TRANSCRIBER’S NOTE
Italic text is denoted by _underscores_.
Bold text is denoted by =equal signs=.
Underlined text is denoted by __double underscores__.
Overlined text is denoted by @at symbols@. Overlines from square and
cube root notations (²√ and ³√, respectively) have been omitted.
Footnote anchors are denoted by [number], and the footnotes have been
placed at the end of the chapter.
A superscript is denoted by ^x or ^{xx}, for example und^r or 36^{th}.
Some minor changes to the text are noted at the end of the book.
This edition of the book did not include a Table of Contents. For
the convenience of the reader one has been created here:
Preface.
Index.
Part I. Arithmetic, Geometry, Drawing, Etc.
Arithmetic.
Some Signs Employed in Calculations.
Powers and Roots.
Proportion.
Elements of Practical Geometry.
Geometrical Drawing.
Drawing Instruments.
Geometrical Drawings.
the Micrometrical Dividing Table.
Other Methods of Dividing Into Equal Parts.
to Subdivide a Circle.
Time.
Part II. Materials Employed in Horology.
Iron.
Cast Iron.
Steel.
General Observations.
Special Observations.
Determination of the Qualities of Steel.
Preparations of Steel.
Hardening.
Tempering.
to Whiten and Blue Steel.
Case-hardening.
Influence of Foreign Metals and Metalloids on the
Qualities of Iron and Steel.
Copper.
Zinc.
Brass.
Hammer Hardening of Brass.
to Anneal Brass.
Cast Brass.
Tin.
Bronze.
Sterro.
Lead.
Nickel.
German Silver.
Gold.
Silver.
Aluminium and Aluminium Bronze.
Mercury.
Platinum.
Palladium.
Characteristic Properties of Alloys.
Soldering.
Methods of Soldering.
Bronzing.
Gilding.
Acids and Salts.
Oil.
Alcohol.
Benzine, Etc.
Polishing Materials.
Preparation of Polishing Materials.
Smoothing.
Smoothing of Brass.
Smoothing of Steel.
Polishing.
to Polish Steel.
Cement, Wax, Resin, Etc.
Enamel.
Precious Stones.
Working in Precious Stones.
Part III. Health and Manipulation.
Preservation of Health.
the Sight.
the Body in General.
Use of the File and Graver.
to File Flat and Square With Both Hands at Once.
to File Flat With One Hand.
to Turn Cylindrical Pivots, Etc., and Square Shoulders.
Part IV. Tools and Appliances.
Workshop Fittings.
the Lathe.
the Foot Wheel.
the Bench.
Idlers.
Chucks.
the Slide Rest.
Gravers and Other Hand-turning Tools.
Drills.
Lathe Attachments.
Miscellaneous Small Tools.
Accessories and Miscellaneous Operations to Be Performed in the
Universal Head.
to Center an Object.
Uprighting and Drilling.
Production of Screw Threads. Screw Plates and Taps.
Taps.
Methods of Tapping Holes.
Rapid Mode of Making a Screw.
Screw-head Tools.
Tools For Cutting and Rounding-up the Teeth of Wheels.
Wheel-cutting Engine.
Cutters For Forming the Teeth of Brass Wheels.
Mill Cutters For Steel.
Tool For Making Cutters.
Tools For Correcting the Form of Teeth.
to Test the Accuracy of Certain Tools.
Part V. Repairing and Examining Watches.
Method.
External Examination of the Watch.
to Examine a Geneva Movement.
Accessories For Beginners.
Cleaning the Watch.
Putting the Watch Together.
to Examine English Or American Movements.
to Rapidly Time a Watch Or Clock.
Timing in Positions. Horizontal and Vertical.
Note on the Proportions of Balances.
Demagnetizing.
Part VI. Practical Recipes.
the Plate.
the Barrel. Including Arbor, Stopwork, Mainspring, Etc.
the Mainspring.
the Fusee.
Chain.
Wheels.
Pinions.
Set-hands Square.
Pivots.
Bushing Pivot Holes, Etc.
Depths.
on the Application of the Geometrical Laws of Depths to Practice.
Pallets.
Cylinder.
Balance Spring.
Dial Plate.
Metal Dials.
Hands.
Glasses.
Broaching.
Solid and Hollow Squares.
to Straighten a Rod, Plate Or Wheel.
American Watch Tool Co.
STONEY BATTER WORKS,
Chymistry District, WALTHAM, MASS.
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Watches, Clocks, Diamonds,
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THE
WATCHMAKERS’ HAND BOOK
INTENDED AS A WORKSHOP COMPANION FOR THOSE ENGAGED
IN WATCHMAKING AND ALLIED MECHANICAL ARTS
BY CLAUDIUS SAUNIER
ILLUSTRATED AMERICAN EDITION
REVISED AND ENLARGED BY
HENRY G. ABBOTT
WITH SPECIAL REFERENCE TO THE WANTS OF AMERICAN WORKMEN
CHICAGO:
GEO. K. HAZLITT & CO., PUBLISHERS.
1894.
COPYRIGHTED 1892, BY GEO. K. HAZLITT & CO.
PREFACE.
M. Saunier’s writings occupy an unique position in Europe, as works of
reference for all who are engaged in watchmaking, and it may not be out
of place to observe that the American edition of his treatise promises
to take a similar place among English-speaking communities.
A glance at the index will show that the information given is
essentially practical in its character, and such as will be of use to
the watchmaker in his daily work. The volume is thus in no sense an
abridgment of the original edition, but on the contrary, it contains
much more matter than the original work.
In recent years the work of the ordinary watch repairer has undergone
considerable change. The apprenticeship he serves, if indeed it can be
called a real apprenticeship, is shorter than formerly. The immense
number of badly constructed watches that he is called upon to put in
going order for a trifling remuneration, compels him to replace the
older methods of procedure by others, whenever by so doing time can
be saved. From this point of view, then, the value of the present
Hand Book can hardly be over-estimated, since it contains, in a
readily accessible form, many details as to the working of metals, and
descriptions of various practical operations, new and improved forms of
tools, etc.
But the volume will be found of daily use to a wider circle of workers
than those above referred to. We believe that there is hardly a
branch of the watchmaking trade which may not benefit by the numerous
practical details that are given; indeed, although the work is
specially designed for the use of watchmakers, a great portion of it
will be seen to be no less applicable to other mechanical arts.
M. Saunier’s original works, the _Guide-Manuel de l’Horloger_, and
_Recueil des Procedes Pratiques_, which appeared as separate volumes,
have been incorporated in this American edition, a large amount of
additional information being at the same time added. As the second
contained many details that more properly belonged to the first, and
_vice versa_, it has been thought desirable to remodel the whole, and,
as will be seen from an examination, the six parts that constitute the
work are distinct in their character.
With a view to further facilitate the use of this Hand Book as a work
of daily reference, an unusually full index has been added, which the
reader is recommended in all cases to consult, and no effort has been
spared to make the cross references in the body of the work as complete
as possible.
It seems desirable here to give some explanation in regard to several
of the practical methods described. Every watchmaker will at once
recognize that recipes are included which are of the nature of
makeshifts, and that it would in many cases be better to replace a
piece by a new one rather than to repair it in the manner indicated.
But, on the other hand, it has been felt that the work will often be
appealed to by those who, from various circumstances, may be prevented
from making or procuring a new part that will suit the watch under
repair, while those who are not so situated will have no difficulty in
deciding for themselves as to which method to adopt.
The original work of M. Saunier, and the English edition, both had the
illustrations in the back of the work, which necessitated considerable
inconvenience when consulting them. In this edition we have ventured
to insert the illustrations in the text that describes them. The
antiquated tools, like the bow, bow-lathe, the turns and mandril,
have been omitted, and the modern foot-wheel, lathe and accessories
substituted.
HENRY G. ABBOTT
INDEX.
_NOTE—The references are in all cases to the pages, not to the
paragraphs._
A
Accuracy of tools, to test, 366
” ” rules, to test, 33
Acids, 127
Adjusting rod, to use, 433
Agate, 164
Akerman on hardening, 77
Alcohol, 132
” and glycerine on oilstones, 192
Alloys, 57
” properties of, 108
” that melt at certain temperatures, 82
Alum, 128
Aluminium, 103, 110
” bronze, 104, 110
” solders, 113, 116
Amalgams, 105
American chucks, 224
Ammonium, chloride of, 128
Angle of cutting edges, 235
Angles of escapement, to measure, 465
” measurement of, 26
” to construct equal, 44
” to subdivide, 44
Annealing brass, 96
” steel, 72
Anvil, to harden face of, 88
Appliances for watchmakers, 189
Aqua Regia, 127
Arbor, barrel, to make, 409, 420
” ” ” repair, 428
” ” ” true, 429
” chuck for wheel-cutting, 224, 337
” to straighten, 505
” nut, to drill, 425
” ” diameter of, 425
” set-hands, to adjust, 451
” ” ” tighten, 294
Area, measurement of, 25
Arithmetic, importance of, 17
Arithmetical ratio, 22
Automatic blow-pipe, 201
Ayr stone, 135
Axis of crystallization, 166
B
Back rests, 253
Balance, pivots, form of, 400
” poise of, 401
” protector, 265
” spring stud index, 280
” to make plain, 436
” vibrations of, 460
Balances, proportions of, 401
Balance spring, action of, 473
” ” collet tool, 286, 474
” ” gauge, 475
” ” to center, 377
” ” ” clean, 384
” ” ” draw, 47
” ” ” flatten, 477
” ” ” select, 478
” ” ” weaken, 477
Balance-springs, palladium, 107
” ” platinum, 107
Balas ruby, 162
Bands, 219
Barrel-arbor, to make, 409, 420
” ” ” repair, 428
” ” ” true, 429
” cover, to make, 410
” going, to put together, 385
” hole to bush, 412
” hook, to fix, 411
” clock, to make, 420
” to examine, 379
” ” make, 409
” ” oil, 388
” ” repair, 379, 412
” ” true, 379, 413
” ” upright, 413
Bars, to make, 409
Baths for hardening, 78
Beaupuy burnishers and files, 195
Beat block, 266
Beginners, aids for, 381
Bell-metal, 98
Belts for lathes, 219
Bench, 189, 221
Bent pivot, to redress, 454
Benzine, 133, 384
Berlioz rounding-up cones, 362
Berzelius pastile, 498
Black polish on steel, 154
Bleaching silver dials, 487
Blow-pipes, 199
Blow-pipe, gas, 201
” automatic, 201
Blue polishing stone, 135
Blueing steel, 86
Board, arrangements of, 177, 189
“Body” of steel, 70
Borax as flux for soldering, 128
Bouchons, movable, 456
” to make, 298, 456
” ” rivet, 456
Brass, 90, 110
” cast, 86, 110
” dials, to silver, 491
” etc., drill blade for, 241
” influence of impurities on, 91
” plates, to harden, 92
” rods, to harden, 95
” to anneal, 96
” ” bronze, 121
” ” polish, 149
” ” select, 91
” ” smooth, 139
” wheel teeth, observations on cutting, 346
” wheels, to polish, 149, 135
Brazing, 119
Breguet spring, to set, 476
Broaches, 199
Broaching, 499
” a hole round, 499
Brocot method of hardening brass rods, 95
Broken screw extractor, 262
” ” to remove with alum, 128
Bronze, 98, 110
” aluminium, 103, 110
” to render malleable, 98
Bronzing, 120
Brush, to clean with, 383
Brushes, preparation of, 382
Buff leather, 133
Burnishers, 198
” Beaupuy, 195
” to re-face, 198
Burnishing brass wheels, 135, 149
Burnt iron, 58
Bushing barrel holes, 412
” pivot-holes, 456
C
Caliper, figure-of-8, 197
” for heights, etc., 197
” micrometer, 270
” pallet, 466
” rest, jeweling, 249
” vernier, 278
Camphorated oil, 131
Cannon pinion, to tighten, 449
Carbon in cast iron and steel, 60
Carnelian, 164
Case hardening, 88
Case, to examine, 389
Cast brass, 90
” iron, 60
” steel, 61
Castings, malleable, 61
Catgut bands, to join, 219
Cement, application of, 156
” steel, 62
Center of circle, to find, 44
Centers, to test truth of, 366
” various, 266
Center wheel, 379
” ” to oil, 388
Centering attachments, 253
” from circumference, 307
” in wax, 157, 314
” methods of, 305
” rods, 285, 385
Chain, fusee, to ease, 434
Chain hook, to rivet, 434
Chalk for cleaning, 382
Chamfering tools, 281, 283
Charcoal for polishing, 133
” used for smoothing, 133
Chinese bronzing, 121
Chloride of ammonium, 128
” of zinc as flux for soldering, 118
Chrysolite, 163
Chucks, 224
Circle, divisions of, 24
” sub-division of, 52
” to find center of, 44
Circular oil stones, 191
Circumference, ratio to diameter, 25
” to center from, 307
Cleaning brushes, 383
” files, 193
” gold or gilt objects, 125
” metal dials, 491
” nickel movements, 101
” rough steel, 75
” watches, 382
Clock barrel, to make, 420
” dials, to clean, 491
” hands, to blue, 86
” pinions, to polish, 443
” to time rapidly, 394
Closing barrel holes, etc., 295
Cocks, to make, 409
Colcothar of vitriol, 135
Cold gilding without mercury, 122
” hammering, 74, 92
Collet, balance spring, tool, 286
Coloring gold dials, 493
Compass, proportional, 37
Compasses, 196
Conductivity, 110, 112
Copper, 89
” alloys with gold, 102
” to bronze, 120
Cord of lathe, 219
Countershaft, 220
Counting vibrations, 394
Cover of barrel, groove for, 410
” ” ” to make, 410
Crocus, 135
Crossing out a wheel, 435
Crossings of wheel, to mark, 298
” to renew broken, 440
Crystallization, axis of, 166
Cube root, 20
Curb, to adjust, 415
Cutters, to adjust form of, 353
” for jewels, 165
” ” making grooves, etc., 312
” ” slide rest, 286
” ” steel wheels, 347
” ” teeth, various kinds, 339
” rose and star, 349
” rounding up, to make, 331
” tool for making wheels, 353
” to sharpen slide rest, 287
” wheel, to make, 339
Cutting a file, 194
” edge, angles of, 235
” faces of gravers, forms of, 239
” glass, 498
” wheels, observations on, 310, 348
Cylinder escape wheel, to test, 290
” to oil, 388
” ” pivot a, 454
” ” polish mechanically, 468
D
Dead smoothing, 143
” surface of gold, to restore, 126
Decantation, 136
Definitions, geometrical, 23
Degrees of circle defined, 24
Demagnetizing watches, 403
Density, 108
Depth caliper, 197
Depths, application of laws of, 458, 460
” in stem-wind work, 462
” theoretical and practical, 458, 460
” to ease, 365
” ” secure good, 457
” visible and invisible, to test, 377
Depthing tool, to test accuracy of, 368
Design, to transfer, 30
Dial, enamel, to cut hole in, 484
” ” ” drill, 480
” ” ” reduce diameter of, 486
” ” ” remove figure from, 486
” plate, to cut, 479
” to file hole in, 481
” ” remove enamel from back of, 482
Dials, enamel for, 160
” fixed by feet, 483
” ” ” screws, 482
” gold, to restore, 493
” metal, to clean, 491
” silver, to restore, 487
” to paint hours on, 495
” ” repair, 161
Diamond, 161
” drill, to use, 169
” ” and gravers, 162, 173
” for polishing, 162
” powder, drilling with, 168
” ” to prepare, 135
Diamantine for polishing, 134, 154
Diameter, ratio to circumference, 25
Dies for screw-making, 315
” ” wheel-making, 438
Dividing plate, 298
” scales, etc., 48
” table, micrometrical, 48
Dome, freedom of, 389
Douzieme gauge, 196
Draw-plate, 298
Drawing, geometrical, 28, 40
” instruments, 32
” scales, 33
” to reduce a, 40
Drawn steel, advantages of, 71
Drifting, 296, 415, 502
Drifts, to make, 296, 422
Drill, diamond, 162, 173
” rest, 267
Drilling a barrel-arbor, 423
” enamel dial, 480
” glass, 498
” in the lathe, 308
” lubricants for, 480
” precautions in, 242, 244
” tool, to test accuracy of, 366
” with diamond powder, 168
Drills for oil-cups, 281
” to make, 244
” ” mount diamond, 173
” various, 241
Duplex ruby-roller, to make, 173
E
Elasticity of steel, 70, 84
Electro-gilding, 124
Elements of practical geometry, 23
Elevation defined, 28
Ellipse, to draw, 46
Emery, 134
” paper, to make, 134
” wheels and sticks, 134
Enamel, 159
” false, 161
” to apply cold, 160
” ” remove from back of dial, 482
” dial, to cut hole in, 484
” ” ” drill, 480
” ” ” erase figure from, 486
” ” ” reduce diameter of, 486
” dials fixed by feet, 483
” ” ” by screws, 482
Endshake, 375
Endstone, to fix, 159
” ” make, 171
Engine, wheel-cutting, 330
English movement, to examine, 389
Engraved design, to transfer, 30
Epicycloidal depths, 459
Equaling file, to cut, 194
Erasing figure from enamel dial, 486
Escapements, action of, 376
” gauges for, 465
” play of pivots in, 452
” to examine, 391
” ” measure angles of, 465
” ” oil, 387
Escapewheel, clip for holding, 289
” to test, 290
” cock passage, to turn, 312
Essences for cleaning, 133, 384
Examining watches, 371, 389
Expansion, 109
Extracting broken screws, 128, 262, 317
” square root of numbers, 21
Eyesight, to preserve, 175
F
Face-plate, 257
” universal, 257
False ruby, 163
Figure-of-8 calliper, 197
File, to cut equaling, 194
” ” round-up teeth with, 363
” use of, 178, 193
Files, Beaupuy, 195
” to clean, 193
” ” renew, 193
” ” set in handles, 193
Filing block, 269
” fixture, 268
” flat, 179
” rest, 269
” square by hand, 179, 182, 500
Finger-piece, to make, 418
Flat filing, 179
” ” device for, 182
” polishing, tool for, 302
Fletcher furnaces, 201
Fluxes for soldering, 117
Foot-wheel, 218
Fourth wheel, uprighting of, 379
Fraise, Ingold, 362
Freedom of various parts, 378, 375, 391
Frosted surface on steel, to produce, 144
Frosting silver dial, 487
Furnace temperatures, table of, 78
Furnaces, wind and muffle, 201
Fusee, advantages of, 432
” to adjust, 433
” ” snail, 145
” watch, to examine, 389
Fusion, points of, 110, 112
G
Gauges, 196
” for balance-springs, 475
” ” escapements, 465
” movement, table of, 462
” registering, 274
” staff, 275
Geneva movement, to examine, 374
Geometrical drawing, 28
” proportion, 22
” ratio, 20
Geometry, elements of practical, 23
” importance of, 17
German silver, 100, 110
Gilding, 122
” bath, to prepare, 124
Gilt objects, to clean, 125
Glass, to cement, 157
” ” cut, 498
” ” drill, 498
” use of, 175
Glue, application of, 156
Glycerine and alcohol on oilstones, 191
Gold, 101, 110
” to clean, 126
” ” deaden surface, 126
” ” prepare in powder, 128
” copper alloys, 102
” ” dials, to restore, 493
” solders, 114, 116
” springs, 102, 478
Graduations of scales, etc., 49
Grain of steel, 68
Graining prior to gilding, 125
” steel, 144
Grammaire, 298
Gravers, diamond, 162
” to mount diamond, 162
” use of, 178
” various, 238
Grinder, traverse spindle, 258
Grindstone, 192
Groove, circular, to cut, 311
” for barrel-cover, to make, 410
” straight, to make, 312
” to turn, 312
H
Hairspring, _see_ Balance spring.
” stud index, 280
Hairspringing collet tool, 286
Hammer hardening, 68, 74, 92
Hand fitting pliers, 288
” holder, 287
” to enlarge hole in, 287
” ” set in position, 497
” turning tools, 238
Hands, freedom of, 373
” to blue, 86
” ” to redden, 497
” ” tighten, 287
Handles, to set tools in, 198
Hard steel, to drill, 243
” ” ” turn, 185
Hardening brass, 92
” gold spring, 478
” solutions, 77
” steel, methods of, 77
” ” precautions in, 79
” ” temperature for, 68, 81
Hardness of jewels, to test, 166
” ” steel, 82
” scale of, 111
Health, preservation of, 175
Heat, mode of applying, to temper, 84
” of furnaces, table of, 78
Homogeneity of steel, 68
Hone slates, 134
Holder, escape wheel, 289
” for watch-hand, 287
Holes, methods of tapping, 323
Hook, chain, to rivet, 484
” to fix in barrel, 411
Hooks for catgut bands, 219
Hooked gravers, 288
Hours, to paint on dials, 495
” wheel, freedom of, 374
Hydrochloric acid, 127
” ” flux for soldering, 118
Hydrofluoric acid, 127
Hydrogen in palladium, 107
I
Idlers, 222
Impurities in brass, their influence, 90
” ” steel ” ”, 88
Index, to ease, 478
Ingold fraise, 362
Inks for painting dials, 496
Instruments, drawing, 82
Involute depths, 458
Iridium, alloy with platinum, 107
Iron, cast, 60, 110
” for soldering, 118
” its properties and uses, 57
” oxides of, for polishing, 185
” to case harden, 88
” ” distinguish from steel, 57
” ” restore, 59
” wrought, 57, 110
J
Jewel in pallet, to advance, 463
” resetting tool, 299
” setting, cutters for, 137
” to find axis of, 166
” holes, thickness of, 400
” ” to make, 168
” ” ” set, 171, 298
” ” ” smooth and polish, 169
Jeweling caliper rest, 249
Jewelry, alloys used for, 102
” to clean after soldering, 126
Jewels, 161
” to select, 166
” working in, 165
K
Keyless pinion, to drift, 503
” winding depths, 392, 462
” ” wheel, to make, 441
” work, need of oil, 383
” ” to examine, 392
L
Lamps, 190
Laps for jewelers, 165
Lathe, 210
” attachments, 247
” belts, 219
” care of, 213
” simple form of, 211
” for jewel making, 165
” large size, 216
” to center rods in, 285, 305
” band, 219
Lead, 99, 110
” use of, in brass, 90
Leaves of pinions, to polish, 445
Leclerre’s vibration counter, 395
Left-handed screws, to make, 320
Lever escapement, to examine, 391
Lift, to measure, 465
Lines, to connect up, 45
” ” draw parallel, 43
” ” subdivide, 43
Liquation, 92
Local time, 54
Locking stones, to make, 172
Longitude, value in time, 55
Lubrication of whetstones, 191
M
Magnetism, to remove, 403
Mainspring, setting up, 385
” to blue, 86
” ” make eye, 431
” ” reduce height of, 432
” ” select, 432
Malleability, etc., of steel, 70
Malleable bronze, 98
” castings, 61
” nickel, 100
Materials used in horology, 57
Measurement, exact, 38
” of angles, 26
” ” triangles, squares, etc., 26
” ” various solid bodies 28
Melting points, 110, 112
Mercury, 105, 110
” to purify, 106
Metals, 57
” to cement, 156
Methylated spirit, 132
Micrometers, 196, 270
Micrometer screw, 39, 48
Micrometrical dividing table, 48
Mill cutters for steel teeth, 347
Milling fixture, 258
Minute wheel, freedom of, 374
” ” to cut, 310
Mixed oils, 131
Mortar for making diamond powder 138
Motion work, to examine, 374
Motive force, effect of variations in, 402
Movement, to examine English and American, 389
Movement, to examine Geneva, 374
Muffle furnace, 201
N
Natural steel, 62, 65
Nickel, 99, 110
” movements, 100
” to surface, 147
” ” render malleable, 100
” silver, 100, 110
Nitric acid, 127
O
Oil, 128
” application of, 387
” necessary in keyless work, 393
” retention of, at center pivots, 378, 386
” ” ” on acting surfaces, 132
” tests of quality of, 129
” to secure permanency in, 130
Oil-cup chamferers, 281
Oil-cups, observations on, 282
” to polish, 153
Oilsinks, 131
Oilstone, 135, 190
Oilstones, circular, 191
Oilstone dust, to prepare, 137
Oriental chrysolite, 163
” ruby, 162
” sapphire, 163
Oxides of iron for polishing, 135
P
Paillard’s balance-springs, 108
Painting dials, 495
Palladium, 107
” balance-springs, 108
Pallets, 463
Pallet-opening calliper, 466
” stone, to alter, 464
” ” ” cement, 159
” ” ” make, 172
” ” ” move, 463
Pallets, verge, to open or close, 467
” ” ” measure, 466
Paper used in cleaning watches, 385
Parallel lines, to draw, 42
Parallelogram, 27
Pendulum, counting vibrations of, 394
” spring, _see_ Balance-spring.
” vibrations of, 460
Permanency in oil, to secure, 130
Perpendicular, to erect, 41
Petroleum, use for cleaning, 384
Pewter, composition of, 99
Physical properties of alloys, 108
Pickling dials, 488
Pinion, cannon, to tighten, 449
” hollow, to replace pivot of, 453
” leaves, to polish, 445
” riveting tool, 294
” to drift keyless, 297
” ” increase or decrease, 444
Pinions, etc., cutters for, 349
” high and low numbered, 460
” sizes of, 443
” to make, 442
Pivot-holes, cleaning, 382
Pivot polisher, 250
” to bush, 456
Pivots, balance, length of, 377
” ” form of, 400
” center, 378
” play of, 378, 452
” replacing broken, 453
” to polish mechanically, 454
” ” redress bent, 454
” ” turn, 183
Pivoting a cylinder, 454
Plate, watch, to make, 407
Plates, brass, to harden, 92
” for screw-making, 315
” to straighten, 504
Plating, gold, 124
” silver dials, 491
Platinum, 106
” balance-springs, 108
” to solder, 131
Pliers, 196
” hand-fitting, 288
Poise of balance, 401
Polish, black, on steel, 154
Polisher, form of, 155
Polishing brass, 149
” circular grooves, 428
” cylinder lips, tool for, 468
” escape-wheel teeth, 153
” flat objects, 302
” jewel-holes, 168
” machine for pinions, 445
” materials, 133
” pivots in lathe, 454
” powders, to prepare, 136
” ratchet teeth, 427
” sinks and oil-cups, 153
” steel, 153
” stones, Cadot’s, 136
Positions, timing in, 399
Post, to adjust, 415
Powers of numbers, 20
Precious stones, 161
” ” working in, 164
Preservation of health, 175
Prime numbers, 18
Proportion of numbers, 22
” ” balances, 401
Proportional compass, 37
Protractor, 33
Puddled steel, 62
Pulleys, 222
Pumice-stone, 135
Punch, centering, 285
” riveting, 198
Putty powder, 135
R
Radius defined, 24
Railway time, 55
Ratchet, to renew barrel-arbor, 425
” ” smooth, 140
” ” snail, 141
” teeth, to cut in lathe, 310
” to polish, 427
Ratio, 25
Receipts for watchmakers, 407
Red-stuff, 135
Re-facing burnishers, 198
Registering gauge, 273
Regulating a clock, 398
” ” watch, 396
Renewing files, 193
Repairing watches, 371
Resetting jewels, 300
Resin as flux for soldering, 117
Resist for use in gilding, 125
Retention of oil on acting surfaces, 132
Riveting stake and punch, 198
Rods, brass, to harden, 95
” to straighten, 504
Roller remover, 264
” to make duplex, 173
Roots of numbers, 20
Rose cutters, 349, 357
Rottenstone, 135
Rouge, 135
Rough steel, to clean, 75
Roughing files, 193
Round hole, to broach, 499
Rounding-up attachment, 259
” ” cones, 362
” ” cutters, to make, 331
” ” teeth by hand, 363
” ” tools, 330, 359
Rubitine, 134
Ruby, 162
” false, 163
” roller, to make duplex, 173
Rust, to remove and to prevent, 58
S
Sal-amoniac as flux for soldering, 128
Salts, 128
Sapphire, 163
Sapphirine, 134
Scale, to remove, from steel, 75
Scales used in drawing, 33
Scratch-brushing, 125
Screw dies, 315
” head sinking tools, 260
” ” to smooth, polish and slit, 143, 301
” micrometer, 39
” plates, to make, 316
” ” and taps, 315
” rapid mode of making, 326
Screws, double and treble threaded, 381
” left-handed, 320
” to blue, 86
” ” make internal and external, 315
” ” extract broken, 128, 262, 317
” tray for, 381
Sealing-wax as cement, 155
Seconds circle, to fix, 484
” ” ” make hole for, 484
Sector, 36
Semi-cylindrical drill, 245
Series of holes, to drill, 309
Set-hands arbor, to adjust, 451, 294
” nut, to make, 449
” square, to make, 422, 449, 500
Setting jewel-holes, 171
” ” cutters for, 236
Set-square, to center with, 286
Sharpening cutters, 237
Shear steel, 62, 66
Shellac as cement, 155
Shoulders, cutters for making, 236
” gravers for turning, 239
” to polish square, 155
” ” turn square, 183
Sight, to preserve, 175
Signs used in calculations, 19
Silver, 103, 110
” dial, to restore, 487
” German or nickel, 100, 110
” solders, 114
” solution for dials, 490
” for plating, to apply, 492
” ” ” ” prepare, 491
Sinks in watch plate, 131, 281
” to polish, 153
Sinking tools, 281
Slates, hone, 134
Slide rest, 233
” cutters, 236
” ” to sharpen, 237
” turning with, 234
Sliding tongs, 196
Smoothing brass wheels, 149
” metallic surfaces, 139
Snailing, circular, 141
” steel, 145
” tool for, 146
Soap, application of, 383
Solders for various metals, 113
” hard and soft, 113
Soldering, 112
” fluid, 118
” fluxes used in, 117
” iron, 118
” methods of, 115
” to clean jewelry after, 126
Solid content, measurement of, 28
” geometry, 23
” squares, 500
Specific gravity, 110, 111
” heat, 110, 111
Spelter solders, 114
Spinel ruby, 162
Spiral, to draw, 47
Split chucks, 224
Spotting, circular, 141
” machine, 141
Spring, balance, 473
Springs, to blue, 86
Square hole, to drift a, 502
” measurement of, 26
” root, 21
” set hands, to make, 422, 449, 500
” shoulders, to turn, 183
” to file by hand, 500
” winding, to renew, 430
Squares, solid and hollow, 500
” to make, 422, 449, 500
Staff gauge, 275, 277
” to straighten, 505
Stake, riveting, 198
Staking tool, 293
Star cutters, 348, 357
” wheel, to true, 417
” ” sink, to make, 410
Steel, 61, 110
” characteristics of good, 63
” drawn, 70
” drill blade for, 242
” influence of impurities on, 88
” maximum elasticity of, 82
” precautions in hardening, 79
” preparation of, 72
” to anneal, 72
” ” braze, 119
” ” clean rough, 75
” ” determine qualities of, experimentally, 66
” ” distinguish from iron, 58
” ” polish, 153
” ” restore, 59
” ” snail, 145
” ” smooth, 143
” ” temper, 81
” ” turn hard, 185
” ” whiten, 84
” wheels, cutters for, 347
Stem wind pinion, to drift, 503
” ” wheels, to make, 441
Stepping device, 227
Sterro, 99, 110
Stone, pallet, to cement, 159
Stones for polishing, Cadot’s, 136
” precious, 161
” to make pallet, 172
Stool, 190
Stopping barrel holes, 412
” pivot-holes, 456
Stopwork, proportions of Geneva, 416
” to examine English, 389
” ” examine Geneva, 374
Straight groove, to make, 312
Straightening rods, etc, 504
Stud index, hairspring, 280
Studs, to remove, 291
Subdivision of angles, 44
” ” circle, 52
” ” lines, 43
Sulphuric acid, 127
Surfaces, smoothing of metallic, 139
Surfacing nickel movements, 147
Synchrometer, Guilmet’s, 398
T
Tailstock, 247
Tap, to increase diameter of, 323
Tapping holes, various methods of, 323
Taps for screw-making, 317
” left-handed screw, 320
Teeth, forms of, 365
” of escape wheels, to test, 290
” ” lever escape wheel, to polish, 153
” ” to cut, 330
” precautions in cutting, 341, 346
” steel, to polish, 427
” to renew in wheel, 439
” ” round up, 363
” ” true, 440
Temper, as indicated by blueing, 87
Temperatures, determined by melting alloys, 82
” determination of furnace, 82
” of furnace, approximate, 73
” of temper colors, 82
Tempering steel, 82
Tightening cannon pinions, 293, 449
” hands, 294
” set hands arbor, 294
Time, 54
” standard, 55
” to ascertain true, 54
Timing in positions, 399
” watches and clocks, 394
Tin, 97, 110
” solders, 114, 115
” use of, in brass, 90
Tinning surfaces, 119
Tool, form of, for turning, 238
” for balance spring collet, 286, 474
” ” closing barrel holes, etc., 293
” ” drifting, 296
” ” flat polishing, 301
” ” measuring verge pallets, 466
” ” polishing cylinder lips, 468
” ” ” pinion leaves, 445
” ” ” pivots, 454
” ” resetting jewels, 299
” ” riveting pinions, 293
” ” snailing, 146
” ” spotting, 141
” ” tightening cannon pinions, 293
” ” ” hands, 293
” ” ” set hands arbor, 293
Tools, ordinary small, 260
” need of good assortment, 187
” screw head, 329
” for hand-turning, 238
” ” making jewels, 165
” ” rounding up teeth, 359
” ” sinking screw heads, 260
” ” watchmakers, 189
” to set in handles, 193
” ” sharpen turning, 237
” ” test accuracy of, 366
” ” ” escape wheels, 290
Tool set, 237
Tongs, sliding, 196
Topaz, 164
Tourmaline plates, 167
Tracing, 30
” from dial, 495
Train, play of pivots of, 378
Trains of watches, usual, 394
Transferring, 30
Traverse spindle grinder, 258
Triangles, 26
Tripoli, 135
Trueing a barrel, 412
” ” ” arbor, 429
” ” star-wheel, 417
” ” wheel, 440
Truth of tools, to test, 366
Turning sphere, tool for, 241
” by hand, 238
” tools, forms of, 238
” ” to grind, 237
” velocity in, 235
” with either hand, 186
” ” slide-rest, 233
Turpentine, use of, in drilling, 480
” Venice, as flux for soldering, 117
Tweezers, 196
” for removing studs, 292
U
Universal face-plate, 256
” head, 256,303
Unlocking pallets, to make, 172
Uprighting a barrel, 413
” of center wheel, 378
” tool, to test accuracy of, 367
V
Velocity in turning, 235
Verge pallets, to measure, 466
” ” ” open or close, 467
” to straighten, 505
Vernier, 37
” caliper, 278
” gauge, 278
Vibrations of balance or pendulum, 460
” to count, 394
W
Watch-glass, to reduce, 499
” hand holder, 287
” ” to blue, 86
” ” ” tighten, 293
” plate, to make, 407
” to de-magnetize, 403
” ” time rapidly, 394
Watches, to clean, 382
” ” put together, 385
” ” repair and examine, 371, 389
” usual trains of, 394
Watchmakers’ bench, 221
” receipts, 407
Water, annealing steel in, 73
” of Ayr stone, 135
Watered surfaces on nickel, 147
” surfaces, to produce, 139
Wavy surfaces, to produce, 139
Wax chucks, 157, 230
” to set objects in, 157
Weight of balance, 401
Wheel, cutters, 258
” ” or punch, 438
” ” tool for making, 354
” ” to make, 340
” cutting engine, 330
” ” ” to divide a rule on, 50
” ” ” to polish in, 427
” foot, 218
” mode of holding in cutting, 337
” teeth, to cut in lathe, 336
” tooth, to renew, 439
” to divide, 333
” ” make steel, 441
” ” mark crossings of, 298
” ” rough out, 435
” ” straighten, 504
” ” true, 440
Wheels, arbors for cutting, 337
” identical, 438
” observations on cutting, 346
” steel, cutters for, 347
” to ease depths of, 365
” ” polish brass, 149
” ” repair, 439
Whetstones, 190
White smoothing, 143
Whiting, 136
Wind furnace, 201
Winding square, to make, 422
” ” ” renew, 430
Wire-drawing plate, 298
” gauges, 272
Work-bench, arrangement of, 189
Y
Yellow copper or brass, 90
Z
Zinc, 89
THE WATCHMAKERS’ HAND BOOK.
PART I.
ARITHMETIC, GEOMETRY, DRAWING, ETC.
ARITHMETIC.
=1.= We often hear the theory advanced that in this country, at the
present day, it is not necessary to have a knowledge of arithmetic,
geometry, drawing, etc., because our interchangeable system of
manufacturing watches, makes all knowledge in these lines superfluous,
and that without any knowledge of arithmetic or geometry a man may
become a thorough master of watchmaking. This is a mistake that too
many of our young men make. The fact that the leading watch factories
of the United States have adopted the interchangeable system, of course
lessens the number of parts which the repairer will have to make and
fit, but it by no means alters the situation as regards the repairing
of foreign-made watches, nor even the changing of American watches from
key to stem-winders. Without a thorough knowledge of arithmetic and,
at least, an insight into the principles of geometry, no young man can
hope to become a first-class watchmaker in the true sense of the word.
Without these accomplishments he will be deprived of the pleasure of
reading understandingly the best literature of the day, the works of
those who are best fitted to impart knowledge to the members of the
trade.
With a knowledge of geometry he will be able to comprehend the works
of the best authors, to ascertain the dimensions of solid bodies, and
be in a position to apply the rules that form the basis of linear
drawing. Every watchmaker, worthy of the name, should be able to make
and understand the drawing of any machine, or of any horological
instrument. Many inventors, and even ordinary workmen, would avoid a
large amount of hard work, often useless, and occupying much time, if,
instead of at once putting an idea into practice with brass and steel,
they were able as a preliminary, to make for themselves a correct
design drawn to scale.
=2.= It is taken for granted that the reader is familiar with the rules
of arithmetic at least, and we will touch upon some points in algebra
and geometry that it will be well to mention. Should the reader have
no knowledge of arithmetic, algebra and geometry, we would advise
him to take up these studies during his leisure hours, using some of
the standard text books for that purpose.[1] Besides possessing a
knowledge of prime numbers, numbers which have no divisors but unity
and themselves, the watchmaker should be able to determine the greatest
common measure of several numbers, a rule which is of great importance
in calculating a train of wheels that is complicated.
We shall confine our attention to the methods of extracting square
roots and proportions, the rules for which may have been forgotten,
owing to their being less frequently employed than the more common
rules of arithmetic; they are of frequent use in horology.
SOME SIGNS EMPLOYED IN CALCULATIONS.
=3.= The sign of addition is an erect cross, +, called _plus_, and when
placed between two quantities it indicates that the second is to be
added to the first. Thus, 5 + 3 equals 8.
The sign of subtraction is a short horizontal line, -, called _minus_,
and when placed between two quantities it indicates that the second is
to be subtracted from the first. Thus, 8 - 3 equals 5.
The double sign, ±, is sometimes written before a quantity to indicate
that in certain cases it is to be added and in others it is to be
subtracted. Thus 5 ± 3 is read 5 _plus or minus_ 3.
The sign of multiplication, ×, when placed between two quantities
indicates that the first is to be multiplied by the second. Thus, 3 × 5
equals 15.
The sign of division is a short horizontal line with a point above and
one below, ÷, and when placed between two numbers or quantities it
indicates that the first is to be divided by the second. Thus, 6 ÷ 2
equals 3.
The sign of equality is two short, horizontal, parallel lines, =,
representing the words equal to. Thus, 6 ÷ 2 = 3.
Inequality is denoted by the angle >, the opening always being toward
the larger number or quantity; thus, in 12 + 7 > 14, the sign >,
indicates that the sum of 12 and 7 is greater than 14, and the whole
expression is read, 12 plus 7 is greater than 14. The expression 9 < 4
+ 7 is read, 9 is less than 4 plus 7.
A parenthesis, (), denotes that the several numbers or quantities
included within it are to be considered together, and subjected to the
same operation. Thus, (10 + 4) × 3 indicates that both 10 and 4, or
their sum, are to be multiplied by 3.
A horizontal vinculum, ------, placed over the numbers or quantities,
is frequently used instead of the parenthesis. Thus, @2 + 4 + 6@ × 7 is
equivalent to (2 + 4 + 6) × 7.
Division is more usually indicated by a line between the two figures,
the dividend being written above and the divisor below the line. Thus,
¹⁶⁄₈ indicates division, the same as 16 ÷ 8.
Algebra is that branch of mathematics in which the operations are
indicated by signs or symbols, and the quantities are represented by
letters.
The sign of ratio consists of two points like the colon, :, placed
between the quantities compared. Thus, the ratio of _a_ to _b_ is
written _a_ : _b_.
The sign of proportion consists of a combination of the signs of ratio.
Thus,: :: :. The first two and last two dots are read _is to_, while
the four in the middle are read _as_. Thus, if _a_, _b_, _c_, and _d_,
are four quantities which are proportional to each other, we say _a_ is
to _b_ as _c_ is to _d_, and this is expressed by writing them thus:
_a_ : _b_ :: _c_ : _d_.
POWERS AND ROOTS.
=4.= The power of a number is the product formed by successive
multiplication of the same number by itself. Thus,
2 × 2 = 4, the second power or square of 2.
2 × 2 × 2 = 8, the third power or cube of 2.
2 × 2 × 2 × 2 = 16, the fourth power of 2, etc.
An exponent is a number written above a quantity, at the right-hand, to
indicate how many times the quantity is to be taken as a factor, as
6^3 = 6 × 6 × 6.
The root of a quantity is a factor which, multiplied by itself a
certain number of times, will produce the given quantity. Thus, in the
above examples 2 is the square root of 4 and the cube root of 8.
The radical sign, √, indicates that the root of the quantity placed
under it is to be taken, and the index of the root is expressed by a
little figure placed outside of the bend. If a square root the index
figure is usually omitted.
√4 = 2 or ²√4 = 2 and ³√8 = 2.
=5. Extracting the square root of whole numbers.= [2]
I. Point the given number off into periods of two figures each,
counting from the units place to the left. For example, we wish to find
the square root of 399424, we point it off thus: 39,94,24.
II. Find the greatest perfect square in the left-hand period, and write
its root for the first figure in the required root; subtract the square
of this figure from the first period, and to the remainder bring down
the next period for a dividend. Thus:
39,94,24(6
36
--
394
III. Double the root already found, and write the result on the left
for a divisor; find how many times this divisor is contained in the
dividend, exclusive of the right-hand figure, and place the result in
the root and at the right of the divisor. Thus:
39,94,24(63
36
--
123 394
IV. Multiply the divisor thus completed by the last figure of the root;
subtract the product from the dividend, and to the remainder bring down
the next period for a new dividend. Thus:
39,94,24(63
36
--
123 394
369
---
2524
V. Double the right-hand figure of the last complete divisor for a new
divisor, and continue the operation as before. Thus:
39,94,24(632
36
--
123 394
369
---
1262 2524
2524
PROPORTION.
=6.= It is often convenient to express the relations of qualities in
the form of a proportion and from the proportion derive an equation.
Ratio is the quotient of one number divided by another. Thus the ratio
of 30 to 6 is ³⁰⁄₆.
=7.= Proportion is the equality of ratios: Thus if ³⁰⁄₆ = 5 and ⁴⁰⁄₈ =
5 then we may state that ³⁰⁄₆ = ⁴⁰⁄₈ or the proportionality is usually
expressed (=3=) thus:
30 : 6 :: 40 : 8
and this constitutes what is called a geometrical proportion, and 30
and 8 are called the extremes and 6 and 40 the means.
=8.= The product of the extremes is always equal to the product of the
means. Thus: 30 × 8 = 6 × 40 = 240. Hence it follows that if we only
know three terms we can always determine the fourth, or unknown term,
which is usually represented by the letter _x_. Thus in the proportion
12 : 3 :: 16 : _x_
we find the product of the means, or 3 × 16 = 48; this product divided
by 12, the known extreme, gives us the value of _x_, or the unknown
extreme, as equalling 4.
=9.= If we know the two extremes and only one of the means the same
rule is applied. Thus in the proportion
20 : 5 :: _x_ : 25,
we have: 20 × 25 = 500. ⁵⁰⁰⁄₅ = 100, the value of _x_.
ELEMENTS OF PRACTICAL GEOMETRY.
=10.= The object of geometry is to measure the extent of bodies. A body
has three dimensions, _length_, _breadth_ and _thickness_, and one of
these latter is sometimes termed _weight_ or _depth_.
Either dimension taken by itself is measured by a straight line.
When the extent of a body is expressed by combining any two dimensions,
it is termed _area_ or _surface_, and when three are employed we obtain
the solid measure or _volume_.
_Plane_ geometry only takes cognizance of figures situated in one plane
or surface, and therefore only possessing two dimensions; _solid_
geometry, however, regards bodies as having all three dimensions.
Two lines are parallel when their distance apart is the same at all
points. The same is also applicable to parallel planes.
Two lines or planes meeting each other will form an _angle_. The point
at which they meet or intersect is termed the _apex_ or _summit_ of the
angle.
A straight line is _perpendicular_ or _at right angles_ to another
straight line, or to a plane, when all the angles which it makes with
that line or plane are equal.
A _circumference_ of a circle is a curved line _l c d f_ (fig. 1,)
such that all its points are equally distant from an internal point,
_o_, termed the _center_. The _circle_ is the space enclosed by the
circumference.
It will be noticed that in geometry these two words are distinguished,
although they are frequently referred to as identical. Thus, the rim of
a wheel or balance is generally termed a circle.
Two circles (_l c d f_ and _b r a_, fig. 1,) described from the same
center are said to be _concentric_. When their centers do not coincide
they are called _excentric_ with regard to one another.
Any portion of a circumference, such as _f n d_, is termed an _arc_ of
the circumference, or, more commonly an _arc of a circle_.
A _chord_ is a straight line, _f d_, which unites the two extremities
of an arc. When the chord passes through the center of a circle it is
termed a _diameter_.
The _radius_ of a circle or circumference is a straight line drawn from
the center to the circumference; and all the radii that can be thus
drawn are equal. A diameter is, then, always double the radius, and,
conversely, the radius, is always half the diameter.
A _tangent_ is a straight line that only touches a circumference at one
point, as _g l_ (fig. 1); whereas a _secant_ cuts the circle, as _i j_.
A circumference is assumed to be divided into 360 equal parts, termed
_degrees_. The degree is subdivided into sixty equal parts, or
_minutes_, and the minute into 60 _seconds_. These are respectively
symbolized by the marks ° ′ ″ placed at the right-hand top corner of
the figure.
Such an expression as 18° 30′ 15.5″ would, then, be read 18 _degrees_,
30 _minutes_, and 15.5 _seconds_.
=11.= =Ratio of the circumference to the diameter.= The diameter of
a circle is to the circumference as 7: 22; or, employing decimal
fractions, as 1: 3.14159 (a number which, in algebra, is always
represented by the Greek letter π.)
[Illustration: _Fig. 1._]
Knowing a diameter (D), the circumference, _x_, can be ascertained from
the proportion:—
1 : 3.14159 :: D : _x_.
Knowing a circumference (C), the diameter, _x_, can be determined from
the proportion:—
3.14159 : 1 :: C : _x_.
The latter proportion will also give the value of the radius, which is
half a diameter.
=12.= The _superficial area of a circle_ is equal to the circumference
multiplied by half the radius, or to the square of the radius
multiplied by 3.14159.
A _sector_ is the circle enclosed between an arc and two radii bounding
it, as _b_ O _r k_ (fig. 1.)
The area or surface of a sector can be ascertained by multiplying the
length of the arc by half the radius.
A _segment_ of a circle is the portion intercepted between an arc and
its chord, as _f d n_ (fig. 1.)
The surface of a segment, as _b k r s_, can be obtained by subtracting
from the area of the sector O _b k r_, the area of the triangle, _b r_
O (=15=).
=13.= _Ring._ To determine the surface of a flat ring, the area of the
inner circle must be subtracted from that of the outer circle; in other
words, take the difference between the areas of the two circles that
fix the inner and outer diameters of the ring.
The area of a flat ring can also be calculated by adding together the
internal and external diameters; then multiply the number so obtained
by their difference and by the decimal fraction 0.7854 (that is,
3.1415/4). The product will be the required area.
=14.= _Angles and their measurement._ When two lines meet one another,
they form an angle, as we have already seen. If we take the apex as the
center of a circle, the number of degrees intercepted between the two
straight lines gives a measure of this angle.
The angle measured by a quarter of a circumference, or 90°, is termed a
_right angle_.
An _obtuse_ angle is greater than a right angle, and an angle that is
less is termed an _acute_ angle.
=15.= =Triangles, squares, etc., and their measurement.= The triangle
or plane area enclosed within three straight lines joined two and two
together (A, B, C, fig. 2,) is said to be _rectangular_ when one of its
angles is a right angle; it is _equilateral_ when the three sides are
equal, under which circumstances the three angles are also equal; and
_isosceles_ when only two sides are of equal length.
The sum of the three angles of a triangle is always equal to two
right angles. If only two of the angles are known, it is thus easy to
determine the third.
_Similar_ triangles are characterized by the fact that their homologous
sides (that is, the sides opposite to equal angles) are proportional.
_Peculiarity of the right-angled triangle._ The square described on the
longest side, termed the _hypothenuse_ (B, fig. 2,) is equal to the
sum of the squares described on the two other sides. Hence, it follows
that, if the lengths of the two shorter sides are known, that of the
hypothenuse can be ascertained by extracting the square root of the
number formed by adding together the squares formed on these two sides
(=5=).
[Illustration: _Fig. 2._]
If the hypothenuse is known and one of the shorter sides, the third
can be determined by extracting the square root of the number formed
by subtracting the square of the known side from the square of the
hypothenuse.
The _surface of a triangle_ is determined by multiplying one of the
sides by half the perpendicular height of the angle opposite to this
side.
=16.= The _surface of a square_ or of an oblong or _rectangle_ (_a b
c d_, fig. 3) is equal to the product of the base multiplied by the
height.
The sum of the squares described on the four sides is equal to twice
the square described on a diagonal. This diagonal divides the rectangle
into two equal rectangular triangles.
=17.= The _surface of a parallelogram or lozenge_, a plane figure with
four sides, opposite pairs of which are parallel (_c f g d_ and _c i
j d_, fig. 3,) is equal to the product of one side multiplied by the
perpendicular height of the figure.
[Illustration: _Fig. 3._]
The sum of the squares described on the four sides of a parallelogram
is equal to the sum of the squares described on the two diagonals.
=18.= =Measures of various solid bodies.= The volume of a cube of
parallelopiped (that is, a body bounded by six four-sided figures,
every opposite two of which are parallel) is obtained by multiplying
the surface of the base by the height.
The _volume_ of a straight cylinder is the product of the surface of
the circle which forms its base into the height of the cylinder.
The _area_ of the curved surface of a cylinder is obtained by
multiplying the circumference of the circle forming its base by the
height.
The _volume_ of a tube or cylindrical ring of rectangular section, such
as the arbor-nut of a barrel, or the rim of a circular balance, etc.,
is equal to the product of the plane surface of its base (=13=) into
its height.
The _volume_ of a _right cone_ or of a _regular pyramid_ is the product
of the base into a third of the height.
The _surface_ of a _sphere_ may be determined by multiplying the square
of the diameter by 3.1416 (=11=).
The _volume_ of a sphere is equal to this surface multiplied by a third
of the radius.
GEOMETRICAL DRAWING.
=19.= An elementary knowledge of the art of drawing, an ability to
represent the outlines of objects by simple lines, is of the first
importance to the watchmaker.
Such a design is obtained by projecting on to one plane all the visible
points of the object represented.
Projection on a vertical plane gives an _elevation_; the object is
looked at from one side.
Projection on to a horizontal plane produces a _plan_; the object is
observed from above, thus giving a bird’s-eye view.
The projection of a point on a vertical or horizontal plane is the foot
of the perpendicular, from the given point on to the plane. Assume the
line _n m_ (fig. 4), to be fixed in space; its horizontal projection
will give _c d_, and its vertical projection, _r s_.
[Illustration: _Fig. 4._]
_Miscellaneous details._ When one portion of the object to be
represented is found to pass behind other pieces so that it cannot be
seen, the continuation is frequently indicated by dotted lines.
Surfaces that are situated in planes one behind the other are shaded,
the more deeply according as they are farther back. This shading
is produced by a number of parallel lines which may be vertical or
horizontal.
Parts that are in relief are indicated by projected shadows, or by
increasing the thickness of a line that would cast a shadow.
In order to distinguish the several shadings or to emphasize the lines
by which they are separated, it is a very usual, though not invariable
practice to assume the light to be coming from the left-hand upper
corner.
When drawing a square in relief, such as _a b c d_ (fig. 3), the lines
_c d b d_, will be made darkest; but if it is a recess, the lines _a
b_, _a c_ should be brought into prominence by means of dark lines.
[Illustration: _Fig. 5._]
These several directions will be found useful when a hole, any cavity,
a pin, a round object etc., has to be depicted, as in fig. 5. As a
general rule, the thick lines should indicate the position at which a
shadow would form, the light being assumed to fall on the drawing in
the manner indicated above.
A _section_ shows a body as it would appear if cut in two, and one
portion removed in order to expose the interior, as in fig. 6. A
section is indicated by a series of parallel lines drawn close together
and at an inclination of about 45° to the vertical.
[Illustration: _Fig. 6._]
In order to leave more room for important details, or to show objects
that are situated behind, a piece is often broken off by an irregular
line, as shown in that drawing.
Lines formed by a series of detached points sometimes serve as a means
of associating several figures representing the same object looked as
in different directions.
=20.= =Tracing and transfering.= These two operations are resorted to
when it is required to obtain one or more copies of a picture or design
already drawn.
Tracing consists in laying a piece of tissue or other translucent paper
over the drawing and copying it by following over the lines that are
visible with a pencil. Or ordinary paper can be used for the purpose,
providing it is not too thick, if the picture be placed against the
pane of a window or, what is more convenient, on a sheet of glass used
as a desk and illuminated from below. When either sheet of paper is too
thick to allow sufficient light to pass, one or other of the methods of
transfering indicated below must be resorted to.
=21.= This operation consists in reproducing a tracing on a separate
sheet of paper or on metal that is to be engraved. Either of the
following methods may be adopted:
1. The picture to be transferred is fixed to a table or drawing board
if tracing paper is to be used, or to a sheet of glass if only ordinary
paper is available. The lines are then traced with a black lead pencil
that must not be too hard. When this is finished it is laid, face
downwards, on a sheet of white paper, taking care that both sheets are
so fixed that they shall not slip. Apply pressure to the upper surface
by tapping with a small pad made on purpose and, at the same time,
gently rubbing. Experience will very soon show how hard the pad should
be. Now remove the tracing, still taking care to avoid any slipping,
and a faint reproduction of the design will be found on the lower sheet
of paper. It is only necessary to follow over the lines with India ink.
The figures will be reversed but a transfer with it in the original
direction may be obtained by inverting the picture and laying it on
glass so as to make a reverse tracing.
2. Lay the picture on a desk or drawing board and trace it with ink
on a very transparent sheet of paper. When the ink is dry, invert the
tracing and blacken the back with a No. 2 pencil. Now lay the tracing,
with the ink side uppermost, on a sheet of clean paper, taking care to
avoid slipping, and go over the several lines with an agate or metal
style, avoiding excessive pressure on account of the risk of tearing
the paper. On removing the upper sheet an impression will be found not
reversed. Go over all the lines with India ink and clean the paper with
India-rubber or stale bread.
_Observations._ The choice of paper and pencil is not a matter of
indifference. All kinds of paper do not receive an impression equally
well, neither do all pencils transfer with equal facility. The Faber
pencil No. 3, will generally be found best suited to such work.
In preparing drawings which you desire to preserve, as drawings of
escapements, etc., a good quality of light weight bristol board will
be found more desirable than the best drawing papers. White wedding
bristol, about two-ply in thickness, answers admirably, and India ink
lines drawn upon it will not spread as they often do on drawing papers.
The prepared liquid India inks now on the market are superior to any
you can prepare by grinding the sticks.
=22.= =To transfer an engraved design.= This method is available when
it is desired to obtain an impression, for example, of the engraved
surface of a watch case.
Procure some of the inks used by copper-plate engravers, or, in its
absence, ordinary stencil ink may be used. Taking a small quantity on
the end of the finger, tap it on the surface of a glass plate, in order
that the ink may be distributed, leaving only a small quantity evenly
spread over the finger: tap with the finger thus prepared over the
watch case long enough to make sure that all the surface in relief has
received some ink; take a piece of writing paper and, after slightly
moistening it, spread it over this surface. Lay above this a piece of
paper folded in four and pass over it in all directions any round body,
such as a small tool handle, and with some pressure; then raise the
papers without allowing them to slide.
If the operation has been carefully performed a very clear impression
will thus be obtained of the engraved surface. The relief will be black
and the hollows white, but, of course, the figure is reversed like
that in a looking-glass. If required in the right direction it must be
traced through to the other side of the paper.
DRAWING INSTRUMENTS.
=23.= It is needless here to describe the rule, set-square, =T=-square,
bow-compass, etc., as every one knows them.
_To verify the accuracy of a rule._ On a perfectly flat smooth surface
carefully draw, with the rule in question, a fine straight line. Then
turn the rule over, hinging it as it were on the line just drawn; if
quite straight the edge of the rule will exactly coincide with the
line, in this new position, throughout its entire length. Each edge
should be thus examined.
_To verify the accuracy of a set-square._ Having fixed an accurate
rule on a smooth surface, place one edge of the set-square against it,
and draw a line along the edge perpendicular to the rule; then, having
turned the set-square, hinging it on the line just drawn, bring it
against the rule and along the line. If the square is true the edge and
line will coincide throughout their length.
=24.= =The Protractor.= Fig. 7 represents a common form of this
instrument. It is made of horn, or, if of metal, the inner portion is
cut away, leaving only a base and a semicircular arc, which is divided
into 180 equal parts or degrees; a complete circle would therefore
consist of 360 such degree. The point indicating the center of the arc,
should be very small in order to facilitate the exact setting of it at
the apex of an angle.
[Illustration: _Fig. 7._]
When an angle has to be drawn with accuracy, the protractor is
unsuitable; it will be better to adopt one of the methods described at
paragraph =37=, or trigonometrical methods.
=25.= =Drawing scales.= When an object is represented by a drawing, if
the dimensions are the same as those of the object itself, or, rather,
as they would project on to a horizontal or vertical plane, the drawing
is said to be full size; but the object is generally represented either
on an increased or diminished scale, which is defined, the proportions
between all the parts being still, however, maintained the same.
With a view to avoid the many calculations that such a change would
involve, it is usual to employ drawing scales. The following notes will
sufficiently explain their construction and use.
Let it be required to reproduce a large drawing on a small scale, in
such a proportion that the dimensions are reduced in the ratio of 10 to
1.
[Illustration: _Fig. 8._]
Take a straight line of indefinite length, _a b_, fig. 8, and mark out
on it spaces equal to _a_ 1, which represents any measurement taken on
the original object; at _c_, the 10th division, draw a perpendicular,
and on it measure _c g_ equal to _a_ 1, or one-tenth of _a c_ and join
_a g_.
Through the points indicating the divisions into tenths draw lines
parallel to _c g_, and you will thus have a series of triangles, _d
a_ 1, _d′ a_ 2, _d″ a_ 3, etc., similar to the triangle _g a c_. In
virtue of a well-known property of such triangles (=15=), _d_ 1 will be
one-tenth of _a_ 1; _d′_ 2 one-tenth of _a_ 2; and so on.
Thus, if a measurement taken on the object, or on a large drawing, is
equal to _a x_, it will only be needful to turn the compass on the
point _x_ as a center, and to observe accurately the perpendicular
height, _x z_, to ascertain the corresponding measurement on the
reduced scale.
Such a scale can be employed to measure meters and decimeters, or feet
and inches (but in this latter case, it would have been necessary to
mark off 12 instead of 10 divisions from _a_). Since _a_ 1 might be
made to represent one metre; _a_ 2, two meters, etc., in virtue of
the principle of the triangle already referred to, _d_ 1 will be the
tenth of _a_ 1, and will therefore represent a decimeter; _d′_ 2 will
represent 2 decimeters, etc. The length required to represent, say 5.3
will be ascertained by taking the distance _a_ 5, to which the distance
_d″_ 3 is added. Similarly 6 feet 2 inches would be given by _a_ 6, to
which _d′_ 2 is added on a 12-division scale.
=26.= The following description of one of these decimal scales, which
is engraved on metal or ivory, and often included in cases of drawing
instruments, will suffice to enable any one to construct a scale on
this principle, that goes to a still further degree of accuracy,
measuring, for example, meters, decimeters and millimeters, or yards,
feet and inches.
[Illustration: _Fig. 9._]
Let A B, fig. 9, be a flat rectangular rule, divided throughout its
length by parallel equidistant lines into ten strips. At right angles
to these are the lines _o o′_, _a a′_, etc., separated from one another
by a distance of one centimeter (doubled in the drawing in order to
make the details more clear.) The first centimeter is subdivided along
the two edges, A _o_, _n′ o_, into 10 equal parts or millimeters, and
the division, _o_, on the upper edge is joined by an oblique line with
the 1 on the lower edge, and the others by parallel oblique lines as
shown in the figure. Thus _c i_ will be one-tenth of a millimeter, _s
j_ two-tenths, and so on.
If the compass is opened so as to reach from _x_ to _z_, it will
be seen that it covers a space of 16 millimeters and 2-10ths of a
millimeter, for there are one large division (or 10 mm.), 6 smaller
divisions (or millimeters) plus a fraction of a millimeter equal to _s
j_ or 2-10ths of a millimeter.
=27.= =Sector.= When it is required to reduce the scale of a drawing,
subject to the condition that the dimensions shall be all diminished in
the ratio of two given lines, we may state the problem thus:
The longer of the two given lines is to the shorter, as any given
dimensions of the old drawing is to _x_. The value of _x_ thus
determined will be the corresponding dimension of the new figure.
Such a rule of three proposition would involve a considerable amount of
work, and the required result can be arrived at with greater facility
by the geometrical methods which forms the basis of the scale just
described, or, better still, by using the sector shown in fig. 10. It
consists of two brass or ivory legs hinged about a center _m_ which is
at the apex of the angle _n m c_ formed by two straight lines similarly
divided into equal parts.
[Illustration: _Fig. 10._]
It is employed as follows: Let us assume that a drawing has to be
reduced in the ratio of the line A to the line B; set off the length
A along _m n_, and suppose its extremity to be at _s_, where division
number 5 occurs. Open a compass to a distance equal to B, and placing
one point on _s_, open the two legs of the scale until the second point
coincides exactly with the corresponding division _t_, that is, with
the 5 on the other leg, _m c_. Maintaining the scale open to this
amount, it is only needful, after measuring a distance on the original
drawing or object, to set it off along _m n_, and to measure the
distance between its extremity and the corresponding point on the other
leg; this distance will be the dimension on the reduced scale.
=28.= =Proportional compass.= This consists of two equal stems
terminating with points, fig. 11. They are cut through for a portion of
their length, and provided with a slide forming a hinge, that can be
clamped by a screw _a_ in any position. Graduations on the two slots
and a mark on the slide indicate in what position of the slide _a_,
the length _a b_ (equal to _a g_) is equal to ½, ⅛, ¼, etc., of _a d_;
and thus show what is the ratio of _g b_ to _c d_, a ratio which is
independent of the extent to which the arms are opened.
[Illustration: _Fig. 11._]
=29.= =The vernier.= The vernier consists of a small graduated slide
which is adapted to a graduated rule or circular arc with a view to
ascertain the value of small fractional parts of the divisions marked
on the rule or arc.
Let A B, in fig. 12, be a rule divided into millimeters (the
proportions are enlarged in the drawing so as to avoid confusion among
the lines), and let it be required to determine a length to within the
tenth of a millimeter.
As the measurement is required to the tenth, take ten less one or nine
of the divisions of the scale; they will extend from O to IX, and this
represents the acting length of the vernier.
Subdivide the vernier into ten equal parts; it is manifest that each
graduation of the vernier differs from the original subdivisions of
the rule by 1-10th of a graduation of the latter. In other words, unity
on the vernier is equal to 9-10ths of unity on the rule.
[Illustration: _Fig. 12._]
When the rule and vernier are placed as shown in fig. 12, so that the o
on both scales coincide, the successive divisions on the rule (marked
with Roman numerals for distinction) will be progressively more and
more in advance of the corresponding divisions on the vernier in the
following proportion:—
The marks I and 1 are 1-10th apart; the marks II and 2, 2-10ths; III
and 3, 3-10ths; and so on, the mark X being 10-10ths, or one complete
division in advance of 10, this division being a unit on the scale.
Thus if the vernier is caused to slide along the edge of the rule, when
1 coincides with I the vernier has advanced 1-10th; when 2 coincides
with II, it has advanced 2-10ths; and so on.
Let it be required to determine the distance P _d_, fig. 13. The
division 6 on the vernier coincides with a division of the scale; hence
it follows that the extremity _d_ of the vernier is at a distance of
6-10ths millimeters from III, the next division of the scale to the
left. The distance between P and _d_ is thus 3.6 millimeters.
[Illustration: _Fig. 13._]
With a vernier showing tenths, if two consecutive divisions of the
vernier fall between two divisions on the rule, and there does not
appear to be a tendency towards one side more than towards another,
even when observed with a strong glass, it is possible to take an
approximate reading to the twentieth.
[Illustration: _Fig. 14._]
In measuring circular arcs a curved vernier is used in place of a
straight one, and its graduations are made to correspond with those on
the circle as shown in fig. 14.
=30.= =Micrometer screw.= By employing a micrometer screw it is
possible to measure infinitesimal amounts, but the screw must be
perfectly accurate, and must work without appreciable backlash or loss
of time.
[Illustration: _Fig. 15._]
Assume V, fig. 15, to be such a screw, having a pitch of 1 millimeter.
It will advance by this amount with each complete rotation.
To the head of the screw is attached a disc of such a size that its
rim can be divided into a number of equal parts, say a hundred. These
graduations may be marks on the edge or notches cut in it when an index
is required to stop in them; but the index is less frequently met with
than a simple divided straight-edge almost in contact with the disc.
The divisions round the disc are numbered in ascending order as the
points _c_ and _a_ separate, so that zero comes under the index or rule
when these points are in contact. Readings of the numbers will thus
afford a measure of the displacement of the point of the screw.
When the disc is rotated the point a will move towards or from _c_
by 1-100th of a millimeter for each division passing under the
straight-edge, and one millimeter for each complete rotation. It is
thus possible to obtain the dimensions of an object when it enters
without play between the two jaws to within an error of about 1-100th
of a millimeter if the instrument is accurately made.
If, instead of passing the object between the two jaws, it is gripped
by them, the measurement will be less exact, as no account is taken of
the pressure exerted and of the elasticity. (=44.=)
GEOMETRICAL DRAWINGS.
=31.= =Sketches.= It is advisable from an early age to accustom oneself
to make rapid freehand sketches of objects as they present themselves
to the eye. Such a sketch will help in the preparation of a more exact
drawing, which involves a knowledge of the several geometrical methods
given below.
A drawing may be transferred, reduced or enlarged as follows:
Draw across the original picture a number of equidistant vertical and
horizontal lines, forming perfect squares, and number the two sets of
lines in succession. Then draw a similar series of lines on a clean
sheet of paper, setting the lines at an equal, less or greater distance
apart, and copy in succession the parts of the figure that are enclosed
within the several squares.
As it is not always possible to draw lines across a figure, they may
be replaced by a frame carrying fine threads or wires stretched in the
two directions. The frame is laid over the original drawing, which can
then be copied, as above explained, on a sheet of paper divided into
squares (fig. 16).
[Illustration: _Fig. 16._]
The frame may, moreover, afford assistance in the drawing of solid
objects. Having placed it above or in front of the object and
in contact with it, copy on to the sectional paper the contents
of each corresponding square, taking care to look at the object
perpendicularly. With a little practice, and by placing the eye in
the correct position and always at the same distance from the frame
(a distance which may be regulated by a glass), a sketch in fair
proportion may easily be obtained.
[Illustration: _Fig. 17._]
=32.= =To erect a perpendicular on a straight line.= Either the compass
or a set-square can be employed; the use of the latter instrument is so
simple that no further reference need be made to it. Assume _a_, fig.
17, to be the point in the line _n m_ at which a perpendicular is to be
drawn. On either side of _a_ measure off equal distances _a n_, _a m_;
from _n_ and _m_, with a radius about equal to the distance _n m_, draw
two circular arcs cutting one another. If their point of intersection
_b_ be joined to _a_, the line _a b_ will be the required perpendicular.
[Illustration: _Fig. 18._]
=33.= =To erect a perpendicular at the extremity of a line.= From the
extremity _c_, fig. 18, mark off four equal parts towards _s_. From
_s_, with a radius equal to five such parts, describe a circular arc,
and from _c_, with a radius of three such parts, describe another
arc cutting the first at _d_. The line joining _c_ and _d_ will be
perpendicular to _s c_.
For the square of 5 is 25, and this is equal to the square of 4 or
16 plus the square of 3 or 9. Thus the triangle _s c d_ must be
right-angled (=15=).
[Illustration: _Fig. 19._]
Or the following method may be adopted: With any center _i_ and radius
_i g_ (fig. 19), as large as possible, describe a circumference passing
through _g_. From the point _p_, where the circle cuts the line, draw
the diameter _p i h_. If the point _h_ be joined to _g_, it is the
required perpendicular; for, by a property of the semicircle, the angle
_h g p_ is a right angle.
[Illustration: _Fig. 20._]
=34.= =To let fall a perpendicular on a straight line.= In order to
let fall a perpendicular from the point _a_ on to the line _b c_ (fig.
20), describe from _a_ as a center, a circular arc sufficiently large,
cutting the straight line in two points, _b_ and _c_. From these two
points, with the same opening of the compass, draw on the under side of
the line two arcs that intersect. The point of intersection _o_ joined
to _a_ gives the required perpendicular.
[Illustration: _Fig. 21._]
=35.= =To draw parallel straight lines.= Having fixed a good
straight-edge over the drawing, as many parallel lines as are required
may be drawn with the aid of a set-square which is caused to slide
along the rule. They will be vertical, horizontal or inclined,
according to the position of the rule, which must be set exactly
perpendicular to the direction in which the parallel lines are to be
drawn (fig. 21).
[Illustration: _Fig. 22._]
_To draw, from a given point, a line parallel to a given line._ Let _d_
be the given point, and _a b_ the given line (fig. 22). From _d_ draw
the circular arc _a c_, and from _a_ where it cuts _a b_, with the same
radius describe the arc _d b_. From _a_ set off on _a c_, a distance
equal to _d b_. The line joining _d_ and _c_ is the required parallel.
[Illustration: _Fig. 23._]
=36.= =To subdivide a line into equal parts.= Let it be required to
divide the line _p v_ (fig. 23), into five equal parts. Draw a line _p
q_ inclined at any angle, and mark off on this line five equal parts of
any length; join _q_, the extremity of the five lengths, and _v_, and
through the points _a_, _b_, _c_, _d_, draw lines parallel to _q v_. In
virtue of a property of similar triangles these lines will divide _p v_
into equal parts. It is advisable that the lines _p v_ and _p q_ should
not differ very considerably in length, as, otherwise the inclination
of the parallel lines to _p q_ will render it difficult to observe the
exact point of intersection.
[Illustration: _Fig. 24._]
_To divide a line into proportional parts._ The proposition can be
solved in a similar manner. Let it be required to divide a line _t r_
(fig. 24), into two sections that are to one another in the proportion
of 5 to 3. On _t s_ mark off a series of equal parts given by the
addition of these numbers together, that is 8; and join the last point
_s_ to _r_. Then draw through _c_, the fifth division, a line parallel
to _s r_. This line _c d_ will cut _t r_ into two parts, which are to
one another in the proportion of 5 to 3.
By an analogous construction a fourth proportional can be graphically
obtained, as already indicated in articles =25=-=27=.
=37.= =To construct an angle equal to a given angle.= The angle may be
measured by means of the protractor (=24=), which then enables us to
draw a similar angle; but greater accuracy is obtainable by using the
compass.
[Illustration: _Fig. 25._]
Let it be required to construct at _m_ on the line _m p_ (fig. 25), an
angle equal to _b a d_. With as large a radius as possible, draw from
the points _a_ and _m_ the arcs _b d_ and _n p_. Measure the distance
_d b_ and mark it off with the compass from _p_ on the arc _p n_. A
line drawn through _m_ to the intersection of the two arcs will give
the required angle equal to _b a d_.
=38.= =To subdivide an angle into 2, 4, or 8 equal parts.= In addition
to the use of the protractor, the following graphic method is often
given in works on geometry.
[Illustration: _Fig. 26._]
An angle _e f g_ being given (fig. 26), from its apex _f_ as a center
describe the arc _e g_, and from its two points of intersection with
the sides, with a radius greater than half their distance apart, draw
two short arcs cutting each other at _s_. A line drawn from _f_ through
the intersection _s_ will divide the angle into two equal parts.
If four divisions are needed, repeat the process on the two angles _s f
e_, _s f g_, and so on for a further sub-division.
The line that divides the angle into two equal parts will also bisect
or divide into two equal parts the chord and the arc _e g_.
[Illustration: _Fig. 27._]
=39.= =To find the center of a circle or of a circular arc.= Take on
the circumference, or on the arc, three points _b c r_ (fig. 27).
Join _b_ to _c_ and _c_ to _r_. At the middle point of each of these
lines[3] erect a perpendicular. The point of intersection of these
perpendiculars is the required center.
A similar method should be resorted to when it is desired to describe a
circle passing through three given points.
[Illustration: _Fig. 28._]
=40.= =To connect up or associate lines.= In order to join up a
straight line, such as _i j_ (fig. 28), with the curve _l p_, erect a
perpendicular at _j_, and through the middle point of a chord, _l p_,
draw a second perpendicular cutting the first in _k_. This point will
be the center from which the curve uniting the two lines should be
struck.
[Illustration: _Fig. 29._]
To unite a curve _a b_ (fig. 29), with another curve, _c x_ or _c
z_, at the point _c_, first find _o_, the center of the curve _a b_,
draw the line _a o_, continuing it beyond the center; join _a_ and
_c_, and erect a perpendicular at the middle point of this chord. The
intersection of this perpendicular with _a o_, produced if necessary,
should be taken as the center for a curve uniting _b a_ with _c_.
[Illustration: _Fig. 30._]
To join up two lines inclined to each other or parallel lines of
unequal length, such as _a r_, _b s_, fig. 30, draw midway between the
two another line, _z d_; join the two extremities _r_ and _s_, and from
these points let fall perpendiculars _r i_ and _s c_; then from _d_
draw a line perpendicular to _s r_. The point _o_ thus obtained will be
the center of the arc _r d_, and _c_ will be the center for _d s_.
=41.= =To describe an ellipse.= Let _a b_ (fig. 31), be the major axis
of the ellipse; divide it into three equal parts, and from the two
points, _c_ and _i_, at which it is divided, with a radius equal to _i
c_, draw (in pencil) two circles, intersecting in the points _x_ and
_z_. Through these points draw the lines _x i g_, _x c h_, _z i f_, _z
c d_.
With the center _z_ describe the arc _d f_, and from _x_ draw _h g_;
the ellipse will be completed by the two arcs, _f b g_, _d a h_, of the
primitive circles.
[Illustration: _Fig. 31._]
If it be required to describe an ellipse that shall have a shorter
minor axis, divide the major axis into four equal parts, thus obtaining
three points of sub-division. With each point as a center and with a
radius equal to one of the spaces describe circles. Those to the right
and left will determine the extremities of the ellipse, and the central
circle will intersect the minor axis in two points which must be taken
as centers for describing the top and bottom portions of the figure.
[Illustration: _Fig. 32._]
When the length of the long and short axes are given, proceed as
follows (fig. 32): From the center _a_, where they intersect at right
angles, mark off the distances _a n_, _a o_, equal to the difference in
the length of two semi-axes. Join _n o_, and add one half of _n o_ to
_a o_ measured in the direction of _a v_, thus obtaining the point _k_;
with the radius _a k_ describe a circle. On this circumference will lie
the four centers; _k_ for the arc _r u s_, _m_ for the arc _p v q_, _t_
and _i_ for the short arcs _q j s_, _p e r_.
The figures obtained by the methods here given closely resemble the
ellipse, but are not of the strict mathematical form. It is well
to acquire some facility in drawing ellipses, for the projection
of a circle on a plane, when the two are neither parallel nor
perpendicular, is an ellipse, and one often has occasion to describe it.
=42.= The following may be added as a mode of describing an ellipse:
The major axis and the two foci (points in this axis) being known, fix
two pins in these foci. Then tie a piece of string into a loop and
place it over the pins; stretch it with a pencil, the point of which
is on the paper, and on moving this around in a circular direction,
the string being maintained stretched, an ellipse will be described.
When the string is so stretched that it lies along the major axis, the
length should be such that the pencil is exactly at its end.
[Illustration: _Fig. 33._]
=43.= =To draw a spiral curve.= Draw four lines forming a small square
(fig. 33). The point _o_ is taken as the center of the first arc, _i
j_; _s_ is the center of _j k_; _u_ of _k l_; _i_ of _l n_. Then,
to continue the curve, _o_ is again taken as the center for _n p_,
and so on. This method produces a volute in which the coils are at a
considerable distance apart, such as has no special applicability to
horology.
[Illustration: _Fig. 34._]
As the balance-spring of a watch is partially concealed by other
pieces, it is generally sufficient to represent the parts that show
themselves by concentric circular arcs, or arcs described from two
centers. If a more accurate representation be required, the following
method may be resorted to: when working on a small scale it involves
the use of the eyeglass, for the figure (fig. 34,) here given is
exaggerated in order to avoid confusion in the lines, numbers, letters,
etc.
A small circle having been described, it is divided into an even
number of equal parts, say four; a less number than this should never
be adopted. From the same center describe another circle as small as
possible, which will be cut by the two diameters drawn between opposite
points of division numbered 1, 2, 3, 4.
Assuming _a_ to represent the starting-point of the curve, from the
center 1 with radius 1 _a_ draw the arc _a b_; from 2 with radius 2 _b_
draw the arc _b c_; from 3 with 3 _c_ draw _c d_; from 4 with 4 _d_
draw _d s_; then recommencing with 1 and the radius 1 _s_ draw _s f_,
and so on.
The less the radius of the small circle and the greater its number of
divisions, the closer will the successive coils be together. To secure
accuracy when working on a small scale, it is advisable that the center
and the several points be in a thin brass or horn plate, which is
maintained in position by steady pins.
THE MICROMETRICAL DIVIDING TABLE.
=44.= This instrument is no more than a simple application of the screw
to dividing straight lines, but it will suffice to enable the reader to
understand the principles on which the more complicated instruments are
based.
[Illustration: _Fig. 35._]
A plate, P, fig. 35, supports a bracket _a_, in which a screw, similar
to the one described in paragraph =30=, is engaged by means of a
collet; it rotates, being supported between this bracket and the small
bearing _b_, that receives the pivot at the end of the screw.
The screw is fitted carefully into a nut _n_, which is rigidly attached
to the small plate _h_; this carries a fine marker, movable on an axis,
and terminating with a chisel edge or a fine diamond point, according
as the instrument is to be used for engraving metal or glass; or it
may be provided with a fine pencil if the object is merely to make
subdivisions on a drawing.
This being understood, it will be evident that, if a rule or rod of
any form be fixed by screws or otherwise between _f_ and _g_, it can
be graduated by means of the marker, the screw being made to advance;
the millimeter screw can be used for dividing into millimeters and
fractions direct, or, with a little calculation, into fractions of an
inch. Each complete rotation of the head means a displacement of the
marker by a millimeter; a half turn will be half a millimeter, etc.
OTHER METHODS OF DIVIDING INTO EQUAL PARTS.
[Illustration: _Fig. 36._]
=45.= =First method.= Having fixed a sheet of drawing paper on a smooth
board, draw the line M N, fig. 36, longer than the rule which is
required to be divided. Then, with a compass or graduated scale, mark
off a series of equidistant points, commencing at N, equal in number
to the required series on the rule, and let M be the last division.
With the center N and radius N M describe the circular arc _p v_, and
with M as a center and the same radius, describe a second arc _r s_,
intersecting the first at O. Join O with M and N. Assume _a c_ to be
the rule that is to be divided into equal parts; slide it on the paper
parallel to M N until the extremities, _a_ and _c_, coincide with the
lines O M, O N, and are equidistant from O. This position can be easily
found by the aid of a compass with one of its centers at O. Now fix
the rule in position with sealing-wax, or by some other means, and,
with a firm upright pin, center the brass rule R at O, so that it can
rotate round this center on the pin as a pivot. It now only remains to
trace a series of lines O _k_, O _b_, O _d_, etc., with the rule, to
the division points of the line M N. The line _a c_ is thus divided
into as many equal parts as the line M N. The graduations will be all
the more exact according as the divisions of the line M N are longer.
=46.= =Second method.= By the side of the chuck of a wheel-cutting
engine, arrange a horizontal slide _y f_ fig. 37, that can travel
easily in a direction perpendicular to _a_ T. A watch fusee-chain, or
a very flexible spring, is fixed by one end to the chuck, and by the
other to the slide at _d_. The chain or spring is kept stretched by a
weight which tends to draw the slide from _f_ towards _d_.
[Illustration: _Fig. 37._]
The rule to be graduated, _a_, is now fixed on the slide, and an
initial division is marked on it with a pointed rotating cutter in the
position usually occupied by the wheel cutter, or else by striking a
small pointed or flat-edged chisel arranged for the purpose, in such a
manner as not to be liable to derangement.
Rotate the table through a definite distance; the rule _a_ will advance
through the same distance; mark the second division; then having moved
the division-plate through a distance equal to its first displacement,
mark the third graduation, and so on. Suppose, for example, that it be
required to make 30 divisions on the rule between _f_ and _d_; select
on the plate the circle corresponding to twice or thrice this amount,
so that the radius of the chuck may not be relatively too short, and
that the chain or spring may not act at a disadvantage; take the number
60 for example:
The two marks at _d_ and _g_ on the spring indicate the length that
corresponds to the straight line to be divided.
The chuck is placed in the lathe and reduced in diameter until the half
circumference is exactly equal to the distance between these two marks
on the spring, which thus fall on a diameter, _i g_, of the chuck.
The spring having been fixed by its two extremities, the slide with the
rule attached is placed in position, so that the mark _g_ is on the
line _a_ T; it will be evident from the figure that each displacement
of the division-plate through one-sixtieth of its circumference will
cause _a_ to advance through one-thirtieth of the space between _f_ and
_d_.
_Remarks._—Knowing the relation of a diameter to the circumference (as
1 : 3.1416), we can determine the diameter of the chuck at once by
calculation.
Its form should be a true cylinder, and it is well to place guides that
will prevent the spring or chain from assuming a helical position.
The side that carries the rule should be strictly perpendicular to _a_
T; and the portion of the spring that is not coiled on the chuck should
always be parallel to this slide.
The chuck and spring must be quite clean and smooth, and the latter
should be very pliable. A greater weight will be needed to keep
the spring stretched than will suffice for a chain, and it must be
increased as the strength of springs is greater.
The slide, _y f_, may simply travel over a horizontal surface between
pins planted in two parallel lines. But it would be preferable to adopt
some other method, for instance, to make this piece (F, fig. 37),
travel with a little friction along a perfectly true cylindrical rod.
=47.= =Third method.= This is merely an application of the arrangement
mentioned in paragraph =44=. The lathe can be employed for marking
off a series of equidistant points in a straight line. Knowing the
pitch of the slide-rest screw, determine the distance apart in, say,
millimetres, of the required divisions, and fix the rule perfectly flat
on the face-plate, which must be rendered immovable by any convenient
means. Then mark the first point with the drill-stock. Advance the
screw by the amount previously determined upon and mark the second
point. After withdrawing the drill, again advance by the same amount
and mark the third point, etc. Always be careful, before making the
first mark, that the screw has already traveled some distance in the
direction it will continue to move, so as to avoid backlash, or loss of
time.
TO SUBDIVIDE A CIRCLE.
=48.= =To divide the circumference into equal parts.= After having
drawn the circle, A, fig. 38, draw two diameters, _d a_, _b c_, at
right angles to each other, dividing the circle into four equal parts.
Join the points, _c a_, and divide the line, _c a_, accurately into
nine equal parts.
Draw a series of circles concentric with the first, at distances apart
equal to one of the divisions of _c a_, and to the number of one, two,
three, etc., according as it is required to subdivide the circle, say
for a pinion, into seven, eight, nine, etc., equal parts.
[Illustration: _Fig. 38._]
With a fine-pointed compass, measure off the radius of the initial
circle A. Placing one point of the compass at _c_, the other point
will give the position of the next leaf, and so on, all around the
circumference. If the innermost circle A be selected for sub-division,
six divisions will be obtained, and there will be one more division for
each larger circle.
The operation will be facilitated by selecting the first circle, so
that the line _a c_ contains exactly nine divisions equal to those of
some scale that is accessible. Such a circle can be easily found, by
first drawing the two diameters, laying the scale in the direction _c
a_, and determining by trial the radius for which the first and ninth
divisions correspond to _a_ and _c_ respectively.
=49.= =To divide a surface into rings of equal or proportional
superficial area.= The following solution is due to M. Brocot:
Let _a d_ be the radius of a circle (fig 39), that is required to
be subdivided into four rings of equal area by concentric circles.
Taking _a d_ as a diameter, draw the semicircumference, _a b d_;
accurately divide _a d_ into four equal parts, and at each point so
obtained, erect a perpendicular. Through the intersections of these
perpendiculars with the semicircle, draw a series of concentric
circles; they will trace out rings, 1, 2, 3, 4, that have equal
superficial areas.
If it be required to divide the surface in a given proportion, divide
the line _a d_, according to that proportion.
[Illustration: _Fig. 39._]
The right-hand side of fig. 39 gives a special application of this
method to the division into two equal areas of the interior of a barrel
exclusive of the space occupied by the arbor-nut. If the mainspring
accurately covers _i c_ when wound up, and _i j_ when unwound, it will
give the greatest possible number of turns.
TIME.
=50.= Solar time is taken from the revolutions of the earth, and the
watchmaker can easily get the exact solar time of any point at which
he may happen to be by a little calculation from known standards.
These standards are: 1. The zenith. 2. The longitude of the point of
observation. 3. The difference between noon at the point of observation
and noon of a known meridian either east or west of the point of
observation. The zenith is that point in the heavens where the rays
of the sun are in a plane exactly perpendicular to the surface of the
earth at the point of observation, and when the rays of sunlight are
in this plane it is noon at that point. The circumference of the earth
is divided into 360 degrees or meridians of longitude, so that as the
earth revolves once every twenty-four hours, each of these meridians
will pass the zenith, or fixed point, in that time. In twenty-four
hours there are 24 × 60 = 1,440 minutes, so that the interval between
the passage of one meridian and the next will be 1,440 ÷ 360 = 4
minutes. A degree of longitude is divided like an hour, into minutes
and seconds, so that
1 degree of longitude equals 4 minutes of time.
1 minute ” ” ” 4 seconds ” ”
1 second ” ” ” ¹⁄₁₅ or .066 seconds of time.
=51.= Thus it happens that, at a town one degree east of a given
point the sun will be visible four minutes sooner, and if to the
west, four minutes later than at that point. The “local,” or solar
time, therefore, will be four minutes earlier at the first town, and
four minutes later at the second town, than it is at the point of
observation.
=52.= It will be readily seen that, having any two of the three factors
given above, the other can be readily found. Thus having the time of
a given meridian and the local noon or meridian time, the longitude
can be readily found; or, having the longitude (which can be readily
obtained from a surveyor) and the time of a given meridian, “noon,”
can be calculated, etc. The first method is followed in calculating
distances at sea; the chronometer keeping Greenwich time, and the local
noon giving the longitude.
When great accuracy is necessary, however, a fixed star is used as a
means of observing the exact time when a revolution of the earth is
completed, as the revolution of the sun in its orbit causes a slight
variation during the year. For further information on this point the
reader is referred to works on astronomy.
To obviate the constantly varying time in running east or west, the
railroads use the time of a given meridian over each fifteen degrees
of longitude, and as each degree of longitude equals four minutes of
time, it follows that only the hour is changed in changing from one
standard to another. In Europe the zero of longitude, or the time of
the meridian of Greenwich is used. In the United States the time of
the 75th degree, which passes through Philadelphia, is used from the
67th to the 80th degree, which comprises the territory from Princeton,
Maine, to a line drawn north and south, passing through Erie and
Pittsburg, Pa., and is called Eastern time. The time of the 90th
meridian is used from the 80th to the 102d meridian, and is called
Central time. The time of the 105th meridian is used from the 102d to
the 114th meridian, and is called Mountain time. The time of the 120th
meridian is used from the 114th meridian to the coast (which ends at
about the 124th meridian) and is called Pacific time. The time of the
various standards is telegraphed through their various territories at
noon each day, and furnishes an accurate standard of comparison to all
watchmakers.
=53.= In very many cities the actual or solar noon has been discarded
and the railway standard adopted, thus making but one standard and
removing the source of confusion and annoyance to many people. In
others, however, the two standards are still used, and it becomes
necessary for the watchmaker to be able to calculate both standards,
in case of accident or irregularity in his regulator. Hence he should
calculate his longitude within one second by means of the difference
between railroad and local noon, and have the nearest surveyor correct
his reckoning; then, by means of the accurate longitude and the
railroad time, correct the solar time; then by means of the solar
noon and the longitude calculate the railroad time. When all these
calculations check each other perfectly, he possesses all the time data
he needs for that place, and can correct his standard or regulator
if at any time it should become irregular. The calculations are very
simple, and can be easily performed from the data given above.
FOOTNOTES:
[1] Loomis’ Treatise on Arithmetic.
Loomis’ Treatise on Algebra.
Loomis’ Elements of Geometry.
Robinson’s Algebra and Geometry.
[2] Adapted from Robinson’s Algebra.
[3] Determined in the manner explained for erecting a perpendicular in
par. 32 except that intersecting arcs are described on both sides of
the line (_n m_, fig. 17); the perpendicular will be a line joining
these points of intersection.