regulator.-The steamdome, or separator, from the upper part of which the supply of steam is obtained, is now generally placed over the fire-box; and in Bury's and Stephenson's engines it forms a part of the external shell of the fire-box; whilst in the engines used on the Grand Junction Railway, it consists of an independent cylindrical vessel, attached to the low roof of the fire-box. Either plan, this latter or Bury's, is perfectly safe and strong, without the addition of stay-rods; but Stephenson's dome presents a large extent of flat surface, from the roof of the internal FRAMING. 255 fire-box up to the arched roof of the external firebox; and this flat surface requires to be powerfully stayed by angle-irons and tension-rods. We remember an instance in which the accidental omission of one of the numerous tensi6n-rods led to the forcing out and partial explosion of the side of the firebox, showing how much depends on the circumstances of these rods, with their joints and pins, remaining sound and uninjured from- corrosion or other source of injury or decay. In this respect the round fire-box, with its dome, has the advantageof superior strength and safety. A large steamdome is found to be the most efficacious mode yet tried for preventing the evil of priming or damp steam; but no height of dome will entirely prevent it if there be not space enough left above the tubes in the cylindrical part of the boiler to allow the free passage of the steam along to the fire-box and dome, while an excessive height of dome is also found to produce an unsteady motion of the engine, by causing the machine to be top-heavy. A height of about 2 feet 6 inches above the cylindrical part of the boiler is found to give satisfactory results in practice, and to lead to the production of as pure steam as any greater altitude could secure. In some engines the steam is withdrawn from a dome placed at the smoke-box end of the boiler, into which the steam-pipe rises. It is thought that the ebullition being less violent at this point, the steam 256 LOCOMOTIVE ENGINES. will thus be more effectually dried. The steam. pipes are made either of iron or copper; and of these, iron best withstands the high temperature of the smoke-box and the impact of the cinders, but it is liable to internal corrosion. The steam-pipe, after entering the smoke-box, divides into two branches, one passing down each side of the smokebox so as to leave a free space for cleaning the tubes, and also to avoid as much as possible the impact of the hot air and cinders; but in some engines the steam-pipe descends vertically, which is somewhat inconvenient in practice. The area of the steam-pipe is one-sixth to one-eighth of the area of cylinder, and the branch steam-pipes are each about one-tenth of the area of cylinder. The admission of the steam from the boiler to the cylinders is regulated by a valve or regulator, which is generally placed immediately above the internal fire-box, and is connected with two copper pipes, one conducting steam from the highest point of the dome down to it, and the other conducting the steam that has passed through it along the boiler to the upper part of the smoke-box. Regulators may be divided into two sorts, viz. those with sliding valves and steam ports, and those with conical valves and seats, of which the latter kind are the best. The former kind have for the most part hitherto consisted of a circular valve and face, with radial apertures, the valve resembling the out SAPETY-VALVES AND FUSIBLE PLUGS. 257 stretched wings of a butterfly, and being made to revolve on its central pivot, by connecting links between its outer edges or by a central spindle. In some of Stephenson's engines with variable expansion geer, the regulator consists of a slide-valve covering a port on the top of the valve-chests. A rod passes from this valve through the smoke-box below the boiler, and, by means of a lever parallel to the starting lever, is brought up to the engineer's reach. Cocks were at first used as regulators, but were given up, as they were found liable to stick fast. A gridiron slide-valve has been used by Stephenson, which consists of a perforated square plate moving upon a face with an equal number of holes. This plan of a valve with a small movement gives a large area of opening. In Bury's engines a sort of conical plug is used, which is withdrawn by. turning the handle in front of the fire-box; a spiral groove of very large pitch is made in the valve-spindle, in which fits a pin fixed to the boiler, and by turning the spindle an end motion is given to it which either shuts or opens the steam passage according to the direction in which it is turned. The best regulator would probably be a valve of the equilibrium description, such as is used in the Cornish engines. Safety-valves and fusible plugs. —The safety-valves are placed upon the dome, in Bury's and Stephenson's engines; but it has been found much 22* 258 LOCOMOTIVE ENGINES. better to place them on the cylindrical part of the boiler, because when an engine commences to prime, the water projected from the blast-pipe generally causes an unusual generation of steam, which escapes at the safety-valve, and in its passage'of course accumulates and lifts the surfacewater and foam at whatever point of the boiler the safety-valves are situated; thus the further they are placed from the steam-dome the better, as they will then diminish the evil of priming, which, if placed upon the steam-dome, they would only aggravate. Indeed, if the safety-valves are properly situated, an engineman has the great advantage of being able to check or stop the priming of the boiler on the instant, by causing his safety-valves to blow off strongly. It is requisite to place the safety-valves upon a tubular pillar, of such altitude as to prevent the escaping cloud of steam from obscuring the look-out of the engineman. Bury's 14-inch engine contains a pair of safety-valves of 24 inches diameter, exclusive of the mitre; and Stephenson's 15inch engine contains a pair of 4-inch diameter. The latter dimension is preferable, as large safetyvalves are much less liable to adhere to their seats than small ones. Safety-valves require to be tested occasionally; and the best method consists in attaching the valve joint-pin to one end of an ordinary pair of scales, when the overbalancing weight at the reverse end will indicate the real pressure upon SAFETY-VALVES AND FUSIBLE PLUGS. 259 the valve, which exceeds the nominal pressure by the weight and friction of the lever, with its joints and spring balance, and the adhesion of the valve to its seat. To bring this adhesion to a minimum, it is a good plan to make the lip of the valve-seat somewhat flatter than a mitre, that is, at a less angle than 45~ with the horizon: 30~ answers very well. Fig. 55. The safety-valve is pressed down by means of a lever, and a screw at its extremity is attached to a spiral spring balance. To find the pressure per square inch, multiply the weight indicated on the scale, by the ratio of the two arms of the lever, and divide the product by the number of square inches in the area of the valve; but to save the trouble of 260 LOCOMOTIVE ENGINES. calculation, the ratio of the arms of the lever is made so as to be expressed by the number which represents the area of the valve, so that the weight marked on the balance is the pressure per square inch upon the valve. Some allowance must be made for the weight of the valve itself, and part of that of the lever. It is expedient to put a stop upon the screw by which the lever is screwed down or the tension of the spring increased, so as to prevent the pressure from exceeding a safe amount. Lock-up valves, which were intended as a precaution against the recklessness or neglect of the engineer, have fallen into disfavour, as from such valves being inaccessible and seldom being.required to act, they became fixed in their seats; but it is an easy thing to make a valve which can be raised, but cannot be forced down by the engineer, and such valves are in general use in steam vessels. In the engines of Cave, Hick, and Jackson, one of the valves is permanently loaded a little above the usual pressure, and enclosed in a chest; it is usually made with bent, fiat, steel springs, pressing against one another, and guided by standards screwed to the valve-seat. One of these valves is shown by fig. 55. A plug of lead is usually fixed in the furnace crown, which wmelts if the boiler becomes short of water, and gives notice of the danger. In some engines a cock is attached to the top of the steamdome, against which a small disk of fusible metal is CYLINDERS AND VALVES. 261 retained by a ring of brass bolted to the cock, and which is intended as an antidote to explosions. When the cock is opened, the steam has access to the under side of the fusible plate, which when melted is forced through the small hole in the retaining plate; and the engineer being thus warned of the undue pressure, can shut the cock and take measures to reduce the pressure. This, however, is altogether a futile expedient, for the steam would be too much cooled in passing through this cock and small pipe to melt the metal: and even if that defect were remedied, the objections still remain, as applying to all fusible plugs, and the danger is increased by leading the engineer to trust to a measure of safety that is inoperative in the hour of danger. Steam gages have not been applied hitherto to locomotives, on account of the inconvenient height of the column of mercury requisite to balance the steam. But it would be an easy thing to make a steam gage of moderate dimensions, by making the tube, whether straight or siphon, of glass, closed at the top, so that the mercury in its ascent would have to compress the air above it; and the graduations would be equal, or nearly so, if the tube were made taper. Cylinders and valves. —The cylinders are made of cast iron, about three-quarters of an inch thick, and should be of hard metal, so as to have but little tendency to wear oval from the weight and friction 262 LOCOMOTIVE ENGINES. of the piston. The ends of the cylinder are made about one inch thich, and both ends are very generally made removable. At each end of the cylinder there is generally about half an inch of clearance. The valve is invariably of the three-ported description: it is made of brass, and is not pressed upon by the valve-casing, as it is necessary in the absence of cylinder escape-valves that the steam-valve should be capable of leaving the face to enable the steam or air shut within the cylinder to escape when the train is carried on by its momentum, and also to afford an escape for the water carried over by the steam when priming takes place. The operation of priming upon the cylinders and valves is very injurious, as the grit and sediment then carried over with the steam wears the pistons, cylinders, and valve faces very rapidly; so that if the water be sandy and the engine addicted to priming, the pistons and valves may be worn out and the cylinders require reboring in the course of a few months. The valve-casing is sometimes cast on the cylinder: the face of the cylinder on which the valve works is raised a little, so that any foreign matters deposited upon it may be pushed off to the less elevated parts by the valve. The area of the steamports is in some cases one-ninth, and in others onetwelfth or one-thirteenth of the area of the cylinder; and the eduction one-sixth to one-eighth of the area CYLINDERS AND VALVES. 263 of the cylinder,-proportions which allow at mean speeds of twenty-five to thirty miles per hour, a pressure little different from that of the steam in the steam-pipes; for higher speeds the ports should be larger in proportion. The valve-casing is covered with a door, which can be removed to inspect the valves or the cylinder face. Some valve-casings have covers upon their front end as well as their top, which admits of the valve and valve bridle being more readily removed. A cock is placed at each end of the cylinder to allow the water to be discharged which accumulates there from priming and condensation. The four cocks of the two cylinders are connected, so that by working a handle the whole are opened or shut at the same time. In Stephenson's engines with variable expansion, there is but one cock, which is on the bottom of the valve chest. The valve lever is usually longer than the eccentric lever, to increase the travel of the valve. The pins of the eccentric lever wear quickly. Stephenson puts a ferule of brass on these pins, which being loose and acting as a roller, facilitates the throwing in and out of gear, and when worn can easily be replaced; so that there need be no material derangement of the motion of the valve from play in this situation. The starting lever travels between two iron segments, and can be fixed at the dead point or for the forward or backward motions. This is done 264 LOCOMOTIVE ENGINES. by a small catch or bell crank jointed to the bottom of the handle at the end of the lever, and coming up by the side of the handle, but pressed out from it by a spring. The smaller arm of this bell crank is jointed to a bolt which shoots into notches made in one of the segments between which the lever moves. By pressing the bell crank against the handle of the lever, the bolt is withdrawn, and the lever may be shifted to any other point; when the spring being released, the bolt flies into the nearest notch. The pistons which consist of a single ring and tongue piece, or of two single rings set one above the other so as to break joint, are preferable to those which consist of many pieces. In Stephenson's pistons, the screws are liable to work slack and the springs to break. The piston-rods are made of steel, the diameter being from one-seventh to one-eighth of the diameter of the cylinder. They are tapered into the piston, and secured there with a cutter. The top of the piston-rod is secured by a cutter into a socket with jaws, through the holes of which a cross-head passes, which is embraced between the jaws by the small end of the connecting-rod, while the ends of the cross-head move in guides. Between the piston-rod clutch and the guide blocks, the feedpump rod joins the cross-head in some engines. The guides are formed of steel plates attached to the framing, between which work the guide blocks, fixed on the ends of the cross-head, and which have FEED APPARATUS. 265 flanges bearing against the inner edges of the guides. Steel or brass guides are better than iron ones. Stephenson and Hawthorn attach their guides at one end to a cross-stay,-at the other to lugs upon the cylinder cover; and they are made stronger in the middle than at the ends. Stout guide-rods of steel encircled by stuffing-boxes on the ends of the cross-head would probably be found superior to any other arrangement. The stuffing-boxes might contain conical bushes cut spirally, in addition to the packing; and a ring cut spirally might be sprung upon the rod and fixed in advance of the stuffingbox with lateral play, to wipe the rod before entering the stuffing-box, and prevent it from being scratched by the adhesion of dust. PEeed apparatus.-The feed-pumps are made of brass, but the plungers are sometimes made of iron, and are generally attached to the piston-rod crosshead, though in Stephenson's engines they are worked by rods attached to eyes on the eccentric hoops. There is a ball valve between the pump and the tender, and two usually in the pipe leading from the pump to the boiler, besides a cock close to the boiler, by which the pump may be shut off from the boiler in the case of accident to the valves. The ball valves are guided by four branches which rise vertically and join at top in a hemispherical form, as shown in fig. 56. The shocks of the ball against this have in some cases broken it after a week's 23 266 LOCOMOTIVE ENGINES. Fig. 56. work, from the top of the cage having been made flat, and the branches not having had their junction at top properly filleted. These valve guards are attached in different ways to the pipes; when one occurs at the junction of two pieces of pipe it has a flange, which, along with the flanges of the pipes and that of the valve seat, are held together by a union joint. It is sometimes formed with a thread at the under end, and screwed into the pipe. The balls are cast hollow, to lessen the shock as well as to save metal: in some cases, where the feed-pump plunger has been attached to the cross-head, the piston-rod has been bent by the strain; and that must in all cases occur if the communication between the pump and boiler be closed when the engine is started, and there be no escape valve for the water. Spindle valves have in some cases been used instead of ball valves, but they are more subject to derange WHEELS. 267 ment. Slide valves might easily be applied, and would probably be found preferable to either of the other expedients. The pipes connecting the tender with the pumps should allow access to the valves and free motion to the engine and tender. The feed-pipe of many engines enters the boiler near the bottom, and about the middle of its length. In Stephenson's, the water is let in at the smoke-box end of the boiler, a little below the water level. By this means, the heat is more effectually extracted from the escaping smoke; but the arrangement is of questionable applicability to engines of which the steam-dome and steam-pipe are at the smoke-box end, as in that case the entering cold water would condense the steam. Wheels.-The driving wheels are made large to increase the speed; the bearing wheels also are easier on the road when large. In freight engines, the driving wheels are smaller than in passenger engines, and are generally coupled together. Wheels are made in various ways; they are frequently made with cast-iron naves, and with the spokes and rim of wrought-iron. The spokes are forged out of flat bars with T-formed heads; these are arranged radially in the founder's mould, while the cast-iron centre is poured around them; the ends of the T heads are then welded together to constitute the periphery of the wheel or inner tire, and little wedge-form pieces are inserted where there is any 268 LOCOMOTIVE ENGINES. deficiency of iron. In some cases, the arms are hollow, though of wrought-iron, the tire of wroughtiron, and the nave of cast-iron; and the spokes are turned where they are fitted into the nave, and are secured in their sockets by means of cutters. Hawthorn makes his wheels with cast-iron naves, and wrought-iron rims and arms, but instead of welding the arms together, he makes palms on their outer end, which are attached by rivets to the rim. These rivets, however, unless very carefully formed, are apt to work loose; and we think it would be an improvement if the palms were to be slightly indented into the rim, in cases in which the palms do not meet one another at the ends. When the rim is turned, it is ready for the tire, which is now often made of steel. The materials for wheel tires are first swaged separately, and then welded together under the heavy hammer at the steel-works, after which they are bent to the circle, welded, and turned to certain gauges. The tire is now heated to redness in a circular furnace; during the time it is getting hot, the iron wheel, previously turned to the right diameter, is bolted down upon a face-plate or surface; the tire expands with the heat, and when at a cherry-red, it is dropped over the wheel, for which it was previously too small, and it is also hastily bolted down to the surface-plate; the whole load is quickly immersed by a swing crane into a tank of water about five feet deep, and hauled up and down WHEELS. 269 until nearly cold; the tires are not afterwards tempered. It is not indispensable that the whole tire should be of steel, but a dovetail groove turned out of the tire at the place where it bears most on the rail, and fitted with a band of steel, which may be put in in pieces, is sometimes adopted, though at the risk of being thrown off in working. The steel, after being introduced, is well hammered, which expands it sideways, until it fills the dovetail groove, but it has sometimes come out. The tire is attached to the rim by rivets with countersunk heads, and the wheel is then fixed on its axle. The tire is turned somewhat conical, to facilitate the passage of the engine round curves-the diameter of the outer wheel being virtually increased by the centrifugal force, and that of the inner wheel correspondingly diminished, whereby the curve is passed without the resistance which would otherwise arise from the inequality of the spaces passed over by wheels of the same diameter fixed upon the same axle. The rails, moreover, are not set quite upright, but are slightly inclined inwards, in consequence of which the wheels must either be conical or slightly dished, to bear fairly upon them. One benefit of inclining the rails, in this way and coning the tires is, that the flange of the wheel is less liable to bear against the side of the rail, and with the same view the flanges of all the wheels are made with large fillets in the corners. Wheels have been tried loose 23-: 270 LOCOMOTIVE ENGINES. upon the axle, but they have less stability, and are not now much used. In all locomotives there is a very material loss of power from the contraction of blast-pipe necessary to maintain the blast; at high speeds one-half of the power of the engine is lost by the inadequate area of the steam passages, of which the greatest loss is that arising from the contraction of the blast-pipe. Tenders are now made larger, to obviate the necessity of so many fuel and water stations. Tenders can be put on any number of wheels, so that inconvenience is not likely to arise from their size and weight. Cranked axle.-The cranked axle is made of wrought-iron, with two cranks forged upon it, towards the middle of its length, at a distance from each other answerable to the distance between the cylinders; bosses are made on the axle for the wheels to be keyed upon, and there are bearings for the support of the framing. The axle is usually forged in two pieces, which are then welded together. Sometimes the pieces for the cranks are put on separately, but those so made are liable to give way. In engines with outside cylinders the axles are straight, the crank-pins being inserted in the naves of the wheels. The bearings to which the connecting-rods are attached are made with very large fillets in the corners, so as to strengthen the axle in that part, and to obviate side play in CONNECTING-RODS. 271 the connecting-rod. In engines which have been in use for some time, however, there is generally a good deal of end play in the bearings of the axles themselves, and this slackness contributes to make the oscillation of the engine more violent. Connecting-rods. —It is very desirable that the length of the connecting-rod should remain invariable, in spite of the wear of the brasses, for there is a danger of the piston striking against the cover of the cylinder, if it be shortened, as the clearance is left as small as possible, in order to economize steam. In some engines the strap encircling the crank-pin is fixed immovably to the connecting-rod by dovetailed keys, as shown in fig. 58, and a bolt passes through the keys, rod, and strap, to prevent the dovetail keys from working out. The brass is tightened by a gib and cutter, which is kept from working loose by three pinching screws, and a cross-pin or cutter through the point. The effect of this arrangement is to lengthen the rod, but at the cross-head end of the rod the elongation is neutralized, by making the strap loose, so that in tightening the brass the rod is shortened by an amount equal to its elongation at the crank-pin end. The tightening here is also effected by a gib and cutter, which is kept from working loose by two pinching screws pressing on the side of the cutter. Both journals of the connecting-rod are furnished with oil-cups, having a small tube in the 272 LOCOMOTIVE ENGINES. centre, with siphon wicks. The connecting-rod, represented in figs. 58, 57, is a thick flat bar, with its edges rounded. Stephenson's connecting-rod is Fig. 58. Fig. 57. made at the crank end; a strap of round iron passes over both brasses, and is attached to the T end of the connecting rod by means of nuts upon the ends of the bent iron, which is made thickest in the middle, to resist the strain. This plan has the defect of shortening the connecting-rod when the ECCENTRICS AND ECCENTRIC-ROD. 273 brasses are screwed up, and the brasses require to be very strong and heavy. Hawthorn's connectingrod has a strap at each end, tightened by a gib and cutter; but, to obviate the tendency to shorten the rod, the piston-sod end is furnished with a cutter for tightening the brass outwards. The point of the cutter is screwed, and goes through a lug attached to the gib, and is tightened by a nut. It would be preferable to attach the lug to the cutter and the screw to the gib, as the projection of the screw, when the cutter is far in, would not then be so great. In the engines on the Rouen Railway the piston-rod end of the connecting-rod has neither strap nor brass, but simply embraces the crosshead, while the crank end is hollowed out to admit brasses, which are tightened by a gib and cutter. The length of the connecting-rod varies from four times the length of the crank to seven times. The long connecting-rod has the advantage of diminishing the friction upon the slides. Eccentrics and eccentric-rod.-The eccentrics are made of cast iron; and when set on the axle between the cranks, they are put on in two pieces held together by bolts, as shown in figs. 59, 60: but in straight-axle engines they are cast in a piece, and are secured on the shaft by means of a key. The eccentric, when in two pieces, is retained at its proper angle on the shaft by a pinchingscrew, which is provided with a jam-nut to prevent 274 LOCOMOTIVE ENGINES. it from working loose. A piece is left out of the eccentric in casting it, to allow of the screw being inserted, and the void is afterwards filled by inserting a dovetailed piece of metal. Stephenson and Hawthorn leave holes in their eccentrics on each side of the central arm, and they apply pinchingscrews in each of these holes. The screws sometimes slacken and allow the eccentric to shift, unless they are provided with jam-nuts. In the Rouen engines with straight axles, the four eccentrics are cast in one piece. Fig. 60. Fig. 59. Eccentric straps are best made of wrought iron, as inconvenience arises from the frequent breakage of brass ones. When made of malleable iron, onehalf of the strap is forged with the rod, the other half being secured to it by bolts, nuts, and jamnuts. Pieces of brass are in some cases pinned within the malleable iron hoop, but it appears to be preferable to put brasses within the strap to encircle the eccentric, as in the case of any other bearing. VALVE MOTIONS. 275 When brass straps are used, the lugs have-generally nuts on both sides, so that the length of the eccentric-rod may be adjusted; but it is better for the lugs of the hoops to abut against the necks of the screws, and if any adjustment is necessary from the wear of the straps, washers can be interposed. In some engines the~adjustment is effected by screwing the valve-rod, and the cross-head through which it passes has a nut on either side of it by which its position upon the valve-rod is determined. The forks of the eccentric-rod are steel. The length of the eccentric-rod is the distance between the centre of the crank axle and the centre of the valve-shaft. Valve motions.-In locomotives the eccentrics are now always fixed upon the axle, and two are used, one for the forward, the other for the backward motion: the loose pulleys have been given up on account of their liability to get out of order from the shocks to which they were subjected by sudden change of direction when worked at a quick speed. The arrangement whereby the motion of the eccentric is transmitted to the valve, is either direct or indirect. In cases of indirect attachment the motion is given through the intervention of levers, and there is some variety in the arrangements by which the reversing is accomplished. Alcard and Buddicome use a pair of eccentrics at the end of the axle, which is straight; the reversing shaft is placed below the level of the piston-rod, and to a lever keyed 276 LOCOMOTIVE ENGINES. upon it are attached links of unequal length, connected at their upper extremities with the ends of the eccentric-rods, one of which is above and one below the studs on the lever of the valve-shaft, so that the upper eccentric-rod, being in gear, gives the forward motion, and the lower gives the backward motion. In other engines, forks are situated above and below the stud of the eccentric levers; the forward eccentric-rod is lifted up out of gear by a link depending from the lever on the reversing shaft, and by the same movement the backing eccentric is lifted into gear by a longer link connecting it to a lever, not upon the reversing shaft, but upon a shaft below it. Stephenson and Hawthorn have both used a similar arrangement, but admitting of the eccentric-rods being both under the studs of the lever on the valve-shaft, so that there is no danger, in the event of a disengaged rod falling down, or of any part of the gearing being bent or twisted by both rods being in gear at the same time. The motion of the eccentrics is now frequently transmitted directly to the valves. In Pauwel's arrangement of valve gearing, the valve works on the side of the cylinder, and the valve-rod is prolonged in the form of a deep flat blade of a lozenge section, on each side of which a stud is fixed,-one being intended for the notch of the forward eccentric-rod, and the other for that of the reversing eccentric. Above them is fixed the reversing shaft, from a VALVE MOTIONS. 277 lever on which depend two links of unequal length, which are jointed to the ends of the eccentric-rods. By working this lever up or down, the eccentricrods will be alternately engaged and disengaged, and will communicate their respective motions to the valve; or if the lever be kept in its mid position, both eccentrics will be out of gear, and the valve of course will remain stationary. Pauwel's engines are difficult to work, and are subject to shocks from going suddenly into gear: this arises from the whole weight of levers aid rods being on the front of the reversing shaft, but the evil might be remedied by attaching a counterbalance to the shaft. Valves situated upon the sides of the cylinders are in many cases more easily connected with the eccentric, but they require springs to keep them up to the face, so that it appears preferable to make the faces of the two cylinders inclined to one another rather than upright, if valves on the sides of the cylinders are preferred. Stephenson's link motion is the most elegant, and one of the most eligible modes of connecting the valve with the eccentric yet introduced. The nature of this arrangement will be made plain by a reference to fig. 61, where e is the valve-rod which is attached by a pin to an open curved link connected at the one end with the driving eccentric-rod d, and at the other with the backing eccentric-rod d'. The link with the eccentric-rods is capable of being moved up or 24 278 LOCOMOTIVE ENGINES. down by the rod f and bell crankf", situated on the shaft g, while the valve-rod remains in the same Fig. 61. horizontal plane. It is very clear that each end of the link must acquire the motion of the eccentricrod in connection with it, whatever course the central part of the link may pursue, and the valve-rod will partake most of the motion of the eccentric-rod that is nearest to it. When the link is lowered down, the valve-rod will acquire the motion of the upper eccentric-rod, which is that proper for going ahead; when raised up, the valve-rod will acquire the motion of the reversing eccentric, while in the central position the valve-rod will have no motion, or almost none. The link motion therefore obviates the necessity of throwing the eccentric-rod out of gear; it also enables the engine to be worked to a certain extent expansively, though as a contrivance for working expansively, we cannot hold it as de VALVE MOTIONS. 279 serving of much commendation. The dead point of the link motion is where the line of the valve-rod bisects the angle formed by the eccentric-rods. The maximum forward motion is when the rods are as figured, and the maximum backward motion when the rods d and d' are in the position h" and h'. The best forms of the link motions have side studs, to which the eccentric-rods are connected, and these are placed so that at the greatest throw, whether Fig. 62. backward or forward, the valve-rod and eccentricrod are in the same straight line, and the valve receives the full throw of the eccentric. A counterweight is also attached to the shaft to balance the weight of the link and rods. The second eccentric and eccentric-rod of the link motion'might, it appears to us, be beneficially dispensed with by placing the shaft g in the plane of the valve-rod, and attaching a pin to the centre of the link, which would work in the eye of the horizontal arm of the lever f. This lever would in such case require to be made much stronger than at present, as it would have to withstand the thrust of the eccentric, and 280 LOCOMOTIVE ENGINES. the link would then virtually be a double-ended lever with a movable centre. Where more convenient, the pin in the centre of the link might be moved in vertical or curved guides, instead of being attached to the lever f. The act of raising the link, and with it the eccentric-rod, would in effect alter the position of the eccentric on the shaft, and, if the eccentric-rod were properly proportioned in length, would make the lead right on the reversing side. How to set the valves of locomotives.-When the cylinder is horizontal, the crank is horizontal at the ends of the stroke; but it is not vertical when the piston is at the middle of its stroke, owing to the deviation from parallelism introduced from the connecting-rod being compelled to move at one of its extremities in a straight line. When the piston is at the end of the bottom stroke, and is gradually advanced towards the middle of the stroke, the end of the connecting-rod is carried round by the crank in a curve opposed. to that which it would naturally describe round the cross-head as centre; but when the piston has approached the end of the top stroke, the curvature of the path in which the end of the connecting-rod is moved by the crank is in the same direction as that of the circle which it would describe round the cross-head, and these curves would coincide if the connecting-rod were equal in length to the crank: it will be easily seen, therefore, that at HOW TO SET THE VALVES. 281 the top stroke the piston-rod requires but a small movement to enable the end of the connecting-rod to traverse a large portion of the circle of the crank, while at the bottom stroke the piston has to travel farther to allow of an equal arc being described by the crank. From these considerations it follows, that the motion of the crank being nearly uniform, there must be considerable inequalities in the speed of the piston; and more than a half circle will be described by the crank during the top half of the stroke, and less than a half circle in the bottom half of the stroke. The length of the connecting-rod is the distance from the cross-head at half stroke to the centre of the shaft; and it is clear, therefore, that at mid-stroke the crank cannot be vertical. The motion of the valve partakes of the same species of irregularity; but as the eccentric-rod is much longer in proportion to the radius of the eccentric than the connecting-rod, that inequality only may be noted which arises from the relation between the circumference of a circle and its diameter. The irregularity arising from the angle of the connecting-rod also affects the valve, but not to an injurious extent in ordinary cases. In fig. 63 we have shown the direct connection, as used in some of Stephenson's locomotives, A E B F representing the crank circle, and the inner circle that of the eccentric. Supposing, now, that the total length of the valve face were equal to the distance between the extreme edges of 24* 282 LOCOMOTIVE ENGINES. the steam ports, the valve would be without lap; and leaving the question of lead out of consideration for the present, that is, supposing that the steam were admitted exactly at the ends of the stroke, the eccentric would be fastened upon the shaft at right angles to the crank; in other words, the small crank which constitutes the eccentric would be at right angles to the large crank, which is attached to the piston-rod. In this way, the valve would be in the middle of its stroke when the piston was at either end of its stroke, so as to close both the steam and eduction passages, and to be ready with the slightest possible advance to open both for the return stroke of the piston. It has been found advantageous, however, to make the valve face longer than the distance between the extreme edges of the steam ports, so that when it is in the middle of its stroke it projects or overlaps the ports at both ends; and hence it requires to move through a space equal to the overlap before it is in a condition to open the steam port for the return stroke of the piston. To effect this, it is only necessary to move the eccentric forward in its path, until, at the end of the stroke of the piston, the valve is on the edge of the steam port, ready, as before, upon the slightest farther advance, to admit the steam to the cylinder. Now, as the valve is thus required to move through a part of its travel or throw equal to the overlap at each end, and as the throw is equal to the diameter of HOW TO SET THE VALVES. 283 the circle which the eccentric describes, it follows that, to give the requisite advance, that distance must be measured upon the diameter of the circle, and the corresponding position of the centre of the eccentric is that of which we are in search. Fig. 63. On the remote side of the centre of the crankshaft, and on the line of centres, mark off D C, the amount of overlap at each end of the valve, and draw a line parallel to E F, the vertical centre line of the crank-shaft; the arc of the eccentric circle intercepted between these parallel lines is that through which the eccentric must move, in order to draw the valve through a portion of its stroke equal to the overlap D C; and the point in which the line intersects the circle of the eccentric is, therefore, the position which the centre of the eccentric should occupy when the piston is at the end of its downstroke, and on the very point of beginning its up 284 LOCOMOTIVE ENGINES. stroke. In practice, however, the valve is not so set as to open simultaneously with the commencement of the stroke of the piston, but is set so that the steam commences to flow into the cylinder a very little before the beginning of the stroke; and hence, when the piston actually commences its stroke, the valve has already partially opened the port. To make this adjustment, an additional advance must be given to, the valve, and of course in the same direction; and the amount of lead, or opening, which the port has at the commencement of the stroke of the piston, must be added to the lap, their sum from C to D being treated the same in every respect as if the whole were lap; and so, for the sake of brevity, we may treat it. Let us suppose now that it was required to find the length that the eccentric-rod should be:-Place the crank horizontal, so that it may have the piston at the bottom of its stroke; bring round the eccentric to the corresponding position which we find it should occupy, and measure the distance from that point to the centre of the joint by which the eccentric-rod is to be attached to the valve-rod; this will be the length of the eccentric-rod. When the length of the eccentric-rod is known, either the valve or eccentric may be put in its proper place, if one of them be already set: thus, if the valve be set, as in the drawing, and the eccentric-rod connected also with the eccentric, it will bring the latter into its HOW TO SET THE VALVES. 285 place, where it may be fixed; but if the valve could not be conveniently set, it would then be necessary to take the following method, which requires the knowledge of the amount of lap, and the length of the eccentric-rod. Find, as before, the position of the eccentric, attach the rod, and the valve must come into connection in the proper position. In practice, the most convenient method of finding the position of the eccentric with a given lap is to draw a circle, such as H K, representing the crank-shaft, upon a board or a piece of sheet-iron, and another equal to the circle of the eccentric, and draw two diameters perpendicular to each other; mark off from the centre of the crank-shaft, and upon one diameter, the amount of lap C D; through this point draw a line parallel to E F, the other diameter; the points in which this line cuts the circle of the eccentric are the positions of the forward and backward eccentrics. Through these points, and from the centre of the crank-shaft, draw lines C M, C N, which will intersect the circumference of the crankshaft; upon this circumference measure with a pair of compasses the chord of the arc intercepted between either Voint of intersection and that of the vertical diameter E F; and the lines of diameters being first drawn upon the shaft itself, then, by transferring with the compasses the distance found upon the diagram, the proper position of the eccen tric at the end of the stroke of the piston is at once 286 LOCOMOTIVE ENGINES. determined; and this being marked upon the shaft, the eccentric can at any time be set, by bringing it round to that mark. Before leaving this figure, we may remark, that as the valve in Stephenson's locomotives of this kind is on the side of the cylinder, the cylinder face should be towards us in the drawing. As this arrangement, however, would have afforded a less easy explanation, we have adopted the present one. It will also be observed that the crank-shaft and cylinder are too close, and are not in a line with each other; but this, while it could not be easily avoided, is at the same time of no importance in considering the respective motions of the piston and valve, crank and eccentric, which are shown in their true relative positions. The crank is upon the centre, and the piston, consequently, at the end of the bottom stroke; the eccentric and valve being put in advance of the piston by the lap, have shut off the steam before the end of the stroke, and have also opened the eduction in readiness for the up-stroke; whereas, without lap, the valve would shut off the steam at one end, and open the eduction at the other, simultaneously with the termination of the stroke of the piston. In fig. 64 we have a different kind of valve-gearing, there being levers which reverse the direction of the motion; that is, while the eccentric-rod and lever are moving in one direction, the valve-rod and lever, being on the opposite side of the weigh-bar HOW TO SET THE VALVES. 287 shaft, are moving in the opposite direction. In the former case there were no levers, and therefore no Fig. 64. reversal of the motion. Hence, in order to give the valve the same motion as before, in relation to the crank, it is necessary to throw the eccentric to the opposite side of the crank-shaft, so that its motion may be in the reverse direction, to compensate for the reversing action of the levers. For whereas, when upon one side of the shaft they caused the valve to move in the same direction as themselves by means of the eccentric-rods, now that the levers are introduced, the eccentrics must themselves move in an opposite direction, to give the valves the same motion as heretofore. And this 288 LOCOMOTIVE ENGINES. can only be done by putting the eccentrics on the opposite side of the crank centre, round which they move, and, of course, in an opposite direction. Fig. 65. Fig. 65 is intended to illustrate the valve connection of the common locomotive, in which the motion of the eccentric is communicated through levers to the valve, and generally with an increase of throw. In this figure we have the cylinder face, with the valve upon it, at one end of its travel. Measure off the length of the valve throw, from the end of the valve face, in the direction of its travel. The throw of the valve may best be found by adding the lap to the breadth of the steam-port, and doubling their sum. If there were no levers intervening between the valve and eccentric, the line thus measured, which is the throw of the valve, would be HIOW TO SET THE VALVES. 289 the diameter also of the circle described by the centre of the eccentric pulley; but the use of levers interferes with this proportion unless the levers be made of equal length. The effect of levers of unequal length, in making a proportional inequality between the throw of the valve and of the eccentric, will be readily seen by reference to a diagram. From the centre A of the diameter, representing the throw of the valve, draw a line perpendicular to the valve face; and from the same point measure off, upon that line, the length of the lever A B, which is to be attached to the valve-rod, and which, for distinction, we shall call the valve lever. From the point B thus found as a cenltre with the radius B A, describe a portion of a circle intersecting perpendiculars drawn from C and D, the extremities of the line which represents the throw of the valve; from those points in the circumference of the circle produce lines through the centre B. On either side of the centre line A E, and at a distance from it equal to the radius of the eccentric, draw a parallel line. From B as a centre, with the distance from the centre B to the points H K,-in which the parallels intersect the produced lines of the lever, as radius,-describe an arc of a circle; the radius of this circle is the length which the eccentric lever must be, in order to give the requisite throw to the valve. It will be evident from the inspection of this diagram, that if it be desired to give a smaller 25 290 LOCOMOTIVE ENGINES. throw to the valve than that of the eccentric, it is necessary to make the valve lever shorter than the eccentric lever; and if it were desired to make the valve throw greater than the eccentric throw, it is indispensable that the valve lever should be made proportionally longer than the eccentric lever. If, for example, the throw of the valve is to be made twice the throw of the eccentric, then this can only be accomplished by making the valve lever twice the length of the eccentric lever. Hence the relations between these quantities are expressed by simple proportion; and any three being given, we can readily find the remaining one. For the sake of clearness, we shall state the various forms which the proportion will assume. First.-Given the throw of the valve, the throw of the eccentric, and the length of the lever attached to the valve-rod, to find the length of the eccentric lever; we have then the proportion:RULE.-As the throw of the valve is to the throw 4, the eccentric, so is the length of the valve lever to the length of the eccentric lever. If we represent the throw of the valve by T, that of the eccentric by t, the valve lever by L, and the eccentric lever by 1, we will have the proportion in a condensed algebraic form, thus,-T: t:: L: 1; or taking the actual dimensions in inches of the engine before us, 45: 3:: 9: 6. HOW TO SET THE VALVES. 291 Secondly.-Given the throw of the valve, the throw of the eccentric, and the length of the eccentric lever, to find the length of the valve lever. Then, RULE.-AS the throw of the eccentric is to the throw of the valve, so is the length of the eccentric lever to the length of the valve lever: Or, algebraically, t: T:: 1: L; or, as before in actual dimensions, 3: 4'5:: 6: 9. Thirdly. —Given the throw of the valve and the lengths of the levers, to find the throw of the eccentric. RULE.-As the valve lever is to the eccentric lever, so is the valve throw to the eccentric throw; Or thus, L: 1:: T: t; or, 9:6:: 45: 3. Fourthly.-Given the eccentric lever, the valve lever, and the eccentric throw, to find the valve throw. RULE.-As the eccentric lever is to the valve lever, so is the eccentric throw to the valve throw; Or,: L::t:T; or, 6:9::3:4'5. We formerly explained how the reversing action of the levers rendered it necessary to set the eccentric on that side of the crank-shaft centre nearest to the cylinder; whereas, in the case of the direct valve connection, it was set on the side remote from the cylinder. Having now found the means of ascer 292 LOCOMOTIVE ENGINES. taining the lengths of the levers to be employed with a given throw of valve and eccentric, the next step necessary is to determine the true position of the eccentric upon the shaft, in reference to the crank. Place the crank-pin in the dead point nearest the cylinder; that is, place the centres of the crankshaft and crank-pin in a line with the centre line of the piston-rod. Upon this line of centres A G, raise a perpendicular L M, through the point F. From F draw a circle, the diameter of which is equal to the throw of the eccentric, and another equal to the cranked axle. If the levers are equal, mark off from F, upon the line of centres and on the cylinder side, the amount of lap, and draw a line parallel to L M, cutting the eccentric circle in the points N 0. From F draw lines through N and O to the circumference of the cranked axle. The points N and 0 are the positions of the centres of the eccentric pulleys for the forward and backward gear, only one of which is necessary for going one way. In practice it is convenient to make marks at P R, as the points N and 0 are inaccessible. If there were no lap upon the valve, there would be nothing to set off from the centre line L M, and therefore that line would give the positions of the eccentrics. The intersections of the perpendicular A G would give the positions of the eccentrics on the shaft if HOW TO SET THE VALVES. 293 the connecting-rod were infinitely long; but inasmuch as the shortness of the connecting-rod introduces irregularity, the true position of the crank at the middle of the stroke of the piston must be taken. If- the lengths of the levers be unequal, the throws of the eccentric and valve will also be unequal; and if the valve lever be the longer, as in the case we have taken, the eccentric throw is less than the valve throw in the same proportion as the eccentric lever is less than the valve lever; and therefore, since the eccentric throw is thus less than the valve throw, by reason of the levers, it follows that the lap, which we set off from F, and which is part of the valve throw, must also be diminished in the same proportion as the whole throw, in order to set off the proper quantity from F. The simplest way of accomplishing this is, by marking off the lap from the line of centres, fig. 65, at the point A, at'the same end as we formerly marked off half the valve throw. This distance will be from A to the edge of the port, that being the overlap; then from the edge of the port draw a parallel to A G; and from the point in which this parallel cuts the arc of the longer lever, draw a line through the centre B, and produce it till it cuts the are K H; the perpendicular from this point to the line E A is the reduced amount of lap, which is to be set off from point F. 25i 294 LOCOMOTIVE ENGINES. Another useful problem is the method of finding the length of the eccentric-rod, the positions of the crank-shaft, and the weigh-bar shaft, and the length of the eccentric lever being given. From the centre of the weigh-bar shaft, with the length of the eccentric lever as radius, describe an arc; draw a tangent from this to the centre of the crank-shaft; from the centre of the weigh-bar shaft drop a perpendicular to the tangential line; the distance from the point of intersection to the centre of the crankshaft is the length of the eccentric-rod, and the perpendicular is the line of the eccentric lever, when the valve lever is perpendicular to the line of the valve-rod: this gives, therefore, the positions in which these levers must be keyed upon the weighbar shaft. In fig. 65 the mid-line of the eccentric-rod was the same as the line of the piston-rod; but in fig. 64 it is thrown down below that of the piston-rod, forming an angle with it, the vertex of which is the centre of the crank-shaft. In this case the centres of the eccentric pulleys must, consequently, be moved downwards as many degrees as the central line. In order to facilitate this adjustment, we may briefly explain, that every circle is supposed to have its circumference divided into 360 equal parts, called degrees; and if two diameters be drawn in it at right angles to each other, they will divide the circumference into four equal parts, each MISCELLANEOUS REMARKS. 295 of which contains 90 degrees. This, therefore, is the means by which the angle is measured; nor will it matter, although the circle be of any size whatever, for it is still equally divided by the two diameters. Hence, if the number of degrees contained in the angle which the mid-line of the eccentric-rod makes with the line of the piston-rod, be measured upon any circle described from the- centre of the crankshaft, and the angle be laid down upon a board, and if from the vertex of the angle a circle be described equal to the diameter of the crank-shaft, the chord of the arc of this circle intercepted between the lines containing the angle, is the distance to be transferred upon the crank-shaft, and through which the eccentric pulley must be moved round, in order to compensate for the obliquity of the eccentric-rod. In the example, fig. 64, the mid-line of the eccentric-rod, when in gear, lies at an angle of five degrees with the line of the piston-rod; and in all such cases this line is to be taken when reference is made to the valve motion; and the pistonrod line is to be taken when reference is made to the motion of the piston. In the case of fig. 65, these lines were made to coincide, for the sake of simplicity. Miscellaneous remarks respecting locomotives.The tractive force requisite for drawing carriages over well-formed and level common roads is about ag of the load, at low speeds. On railways, the 296 LOCOMOTIVE ENGINES. tractive force has generally been rated at about of the load, or 7~ pounds per ton, at low speeds; but in well-formed railways the tractive force is probably less than this, to keep the train moving slowly. The resistance of railway trains, however, increases rapidly with the- speed, on account of the resistance of the atmosphere; and the resistance occasioned by the atmosphere may be taken at 15 pounds per ton, with an ordinary passenger train moving at the rate of 30 miles an hour. The friction of the engine and the resistance of the rails vary simply as the velocity, if the power of the engine remains the same; but the resistance of the atmosphere varies as the square of the velocity, and the power requisite for overcoming that resistance as the cube of the velocity: so that by doubling the speed of a train, by diminishing the load without increasing the power, the friction is doubled, the atmospheric resistance is made four times greater than before, and the power requisite to overcome that resistance eight times greater. This shows the extravagance of high speeds, even if the power were as economically produced at high speeds, which is by no means the case. In moderately light trains, upwards of 50 per cent. of the power is expended in overcoming atmospheric resistance, in speeds of about 35 miles per hour; and the loss will be greater if the trains be very light, and present a large frontage. MISCELLANEOUS REMARKS. 297 We have already stated that in low-pressure condensing engines the evaporation of one cubic foot of water from the boiler may be taken to represent a horse power. In high-pressure engines, working without expansion, the mechanical efficacy of a cubic foot of water raised into steam will be somewhat less, on account of the resistance to the motion of the piston, occasioned by the pressure of the atmosphere; but in locomotive engines, where the working pressure is very high, the resistance due to the pressure of the atmosphere becomes relatively nearly as small as the resistance due to the rare vapor within the condenser of a condensing-engine; and it will not, therefore, be a material deviation from the truth if, in locomotive engines, working without priming, we reckon a cubic foot of water evaporated per hour as equivalent to a horse power. An engine evaporating 200 cubic feet of water per hour, and therefore exerting about 200-horse power, draws about 110 tons at thirty miles an hour; but if there were no loss from the resistance of the atmosphere, or of the blast-pipe, and no increased friction upon the engine from the increased power requisite for high speeds, the tractive force, if taken at 8 pounds per ton, would only require to be 70'4-horse power for 110 x 8 x 2640, the number of feet travelled per minute at 30 miles an hour, -. 33000 = 70'4horse power. The friction of the train, however, at 30 miles an hour, including that of an engine of 298 LOCOMOTIVE ENGINES. 200-horse power, cannot be taken at much less than 10 pounds per ton; for the friction of an engine increases with the power exerted, which determines the pressure upon its moving parts; and the friction of the carriages is also increased at high speeds, in consequence of the draw-bars being attached below the centre of effort of the frontage exposed to the wind, whereby the carriages are pressed down more firmly on the rails. If the traction be taken at 10 pounds per ton, then the power requisite for propulsion of a train, setting aside the resistance of the atmosphere, will be about 90-horse power, and the remaining 110-horse power is absorbed in overcoming the resistance of the atmosphere and of the blast-pipe. If the speed be increased from 30 to 60 miles an hour, about 200-horse power will be required for overcoming the friction of the train, and 880-horse power will be required to overcome the atmospheric resistance; making 1080-horse power, which will be necessary to propel a train of 110 tons at 60 miles an hour. The evaporation of a locomotive boiler is greatest when the speed is at its maximum, as the blast-pipe then produces its greatest effect; and the power of the engine varies nearly as the rate of evaporation, provided the blastpipe be not unduly contracted. At ordinary railway speeds, the power of the boiler is seven or eight times greater than it would be without the blast, though, indeed, such a comparison hardly holds, as MISCELLANEOUS REMARKS. 299 without the blast the fire of a locomotive boiler would not draw at all. At a speed of 20 miles an hour, a locomotive boiler boils off from 10 pounds to 14 pounds of water per square foot of heating-surface, and the rate of evaporation varies nearly as the V of the speed. The adhesion of the wheels upon the rails is about one-fifth of the weight when the rails are clean, and either perfectly wet or perfectly dry; but when the rails are half wet or greasy, the adhesion is not more than one-tenth or one-twelfth of the weight. The weight of locomotive engines varies from 15 to 20 tons. A powerful locomotive engine and tender, such as is suitable for high speeds, will weigh about 25 tons. The consumption of power by the locomotive itself is very great at high speeds, chiefly in consequence of the resistance occasioned by the blast-pipe to the free escape of the steam. Mr. Stephenson considers that, at ordinary railway speeds, a locomotive engine will absorb as much power as 15 loaded carriages, weighing 60 tons; so that in a train of 15 carriages, half the power is consumed by the engine. These determinations, however, are all very indefinite, and experiments are yet wanting to show the power produced and consumed by locomotives under different circumstances. Locomotive engines in England cost from $9,000 to $11,000 each. They run, on an average. about 130- miles a day, at a cost for repairs of about 300 LOCOMOTIVE ENGINES. 5 cents per mile; and the cost of locomotive power, including repairs, wages, oil, and tallow, and coke, may be taken at 12 cents per mile, on economically managed railways. This does not include a sinking fund for the renewal of engines which may be worn out, and which may be taken at 10 per cent. on the original cost of the locomotives. On second class railways, the expense of locomotives, and workshops, and tools for repairing them, may be set down at $10,000 per mile. Economy of fuel in locomotives is materially promoted by working expansively; but all attempts at economizing fuel in locomotives should begin with an increase in the area of the fire-grate, so that the power of the engine may not suffer so large a diminution by the creation of the necessary draft. Every locomotive engine should be furnished with efficient expansion apparatus of some kind or other; as, setting aside the economy of fuel accomplishable by expansion, it is clear that expansion acts beneficially by diminishing the weight of the boiler, which may be made smaller at every increase of the efficiency of the steam. When the draft is strong, a great loss of effect is caused by opening the furnace door, from the refrigeration due to the large volume of air admitted; and it would be a material improvement if the furnace could be fed by some such mechanism as the revolving grate. The use of sediment-collectors in locomotive boilers MISCELLANEOUS REMARKS. 301 also appears expedient, as, if judiciously applied, they will effectually prevent the formation of scale upon the tubes, and will also operate as an antidote to priming in many cases. The form of collector best adapted for a locomotive boiler, will depend in a great measure upon the peculiar structure of the boiler; but generally any form will answer which communicates with the water level, and contains water within it in a tranquil state. The V-shaped cuts for establishing the communication between the exterior and interior of the vessel, have been found preferable to holes of any other form; for a subsiding particle, so soon as it falls in a slight degree, gets behind the case of the collecting vessel, and cannot afterwards escape. A very important appendage to a locomotive, which I have never seen. and one which I consider indispensable, is a vacuum valve, to prevent the leakage of the tubes and an undue strain upon the boiler. No one would suppose the enormous pressure that is brought to bear upon the tubes when the boiler is blown out and a vacuum formed inside. For example, a tube is generally 10 feet long and 2 inches diameter, which equals in surface 753 square inches: now multiply that number by the pressure of the atmosphere on a square inch, which is 15 pounds, and we have the enormous pressure of 11,293 pounds; deduct one-half as not being a perfect vacuum, we have a pressure on each tube 26 302 LOCOMOTIVE ENGINES. of 5,647 pounds, sufficient to start the joints and brazing. A simple valve opening inwards and held in its place by a spiral wire spring, will remedy the evil; as the vacuum is formed, the atmosphere will press the valve inwards and the atmosphere fill the boiler. I consider it highly important to have it on all locomotive boilers. THE END. PUBLICATIONS HENRY CAREY BAIRD, SUCCESSOR TO E. L. C&REY, No.7 Hart's Building, Sixth Street above Chestnut, Philadelphia SCIENTIFIC AND PRACTICAL. THE PRACTICAL MODEL CALCULATOR, FOR the Engineer, Machinist, Manufacturer of Engine Work, Naval Architect, Miner, and Millwright. By OLIVER BYRNE, Compiler and Editor of the Dictionary of Machines, Mechanics, Engine Work and Engineering, and Author of various Mathematical and Mechanical Works. Illustrated by numerous Engravings. Now Complete, One large Volume, Octavo, of nearly six hundred pages..................................................... $3.50 It will contain such calculations as are met with and required in the Mechanical Arts, and establish models or standards to guide practical men. The Tables that are introduced, many of which are new, will greatly economize labour, and render the every-day calculations of the practical man comprehensive and easy. From every single calculation given in this work numerous other calculations are readily modelled, so that each may be considered the head of a numerous family of practical results. The examples selected will be found appropriate, and in all cases taken from the actual practice of the present time. Every rule has been tested by the unerring results of mathematical research, and confirmed by experiment, when such was necessary. The Practical Model Calculator will be found to fill a vacancy in the library of the practical working-man long considered a requirement. It will be found to excel all other works of a similar nature, from the great extent of its range, the exemplary nature of its well-selected examples, and from the easy, simple, and systematic manner in which the model calculations are established. NORRIS'S HAND-BOOK FOR LOCOMOTIVE ENGINEERS AND MACHINISTS: Comprising the Calculations for Constructing Locomotives Manner of setting Valves, &c. &c. By SEPTIMUS NORRIS, Civil and Mechanical Engineer. In One Volume, 12mo., with illus trations-................................-................................. $1.5( 2 THE ARTS OF TANNING AND CURRYING Theoretically and Practically considered in all their details. Being a full and comprehensive Treatise on the Manufacture of the various kinds of Leather. Illustrated by over two hundred Engravings. Edited from the French of De Fontenelle and Malapeyere. With numerous Emendations and Additions, by CAMPBELL TMORFIT, Practical and Analytical Chemist. Complete in one Volume, octavo....................................... $5.00 This important Treatise will be found to cover the whole field in the most masterly manner, and it is believed that in no other branch of applied science could more signal service be rendered to American MIanufacturers. The publisher is not aware that in any other work heretofore issued in this country, more space has been devoted to this subject than a single chapter; and in offering this volume to so large and intelligent a class as American Tanners and Leather Dressers, he feels confident of their substantial support and encouragement. THE PRACTICAL COTTON-SPINNER AND MANUFACTURER; Or, The Manager's and Overseer's Companion. This works contains a Comprehensive System of Calculations for Mill Gearing and Machinery, from the first moving power through the different processes of Carding, Drawing, Slabbing, Roving, Spinning, and Weaving, adapted to American Machinery, Practice, and Usages. Compendious Tables of Yarns and Reeds are added. Illustrated by large Working-drawings of the most approved American Cotton Machinery. Complete in One Volume, octavo.....................................................................$3.50 This edition of Scott's Cotton-Spinner, by OLIVER BYRNE, is designed for the American Operative. It will be found intensely practical, and will be of the greatest possible value to the Manager, Overseer, and Workman. THE PRACTICAL METAL-WORKER'S ASSISTANT, For Tin-Plate Workers, Brasiers, Coppersmiths, Zinc-Plate Ornamenters and Workers, Wire Workers, Whitesmiths, Blacksmiths, Bell Hangers, Jewellerg, Silver and Gold Smiths, Electrotypers, and all other Workers in Alloys and Metals. 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There is no description of turning or lathe-work that this elegant little treatise does sot describe and illustrate.- Western Lit. ALfessensr. 5 THE PAINTER, GILDER, AND VARNISHER'S COMPANION: Containing Rules and Regulations for every thing relating to the arts of Painting, Gilding, Varnishing, and Glass Staining; numerous useful and valuable Receipts; Tests for the detection of Adulterations in Oils, Colours, &c., and a Statement of the Diseases and Accidents to which Painters, Gilders, and Varnishers are particularly liable; with the simplest methods of Prevention and Remedy. In one vol. small 12mo., cloth. 75cts. Rejecting all that appeared foreign to the subject, the compiler has omitted nothing of real practical worth.-Hunt's lMerchant's Magazine. An excellent practical work, and one which the practical man cannot afford to be without.-Farmer and elechanic. It contains every thing that is of interest to persons engaged in this trade. -Bulletin. This book will prove valuable to all whose business is in any way connected with painting.-Scott's Weekly. Cannot fail to be useful.-N. Y. Commercial. THE BUILDER'S POCKET COMPANION: Containing the Elements of Building, Surveying, and Architecture; with Practical Rules and Instructions connected with the subject. By A. C. STUEATON, Civil Engineer, &c. In one volume, 12mo. $1. CONTENTS:-The Builder, Carpenter, Joiner, Mason, Plasterer, Plumber, Painter, Smith, Practical Geometry, Surveyor, Cohesive Strength of Bodies, Architect. It gives, in a small space, the most thorough directions to the builder, from the laying of a brick, or the felling of a tree, up to the most elaborate production of ornamental architecture. It is scientific, without being obscure and unintelligible, and every house-carpenter, master, journeyman, or apprentice, should have a copy at hand always.-Evening Bulletin. Complete on the subjects of which it treats. A most useful practical work. -Balt. American. It must be of great practical utility.-Savannah Republican. To whatever branch of the art of building the reader may belong, he will find in this something valuable and calculated to assist his progress.-Farmer and Miechanic. This is a valuable little volume, designed to assist the student in the acquisition of elementary knowledge, and will be found highly advantageous to every young man who has devoted himself to the interesting pursuits of which it treats.-Va. IHerald. 1* THE DYER AND COLOUR-MAiER'S COMPANION: Containing upwards of two hundred Receipts for making Co lors, on the most approved principles, for all the various styles and fabrics now in existence; with the Sc'ouring Process, and plain Directions for Preparing, Washing-off, and Finishing the Goods. In one volume, small 12mo., cloth. 75 cts. This is another of that most excellent class of practical books, which the publisher is giving to the public. Indeed we believe there is not, for manufacturers, a more valuable work, having been prepared for, and expressly adapted to their business.-Farmer and Mechanic. It is a valuable book.-Otsego Republican. We have shown it to some practical men, who all pronounced it the completest thing of the kind they had seen-N. Y. Nation. THE CABINET-MAKER AND UPHOLSTERER'S COMPANION: Comprising the Rudiments and Principles of Cabinet Making and Uphbolstery, with familiar instructions, illustrated by Examples, for attaining a- proficiency in the Art of Drawing, as applicable to Cabinet Work; the processes of Veneering, Inlay ing, and Buhl Work; the art of Dyeing and Staining Wood. Ivory, Bone, Tortoise-shell, etc. Directions for Lackering, Japanning, and Varnishing; to make French Polish; to prepare the best Glues, Cements, and Compositions, and a number of Receipts particularly useful for Workmen generally, with Explanatory and Illustrative Engravings. By J. STOKES. In one volume, 12mo., with illustrations. Second Edition. 75 cts. THE PAPER-HANGER'S COMPANION: In which the Practical Operations of the Trade are system atically laid down; with copious Directions Preparatory to Pa pering; Preventions against the effect of Damp in Walls; the various Cements and Pastes adapted to the several purposes of the Trade; Observations and Directions for the Panelling and Ornamenting of Rooms, &c. &c. By JA.1;ES AnruowsrTITn. In'One Volume, 12mo. 75 cts. THE ANALYTICAL CHEMIST'S ASSISTANT: A Manual of Chemical Analysis, both Qualitative and Quantitative, of Natural and Artificial Inorganic Conpounds; te which are appended the Rules for Detecting Arsenic in a Case of Poisoning. By FREDERIE WEaHLER, Professor of Chemistry in the University of Gottingen. Translated from the German, with an Introduction, Illustrations, and copious Additions, by OSCAR M. LIEBER, Author of the "Assayer's Guide." In one Volume, 12mo. $1.25. RURAL CHEMISTRY: An Elementary Introduction to the Study of the Science, in its relation to Agriculture and the Arts of Life. By EDWARD SOLLEY, Professor of Chemistry in the Horticultural Society of London. From the Third Improved London Edition. 12mo. $1.25. THE FRUIT, FLOWER, AND KITCHEN GARDEN. By PATRICK NEILL, L.L.D. Thoroughly revised, and adapted to the climate and seasons of the United States, by a Practical Horticulturist. Illustrated by numerous Engravings. In one volume, 12mo. $1.25. HOUSEHOLD SURGERY; OR, HINTS ON EMERGENCIES. By J. F. SOUTH, one of the Surgeons of St. Thomas's Hospital. In one volume, 12mo. Illustrated by nearly fifty Engravings. $1.25. HOUSEHOLD MEDICINE. In one volume, 12mo. Uniform with, and a companion to, the above. (fn immediate preparation.) THE COMPLETE PRACTICAL BREWER; Or, Plain, Concise, and Accurate Instructions in the Art of Brewing Beer, Ale, Porter, &c. &c., and the Process of Making all the Small Beers. By M. LAFAYETTE BYRN, M. D. With Illustrations, 12mo. $1. THE COMPLETE PRACTICAL DISTILLER; By M. LAFAYETTE BYRN, M. D. With Illustrations, 12mo. $1. THE ENCYCLOPEDIA OF CHEIHISTRY, PRACTI. CAL AND THEORETICAL: Embracing its application to the Arts, Metallurgy, Mineralogy, Geology, Medicine, and Pharmacy. By JAMES C. BOOTH, Melter and Refiner in the United States Mint; Professor of Applied Chemistry in the Franklin Institute, etc.; assisted by CAMPBELL MORFIT, author of "Chemical Manipulations," etc. Complete in one volume, royal octavo, 978 pages, with numerous wood cuts and other illustrations. $5. It covers the whole field of Chemistry as applied to Arts and Sciences. * * * As no library is complete without a common dictionary, it is also our opinion that none can be without this Encyclopedia of Chemistry.-Scientific American. A work of time and labour, and a treasury of chemical information.-North American. By far the best manual of the kind which has been presented to the American public.-Boston Couricr. PERFUMERY; ITS MANUFACTURE AND USE; With Instructions in every branch of the Art, and Receipts for all the Fashionable Preparations; the whole forming a valuable aid to the Perfumer, Druggist, and Soap Manufacturer. Illustrated by numerous Wood-cuts. From the French of Celnart, and other late authorities. With Additions and Improvements by CAMPBELL MORFIT, one of the Editors of the " Encyclopedi. of Chemistry." In one volume, 12mo., cloth. $1.50 9 A TREATISE ON A BOX OF INSTRUMENTS, And the SLIDE RULE, with the Theory of Trigonometry and Logarithms, including Practical Geometry, Surveying, Measur* ing of Timber, Cask and Malt Gauging, Heights and Distances. By THOMAS KENTISH. In One Volume, 12mo. $1. THE LOCOMOTIVE ENGINE: Including a Description of its Structure, Rules for Estimating its Capabilities, and Practical Observations on its Construction and Management. By ZERAH COLBURN, 12mo............. 75 cts. SYLLABUS OF A COMPLETE COURSE OF LECTURES ON CHEMISTRY: Including its Application to the Arts, Agriculture, and Mining, prepared for the use of the Gentlemen Cadets at the Hon. E. I. Co.'s Military Seminary, Addiscombe. By Professor E. SOLLY, Lecturer on Chemistry in the Hon. E. I. Co.'s Military Seminary. Revised by the Author of "Chemical Manipulations." In one volume, octavo, cloth. $1.25. THE ASSAYER'S GUIDE; Or, Practical Directions to Assayers, Miners, and Smelters for the Tests and Assays, by Heat and by Wet Processes, of the Ores of all the principal Metals, and of Gold and Silver Coins and Alloys. By OSCAR M. LIEBER, late Geologist to the State of Mississippi. 12mo. With Illustrations. 75 cts. THE BOOKBINDER'S MANUAL. Complete in one Volume, 12mo. (in press.) 10 ELECTROTYPE MANIPULATION: Being the Theory and Plain Instructions in the Art of Working in Metals, by Precipitating them from their Solutions, through the agency of Galvanic or Voltaic Electricity. By CHARLES V, WALKER, lion. Secretary to the London Electrical Society, eto Illustrated by Wood-cuts. A New Edition, from the Twentyfifth London Edition. 12mo. 75 cts. PHOTOGENIC MANIPULATION: Containing the Theory and Plain Instructions in the Art of Photography, or the Productions of Pictures through the Agency of Light; including Calotype, Chrysotype, Cyanotype, Chromatype, Energiatype, Anthotype, Amphitype, Daguerreotype, Thermography, Electrical and Galvanic Impressions. By GEOROE THOiAS FISHER, Jr., Assistant in the Laboratory of the London Institution. Illustrated by wood-cuts. In one volume, 24mo., cloth. 62 cts. MATHEMATICS FOR PRACTICAL MEN: Being a Common-Place Book of Principles, Theorems, Rules, and Tables, in various departments of Pure and Mixed Mathematics, with their Applications; especially to the pursuits of Surveyors, Architects, Mechanics, and Civil Engineers, with nu-' merous Engravinravings. By OLINTIUS GREGORY, L. L.D. $1.50. Only let men awake. and fix their eyes, one while on the nature of things, another while on the application of them to the use and service of mankind. -Lord Bacon EXAMINATIONS OF DRUGS, MEDICINES, CHEMICALS, &c. As to their Purity and Adulterations. By C. H. PEIRCE, M.D., Translator of "Stockhardt's Chemistry," Examiner of Medicines for the Port of Boston, &c. &c. 12mo, cloth................. 1.25 11 SHEEP-HUSDANDRY IN TH], SOUTH: Comprising a Treatise on the Acclimation of Sheep in the Southern States, and an Account of the different Breeds. Also, a Complete Manual of Breeding, Summer and Winter Management, and of the Treatment of Diseases. With Portraits and other Illustrations. By IIExRY S. RANDALL. In One Volume, octavo.......................................................$1.25 ELWOOD'S GRAIN TABLES: Showing the value of Bushels and Pounds of different kinds of Grain, calculated in Federal Money, so arranged as to exhibit upon a single page the value at a given price from ten cents to two dollars per bushel, of any quantity from one pound to ten thousand bushels. By J. L. ELWOOD. A new Edition. In One Volume, 12mo.........$1.............................................................. $1 To Millers and Produce Dealers this work is pronounced by all who have it in use, to be superior in arrangement to any work of the kind published-and unerring accuracy in every calculation may be relied upon in every instance. J6l- A reward of Twenty-five Dollars is offered for an error of one cent found in the work. MISS LESLIE'S COMPLETE COOKERY. Directions for Cookery, in its Various Branches. By Miss LESLIE. Forty-seventh Edition. Thoroughly Revised, with the Addition of New Receipts. In One Volume, 12mo, half bound, or in sheep..................................................................$1 In preparing a new and carefully revised edition of this my first work on cookery, I have introduced improvements, corrected errors, and added new receipts, that I trust will on trial be found satisfactory. The success of the book (proved by its immense and increasing circulation) affords conclusive evidence that it has obtained the approbation of a large number of my countrywomen; many of whom have informed me that it has made practical housewives of young ladies who have entered into married life with no other acquirements than a few showy accomplishments. Gentlemen, also, have told me of great improvements in the family table, after presenting their wives with this manual of domestic cookery, and that, after a morning devoted to the fatigues of business, they no longer find themselves subjected to the annoyance of an ill-dressed dinner.-Preface. MISS LESLIE'S TWO HUNDRED RECEIPTS IN FRENCH COOKERY. A new Edition, in cloth................................26 cts. 12 THE DYER'S INSTRUCTOR, Comprising Practical Instructions in the Art of Dyeing Silk, Cotton, Wool and Worsted and Woollen Goods, &c., containing nearly 800 Receipts, to which is added the Art of Padding and the Printing of Silk Warps, Skeins, and Handkerchiefs, and the various Mordants and Colours for the different styles of such work. By DAVID SMITH, Pattern Dyer, 1 vol. 12mo, (just published).............................................................. $1.50 TWO HUNDRED DESIGNS FOR COTTAGES AND VILLAS, &c. &c. Original and Selected. By THOMAS U. WALTER, Architect of Girard College, and JOHN JAY SMITH, Librarian of the Philadelphia Library. In Four Parts, quarto........................ $10 GUIDE FOR WORKERS IN METALS AND STONE. Consisting of Designs and Patterns for Gates, Piers, Balcony and Cemetery Railing, Window Guards, Balustrades, Staircases, Verandas, Fanlights, Lamps and Lamp Posts, Palisades, Monuments, Mantles, Gas Fittings, Stoves, Stands, Candlesticks, Silver and Plated Ware, Chandeliers, Candelabras, Potters' Ware, &c. &c. By T. U. WALTER, Architect, and JOHN JAY SMITH, 4 vols. 4to, plates........................................... $10 FAMILY ENCYCLOPEDIA Of Useful Knowledge and General Literature; containing about Four Thousand Articles upon Scientific and Popular Subjects. With Plates. By JOHN L. BLAKE, D. D. In One Volume, 8vo, cloth extra................................................$3.50 THE PYROTECHNIST'S COMPANION; Or, A Familiar System of Recreative Fire-Works. By G. W. MORTIMER. Illustrated by numerous Engravings. 1 2mo. 75cts 13 STANDARD ILLUSTRATED POET[Y. THE TALES AND POEMS OF LORD BYRON: Illustrated by HENRY WARREN. In One Volume, royal 8vo with 10 Plates, scarlet cloth, gilt edges..............................$5 Morocco extra..................$....................7........$7 It is illustrated by several elegant engravings, from original designs by WARREN, and is a most splendid work for the parlour or study.-Boston Evening Gazette. CHILDE HAROLD; A ROMAUNT BY LORD BYRON: Illustrated by 12 Splendid Plates, by WARREN and others. In One Volume, royal Svo., cloth extra, gilt edges..................$5 Morocco extra.................$.........7.................... $7 Printed in elegant style, with splendid pictures, far superior to any thing of the sort usually found in books of this kind.-N. Y. Courier. THE FEMALE POETS OF AMERICA. By RuFus W. GISWOaLD. A new Edition. In One Volume, royal 8vo. Cloth, gilt................................................$2.50 Cloth extra, gilt edges............................................ $3 Morocco super extra...............................................$4.50 The best production which has yet come from the pen of Dr. GRISWOLD, and the most valuable contribution which he has ever made to the literary celebrity of the country.-N. Y. )Tribune. THE LADY OF THE LAKE: By SIR WALTER SCOTT. Illustrated with 10 Plates, by CORnBOULD and MEADOWS. In One Volume, royal 8vo. Bound in cloth extra, gilt edges....................................................$5 Turkey morocco super extra........................................$7 This is one of the most truly beautiful books which has ever issued from the American press. LALLA ROOKH; A ROMANCE BY THOMAS MOORE: Illustrated by 13 Plates, from Designs by CORBOUILD, MEADows, and STEPHANOFF. In One Volume, royal 8vo. Bound in cloth extra, gilt edges........................................... $5 Turkey morocco super extra......................................7 This is published in: a stfjle uniforlm with the' Lally of thie Lake." 2 14 THE POETICAL WORKS OF THOMAS GRAY: With Illustrations by C. W. RADCLIFF. Edited with a Memoir, by tIENRY RE:EI), Professor of English Literature in the University of Pennsylvania. In One Volume, 8vo. Bound in cloth extra, gilt edges.......................................................3.50 Turkey morocco super extra....................................... $5.50 In One Volume, 12mb, without plates, cloth................... 1.25 Do. do. do. cloth, gilt edges....$1.50 We have not seen a specimen of typographical luxury from the American press which can surpass this volume in choice elegance.-Boston Courier. It is eminently calculated to consecrate among American readers, (if they have not been consecrated already in their hearts,) the pure, the elegant, the refined, and, in many respects, the sublime imaginings of THOMsAs GRAY.Bichihsond W'ig. THE POETICAL WORKS OF HENRY WADSWORTH LONGFELLOW: Illustrated by 10 Plates, after Designs by D. HUNTINGDON, with a Portrait. Ninth Edition. In One Volume, royal 8vo. Bound in cloth extra, gilt edges.......................................$5 Morocco super extra......................................................$7 This is the very luxury of literature-LoNoFELLOW'S charming poems presented in a form of unsurpassed beauty.-Neal's Gazette. POETS AND POETRY OF ENGLAND IN THE NINETEENTH CENTURY. By RUFUS W. GRISWOLD. Illustrated. In One Volume, royal 8vo. Bound in cloth.........$.............................................$3 Cloth extra, gilt edges................................................$3.50 Morocco super extra.....................................................$5 Such is the critical acumen discovered in these selections, that scarcely a page is to be found but is redolent with beauties, and the volume itself may be regarded as a galaxy of literary pearls.-Democratic Review. THE TASK, AND OTHER POEMS. By WILLIAMI COWPER. Illustrated by 10 Steel Engravings. In One Volume, 12mo. Cloth extra, gilt edges................... 2 Morocco extra...............................................................3 "The illustrations in this edition of Cowper are most exquisitely designed and engraved." 15 THE FEMALE POETS OF GREAT BRITAIN. With Copious Selections and Critical Remarks. By FREDERIC ROWTON. With Additions. Illustrations. 8vo, cloth...... $2.50 Cloth extra, gilt edges................................................$3.00 Turkey morocco, super.............................................. $4.50 Mr. ROWTONe has presented us with adnlirably selected specimens of nearly one hundred of the most celebrated fiemale poets of Great Britain, from the time of Lady Juliana Bernres, the first of whom there is any record, to the Mitfords, the Ilewitts, the Cooks, the lBarretts, and others of the present day.Hunt's Mlerchants' oaliainse. SPECIMENS OF THE BRITISH POETS. From the time of Chaucer to the end of the Eighteenth Century. By TiIoMAs CA~IIPBELL. In One Volume, royal 8vo. (In press.) THE POETS AND POETRY OF THE ANCIENTS: By WVILLIAM PETER, A. MI. Comprising Translations and Specimens of the Poets of Greece and Rome, with an elegant engraved View of the Coliseum at Rome. Bound in cloth...... $3 Cloth extra, gilt edges................................................$3.50 Turkey morocco super extra........................................... $5 It is without fear that we say that no such excellent or complete collection has ever been made. It is made with skill, taste, and judgment.-Charleston Patriot. THE POETICAL WORKS OF N. PARKER WILLIS, Illustrated by 16 Plates, after designs by E. LEUTZE. In One Volume, royal 8vo. A new Edition. Bound in cloth extra, gilt edges.....................................................................$5 Turkey morocco super extra.......................................... $7 This is one of the most beautiful works ever published in this country.Courier and.Inquirer. Pure and perfect ill sentiment, often in expression, and many a heart has been won from sorrow or roused from apathy by his earlier melodies. The illustrations are by LEUTZE,-a sufficient guarantee for their beauty and grace. As for the typographical execution of the volume, it will bear compa:ison with any English book, and quite surpasses most issues in America.-Neal's Gazette. The admirers of the poet could not have his gems in a better form for hol, day presents. —W Continent. 16 MISCELLANEOUS. JOURNAL OF ARNOLD'S EXPEDITION TO QUEBEC, IN 1775. By ISAAC SENTER, AI. D. 8vo, boards......................62 cts ADVENTURES OF CAPTAIN SIMON SUGGS; And other Sketches. By JOIINSON J. HIOOPER. With Illustrations. 12mo, paper...................................................50 cts. Cloth..................................................................... 75 cts. AUNT PATTY'S SCRAP-BAG. By Mrs. CAROLINE LEE HENTZ, Author of "Linda." 12mo. Paper covers.......................................................... 50 cts. Cloth..................................................................... 75 cts. BIG BEAR OF ARKANSAS; And other Western Sketches. Edited by W. T. PORTER. In One Volume, 12mo, paper....................................50 cts. Cloth..................................................................... 75 cts. COMIC BLACKSTONE. By GILBERT ABBOT A' BECKET. Illustrated. Complete in One Volume. Cloth......................................................... 75 cts. GHOST STORIES. Illustrated by Designs by DAIRLEY. In One Volume, 12mo, paper covers.......................................................... 50 cts. MODERN CHIVALRY; OR, THE ADVENTURES OF CAPTAIN FARRAGO AND TEAGUE O'REGAN. By II. H. BRACKENRIDGE. Second Edition since the Author's death. With a Biographical Notice, a Critical Disquisition on the Work, and Explanatory Notes. With Illustrations, from Original Designs by DARLEY. Two volumes, paper covers75cts. Cloth or sheep......................00.............1.00 17 COMPLETE WORKS OF LORD BOLINGBROKE With a Life, prepared expressly for this Edition, containing Additional Information relative to his Personal and Public Character, selected from the best authorities. In Four Volumes, 8vo. Bound in cloth.................................................$7.00 In sheep........................................... 8.00 CHRONICLES OF PINEVILLE. By the Author of "Major Jones's Courtship.' Illustrated by DARLEY. 12mo, paper........................................... 50 cts. Cloth.....................................................................75 cts. GILBERT GURNEY. By THEODORE HOOK. With Illustrations. In One Volume, 8vo., paper........................................................... 50 cts. MEMOIRS OF THE GENERALS, COMMIODORES, AND OTHER COMMANDERS, Who distinguished themselves in the American Army and Navy, during the War of the Revolution, the War with France, that with Tripoli, and the War of 1812, and who were presented with Medals, by Congress, for their gallant services. By THOMAs WYATT, A. M., Author of "History of the Kings of France." Illustrated with Eighty-two Engravings from the Medals. 8vo. Cloth gilt.................................................................$2.00 Half morocco..$....................................... $2.50 GEMS OF THE BRITISH POETS. By S. C. HALL. In One Volume, 12mo., cloth............ $1.00 Cloth, gilt....................................................... $1.25 VISITS TO REMARKABLE PLACES: Old Halls, Battle Fields, and Scenes Illustrative of striking passages in English History and Poetry. By WILLIAM1 HOWITT. In Two Volumes, 8vo, cloth.....................................00 2* NARRATIVE OF THE ARCTIC LAND EXPEDITION. By CAPTAIN BACK, RI. N. In One Volume, 8vo, boards...$2.00 THE MISCELLANEOUS WORKS OF WILLIAM HAZLITT. Including Table-talk; Opinions of Books, Men, and Things; Lectures on Dramatic Literature of the Age of Elizabeth; Lectures on the English Comic Writers; The Spirit of the Age, or Contemporary Portraits. Five Volumes, 12mo., cloth.......00 Half calf.......................$.................................6.25 FLORAL OFFERING A Token of Friendship. Edited by FRANCES S. OSGOOD. IllUStrated by 10 beautiful Bouquets of Flowers In "One Volume, 4to, muslin, gilt edges...................................$3.50 Turkey morocco super extra.........................................$5.50 THE HISTORICAL ESSAYS, Published under the title of "Dix Ans D'Etude tistorique," and Narratives of the Merovingian Era; or, Scenes in the Sixth Century. With an Autobiographical Preface. 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MONTAGUE. 12mo, cloth........................................ $1 MY DREAMS: A Collection of Poems. By Mrs. LOUISA S. McCORD. 12mo, boards................................................ 75 cts AMERICAN COMEDIES. By JAMES K. PAULDING and WM. IRVING PAULDING. One Volume, 16mo, boards.....................5.........................0 cts. 20 RAMBLES IN YUCATAN; Or, Notes of Tra;vei th:ough the Peninsula: including a Visit to the Remarkable'Ruinss of Chi-cleui, Kabah, Zayi, and Uxmal. With numerous Illustrations By B. M1. NORMAN. Seventh Edition. In One Volume, octavo, cloth..................................$2 THE AMERICAN IN PARIS. By JOHN SANDERSON. A New Edition. In Two Volumes, 12mo, cloth................................................. $1.50 This is the most animated, gracefill, and intelligent sketch of French manners, or any other, that we have had for these twenty years.-London Monthly,Magazine. ROBINSON CRUSOE. A Complete Edition, with Six Illustrations. 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