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QUARTZ OPERATOR'S 
 
 HAND BOOK. 
 
 WHEELER & RANDALL. 
 
 SAN FRANCISCO: 
 
 Mining and Scientific Press Job Printing Office. 
 
 1865. 
 

I 
 
 Ibwry 
 
 QUARTZ OPERATOR'S HAND ROOK. 
 
 In the preparation of this Hand Book, the object 
 has been to make it a reliable and practical guide to the 
 Quartz Operator. 
 
 No pains have been spared in gathering the material 
 and in rendering the subjects discussed as concise and 
 plain as their nature would admit. 
 
 Mindful that science and practice go hand in hand, 
 the authors have confined their investigations to useful 
 and established facts, leaving untouched all that is 
 doubtful and chimerical. 
 
 No claim is made to originality, unless it may be in 
 regard to the discussion of the tractory and the grind- 
 ing effects of differently formed plates — a subject of 
 great importance to every quartz miner. 
 
 To what extent the authors have succeeded in their 
 object, is submitted with no little diffidence to the decis- 
 ion of the public. 
 
 WHEELER & RANDALL. 
 
 San Francisco, April 26, 1865. 
 
4 QUARTZ OPERATOR'S 
 
 BLOWPIPE. 
 
 The Blowpipe is an instrument used for directing, by 
 a current of air, the flame of a lamp or candle upon a 
 mineral substance to fuse or oxydize it. The flame 
 consists essentially of two parts — the oxydizing and 
 reducing. 
 
 1st. The oxydizing part is the outer and slightly 
 luminous flame. 
 
 2nd. The reducing part, which is hottest, is the inner 
 blue flame. 
 
 The reagents mostly used in making blowpipe tests 
 are charcoal, carbonate of soda, cyanide of potassium 
 and borax. 
 
 The charcoal performs the part of a cupel as well as 
 that of a reagent. The best charcoal is made of 
 young hard wood. 
 
 The cupel or support consists of a sound piece of 
 coal sawed or broken lengthwise, having a small cavity 
 made in its plain side near the edge to hold the sub- 
 stance to be tested. 
 
 The borax is prepared by being melted or vitrified 
 and pulverized. 
 
 (A) Blowpipe Assay of Silver Ores containing sul- 
 phur and arsenic. 
 
 1st. Roast the pulverized ore within a shallow cavity 
 on the coal support. To do this, direct by the blowpipe 
 the oxydizing flame, that is, the extreme point of the 
 outer flame upon the powdered ore; turn the specimen 
 
HAND BOOK. 5 
 
 occasionally to expose all of its parts to oxydation. 
 Let the blast at first be gentle, then increased and con- 
 tinued till the sulphurous odors cease. 
 
 2nd. Pulverize the roasted ore and put it upon the 
 charcoal within a prepared cavity ; cover it over with 
 carbonate of soda and direct upon it the reducing flame, 
 that is, the inner blue flame; let the blast at first be 
 gentle, and chiefly employed in fusing and bringing to- 
 gether all parts of the flux, then increase the blast, 
 shaking the support or coal, so as to gather the metal 
 in one globule, which cool and extract from the slag. 
 
 3rd. Cupel the button thus obtained, adding to it three 
 parts of lead in a bone-ash cupel. To do this, employ 
 the oxydizing flame by using a very gentle blast. 
 
 Cool, weigh the button and compare its weight with 
 that of the original ore. 
 
 (B) Place within the cavity, on the charcoal, a mix- 
 ture of 
 
 Powdered Ore, (not necessarily roasted) . . 1 part. 
 
 Borax- Glass, (pulverized) 1 " 
 
 Lead, (Litharge is often preferable) 3 " 
 
 and direct upon it the reducing flame till all the metal 
 is gathered in one button. Let the blast at first be 
 gentle and chiefly directed to the borax, then increased 
 and continued till the object is attained. Extract and 
 cupel the button as in example (A.) 
 
 BLOWPIPE TESTS. 
 
 " When common borax of the shops is heated, it 
 
 3a 
 
6 QUARTZ OPERATOR'S 
 
 forms, at first, a white spongy mass, and if we continue 
 to heat it, pressing it down into a clay or Hessian 
 crucible, it melts at a brown-red heat into a clear trans- 
 parent glass. This glass, when powdered fine, is the 
 substance which we make use of in blowpipe experi- 
 ments. When we heat a platinum wire to redness, and 
 dip it into powdered borax, it will, when heated again, 
 form a globule of borax-glass at its lower extremity. 
 This globule is brought in contact with the hot specimen 
 to be tested, and the flame applied ; it will now absorb 
 some of the hot specimen, dissolve it, and show by its 
 color what kind of mineral is under examination. 
 
 " This test must be made by means of the spirit lamp 
 or alcohol flame. The color of the glass formed, when 
 cold or warm, when in the oxydizing part of the flame, 
 or in the reducing flame, is, in many cases, a decisive 
 test, if it forms metal in the latter, or merely changes 
 its color. 
 
 "Lime is infusible by' itself, but gives a strong 
 light ; with borax, it melts into a colorless glass. It is 
 not soluble in carbonate of soda. Both, when heated 
 together, are absorbed by the charcoal. Magnesia acts 
 like lime. Alumina is infusible by itself; it melts with 
 borax into a colorless glass, and if much alumina is 
 present the glass is opaque ; carbonate of soda does not 
 dissolve this substance. 
 
 " Oxyd of Antimony sublimes by itself, and colors 
 the flame a greenish-blue ; with borax it forms a yellow 
 glass when hot, and colorless when cold, in the oxydiz- 
 
HAND BOOK. 7 
 
 ing flame. In the reducing flame, it forms a gray, 
 vitrious mass and shrinks. With carbonate of soda, or 
 a mixture of cyanide of potassium and soda, it forms a 
 metal which easily evaporates in white smoke in the 
 flame. The metali s very fusible and brittle. 
 
 • " Oxyd of Bismuth is fusible and forms on wire a 
 dark-brown glass while hot, and a yellow glass when 
 cold ; it forms metal on charcoal, and with borax it 
 melts into a colorless bead. With carbonate of soda, it 
 forms metal on a coal support. 
 
 " Oxyd of Chrome does not change in the flame when 
 alone ; it forms with borax a bead which is red while 
 hot, but changes into green when cold and under the 
 oxydizing influences ; when the mixture is exposed to 
 the reducing flame it is always green. With carbonate 
 of soda it melts to a dark-orange glass when hot and 
 oxydized, and becomes opaque when cold. In the 
 reducing flame it is always opaque and orange, changing 
 to green when cold. 
 
 " Oxyd of Cobalt is unchangeable by itself, but it 
 forms a characteristic deep-blue glass with borax. It 
 is reduced on charcoal when mixed with carbonate of 
 soda, forming a gray magnetic powder. 
 
 " Oxyd of Copper fuses in the oxydizing flame ; in 
 the reducing flame it forms metal ; with borax it 
 forms a green glass in the oxydizing flame, and in the 
 reducing flame brown-red; with soda it is reduced on 
 charcoal, giving malleable metal. 
 
 " Per- Oxyd of Iron is unchangeable in the oxydizing 
 
 4a 
 
8 QUARTZ OPERATOR'S 
 
 flame; in the reducing fire it blackens and becomes 
 magnetic oxyd. It forms a bright-red glass when heated 
 with borax, which changes to a pale dirty-red when 
 cold, in the oxydizing flame ; in the reducing flame it 
 forms a bottle-green, often black-green glass. With 
 soda or alkaline flux, it forms metal on the charcoal 
 support, which appears as a dark magnetic powder. 
 
 " Oxyd of Lead shows at first a clear blue flame, after 
 which it fuses to an orange-yellow glass ; on a charcoal 
 support it is reduced to metal. With borax it forms a 
 yellow glass when hot, which is nearly colorless when 
 cold. When this oxyd is mixed with alkaline flux, it 
 forms metal instantly on charcoal ; in the alcohol flame, 
 on wire, it forms a transparent yellow glass with car- 
 bonate of soda. 
 
 " Oxyd of Manganese is infusible alone, and changes 
 its colors to brown ; with borax it melts to a black glass 
 when much manganese is used ; when little oxyd and 
 much borax are melted together in the presence of 
 carbon, the result is an amethyst colored glass, and if 
 brought within the reducing flame on charcoal, it is 
 colorless, and remains so when quickly cooled. With soda 
 it melts to an opaque green glass in the oxydizing fire, 
 and on a foil of platinum. This test is characteristic of 
 manganese. 
 
 " Oxyd of Nichle is not changed by heat; with borax 
 it melts to an orange-red glass, which is almost colorless 
 when cold. In the reducing flame, on charcoal, this 
 glass becomes gray; this color is caused by reduced 
 
HAND BOOK. 9 
 
 metal. With soda, it forms a magnetic powder of metal 
 on charcoal. 
 
 " Oxyd of Silver is instantly reduced to metal when 
 brought within the flame. It forms a white opaque 
 glass with borax, and is partly reduced to metal in all 
 instances ; with alkaline fluxes it forms metal directly, 
 when brought in the flame. 
 
 "Oxyd of Tellurium imparts to the flame a green 
 color, fuses and sublimes ; on charcoal it is easily re- 
 duced to metal. With borax, it melts to a colorless 
 glass in the oxydizing flame ; in the reducing flame the 
 glass is gray. With carbonate of soda it acts as with 
 borax but less distinctly. 
 
 " The Oxyds of Tin are converted into sesqui-oxyd, 
 becoming dirty-yellow in the oxydizing flame ; it forms 
 metal after protracted heating on the charcoal support 
 and in the reducing flame. With borax it forms a clear 
 glass, and with alkaline fluxes it is easily reduced to 
 metal on charcoal. 
 
 "Titanic Acid is not altered in the flame when ex- 
 posed to it ; with borax it melts to a colorless glass, 
 which becomes opaque in cooling. In the reducing 
 flame it becomes first yellow, then amethyst, and dark- 
 ens in cooling. With carbonate of soda it dissolves 
 with effervesence, forming a faint yellow glass, which 
 becomes gray in cooling. It forms no metal on char- 
 coal. 
 
 "Zinc. — The oxyd of this metal forms a strong whitish- 
 green flame ; it is slightly vellow when hot but turns 
 
 5a 
 
10 QUARTZ OPERATOR'S 
 
 white in cooling. With borax it forms a transparent 
 glass, which becomes milky by an intermittent flame 5 
 in the reducing flame it forms metal .which is quickly 
 evaporated. Alkaline fluxes do not alter it in the 
 oxydizing flame ; it is reduced on charcoal, and in the 
 reducing flame. The metal burns readily and forms a 
 white floculent oxyd, which is yellow when hot." — 
 Overman's Treaties on Metallurgy, pages 154 — 157. 
 
 CHEMICAL TESTS. 
 
 Reagents or tests, usually in liquid form, are sub- 
 stances for indicating the presence of other bodies. In 
 the following examples the reagents are arranged at the 
 left hand side of the page, and the precipitates or pro- 
 ducts at the right. The proper solvents are indicated 
 by the name of the solution holding the substances 
 sought : 
 
 TESTS FOR GOLD IN SOLUTION WITH AQUA-REGIA. 
 Sulphate of Iron gives .... Metallic Gold as a purple 
 
 powder. 
 
 Oxalic Acid gives Metallic Gold in large flakes 
 
 Potash u Yellow Precipitate. 
 
 Soda " " " 
 
 TESTS FOR SILVER IN SOLUTION WITH NITRIC ACID. 
 
 Potash gives Dark-Olive Precipitate. 
 
 Soda " " " " 
 
 Plate of Copper gives .... Metallic Silver. 
 Muriatic Acid " White, Curdy Precipitate. 
 
HAND BOOK. 11 
 
 Common Salt gives .... White Curdy Precipitate. 
 Tincture of Nut-Gail gives, Brown Precipitate. 
 
 TESTS FOR COBALT IN SOLUTION WITH NITRIC ACID. 
 
 Potash gives Blue Precipitate. 
 
 Soda " " " 
 
 Ferro-Prusiate of Potash 
 
 gives Green " 
 
 Carbonate of Potash " Red " 
 
 TESTS FOR BISMUTH IN SOLUTION WITH NITRIC ACID. 
 
 Pure Water gives White Precipitate. 
 
 Gallic Acid " Greenish Yellow. 
 
 Potash " White Precipitate. 
 
 Soda " " " 
 
 TESTS FOR LEAD IN SOLUTION WITH NITRIC ACID. 
 Sulphate of Soda gives . . . White Precipitate. 
 Sulphuric Acid "... " " 
 
 Infusion of Nut- Gall gives, " * 
 
 TESTS FOR COPPER IN SOLUTION WITH NITRIC ACID. 
 
 Plate of Iron gives Metallic Copper. 
 
 " " Zinc " " " 
 
 Potash " Green Precipitate. 
 
 Ammonia " Azure-Blue Color. 
 
 Infusion of Nut- Gall gives, Brown Precipitate. 
 
 TESTS FOR ANTIMONY IN SOLUTION WITH FOUR PARTS 
 OF MURIATIC ACID AND ONE PART OF NITRIC 
 ACID. 
 
 Pure Water gives White Precipitate. 
 
 6a 
 
12 QUARTZ OPERATOR'S 
 
 Plate of Iron gives Black Powder of the Metal. 
 
 TESTS FOR MERCURY IN SOLUTION WITH NITRIC OR 
 
 MURIATIC ACID. 
 Plate of Copper gives .... Metallic Mercury. 
 
 " " Iron " .... Dark Powder. 
 
 Gallic Acid " . . . , Orange Yellow. 
 
 TESTS FOR IRON IN SOLUTION WITH MURIATIC ACID. 
 Infusion of Nut- Gall gives, Black Precipitate. 
 Ferro-Prusiate of Potash 
 
 gives Blue " 
 
 Ammonia gives Brownish Red Precipitate. 
 
 EXPLANATION OF CHEMICAL TERMS. 
 Aqua Regia. — A fuming liquid composed of nitric 
 acid and muriatic acid, viz : One part of the former 
 and two of the latter. This mixture readily dissolves 
 gold and platinum. 
 
 Sulphate of Iron ) c Green vitrioL 
 
 Proto- Sulphate of Iron . . ) rr ■ 
 
 Sulphate of Copper Blue \ itriol, Blue- Stone, 
 
 Blue Copperas. 
 Nitrate of Potassa ....... Nitre, Saltpetre. 
 
 Sulphate of Soda Glauber Salts. 
 
 Chloride of Sodium Sea Salt, Common Table 
 
 Salt. 
 
 Nitric Acid Aquafortis. 
 
 Sulphuric Acid (Concen-) 0il f y itriol 
 
 trated) j 
 
 Hgdrochloric Acid Muriatic Acid. 
 
HAND BOOK. 13 
 
 Oxalic Acid. — Sorrel Acid. — It is commonly manu- 
 factured by the action of nitric acid upon saccharine and 
 farinaceous substances. 
 
 Gallic Acid. — An acid obtained from nut-galls or oak- 
 apples. It is also obtained from several other vegetable 
 astringents. 
 
 Nut- Galls. — Oak apples, which are excrescences pro- 
 duced by small insects depositing their eggs in the 
 tender shoots of a species of oak. 
 
 Potassa. — Pure Potash, when refined by heat, is 
 called pearlash. It is a vegetable fixed alkali. 
 
 Aqua- Vitae. — A liquid much used for precipitating 
 hombres, properly called aquamortis, alcohol. 
 
 Chloride of Ammonium. — Sal-Ammoniac. 
 
 Catechu. — A dry, brown astringent extract obtained 
 by decoction and evaporation from the acacia catechu in 
 India. It contains a large portion of tannin or tannic 
 acid. 
 
 ASSAY. 
 
 Assays are three kinds — Mechanical, dry and humid. 
 
 I. Mechanical Assays consist in washing or otherwise 
 freeing, without the aid of chemical agents, the metallic 
 substances from sands and other impurities. " Panning 
 out" or separating the gangue (earthy matter) from the 
 metallic substances, by washing in common mining pans, 
 also, " winnowing" as practiced on rich, dry sands, are 
 familiar examples of mechanical assays, and require no 
 explanations ; the former of which often furnishes safer 
 
14 QUARTZ OPERATOR'S 
 
 and more practical data for extensive operations, espe- 
 cially in gold mining, than either the dry or humid way. 
 II. The Dry way of assaying ores, usually requires 
 fluxes for separating the gangue (earthy matters) from 
 the metallic substances. 
 
 Assay of Galena. — Fuse in an earthen crucible, at a 
 bright red heat. 
 
 Powdered ore 6 parts. 
 
 Black Flux, 9 " 
 
 Iron, in small pieces, 2 " 
 
 Extract from the slag, and weigh the button of lead 
 thus obtained. 
 
 Assay of Iron. — Fuse in a covered crucible, about 
 one hour, a well triturated mixture of 
 
 Powdered and roasted ore, 2 parts 
 
 Fluor Spar, 1 " 
 
 Charcoal,: 1 " 
 
 Common Salt, » 4 " 
 
 Extract and weigh the button of cast iron thus ob- 
 tained. 
 
 Various other fluxes, as lime, clay, etc., may be em- 
 ployed instead of the above. No general formulae can 
 be given, as their application depend upon the nature 
 of the ore. 
 
 Assay of Copper Ores, containing no other metals 
 besides iron and copper : 
 
 Heat gradually, at first, in an earthen crucible, and 
 afterward increase the heat to bright red, which continue 
 fifteen minutes. 
 
HAND BOOK. 15 
 
 Powdered Ore, 1 part. 
 
 Black Flux, 3 " 
 
 Extract from the slag, and weigh the button of cop- 
 per thus obtained. 
 
 Assay of Copper Ores containing sulphur, but other- 
 wise similar to the above : 
 
 Fuse in an earthen crucible, at a dull red heat, equal 
 parts of the powdered ore and dried borax. Extract 
 from the slag the matte (crude copper) button, which 
 pulverize ; roast slowly in an earthen crucible, and stir, 
 in the meantime with a steel rod, till sulphurous acid 
 ceases to be evolved ; then increase the temperature to 
 a white heat, which continue for several minutes. Next 
 mix in the same crucible : 
 
 The Roasted Matte, 1 part 
 
 Black Flux, from ... 3 to 4 " 
 
 Cover the mixture with a layer of fused borax, and 
 subject it to a cherry heat for twenty minutes, in a wind 
 furnace ; then extract and weigh the button of copper. 
 
 Assay of Copper Ores containing arsenic and various 
 other metals : 
 
 Obtain and pulverize the matte as in the preceding 
 case, then roast with it powdered charcoal, till the gar- 
 lic odors of arsenic cease to be exhaled. 
 
 Reduce the matte thus obtained, as in the last case, 
 with black flux and borax. 
 
 Cupel the button in a» bone-ash cupel, with pure lead. 
 Throw a little borax glass over the globule when its 
 rotation ceases and brightening occurs ; cool and weigh 
 the button of copper. 
 
16 QUARTZ OPERATOR^ 
 
 ASSAY OF GOLD OR SILVER, OR GOLD AND SILVER ORES 
 
 Fuse in an earthen crucible : 
 
 Powdered Ore, 4 parts. 
 
 Litharge, 4 " 
 
 Black Flux, 3 " 
 
 If the ores contain much oxyd of lead, add only 
 black flux. 
 
 If the ores are very rich in pyrites, add litharge and 
 nitre. 
 
 If the button obtained be an alloy — for instance, of 
 gold, silver, copper and lead — make additions to it of 
 silver and lead, so that the prepared alloy shall contain 
 as near as may be, of 
 
 Gold, 1 part. 
 
 Silver, 3 " 
 
 Lead, 16 " 
 
 First fuse the lead in a bone-ash cupel, within a muf- 
 fle ; then add the gold and silver inclosed in a piece of 
 paper, and continue the heat till the button brightens 
 and becomes tranquil. Cool and weigh the button. To 
 separate the gold from the silver, called " parting of 
 gold," anneal, beat the button into a thin plate, make it 
 into a roll, which is termed a cornet. First heat this 
 plate or cornet in dilute nitric acid as long as the acid 
 acts upon it, then in concentrated nitric acid till all of 
 the silver is dissolved. Thoroughly wash, dry and ignite 
 the cornet. The weight of silver is equal to the weight 
 of the button before " parting," less that of the refined 
 cornet. 
 
HAND BOOK. 17 
 
 HUMID WAY OF ASSAY — ASSAY OF GALENA. 
 
 1. — Digest the powdered ore in equal parts of nitric 
 acid and pure water. 
 
 2. — Filter and digest the residual several hours with 
 a strong solution of carbonate of soda. 
 
 3. — Filter and digest the second residual in dilute 
 nitric acid, and again filter. 
 
 4. — Add either a solution of the sulphate of soda, or 
 sulphuric acid to the collected filtrates, as long as any 
 precipitate takes place. 
 
 5. — Filter, then wash and dry the residual. 
 
 6. — Reduce the residual, with powdered charcoal, in 
 an earthen crucible ; cool and weigh the button. 
 ASSAY OF COPPER ORES. 
 
 1. — Digest the powdered ore in dilute nitro-muriatic 
 acid. 
 
 2. — Filter the solution. 
 
 3. — Add ammonia in excess to the filtrate. 
 
 4. — Filter and wash residual in ammonia. 
 
 5. — Evaporate the filtrate to dryness. 
 
 6. — Dissolve the dried filtrate in muriatic acid. 
 
 7. — Add clean iron or zinc plates to the solution much 
 diluted. 
 
 8. — Wash dry and weigh the copper precipitate. 
 ASSAY OF SILYER [ORES.* 
 
 1. — Digest the pulverized ore in nitric acid. 
 
 * Chloride of Silver,^ as found native, is called horn-silver ; it is 
 completely insoluble in nitric acid. It is readily dissolved by 
 
18 QUARTZ OPERATOR'S 
 
 2. — Add muriatic acid or solution of common salt to 
 the silver solution as long as any precipitate takes 
 place. 
 
 3. — Filter and dry the residual. 
 
 4. — Reduce the dry residual with carbonate of soda, 
 or black rosin in an earthen crucible, then cool and 
 weigh the button of silver. It also may be reduced 
 with chalk and charcoal. 
 
 ASSAY OF GOLD ORES. 
 
 1. — Digest the pulverized ores in one part of nitric 
 acid, and four parts hydrochloric acid. 
 
 2. — Dilute, filter and evaporate the filtrate to dry- 
 ness. 
 
 3. — Digest the dried filtrate in pure water, then boil 
 the solution with a solution of sulphate of iron, which 
 precipitates the gold as a dark-purple powder. 
 
 4. — Filter and heat the residual with hydrochloric 
 acid. 
 
 ammonia, and can be precipitated from this solution by the ad- 
 dition of nitric acid. It is also soluble in a strong, hot solution 
 of common salt, (see Augustin process) from which it may be 
 precipitated in its metallic state by a clean plate of copper. 
 
 Quicksilver partially decomposes the chloride of silver forming 
 a silver amalgam; this is attended, however, with a loss of 
 quicksilver, and should be avoided in practical operations. 
 Silver may be revived from its chloride state by being kept from 
 twelve to twenty -four hours in contact with clean iron, copper or 
 zinc plates. 
 
 Bromide of Silver, in almost every respect, resembles chloride 
 of silver ; it is, however, less soluble in ammonia. 
 
 Iodide of Silver, found also native, is readily converted into 
 chloride of silver by muriatic acid, and then may be treated as 
 above described. 
 
HAND BOOK. 19 
 
 5. — Filter, wash, dry and weigh the gold powder. 
 Oxalic acid substituted for the sulphate of iron pre- 
 cipitates the gold in large flakes. 
 
 ASSAY OR ANALYSIS OF IRON ORES CONTAINING 
 
 MANGANESE. 
 
 1. — Digest the roasted and pulverized ore in dilute 
 hydrochloric acid. 
 
 2. — Filter, wash residual and add washings to the 
 filtrate. 
 
 3. — Add muriate of barytes until no farther precipi- 
 tation takes place. 
 
 4. — Filter, wash and add washings to the filtrate. 
 
 5. — Evaporate the filtrate nearly to dryness and to it 
 add sufficient nitric acid to transform the sulphate of 
 iron to per-oxyd. 
 
 6. — Add solution of caustic ammonia in excess to the 
 solution. 
 
 7. — Filter and reduce the iron to the magnetic state 
 by heating the residual with resin in an iron crucible — 
 then cool and weigh. 
 
 8. — Precipitate the oxyd of manganese from the 
 filtrate, by expelling the excess of ammonia with heat. 
 
 When the ores contain much alumina or silex, flux 
 them with three or four times their weight of caustic 
 potash, then digest in hydrochloric acid and proceed as 
 above. 
 
20 QUARTZ OPERATOR'S 
 
 ASSAY OR ANALYSIS OF ORES CONTAINING GOLD 
 SILVER, COPPER, LEAD, IRON AND SULPHUR. 
 
 1. — Digest well the pulverized ore in nitric acid. 
 
 2. — Filter, wash residual (1) and add washings to 
 filtrate (1). 
 
 3. — Add to filtrate (1) hydro-chloric acid, or a solu- 
 tion of common salt, which precipitates the silver as a 
 chloride. 
 
 4. — Filter, and digest residual (2) in hydro-chloric 
 acid. 
 
 5. — Filter, wash residual (3) in warm water, and to 
 filtrate (3) with the washings add filtrate (2). 
 
 6. — Reduce the chloride of silver with carbonate of 
 soda by fusion, and weigh the button of silver. 
 
 7. — Add to filtrate (3), sulphate of soda-in solution, 
 which precipitates the lead as a sulphate. 
 
 8. — Filter, and add the residual to residual (1). 
 
 9. — Evaporate filtrate (4) to any desirable extent. 
 
 10. — Add, in excess, to concentrated filtrate (4) am- 
 monia, which precipitates sesqui-oxyd of iron. 
 
 11. — Filter, wash residual and add washings to 
 filtrate (5). 
 
 12. — Dry and heat the residual in hydrogen gas 
 within a glass tube as long as any vapor of water is dis- 
 engaged, then weigh the iron. This powder, with glass 
 as a flux at a high heat, becomes a button of iron. 
 
 13. — Treat filtrate (5), after evaporating it to dryness 
 with hydro-chloric acid, then add clean iron or zinc 
 
HAND BOOK. 21 
 
 plates to the solution diluted. Wash, dry and weigh the 
 copper precipitate. 
 
 14. — Treat residual (1), first with a strong solution 
 of carbonate of soda, then with dilute nitric acid ; and to 
 the combined filtrates add -sulphuric acid, or a solution 
 of the sulphate of soda. Wash, dry and reduce the pre- 
 cipitate with powered charcoal in an earthen crucible ; 
 then cool and weigh the button of lead. 
 
 15. — Digest the last residual in nitro-muriatic acid ; 
 add chloride of sodium in solution, filter, precipitate the 
 gold from its solution by the addition of sulphate of iron 
 in solution ; wash, dry and weigh the gold. 
 
 16. — If the gold may be alloyed with silver and 
 copper, precipitate the copper from the last filtrate by 
 the addition of iron or zinc plates ; wash, dry and add 
 the weight of the precipitate to the copper already 
 obtained. 
 
 Heat the residuary ore in a strong solution of 
 chloride of sodium, filter and precipitate the silver with 
 a clean copper plate ; wash, ignite and add the silver to 
 that already obtained. 
 
 17. — Burn off the sulphur and weigh residuary ore. 
 The sum of the weights of the gold, silver, copper, lead, 
 iron and calcined ore taken from the weight of the 
 original ore, leaves the weight of the sulphur. 
 
 EECIPES. 
 Mack Mux. — Black Flux is prepared by introducing 
 gradually in small quantities, into a crucible heated to a 
 
22 QUARTZ OPERATOR'S 
 
 very dull redness, a mixture of either two parts of cream 
 of tartar and one of nitre ; or equal parte of cream of 
 tartar and nitre. White Flux is similarly prepared ex- 
 cept that the mixture consists of one part of cream of 
 tartar and two parts of nitre. 
 
 Iron Rust Cement. — To one hundred parts of pow- 
 dered and sifted iron borings, add one part of sal-am- 
 moniac. Moisten the mixture with water to a pasty con- 
 sistency for use. 
 
 Lead Cement — Red or white lead in oil, four parts ; 
 iron borings, two to three parts. Makes a good cement 
 for steam boilers, steam pipes, etc. 
 
 Solders, for Lead. — Melt one part of block tin, and 
 when in a state of fusion add two parts of lead. Resin 
 should be used with this solder. 
 
 For Tin. — Pewter four parts, tin one part, and bis- 
 muth one part ; melt them together. Resin is also used 
 with this solder. 
 
 For Iron. — Tough brass with a small quantity of 
 borax. 
 
 For Iron, Copper and Brass. — Spelter, that is an alloy 
 of zinc and copper in nearly equal parts, is used. 
 
 Quicksilvering of Copper Plate. — First thoroughly 
 cleanse the surface of the plate, and rub it over 
 with quicksilver, or with the nitrate of mercury. — 
 The surface is sometimes cleansed by simply scouring 
 it with wood ashes, brick dust, or fine sand ; and some- 
 times by washing it with dilute acid or strong alkali. 
 When acid is employed, its corrosive qualities should 
 
HAND BOOK. 23 
 
 be neutralized before the application of the quicksilver. 
 Nitrate of mercury, when crystalized, is readily con- 
 verted to a liquid by heat, in which state it may be ap- 
 plied as a wash to the plate. 
 
 ROASTING. 
 
 Boasting is employed to dissipate the volatile parts 
 of ore by heat, and is effected in heaps or furnaces. 
 
 In Heaps. — Alternate layers of fuel and ore, usually 
 as it comes from the mine, are heaped up to the depth of 
 several feet. The lowest or ground layer is of wood, 
 arranged by cross-piling so as to afford a free circulation 
 of air. The upper layers may be of wood or coal. 
 
 The ratio of fuel in volume to that of ore varies from 
 1 to 6 to 1 to 18. Fine ores and those rich in 
 sulphur require less than coarse ores and poor in sul- 
 phur. The fire is kindled through vertical openings or 
 chimnies which extend to the ground layer. These 
 openings are closed when the fuel has well taken fire. 
 The roasting should be slow and uniform in all parts of 
 the heap. The heat may be regulated by opening or 
 closing the draft holes and chimnies. Several days and 
 even months, sometimes, are required for roasting one 
 heap. Ores similarly piled with fuel are sometimes 
 roasted in walled inclosures provided with side openings. 
 
 Furnaces, — There are a great variety of furnaces. 
 Those mostly approved for the roasting of ores embrac- 
 ing also calcining and chloridvzing are the reverberatory. 
 The interior walls of the furnace should be of the best 
 fire brick laid edgewise ; the outer walls may be of com- 
 
24 QUARTZ OPERATOR'S 
 
 mon building brick or stone. The furnace must be well 
 tied with iron rods, and carefully dried before being 
 used. 
 
 The Reverbreatory Furnace is constructed sometimes 
 with one and sometimes with two hearths or soles one 
 above the other. In the double hearth furnace, for in- 
 stance, in the treatment of silver ores the roasting and 
 sulphatization are effected on the upper sole, and the 
 calcining and chloridizing on the lower. The ore pul- 
 verized fine, is charged upon the upper sole to the depth 
 of from two to four inches, and is kept well stirred du- 
 ring the roasting. The heat should be at a low temper- 
 ature, not exceeding brown or dull red. The access of 
 air should be free. A small jet of steam into the fur- 
 nace assists in regulating the temperature and also fa- 
 cilitates oxydation. The addition of powdered charcoal 
 in small quantities may be made to advantage when the 
 ores contain arsenic. If the ores are poor in sulphur add 
 from two to three per cent, of the sulphate of iron. The 
 first operation of roasting and sulphatizing is accom- 
 plished in four or five hours. Then through an opening 
 in the upper hearth the ore is let fall upon the lower, 
 where it is heated for some time at a temperature not 
 much higher than that above. The heat is then gradu- 
 ally increased to cherry red, at which it is kept during 
 the time required for calcining and chloridizing. The 
 heat should never exceed bright red. The ore is fre- 
 quently stirred. When calcination is complete a mix- 
 ture of common salt melted and pulverized and seven 
 
HAND BOOK. 25 
 
 parts of cold calcined ore are added to the hot ore, esti- 
 mated at fifteen parts, and quickly and thoroughly mixed 
 with it by stirring. Calcination is usually effected in four 
 or five hours, and chlorination in fifteen or twenty 
 minutes. 
 
 PURIFICATION OF MERCURY. 
 
 Mercury for the purposes of amalgamation should be 
 pure. Any foreign substance such as lead, tin, zinc, or 
 bismuth diminishes its properties of combining with 
 gold and silver. To free from these and other impuri- 
 ties, 
 
 1st. Distil the impure mercury. A retort for this 
 process may readily be made of a common quicksilver 
 flask and iron pipe of syphon form. The short leg of 
 the pipe, a few inches long, is attached to the flask i n 
 the place of the removed stopper. 
 
 The long leg, three or four feet in length, inclines 
 downward from the bend. The retort should not be 
 jver two thirds filled with mercury. The heat ought 
 first to be applied to the short leg of the pipe and upper 
 part of the retort, then to all parts of the flask alike. 
 The long leg of the pipe must be kept cold. This may 
 be effected by wrapping it with cloths and pouring on 
 cold water. The discharge end may also be immersed 
 in cold water, kept in the receiver. The heat should be 
 uniform, and the distillation slow. The common covered 
 retort is far preferable to the one described. 
 
 2. Heat and frequently agitate the distilled mercury 
 in thin sheets, with one part of nitric acid and two parts 
 
26 QUARTZ OPERATOR'S 
 
 of pure water. The heat should be kept at 120 degrees 
 Fahrenheit, for several hours. Repeat these operations 
 until satisfactory results are obtained. Then pour off 
 the mercury for use. 
 
 3. Digest the crust (nitrate of mercury and impuri- 
 ties) in nitric acid. Then dilute the solution, filter, pre- 
 cipitate the mercury by metallic copper, and add it to 
 the mercury already obtained. Or the nitrate of mercu- 
 ry may be converted to a liquid, simply by heat, and 
 the metal then precipitated by copper plate. 
 
 EXTRACTION OF GOLD BY THE PAN PROCESS. 
 
 1. The rock, as it comes from the mines, is usually 
 crushed wet by stamps, to a fine granular state, and run 
 into large tanks. 
 
 2. Charges of the reduced ore, with sufficient water 
 to form a thin paste, are thoroughly ground in iron pans. 
 As gold found in rock exists almost without exception in 
 a metalic state, friction alone is required to fit it for 
 amalgamation. 
 
 3. Quicksilver is ordinarily added to the pulp, as the 
 pans commence running. To avoid grinding the quick- 
 silver excessively, the addition is sometimes made with 
 the muller slightly raised, after the reduction of the ores 
 
 4. The charge is then drawn off and washed, leaving 
 the amalgam in the separators. 
 
 5. The proportions usually observed, for instance, in 
 the Wheeler & Randall grinders and amalgamators, are 
 
 Ore to the charge, 1,200 pounds. 
 
 Quicksilver to the charge of ore, .... 75 " 
 
HAND BOOK. 27 
 
 Revolutions of muller, 60 to 75 
 
 Time of reducing, 2 to 3 hours. 
 
 As gold-bearing rock is seldom found sufficiently rich 
 to render it advisable to treat the entire mass in pans, 
 the above method is subject to various modifications, of 
 which the following are a few : 
 
 1. The heavier and richer portions of the rock, as 
 crushed, are concentrated by revolving-blankets, buddies 
 or other machinery, and then pulverized and amalga- 
 mated in pans. 
 
 2. Amalgamation is commenced in the batteries 
 during the crushing operation, and is carried on through 
 a series of shaking tables, riffles, and copper plates. The 
 richer portions of the tailings are then concentrated and 
 treated in pans. 
 
 3. Grinding and amalgamating are effected in pans 
 while the reduced ores are flowing continuously through 
 them. 
 
 4. The sulphurets or concentrated tailings are some- 
 times roasted in a reverberating furnace, before being 
 ground and amalgamated. 
 
 5. Thin layers of the concentrated sulphurets or tail- 
 ings are spread in inclosures open to the sky, and 
 allowed to remain a long time, for instance, a year. The 
 tailings are occasionally turned with shovels and the 
 lumps broken, so as to expose as much surface as possi- 
 ble to the action of the air. Common salt mixed with 
 the tailings assists in their oxydation. When quite 
 
28 QUARTZ OPERATOR'S 
 
 thoroughly oxydized, they are treated in pans. This 
 is very economical and effectual, and by it the yield of 
 gold, (especially if very fine) to the ton is frequently 
 much greater than was obtained at first from the same 
 ores. 
 
 EXTRACTION OF GOLD BY CHLORINATION. 
 
 1. Pulverized Ores, containing gold, having been well 
 roasted, cooled and moistened with water, are put into 
 closely covered wooden cisterns, whose bottoms are so 
 constructed that chlorine gas can permeate the mass 
 from underneath. 
 
 2. Chlorine gas produced by heating sulphuric acid, 
 per oxyd of manganese and common salt, in a suitable 
 generator, is caused to enter the cisterns at the bottom? 
 through leaden pipes. The effect of the chlorine on the 
 gold, is to produce terchloride of gold. 
 
 3. Pure water, after the chloride has done its duty> 
 which takes from ten to fifteen hours, the covers being 
 removed, is added sufficient to keep the cisterns even 
 with the mass. The effect of the water is to dissolve 
 the terchloride of gold. The solution is then drawn off 
 into glass vessels. 
 
 4. Sulphate of iron, in solution, is used to precipitate 
 the gold, which may then be gathered as a powder. 
 
 EXTRACTION OF SILVER BY THE PATIO PROCESS. 
 
 1. Patio signifies a yard. For amalgamating pur- 
 poses, the floor of the yard is made level, paved with 
 brick or granite blocks, surrounded by high walls, and 
 
HAND BOOK. 29 
 
 usually left open to the sky. On this floor circular 
 batches of silver ore, reduced to an inpalpable paste by 
 stamps and arastras, or other machinery, are spread to 
 the depth of seven to twelve inches, and inclosed by low 
 close curbs. 
 
 2. Salt, varying in quantity according to its quality 
 and the richness of the ore, is well mixed with the pulp 
 by treading it with horses, mules, or oxen, and turning 
 it with shovels. The effect of the salt is to desulphur- 
 ize the sulphurets, and produce chloride of silver. The 
 batch is then left one entire day. 
 
 3. Magistral, that is, roasted and pulverized copper 
 pyrites, varying in quantity with its quality, the richness 
 of the ores and season, is well mixed with the pulp after 
 it has been subjected to the treading and turning opera- 
 tion one hour. The ultimate effect of the magistral is 
 to revive the silver by depriving it of its chlorine. 
 
 4. Quicksilver is added, usually in three charges to 
 the mass, by being sprinkled in minute particles through 
 cloth or other porous substance. After the addition of 
 the first charge of quicksilver, the batch is thoroughly 
 mixed, thrown into heaps of about one ton each, smooth- 
 ed and left at rest one whole day. The treading, turn- 
 ing and heaping operation is performed every other day, 
 occupying five or six hours, and is found much more 
 effective in a morning than an evening. The second 
 charge of quicksilver is added and similarly treated 
 when it is ascertained by washing a small quantity of 
 the mixture, that the first has been well incorporated. 
 
 b3 
 
30 QUARTZ OPERATOR'S 
 
 After the second charge has performed its work, the 
 third charge is added to take up any stray particles of 
 silver, and to fit the amalgam better for separation. 
 
 5. Lime is added to cool, and magistral to heat the 
 mass, according as it may be too hot or too cold. Too 
 much heat is indicated by the quicksilver becoming ex- 
 tremely divided, and of a dark color, with occasional 
 brown spots upon its surface. Too little heat is indica- 
 ted by the quicksilver retaining its natural color and 
 fluidity. A proper degree of heat is indicated by the 
 amalgam's being of a greyish-white color, and yielding 
 readily to a slight pressure. 
 
 6. The proportions to the ton of ore, valued at fifty 
 dollars, are : 
 
 Sea Salt, of good quality, , . . . . 80 pounds. 
 
 Magistral. — When containing ten per cent., 
 of the sulphate of copper, 
 
 in summer, 20 " 
 
 in winter, 10 " 
 
 Quicksilver — First charge, ... 14 " 
 
 Second charge, 5 " 
 
 Third charge, 7 " 
 
 Lime. — More or less, see section 5th,. . .15 " 
 
 An excess of magistral, quicksilver, or lime is inju- 
 rious. An excess of salt causes a loss of quicksilver 
 but is not otherwise injurious. 
 
 The time employed in treating a batch of ore varies 
 from twelve to sixty days. Light and good weather 
 greatly facilitate operations. 
 
 7. The separation is accomplished by agitating the 
 
HAND BOOK. 31 
 
 pulp or mixture with abundance of water, in a large, 
 deep, circular vessel, and causing the lighter portions of 
 the mass to flow slowly off, until the amalgam is gath- 
 ered by itself. 
 EXTRACTION OF SILVER BY THE FREYBERG PROCESS. 
 
 1. This process takes its name from Freyberg, a 
 place in Germany, where it was first practiced. The 
 ores, if possible, are assorted so as to contain not less 
 than twenty-five per cent, of sulphurets. When they 
 contain less, the sulphate of iron is added to make up the 
 deficiency. When more, then a sufficient quantity of 
 the richest in sulphurets is roasted without sea-salt to 
 make good the ratio : the ores are crushed dry. 
 
 2. Sea-Salt and crushed ores are thoroughly mixed 
 together, roasted in a reverberatory furnace, and then re- 
 duced to an impalpable powder in a suitable mill. The 
 salt and heat transform the sulphurets of silver to 
 chloride of silver. 
 
 3. Wrought Iron, in small pieces, with a pasty mixt- 
 ure of the reduced ores and water are put into German 
 barrels, which, making twenty revolutions a minute, are 
 run two hours. The effect of the iron is to revive the 
 silver to its metallic state. 
 
 4. Quicksilver is then poured into the barrels, after 
 which they are run sixteen hours continuously, except 
 the time taken to regulate the consistency of the pulp, 
 by the addition of ore or water. At the end of the time 
 run, the casks are filled with water and revolved quite 
 
 b4 
 
32 QUARTZ OPERATOR'S 
 
 slowly for one or two hours, when the mass is dis- 
 charged into lai ge vats and the amalgam separated by 
 washing. 
 
 5. The Proportions to the ton of ore valued at $75.00 
 per ton, are 
 
 Sea-Salt, added before the roasting process, 200 lbs. 
 
 Wrought Iron, added to the ton of roasted 
 ore 200 " 
 
 Quicksilver, added to the ton of roasted 
 ore.. 1000 " 
 
 EXTRACTION OF SILVER BY THE VEATCH PROCESS. 
 The only essential difference between this and the 
 Freyberg process consists in the employment of tubs 
 instead of barrels, and the use of steam directly in the 
 pulp. Vertical plates of iron or copper, for reviving the 
 silver from its chloride state, are fastened to the muller 
 arms, so as to revolve edgewise through the pulp or 
 mass. The operations are greatly hastened by the 
 application of steam, so that not more than five or six 
 hours are required for the treatment of a charge of ore. 
 
 EXTRACTION OF SILVER BY THE PAN PROCESS. 
 
 1. The Ores, as they come from the mines, are 
 usually crushed wet to a granular state by stamps, and 
 run into a series of large settling tanks. To crush wet, 
 and at the same time fine, is very objectionable, as 
 much silver thereby is carried off by the water. 
 
 2. Charges of the reduced ores, with sufficient water 
 to form a soft, pasty mass, are put into iron pans con- 
 
HAND BOOK. 33 
 
 structed as grinders, which are run from two to six 
 hours, according to their reducing properties. Water is 
 occasionally added during the grinding process, as the 
 condition of the pulp may require. 
 
 3. Quicksilver is ordinarily poured into the pans as 
 they commence running. Sometimes, to avoid grinding 
 it excessively, the muller is slightly raised and the ad- 
 dition made after the reduction of the ores. 
 
 4. Chemicals, differing in kind and proportions, to 
 almost an indefinite extent are employed. As to their 
 practical value, a diversity of opinion prevails among 
 the most experienced and intelligent amalgamators 
 and mill-men. In pans of slow motion and of little 
 grinding capacity, certain chemicals, in the treatment of 
 some ores, have been used to advantage. Their em- 
 ployment and proportions, in all cases, depend upon the 
 composition and character of the ores. Experience 
 thus far, chiefly goes to show that the chemicals in pans, 
 which grind rapidly, are not only valueless but in many 
 instances injurious to amalgamation. In pans of this 
 character, the sulphurets of silver ores become not only 
 mechanically divided, but chemically decomposed. The 
 heat of the steam contributes to the attainment of this 
 desirable object ; the iron of the pans serves also to 
 revive any silver existing as a chloride. 
 
 The proportions usually observed in operating the 
 Wheeler & Randall grinders and amalgamators, are : 
 
 5b 
 
34 QUARTZ OPERATOR'S 
 
 Ore to the charge, 1,200 lbs 
 
 Quicksilver to the charge of ore valued 
 
 at $50 per ton, 200 " 
 
 Revolutions of muller per minute, 60 to 75 
 
 Heat, to nearly the boiling point of water. 
 
 Time of working charge, , 2 to 3 hs. 
 
 The charge is next drawn off and separated by wash- 
 ing. The grinders and amalgamators should be well 
 cleansed before being recharged. 
 
 6. A few of the chemical recipes employed in Nevada, 
 to the ton of ore, valued at $50 : 
 recipe I. 
 
 Sulphuric Acid, 3 pounds. 
 
 Sulphate of Copper, 2 " 
 
 Common Salt, 15 " 
 
 rec. ii. 
 
 Sulphuric Acid, 2 " 
 
 Alum, 2 
 
 Sulphate of Copper 1.5 " 
 
 REC. III. 
 
 Sulphate of Copper, 1.2 " 
 
 Sulphate of Iron, 1.0 " 
 
 Sal Ammoniac, 0.8 " 
 
 Common Salt, 2.0 " 
 
 REC. IY. 
 
 Catechu, 2 " 
 
 Sulphate of Copper, 2 " 
 
 Common Salt, 25 " 
 
HAND BOOK. 35 
 
 REC V. 
 
 Sulphate of Iron, 1.5 " 
 
 Nitric Acid, 1.5 " 
 
 Common Salt, 15.0 " 
 
 KEC. VI. 
 
 Muriatic Acid, 30 ounces. 
 
 Peroxyd of Manganese, ... 8 " 
 
 Sulphate of Copper, 10 " 
 
 Sulphate of Iron, 10 " 
 
 The salt is applied half an hour before the other 
 chemicals. 
 
 SEPARATION OF SILVER FROM LEAD BY THE PATTIN 
 SON PROCESS. 
 
 1. This process is founded on these facts: If a 
 melted alloy of silver and lead is stirred while cooling 
 slowly, crystals of lead form and sink, which may be 
 removed with a drainer. A large portion of the lead 
 may thus be separated from the silver. 
 
 2. Cast-iron pans, capable of holding about five tons 
 each, and provided with fire places, are arranged in a 
 series, as A, B, C, D, E, F, G, in a straight line. 
 
 3. The metal of ores containing silver and lead as it 
 comes from ordinary smelting works, is melted, for in- 
 stance, in pan D, and then allowed to cool very slowly. 
 The metal while cooling is stirred, especially near the 
 edges of the pan, with an iron bar. As soon as crystals 
 form and sink to the bottom, they are taken out with an 
 iron drainer raised to a temperature somewhat higher 
 
 b6 
 
36 QUARTZ OPERATOR'S 
 
 than that of the metal bath. From one half to two 
 thirds of the charge is thus removed to pan E, and the 
 balance taken to pan C. Other charges of D, are simi- 
 larly treated and disposed of. The charges of C and E 
 are treated and disposed of in like manner, except that 
 the crystals of E go to F, and the balance to D, and the 
 crystals of C go to D, and the balance to B. Thus, after 
 successive meltings and drainings, the alloys rich in 
 silver pass to A, while the lead, almost entirely de- 
 prived of silver, goes to G. The alloys obtained in pan 
 A are then subjected to cupellation. An alloy contain- 
 ing over six hundred dollars of silver to the ton should 
 not be treated by this process. 
 
 SEPARATION OF SILVER FROM COPPER BY THE LIQUA- 
 TION PROCESS. 
 
 1. This process is founded on these facts : Lead and 
 copper fused together form an alloy, which, if rapidly 
 cooled, maintains an intimate admixture, but if slowly 
 cooled, separates. An alloy of lead and copper slowly 
 heated to near its point of fusion, also separates. Silver, 
 if contained in the alloy, goes with the lead. 
 
 2. Either an alloy of copper or silver, or matte (crude 
 black copper reduced, but not refined from sulphur and 
 other impurities), containing silver, as it comes from the 
 smelting furnace, is melted with lead of about four 
 times its weight, in a cupola furnace, and cast into plain 
 circular plates which are suddenly cooled. These plates, 
 called " liquation cakes," are arranged on their edges, 
 
HAND BOOK. 37 
 
 with alternate layers of charcoal, in a liquation furnace. 
 The charcoal is then ignited, and a degree of heat pro- 
 duced somewhat below that of the fusing point of cop- 
 per. The lead and silver melt and flow into a receiver, 
 while the copper, in a porous state, retains the forms of 
 the original cakes. If the separation may have been 
 imperfect, the cakes are farther treated by being raised 
 to a higher degree of heat in the " sweating furnace." 
 The silver is then separated from the lead by cupella- 
 tion. 
 
 SEPARATION OF SILVER FROM LEAD BY THE PARKE 
 PROCESS. 
 
 1. Lead, containing silver, is fused in large cast-iron 
 pots. Melted zinc is added and well stirred in the alloy. 
 The fire being withdrawn from under the pot, the whole 
 is left at rest for a short time. 
 
 2. The silver and zinc separating from the lead, form 
 an independent alloy, which is skimmed from the sur- 
 face of the metal bath, as long as it rises. 
 
 3. This scum alloy, containing some lead, is heated in 
 a liquation retort. The silver and lead fuse, and, to a 
 great extent, flow into prepared moulds. The alloy thus 
 run off is then cupelled; the alloy of zinc and silver 
 remaining in the retort, are partially separated by dis- 
 tillation. The silver thus obtained is freed of its im- 
 purities by cupellation. 
 
 4. The proportions are : 
 
38 QUARTZ OPERATOR'S 
 
 Charge of argentiferous lead to the pot 
 
 usually from 6 to 7 tons. 
 
 Charge of zinc to the ounce, of silver 
 
 by estimation, 1.5 to 2 pounds 
 
 Quantity of silver to the ton of lead. . 10 to 15 ounces 
 Time of stirring alloy, after the addition 
 
 of zinc, from 10 to 15 hours. 
 
 The alloy prepared for cupellation contains, of silver, 
 to the ton, about 1,000 ounces. 
 
 SEPARATION OF SILVER FROM LEAD BY CUPEL- 
 LATION. 
 
 1. The Alloy of silver and lead is melted in a circular 
 reverberatory furnace provided with openings through its 
 sides for the admission of metal, heat, currents of air, 
 and for the escape of vapors or litharge. The escape is 
 opposite the blast opening. The roof or top of the fur- 
 nace is of dome-form and movable. At each cupella- 
 tion, the hearth, usually of hollow form, is broken up 
 and replaced by one made of clay, sand and carbonate 
 of lime. 
 
 2. Blasts, or currents of air, are blown continually 
 during the operation upon the surface of the fused alloy, 
 promoting oxydation of the lead and causing the litharge 
 to pass out through the escape opening. The gate-way 
 of this opening is kept level w T ith the surface of the 
 metal within. The silver thus separated from the lead 
 remains on the hearth of the furnace in nearly a pure 
 state. It is deprived of what lead it may contain by 
 the humid way of assay. 
 
HAND BOOK. 30 
 
 EXTRACTION OF SILVER BY THE AUGUSTIN PROCESS. 
 
 1. This Process, employed thus far chiefly in the 
 treatment of matte, (impure copper) containing silver, is 
 founded on the solubility of chloride of silver in a hot, 
 concentrated solution of common salt. 
 
 2. The Matte, as it comes from the cupola or high 
 furnace, is crushed dry by stamps, pulverized in suit- 
 able mills and bolted. The coarser portions thus 
 obtained are taken to the copper works. 
 
 3. The Roasting of the powdered matte in a reverber- 
 atory furnace is commenced at a low temperature with 
 a free access of air. By careful, uniform roasting, at a 
 dull red heat, the sulphurets of silver, iron and copper 
 are produced. The heat is then increased to cherry-red 
 which decomposes the sulphates of iron and copper, but 
 not the sulphate of silver. 
 
 4. Salt, previously melted, pulverized and mixed 
 with cold calcined matte, is added to the hot matte in 
 the furnace and thoroughly mixed with it by stirring. 
 The sulphate of silver is thus transformed to chloride of 
 silver. 
 
 5. The Apparatus for the humid operations consist of 
 a large heating reservoir, a series of dissolving tubs, two 
 large settling cisterns, four precipitating tubs to each one 
 of the dissolving tubs, and two large receptacles, 
 arranged in the order here given on descending steps. 
 The dissolving and precipitating tubs are nearly cylin- 
 drical. They are provided with filters made of small 
 
40 QUARTZ OPERATOR'S 
 
 sticks and straw, covered with cloth ; a vertical parti- 
 tion, resting on the filter, divides each tub into two 
 unequal compartments. 
 
 6. The Ghloridized Matte being put into the larger 
 compartments of the dissolving tubs, sufficient of the hot 
 salt solution from the heating reservoir above to com- 
 pletely immerse the matte, is let into the tubs ; they are 
 then left at rest one hour. The discharge cocks of the 
 heating reservoir and tubs then being opened, the hot 
 salt solution is filtered through the contents of the tubs, 
 and run off from the smaller compartments, at openings 
 at first above the level of the matte, afterwards at open- 
 ings near the bottoms of the tubs, into the settling 
 cisterns, until a test with clean copper plate shows no 
 trace of silver in the filtered solution. 
 
 7. Copper (copper cement) is put into each of the 
 upper two precipitating tubs in the several series of 
 four, and iron (wrought scrap iron) into each of the 
 lower two. The chloride solution from the settling 
 cisterns is then slowly filtered through the several series 
 of precipitating tubs, and the filtered solution run into 
 the large receptacles below. The silver is precipitated 
 by the copper in the upper tubs, and the copper in solu- 
 tion is precipitated by the iron in the lower tubs. The 
 silver is taken twice a week from the precipitating tubs 
 and refined. The copper precipitated in the lower tubs 
 is transferred to the upper tubs. The filtered matte is 
 washed and taken to the copper works. The filtered 
 
HAND BOOK. 41 
 
 solution, in the receptacles, is pumped into the heating 
 reservoir and used again. 
 
 8. The Proportions usually observed are : 
 
 Matte, before roasting, should con- 
 tain of sulphur, not less than, 20 per ct. 
 
 Charge of Matte, to the furnace, 
 
 for roasting and calcining,. . 500 pounds. 
 
 Charge.— Melted Salt, 35 " 
 
 Roasted Matte, 220 " 
 
 Add same for chloridizing. 
 
 Time. — Roasting on the upper sole 
 
 of Furnace 4 to 4J hs. 
 
 Calcining on lower sole of 
 
 Furnace 4 to 4£ " 
 
 From 8 to 9 " 
 
 Time of chloridizing, from 15 to 20 minutes. 
 
 Charge of chloridized matte to the 
 
 tub, from 1000 to 1200 lbs. 
 
 Salt — In the solution, in reference 
 
 to the water, from 20 to 25 per ct. 
 
 Degrees of Heat of salt solution. . 131° Fahr. 
 
 Time of dissolving and precipita- 
 ting, from ... 20 to 24 hours. 
 
 Solution of Sah y run through each 
 tub to 1000 pounds of matte, 
 from 200 to 250 cubic ft. 
 
 Depth of Copper in precipitating 
 
 tubs, about 6 inches. 
 
 Depth of Iron in precipitating 
 
 tubs, about ... 6 " 
 
42 QUARTZ OPERATOR'S 
 
 EXTRACTION OF SILVER BY THE ZIERVOGEL PORCESS. 
 
 1. This Process, employed thus far chiefly in the 
 treatment of matte (impure copper) containing silver, 
 is founded on the solubility of sulphate of silver in hot 
 water. 
 
 2. The Matte, as in the Augustin process, having 
 been thoroughly pulverized, is carefully roasted and 
 calcined till the sulphates of iron and copper are com- 
 pletely decomposed, but none of the sulphate of silver. 
 When small quantities of the roasted matte, thrown hot 
 into water, give only a very slight blue color, the cal- 
 cination is regarded complete. 
 
 3. The Sulphatized Matte is then treated, in all respects, 
 the same as the chloridized matte, (see sec. 6, page JJ^) 
 in the Augustin process, except that pure water is em- 
 ployed instead of solution of salt. 
 
 4. The Proportions usually observed are : 
 Matte, before roasting, should con- 
 tain of sulphur, not less than, 20 per ct. 
 
 Charge of Matte to the Furnace, . 500 lbs. 
 
 Time. — Roasting on upper sole of 
 
 Furnace, « 4 to 4J hours. 
 
 Calcining on lower sole of 
 
 Furnace, . . ............. 4 to 4£ hours. 
 
 From 8 to 9 " 
 
 Charge of sulphatized matte to 
 
 the tub, from 1000 to 1200 lbs. 
 
 Degrees of Heal of the water for 
 
HAND BOOK. 43 
 
 dissolving 149° Fahr. 
 
 Time of dissolving and precipita- 
 ting, from 20 to 24 hours. 
 
 Hot Water run through each tub, 
 
 from 200 to 250 cubic ft. 
 
 Depth of Copper in upper precipi- 
 tating tubs, 6 inches. 
 
 Depth of Iron in lower precipita- 
 ting tubs, « " 
 
 EXTRACTION OP SILVER BY THE PATERA PROCESS. 
 
 1. In this Process the ores are thoroughly pulverized 
 and chloridized by roasting with common salt. 
 
 2. Hot Water to dissolve the chlorides of various base 
 metals is filtered through the chloridized ores put in 
 tubs similar to the dissolving tubs in the Augustin pro- 
 cess. The ores are then cooled and transferred to 
 similar, but smaller tubs. 
 
 3. Hyposulphite of Soda, in cold solution, is then 
 filtered through the ores and run into precipitating tubs 
 until all the chloride of silver is completely dissolved. 
 
 4. Poly sulphide of Sodium, sufficient to produce a 
 neutral liquor, is then added, which precipitates the 
 silver as a sulphide in sacks fitted to the inside of the 
 tubs. This neutral liquor is preserved for lixiviating 
 purposes. 
 
 5. The Sulphide of Silver thus obtained, after being 
 washed in warm water, pressed and dried, is heated 
 under muffles with free access of air till nearly all the 
 
44 QUARTZ OPERATOR'S 
 
 sulphur is expelled. The metallic silver is then re- 
 refined. 
 
 MECHANICS. 
 
 Force. — That which produces or tends to produce mo- 
 tion, or change of motion, is termed force. 
 
 Work. — The product of force and the distance through 
 which it is exerted, is termed icork — mechanical work. 
 
 Units. — The units of force, distance, and time are re- 
 spectively one (1) pound, one (1) foot, and one (1) 
 minute. 
 
 Horse Power. — Thirty three thousand (33,000) units 
 of work constitute one (1) " horse power," that is, thirty- 
 three thousand pounds raised vertically one (1) foot in 
 one (1) minute, or its equivalent. 
 
 TO FIND THE HORSES' POWER IN A GIVEN TIME. 
 
 Rule. — Divide the product of the weight in pounds 
 and the vertical distance in feet through which the 
 weight is to be raised, by the product of the time in 
 minutes and thirty-three thousand. (33,000.) 
 
 Ex. 1. — Required the horses' power necessary to drive 
 forty-five (45) stamps, each stamp weighing six hundred 
 and forty (640) pounds, falling ten inches, and making 
 seventy-seven (77) drops per minute, allowing twenty- 
 five per cent, for friction. 
 
 Cat 45X640X77X10--12=1848000. 
 
 1848000X1.25=2310000. 
 
 2310000h-33000=70 horses' power. Ans. 
 
 Ex. 2. — How many horses' power are required to 
 
HAND BOOK. 45 
 
 raise water three hundred (300) feet by a single acting 
 pump seven (7) inches diameter, thirty (30) inches 
 stroke, making fifteen (15) lifting strokes per minute, 
 allowing thirty -five (35) per cent, for friction ? 
 
 Gal. 15 X 30— 450 inches, column of water raised. 
 
 450-=- 12=37.5 feet, column of water raised. 
 
 7X7X,7854-f-144=,2672 area of end of column of 
 water. 
 
 ,2672X37.5=10.02 solid inches in column of water. 
 
 10.02X62. 5=626.25 pounds in column of water. 
 
 626.25X1.35=845.4375 pounds with friction added. 
 
 845.4375 X 300-^-330000=7.68 horses* power. Ans. 
 
 Ex, B. — The slant depth of the hoisting shaft of the 
 Eureka Mine, Sutter Creek, is one thousand (1000) 
 feet ; the dip of the lode being sixty (60°) degrees ; how 
 many horses' power are required to raise one ton (2,000 
 pounds) of rock to the surface in five minutes, allowing 
 fifty per cent, for friction ? 
 
 90°— 60°=30° complement. 
 
 1000^-2=500 feet side of right angled triangle op- 
 posite 30° 1000X1000=100000; 500X500=250000; 
 1000000—250000=750000. 
 "l/750000= : 866.0254 perpendicular depth of mine. 
 
 866.0254X1.50=1299.0376, with friction added. 
 
 1299.0376X2000=2598075 ; 33000X5=165000. 
 
 2598085-^-165000=15.75 horses' power. Ans. 
 
 Remarks. — 1. It is found by experience that it re- 
 quires to reduce by stamps hard quartz rock (frequently 
 
46 QUARTZ OPERATOR'S 
 
 called by miners, live rock,) from the size usually fed 
 into batteries to the ordinary granular size coming from 
 the same, about one horse power to the ton in twenty- 
 four hours ; and to farther reduce it from the granular 
 state by the common pan-grinders and amalgamators to 
 a " slum " or slime of economical fineness, about one 
 horse power in the same length of time. 
 
 2. That the Wheeler & Randall Grinder and Amal- 
 gamator, four feet diameter, making sixty-five revolu- 
 tions per minute, will reduce from the granular to the 
 slime condition, five tons of rock per twenty-four hours. 
 
 Note. — It is estimated by those using the Wheeler and 
 Randall Grinder and Amalgamator, that it requires to 
 run each four foot machine sixty-five revolutions per 
 minute, three (3) horse power. This is somewhat less 
 than the inventors were ready to believe, on account of 
 the extraordinary reducing properties of their invention. 
 
 3. That it requires to run a Separator seven feet di- 
 ameter, (not grinding,) about one half of one horse 
 power. 
 
 VARIED MOTION. 
 Laws of Uniformly Varied Motion. — 1. In uniformly 
 varied motion, the path described at the end of any time 
 is half that which the body would describe in the same 
 time if it were to move uniformly with the velocity ac- 
 quired during this time. 
 
 2. In uniformly accelerated motion, the paths de- 
 scribed at the end of any two times, are to each other as 
 the squares of these times. 
 
 3. That these paths are to each other as the square 
 
HAND BOOK. 47 
 
 of the velocities acquired at the end of the correspond- 
 ing times. 
 
 Let one second be a unit of time ; then will the times 
 of a falling body, in vacuo, be 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, etc., etc. 
 
 The corresponding fall during each second, will be 
 
 1, 3, 5, 7, 9, 11, 13, 15, 17, etc., etc. 
 
 The fall during any number of seconds, will be 
 
 1, 4, 9, 16, 25, 36, 49, 64, 81, etc., etc. 
 
 And the velocities acquired at the end of each second, 
 will be 
 
 2, 4, 6, 8, 10, 12, 14, 16, 18, etc., etc. 
 
 Now, a body at the equator falls, in vacuo, as deter- 
 mined by experiments, 16.0904 feet in one second of 
 time. The resistance of the atmosphere does not much 
 retard the velocity of heavy falling bodies. Let then, 
 sixteen feet represent the distance which a falling body 
 will describe in one second of time, when impressed by 
 gravity alone. The velocity acquired at the end of 
 the first second of time, is called the "initial veloc- 
 ity," and is found to be 32.1808 feet; that is, twice 
 16.0904 feet. This velocity, due to the force of gravity, 
 is usually denoted by g; that is, g=32.1808, which, 
 for most practical purposes, may be taken at 32 feet. 
 TO FIND THE DISTANCE A BODY WILL FALL IN TERMS 
 OF THE VELOCITY. 
 
 Rule 1. — Divide the square of the velocity by sixty- 
 four (64). 
 
48 QUARTZ OPERATOR'S 
 
 Example. — The velocity is 256 feet, what distance 
 has the body fallen ? 
 
 Calculation. 256X256-f-64=1024 feet. Ans. 
 TO FIND WHAT DISTANCE IN FEET A BODY WILL FALL 
 IN A GIVEN TIME. 
 
 Rule 2. — Multiply the square of the time in seconds 
 by si teen (16). 
 
 Example. — What distance will a body fall in one 
 minute ? 
 
 Calculation. 60X60X16=57600 feet. Ans. 
 
 TO FIND THE VELOCITY IN FEET, IN TERMS OF THE 
 TIME. 
 
 Rule 3. — Multiply the time in seconds by thirty-two. 
 
 Example. — What velocity does a falling body acquire 
 in seven seconds (7)? 
 
 Calculation. 32X7=224 feet. Ans. 
 
 TO FIND THE VELOCITY IN TERMS OF THE DISTANCE. 
 
 Rule 4. — Multiply the square root of the distance by 
 eight (8). 
 
 Example. — What velocity will a body acquire by 
 falling one hundred and ninety-six feet (196) ? 
 
 Calculation. 8|/i96=112 feet. Ans. 
 TO FIND THE TIME FALLEN, THE VELOCITY BEING 
 GIVEN. 
 
 Rule 5. — Divide the velocity by thirty-two (32). 
 
 Example. — The velocity is 1920 feet, what time has 
 it fallen? 
 
 Calculation. 1920-7-32—60 seconds. Ans. 
 
HAND BOOK. 49 
 
 TO FIND THE TIME A BODY HAS FALLEN, THE DIST- 
 ANCE BEING GIVEN. 
 
 Rule 6. — Divide the square root of the distance fal- 
 len by four (4). 
 
 Example. — How long will it take a body to fall one 
 hundred and forty-four feet (144) ? 
 
 Calculation, j/ 144 -^-4— 3 seconds. Ans. 
 WATER POWER. 
 
 The theoretical velocity with which a liquid issues 
 from an orifice in the bottom or side of a vessel that is 
 kept full, is equal to that which a heavy body would 
 acquire by falling from the level of the surface to the 
 level of the orifice. The rules, therefore, under the 
 head of " Varied Motion," apply equally well to falling 
 bodies and to hydraulics. 
 
 The practical velocity estimated for the entire open- 
 ing, is considerably less than the theoretical velocity 
 owing to oblique currents and to friction. These oblique 
 currents produce a contraction in the vein or stream. 
 The minimum transverse section of the contracted vein, 
 is the plane at which the velocity is nearly equal to the 
 theoretical velocity. The quantity of water which will 
 be discharged in a certain time, depends upon the form 
 of the opening, as well as upon the head. Tlius, by 
 means of a conical tube of the form of the contracted 
 vein, the velocity at the opening or smaller end of the 
 tube, is nearly equal to the theoretical velocity. The 
 theoretical velocity per second (rule 4, varied motion), 
 
50 QUARTZ OPERATOR'S 
 
 is eight times the square root of the head in feet. The 
 actual velocity estimated for the entire opening, as ordi- 
 narily constructed, is five and four-tenths the square 
 root of the head. 
 
 Water, as a power or force, is exerted on water 
 wheels by its weight and by its impulse. Weight and 
 impulse are combined on the overshot and breast wheels. 
 The theoretical work accomplished by weight is the 
 product of its force and the vertical distance through 
 which it is exerted. The theoretical work accomplished 
 by impulse, is the product of the force produced by 
 weight of the flow of water, and the vertical height or 
 head necessary to produce the velocity with which the 
 weight moves. 
 
 The available work depends not only upon the mag- 
 nitude of the force exerted, but upon the direction of 
 that force in reference to the direction given to the re- 
 sistance ; also upon the form of the floats or buckets of 
 the wheel, friction, losses by leakage, etc. 
 
 The average efficiency of various water wheels, run- 
 ning under favorable circumstances, as found by expe- 
 rieuce, is as follows, to wit : 
 
 Undershot, having flat, radial floats 33 
 
 Poncelet, improved undershot, , 60 
 
 Turbine, for example, the " Jouval," 68 
 
 Reaction, for example, the Scotch Turbine, .... 66 
 Overshot and Breast, the efficiency of that part 
 of the fall acting by weight, is about 78 
 
HAND BOOK. 51 
 
 And of that part acting by impulse, 40 
 
 The best velocities of the various water wheels, 
 as compared with the supply velocities, are 
 as follows : 
 Undershot and Low Breast, at circumference, . . 50 
 
 Turbines, at the middle of ring of buckets, 65 
 
 Reaction, at circumference, 97 
 
 Overshot, at circumference, .50 
 
 The velocity of the overshot wheel at its circumfer- 
 ence, should be about six feet ; which is due a head of 
 2.25 feet. 
 
 Let the vertical distance from the centre of the open- 
 ing in the gate, to the surface of the water in the flume 
 or reservoir, be termed the head, and the vertical dis- 
 tance from the centre of the opening in the gate to the 
 lower edge of the wheel, the fall 
 
 2c 
 
52 
 
 QUARTZ OPERATOR'S 
 
 TABLE OF COEFFICIENTS FOR ESTIMATING THE 
 HORSES' POWER OF WATER WHEELS. 
 
 Head. 
 
 
 Head. 
 
 
 Head. 
 
 
 Head. 
 
 
 ft. in. 
 
 coefficient. 
 
 ft. in. 
 
 coefficient 
 
 ft. in 
 
 coefficient. 
 
 ft. in 
 
 coefficient. 
 
 r 
 
 12 
 
 1 7 
 
 54" 
 
 3 2 
 
 76" 
 
 9. 
 
 128 
 
 2 
 
 17 
 
 1 8 
 
 55 
 
 3 4 
 
 78 
 
 10. 
 
 135 
 
 3 
 
 21 
 
 1 9 
 
 56 
 
 3 6 
 
 80 
 
 12. 
 
 148 
 
 4 
 
 25 
 
 1.10 
 
 58 
 
 3 8 
 
 82 
 
 14. 
 
 160 
 
 5 
 
 28 
 
 1.11 
 
 59 
 
 3 10 
 
 84 
 
 16. 
 
 171 
 
 6 
 
 30 
 
 2 
 
 60 
 
 4 
 
 85 
 
 20. 
 
 191 
 
 7 
 
 33 
 
 2 1 
 
 62 
 
 4 3 
 
 88 
 
 25. 
 
 213 
 
 8 
 
 35 
 
 2 2 
 
 63 
 
 4 6 
 
 90 
 
 30. 
 
 233 
 
 9 
 
 37 
 
 2 3 
 
 64 
 
 4 9 
 
 93 
 
 36. 
 
 256 
 
 10 
 
 39 
 
 2 4 
 
 65 
 
 5 
 
 95 
 
 49. 
 
 298 
 
 11 
 
 41 
 
 2 5 
 
 66 
 
 5 4 
 
 98 
 
 64. 
 
 341 
 
 1 
 
 43 
 
 2 6 
 
 67 
 
 5 8 
 
 101 
 
 81- 
 
 384 
 
 1 1 
 
 44 
 
 2 7 
 
 69 
 
 6 
 
 104 
 
 100. 
 
 426 
 
 1 2 
 
 46 
 
 2 8 
 
 70 
 
 6 6 
 
 109 
 
 121. 
 
 469 
 
 1 3 
 
 48 
 
 2 9 
 
 71 
 
 7 
 
 113 
 
 144. 
 
 511 
 
 1 4 
 
 49 
 
 2.10 
 
 72 
 
 7 6 
 
 117 4 
 
 169. 
 
 554 
 
 1 5 
 
 51 
 
 2.11 
 
 73 
 
 8 
 
 121 
 
 196. 
 
 597 
 
 1 6 
 
 52 
 
 3 
 
 74 
 
 8 6 
 
 124 
 
 225. 
 
 639 
 
 TO FIND THE HORSES 7 POWER FOR VARIOUS WATER 
 WHEELS. 
 
 Rule, — Multiply the product of the tabular coefficient 
 opposite the given head, the area of the opening in the 
 gate in square inches, the entire head in feet (head and 
 fall in case of overshot and breast wheels), by the effi- 
 ciency of the class of wheel, pointing off six figures as 
 decimals. 
 
 Example 1. — The dimensions of a stream are two 
 inches by two hundred inches. What is its horses' 
 
HAND BOOK. 53 
 
 power, applied to a breast wheel affording a fall of ten 
 feet? 
 
 Calculation. 2 inches by 200 inches=400 square 
 inches opening. 
 
 Tabular coefficient opposite 2 feet 3 inches, .... 64 
 
 Efficiency of wheel, arising from impulse 40 
 
 Efficiency of wheel arising from weight, 78 
 
 Head, 2 feet 3 inches =2.25 
 
 2.25X40=90, product of efficiency and head. 
 
 lOX'8—780 product of efficiency and fall. 
 
 90+780=870 sum of products. 
 
 870X400X64=22.27 horses' power. Ans. 
 
 Example 2. — The dimensions of the stream are ten 
 inches square, the head twenty-five feet, what is its 
 horses' power applied to a good turbine ? 
 
 Calculation. 10X10=100 square inches opening. 
 
 Tabular coefficient opposite head of twenty-five 
 
 feet, =213 
 
 Efficiency of turbine =68. 
 '100X213X68X25=36.21 horses' power. Ans. 
 
 STEAM POWER. 
 Steam, as a force, acts by elastic pressure. The law 
 " that in compressing the same quantity of air, or of a 
 - perfect gas into smaller spaces, the volumes occupied by 
 it are inversely proportioned to the pressures," does not 
 hold good in relation to saturated steam. In the follow- 
 ing table, P denotes the total pressure in pounds per 
 square inch ; T the corresponding temperature, and V 
 
 3c 
 
54 
 
 QUARTZ OPERATOR S 
 
 the volume of the steam compared to the volume of the 
 water that has produced it. 
 
 TABLE OF PRESSURES, TEMPERATURES AND VOLUMES. 
 
 p- 
 
 T. 
 
 1 
 
 i02.r 
 
 5 
 
 162.3 
 
 10 
 
 193.3 
 
 14.7 
 
 212.0 
 
 15 
 
 213.1 
 
 20 
 
 228.0 
 
 25 
 
 240.1 
 
 30 
 
 250.4 
 
 35 
 
 259.3 
 
 40 
 
 267.3 
 
 45 
 
 274.4 
 
 50 
 
 281.0 
 
 V. 
 
 P. 
 
 T- 
 
 20582 
 
 60 
 
 292.7^ 
 
 3813 
 
 65 
 
 298.0 
 
 2358 
 
 70 
 
 302 9 
 
 1642 
 
 75 
 
 307.5 
 
 1610 
 
 80 
 
 312.0 
 
 1229 
 
 90 
 
 320.2 
 
 996 
 
 100 
 
 327 9 
 
 838 
 
 110 
 
 334.6 
 
 726 
 
 120 
 
 341.1 
 
 640 
 
 135 
 
 350.1 
 
 572 
 
 150 
 
 358.3 
 
 518 
 
 165 
 
 366.0 
 
 V. 
 
 P. 
 
 T. 
 
 437 
 
 180 
 
 372.9 
 
 405 
 
 190 
 
 377.5 
 
 378 
 
 200 
 
 381.7 
 
 353 
 
 210 
 
 386.0 
 
 333 
 
 220 
 
 389,9 
 
 298 
 
 230 
 
 393.8 
 
 270 
 
 240 
 
 397.5 
 
 247 
 
 250 
 
 401.1 
 
 227 
 
 260 
 
 404.5 
 
 203 
 
 270 
 
 407.9 
 
 184 
 
 280 
 
 411.2 
 
 169 
 
 300 
 
 417.5 
 
 V. 
 
 155 
 148 
 141 
 135 
 129 
 123 
 119 
 114 
 110 
 106 
 102 
 96 
 
HAND BOOK. 
 
 55 
 
 TABLE FOR ESTIMATING 
 STEAM FOR A GIVEN 
 
 THE MEAN PRESSURE OF 
 CUT-OFF OF STROKE. 
 
 TJNJACKETED 
 
 CYLINDER. 
 
 JACKETED CYLINDER. 
 
 cut-cff 
 
 coefficient, correction. 
 
 cut-off. 
 
 coefficient. 
 
 correction. 
 
 A 
 
 .177 
 
 12.098 
 
 A 
 
 .186 
 
 11.966 
 
 A 
 
 .244 
 
 11.113 
 
 A 
 
 .254 
 
 10.966 
 
 A 
 
 .303 
 
 10.246 
 
 A' 
 
 .314 
 
 10.084 
 
 i 
 
 .356 
 
 9.467 
 
 i 
 
 .370 
 
 9.261 
 
 A 
 
 .407 
 
 8.717 
 
 A 
 
 .417 
 
 8.570 
 
 i 
 
 .496 
 
 7.409 
 
 i 
 
 .505 
 
 7.297 
 
 i 
 
 .572 
 
 6.290 
 
 i 
 
 .582 
 
 6.145 
 
 A 
 
 .639 
 
 5.307 
 
 A 
 
 .648 
 
 5.174 
 
 A 
 
 .697 
 
 4.454 
 
 A 
 
 .707 
 
 4.307 
 
 t 
 
 .748 
 
 3.704 
 
 1 
 
 .756 
 
 3.587 
 
 A 
 
 .797 
 
 2.984 
 
 A 
 
 .800 
 
 2.940 
 
 T 
 
 .833 
 
 2.455 
 
 T 
 2 
 
 .840 
 
 2.352 
 
 H 
 
 .869 
 
 1.926 
 
 11 
 
 ¥0 
 
 .874 
 
 1.852 
 
 9 
 
 3" 
 
 .894 
 
 1.558 
 
 I 
 
 .900 
 
 1.470 
 
 It 
 
 .923 
 
 1.132 
 
 it 
 
 .929 
 
 1.044 
 
 A 
 
 .945 
 
 0.808 
 
 7 
 
 ITT 
 
 .945 
 
 0.808 
 
 1 
 
 .960 
 
 0.588 
 
 3 
 
 4 
 
 .960 
 
 0.588 
 
 t 
 
 .976 
 
 0.353 
 
 t 
 
 .976 
 
 0.353 
 
 ft 
 
 .986 
 
 0.206 
 
 H 
 
 .986 
 
 0.206 
 
 A 
 
 •997 
 
 0.044 
 
 A 
 
 .997 
 
 0.044 
 
 c4 
 
56 QUARTZ OPERATOR'S 
 
 By the table of pressures, temperature and volumes, it 
 will be seen that the volume of steam under a pressure 
 of thirty pounds to the square inch produced from a 
 cubic inch of ice-cold water is 838 cubic inches ; while 
 under a pressure of ninety pounds to the square inch, 
 the volume is 298 cubic inches. Thus the ratio of the 
 two pressures is as 30 to 90, or as 1 to 3, while the 
 inverse ratio of the respective volumes of steam is as 1 
 to 2.81. The mechanical effect deduced from the 
 above data is as follows: 838X30-^-12=2095, and 
 298X90—12=2235. Then 2235—2095=140 differ- 
 ence of mechanical effects, and 140-^-2095=,0668 ; 
 showing an advantage of nearly seven per cent, in 
 favor of using steam at the higher pressure. 
 
 By the table for estimating the mean pressure of 
 steam for a given cut-off of stroke, the coefficients for 
 one-fourth (J) cut-off are 572 in the unjacketed. cylin- 
 der, and 582 in the jacketed cylinder. 
 Then 582—572=10, 
 and 10--572=,0175; 
 showing an advantage of one and three-fourths of one 
 per cent, in favor of the jacketed cylinder for the given 
 cut-off of stroke. 
 
 The back pressure of steam in the cylinder of an 
 engine of ordinary structure is found, by experience, to 
 be about four pounds to the square inch above the 
 atmospheric pressure ; the velocity of piston being three 
 hundred (300) feet per minute. It is also found by ex- 
 
HAND BOOK. 57 
 
 perience that the excess of the back pressure above the 
 atmospheric pressure varies nearly as the square of the 
 velocity of the piston. 
 
 Thus, if the velocity of the piston be four hundred 
 (400) feet per minute, the back pressure will be 7.11 
 pounds. 
 
 Calculation. 300X300=90000; 400X400=160000; 
 4X160000^-90000=7.11 pounds. 
 
 TO FIND THE MEAN PRESSURE OF STEAM FOR A 
 GIVEN CUT-OFF OF STROKE. 
 Rule. — Multiply the excess of the pressure of steam 
 above the atmospheric pressure per square inch as it 
 enters the cylinder by the tabular coefficient opposite 
 the given cut-off, pointing off three figures as decimals, 
 and deduct from the product the tabular correction for 
 the same cut-off. 
 
 Example. — What is the mean pressure of steam 
 entering the cylinder at a pressure of ninety pounds to 
 the square inch, and cut-off at three- tenths stroke? 
 Calculation. — Tabular coefficient, unjacketed 
 
 cylinder, for, T \ stroke=,639 
 
 Correction for same, =5.307 
 
 Then, ,639X90=57.510 
 
 57.510—5.307=52.203 pounds. Ans. 
 And, jacketed cylinder, 
 
 ,648X90=58.320 
 58.320—5.174=53.146. 
 
 c5 
 
58 QUARTZ OPERATOR'S 
 
 TO FIND THE EFFECTIVE HOKSES ; POWER OF A NON- 
 CONDENSING STEAM ENGINE. 
 
 Rule. — Multiply four times the square of the diame- 
 ter of the piston in inches by the product of the number 
 of revolutions, length of stroke in feet, and the differ- 
 ence between the average forward pressure and the 
 back pressure of steam in pounds per square inch? 
 pointing off five figures as decimals. 
 
 Example. — What is the effective horses' power of an 
 engine, the diameter of the piston being sixteen inches) 
 the length of stroke three feet, the number of revolu- 
 tions fifty per minute, and the average forward pressure, 
 above the atmospheric pressure, seventy-five pounds per 
 square inch? 
 
 Gal. — The back pressure =4 pounds. 
 75—4=71 pounds. 
 
 Then, 
 
 16X16X4X50X3X71=109.05 horses' power. Ans. 
 
 Ex. 2. — What is the effective horses' power of an 
 engine, the diameter of the piston being twelve inches, 
 the length of stroke two feet, the pressure of steam, as 
 it enters the cylinder, sixty pounds in excess of atmos- 
 pheric pressure, cut-off at one-half stroke, and making 
 seventy-five revolutions per minute? 
 
 Gal. — Back pressure 4 pounds. 
 
 Tabular coefficient for J stroke=,833. 
 " correction for same =2.455. 
 
 833x60—2.455—4=43.525 effective pressure. 
 
HAND BOOK. 59 
 
 Then, 12X12X4X43.525X2X75=37.60 horses' 
 power. Ans. 
 
 MECHANICAL POWERS. 
 
 There are three classes of mechanical powers, viz : — 
 the lever, pulley and inclined plane. 
 
 The wheel and axle belong to the first class, and are 
 sometimes termed the perpetual lever. 
 
 The wedge and screw belong to the third class. 
 
 Weight signifies the resistance to be overcome, and 
 power the force which overcomes or tends to overcome 
 the resistance. 
 
 THE LEVER. 
 
 The arms of a lever are the portions of it which are 
 intercepted between the power and fulcrum, and be- 
 tween the weight and fulcrum. 
 
 There are three kinds of levers, in which : — 
 
 1. The fulcrum is between the power and weight. 
 
 2. The weight is between the power and fulcrum. 
 
 3. The power is between the fulcrum and weight. 
 Assuming the lever itself to have no weight, and no 
 
 friction, the condition of equilibrium or balance, is as 
 follows : 
 
 The product of the power and the length of arm to 
 which it is applied, is equal to the product of the weight 
 and the length of arm to which it is applied. 
 
 Ex. 1. If one arm of a lever be ten feet, and the 
 
 c6 
 
60 QUARTZ OPERATOR'S 
 
 other two feet, what power must be applied to the 
 longer arm to balance a weight of 1000 pounds? 
 
 Gal. 1000X2-^-10=200 pounds. Ans. 
 
 Ex, 2. If the radius of the axle, of " the wheel and 
 axle," be six inches, and the radius of the wheel forty- 
 eight inches, what power must be applied to the circum- 
 ference of the wheel to balance 2400 pounds at the 
 circumference of the axle? 
 
 Gal. 2400 X 6^48=300 pounds. Ans. 
 THE PULLEY. 
 
 Assuming the pulley (as commonly arranged) to be 
 without weight and free from friction and stiffness of 
 cordage, the condition of equilibrium or balance is, that 
 the weight is equal to the product of the power and the 
 number of cords at the movable block. 
 
 Ex. — What weight will be balanced by a power of 
 one hundred pounds, there being three movable pulleys, 
 or, in other words, six cords at the movable block? 
 
 Gal. 100X6=600 pounds. Ans. 
 THE INCLINED PLANE. 
 
 Assuming that there is no friction, and that the direc- 
 tion of the power applied is parallel to the plane, the 
 conditions of equilibrium of a body sustained by any 
 force on an inclined plane, is as follows : 
 
 The product of the power and the length of the plane 
 is equal to the product of the weight and the height of 
 the plane. 
 
 Ex. — What power would be necessary to sustain a 
 
HAND BOOK. 61 
 
 rolling weight of 1,200 pounds upon an inclined plane 
 of 10 feet length and 6 feet perpendicular height? 
 
 Gal— 1200X6-^-10=720 pounds. Ans. 
 
 If the power acts parallel to the base of the plane, 
 then the product of the power and the length of the 
 base is equal to the product of the weight and the height 
 of the plane. 
 
 Ex. — What power would be necessary to sustain a 
 rolling weight of 1200 pounds upon an inclined plane 
 whose base is 8 feet and height 6 feet ? 
 
 Gal 1200X6^-8=900 pounds. Ans. 
 
 Ex. 3d. — Omitting the consideration of friction, what 
 power applied to the back of a wedge of the form of 
 either a single or double inclined plane, and in the di- 
 rection of the base of the inclined plane, will raise a 
 weight of 2400 pounds, the back of the wedge being 3 
 inches thick, and the base being 48 inches long? 
 
 Gal 2400X3-^-48=150 pounds. Ans. 
 
 Ex. 4. — Omitting the consideration of friction, if the 
 threads of a screw be 2 inches apart, and a power of 
 500 pounds be exerted at the end of a lever 84 inches 
 long, what weight or force will be produced at the end 
 of the screw ? 
 
 Gal. 84X2X- 2 T 2 -=528 inches, base of inclined plane. 
 
 Distance of threads apart=2 inches, height of in- 
 clined plane, 528X500-^2=132000 pounds. Ans. 
 
 THIN CYLINDERS. 
 To determine the thickness of a thin hollow cylinder ; 
 
62 QUARTZ OPERATOR'S 
 
 the internal radius, pressure, and the tenacity of the 
 material being given. 
 
 Rule. Multiply the internal radius in inches by the 
 fluid pressure in pounds, and divide the product by the 
 tenacity per square inch of the material. 
 
 Ex. — The internal radius of a cylinder being 30 
 inches, the fluid pressure 250 pounds to the square inch, 
 and the tenacity of the material of the cylinder twelve 
 thousand pounds per square inch, what is the thickness 
 of the cylinder ? 
 
 Cal. 250X30=7500. 
 
 7500^12000=| inches. Ans. 
 To determine the fluid pressure, the internal radius, 
 
 thickness of cylinder, and tenacity of material being 
 
 given. 
 
 Rule. — Divide the product of the thickness of the 
 cylinder and tenacity of the material, by the internal 
 radius. 
 
 Ex. — The thickness of the cylinder being one-fourth 
 of an inch, the tenacity eighteen thousand pounds., and 
 the radius six inches, what fluid pressure will the cylin- 
 der withstand per square inch ? 
 
 Gal. 18000X^6=750 pounds. Ans. 
 THICK HOLLOW CYLINDERS. 
 
 To determine the thickness of thick, hollow cylinders, 
 the internal radius, the fluid pressure and the tenacity 
 of the material of the cylinder being given. 
 
 Rule. — Subtract one from the square root of the quo- 
 
HAND BOOK. 63 
 
 tient of the sum and difference of the tenacity per 
 square inch of the material of the cylinder, and the 
 fluid pressure per square inch, and multiply this differ- 
 ence by the internal radius. 
 
 Ex. 1. — The internal radius of a thick, hollow cylin- 
 der being nine inches, the tenacity of the material of 
 the cylinder ten thousand pounds per square inch, what 
 is the requisite thickness of the cylinder ? 
 
 Cal. Sum of tenacity and 
 
 fluid pressure, 10000-^8000=18000 
 
 Difference of tenacity and 
 
 fluid pressure, 10000—8000=2000 
 
 Quotient of sum and difference, 18000-^2000=9 
 
 Square root of quotients, -j/9— 3 
 
 Difference between root and one, 3 — 1=2 
 
 Product of radius and this differ- 
 ence 9X2=18 inches. Ans. 
 
 To determine the fluid pressure per square inch, 
 which a thick, hollow cylinder will withstand, the inter- 
 nal and external radii, and the tenacity of the material 
 of the cylinder being given. 
 
 Rule, — Divide the difference of the squares of the 
 radii, and multiply the quotient by the tenacity per 
 square inch of the material of the cylinder. 
 
 Ex, 2. — The internal and external radii of a thick, 
 hollow cylinder being respectively nine inches and 
 twenty-seven inches, and the tenacity per square inch 
 of the material of the cylinder ten thousand pounds, 
 
64 QUARTZ OPERATOR'S 
 
 what fluid pressure per square inch will it withstand ? 
 Gal. Square of external radius, 27X27=729 
 
 Square of internal radius, 9X9=81 
 
 Difference of squares, 729 — 81=648 
 
 Sum of squares, 729 + 81=810 
 
 Then 10000X648-^810=8000 pounds. Ans. 
 
 SUSPENSION RODS OF UNIFORM STRENGTH. 
 
 To determine the transverse section at any point of a 
 suspension rod of uniform strength : 
 
 Rule 1. — Divide the constant weight to be raised by 
 the uniform tension per square inch, due the tenacity of 
 the material in the rod, and multiply the quotient by 
 2,71828, raised to a power equal to the product of the 
 length of the rod in inches, and the weight of one cubic 
 inch of the rod, divided by the intensity of the tension 
 per square inch. 
 
 Ex. — The weight to be raised being 27000 pound?, 
 the intensity per square inch of the working tension 
 3,000, the weight of rod per cubic inch j\ of a pound, 
 and the length of rod 600 feet. Required the trans- 
 verse section at the upper end of rod ? 
 
 GaL — Weight divided by intensity of tension, 
 
 27000^-3000=9 
 Product of length in inches and 
 
 weight of cubic inch, 7200X A=2000 
 
 This product divided by intensity 
 
 of tension, 2000—3000— =§ 
 
HAND BOOK. 65 
 
 Product of 434295 and last quotient, ,434295 X§= 
 
 ,289530. 
 
 Number corresponding to loga- 
 rithm, 289530=1,94887 
 
 And 1,94887X9=17,5983 inches. Ans. 
 
 TO DETERMINE THE WEIGHT OF THE ROD THE SAME 
 DATA BEING GIVEN AS IN EXAMPLE UNDER RULE 
 ONE. 
 
 Rule 2. — Multiply the transverse section by the in- 
 tensity of the tension, and subtract from the product the 
 constant weight to be raised. 
 
 Ex. — The transverse section being 17,5983 square 
 inches, as determined by solution of example one, the 
 intensity of tension being 3,000 pounds per square inch, 
 and the constant weight to be raised 27,000 pounds, 
 what is the weight of the rod ? 
 
 Cal 17,5983X3000=52794.9 
 
 52794.9—27000.0=25794.9 pounds. Ans. 
 
 WATER PIPES. 
 To determine the velocity of water per second, flow- 
 ing through long pipes, the head or hight of reservoir 
 above the point of delivery, the length and diameter of 
 the pipe being given. 
 
 Rule. — Multiply the product of the head and diame- 
 ter of the pipe in feet by twenty-three hundred. Divide 
 this product by the sum of once the length and fifty-two 
 
66 QUARTZ OPERATOR'S 
 
 times the diameter of the pipe, and extract the square 
 root of the quotient. 
 
 Ex. — The head is six hundred feet ; the diameter of 
 pipe nine inches ; the length of pipe six thousand feet ; 
 what is the velocity of the water per minute ? 
 
 Cal. — Diameter of pipe =9 inches=,75 feet. 
 
 Product of head, diameter, etc., 600X2300X>75= 
 1035000,00. 
 
 Sum of length and product, 6000+52X,75==6039 
 
 Quotient, 1035000-^6039=171,386. 
 
 Square root ^/i7i,386=13,09 feet, velocity per sec. 
 
 Per minute, 13,09 X 60=785,4 feet. Ans. 
 
 Plane Circular Plates. — To determine the grinding 
 effects by one revolution of a plane circular plate of the 
 usual ring form, of uniform hardness, about its axis per- 
 pendicular to the grinding plane — the greater and less 
 diameters and pressure per square inch being given. 
 
 Rule. — Multiply the difference of the fourth powers 
 of the radii by the product of the pressure per square 
 inch, and the square of the ratio between the radius 
 and circumference of a circle, and divide the product by 
 the greater radius. Deduced from formulas 4 and 5 
 Discussion of Tractory, etc. 6 
 
 Ex. — The greater diameter of a plane circular plate 
 (muller) of the usual ring form, and of uniform hard- 
 ness, being forty inches; the less diameter sixteen inches, 
 and the pressure per square inch five pounds, what is 
 the grinding effect by one revolution ? 
 
HAND BOOK. 67 
 
 Cal. 48-v-2=24 greater radius. 
 
 16-=-2=8 less radius. 
 24X24X24X24X=331776 fourth power of greater 
 radius. 
 
 8X8X8X8^=4096 fourth power of less radius. 
 331776—4096=327680 difference of fourth powers. 
 
 22 22 
 
 327680XyXyX5--24=674307.47.* Ans. 
 
 Conical Plates. — To determine the grinding effect by 
 one revolution of a conical plate (muller) of the usual 
 ring form, of uniform hardness, about its axis perpen- 
 dicular to the plane of the base, the greater and less 
 diameters of the frustum, the height of the cone and the 
 pressure per square inch, parallel with the axis of revo- 
 lution being given. 
 
 Ride. — Multiply the square root of the sum of the 
 squares of the greater radius, and height of the cone by 
 the difference of the fourth powers of the radii, and this 
 product by the product of the pressure per square inch, 
 and the square of the ratio between the diameter and 
 circumference of a circle, divided by the square of the 
 greater radius. Deduced from formulas 9 and 10. 
 
 7 7 
 
 Ex. — The greater diameter of a conical plate (muller) 
 of the usual ring or frustum form being forty-eight 
 
 Remark. — The unit of grinding effect here taken is the result 
 of a body impressing one square inch of another body, with on 
 pound pressure, and moving, under these circumstances, on 
 inch. See Laws of Grinding, Discussion of Tractory, etc. 
 
68 QUARTZ OPERATOR'S 
 
 inches, the less diameter sixteen inches, the height of the 
 cone twelve inches, the pressure per square inch, parallel 
 with the axis of revolution, five pounds, what is the 
 grinding effect by one revolution ? 
 
 Cat 48-f-2=24 greater radius. 
 
 16-^2=8 less radius. 
 
 24X24=576 square of greater radius. 
 
 12X12=144 square of hight of cone. 
 
 j/ 576 + 144=26.8328 square root of sum.* 
 
 24X24X24X24=331776 fourth power of greater ra- 
 dius. 
 
 8X8X8X8=4096 fourth power of less radius. 
 
 331776—4096=327680 difference of fourth powers. 
 
 327680X26.8328X 2 y 2 X 2 y 2 X5--576=753875.77. 
 
 Ans. 
 
 Tractory Conoidal Plates. — To determine the grind- 
 ing effect by one revolution of a tractory conoidal plate 
 (muller) of frustum form, of uniform hardness, about its 
 axis perpendicular to the plane of its base, the greater 
 and less diameters of the frustum, and the pressure 
 per square inch parallel with the axis of revolution be- 
 ing given. 
 
 Rule. — Multiply the difference of the squares of the 
 radii of the frustum by twice the product of the greater 
 radius, pressure per square inch, and the square of the 
 ratio between the diameter and circumference of a 
 circle. Deduced from formula 5. 
 
 4 
 
 Ex. — The greater diameter of a tractory conoidal 
 
HAND BOOK. 69 
 
 plate (mullerj being forty-eight inches, the less diame- 
 ters sixteen inches, and the pressure, per square inch ? 
 five pounds ; what is the grinding effect by one revo- 
 lution. 
 
 Cat. 48-4-2=24 greater radius. 
 
 16-^-2=8 less radius. 
 
 24X24=576 square of greater radius. 
 
 8X8=64 square of less radius. 
 
 576 — 64=512 difference of squares. 
 
 5l2X24X5X 2 7 2 X 2 T 2 X2=1213753.47. Ans. 
 
 Remark. — Since, in this style of plate, the wear is 
 perfectly uniform, the grinding effect will be the same, 
 whether the central opening be large or small, providing 
 the weight of the muller be the same in each case. 
 
 MENS UEATION. 
 
 Prop. 1. — The square root of the sum of the squares 
 of the base, and perpendicular of a right-angled triangle 
 is equal to the hypothenuse. 
 
 Ex. — The base of a right-angled triangle is eight feet, 
 the perpendicular fifteen feet, what is the hypothenuse ? 
 
 Gal. 8X8=64 15X15=225 
 
 64+225=289 i/att±±l 7 feet. Ans. 
 
 Prop. 2. — The square root of the difference of the 
 squares of the hypothenuse, and one of the sides of a 
 right angled triangle is equal to the other side. 
 
70 QUARTZ OPERATOR'S 
 
 Ex. 1. — The hypothenuse is thirteen feet, the perpen- 
 dicular twelve feet ; what is the base ? 
 
 Cal 13X13=169 12X12=144 
 
 169—144=25 j/?7=5feet. Ans. 
 
 Ex. 2. — The hypothenuse is thirty-seven feet, the 
 base twelve feet ; what is the perpendicular ? 
 
 Cal. 37X37=1369 12X12=144 
 
 1369—144=1225 1 /ITiT=35 feet. Ans. 
 
 Prop. 3. — The circumference of a circle is equal to 
 twenty-two times the diameter, divided by seven ? 
 (nearly). 
 
 Remark. — The true circumference of a circle lies 
 between 3|J and 3}f times the diameter, which is 
 nearly 3.1416. This ratio is usually represented in 
 formulas by the character n. 
 
 Ex. — The diameter of a circle being ten feet, what is 
 the circumference ? 
 
 Cal 10X22--7=31.428, or 10X3.1416=31.416. 
 
 Ans. 
 
 Prop. 4. — The length of an arc of a circle containing 
 any number of degrees is equal to the product of the 
 number of degrees in the arc, diameter of the circle and 
 3.1416 divided by three hundred and sixty. 
 
 Ex. — What is the length of an arc containing seventy- 
 five degrees, the diameter of whose circle is twenty-one. 
 
 Gal. 75X21X3.1416--360=13.74feet. Ans. 
 • Prop. 5 — The length of an arc of a circle is equal to 
 
HAND BOOK. 71 
 
 one-third of the difference between eight times the chord 
 of half the arc, and once the chord of the arc. 
 
 Ex. — Required the length of an arc whose chord is 
 ten feet, and the chord of one-half the arc is 5.177 feet ? 
 
 Cal. 5.177X8—10=31.416. 31.416-5-3=10.472 
 feet. Ans. 
 
 Prop. 6 — The diameter of a circle is equal to the sum 
 of the hight of any segment of the circle, and one third 
 the square of half the base of that segment. 
 
 Ex. — The base of a segment of a wheel is fifty -four 
 inches ; the hight of the segment three inches ; what is 
 the diameter of the wheel. 
 
 Cal. 54--2=27 27X27-5-3=243 
 243 + 3=246 inches. Ans. 
 
 Prop. 7. — The length of the common cycloid is equal 
 to four times the diameter of the generating circle. 
 
 Ex. — What is the length of a common cycloid, the 
 length of whose generating circle is five feet ? 
 
 Cal. 5X4=20 feet. Ans. 
 
 Prop. 8. — The hight of a common parabola is equal to 
 the square of half the base, or the square of the ordinate 
 divided by twice the parameter. 
 
 Ex. — The base of a common parabola is eight feet ; 
 the perameter four thirds of a foot ; w T hat is its hight ? 
 
 Cal 8-5-2=4 4X4=16 
 
 4X2-^-3=8-r-3 16X3-5-8=6 feet. Ans. 
 
72 QUARTZ OPERATOR'S 
 
 Prop. 9. — The length of an arc of the common para- 
 bola, measured from its vertex, is as follows : 
 
 Length of arc. =y |/ p2 + y2 +p_log ( y + i/p2+y2 -) 
 ~2p "2 , ( p 
 
 Ex. — The half base or coordinate (y) being twelve 
 feet, and the half parameter (p) being five feet, what is 
 the length of the arc of the parabola, measured from the 
 vertex ? 
 
 Cat. 12X12=144 5X5=25 
 144+25=169 j/I^IS 
 13X12-5-10=15.6 13 + 12=25 
 25-1-5=5 Naperian log, 5-.1.61 
 1,61X5-^-2=4.02 15.6 + 4.02= 
 Prop. 10. — An ordinate of an ellipse is equal to the 
 product of the minor axis, and the square root of the 
 difference of the squares of the major axis, and the co- 
 ordinate divided by the major axis. 
 
 Ex. — The axes of an ellipse are seven and ten feet 
 An ordinate is six feet ; what is the coordinate. ? 
 Cal. 10X10=100 6X6=36 
 100—36=64 !/64=8 
 8X7-^-10=5.6 feet. Ans. 
 Prop. 11. — The circumference of an ellipse is as 
 follows : 
 
 2 4 6 
 
 Circumference =2 11 A (1 — f — }f~ jftfgSr— etc )- 
 
 Remark. — In the above formula, A denotes the semi- 
 major axis ; E the eccentricity ; that is, the distance 
 
HAND BOOK. 73 
 
 between the centre and one of the foci of the ellipse, 
 
 divided by the semi-major axis. 
 
 2 4 
 
 Ex. E-h4=,01. EX3-^64=,00G075. 
 
 EX45-t-2304=,000001. 
 
 .01 +000075 + ,000001= 010076 
 1_.010076=,989924 
 
 .989924X2X20—2X3.1416=62.1989 feet. Ans. 
 
 Prop. 12. — An ordinate of an hyperbola is equal to 
 the product of the conjugate axis, and the square root 
 of the difference of the squares of the co-ordinate and 
 transverse axis, divided by the transverse axis. 
 
 Ex. — The transverse axis of an hyperbola is four 
 feet ; the conjugate axis three feet, and an ordinate on 
 the transverse axis and hight of the hyperbola ten 
 feet ; what is the co-ordinate ? 
 
 Cat. 10X10=100 4X4=16 100—16=84 
 
 y'(84)=9.165 9.165X3^-4=6.874 feet. Ans. 
 
 Prop 13. — The length of an hyperbola is as follows, 
 to wit : 
 
 =iiA[l-i(2-E 2 )-A(2-EV-rffe(2-E 2 )* ] 
 Remark. — In the above formula, A represents the 
 
 semi-transverse axis ; E the eccentricity ; that is, the 
 
 distance of the center from one of the foci divided by 
 
 the semi-transverse axis. 
 
 Ex. — The difference of the distances of a point in the 
 
 curve, from the foci, being twenty feet, and the distance 
 
 D 
 
74 QUARTZ OPERATOR'S 
 
 from the center from one of the foci, fifteen feet, what is 
 the length of the hyperbola ? 
 
 Col. 20-r2=10. 15-^10=E. Eccentricity. 
 — (2— E )-Ht = + ,1875 ; 3(2-E 2 ) 2 --64=-,0264 
 
 2 3 
 
 —45 (2-E ) -^2304= + .0082 1.+.1875+.0082 
 —,0264=1.1693 
 
 1.1693X10X3.1416=36.7347 feet. Ans. 
 
 Prop. 14. — The length of the spiral of Archimedes is 
 as follows : 
 
 Length.^ <J t V (1+t )+log[i/(l+/)+<] )■ 
 
 JRemarh. — In the above formula, (t) represents the 
 measuring arc, and (A) the relation between the radius 
 vector and measuring arc. 
 
 Ex. — The measuring arc being the entire circle, and 
 the radius vector unity, what is the length of the spiral 
 or spire ? 
 
 Cal. 15 = 112=6,2832. a = -^ 1 - l = ^^ 
 £l/l+4X(3,1416) 2 ==3,1811 
 
 , log [i/l+4X {3.1416)2 + 2X3.1416]~2019 
 
 4X3,1416 
 
 3.181 1 -+-,2019=3.383. Ans. 
 
 Ex. 2. — The measuring arc being twice the circumfer- 
 ence of a circle whose radius is unity, what is the length 
 of the spiral, and what is the length of the second spire ? 
 
HAND BOOK. 75 
 
 Cal. t =4n ; flt=r=jjj a constant quantity. 
 
 V / l + l6X(3.l4i6)2^-12.6060 
 4X§^4i6 lo g Cv / i + i6X(3.i4i6)2 + 4X3.1416]= ? 2567 
 
 12,6060+, 2567=12,8627 length of spiral. Ans. 
 
 The first spire, as found, for example one, is 3.383. 
 
 Hence, 12.8627—3.383=9,4797 length of second 
 spire. Ans. 
 
 Prop. 15. — The area of a square, a rectangle, or a 
 parallelogram, is equal to the product of its base and 
 altitude. 
 
 Ex. — What is the area of a fljor forty-five feet by 
 one hundred and twenty-five feet ? 
 
 Cal. 125X45=5625 square feet. Ans. 
 
 Prop. 16. — The area of a triangle is equal to half 
 the product of the base and perpendicular. 
 
 i&r.— What is the area of a triangle whose base is 
 twelve feet, and altitude eighteen feet ? 
 
 Cal. 12X18-^2=108 square feet. Ans. 
 
 Prop. 17. — The area of a trapezoid is equal to half 
 the product of the sum of the two parallel sides and the 
 altitude. 
 
 Ex. — The parallel sides being thirteen feet and sev- 
 enteen feet, and their distance apart seven feet, what is 
 the area of the trapezoid ? 
 
 Cal. 134-17=30. 30X7^-2=105 square feet. 
 
 Ans. 
 
 Prop. 18. — The area of an irregular polygon is equal 
 
 d2 
 
76 
 
 QUARTZ OPERATOR'S 
 
 to the sum of the areas of the separate triangles com- 
 posing it. 
 
 TABLE. 
 
 Triangle, 3 ,43301 27 
 
 Square 4 1,0000000 
 
 Pentagon, 5 1 ,7204774 
 
 Hexagon, 6 2,5980762 
 
 Heptagon, . . .7 3,6339124 
 
 NAMES. SIDES. 
 
 AREAS. 
 
 Octagon 8 4,8284271 
 
 Nonagon 9 6,1818242 
 
 Decagon,.. 10 7,6942088 
 
 Undecagon .11 9,3656399 
 
 Dodecagon.. 12 11,1961524 
 
 Prop. 19. — The area of a regular polygon is equal to 
 the square of one of its sides multiplied by the area of a 
 polygon of the same number of sides, and whose side 
 are unity. 
 
 Ex. — What is the area of a nonagon whose side is 
 ten feet ? 
 
 pal 10X10=100. Tabular area =6,1818242 
 
 6,1818242X100=618,18242 square feet. Ans. 
 
 Prop. 20. — The area of a circle is equal to one fourth 
 the product of the diameter and circumference ; also 
 equal to the product of the square of the diameter and 
 ,7854 ; and also equal to the square of the diameter and 
 eleven divided by fourteen. 
 
 Ex. — What is the area of a circle whose diameter is 
 forty feet ? 
 
 Gal. 40X40X11-5-14=1257,14 square feet. Ans. 
 
 Prop. 21. — The area of a sector of a circle is equal 
 to half the arc of the sector multiplied by half the di- 
 ameter of the circle. 
 
HAND BOOK. 77 
 
 Ex, — What is the area of a sector whose arc is forty- 
 degrees, and the diameter of the circle thirty feet. 
 
 Cal. 3.1416X40--360X30=10.472 
 
 10.472-=-2X 15=78.54 square feet. Ans. 
 
 Prop, 22. — The area of a circuWring is equal to the 
 difference of the squares of the diameters multiplied by 
 ,7854. 
 
 Ex. — What is the area of a circular ring whose great- 
 er diameter is forty-two feet, and less diameter seven 
 feet? 
 
 Cal 42X42=1764. 7X7=49. 
 
 1764—49=1715. 171oX,7854=1346,96 square 
 feet. Ans. 
 
 Prop. 23. — The area of the common cycloid is equal 
 to three times the area of the generating circle. 
 
 Ex. — What is the area of a common cycloid, the di- 
 ameter of whose generating circle is seven feet ? 
 
 Cal. 7X7X11-M 4X3=115.5 square feet. Ans. 
 
 Prop. 24. — The area of the common parabola is equal 
 to two-thirds the product of the base and altitude. 
 
 Ex. — The base of a common parabola is twelve feet, 
 and the hight fifteen feet, what is the area ? 
 
 Cal. 12X15X2^-3= 120 square feet. Ans. 
 
 Prop. 25. — The area of an ellipse is equal to the pro- 
 duct of the semi-diameters multiplied by 3.1416. 
 
 Ex. — What is the area of an ellipse, the semi-diame- 
 ters being ten feet and eight feet ? 
 
 Cal. 10X8X3.1416=251.328 square {eet. Ans. 
 
 b3 
 
78 QUARTZ OPERATOR'S 
 
 Prop, 26. — The area of an hyperbola is as follows : 
 
 Area =ay—AXB log (^•+-g) 
 
 Remark. — In this formula, A represents the semi- 
 transverse axis, B the conjugate axis, x and y the gen- 
 eral co-ordinates. 
 
 Ex. — Given the base, 13,748 feet, the hight six feet, 
 transverse axis eight feet, and the conjugate axis six 
 feet, what is the area of the hyperbola ? 
 Cal. A=8-i-2=4 semi-transverse axis. 
 B=6-i-2=3 semi-conjugate axis. 
 #=6 + 4—10 co-ordinate. 
 #=13.748-^-2=6.874 co-ordinate. 
 Then area=68.74— 12 log 4.791=49.88 square feet. 
 
 Ans. 
 Prop. 27. — The area of the equable spiral or spiral 
 of Archimedes, is equal to one-third the difference be- 
 tween the cube of the number of revolutions and the 
 cube of a number one less than that of the revolutions, 
 multiplied by 3.1416. 
 
 Ex. — What is the area of an equable spiral, whose 
 radius vector is seven feet, described by seven revolu- 
 tions ? 
 
 Cal. 7X7X7=343 cube of revolutions =n 3 
 
 6X6X6=216 cube of number one less=(w-l) 3 
 Then 343=216=127 difference =n 3 — (n-1) 3 
 
 127X3,14.16-^-3=132.99 square feet. Ans. 
 Prop. 28. — The area of an irregular plane surface is 
 equal to one- third the distance between any two con- 
 
HAND BOOK. 79 
 
 secutive ordinates multiplied by the sum of the extreme 
 ordinates, increased by four times that of the even or- 
 dinates, and twice that of the uneven ordinates. 
 
 Remark. — The entire number of ordinates or perpen- 
 diculars is to be uneven, and at equal distances apart. 
 Fig.!. Ex. Having given 
 
 - i as in figure 1, the base 
 
 P AB, equal to o40 feet> 
 and having found the 
 perpendiculars AD = 
 50, a 2 e 3 = 55, a s e 3 = 80, a 4 e 4 = 100, a 5 e 5 = 120, 
 a G e G = 110, a 7 e 7 = 115, a 8 e 8 = 130, a 9 e 9 = 128, 
 «io*io = 90 > 011*11= HO, a u e u = 108, a iz e u = 75 
 feet, what is the area of ABCD ? 
 
 Cal. 540-^-12=45 distance apart of perpendiculars. 
 
 45-^3=15 one-third distance between any two per- 
 pendiculars. 
 
 50+75=125 sum of extremes. 
 
 55+100+110+130+90+108=593 sum of even 
 ordinates or perpendiculars. 
 
 80+120+115+128+110=553 sum of uneven or- 
 dinates or perpendiculars. 
 
 593X4=2372; 553X2=1106. 
 
 125+2372+1106=3603. 
 
 3603xi5=Area of ABCD,=54045 square feet. 
 Ans. 
 
 Prop. 29. — The area or convex surface of a right 
 
 d4 
 
80 QUARTZ OPERATOR'S 
 
 prism, or of a cylinder, is equal to the primeter of the 
 base multiplied by the altitude. 
 
 Ex. — What is the convex surface of a cylinder whose 
 diameter is seven feet and length ten feet ? 
 
 Col. 7X22-^7=22; 22X10=220 square feet. 
 Ans. 
 
 Prop. 30. — The area or convex surface of a right 
 pyramid or cone is equal to the perimeter of the base 
 multiplied by one half of the slant height. 
 
 Ex. — What is the area or convex surface of a cone 
 whose slant height is twenty feet, and the diameter of 
 whose base is seven feet ? 
 
 Gal. 7 X 22—7=22 ; 22 X 20-r-2=220 square feet. 
 Ans. 
 
 Prop. 31. — The area or convex surface of a spheri- 
 cal zone is equal to the altitude of the zone multiplied 
 by the circumference of a great circle of the sphere. 
 
 Ex. — The height of a zone is eight feet, the diameter 
 of the sphere twenty-five feet. What is the area of the 
 zone ? 
 
 Cat. 8X25X3.1416=628,32 square feet. Ans. 
 
 Prop. 32. — The area of a sphere is equal to the pro- 
 duct of the diameter and circumference. 
 
 Ex. — What is the area of a ball ten inches in diam- 
 eter ? 
 
 Cat. 10X3.1416=31.416; 31.416X10=314.16 
 square feet. Ans. 
 
HAND BOOK. 81 
 
 Prop. 33. — The area or surface described by revolv- 
 ing a cycloid about its base is sixty-four thirds of the 
 generating circle. 
 
 Ex. — What is the area or surface described by the 
 revolution of a cycloid about its base, the diameter of 
 the generating circle being seven feet ? 
 
 Cat. 7X22-^4=38.5; 38.5x64-^3=821.33 square 
 feet. Ans. 
 
 Prop. 34. — The area or convex surface of a parabo- 
 loid is as follows : 
 
 Area=i^jA+^Y--il} 
 3 IV 4A 2 y 8A 2 ) 
 
 Remark. — In this formula b represents one-half the 
 diameter of the base, h the height of the paraboloid, 
 and ii=3.1416. 
 
 Ex. — The diameter of the base of a paraboloid being 
 six feet, and the height two feet, what is the surface of 
 revolution ? 
 
 Cat. 4X6-f-2x2X3.1416-f-3=25.1328. 
 
 I v 3. 
 
 (-4) 
 
 125 27 27 125 27 98 
 
 64 8X8 64 64 64 64 
 25.1328 X98-v-64=38.48 square feet. Ans. 
 Prop. 35. — The area of an ellipsoid described by re- 
 volving an ellipse about its major axis, is as follows : 
 
 2 4 6 
 
 Arearz4nAB ( 1 -±-±—±— &C.) 
 
 \ 6 40 113 / 
 
 »5 
 
82 QUARTZ OPERATOR'S 
 
 Ex. — The major axis being twenty feet, and the 
 eccentricity four-tenths of a foot, what is the area of 
 the ellipsoid? 
 
 Gal. B^zj/(100 — lG)n9.1G5 semi-conjugate axis, 
 4X3.1416X10X9.16SX(1— .0167— .0064— &c.)= 
 1113.59 square feet. Ans. 
 
 Prop. 36. — The convex surface of a hyperboloid is 
 as follows : 
 
 Ayw=— J x j/( x* — a 2 ) — A !/( A 2 — a 2 ) 
 +aHog [ A+ 1 /(A 2 - a_ 2 ) | 
 
 Remark. — In the above formula, (A) represents the 
 transverse axis} (B) the conjugate axis; (x) an ordi- 
 nate on the axis of revolution; and, (a) equal to the 
 square root of A 4 
 
 A 9 +B a " 
 
 Ex. — The transverse axis (A) is four feet, the conju- 
 gate axis (B) three feet, and the hight six feet ; what is 
 the area of the hyperboloid ? 
 
 Gal. 0=4+6=10. a=J£. a 2 =W. 
 15X3.1416 (94.7 4 2.-9.6— 11.397)=217.31 square 
 
 feet. Ans. 
 
 Prop. 37. — The solid contents of a prism, or of a 
 
HAND BOOK. 83 
 
 cylinder, are equal to the area of the base multiplied by 
 the altitude. 
 
 Ex. — The diameter of a cylinder is twenty-one inches 
 and its length forty inches ; what are the solid contents ? 
 
 Gal. 21--7X22=66; 6 6X21 --4:= 346.5 
 346.5X40.-^13860 solid inches. Ans. 
 
 Prop. 38. — The solid contents of a pyramid, or of a 
 cone, are equal to the base multiplied by one-third of 
 the altitude. 
 
 ]$x. — The square inches in the base of a cone are 
 346.5, and the hight forty inches; what are the solid 
 contents ? 
 
 Gal 346.5X40-^-3=^4620 solid inches. Ans. 
 
 Prop. 39. — The solid contents of a frustum of a pyra- 
 mid, or a cone, are equal to one-third of the altitude of 
 the frustum, multiplied by the sum of the two bases, in- 
 creased by a mean proportional between them. 
 
 J£x. — What is the solidity of the frustum of a cone 
 whose lower diameter is ten inches, upper diameter 
 eight inches, and hight twenty-four inches ? 
 
 Gal. 10X10=100 8X8=64 10X8=80 
 100 + 64+80=244; 244X,?584X24--3=1 533.10 solid 
 inches. Ans. 
 
 Prop. 40. — The solidity of a sphere is equal to one- 
 third of the area of a great circle, multiplied by the 
 radius ; or, it is equal to the cube of the diameter, mul- 
 tiplied by ,5236. 
 
 d6 
 
84 QUARTZ OPERATOR'S 
 
 Ex. — What is the solidity of a ball fourteen inches 
 in diameter? 
 
 Gal. 14x22-^-7=44; 14X44X14-^-6=1437.33 
 cubic inches. Ans. 
 
 Prop. 41. — The solidity of a spherical segment is 
 equal to one-half the hight of the segment multiplied 
 by the sum of the areas of the two basis ; and this pro- 
 duct increased by the solid contents of a sphere whose 
 diameter is equal to the hight of the segment. 
 
 Remark. — If the segment has but one base, the other 
 is to be regarded equal to (zero). 
 
 Ex. — What is the solidity of a spherical segment, the 
 diameter of the sphere being forty inches, and the dis- 
 tances from the center to the bases sixteen inches, and 
 ten inches ? 
 
 Cal. 20X20=400; 16X16=256; 10X10=100; 
 400—256=144; 400—100=300; 144+300=444; 
 444X4=1776; 1776X,7854=1394.8704; 1394,8704 
 X3=4184.6112 ; 6X6X6X,5236=113.0976 ; 4184 
 ,6112+113.0976=4297.7088 solid inches. Ans, 
 
 Prop. 42. — The solidity of a regular polyedron is 
 equal to the cube of one of its edges multiplied by the 
 solidity of a similar polyedron whose edge is one. 
 
HAND BOOK. 85 
 
 Table of Regular Polyedrons "whose edges are one. 
 
 Names. 
 
 No. of 
 Faces. 
 
 Surface. 
 
 Solidity. 
 
 Tetraedron. . . 
 Hexaedron. . . 
 Octaedron. . . 
 Dodecaedron . . 
 Icosaedron. . . . 
 
 4 
 
 6 
 
 8 
 
 12 
 
 20 
 
 1.7320508 
 6.0000000 
 3.4641016 
 20.6457288 
 8.6602540 
 
 0.1178513 
 1.0000000 
 0.4714045 
 7.6631189 
 2.1816950 
 
 Ex. — What is the solidity of an icosaedron whose 
 edge is twenty inches ? 
 
 Cal. 20X20X20X2.1816950=17453.56 solid inches. 
 
 Ans. 
 
 Prop. 43. — The solidity of a solid, generated by the 
 revolution of the cycloid about its base, is equal to five- 
 eighths of a cylinder whose length is equal to the base 
 of the cycloid, and whose diameter is twice that of the 
 generating circle. 
 
 Ex. — What is the solidity of a solid, generated by the 
 revolution of a cycloid about its base, the diameter of 
 whose generating circle is seven feet ? 
 
 Cal 7X22-^7=22 length of solid, 7X2=14 diameter, 
 7X22=154; 154X22X5--8=2117.5 solid feet. Ans. 
 
 Prop. 44. — The solidity of a paraboloid is equal to 
 one-half of the solid contents of a cylinder of the same 
 hight, and same base as the paraboloid. 
 
86 QUARTZ OPERATOR'S 
 
 Ex. — What are the solid contents of a paraboloid, the 
 diameter of whose base is twenty-eight inches, and 
 height forty inches ? 
 
 Gal. 28X22^-7=88; 88X28--4=616; 
 616x40-^-2=12320 solid inches. Ans. 
 
 Prop. 45. — The solidity of a spheroid is equal to 
 two-thirds the solidity of a circumscribing cylinder. 
 
 Ex. 1. — What is the solidity of an oblate spheroid 
 whose major diameter is twenty-eight inches, and 
 minor diameter twenty-one inches ? 
 
 Gal (28X88--4)X(21X2--3)=8624 solid inches. 
 
 Ans. 
 
 Ex. 2. — What is the solidity of a prolate spheroid, 
 the diameters being as in example 1. 
 
 Gal. (2lX66-=-4)X(28X2--3)=6468 solid inches. 
 
 Ans. 
 
 Prop. 46. — The solidity of an hyperboloid is as 
 follows : 
 
 Solidity.^ (—^- A2a 
 
 Remark. — In this formula the transverse and conju- 
 gate axes are respectively represented by (A) and (B), 
 and the ordinate on the axis of revolution by (#). 
 
 Ex. — What is the solidity of an hyperboloid whose 
 transverse axis is four feet, conjugate axis three feet, 
 and hight six feet ? 
 
HAND BOOK. 87 
 
 Cat. 6 + 4=10; 3.1416X94-16=1.76715; 1000 
 + 128=1128 ; 1128--3=376 ; 376—160=216 ; 
 1.76715X216=381.7 solid feet Ans. 
 
 Remark. — The relations of the co-ordinates to each 
 other in the tractory curve, the length of the curve, the 
 quadrature of the meridian plane coinciding with the 
 axis, the surface of revolution of the tractory conoid ; 
 and the solid contents of that solid will be found under 
 the heading " Discussion of the Tractory and differently 
 formed Grinding Plates." 
 
 INVOLUTION. 
 
 Involution is the raising of quantities to any proposed 
 power. 
 
 The power of any quantity is that quantity multi- 
 plied any number of times by itself. 
 
 Thus, 3X3=9 is the second power of 3 ; 
 
 5X5X5=125 is the third power of 5. 
 
 The power is sometimes expressed by writing the 
 quantity with the number of the power a little above it, 
 and at the right hand. 
 
 Thus, to express the power, write 
 3 2 =3X3=9 square of 3. 
 53=5X5X5=125 cube of 5. 
 27=2X2X2X2X2X2X2=128 seventh power of 2. 
 
 The number of the power, as 2, 3, 7 above, is termed 
 the exponent or index. 
 
 EVOLUTION. 
 Evolution is the extracting of roots. 
 
88 QUARTZ OPERATOR^ 
 
 The root of any quantity is a quantity which, if mul- 
 tiplied by itself a certain number of times, produces the 
 original quantity; and is called the second or square 
 root, the third or cube root, etc., according to the num- 
 ber of multiplications. 
 
 Thus the second or square root of 9 is a quantity 
 whose square or second power produces 9 — that is 
 3. And the seventh root of 1 28 is a quantity whose 
 seventh power produces 128 — that is 2. 
 
 Roots are sometimes represented by the symbol j/, 
 with the number of the root written within the angle of 
 the symbol, and sometimes by exponents or indices. 
 
 Thus the square root is represented 
 
 2 — 1 
 
 |/9— 9^=3 and is read square root or half power 
 
 of 
 
 i 
 
 j!K9i 25=1 25^=5 and is read cube root or one-third 
 
 power of 125. 
 
 7 1 
 
 -j/ 128=1 28 =2 and is read seventh root or one- 
 seventh power of 128. 
 
 The number 2 within the angle of the symbol in ex- 
 pressing the square root, is usually omitted. 
 
 Both involution and evolution are sometimes expressed 
 by fractional exponents or indices. 
 
 3 
 
 Thus 32^ The numerator shows that the quantity 
 is to be raised to the third power, 32X32X32=32768, 
 and the denominator shows that the fifth root of that 
 
 5 1 
 
 power is to be extracted — ^32768=32768 =8. 
 
HAND BOOK. «y 
 
 Or the denominator shows that the fifth root of the 
 
 quantity is to be extracted j/32=32 F =2. 
 
 And the numerator shows that the root thus obtained 
 is to be cubed or raised to the third power, 2 X 2 X 2=8. 
 TO EXTRACT ANY ROOT OF A POWER OR QUALITY. 
 
 Rule. — 1st. Point off the given power or quantity 
 into periods, containing each, except the left-hand period, 
 as many figures as the required root indicates, beginning 
 at the units place and pointing to the left in integers and 
 to the right in decimals. 
 
 2d. Find by trial the first figure of the root and set it 
 to the right of the quantity or power in the quotient's or 
 root's place. Also place its first power at the head of 
 a column on the extreme left, and its successive higher 
 powers (regularly increasing the exponents by onej at 
 the heads of columns following in order toward the right ; 
 thus forming as many columns as there are units in the 
 exponent of the power whose root is sought. The 
 highest power of the first root figure falls under the left- 
 hand period of the quantity, and is to be subtracted 
 therefrom, and with the remainder the next period is to 
 be brought down. 
 
 3d. Add the root figure to the first column, and mul- 
 tiply this sum by the root figure, placing the product in 
 the next right hand column and adding it thereto. Mul- 
 tiply this sum by the root figure, placing the product in 
 the next right hand column, which add thereto, and thus 
 proceed adding and multiplying until the number of 
 
90 QUARTZ OPERATOR'S 
 
 additions shall be one less than the exponent of the 
 number whose root is sought. The number thus found is 
 termed the trial divisor. The root figure is to be added 
 to the first column (the successive multiplications and 
 additions following as above, except that one multiplica- 
 tion less is made each time) as many times as there are 
 units in the exponent of the number whose root is 
 sought. 
 
 4th. Find how many times the trial divisor is con- 
 tained in the remainder, with the first left hand figure of 
 the next period brought down, and place the quotient as 
 a second root figure. Also add this root figure, removed 
 one place to the right, to the first column. Multiply 
 the sum thus obtained by this root figure, adding the 
 product, removed two places to the right, to the second 
 column. Thus continue to multiply and add to the suc- 
 cessive columns, removing the product at each addition 
 one figure further to the right, until the product, falling 
 under the quantity whose root is sought, is to be sub- 
 tracted therefrom, and the remainder with the next 
 period brought down as before. 
 
 5th. In a similar manner the successive figures of the 
 root are determined. 
 
HAND BOOK. 91 
 
 Example 1st Required the side of a square inclosure 
 containing 55225 square rods. Ans. 235 rods. 
 
 Cal. 
 
 
 / / / 
 
 
 55225 [235 
 
 2 
 
 4 
 
 2 
 
 152 
 
 43 
 
 129 
 
 3 
 
 
 
 
 2325 
 
 465 
 
 2325 
 
 Ex. 2d. Required the side of a cubical reservoir con- 
 taining 9663597 solid inches. Ans. 213 inches. 
 
 Cal. 
 
 
 9663597 [213 
 
 2 
 
 4 
 
 8 
 
 2 
 
 8 
 
 
 - 
 
 
 1663 
 
 4 
 
 1261 
 
 1261 
 
 2 
 
 62 
 
 
 402597 
 
 61 
 
 1323 
 
 402597 
 
 1 
 
 1899 
 
 
 62 
 
 134199 
 
 
 1 
 
 
 
 633 
 
92 QUARTZ OPERATOR'S 
 
 Ex. 3d. Required the fifth root of 6436343. Ans. 23. 
 
 Cal 
 
 
 
 6436343 [23 
 
 2 
 
 4 
 
 8 
 
 16 32 
 
 2 
 
 8 
 
 24 
 
 64 
 
 4 
 
 12 
 
 32 
 
 80 3236343 
 
 2 
 
 12 
 
 48 
 
 278781 3236343 
 
 6 
 
 24 
 
 80 
 
 1078781 
 
 2 
 
 16 
 
 12 927 
 
 
 8 
 
 40 
 
 92927 
 
 
 2 
 
 309 
 
 
 
 103 
 
 4309 
 
 
 
 DISCUSSION OF THE TRACTORY AND 
 DIFFERENTLY FORMED GRINDING 
 PLATES. 
 
 The discovery of the Tractory Curve was long at- 
 tributed to Huygens, an eminent mathematician of the 
 seventeenth century. That the curve, however, was 
 known long prior to the discovery of Huygens, is proven 
 by recent excavations in the ruins of Pompeii, where has 
 been found a mill whose grinding surfaces are of trac- 
 tory conoidal form. 
 
HAND BOOK. 93 
 
 What properties of the curve Huygens may have in- 
 vestigated does not appear. D'Alembert says that the 
 e volute of the curve is the common catenary. Dr. 
 Peacock says that the Tractory is an inverted semi- 
 cycloid. As the term cycloid here used is not qualified, 
 it must be inferred that the common is meant. Now, 
 D'Alembert and Dr. Peacock cannot both be right ; for 
 the involute of the common cycloid is similar and equal 
 to the evolute, that is, the cycloid itself. But the cycloid 
 is essentially dissimilar and unequal to the catenary. 
 Therefore, the cycloid and the involute of the catenary 
 cannot both coincide with, or be the Tractory. 
 
 Again, as the equations of the Tractory, Catenary and 
 cycloid, are essentially different, it is evident that neither 
 D'Alembert, nor Dr. Peacock has correctly demon- 
 strated and set forth the properties of the Ttractory. 
 
 The discrepancy of the above named authors, and the 
 unsatisfactory and unreliable manner of laying down 
 the curve mechanically, as adopted by C. Schiele, of 
 Oldham, have led to the following discussion, the cor- 
 rectness of the result of which is not only certified to by 
 several distinguished mathematicians, but is farther con- 
 firmed by numerous and carefully made experiments. 
 
 A tractory is defined to be " A curve whose tangent 
 is always equal to a given line." 
 
 The directrix of the tractory here investigated is 
 parallel with one of the axes, and the point of origin of 
 the curve is in a line at right angles to the directrix. 
 
94 
 
 QUARTZ OPERATOR'S 
 
 2'iq.l. 
 
 Let A Fig. 1 be the point 
 of origin. 
 
 Lay off on the axis AY 
 the constant tangent AF=a. 
 
 Draw the directrix FD ; 
 parallel with the axis AX. 
 
 Draw CD=a tangent to 
 the curve ACC at the point C 
 
 Produce BC in a right line 
 intersecting FD in E. 
 
 Let x and y be co-ordinates 
 of the curve at any point C, 
 and let z denote the length of 
 the curve AC. 
 
 Then CE=BE— BC=a— y, and EDS(cB— Cl)* 
 
 : r a 
 
 C'\ 
 
 1. By similar triangles 
 
 2. By similar triangles] 
 
 = {2ay-f) 
 
 d X ^Z^\ 
 
 a—y 
 
 dy 
 
 dz= a ^ 
 
 a—y 
 
 l v Putting u=(2ay- tffi 
 
 2 2 . Squaring Eq. l x , u 2 = 2ay — y 2 
 
 3 r Differentiating Eq. 2 lf 2udu=2ady — 2ydy 
 
 udu 
 
 4 X . Dividing Eq. 8 2 , by 2 (a—y) dy=- 
 
 5 V Transposing Eq. 2 l? y 2 — 2ay~- 
 
 a—y 
 
HAND BOOK. 95 
 
 6^ Completing square to Eq. 5 1? y 1 — 2ay-\-a 2 =a 2 — u 2 
 l v Extracting sq. root Eq. 6 19 y — a=±^/(a 2 — u 2 ) 
 S x . Or, a— y=± v /(a 2 — u 2 ) 
 
 3. Transposing Eq. 7 19 y=a + i/(a 2 — u 2 ) 
 9j. Substituting value of (a — y) Eq. 8 l7 in Eq. 4 l9 
 
 7 udu 
 
 ^~zt 1 /{a 2 -u 2 ) 
 
 4. Substituting values of y / (2ay — y*) Eq. l l9 
 
 (a—y) Eq. 8 lf and dy Eq. 4 1? in Eq. 1, 
 
 7 n 2 du 
 dx= 
 
 a 2 — u 2 
 
 5. Integrating Eq. 4, x= | u 
 
 J a 2 — u 2 
 
 u 2 " 2 
 
 1 2 . By division . = — 1-j- 
 
 
 
 or — vr 
 
 a z — uf 
 
 2, 
 
 Decomposing 
 
 a 2 A 
 a 2 —u 2 a — u 
 
 a-\-u 
 
 3. 2 . 
 
 Reducing to common denominator 
 a 2 Aa-f- Au-\-Ba- 
 
 -Bu 
 
 
 a 2 — u 2 
 
 a 2 — u 2 
 
 
 4J. Clearing Eq. 3 2 of fractions 
 
 a 8 =(A+B)a-j-(A— ■ B)u 
 5 2 . Transposing Eq. 4 2 0=(A-fB)a+(A— B)w— a 3 
 6 2 . Eq. 4 2 being true for all values 
 
 of u, is true when u=0 ; hence A — B=0 ; 
 7 2 . And A+B=a 
 
96 QUARTZ OPERATOR'S 
 
 8 2 . Adding Eqs. 6 2 and 7 2 ; and dividing by 2, A= - 
 
 Z 
 
 9 2 . Subtracting Eq. 6 3 from Eq. 7 2 ; dividing by 2, B=_ 
 
 Z 
 
 10 2 . Substituting values of A and B 
 
 a 2 a a 
 
 in Eq. 2 3 
 
 a 2 — u 2 2 (a — u) 2 (a + u) 
 
 11 2 . Substituting value of 
 
 a 2 . ^ - u 2 1 a a 
 
 in Eq. 1 2 = — 1+- 
 
 a 2 — u 2 a 2 — u 2 2 (a — u) 2 (a-\-u) 
 
 6. Substituting value of 
 
 u 2 . w - C j x C adu i r ^ w 
 in Eq. o ; £== | — du-\- 1 -f- I «, 
 
 a 2 — u" J J 2 (a — u) J 2 (a-|- u) 
 
 When #=0, w=0 also C=0 ; 
 
 7. hence, #= — u — _7(a — w)-f--Z(a-|-#) 
 
 8. which may be rendered x== — u-\-^l ( —£- ) 
 
 9. Integrating Eq. 2, «— f^ =— al{a— y) + C 
 
 J a—y 
 
 When 3/=0> z=0 and C=ala 
 
HAND BOOK. 97 
 
 10. hence, z—ala — at {a — y) 
 
 11. which may be rendered z=al( ) 
 
 \a~ y' 
 
 Z, in the above calculations denotes the Naperian 
 logarithm. 
 
 TO FIND THE AREA OF THE MERIDIAN PLANE COIN- 
 CIDING WITH THE AXIS OF REVOLUTION OF THE 
 TRACTORY CONOID. 
 
 (b) . Differential Equation, c?(area) =ydx 
 
 (b) v Substituting value (dx) Eq. 1; ydx=\/ [2ay—y i )dy 
 (5) 3 . Putting y=a — n; ydx— — \/(a 2 — n?)dn 
 (b) 3 . Decomposing and integrating, 
 
 fydx^—Ccfia*—?! 2 ) ~ %dn+C?i 2 {a*—n 2 ) " %dn 
 (5) 4 . Second term of Eq. (b) 5 ; 
 
 CtficP—tf) " %dn=—n l / (a 2 — n*)+fdn yV— r?) 
 
 (b) 5 . Substituting in Eq. {b) s , —C{a*-rfT$dn= 
 
 ^Ca%a 2 -nT^dn^n(a'^n^+C(a 2 ^n^dn 
 (5) 6 . Transposing, 
 
 E 
 
98 QUARTZ OPERATOR'S 
 
 (b) 7 . But, _ I _— -=-^™s ( - ) 
 
 (b) & . Hence, 
 
 -C(c?-n*fidn==^cos- f 7 ^ -^{cf-r?) + C 
 
 (5) 9 . Restoring value of n, I -|/ \2ay~y i )dy r= 
 
 2 V a J 2 
 
 Making #=0; Then C=0. 
 
 (b) 10 . Uence,j l /(2ay—y i )dy= 
 
 ^cos-( a Ziy\-J^ V {2ay^) 
 2 V a J a 
 
 (5) n . Making y=a. Area=__ (0)=__^_== _ 
 
 (5) 13 . Makingy=2a. Area =-^ cos" (-1)=— X-^=^- T 
 
 4a 
 Making y=—<, as adopted in one of the inventions 
 
 of Wheeler & Randall, viz : in their grinder and amal- 
 gamator having the greater base of its muller upward. 
 Making a =1, or unity. 
 
 Area == lcos~ (,6) — ?(,8)=,2232 half plane of muller. 
 2 2 
 
HAND BOOK. 99 
 
 lence, area =,4464 entire plane of muller. 
 10 FIND THE SURFACE OF REVOLUTION OF THE 
 TRACTORY CONOID. 
 
 (c) v Differential Equation ^(surface) = (a — y)dz 
 (c) 3 . Substituting value of dz JEq. 2, ds=2nady 
 (c) 3 . Integrating s=2nay 
 
 (c) 4 . Making y=a s=2na 2 
 
 Making y=,4, as adopted in that grinder and amal- 
 gamator of Wheeler & Randall, having its greater base 
 upward. 
 
 (c) 5 . Surface of revolution =s=2,5028 
 
 TO FIND THE SOLID CONTENTS OF THE TRACTORY 
 
 CONOID. 
 
 Cj. Differential Equation, d (solid contents) =n(a--yydx 
 
 e z . Substituting val. dx. eq. 1, 
 
 dC =n(a^y)y / (2ay-y 2 )dy 
 
 e s . Put a~y=,n then y=a — n 
 
 e 4 . dC = — iiai/ (a*-^n?)dn 
 
 ds 
 e 5 . Put s=d l — w 2 , then ds= — 2ndn, and — = — ndn 
 
 A 
 € 6 . Substituting in Eq. e 4 dC =~ — - 
 
 e 7 . Integrating Eq. e Q C = 
 
 e2 
 
100 
 
 QUARTZ OPERATOR'S 
 
 e 8 . Restoring (a 2 — ri 2 ) for s, C = — - — . L 
 
 e d . Restoring (a~y) for n C = — — f"^ ' 
 
 o 
 
 e 10 . Making y=a 
 
 n lice 
 C„=— 
 
 e n . Making y = J^ == . ,4 when a=unity C = ,5362 
 
 10 
 ♦ 
 
 The following table is computed by making the tan- 
 gent a — 1, and assigning different values to u, not 
 greater than a as shown in column marked u. That is, 
 these values are substituted in equations 8, 3 and 11, 
 and the co-ordinate values of x and y, and the length of 
 the tractory curve z, thus determined in terms of the 
 tangent or unity : 
 
 TRACTORY TABLE. 
 
 
 
 
 a 
 
 
 I 
 
 
 
 :n ; o 
 
 u 
 ,100 
 
 X 
 
 y r z \ 
 
 ,00501 4 ! 005027 
 
 No 
 14 
 
 u 
 
 x y 
 
 z 
 
 i 
 
 ,000336 
 
 ,725 
 
 193102311251 
 
 368505 
 
 2 
 
 ,150 
 
 ,001136 
 
 ,011316011379 
 
 15 
 
 ,750 
 
 222956338562 
 
 413293 
 
 3 
 
 ,200 
 
 ,002732 
 
 ,020205 029412 
 
 16 
 
 ,775 
 
 2577301368039 
 
 458928 
 
 4 
 
 ,250 
 
 ,005415 
 
 ,031756 032260 
 
 !7 
 
 ,800 
 
 298611 400000 
 
 510827 
 
 5 
 
 ,300 
 
 ,009514 
 
 ,046063j047157 
 
 18 
 
 ,825 
 
 347276434867 
 
 570796 
 
 6 
 
 ,350 
 
 ,015454 
 
 ,063252 065571 
 
 19 
 
 ,850 
 
 406153 473297 
 
 641118 
 
 7 
 
 ,400 
 
 ,023648 
 
 ,0834851087178 
 
 20 
 
 ,875 
 
 479027.515878 
 
 725418 
 
 8 
 
 ,450 
 
 ,034699 
 
 ,106972 113137 
 
 21 
 
 ,900 
 
 572220564109 
 
 830367 
 
 9 
 
 ,500 
 
 ,049304 
 
 ,133976 143842 
 
 22 
 
 ,925 
 
 697603620033 
 
 967651 
 
 10 
 
 .550 
 
 ,068381 
 
 ,164835:180127 
 
 23 
 
 ,950 
 
 881785687751 1,163903 
 
 11 
 
 ,600 
 
 ,093146 
 
 ,200000l223144 
 
 24 
 
 ,975 
 
 1,209723 777795 1,504152 
 
 12 
 
 ,650 
 
 ,125296 
 
 ,240065 274523 
 
 25 
 
 ,99011,656677 858933 1,958518 
 
 13 
 
 ,700 
 
 ,167304 
 
 ,285857'336672 
 
 26 
 
 ,999*2,801 193 955289 3,107543 
 
HAND BOOK. 
 
 101 
 
 TO CONSTRUCT THE TRACTORY BY THE PRECEDING 
 TABLE. 
 
 Draw , the rectangular co-ordinate axes AX, AY, 
 Figure 2. 
 
 On AY, lay off AF, equal to the given tangent, and 
 draw the directrix FD, parallel with the axis AX, 
 through the point F. 
 
 f; 3 .2. 
 
 Multiply any tabular 
 
 A. quantity in column x by the 
 B given tangent, and lay off 
 
 B , the product as AB or AX. 
 „ Draw BE parallel with AF. 
 ,„ Multiply the quantity in 
 
 column y, horizontal with 
 the quantity taken in column 
 x, by the given tangent, and 
 lay off the product as BC on 
 the line BE. The point C 
 is in the curve. 
 
 In a similar manner determine the points C, C", C", 
 C"", etc., etc., in the curve. 
 
 Multiply the quantity in column z horizontal with the 
 quantities thus taken in column x and y, and the pro- 
 duct will be the length of the curve A C, A C C, 
 
 etc., etc. 
 
 e3 
 
 Y 
 
 F 
 
 
 E 
 
 (?/ 
 
 £' 
 
 .z___ 
 
 J?" 
 
 C7 
 
 E !r 
 
 / 
 
 can 
 
 M 
 
 rrm 
 
 I 
 
 1 
 
102 QUARTZ OPERATOR'S 
 
 When the points A, C, C, C", C", etc., are not far 
 apart, the arc of a circle drawn through any three of 
 them taken consecutively, approximates the true curve 
 sufficiently near for most practical purposes. ; . 
 
 Ex. — Let it be required to find the values of x', y\ 
 and z', when the tangent a = 24 for the point C. 
 
 Now since C may be any point of the curve, let the 
 co-ordinate x be taken in the table opposite No. 17. 
 
 Then, 2±x = A B' =24 X ,298611 = 7.166664=^ 
 24 y = B' C =24 X ,400000 = 9.600000 =y' 
 24 z = ACC =24 X ,570827 =12.259848 =*£ 
 
 Laws of Grinding. — 1. The measure of a unit of 
 grinding effect is the "result of a body impressing a unit 
 surface of another body with a unite pressure, and 
 moving over a unite space. 
 
 2. A body impressing a unite surface of another body 
 with a unite pressure, and moving under these circum- 
 stances over a unite space, producing thereby a uniie of 
 grinding effect, (the pressure being uniform and the 
 nature of the surfaces of contact being the same) pro- 
 duces as many units of effect as there are units in the 
 distance passed over. 
 
 3. The wear or distinction to the moving body per- 
 pendicular to the surfaces of contact, bears a constant 
 ratio to the distance passed over. 
 
HAND BOOK. 
 
 103 
 
 M S .3. 
 
 To prove that to grinding plates coinciding, of uniform 
 hardness, and of tractory conoidal form, the wear paral- 
 lel with the axis of revolution is the same at all points of 
 the grinding surface. 
 
 Let A F D" C" represent one 
 half of a meridian plane of a 
 tractory conoid taken through 
 the axis or directrix F D". 
 
 Draw C D =■. a, and C D' 
 = a each tangent to the curve 
 at any points C, C ; also draw 
 the radii CE = r and C E' = 
 r' parallel with A F. 
 
 Let w = o c denote the wear 
 perpendicular to C D, and w f = 
 o' c f the wear perpendicular to 
 C D'. 
 
 Let P = C c and P' = C c denote the wear re- 
 spectively at the points C, C and each parallel with the 
 axis F D". 
 
 h- 
 
 Then by law 2d, 
 
 w' = 
 
 HU T 
 
 r 
 
 2s. 
 
 By similar triangles, 
 
 P = 
 
 a w 
 r 
 
 3 3 . 
 
 By similar triangles, 
 
 P' = 
 
 aw f 
 
 e4 
 
104 QUARTZ OPERATOR'S 
 
 4 2 . Substituting value of w K Eq. 1 3 
 
 a w r 1 
 in Eq. 3 3 P = -^r- 
 
 5 3 . Reducing 2d member of Eq. 4 3 P' = — 
 
 6 3 . Substituting value of — in Eq. 2 3 P' =P 
 7 3 . That is, C'c' = Cc 
 
 (a) Hence, as the wear is the same at any two points 
 as shown- by Eqs. 6 3 and 7 3 , it is the same at all points 
 parallel with the axis, of revolution. 
 
 (b) And hence the curve A' c c' c" is similar and 
 equal to the tractory ACC C". 
 
 Again, plates whose grinding surfaces are of tractory 
 conoidal form, alone passes the property of uniform wear 
 parallel with the axis of revolution. 
 
 To prove this proposition 
 Put (Fig. 3) a' = C D' 
 
 8 3 . Then Eq. 5 3 becomes P' = 
 
 r 
 
 9 3 . From Eq. 2 3 and 8 3 we have P : P' : : — : — 
 
 r r 
 
 10 3 . Reducing 2d couplet P : P' : : a : a f 
 
 Proportion 10 3 shows that the wear P is equal to P' 
 only when tangent a is equal to d ; but when a is equal 
 to a! the line or curve is a tractory. Q. E. D. 
 
HAND BOOK. 
 
 105 
 
 TO DETERMINE THE SOLIDITY OF A HOLLOW 
 TRACTORY CONOID. 
 
 Jfy.4* 
 
 J> &'G' 
 
 e a. 
 
 Jj 
 
 I . 
 
 L C 
 
 ' AL 
 
 JS> 
 
 I' 
 
 .A 
 
 V 
 
 
 C"l 
 
 / / 
 
 
 J) 
 
 A 
 
 C" 
 
 I r" 
 
 r 
 
 Let AFD'C" represent one- 
 half of a meridian plane taken 
 through the axis F D'. 
 
 Draw the curve A' c c' c" c'" 
 similar and equal to the tractory 
 ACC C" C", and at the dis- 
 tance A A', C" c" from it. (See 
 conclusion {b). 
 
 Draw A'EjC'E^E", par- 
 allel with A F, also G c through 
 C, GV, through C, G'V, thro- 
 ugh C" parallel with F D'. 
 Since the curve c' c" is similar 
 and equal to C C", and the line C c' =- C" c", and C I' 
 = c'C", and the angle c'C"c"=the angle CTC"; 
 it follows that the triangle c" c' C"= C" C I', and also 
 similar to it. 
 
 And since the conoidal triangle C" c' C is common to 
 the parallelogram C" c' C V, and to the conoidal section 
 c" c' C C", it follows that the conoidal section c" V C C" 
 is equal to the parallelogram C" c' C I'. 
 
 Again, since c'C' = A / A = LG', and C"c = IL, 
 it follows that the parallelogram c'V = the parallelo- 
 gram L G". Hence the conoidal section c" c' C C" = 
 the parallelogram L G" ; and since its position, in all 
 
 respects, is at the same distance from the axis F D', it 
 
 s5 
 
106 QUARTZ OPERATOR'S 
 
 is evident, if it be revolved about F D' as an axis, it 
 will generate a conoidal ring of the same magnitude as 
 a ring generated by revolving L G" about the same 
 axis. 
 
 In the same manner may it be shown that any oo- 
 noidal ring, under similar circumstances, is equal to the 
 ring generated by revolving the corresponding paral- 
 lelogram taken in ^AFE around the axis F D'. 
 
 Hence the solidity of a hollow tractory conoidal is 
 equal to the solidity of a cylinder having the same base 
 and same height as A' A, c" C", measured parallel with 
 the axis of revolution. 
 
 Tractory Plates, — To determine the grinding effect of 
 hollow tractory conoidal plates of uniform hardness. 
 
 Let, as in Figs. 1 and 2, the tangent or greater 
 radius sss a. 
 
 Let x and y be co-ordinates of any point, C, etc. 
 
 Let z denote the length of curve A C, etc. 
 
 Let P denote the wear parallel with the axis of revo- 
 lution, by one revolution under a unit pressure to a unit 
 surface. » 
 
 Let S denote the grinding surface. 
 
 Let ii denote the ratio of the diameter to the circum- 
 ference of a circle. 
 1 4 . Then, d$ = 2n(a— y)dz 
 
 2 4 . Substituting value of dz of Eq. 2, 
 io Eq. 1^ da » 2uady 
 
HAND BOOK. 107 
 
 3 4 . Then, by one revolution, 
 
 2nP(a — y)ds b 4n 2 Pa 2 <fy — An 2 Faydy 
 4 4 . Integrating Eq. 3 4 , 
 
 2nP(a— y)ds = 4n 2 Pa# — %ii 2 Patf 
 
 5 4 . Resolving 2d member of Eq. 4 4 into factors ; 
 
 f2nP(a — y)ds = 2n 8 Pa(2ay — f) 
 
 i> 
 
 s* 
 
 6 4 . By making y = a in Eq. 5 4 ; 
 
 Grinding effect = 2n 2 Pa 3 
 
 Plane Plates. — To determine the grinding effect of 
 plane circular plates, increasing in hardness from the 
 center to the circumference in the ratio of the increase 
 of the radius. 
 
 Let a =ss= the radius of the plates. 
 
 Let y = any radius of the plates less than a. 
 
 Let P = the wear at the circumference, perpendicular 
 to the grinding surface, under a unit pressure to the 
 unit surface, by one revolution. 
 
 Let ii = ratio of diameter to the circumference of a 
 circle. 
 
 1 5 . Then ds = 2nydy 
 
 2 5 . By one revolution, 2liPyds = 4ii 2 P^ dy 
 
 S 5 . Integrating Eq. 2 5 , f 2nPycfe = - n*Ff 
 
 J 3 
 
 4 5 . By making y = a in Eq. 3 5 ; 
 
 Grinding effect = - n 2 Pa 8 
 3 
 
 sC 
 
108 QUARTZ OPERATOR'S 
 
 Plane Plates. — To determine the grinding effect of 
 plane circular plates of uniform hardness. 
 
 Let a ==s the radius of the plates. 
 
 Let y — any radius less than a. 
 
 Let P === the tendency to wear at the circumference, 
 perpendicular to the grinding surfaces. 
 
 Let S = the surface of which y is the radius. 
 
 Let ii == ratio, etc. 
 1 6 . Then ds = 2 n y dy 
 
 Now, it is evident that when the tendency to wear at 
 the circumference and at the distance a is P, that the 
 tendency to wear at the distance y from the centre 
 
 will be 5? 
 
 2 6 . Then by one revolution 2ll ^ yds = 4n2p ^ 
 
 a a 
 
 3 6 . Integrating Eq. 2 6 fl^yds = *5-3? 1 
 
 4 6 . Reducing 2d member of Eq. 3 6 — ; 
 
 p 
 
 .yds 
 
 a 
 
 5 e . By making y = a in Eq. 4 6 , 
 
 Grinding effect = n 2 Pa 8 
 
 Conical Plates. — To determine the grinding effect of 
 hollow conical plates of uniform hardness. 
 
J3AND BOOK. 
 
 109 
 
 Fig. 5. 
 
 Let, in Fig. 5, the right angled 
 
 triangles ABC and A' B' C> 
 
 similar and equal, and at the 
 
 A' distance AA' = CC apart, be 
 
 revolved about the axis B C 
 
 Then will the solid generated 
 
 by A A' B B' be equal to the 
 
 solid generated by C A' A O 
 
 Let a — the radius B A. 
 
 Let h tsz height of cone 
 
 BC=B' C. % 
 
 Let P = tendency to wear at 
 A, parallel with the axis of rev- 
 olution, by one revolution of the plate under a unit 
 pressure to a unit surface. 
 
 Let x and y be co-ordinates of any point P. 
 Let z = C P the slant height of that cone the 
 radius of whose base is y. 
 
 Let S = the surface whose slant height is z. 
 Let ii = ratio, etc. 
 
 1 7 . Then by similar triangles 
 
 hy 
 
 %. And (Euclid, Book 1, Prop. 47) ar 2 + y 2 = z> 
 
 3 7 . Squaring Eq. 1 7 x 2 rz —JL 
 
 a 
 
 4j. Substituting value of x 2 of Eq. 3 7 in Eq. 2 7 
 
110 QUARTZ OPERATOR'S 
 
 5 7 . Extracting square root of Eq. 4 7 
 
 6 7 . Surface of cone whose slant height is z 
 
 7 7 . Differentiating Eq. 6 10 ds = 2 n -j/ fE + l\y dy 
 
 When the tendency to wear at A is P, it is evident 
 that the tendency to wear at P is ^ parallel with the 
 
 axis. 
 
 8 7 . Hence differential of effect 
 
 1±U? ds = t2L*jL{»+ l\ f dy 
 
 a a \a 2 y 
 
 9 7 . Integrating Eq. 8 7 Cl l lE^ds= l £E(£ + l)y* 
 
 10 7 . By making^ — a in Eq. 9 7 . 
 
 Grinding effect = n 2 P \/f-; + l^a 3 
 
 11 7 . By making h =_ in Eq. 10 7 . 
 
 Grinding effect = 1L _ i/o" 
 2 Y 
 
 12^ Removing surd sign from 2d member of 
 
 Eq. ll r . Grinding effect « 1.118 ll 9 P a 8 
 
HAND BOOK. Ill 
 
 Comparison. — Let a comparison now be instituted 
 between the grinding effects of plane circular plates, in- 
 creasing in hardness from the center to the circumfer- 
 ence in the ratio of the increase of the radius, plane 
 circular plates of uniform hardness, " Randall's Patent 
 Grinding Plates," conical plates of uniform hardness, 
 and tractory conoidal plates, and all of the usual ring, 
 form, same diameter, same weight, and running at the 
 same velocity. 
 
 1st. Plane circular plates, increasing in hardness from 
 the center to the circumference in the ratio of the in- 
 crease of the diameter. 
 
 Making y less than a. For example, y = J and 
 
 o 
 
 substituting this value for y in Eq. 3 5 , and we have 
 
 1 8 . Grinding effect (the radius being f)= J-ii 2 Pa 3 
 
 O Li 
 
 sin S ->) = 
 o 
 
 2 8 . Subtracting Eq. 1 8 from Eq. 4 5 . 
 
 32 
 
 Grinding effect (ring) = — irPa 8 
 
 3 8 . Or expressing decimally. 
 
 Grinding effect (ring) == 1,1852 II 2 P a 3 
 2d. Plane Circular Plates of uniform hardness. 
 
 Making y less than a. For example, y = a ' and 
 
 o 
 
 substituting this value for y in Eq. 4 6 , and we have 
 
 1 9 . grinding effect (radius being 2_) 
 
 ii 3 P a 3 
 
 8 81 
 
112 QUARTZ OPERATOR'S 
 
 2 9 . Subtracting Eq. 1 9 from Eq. 5 6 , 
 
 n . ,. ~ l, . . v 80 ii 2 P a 3 
 Grinding enect ( ring) = 
 
 . 81 
 
 3 9 . Or expressing decimally. 
 
 Grinding effect (ring) == ,9877 n 2 P a 3 
 
 3d. Randall's Patent Grinding Plates. — These plates 
 consist of two or more concentric rings of different hard- 
 ness. The softer plates are arranged nearer the center 
 where there is the less wear, and the harder plates more 
 remote where the wear is greater. This arrangement 
 remedies in a great measure the otherwise fatal defects 
 of plane circular and conical plates. 
 
 For Example. — Let a = the greater radius of the 
 plates. 
 
 Let - rr the radius of the opening. 
 
 2a 
 Let — the greater radius of the inner ring. 
 
 3 
 
 1st. Making y=-, and substituting this value for y 
 o 
 
 in Eq. 4 6 , and we have 
 
 n 2 Pa 3 
 
 1 10 . Grinding effect ( radius - j 
 
 81 
 
 v 2a 
 
 2d. Making y ■zz — , and substituting this value for 
 
 y in Eq. 4 6 , and we have 
 
 2 10 . Grinding effect ( radius -q) — — ^r — 
 
HAND BOOK. 113 
 
 3 10 . Subtracting Eq. 1 10 from Eq. 5 6 ; 
 
 80 
 Grinding effect (ring) ^ — n 2 Pa 3 
 V oJ 81 
 
 4 i0 . Subtracting Eq. 2 10 from Eq. 5 6 
 
 Grinding effect (ring) = - u 2 Pa s 
 
 5 10 . Comparative weight of the outer ring, 
 2 _4a 2 = 5^ 2 
 9 T 
 
 6 10 . Comparative weight of entire plane, less the opening, 
 
 . 3 __ a 2 ^^ 2 
 
 9" ~9~ 
 
 - rp, 5a 2 . 8a 2 65n 3 Po 8 104n 2 Pa 3 
 7 10 . Inen : : : : 
 
 9 9 81 81 
 
 8io- Or grinding effect of outer ring, = — n 2 Pa 3 
 
 81 
 
 9 10 . The loss sustained by plates of uniform hardness, 
 
 104* 2t> . 80 9n , 24 ,„ o 
 
 == n 2 Pa 3 — _ n 2 Pa 3 = _ n 2 Pa 2 
 
 81 81 81 
 
 Let it now be assumed that the inner plates or rings 
 are one-third as hard as the outer plates. 
 10 10 . See Eq. 9 10 . Loss sustained, 
 
 81 81 
 
114 QUARTZ OPERATOR'S 
 
 Subtracting Eq. 10 10 from Eq. 8 10 , gives for "RandaWs 
 Patent Grinding Plates, viz, : 
 
 Grinding effect = . 
 
 81 
 1 2 10 . Expressing decimally ; 
 
 Grinding effect = 1.1852ii 2 P« 3 . 
 Subtracting Eq. 2 9 from Eq. 11 10 , and dividing the 
 remainder by Eq. 2 9 , gives the per centage of " Ran- 
 dall's Patent Grinding Plates," both over plane circular, 
 and conical plates of uniform hardness, as follows : 
 13 10 . Per centage in favor of " Randall's Patent Grind- 
 
 ing Plates," ( 96 nW __80ii W) £_81_ 2Q 
 v 81 81 J 80n 2 Pa 3 
 
 4th. Conical Plates of uniform hardness and of 
 ring form. 
 
 l n . Making y less than a, for example y=-, and substi- 
 
 o 
 
 tuting this value for y in Eq. 9 7 , and we have the grind- 
 ing effect ("the radius °L\ = ^^f *L+ 1 ^S 
 
 2 n . Subtracting Eq. l n from Eq. 10 7 . 
 
 n - v t * f - \ 80n*Pa 3 ,># . ,\ 
 Grinding effect (ring) = j/( — -p 1 J 
 
 8 1 V« 2 J 
 
 3 U . Making h =," in Eq. 2 n . 
 
 Grinding effect (ring) = S ln 2 F aYf-^) 
 81 v4>/ 
 
HAND BOOK. .115 
 
 4 n Expressing decimally. 
 
 Grinding effect (ring) = 1,1042 n 2 P a s 
 
 5th. Tractory Conoidal Plates of uniform hardness. 
 
 2 a 
 Making y less than a. For example y = —'as is 
 
 5 
 
 the case in the Wheeler & Randall muller or grinding 
 
 plates. 
 
 2 a 
 1 12 . Substituting . for y in Eq. 5 4 . 
 
 5 
 
 „. „ . 32n2Pa3 
 
 Kino; effect = J 
 
 25 
 
 2 12 . Making the tractory conoidal plate of the same 
 weight or solidity as the plane circular or conical plates, 
 
 and Eq. 1 13 becomes Grinding effect (ring) = — if P a 5 
 
 3 12 . Expressing decimally. 
 
 Grinding effect = 1,7778 n 2 P a 3 
 
 Recapitulation. — To express the relative grinding 
 effects of* the differently formed plates now considered, 
 the literal factors n 2 P a 3 , common to all their formulas, 
 may be omitted. 
 
 Omitting the literal factors, and the relative grinding 
 effects of differently formed plates become as follows, to 
 wit : 
 
 l l3 . Eq. 3 9 . — Plane circular plates of uniform 
 hardness = ,9877. 
 
116 QUARTZ OPERATOR'S 
 
 2, 3 . Eq. 12 10 . — Randall's Patent Grinding Plates 
 = 1,1752. 
 
 3 13 . Eq. 4 1]L , — Conical plates of uniform hardness 
 = 1,1042. 
 
 4 l3 . Eq. 3 12 . — T ractor j conoid al plates of uniform 
 hardness == 1,7778. 
 
 Hence the conclusion that tractory conoidal plates not 
 only differ materially in form from plane circular and 
 from conical plates, but also differ essentially from, and 
 are greatly superior to them in their grinding properties. 
 
HAND BOOK. 
 
 117 
 
 PRORERTIES OP BODIES. 
 
 NAME. 
 
 Specific Melt'ng Rates 
 gravity points of'h'rd- 
 at ;J2 Q f h. at fahr. nes s . 
 
 Tenacity 
 in lbs. per 
 sqr. inch. 
 
 Crush'ng 
 force in 
 Ibs.sq. in 
 
 Volatile at 
 
 
 20.336 3^80° 
 
 
 
 
 very high ht. 
 very high ht. 
 
 do drawn wire 
 
 21.042 
 22.069 
 19.258 
 19.361 
 
 
 
 34000 
 
 
 do plates .... 
 
 Gold, cast 
 
 do liammered. 
 
 
 
 2016 
 
 1.8 
 
 18000 
 
 
 moderate ht. 
 
 do wire 
 
 
 
 28000 
 36000 
 
 
 
 Silver, cast. . . . 
 do hammered 
 
 10.474 
 10.510 
 
 1873 
 
 2.4 
 
 
 high heat. 
 high heat. 
 
 do wire 
 
 
 
 34000 
 17000 
 30000 ( 
 42000 \ 
 55000 
 4000 
 
 117000 
 
 92000 
 
 103000 
 
 high heat, 
 high heat. 
 
 > high heat. 
 
 Copper, cast. . . 
 do hammered 
 
 8.788 
 8.910 
 
 1996 
 
 2.8 
 
 do sheets . . . 
 
 
 
 do wire 
 
 
 
 
 
 Tin, cast 
 
 do hardened . 
 
 7.291 
 7.291 
 
 442 
 
 1.2 
 
 15500 
 
 
 do wire 
 
 
 
 6000 
 
 
 
 Zinc, cast 
 
 6.200 
 
 7.191 
 
 11.352 
 
 773 
 
 1.6 
 
 
 white heat. 
 
 do sheets 
 
 
 
 
 Lead, cast 
 
 
 
 1600 
 3000 
 2200 
 
 7730 
 
 white heat. 
 
 do milled. . . . 
 
 
 
 
 do wire 
 
 
 
 
 
 
 Mercury 
 
 Iron, cast, gray. 
 do do white. 
 
 13.598 
 
 7.248 
 7.500 
 7.788 
 
 I 
 
 —39 
 2800 
 
 
 
 680° 
 
 any 
 deg. 
 
 4.7 
 
 j 
 
 
 
 very high ht. 
 
 
 
 do forged .... 
 do wire, 1-20 
 
 120000 1 
 182000 \ 
 72000 ) 
 86000 J 
 
 54000 
 
 
 
 
 to 1-30 in. diam 
 Iron, wire, 1-10 
 inch diameter.. 
 Iron bars, Rus- 
 
 
 
 i 
 
 
 
 \ 
 
 
 \ 
 
 
 
 sian, mean 
 
 Iron, American 
 
 gun metal 
 
 Iron, American 
 gun motal.mean 
 Iron. English 
 
 Stirling 
 
 Iron, English 
 Stirling, mean. . 
 
 \ 
 
 
 
 147803 
 
 129000 
 
 119550 
 
 90833 
 
 
 I 
 
 
 
 
 
 I 
 
 1 
 
 
 
 
 
 s 
 
 
 
 
 
 f 
 
 
 
 
 
118 
 
 QUAl'iTZ OPERATORS 
 
 PROPERTIES 
 
 OF 
 
 BODIES. — CONTINUED. 
 
 NAME. 
 
 Specific 
 gravity 
 at 32° f'h. 
 
 M*H'ng Rntes 
 points ofh'ru- 
 at fahr. ness. 
 
 Tenacity Crush'ng 
 in libs, per force in 
 sqr inch, lbs sq, in 
 
 Volatile at 
 
 Iron, American 
 
 1 
 
 . .... 
 
 
 
 
 83500 
 
 65200 
 
 198944 
 
 373041 
 
 295000 
 
 
 wrought, mean. 
 Iron, English, 
 
 1 
 
 
 
 
 
 wrought, mean. 
 Steel, cast, ) 
 
 soft S 
 
 Steel, cast, 1 
 tempered. . . . ) 
 Steel, cast, ) 
 tern p'd, mean ) 
 
 ) 
 
 7.833 
 7.816 
 
 
 
 88000 
 
 
 
 
 
 
 
 
 
 17.600 
 7.824 
 
 
 
 
 155916 
 
 146000 
 164800 
 
 
 
 1900 c 
 
 
 16000 
 
 
 
 
 
 8.396 
 
 
 
 
 
 
 Nickel, cast. .. . 
 
 8.279 
 9.880 
 6.702 
 5.763 
 8.600 
 8.600 
 
 2.716 
 
 2810 
 476 
 932 
 400 
 
 
 
 
 very high ht. 
 white heat. 
 
 Bismith 
 
 2.0 
 
 
 
 Antimony 
 
 Arsenic ....... 
 
 
 
 white heat. 
 
 
 
 
 356° 
 
 Cob 'It 
 
 
 
 
 very high ht. 
 600° 
 
 
 550 
 
 
 
 
 Stone, mar- \ 
 blc, white . . . ) 
 * Stone, marble 
 
 
 8000 
 
 2800 
 10382 
 23917 
 22702 
 12624 
 11200 
 
 5340 
 
 3319 
 3065 
 
 
 
 
 
 Stockbridgg. . . 
 t Stone, marble 
 
 J 
 1 
 
 
 
 
 
 East Chester . . 
 Stone, marble, 
 
 ) 
 \ 
 
 
 
 
 
 Lee, Mass 
 
 Stone, marble, 
 
 ) 
 \ 
 
 
 
 
 
 Italian 
 
 Granite, Pa- ) 
 
 tapsco ) 
 
 t Sand Stone, / 
 Aquia Creek. ) 
 Freestone, Con. 
 
 ) 
 
 2.613 
 
 2.956 
 
 i 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 Limes! one .... 
 
 2 . 386 
 
 
 
 
 
 
 
 
 
 
 
 
 ♦Same as that of Ciiy Mall, Mow York, 
 f Same as that of General Post Office. VVashington. 
 % Same as that of tho Capitol, Treasury Department, and Post Office 
 Department, D. O. 
 
HAND BOOK. 
 
 119 
 
 PROPRTIES OF BODIES — CONTINUED. 
 
 NAME, 
 
 Specific 
 gravity 
 at 328 fh. 
 
 Melt'ng 
 points 
 at fahr. 
 
 Rates 
 ofhr'd- 
 ness. 
 
 Tenacity 
 in lbs. per 
 sqr, inch. 
 
 Crush'ng 
 force in 
 lbs. sq. in 
 
 Volatile at 
 
 Mica. 
 
 2.650 
 
 
 
 
 
 
 Quartz <j 
 
 Serpentine 
 
 Cinnabar 
 
 2.624 
 3.75U 
 2.264 
 6.902 
 2.438 
 2.582 
 .845 
 
 [ 
 
 
 
 
 
 ) 
 
 
 
 
 
 
 
 
 
 
 Felspar 
 
 Flint 
 
 
 
 
 
 
 
 
 
 
 
 Ash 
 
 
 
 
 8663 
 4100 
 5982 
 6484 
 5375 
 6645 
 5768 
 10331 
 
 
 Oak, American 
 
 
 
 
 
 
 do Canadian. 
 
 
 
 
 
 
 do English.. . 
 
 .920 
 
 
 
 
 
 Pine, Yellow. . 
 
 
 
 
 
 Walnut 
 
 
 
 
 
 
 Cedar 
 
 
 
 
 
 
 Kim 
 
 .671 
 .600 
 .862 
 
 
 
 12000 
 10000 
 18000 
 
 
 Fir 
 
 
 
 
 Box 
 
 
 
 
 
 Teak 
 
 
 
 12100 
 
 
120 QUARTZ OPERATOR'S 
 
 MISCELLANEOUS. 
 
 5760 grains = 1 pound troy = 1 pound apothecary. 
 480 grains = 1 ounce troy = 1 ounce apothecary. 
 1 2 ounces = 1 pound troy = 1 pound apothecary. 
 7000 grains = 1 pound avoirdupois. 
 437,5 grains = 1 ounce avoirdupois. 
 1 6 ounces avoirdupois = 1 pound avoirdupois. 
 252,458 grains = 1 cubic inch distilled water, Eng- 
 lish standard 62° Fahr., Barometer at 30 inches. 
 
 252,693 grains = 1 cubic inch distilled water, U. S. 
 standard 30.83° Fahr., Barometer at 30 inches. 
 
 27,7015 cubic inches distilled water = 1 pound 
 avoirdupois. 
 
 1 cubic foot distilled = 62,37929 pounds avoirdupois. 
 321 cubic inches = 8,3388822 pounds avoirdupois = 
 1 gallon U. S. 
 
 277,274 cubic inches = 10 pounds avoirdupois = 1 
 gallon Imperial. 
 
 2150.42 cubic inches =77,627413 pounds avoirdupois 
 = 1 bushel. 
 
 1 grain Gold, 1000 fine = $,0430663 Mint value. 
 1 grain Silver, 1000 fine = ,0026936 Mint value. 
 1 grain Copper, 1000 fine = ,0000595 Mint value. 
 1 ounce Gold, 1000 fine = 20,671791 Mint value. 
 1 ounce Silver, 1000 fine = 1,292929 Mint value. 
 1 ounce Copper,l 000 fine = ,028571 Mint value. 
 23.22 grains Gold, 1000 fine + 2.58 grains alloy = 
 25.8 grains = $1.00. 
 
HAND BOOK. 121 
 
 371.25 grains Silver, 1000 fine + 41.25 grains alloy 
 = 412.5 grains = $1.00. 
 
 16800 grains Copper, 1000 fine = $1.00. 
 
 1 cubic inch Gold, 1000 fine = 10,12883 ounces troy 
 = $209.38. 
 
 1 cubic inch Silver, 1000 fine = 5,50885 ounces troy 
 = $7.13. 
 
 1 cubic inch Copper, 1000 fine = 4,62209 ounces troy 
 = $0,133. 
 
 Gold and Silver, when pure, are said to be 1000 fine ; 
 or, by the old method, 24 carats fine. 
 
 The standard fineness of the United States Coin is 
 900 ; or, by the old method, 24X,900 = 21.6 carats fine. 
 
 gunter's CHAIN, 
 
 7.92 inches =1 link. 
 100 links = 4 rods = 1 chain. 
 5280 feet =320 rods =80 chains = 1 mile. 
 69.77 statute miles = 1 degree of a great circle of the 
 earth. 
 
 160 square rods = 10 square chains =1 acre. 
 640 acres = 1 square mile. 
 
 FRENCH WEIGHTS AND MEASURES. 
 
 1 Metre =39.371 inches. 
 1 Are =3.953 square rods. 
 1 Litre = 61.028 cubic inches. 
 1 Stere =35,31714 cubic feet. 
 1 Gramme = 15,434 grains troy. 
 
122 QUARTZ OPERATOR'S 
 
 The Greek prefixes Deca, Hecto 9 Chilo, and Myria, 
 respectively, signify 10 times, 100 times, 1000 times, 
 and 10000 times. 
 
 And the Latin prefexes Deci, Centi, and Milli, re- 
 spectively, 10th part, 100th part, and 1000th part. 
 
 Thus, 1 Deca-metre = 10 metres, and 1 metre =10 
 Deci-meters. 
 
 Thus, 1 Chilo-gramme =1000 grammes, etc. 
 1 Arroba (Mexican) = 26 pounds avoirdupois. 
 1 Fanega = 1,599 bushels, U. S., =3438.52 cubic 
 inches. 
 
 1 Marc or Marco = 8 ounces troy. 
 1 Vara = 33,384 inches. 
 25 cubic feet of sand = 1 ton. 
 18 cubic feet of earth = 1 ton. 
 
 17 cubic feet of clay =1 ton. 
 
 13 -cubic feet of quartz, unbroken in lode, =1 ton. 
 
 18 cubic feet of gravel or earth, before digging, =27 
 cubic feet when dug. 
 
 20 cubic feet of quartz, broken, (of ordinary fineness) 
 = 1 ton, contract measurement. 
 
HAND BOOK. 123 
 
 Oakland, August 27, 1864. 
 
 Mr. Randall — Sir : — I have carefully examined your 
 demonstration of the Tractory Curve, and of the grind- 
 ing effects of differently formed plates, and find your 
 calculations correct. 
 
 Yours, etc., Francis D. Hodgson, 
 
 Prof. Math. College of California. 
 
 San Francisco, September 7, 1864. 
 
 Mr. M. P. Randall — Dear Sir : — I have to thank 
 you for the opportunity of inspecting the drawing and 
 model of your and Mr. Wheeler's new form of grinding 
 and amalgamating apparatus, in which you have adopted 
 the Tractory Conoid as the form of the grinding surfaces. 
 
 Your mathematical demonstration of the mechanical 
 properties of this curve is, so far as I am informed, 
 original and very interesting, and satisfies perfectly the 
 practical requirements of the problem. The Tractory 
 Conoid is a solid the nature of whose curve is as differ- 
 ent from that of the surface of a cone as is a cycloid from 
 an inclined plane. 
 
 Your mathematical analysis of the problem of uniform 
 grinding, by tractoroidal surfaces, is extremely interest- 
 ing, and furnishes a fine illustration of the value of this 
 method of discussion applied to a case which at first sight 
 
 f2 
 
124 QUARTZ OPERATOR'S 
 
 would seem to be completely beyond the reach of such 
 subtle and exact tests. The practical value of this dis- 
 cussion and of the results which it appears to sustain, are 
 such as commend it to the serious attention of all who 
 are interested in the development of the resources of 
 the Pacific coast in the precious metals. 
 Yours, respectfully, 
 
 B. SlLLIMAN, Jr. 
 
 San Francisco, May 4, 1865. 
 
 Gentlemen : — Having made a careful and critical 
 examination of your " Quartz Operator's Hand Book," 
 it is with extreme pleasure that I certify to the correct- 
 ness of your statements and deductions. It bears the 
 impress of extensive research and thorough investiga- 
 tion. Your discussions of all the various subjects are 
 remarkably clear, concise and rigidly exact ; but permit 
 me more especially to congratulate you upon your 
 masterly discussion jof the Tractory and the grinding 
 effects of differently formed plates — a subject practically 
 of the highest importance to every quartz miner. 
 
 With sentiments of high regard, I remain, 
 Yours, truly, 
 
 W. R. Eckart, Jr., 
 Engineer (late of) U. S. N. 
 
 To Messrs. Wheeler 6? Randall. 
 
HAND BOOK. 125 
 
 San Francisco, May 29, 1865. 
 
 Messrs. Wheeler Sf Randall : — Having examined your 
 " Quartz Operator's Hand Book," I take pleasure in 
 recommending it to miners and millmen, as a work likely 
 to be of great use in properly understanding the nature 
 of their ores, and consequently the treatment necessary 
 to produce favorable results. 
 
 Respectfully, yours, etc., 
 
 W. M. Belshaw, 
 
 Assayer, and Sup't, of the S. T. M. Co. 
 
 f3 
 
INDEX. 
 
 Assay, 13 
 
 Assay Blowpipe, 4 
 
 Assay of Copper — dry way, 14-1 5 
 
 Assay of Copper — humid way, 17 
 
 Assay of Gallena — dry way, 14 
 
 Assay of Gallena — humid way, 17 
 
 Assay of Gold — dry way, 16 
 
 Af say of Gold — humid way 18 
 
 Assay of Iron — dry way, 14 
 
 Assay of Iron Ores containing Manganese, 19 
 
 Assay of Silver — dry way 16 
 
 Assay of Silver — humid way 17 
 
 Assay or Analysis of Ores containing Gold, Silver, Copper, 
 
 Lead, Iron, and Sulphur, 20 
 
 Blowpipe, 4 
 
 Cupellation, 38 
 
 Comparison, Ill 
 
 Cement, Iron Rust, 22 
 
 Chemical Recipes, 84-35 
 
 Chemical Terms, Explanation of, 12 
 
 Discussion of the Tractory and differently formed Grinding 
 
 Plates, 92-1 16 
 
 f4 
 
128 
 
 Evolution, 87 
 
 Extraction of Gold by Chloration, 28 
 
 Extraction of Gold b j the Pan Process, 26 
 
 Extraction of Silver by the Pan Process 32 
 
 Extraction of Silver by the Freyberg Process, 31 
 
 Extraction of Silver by the Patio Process, 28 
 
 Extraction of Silver by the Yeach Process, 32 
 
 Extraction of Silver by the Augustin Process, 39 
 
 Extraction of Silver by the Ziervogel Process, 42 
 
 Extraction of Silver by the Patera Process, . . . * 43 
 
 Flux, Black, 21 
 
 Flux, White, 22 
 
 Involution, 87 
 
 Laws of Grinding, 102 
 
 Miscellaneous Table, 120 
 
 Mensuration, 69 
 
 Mechanics, and Mechanical Problems, 44 
 
 Mechanical Powers, viz. : 59 
 
 The Lever, 59 
 
 The Pulley, 6» 
 
 The Inclined Plane, 60 
 
 Mechanical Problems, 61-69 
 
 Plates, Grinding effects of, viz. : 66-69 
 
 Conical, 107 
 
 Plane Circular, 107-108 
 
 Randall's Patent, 112 
 
 Tractory Conoidal 106 
 
 Purification of Mercury, 25 
 
 Preface, 3 
 
 Properties of Bodies, Table of 117 
 
 Recipes — Chemicals employed in Silver Mining, 34 
 
 Recapitulation of the Grinding effects of differently formed 
 
 Plates, 115 
 
 Quicksilvering of Copper Plate, 23 
 
129 
 
 Roasting, viz. : 23 
 
 In Heaps, 23 
 
 In Furnaces, 23 
 
 In Reverberatory Furnaces, 24 
 
 Steam Power, 53 
 
 Solders, 22 
 
 Separation of Silver from Lead by the Pattinson Process, . 35 
 
 Separation of Silver from Lead by the Parke Process, .... 37 
 
 Separation of Silver from Lead by Cupellation, 38 
 
 Suspension Rods of Uniform Strength, 64 
 
 Separation of Silver from Copper by the Liquation Process 36 
 Table of Coefficients for estimating the Horse Power of 
 
 Water Wheels, 52 
 
 Table showing the Average efficiency of various Water 
 
 Wheels, 50 
 
 Table showing the proper velocity of Water Wheels, 51 
 
 Table of Pressures, Temperatures and Volumes, 54 
 
 Table for estimating the Mean Pressures of Steam for a 
 
 given Cut-off of Stroke, 55 
 
 Table, Tractory 100 
 
 Tests, Chemical '. 10 
 
 To find the Mean Pressure of Steam for a given Cut-off of 
 
 Stroke, 57 
 
 To find the effective Horse Power of a Non- Condensing 
 
 Engine, Rule, 58 
 
 To find the Horse Power for various Water Wheels, Rule. 52 
 
 Thin Cylinders, 61 
 
 Thick Hollow Cylinders, ,62 
 
 To find the Grinding effects of 
 
 Plane Circular Plates, Rule, 66 
 
 Conical Plates, Rule, 67 
 
 Tractory Conoidal, Rule, 68 
 
 Table of Regular Polygons, etc, 76 
 
 Table of Regular Polyedrons, 85 
 
 p5 
 
130 INDEX. 
 
 To find the Area of the Meridian Tractory Plane, etc 97 
 
 To find the Surface of Revolution of the Tractory Conoid. 99 
 
 To find the Solid Contents of the Tractory Conoid, 99 
 
 To construct the Tractory, etc., 101 
 
 To determine the Solidity of a Hollow Tractory Conoid,. . 105 
 
 To extract any Root of a Power or Quantity, 89 
 
 Varied Motion, 46 
 
 Water Power, 49 
 
 Water Pipes, 65 
 
 Weights and Measures, 121 
 
ERRATA. 
 
 17th Page, 26th line. — After the word "acid," add dilute and 
 
 filter. 
 28th page, 17th line. — For "chloride" read chlorine. 
 
 Add the word filled after "cisterns." 
 
 For "Jouval" read Jonval. 
 
 After the word "feet" add the head being 
 2 feet 3 inches. 
 
 For "Quality," read Quantity. 
 
 For "Catenary," read Involute of the 
 Catenary. 
 
 For "or AX," read on AX. 
 
 For "distruction," read destruction. 
 
 For *'uniie," read unit. 
 
 For "passes," read possess. 
 
 For "Conoidal," read Conoid. 
 
 28th " 
 
 19th 
 
 50th " 
 
 25th 
 
 53d " 
 
 2d 
 
 89th " 
 
 5th 
 
 93d " 
 
 13th 
 
 101st " 
 
 11th 
 
 102d " 
 
 23d 
 
 102d " 
 
 10th 
 
 104 th " 
 
 12th 
 
 106th " 
 
 9th 
 
 f6 
 
WHEELER'S 
 
 AMALGAMATOR. 
 
 ^^.TEnSTTEID DEO. 1803- 
 
 This favorite Amalgamator has recently been greatly 
 simplified and improved. 
 
 Over three hundred of these Machines are now in 
 successful operation, and giving entire satisfaction, in 
 California, Nevada, Mexico, Idaho, Colorado and Lower 
 California. 
 
 Further comments are unnecessary. 
 
 MANUFACTURED AT 
 
 MINERS* FOUNDRY San Francisco. 
 
 PACIFIC IRON WORKS 
 
 UNION IRON WORKS 
 
 GOLDEN STATE IRON W'KS . " 
 
 MARYS VILLE FOUNDRY Marysville 
 
 P. W. GATES & CO Chicago, III 
 
 WHEELER & RANDALL. 
 
 SAN FRANCISCO, June 13, 1865, 
 
THE 
 
 WHEELER & RANDALL 
 
 Grinder and Amalgamator. 
 
 In the engraving on the opposite page, A represents the Tiim of the 
 Pan; B, Cross Frame; C, Legs ; D, Gear ; G, Driving Pulley ; H, 
 Muller ; I, Driver ; K, Dies ; L, Shoes ; M, Hand Wheel ; N, Jam 
 Nut, and O, Wiugs or Guide Plates. 
 
 The attention of the Public is respectfully called to these 
 facts : 
 
 1st. That the Mechanical work accomplished by differently 
 formed grinding plates, having the same diameter, weight, hard- 
 ness, and revolving at the same velocity, is as follows, to wit : 
 
 The Mechanical Work of plane, circular plates of the usual 
 ring form, is ninety-eight 98 
 
 The Mechanical work of conical plates of the most approved 
 form, is one hundred and ten 110 
 
 The Mechanical work of Tractory-formed plates, as introduced 
 in the invention of Wheeler & Randall, is one hundred and 
 seventy-seven .177 
 
 That is, the Tractory-formed grinding plates will reduce one 
 hundred and seventy-seven tons of ore, the conical grinding plates 
 one hundred and ten tons, and the plane circular grinding plates 
 ninety-eight tons, to the same degree of fineness in the same time. 
 Those using this invention certify that they thoroughly reduce 
 five tons of ore, as it ordinarily comes from the wet battery, per 
 day in each pan, four feet diameter, the muller making sixty-five 
 revolutions per minute. 
 
 2d. That as a whole, the Wheeler & Randall Grinder and Amal- 
 gamator is one of the most simple, compact, substantial, conve- 
 nient and efficient pans in use. 
 
 K^ Patent applied for. 
 
 MANUFACTURED AT 
 
 GOLDEN STATE IRON WORKS San Francisco. 
 
 UNION IRON WORKS 
 
 MINERS' FOUNDRY " 
 
 SAN FRANCISCO FOUNDRY " 
 
 PRESCOTT & SCHEIDEL Marysville. 
 
 OREGON IRON WORKS Portland, Oregon. 
 
 WHEELEE & RANDALL, Inventors. 
 
 San Francisco, June 13, 1865. 
 
THE EXCELSIOR 
 
 mnm m mmmhm* 
 
 In the engraving; on the opposite page. A represents the Rim of the 
 Pan ; B, Muller ; C, Legs ; D, Cross-Frame; E, Gearing ; F, Screw ; 
 G, Lever ; H, Dash-Boards ; I, Key ; a, Dies ; c, Shoes, and o, 
 Openings, 
 
 » .+. , 
 
 The relative grinding capacities of "The Excelsior" Grinder 
 and Amalgamator, of the Flat Bottomed Pan, and of the Conical 
 Pan when properly constructed, are respectively 177, 98 and 110. 
 
 That is, the respective mailers being of the same diameter, 
 same weight, same hardness, and running at the same velocity, 
 "The Excelsior Grinder and Amalgamator" will reduce one hun- 
 dred and seventy-seven tons of ore, the Flat Bottomed Pan ninety- 
 eight tons, and the Conical Pan one hundred and ten tons to the 
 same degree of fineness in the same time. 
 
 The wear to the Shoes and Dies at their grinding surfaces in the 
 Excelsior Grinder and Amalgamator, is perfectly uniform, thus 
 securing evenness of reduction to the pulp, as well as steadiness 
 of motion to the muller. Uniform wear of the grinding plates 
 has been attained in no other than that of the Tractory form — nor 
 can it be. 
 
 Another property of excellence in this machine is that the metal 
 or substance to be amalgamated passes direct from the grinding 
 surfaces into the quicksilver ; thus excluding the possibility of its 
 becoming coated with any foreign substances, after having been 
 burnished. It is truthfully said "that the Tractory-formed Pan 
 as a Grinder has no equal, and as an Amalgamator no superior." 
 
 As a whole, it is far superior to any other pan in use. 
 
 MANUFACTURED AT THE 
 
 Union Iron Works and Golden State Iron Works, 
 WHEELER & RANDALL, Inventors. 
 
 The undersigned having had several years of experience in practical 
 quartz mining operations, will ever take great pleasure in furnishing 
 parties interested in mining and machinery any desired information 
 which they may possess. 
 
 WHEELER & RANDALL. 
 
 San Francisco, June 13, 1865. 
 
Q. W. PRESCOTT. CHAS. W. SCHEIDEL. 
 
 MARYSYILLE FOUNDRY 
 
 AND 
 
 3JCacliino Shop. 
 
 CORNER FOURTH AND B STREETS, 
 
 MARYSVILLE. 
 
 The above Establishment has been in successful operation for the 
 last twelve years. Having superior tools for manufacturing and 
 finishing, and greatly increased their STOCK OF PATTERNS, the 
 undersigned are now prepared to supply all demands for 
 
 MACHINERY AND CASTINGS, 
 
 3f every description, at as LOW RATES as any Foundry in the State. 
 
 STEAM ENGINES BUILT AND REPAIRED. 
 
 Quartz Mills, Saw Mills, Grist Mills, Threshing Machines 
 
 HORSE POWERS, 
 
 MALT ROLLERS, CAST IRON RIFFLES, AND 
 GEARING OF ALL KINDS. 
 
 We have Sixteen Steam Engines For Snle, of our own manu- 
 facture, from 8 to 100 horse power, and will be furnished with Boilers 
 and Fixtures complete. 
 
 AMALGAMATING MACHINERY, 
 
 Of every description, with all the Latest Improvements. 
 WHEELER & RANDALL PANS, 
 
 WHEELER PANS AND SEPARATORS, 
 AMALGAMATING TUBS, 
 
 Car wheels, Derricks, etc. etc. 
 
 mining :f>tt:m::e* s. 
 
 CORNISH PUMPS of all sizes, with Gearing and Pipes 
 made to order. 
 HOISTING MACHINERY, for Shafts or Inclines of every variety. 
 We are sole manufacturers for the State of California of 
 
 WINHAM'S HYDKAT7LIC COUPLING. 
 All orders promptly filled at the shortest notice, and at reduced 
 prices for cash. 
 
 PRESCOTT & SCIIEIDEC 
 
A. C. GIBBS JOHN NATION E. S MORGAN. 
 
 A. C. GIBBS & CO. 
 
 MANUFACTURERS OF 
 
 STEAM EN(HNES,B0ILER8 
 
 AND ALL KINDS OF MILLS. 
 
 * it * 
 
 ®M Separators 
 
 OF THE MOST APPROVED KINDS. 
 
 COOKING RANGES, PUMPS, ETC., ETC. 
 
 HP 5 * PATTERN MAKING in all its forms connected with the 
 Establishment. 
 
 03*- Plans and Specifications for Mills and all kinds of 
 Machinery furnished to order, by an experienced Draughtsman. 
 
 THE WHEELER & RANDALL 
 
 &riad$r tit) Amalgamator. 
 
GOLDEN STATE 
 
 Iron Works, 
 
 PALMER, KNOX & CO. Proprietors, 
 
 MANUFACTURE ALL KINDS OF 
 
 CASTINGS AND MACHINERY 
 
 Best Adapted to Mining and Milling. 
 
 QUARTZ MILLS, combining all improvements made to 
 date. 
 
 STEAM ENGINES of all sizes. 
 
 WHEELER & RANDALL'S GRINDING AND AMALGAMATING PANS. 
 
 KNOX'S AMALGAMATORS, with Palmer's Steam Chest. 
 HANSCOM'S CRUSHERS, designed to prepare Ore for 
 batteries. 
 
 TYLER'S PATENT IMPROVED WATER WHEEL. 
 DUNBAR'S PATENT PISTON PACKING. 
 
 A FULL SUPPLY of WHITE IRON CONSTANTLY ON HAND. 
 
 Parties desiring anything in our line, are invited to a personal 
 interview, or to communicate by letter. 
 
 Nos. 19, 21, 23, 25 &27 First Street, 
 
 SAN FRANCISCO, 
 
UNION IRON WORKS. 
 
 HENRY J. BOOTH. . .GEO. W. PRESCOTT. . .IRVING M. SCOTT 
 
 H. J. BOOTH & CO. 
 
 MANUFACTURERS OF 
 
 (L<d©©qctj©$w©j> S$atp5m©i> acn^l StftttofUMTjf dKg'S^os. 
 
 FLUE, TUBULAR, CORNISH, AND 
 
 MARINE BOILERS. 
 
 HOISTING MACHINES, 
 PUMPS AND PUMPING MACHINERY, 
 
 ALL KINDS OF 
 
 STAMPS and MORTARS, AMALGAMATING 
 PANS AND SEPARATORS, 
 
 Of the most Improved Patterns. 
 ALL KUN-IDS OF SOIRJBEISrS, 
 
 PATTERNS AND PATTERN MAKING, 
 
 OF ALL KINDS, CARRIED OX. 
 
 PLANS and SPECIFICATIONS for Mills and all kinds of 
 Machinery furnished free of cost. 
 
 H. J. BOOTH & CO. 
 
 FIRST STREET, Between Market and Mission, 
 SAN FRANCISCO. 
 
WHEELER & RANDALL'S 
 
 ZEN.AS WHEELER'S 
 Patent Self-Regulating 
 
 QUICKSILVER DISCHARGE APPARATUS. 
 
 The structure of this Machine is simple and sub- 
 stantial. Its practical workings complete, requiring but 
 little attention. 
 
 ...* Its superior settling properties are readily acknow- 
 ledged by the most intelligent and practical amalga- 
 mators and millsmen. 
 
 WHEELEE & RANDALL, 
 
 Inventors. 
 
fpimjj mi $ tkntxlu gta*; 
 
 PUBLISHED EVERY SATURDAY BY 
 
 PATENT AGENTS, 
 
 No. 505 Clay Street, corner of Sansome, 
 
 SAN FRANCISCO. 
 
 Each number of the Press contains sixteen pages of sixty-four 
 columns, (size of Harper's Weekly,) embracing- a variety of reading 
 of direct value to the progressive citizens of the Pacific Coast. Our 
 columns contain a liberal discussion of the different methods of 
 
 s.A/vi3sra- g-old, 
 
 All new discoveries in the treatment and development of precious 
 ores and metals, and illustrations of new machinery and inventions. 
 Commencement of Eleventh Volume, July 8th, 1865. 
 
 Terms, in Advance.— One year, $5 ; six months, $3. Back vol- 
 umes, to January 1, 1864, furnished at $3 per volume. 
 
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 Is conducted in a competent, thorough, reliable and strictly confiden- 
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 patents made out through our Agency, can sign their papers at once, 
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