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DISTANCES FROM SIDE STAKES FOR CROSS-SECTIONING Roadway of any Width. Side Slopes 1/2 to 1. In the figure below: opposite 7 under "Cut or Fill" and under .3 read 11.0, the distance out from the side stake at left. Also, opposite 11 under "Cut or Fill" and under .1 read 16.7, the distance out from the side stake at right. Slope Stake Side Stake 1/2 to 1 Cut 7/3 Grade Center Stake Side Stake 11/2 to 1 Fill 11/1 Slope Stake 0 .0 .2 .3 .5 .6 .8 .9 1.1 1.2 1.4 0 1 1.5 1.7 1.8 2.0 2.1 2.3 2.4 2.6 2.7 2.9 1 2 3.0 3.2 3.3 3.5 3.6 3.8 3.9 4.1 4.2 4.4 2 3 4.5 4.7 4.8 5.0 5.1 5.3 5.4 5.6 5.7 5.9 3 4 6.0 6.2 6.3 6.5 6.6 6.8 6.9 7.1 7.2 7.4 4 5 7.5 7.7 7.8 8.0 8.1 8.3 8.4 8.6 8.7 8.9 5 6 9.0 9.2 9.3 9.5 9.6 9.8 9.9 10.1 10.2 10.4 6 7 10.5 10.7 10.8 11.0 11.1 11.3 11.4 11.6 11.7 11.9 7 8 12.0 12.2 12.3 12.5 12.6 12.8 12.9 13.1 13.2 13.4 8 9 13.5 13.7 13.8 14.0 14.1 14.3 14.4 14.6 14.7 14.9 9 10 15.0 15.2 15.3 15.5 15.6 15.8 15.9 16.1 16.2 16.4 10 11 16.5 16.7 16.8 17.0 17.1 17.3 17.4 17.6 17.7 17.9 11 12 18.0 18.2 18.3 18.5 18.6 18.8 18.9 19.1 19.2 19.4 12 13 19.5 19.7 19.8 20.0 20.1 20.3 20.4 20.6 20.7 20.9 13 14 21.0 21.2 21.3 21.5 21.6 21.8 21.9 22.1 22.2 22.4 14 15 22.5 22.7 22.8 23.0 23.1 23.3 23.4 23.6 23.7 23.9 15 16 24.0 24.2 24.3 24.5 24.6 24.8 24.9 25.1 25.2 25.4 16 17 25.5 25.7 25.8 26.0 26.1 26.3 26.4 26.6 26.7 26.9 17 18 27.0 27.2 27.3 27.5 27.6 27.8 27.9 28.1 28.2 28.4 18 19 28.5 28.7 28.8 29.0 29.1 29.3 29.4 29.6 29.7 29.9 19 20 30.0 30.2 30.3 30.5 30.6 30.8 30.9 31.1 31.2 31.4 20 21 31.5 31.7 31.8 32.0 32.1 32.3 32.4 32.6 32.7 32.9 21 22 33.0 33.2 33.3 33.5 33.6 33.8 33.9 34.1 34.2 34.4 22 23 34.5 34.7 34.8 35.0 35.1 35.3 35.4 35.6 35.7 35.9 23 24 36.0 36.2 36.3 36.5 36.6 36.8 36.9 37.1 37.2 37.4 24 25 37.5 37.7 37.8 38.0 38.1 38.3 38.4 38.6 38.7 38.9 25 26 39.0 39.2 39.3 39.5 39.6 39.8 39.9 40.1 40.2 40.4 26 27 40.5 40.7 40.8 41.0 41.1 41.3 41.4 41.6 41.7 41.9 27 28 42.0 42.2 42.3 42.5 42.6 42.8 42.9 43.1 43.2 43.4 28 29 43.5 43.7 43.8 44.0 44.1 44.3 44.4 44.6 44.7 44.9 29 30 45.0 45.2 45.3 45.5 45.6 45.8 45.9 46.1 46.2 46.4 30 31 46.5 46.7 46.8 47.0 47.1 47.3 47.4 47.6 47.7 47.9 31 32 48.0 48.2 48.3 48.5 48.6 48.8 48.9 49.1 49.2 49.4 32 33 49.5 49.7 49.8 50.0 50.1 50.3 50.4 50.6 50.7 50.9 33 34 51.0 51.2 51.3 51.5 51.6 51.8 51.9 52.1 52.2 52.4 34 35 52.5 52.7 52.8 53.0 53.1 53.3 53.4 53.6 53.7 53.9 35 36 54.0 54.2 54.3 54.5 54.6 54.8 54.9 55.1 55.2 55.4 36 37 55.5 55.7 55.8 56.0 56.1 56.3 56.4 56.6 56.7 56.9 37 38 57.0 57.2 57.3 57.5 57.6 57.8 57.9 58.1 58.2 58.4 38 39 58.5 58.7 58.8 59.0 59.1 59.3 59.4 59.6 59.7 59.9 39 40 60.0 60.2 60.3 60.5 60.6 60.8 60.9 61.1 61.2 61.4 40 Cut or Fill 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Cut or Fill Distance out from Side or Shoulder Stake KEUFFEL & ESSER CO., N. Y. For Curve Tables see end of book.
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GENERAL OUTCROP SECTION OF THE BLACK HILLS AREA FORMATION SECTION TIME IN FEET DESCRIPTION QUATERNARY SANDS AND GRAVELS [illegible] [illegible] TERTIARY PLIOCENE OGALLALA GROUP [illegible] Light colored sands and silts MIOCENE ARIKAREE GROUP 0 - 500 Light colored clays and silts while age bed at base OLIGOCENE WHITE RIVER GROUP Brule Formation 0 - 600 Light colored clays with sandstone channel fillings and local limestone lenses Chetron Formation PALEOCENE FORT UNION FORMATION TONGUE RIVER MEMBER 0 - 485 Light colored clays and sands, with coal beds former marl CANNONBALL MEMBER 1.1 - 22? Green mottled shales and yellow sandstones, "r", letter often as concretions LUDLOW MEMBER 0 - 350 Soft-gray clays and sandstones with i.r. beds of lignite. UPPER HELL CREEK FORMATION [Lance Formation] 425 Sand-colored soft brown shale and gray sandstone, with thin lignite lenses in upper part. Lower half more sandy, hard, brittle concretions and thin masses of iron carbonate. CRETACEOUS FOX HILLS SANDSTONE 25 - 200 Grayish-white to yellow sandstone. Calc. PIERRE SHALE 1200-2000 Principal horizon of limestone lenses giving teppe buttes Sharon Springs Nsm [illegible] Dark-gray shale containing scattered concretions Niobrara Formation [illegible] Widely scattered limestone masses, giving small teppe buttes. Turner Sand Zone 100-225 Black fissile shale with concretions CARLILE FORMATION Light-gray shale with numerous large concretions and sandy layers. Well Creek Sands 400-750 Dark-gray shale GREENHORN FORMATION [25 - 30] Impure sandy limestone Weathers buff Belle Fourche Shale (200-350) Dark gray calcareous shale, with thin Orms Lake limestone of base Mowry Shale 300-390 Gray shale with scattered limestone concretions Newcastle Sandstone Clay tour bastonite at base. Skull Creek Shale 150 - 250 Light-gray siliceous shale, fish scales and the layers of basaltine Fall River [Dakota (?)] S.S. 20-60 Brown to light yellow and white sandstone Fuson Shale 170-270 Dark gray to black shale. Minnewaste Limestone 10 - 200 Massive to stibby sandstones Lakota Sandstone 10 - 188 Gray to purple shale Thin sandstones. Morrison Formation 0 - 25 Massive gray limestone Unkpapa Sandstone 25 - 485 Coarse, hard, crisscrossed sandstone mostly buff to gray Compartmental locality and Coal of base. Sundance Formation 0 - 220 Green to maroon shale Thin sandstones Gypsum Spring Formation 0 - 225 Massive fine grained sandstones NUGGET (?) FORMATION 250-450 Greenish-gray shales, thin limestone lenses. Diatomite sandstone red in lower middle. 0 - 45 Red siltstone, gypsum and limestone LOWER SPEARFISH FORMATION 0 - 60 Massive crisscrossed salmon sandstones CRETACEOUS Triassic ? 300-700 Red sandy shale, soft red sandstone and siltstone with gypsum and thin limestone layers. Permian ? Opeche Formation [illegible] Gypsum locally near the base. Minnelusa Sandstone 50 - 135 Massive gray, laminated limestone Yellow to red cross-banded sandstone, limestone and calcareous facies at top. Interbedded limestone, dolomite, shale, and conglomerate Red shales with interbedded limestone and sandstone at base. MISSISSIPPIAN PAHASAPA (MADISON) LIMESTONE 300-630 Massive light-colored limestone. Dolomitic in part Englewood Limestone 30 - 60 Poor to buff limestone Shale locally at base White Wood (Red River) Formation [illegible] Bull dolomite and limestone Winnipeg Formation Green shale with siltstone at top and black sandstone at base. Dead Wood Formation 0 - 100 Massive to buff limestone. Greenish phaccerous shale. Clay, dolomite and flat-pebbled impure conglomerate. Sandstone with conglomerate locally at the base ORDOVICIAN [illegible] 10-400 Shales, slates, siltstones, and breccia grits GAMBRIAN METAMORPHIC and intruded by diorite, metamorphosed to amphibolite, and to granite and pegmatite. IGNEOUS ROCKS PRE-CAMBRIAN
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GENERAL OUTCROP SECTION OF THE BLACK HILLS AREA FORMATION SECTION THICKNESS IN FEET DESCRIPTION QUATERNARY SANDS AND GRAVELS [illegible] 0 - 5ft Sand, gravel and boulders TERTIARY PIOCENE OGALLALA GROUP [illegible] 0 - 500 Light colored sands and silts. MIOCENE ARIKAREE GROUP [illegible] 0 - 300 Light brown clays and silts White clay bed at base OLIGOCENE WHITE RIVER GROUP Brule Formation 0 - 600 Light colored clays with sandstone channel Chadron Formation fillings and local limestone lenses [illegible] 0 - 485 Light colored clays and sands, with coal near [illegible] farther north. PALEOCENE TONGUE RIVER MEMBER [illegible] 0 - 225 Green marl shales and yellow sandstone, FORT UNION FORMATION "" latter often as concretions CANNONBALL MEMBER So- gray clays and sandstones with LUDLOW MEMBER thin beds of lignite. CRETACEOUS HELL CREEK FORMATION [illegible] 425 Sando, colored soft brown shale and gray (Lance Formation) (Lance Formation) sandstone, with thin lignite lenses in the upper part. Lower half more sandy. Hay lignite concretions and thin lenses of iron carbonate. FOX HILLS SANDSTONE [illegible] 25 - 200 Greyish-white to yellow sandstone Calc. UPPER PIERRE SHALE [illegible] 1200-2000 Principal horizon of limestone lenses giving tepee buttes Dark gray shale containing scattered concretions. Widely scattered limestone masses, giving small tepee buttes. Black fissile shale with concretions. Sharan Springs Nem NIOBRARA FORMATION [illegible] 100 - 225 Impure chole and calcareous shale. Turner Sand Zone CARLILE FORMATION [illegible] 400-750 Light-gray shale with numerous large Wall Creek Sands concretions and sandy layers. Dark gray shale GREENHORN FORMATION [illegible] (25 - 30) Impure tabby limestone. Weathers buff GRANEROS SHALE (200-350) Dark gray calcareous shale, with thin [illegible] Ormen Lake limestone at base. Belle Fourche Shale [illegible] 300 - 190 Very shale with scattered limestone MOWRY SHALE Clay-sour bentonite at base. NEWCASTLE SANDSTONE Light-gray siliceous shale, fish scales SKULL CREEK SHALE and thin layers of bentonite. Brown to light yellow and white sandstone Dark gray to black shale. LOWER FALL RIVER [DAKOTA (?)] S.S. [illegible] 10 - 200 Massive to sloppy sandstone. FUSON SHALE [illegible] 10 - 188 Gray to purple shale. Thin sandstones. MINNEWASTE LIMESTONE [illegible] 0 - 25 Massive gray limestone Lakota Sandstone Loose, hard, crossbedded sandstone mostly MORRISON FORMATION buff to gray. Conglomerate locally and UNKPAPA SANDSTONE clay at base. SUNDANCE FORMATION Greenish-gray shale, thin limestone lenses GYPSUM SPRING FORMATION Glaucophitic sandstone, red at near middle. NUGGET (?) FORMATION Red siltstone, gypsum, and limestone. JURASSIC SPEARFISH FORMATION [illegible] 0 - 220 Massive fine-grained sandstone. UNKPAPA SANDSTONE 250-450 Greenish-gray shale, thin limestone lenses SUNDANCE FORMATION Glaucophitic sandstone, red at near middle. GYPSUM SPRING FORMATION Red siltstone, gypsum, and limestone. NUGGET (?) FORMATION Massive crossbedded siliman sandstone. TRIASSIC SPEARFISH FORMATION [illegible] 300 - 700 Red sandy shale, soft red sandstone and [illegible] siltstone with gypsum and thin limestone Gypsum locally near the base. Massive gray, laminated limestone. PERMIAN MINNEKAHTA LIMESTONE [illegible] 30 - 50 Red shale and sandstone PENNSYLVANIAN OPECHE FORMATION [illegible] 50 - 135 Yellow to red cross-bedded sandstone, MINNELUSA SANDSTONE limestone, and anhydrite locally of "top [illegible] interbedded sandstone, limestone, dolomite, Red shale with interbedded limestone and siltstone at base. [illegible] MISSISSIPPIAN PAHASAPA (MADISON) LIMESTONE [illegible] 300-630 Massive light-colored limestone ENGLEWOOD LIMESTONE Dolomitic in part. [illegible] Pink to buff limestone. Shale locally at base. [illegible] Shell dolomite and limestone. ORDOVICIAN WHITEWOOD (RED RIVER) FORMATION [illegible] 0 - 60 Green shale with siltstone of top and WINNIPEG FORMATION white sandstone at base. [illegible] Massive to buff sandstone. Greenish DEADWOOD FORMATION glauconitic shale, flaggy dolomite and flat- PRE-CAMBRIAN pebble limestone conglomerate. Sandstone, METAMORPHIC and with conglomerate locally of the base. IGNEOUS ROCKS Schist, slate, quartzite, and arkasic grit intruded by diorite, metamorphosed to amphibolite, and by granite and pegmatite.
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{ "text": "[illegible]\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n [TRANSCRIPTION_TRUNCATED_DUE_TO_LOOP]
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CURVE TABLES. Published by KEUFFEL & ESSER CO. HOW TO USE CURVE TABLES. Table I. contains Tangents and External to a 1° curve. Tan. and Ext. to any other radius may be found nearly enough, by dividing the Tan. or Ext. opposite the given Central Angle by the given degree of curve. To find Deg. of Curve, having the Central Angle and Tangent: Divide Tan. opposite the given Central Angle by the given Tangent. To find Deg. of Curve, having the Central Angle and External: Divide Ext. opposite the given Central Angle by the given External. To find Nat. Tan. and Nat. Ex. Sec. for any angle by Table I.: Tan. or Ext. of twice the given angle divided by the radius of a 1° curve will be the Nat. Tan. or Nat. Ex. Sec. EXAMPLE. Wanted a Curve with an Ext. of about 12 ft. Angle of Intersection or I. P. =23° 20' to the R. at Station 542+72. Ext. in Tab. I opposite 23° 20'=120.87 120.87÷12=10.07. Say a 10° Curve. Tan. in Tab. I opp. 23° 20'=1183.1 1183.1÷10=118.31. Correction for A. 23° 20' for a 10° Cur.=0.16 118.31+0.16=118.47=corrected Tangent. (If corrected Ext. is required find in same way) Ang. 23° =23.33°÷10=2.3333=L. C. 2° 19½'=def. for sta. 542 I. P.=sta. 542+72 4° 49½'=" " " +50 Tan.= 1 .18.47 7° 19½'=" " " 543 B. C.=sta. 541+53.53 9° 49½'=" " " +50 L. C.= 2 .33.33 11° 40'=" " " 543+ E. C.=Sta. 543+86.86 86.86 100-53.53=46.47×3'(def. for 1 ft. of 10° Cur.)=139.41'= 2° 19½'=def. for sta. 542. Def. for 50 ft.=2° 30' for a 10° Curve. Def. for 36.86 ft.=1° 50½' for a 10° Curve.
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II TABLE I. - Tangents and Externals to a 1° Curve. Chord = 100 ft. Int. Tangent External Int. Tangent External Int. Tangent External Angle 1° 50.00 .22 8° 400.66 13.99 15° 754.32 49.44 10' 58.34 .30 10' 409.03 14.58 10' 762.80 50.55 20 66.67 .39 20 417.41 15.18 20 771.29 51.68 30 75.01 .49 30 425.79 15.80 30 779.77 52.82 40 83.34 .61 40 434.17 16.43 40 788.26 53.97 50 91.68 .73 50 442.55 17.07 50 796.75 55.13 2 100.01 .87 9 450.93 17.72 16 805.25 56.31 10 108.35 1.02 10 459.32 18.38 10 813.75 57.50 20 116.68 1.19 20 467.71 19.06 20 822.25 58.70 30 125.02 1.36 30 476.10 19.75 30 830.76 59.91 40 133.36 1.55 40 484.49 20.45 40 839.27 61.14 50 141.70 1.75 50 492.88 21.16 50 847.78 62.38 3 150.04 1.96 10 501.28 21.89 17 856.30 63.63 10 158.38 2.19 10 509.68 22.62 10 864.82 64.90 20 166.72 2.43 20 518.08 23.38 20 873.35 66.18 30 175.06 2.67 30 526.48 24.14 30 881.88 67.47 40 183.40 2.93 40 534.89 24.91 40 890.41 68.77 50 191.74 3.21 50 543.29 25.70 50 898.95 70.09 4 200.08 3.49 11 551.70 26.50 18 907.49 71.42 10 208.43 3.79 10 560.11 27.31 10 916.03 72.76 20 216.77 4.10 20 568.53 28.14 20 924.58 74.12 30 225.12 4.42 30 576.95 28.97 30 933.13 75.49 40 233.47 4.76 40 585.36 29.82 40 941.69 76.86 50 241.81 5.10 50 593.79 30.68 50 950.25 78.26 5 250.16 5.46 12 602.21 31.56 19 958.81 79.67 10 258.51 5.83 10 610.64 32.45 10 967.38 81.09 20 266.86 6.21 20 619.07 33.35 20 975.96 82.53 30 275.21 6.61 30 627.50 34.26 30 984.53 83.97 40 283.57 7.01 40 635.93 35.18 40 993.12 85.43 50 291.92 7.43 50 644.37 36.12 50 1001.7 86.90 6 300.28 7.86 13 652.81 37.07 20 1010.3 88.39 10 308.64 8.31 10 661.25 38.03 10 1018.9 89.89 20 316.99 8.76 20 669.70 39.01 20 1027.5 91.40 30 325.35 9.23 30 678.15 39.99 30 1036.1 92.92 40 333.71 9.71 40 686.60 40.99 40 1044.7 94.46 50 342.08 10.20 50 695.06 42.00 50 1053.3 96.01 7 350.44 10.71 14 703.51 43.03 21 1061.9 97.57 10 358.81 11.22 10 711.97 44.07 10 1070.6 99.16 20 367.17 11.75 20 720.44 45.12 20 1079.2 100.75 30 375.54 12.29 30 728.90 46.18 30 1087.8 102.35 40 383.91 12.85 40 737.37 47.25 40 1096.4 103.97 50 392.28 13.41 50 745.85 48.34 50 1105.1 105.60 Corrections to be Added (T=Tangent. E=External.) Int. Curve 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° Angle 5° T=.02 .03 .05 .06 .08 .10 .11 .13 .15 .16 .18 .20 .21 .23 E=.000 .000 .001 .001 .002 .002 .002 .003 .003 .004 .004 .004 .005 .005 10° T=.03 .06 .09 .13 .16 .19 .22 .25 .28 .31 .34 .38 .42 .46 E=.001 .003 .004 .006 .007 .008 .009 .011 .012 .014 .015 .017 .018 .020 15° T=.04 .10 .14 .19 .24 .29 .34 .39 .45 .51 .53 .58 .63 .68 E=.003 .007 .010 .014 .018 .023 .027 .029 .032 .035 .039 .043 .047 .051 20° T=.06 .13 .19 .26 .32 .39 .45 .51 .58 .65 .72 .79 .84 .90 E=.006 .011 .017 .022 .028 .034 .038 .045 .051 .057 .063 .070 .076 .083 25° T=.08 .16 .24 .33 .40 .49 .58 .67 .75 .83 .90 .99 1.06 1.14 E=.009 .018 .027 .036 .046 .056 .065 .074 .083 .093 .106 .120 .127 .135
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TABLE I. — Tangents and External to a 1° Curve. Chord = 100 ft. III Int. Angle Tangent External Int. Angle Tangent External Int. Angle Tangent External 22° 1113.7 107.24 29° 1481.8 188.51 36° 1861.7 294.9 10' 1122.4 108.90 10' 1490.7 190.74 10' 1870.9 297.7 20 1131.0 110.57 20 1499.6 192.99 20 1880.1 300.6 30 1139.7 112.25 30 1508.5 195.25 30 1889.4 303.5 40 1148.4 113.95 40 1517.4 197.53 40 1898.6 306.4 50 1157.0 115.66 50 1526.3 199.82 50 1907.9 309.3 23 1165.7 117.38 30 1535.3 202.12 37 1917.1 312.2 10 1174.4 119.12 10 1544.2 204.44 10 1926.4 315.2 20 1183.1 120.87 20 1553.1 206.77 20 1935.7 318.1 30 1191.8 122.63 30 1562.1 209.12 30 1945.0 321.1 40 1200.5 124.41 40 1571.0 211.48 40 1954.3 324.1 50 1209.2 126.20 50 1580.0 213.86 50 1963.6 327.1 24 1217.9 128.00 31 1589.0 216.3 38 1972.9 330.2 10 1226.6 129.82 10 1598.0 218.7 10 1982.2 333.2 20 1235.3 131.65 20 1606.9 221.1 20 1991.5 336.3 30 1244.0 133.50 30 1615.9 223.5 30 2000.9 339.3 40 1252.8 135.35 40 1624.9 226.0 40 2010.2 342.4 50 1261.5 137.23 50 1633.9 228.4 50 2019.6 345.5 25 1270.2 139.11 32 1643.0 230.9 39 2029.0 348.6 10 1279.0 141.01 10 1652.0 233.4 10 2038.4 351.8 20 1287.7 142.93 20 1661.0 235.9 20 2047.8 354.9 30 1296.5 144.85 30 1670.0 238.4 30 2057.2 358.1 40 1305.3 146.79 40 1679.1 241.0 40 2066.6 361.3 50 1314.0 148.75 50 1688.1 243.5 50 2076.0 364.5 26 1322.8 150.71 33 1697.2 246.1 40 2085.4 367.7 10 1331.6 152.69 10 1706.3 248.7 10 2094.9 371.0 20 1340.4 154.69 20 1715.3 251.3 20 2104.3 374.2 30 1349.2 156.70 30 1724.4 253.9 30 2113.8 377.5 40 1358.0 158.72 40 1733.5 256.5 40 2123.3 380.8 50 1366.8 160.76 50 1742.6 259.1 50 2132.7 384.1 27 1375.6 162.81 34 1751.7 261.8 41 2142.2 387.4 10 1384.4 164.86 10 1760.8 264.5 10 2151.7 390.7 20 1393.2 166.95 20 1770.0 267.2 20 2161.2 394.1 30 1402.0 169.04 30 1779.1 269.9 30 2170.8 397.4 40 1410.9 171.15 40 1788.2 272.6 40 2180.3 400.8 50 1419.7 173.27 50 1797.4 275.3 50 2189.9 404.2 28 1428.6 175.41 35 1806.6 278.1 42 2199.4 407.6 10 1437.4 177.55 10 1815.7 280.8 10 2209.0 411.1 20 1446.3 179.72 20 1824.9 283.6 20 2218.6 414.5 30 1455.1 181.89 30 1834.1 286.4 30 2228.1 418.0 40 1464.0 184.08 40 1843.3 289.2 40 2237.7 421.4 50 1472.9 186.29 50 1852.5 292.0 50 2247.3 425.0 Corrections to be Added (T=Tangent. E=External.) Int. Angle Curve 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° 20° T=.06 .13 .19 .26 .32 .39 .45 .51 .58 .65 .72 .79 .84 .90 E=.006 .011 .017 .022 .028 .034 .038 .045 .051 .057 .063 .070 .076 .083 25° T=.08 .16 .24 .33 .40 .49 .58 .67 .75 .83 .90 .99 1.06 1.14 E=.009 .018 .027 .036 .046 .056 .065 .074 .083 .093 .106 .120 .127 .135 30° T=.10 .19 .29 .39 .49 .59 .69 .79 .89 .99 1.09 1.20 1.29 1.39 E=.013 .025 .038 .051 .065 .078 .090 .103 .116 .129 .149 .170 .179 .188 35° T=.11 .22 .34 .47 .58 .69 .80 .93 1.05 1.17 1.29 1.42 1.54 1.66 E=.018 .035 .054 .072 .086 .109 .131 .153 .175 .197 .213 .230 .247 .264 40° T=.13 .26 .40 .53 .67 .80 .93 1.06 1.20 1.34 1.49 1.64 1.79 1.94 E=.023 .046 .070 .093 .117 .141 .172 .203 .234 .265 .277 .290 .315 .341 45° T=.15 .30 .44 .60 .76 .91 1.06 1.21 1.37 1.52 1.70 1.87 2.04 2.21 E=.030 .060 .093 .119 .153 .184 .216 .254 .289 .325 .351 .378 .411 .445
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IV TABLE I. — Tangents and External to a 1° Curve. Chord = 100 ft. Int. Tangent External Int. Tangent External Int. Angle Tangent External Angle 43° 2257.0 428.5 50° 2671.8 592.3 57° 3110.9 790.1 10' 2266.6 432.0 10' 2681.9 596.6 10' 3121.7 795.2 20 2276.2 435.6 20 2692.1 600.9 20 3132.6 800.4 30 2285.9 439.2 30 2702.3 605.3 30 3143.4 805.6 40 2295.6 442.8 40 2712.5 609.6 40 3154.2 810.9 50 2305.2 446.4 50 2722.7 614.0 50 3165.1 816.1 44 2314.9 450.0 51 2732.9 618.4 58 3176.0 821.4 10 2324.6 453.6 10 2743.1 622.8 10 3186.9 826.7 20 2334.3 457.3 20 2753.4 627.2 20 3197.8 832.0 30 2344.1 461.0 30 2763.7 631.7 30 3208.8 837.3 40 2353.8 464.6 40 2773.9 636.2 40 3219.7 842.7 50 2363.5 468.4 50 2784.2 640.7 50 3230.7 848.1 45 2373.3 472.1 52 2794.5 645.2 59 3241.7 853.5 10 2383.1 475.8 10 2804.9 649.7 10 3252.7 858.9 20 2392.8 479.6 20 2815.2 654.3 20 3263.7 864.3 30 2402.6 483.4 30 2825.6 658.8 30 3274.8 869.8 40 2412.4 487.2 40 2835.9 663.4 40 3285.8 875.2 50 2422.3 491.0 50 2846.3 668.0 50 3296.9 880.8 46 2432.1 494.8 53 2856.7 672.7 60 3308.0 886.4 10 2441.9 498.7 10 2867.1 677.3 10 3319.1 892.0 20 2451.8 502.5 20 2877.5 682.0 20 3330.3 897.5 30 2461.7 506.4 30 2888.0 686.7 30 3341.4 903.2 40 2471.5 510.3 40 2898.4 691.4 40 3352.6 908.8 50 2481.4 514.3 50 2908.9 696.1 50 3363.8 914.5 47 2491.3 518.2 54 2919.4 700.9 61 3375.0 920.2 10 2501.2 522.2 10 2929.9 705.7 10 3386.3 925.9 20 2511.2 526.1 20 2940.4 710.5 20 3397.5 931.6 30 2521.1 530.1 30 2951.0 715.3 30 3408.8 937.3 40 2531.1 534.2 40 2961.5 720.1 40 3420.1 943.1 50 2541.0 538.2 50 2972.1 725.0 50 3431.4 948.9 48 2551.0 542.2 55 2982.7 729.9 62 3442.7 954.8 10 2561.0 546.3 10 2993.3 734.8 10 3454.1 960.6 20 2571.0 550.4 20 3003.9 739.7 20 3465.4 966.5 30 2581.0 554.5 30 3014.5 744.6 30 3476.8 972.4 40 2591.0 558.6 40 3025.2 749.6 40 3488.3 978.3 50 2601.1 562.8 50 3035.8 754.6 50 3499.7 984.3 49 2611.2 566.9 56 3046.5 759.6 63 3511.1 990.2 10 2621.2 571.1 10 3057.2 764.6 10 3522.6 996.2 20 2631.3 575.3 20 3067.9 769.7 20 3534.1 1002.3 30 2641.4 579.5 30 3078.7 774.7 30 3545.6 1008.3 40 2651.5 583.8 40 3089.4 779.8 40 3557.2 1014.4 50 2661.6 588.0 50 3100.2 784.9 50 3568.7 1020.5 Corrections to be Added (T = Tangent. E = External.) Int. Curve 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° Angle 40° T=.13 .26 .40 .53 .67 .80 .93 .106 1.20 1.34 1.49 1.64 1.79 1.94 E=.023 .046 .070 .093 .117 .141 .172 .203 .234 .265 .297 .330 .361 .394 45° T=.15 .30 .44 .60 .76 .91 1.06 1.21 1.37 1.52 1.70 1.87 2.04 2.21 E=.030 .060 .093 .119 .153 .184 .216 .254 .289 .325 .357 .378 .411 .445 50° T=.17 .34 .51 .68 .85 1.02 1.19 1.36 1.54 1.72 1.91 2.10 2.29 2.48 E=.037 .075 .116 .151 .189 .227 .266 .305 .345 .384 .425 .467 .508 .550 55° T=.19 .38 .57 .76 .95 1.14 1.32 1.52 1.72 1.92 2.14 2.35 2.56 2.77 E=.046 .093 .142 .188 .236 .283 .332 .381 .420 .479 .530 .582 .641 .700 60° T=.21 .42 .63 .84 1.05 1.27 1.49 1.71 1.94 2.17 2.38 2.60 2.88 3.07 E=.056 .112 .168 .225 .283 .340 .398 .457 .516 .575 .636 .697 .774 .851 65° T=.23 .46 .69 .93 1.16 1.40 1.64 1.88 2.13 2.38 2.63 2.88 3.13 3.39 E=.067 .135 .204 .273 .343 .412 .483 .554 .625 .697 .771 .845 .922 1.01
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TABLE I. — Tangents and Externals to a 1° Curve. Chord = 100 ft. Int. Angle | Tangent | External | Int. Angle | Tangent | External | Int. Angle | Tangent | External 64° | 3580.3 | 1026.6 | 71° | 4086.9 | 1308.2 | 78° | 4639.8 | 1643.0 10' | 3591.9 | 1032.8 | 10' | 4099.5 | 1315.6 | 10' | 4653.6 | 1651.7 20 | 3603.5 | 1039.0 | 20 | 4112.1 | 1322.9 | 20 | 4667.4 | 1660.5 30 | 3615.1 | 1045.2 | 30 | 4124.8 | 1330.3 | 30 | 4681.3 | 1669.2 40 | 3626.8 | 1051.4 | 40 | 4137.4 | 1337.7 | 40 | 4695.2 | 1678.1 50 | 3638.5 | 1057.7 | 50 | 4150.1 | 1345.1 | 50 | 4709.2 | 1686.9 65 | 3650.2 | 1063.9 | 72 | 4162.8 | 1352.6 | 79 | 4723.2 | 1695.8 10 | 3661.9 | 1070.2 | 10 | 4175.6 | 1360.1 | 10 | 4737.2 | 1704.7 20 | 3673.7 | 1076.6 | 20 | 4188.5 | 1367.6 | 20 | 4751.2 | 1713.7 30 | 3685.4 | 1082.9 | 30 | 4201.2 | 1375.2 | 30 | 4765.3 | 1722.7 40 | 3697.2 | 1089.3 | 40 | 4214.0 | 1382.8 | 40 | 4779.4 | 1731.7 50 | 3709.0 | 1095.7 | 50 | 4226.8 | 1390.4 | 50 | 4793.6 | 1740.8 66 | 3720.9 | 1102.2 | 73 | 4239.7 | 1398.0 | 80 | 4807.7 | 1749.9 10 | 3732.7 | 1108.6 | 10 | 4252.6 | 1405.7 | 10 | 4822.0 | 1759.0 20 | 3744.6 | 1115.1 | 20 | 4265.6 | 1413.5 | 20 | 4836.2 | 1768.2 30 | 3756.5 | 1121.7 | 30 | 4278.5 | 1421.2 | 30 | 4850.5 | 1777.4 40 | 3768.5 | 1128.2 | 40 | 4291.5 | 1429.0 | 40 | 4864.8 | 1786.7 50 | 3780.4 | 1134.8 | 50 | 4304.6 | 1436.8 | 50 | 4879.2 | 1796.0 67 | 3792.4 | 1141.4 | 74 | 4317.6 | 1444.6 | 81 | 4893.6 | 1805.3 10 | 3804.4 | 1148.0 | 10 | 4330.7 | 1452.5 | 10 | 4908.0 | 1814.7 20 | 3816.4 | 1154.7 | 20 | 4343.8 | 1460.4 | 20 | 4922.5 | 1824.1 30 | 3828.4 | 1161.3 | 30 | 4356.9 | 1468.4 | 30 | 4937.0 | 1833.6 40 | 3840.5 | 1168.1 | 40 | 4370.1 | 1476.4 | 40 | 4951.5 | 1843.1 50 | 3852.6 | 1174.8 | 50 | 4383.3 | 1484.4 | 50 | 4966.1 | 1852.6 68 | 3864.7 | 1181.6 | 75 | 4396.5 | 1492.4 | 82 | 4980.7 | 1862.2 10 | 3876.8 | 1188.4 | 10 | 4409.8 | 1500.5 | 10 | 4995.4 | 1871.8 20 | 3889.0 | 1195.2 | 20 | 4423.1 | 1508.6 | 20 | 5010.0 | 1881.5 30 | 3901.2 | 1202.0 | 30 | 4436.4 | 1516.7 | 30 | 5024.8 | 1891.2 40 | 3913.4 | 1208.9 | 40 | 4449.7 | 1524.9 | 40 | 5039.5 | 1900.9 50 | 3925.6 | 1215.8 | 50 | 4463.1 | 1533.1 | 50 | 5054.3 | 1910.7 69 | 3937.9 | 1222.7 | 76 | 4476.5 | 1541.4 | 83 | 5069.2 | 1920.5 10 | 3950.2 | 1229.7 | 10 | 4489.9 | 1549.7 | 10 | 5084.0 | 1930.4 20 | 3962.5 | 1236.7 | 20 | 4503.4 | 1558.0 | 20 | 5099.0 | 1940.3 30 | 3974.8 | 1243.7 | 30 | 4516.9 | 1566.3 | 30 | 5113.9 | 1950.3 40 | 3987.2 | 1250.8 | 40 | 4530.4 | 1574.7 | 40 | 5128.9 | 1960.2 50 | 3999.5 | 1257.9 | 50 | 4544.0 | 1583.1 | 50 | 5143.9 | 1970.3 70 | 4011.9 | 1265.0 | 77 | 4557.6 | 1591.6 | 84 | 5159.0 | 1980.4 10 | 4024.4 | 1272.1 | 10 | 4571.2 | 1600.1 | 10 | 5174.1 | 1990.5 20 | 4036.8 | 1279.3 | 20 | 4584.8 | 1608.6 | 20 | 5189.3 | 2000.6 30 | 4049.3 | 1286.5 | 30 | 4598.5 | 1617.1 | 30 | 5204.4 | 2010.8 40 | 4061.8 | 1293.6 | 40 | 4612.2 | 1625.7 | 40 | 5219.7 | 2021.1 50 | 4074.4 | 1300.9 | 50 | 4626.0 | 1634.4 | 50 | 5234.9 | 2031.4 Corrections to be Added (T = Tangent. E = External.) Int. Angle | Curve 5° | 10° | 15° | 20° | 25° | 30° | 35° | 40° | 45° | 50° | 55° | 60° | 65° | 70° 60° | T = .21 | .42 | .63 | .84 | 1.05 | 1.27 | 1.49 | 1.71 | 1.94 | 2.17 | 2.38 | 2.60 | 2.83 | 3.07 E = .056 | .112 | .168 | .225 | .283 | .340 | .398 | .457 | .516 | .575 | .636 | .697 | .774 | .851 65° | T = .23 | .46 | .69 | .93 | 1.16 | 1.40 | 1.64 | 1.88 | 2.13 | 2.38 | 2.63 | 2.88 | 3.13 | 3.39 E = .067 | .135 | .204 | .273 | .343 | .412 | .483 | .554 | .625 | .697 | .771 | .845 | .922 | 1.01 70° | T = .25 | .51 | .76 | 1.02 | 1.28 | 1.54 | 1.80 | 2.06 | 2.33 | 2.60 | 2.88 | 3.16 | 3.44 | 3.72 E = .080 | .159 | .240 | .321 | .403 | .485 | .568 | .652 | .735 | .819 | .906 | .994 | 1.08 | 1.17 75° | T = .27 | .56 | .83 | 1.12 | 1.40 | 1.69 | 1.98 | 2.27 | 2.57 | 2.87 | 3.16 | 3.47 | 3.78 | 4.09 E = .095 | .182 | .286 | .383 | .480 | .578 | .678 | .777 | .877 | .977 | 1.07 | 1.18 | 1.29 | 1.39 80° | T = .30 | .61 | .91 | 1.22 | 1.53 | 1.84 | 2.15 | 2.46 | 2.78 | 3.10 | 3.44 | 3.78 | 4.12 | 4.46 E = .110 | .220 | .332 | .445 | .558 | .671 | .787 | .903 | 1.02 | 1.13 | 1.25 | 1.38 | 1.50 | 1.62 85° | T = .33 | .66 | 1.00 | 1.33 | 1.68 | 2.02 | 2.36 | 2.70 | 3.05 | 3.40 | 3.77 | 4.14 | 4.55 | 4.89 E = .128 | .259 | .391 | .524 | .657 | .790 | .926 | 1.06 | 1.20 | 1.34 | 1.47 | 1.62 | 1.76 | 1.91
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VI TABLE I. - Tangents and Externals to a 1° Curve. Chord = 100 ft. Int. Angle Tangent External Int. Angle Tangent External Int. Angle Tangent External 85° 5250.3 2041.7 92° 5933.2 2518.5 99° 6708.6 3092.7 10' 5265.6 2052.1 10' 5950.5 2531.0 10' 6728.4 3107.7 20 5281.0 2062.5 20 5967.9 2543.5 20 6748.2 3122.9 30 5296.4 2073.0 30 5985.3 2556.0 30 6768.1 3138.1 40 5311.9 2083.5 40 6002.7 2568.6 40 6788.1 3153.3 50 5327.4 2094.1 50 6020.2 2581.3 50 6808.2 3168.7 86 5343.0 2104.7 93 6037.8 2594.0 100 6828.3 3184.1 10 5358.6 2115.3 10 6055.4 2606.8 10 6848.5 3199.6 20 5374.2 2126.0 20 6073.1 2619.7 20 6868.8 3215.1 30 5389.9 2136.7 30 6090.8 2632.6 30 6889.2 3230.8 40 5405.6 2147.5 40 6108.6 2645.5 40 6909.6 3246.5 50 5421.4 2158.4 50 6126.4 2658.5 50 6930.1 3262.3 87 5437.2 2169.2 94 6144.3 2671.6 101 6950.6 3278.1 10 5453.1 2180.2 10 6162.6 2684.7 10 6971.3 3294.1 20 5469.0 2191.1 20 6180.2 2697.9 20 6992.0 3310.1 30 5484.9 2202.2 30 6198.3 2711.2 30 7012.7 3326.1 40 5500.9 2213.2 40 6216.4 2724.5 40 7033.6 3342.3 50 5517.0 2224.3 50 6234.6 2737.9 50 7054.5 3358.5 88 5533.1 2235.5 95 6252.8 2751.3 102 7075.5 3374.9 10 5549.2 2246.7 10 6271.1 2764.8 10 7096.6 3391.2 20 5565.4 2258.0 20 6289.4 2778.3 20 7117.8 3407.7 30 5581.6 2269.3 30 6307.9 2792.0 30 7139.0 3424.3 40 5597.8 2280.6 40 6323.3 2805.6 40 7160.3 3440.9 50 5614.2 2292.0 50 6344.8 2819.4 50 7181.7 3457.6 89 5630.5 2303.5 96 6363.4 2833.2 103 7203.2 3474.4 10 5646.9 2315.0 10 6382.1 2847.0 10 7224.7 3491.3 20 5663.4 2326.6 20 6400.8 2861.0 20 7246.3 3508.2 30 5679.9 2338.2 30 6419.5 2875.0 30 7268.0 3525.2 40 5696.4 2349.8 40 6438.4 2889.0 40 7289.8 3542.4 50 5713.0 2361.5 50 6457.3 2903.1 50 7311.7 3559.6 90 5729.7 2373.3 97 6476.2 2917.3 104 7333.6 3576.8 10 5746.3 2385.1 10 6495.2 2931.6 10 7355.6 3594.2 20 5763.1 2397.0 20 6514.3 2945.9 20 7377.8 3611.7 30 5779.9 2408.9 30 6533.4 2960.3 30 7399.9 3629.2 40 5796.7 2420.9 40 6552.6 2974.7 40 7422.2 3646.8 50 5813.6 2432.9 50 6571.9 2989.2 50 7444.6 3664.5 91 5830.5 2444.9 98 6591.2 3003.8 105 7467.0 3682.3 10 5847.5 2457.1 10 6610.6 3018.4 10 7489.6 3700.2 20 5864.6 2469.3 20 6630.1 3033.1 20 7512.2 3718.2 30 5881.7 2481.5 30 6649.6 3047.9 30 7534.9 3736.2 40 5898.8 2493.8 40 6669.2 3062.8 40 7557.7 3754.4 50 5916.0 2506.1 50 6688.8 3077.7 50 7580.5 3772.6 Corrections to be Added (T = Tangent. E. = External.) Int. Curve 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° Angle 85° T=.33 .66 1.00 1.33 1.68 2.02 2.36 2.70 3.05 3.40 3.77 4.14 4.55 4.89 E=.128 .259 .391 .524 .657 .790 .926 1.06 1.20 1.34 1.47 1.62 1.76 1.91 90° T=.36 .72 1.09 1.45 1.83 2.20 2.57 2.94 3.32 3.70 4.10 4.50 4.91 5.32 E=.149 .292 .450 .603 .756 .910 1.07 1.22 1.38 1.54 1.70 1.87 2.03 2.20 95° T=.39 .79 1.19 1.55 2.00 2.40 2.80 3.20 3.61 4.02 4.49 4.98 5.38 5.83 E=.174 .350 .522 .706 .985 1.06 1.25 1.43 1.62 1.80 1.99 2.18 2.38 2.58 100° T=.43 .86 1.30 1.74 2.18 2.62 3.06 3.50 3.95 4.40 4.88 5.37 5.85 6.34 E=.200 .401 .604 .809 1.01 1.22 1.43 1.64 1.85 2.06 2.28 2.50 2.73 2.96 105° T=.46 .94 1.42 1.90 2.38 2.87 3.34 3.84 4.35 4.84 5.35 5.87 6.40 6.93 E=.230 .470 .700 .938 1.17 1.42 1.65 1.90 2.14 2.39 2.64 2.90 3.16 3.41
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TABLE I.—Tangents and External to a 1° Curve. Chord=100 ft. Int. Angle Tangent External Int. Angle Tangent External Int. Angle Tangent External 106° 7603.5 3791.0 111° 8336.7 4386.1 116° 9169.4 5082.7 10' 7626.6 3809.4 10' 8362.7 4407.6 10' 9199.1 5107.9 20 7649.7 3827.9 20 8388.9 4429.2 20 9229.0 5133.3 30 7672.9 3846.5 30 8415.1 4450.9 30 9259.0 5158.8 40 7696.3 3865.2 40 8441.5 4472.7 40 9289.2 5184.5 50 7719.7 3884.0 50 8468.0 4494.6 50 9319.5 5210.3 107 7743.2 3902.9 112 8494.6 4516.6 117 9349.9 5236.2 10 7766.8 3921.9 10 8521.3 4538.8 10 9380.5 5262.3 20 7790.5 3940.9 20 8548.1 4561.1 20 9411.3 5288.6 30 7814.3 3960.1 30 8575.0 4583.4 30 9442.2 5315.0 40 7838.1 3979.4 40 8602.1 4606.0 40 9473.2 5341.5 50 7862.1 3998.7 50 8629.3 4628.6 50 9504.4 5368.2 108 7886.2 4018.2 113 8656.6 4651.3 118 9535.7 5395.1 10 7910.4 4037.8 10 8684.0 4674.2 10 9567.2 5422.1 20 7934.6 4057.4 20 8711.5 4697.2 20 9598.9 5449.2 30 7959.0 4077.2 30 8739.2 4720.3 30 9630.7 5476.5 40 7983.5 4097.1 40 8767.0 4743.6 40 9662.6 5504.0 50 8008.0 4117.0 50 8794.9 4766.9 50 9694.7 5531.7 109 8032.7 4137.1 114 8822.9 4790.4 119 9727.0 5559.4 10 8057.4 4157.3 10 8851.0 4814.1 10 9759.4 5587.4 20 8082.3 4177.5 20 8879.3 4837.8 20 9792.0 5615.5 30 8107.3 4197.9 30 8907.7 4861.7 30 9824.8 5643.8 40 8132.3 4218.4 40 8936.3 4885.7 40 9857.7 5672.3 50 8157.5 4239.0 50 8965.0 4909.9 50 9890.8 5700.9 110 8182.8 4259.7 115 8993.8 4934.1 120 9924.0 5729.7 10 8208.2 4280.5 10 9022.7 4958.6 10 9957.5 5758.6 20 8233.7 4301.4 20 9051.7 4983.1 20 9991.0 5787.7 30 8259.3 4322.4 30 9080.9 5007.8 30 10025.0 5817.0 40 8285.0 4343.6 40 9110.3 5032.6 40 10059.0 5846.5 50 8310.8 4364.8 50 9139.8 5057.6 50 10093.0 5876.1 Corrections to be Added (T=Tangent. E=External.) Int. Angle Curve 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° 100° T=.43 E=.200 .86 .401 1.30 .604 1.74 .938 2.18 1.01 2.62 1.22 3.06 1.43 3.50 1.64 3.95 1.85 4.40 2.06 4.88 2.28 5.37 2.50 5.85 2.73 6.34 2.96 105° T=.46 E=.230 .94 .470 1.42 .700 1.90 .938 2.38 1.17 2.87 1.42 3.34 1.65 3.84 1.90 4.35 2.14 4.84 2.39 5.35 2.64 5.87 2.90 6.40 3.16 6.93 3.41 110° T=.50 E=.260 1.03 .535 1.55 .808 2.08 1.08 2.60 1.36 3.14 1.63 3.66 1.91 4.21 2.19 4.76 2.49 5.31 2.61 5.86 3.05 6.43 3.35 7.01 3.65 7.59 3.95 115° T=.54 E=.307 1.13 .624 1.70 .939 2.29 1.26 2.86 1.57 3.45 1.89 4.03 2.21 4.63 2.54 5.23 2.87 5.83 3.20 6.44 3.53 7.07 3.88 7.70 4.23 8.35 4.58 120° T=.61 E=.339 1.25 .720 1.89 1.08 2.52 1.45 3.16 1.82 3.81 2.20 4.44 2.56 5.11 2.95 5.78 3.33 6.44 3.72 7.11 4.10 7.80 4.50 8.51 4.91 9.21 5.32
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VIII TABLE II. - Radii, Ordinates and Deflections. Chord = 100 ft. Deg. | Radius | Mid. | Tan. | Def. | Def. | | Ord. | Dist.| Dist.| for | ft. | ft. | ft. | ft. | 1 Ft. 0° 10' | 34377. | .036 | .145 | .291 | .05/ 20 | 17180. | .073 | .291 | .582 | .10 30 | 11450. | .109 | .436 | .873 | .15 40 | 8594.4 | .145 | .582 | 1.164| .20 50 | 6875.5 | .182 | .727 | 1.454| .25 1 | 5729.6 | .218 | .873 | 1.745| .30 10 | 4911.2 | .255 | 1.018| 2.036| .35 20 | 4297.3 | .291 | 1.164| 2.327| .40 30 | 3819.8 | .327 | 1.309| 2.618| .45 40 | 3437.9 | .364 | 1.454| 2.909| .50 50 | 3125.4 | .400 | 1.600| 3.200| .55 2 | 2864.9 | .436 | 1.745| 3.490| .60 10 | 2644.6 | .473 | 1.891| 3.781| .65 20 | 2455.7 | .509 | 2.036| 4.072| .70 30 | 2292.0 | .545 | 2.181| 4.363| .75 40 | 2148.8 | .582 | 2.327| 4.654| .80 50 | 2022.4 | .618 | 2.472| 4.945| .85 3 | 1910.1 | .655 | 2.618| 5.235| .90 10 | 1809.6 | .691 | 2.763| 5.526| .95 20 | 1719.1 | .727 | 2.908| 5.817| 1.00 30 | 1637.3 | .764 | 3.054| 6.108| 1.05 40 | 1562.9 | .800 | 3.199| 6.398| 1.10 50 | 1495.0 | .836 | 3.345| 6.689| 1.15 4 | 1432.7 | .873 | 3.490| 6.980| 1.20 10 | 1375.4 | .909 | 3.635| 7.271| 1.25 20 | 1322.5 | .945 | 3.718| 7.561| 1.30 30 | 1273.6 | .982 | 3.926| 7.852| 1.35 40 | 1228.1 | 1.018| 4.071| 8.143| 1.40 50 | 1185.8 | 1.055| 4.217| 8.433| 1.45 5 | 1146.3 | 1.091| 4.362| 8.724| 1.50 10 | 1109.3 | 1.127| 4.507| 9.014| 1.55 20 | 1074.7 | 1.164| 4.653| 9.305| 1.60 30 | 1042.1 | 1.200| 4.798| 9.596| 1.65 40 | 1011.5 | 1.237| 4.943| 9.886| 1.70 50 | 982.6 | 1.273| 5.088| 10.18| 1.75 6 | 955.4 | 1.309| 5.234| 10.47| 1.80 10 | 929.6 | 1.346| 5.379| 10.76| 1.85 20 | 905.1 | 1.382| 5.524| 11.05| 1.90 30 | 881.9 | 1.418| 5.669| 11.34| 1.95 40 | 859.9 | 1.455| 5.814| 11.63| 2.00 7° | 819.0 | 1.528| 6.105| 12.21| 2.10 20' | 781.8 | 1.600| 6.395| 12.79| 2.20 30 | 764.5 | 1.637| 6.540| 13.08| 2.25 40 | 747.9 | 1.673| 6.685| 13.37| 2.30 8 | 716.8 | 1.746| 6.976| 13.95| 2.40 20 | 688.2 | 1.819| 7.266| 14.53| 2.50 30 | 674.7 | 1.855| 7.411| 14.82| 2.55 40 | 661.7 | 1.892| 7.556| 15.11| 2.60 9 | 637.3 | 1.965| 7.846| 15.69| 2.70 20 | 614.6 | 2.037| 8.136| 16.27| 2.80 30 | 603.8 | 2.074| 8.281| 16.56| 2.85 40 | 593.4 | 2.110| 8.426| 16.85| 2.90 10 | 573.7 | 2.183| 8.716| 17.43| 3.00 30 | 546.4 | 2.292| 9.150| 18.30| 3.15 11 | 521.7 | 2.402| 9.585| 19.16| 3.30 30 | 499.1 | 2.511| 10.02| 20.04| 3.45 12 | 478.3 | 2.620| 10.45| 20.91| 3.60 30 | 459.3 | 2.730| 10.89| 21.77| 3.75 13 | 441.7 | 2.839| 11.32| 22.64| 3.90 30 | 425.4 | 2.949| 11.75| 23.51| 4.05 14 | 410.3 | 3.058| 12.18| 24.37| 4.20 30 | 396.2 | 3.168| 12.62| 25.24| 4.35 15 | 383.1 | 3.277| 13.05| 26.11| 4.50 30 | 370.8 | 3.387| 13.49| 26.97| 4.65 16 | 359.3 | 3.496| 13.92| 27.84| 4.80 30 | 348.5 | 3.606| 14.35| 28.70| 4.95 17 | 338.3 | 3.716| 14.78| 29.56| 5.10 18 | 319.6 | 3.935| 15.64| 31.29| 5.40 19 | 302.9 | 4.155| 16.51| 33.01| 5.70 20 | 287.9 | 4.374| 17.37| 34.73| 6.00 21 | 274.4 | 4.594| 18.22| 36.44| 6.30 22 | 262.0 | 4.814| 19.08| 38.16| 6.60 23 | 250.8 | 5.035| 19.94| 39.87| 6.90 24 | 240.5 | 5.255| 20.79| 41.58| 7.20 25 | 231.0 | 5.476| 21.64| 43.28| 7.50 26 | 222.3 | 5.697| 22.50| 44.99| 7.80 27 | 214.2 | 5.918| 23.35| 46.69| 8.10 28 | 206.7 | 6.139| 24.19| 48.38| 8.40 29 | 199.7 | 6.360| 25.04| 50.07| 8.70 30 | 193.2 | 6.583| 25.88| 51.76| 9.00 The middle ordinate in inches for any cord of length (C) is equal to .0012 C² multiplied by the middle ordinate taken from the above table. Thus, if it desired to bend a 30 ft. rail to fit a 10 degree curve, its middle ordinate should be .0012x900x2.183 or 2.36 inches. TABLE III. Deflections for Sub Chords for Short Radius Curves. Degree | Radius | sub chord = sin of ½ def. angle | Length of | 50 | 12.5 Ft. | 15 Ft. | 20 Ft. | 25 Ft. | of arc Curve | | | | | | for 100 ft. 30° | 193.18 | 1° 51' | 2° 17' | 2° 58' | 3° 43' | 101.15 32° | 181.39 | 1° 59' | 2° 25' | 3° 10' | 3° 58' | 101.33 34° | 171.01 | 2° 06' | 2° 33' | 3° 21' | 4° 12' | 101.48 36° | 161.80 | 2° 13' | 2° 41' | 3° 33' | 4° 26' | 101.66 38° | 153.58 | 2° 20' | 2° 49' | 3° 44' | 4° 40' | 101.85 40° | 146.19 | 2° 27' | 2° 57' | 3° 55' | 4° 54' | 102.06 42° | 139.52 | 2° 34' | 3° 05' | 4° 07' | 5° 08' | 102.29 44° | 133.47 | 2° 41' | 3° 13' | 4° 18' | 5° 22' | 102.53 46° | 127.97 | 2° 48' | 3° 21' | 4° 29' | 5° 36' | 102.76 48° | 122.92 | 2° 55' | 3° 29' | 4° 40' | 5° 50' | 103.00 50° | 118.31 | 3° 02' | 3° 38' | 4° 51' | 6° 04' | 103.24 52° | 114.06 | 3° 09' | 3° 46' | 5° 02' | 6° 17' | 103.54 54° | 110.11 | 3° 16' | 3° 54' | 5° 13' | 6° 31' | 103.84 56° | 106.50 | 3° 22' | 4° 02' | 5° 23' | 6° 44' | 104.14 58° | 103.14 | 3° 29' | 4° 10' | 5° 34' | 6° 57' | 104.43 60° | 100.00 | 3° 35' | 4° 18' | 5° 44' | 7° 11' | 104.72
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CURVE FORMULAS T = R tan I/2 T = 50 tan I/2 / Sin. D/2 Sin. D/2 = 50/R Sin. D/2 = 50 tan I/2 / T R = T cot. I/2 R = 50 / Sin. D/2 E = R ex. sec I/2 E = T tan I/4 Chord def. = chord²/R No. chords = I/D Tan. def. = 1/2 chord def. The square of any distance, divided by twice the radius, will equal the distance from tangent to curve, very nearly. To find angle for a given distance and deflection. Rule 1. Multiply the given distance by .01745 (def. for 1° for 1 ft. see Table II.), and divide given deflection by the product. Rule 2. Multiply given deflection by 57.3, and divide the product by the given distance. To find deflection for a given angle and distance. Multiply the angle by .01745, and the product by the distance. GENERAL DATA RIGHT ANGLE TRIANGLES. Square the altitude, divide by twice the base. Add quotient to base for hypotenuse. Given Base 100, Alt. 10. 10²÷200=.5. 100+.5=100.5 hyp. Given Hyp. 100, Alt. 25.25²÷200=3.125. 100-3.125=96.875=Base. Error in first example, .002; in last, .045. To find Tons of Rail in one mile of track: multiply weight per yard by 11, and divide by 7. LEVELING. The correction for curvature and refraction, in feet and decimals of feet is equal to 0.574d², where d is the distance in miles. The correction for curvature alone is closely, 3/8d². The combined correction is negative. PROBABLE ERROR. If d₁, d₂, d₃, etc. are the discrepancies of various results from the mean, and if Σd²=the sum of the squares of these differences and n=the number of observations, then the probable error of the mean= ±0.6745√Σd²/n(n-1) SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for the current year, published by Keuffel & Esser Co., and furnished free of charge upon request, which is 3¼x5⅝ in., with about 90 pages of data very useful to the Surveyor; such as the adjustments of transits, levels and solar attachments; directions and tables for determining the meridian and the latitude from observations on the sun and Polaris; stadia measurements; magnetic declination; arithmetic constants; English and Metric conversions; trigonometric formulas; Natural and Logarithmic Functions; and Logarithms of Numbers. TABLE IV. — Minutes in Decimals of a Degree. 1' .0167 | 11' .1833 | 21' .3500 | 31' .5167 | 41' .6833 | 51' .8500 2 .0333 | 12 .2000 | 22 .3667 | 32 .5333 | 42 .7000 | 52 .8667 3 .0500 | 13 .2167 | 23 .3833 | 33 .5500 | 43 .7167 | 53 .8833 4 .0667 | 14 .2333 | 24 .4000 | 34 .5667 | 44 .7333 | 54 .9000 5 .0833 | 15 .2500 | 25 .4167 | 35 .5833 | 45 .7500 | 55 .9167 6 .1000 | 16 .2667 | 26 .4333 | 36 .6000 | 46 .7667 | 56 .9333 7 .1167 | 17 .2833 | 27 .4500 | 37 .6167 | 47 .7833 | 57 .9500 8 .1333 | 18 .3000 | 28 .4667 | 38 .6333 | 48 .8000 | 58 .9667 9 .1500 | 19 .3167 | 29 .4833 | 39 .6500 | 49 .8167 | 59 .9833 10 .1667 | 20 .3333 | 30 .5000 | 40 .6667 | 50 .8333 | 60 1.0000 TABLE V. — Inches in Decimals of a Foot. 1-16 .0052 | 3-32 .0078 | ½ .0104 | 3-16 .0156 | ¼ .0208 | 5-16 .0260 | ⅜ .0313 | ½ .0417 | ⁵⁄₈ .0521 | ¾ .0625 | ⅞ .0729 1 .0833 | 2 .1667 | 3 .2500 | 4 .3333 | 5 .4167 | 6 .5000 | 7 .5833 | 8 .6667 | 9 .7500 | 10 .8333 | 11 .9167
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X Natural Trigonometrical Functions Angle. Sin. Tan. Sec. Cosec. Cotg. Cosin. 0° 10 .0029 .0029 343.8 343.8 1. 90 20 .0058 .0058 171.9 171.9 .99998 40 30 .0087 .0087 114.6 114.6 .99996 30 40 .0116 .0116 1.0001 85.94 85.94 .99993 20 50 .0145 .0145 1.0001 68.76 68.75 .99989 10 1 .0175 .0175 1.0002 57.30 57.29 .99985 89 10 .0204 .0204 1.0002 49.11 49.10 .99979 50 20 .0233 .0233 1.0003 42.98 42.96 .99973 40 30 .0262 .0262 1.0003 38.20 38.19 .99966 30 40 .0291 .0291 1.0004 34.38 34.37 .99958 20 50 .0320 .0320 1.0005 31.26 31.24 .99949 10 2 .0349 .0349 1.0006 28.65 28.64 .99939 88 10 .0378 .0378 1.0007 26.45 26.43 .99929 50 20 .0407 .0407 1.0008 24.56 24.54 .99917 40 30 .0436 .0437 1.0010 22.93 22.90 .99905 30 40 .0465 .0466 1.0011 21.49 21.47 .99892 20 50 .0494 .0495 1.0012 20.23 20.21 .99878 10 3 .0523 .0524 1.0014 19.11 19.08 .99863 87 10 .0552 .0553 1.0015 18.10 18.07 .99847 50 20 .0581 .0582 1.0017 17.20 17.17 .99831 40 30 .0610 .0612 1.0019 16.38 16.35 .99813 30 40 .0640 .0641 1.0020 15.64 15.60 .99795 20 50 .0669 .0670 1.0022 14.96 14.92 .99776 10 4 .0698 .0699 1.0024 14.34 14.30 .99756 86 10 .0727 .0729 1.0027 13.76 13.73 .99736 50 20 .0756 .0758 1.0029 13.23 13.20 .99714 40 30 .0785 .0787 1.0031 12.75 12.71 .99692 30 40 .0814 .0816 1.0033 12.29 12.25 .99668 20 50 .0843 .0846 1.0036 11.87 11.83 .99644 10 5 .0872 .0875 1.0038 11.47 11.43 .99619 85 10 .0901 .0904 1.0041 11.10 11.06 .99594 50 20 .0929 .0934 1.0043 10.76 10.71 .99567 40 30 .0958 .0963 1.0046 10.48 10.39 .99540 30 40 .0987 .0992 1.0049 10.18 10.08 .99511 20 50 .1016 .1022 1.0052 9.839 9.788 .99482 10 6 .1045 .1051 1.0055 9.567 9.514 .99452 84 10 .1074 .1080 1.0058 9.309 9.255 .99421 50 20 .1103 .1110 1.0061 9.065 9.010 .99390 40 30 .1132 .1139 1.0065 8.834 8.777 .99357 30 40 .1161 .1169 1.0068 8.614 8.556 .99324 20 50 .1190 .1198 1.0072 8.405 8.345 .99290 10 7 .1219 .1228 1.0075 8.206 8.144 .99255 83 10 .1248 .1257 1.0079 8.016 7.953 .99219 50 20 .1276 .1287 1.0082 7.834 7.770 .99182 40 30 .1305 .1317 1.0086 7.661 7.596 .99144 30 40 .1334 .1346 1.0090 7.496 7.429 .99106 20 50 .1363 .1376 1.0094 7.337 7.269 .99067 10 82° Angle. Sin. Tan. Sec. Cosec. Cotg. Cosin. 8 .1392 .1405 1.0098 7.185 7.115 .99027 82 10 .1421 .1435 1.0102 7.040 6.968 .98986 50 20 .1449 .1465 1.0107 6.900 6.827 .98944 40 30 .1478 .1495 1.0111 6.766 6.691 .98902 30 40 .1507 .1524 1.0115 6.636 6.561 .98858 20 50 .1536 .1554 1.0120 6.512 6.435 .98814 10 9 .1564 .1584 1.0125 6.394 6.314 .98769 81 10 .1593 .1614 1.0129 6.277 6.197 .98723 50 20 .1622 .1644 1.0134 6.166 6.084 .98676 40 30 .1650 .1673 1.0139 6.059 5.976 .98629 30 40 .1679 .1703 1.0144 5.955 5.871 .98580 20 50 .1708 .1733 1.0149 5.855 5.769 .98531 10 10 .1736 .1763 1.0154 5.759 5.671 .98481 80 10 .1765 .1793 1.0160 5.665 5.578 .98430 50 20 .1794 .1823 1.0165 5.575 5.485 .98378 40 30 .1822 .1853 1.0170 5.488 5.396 .98325 30 40 .1851 .1883 1.0176 5.403 5.309 .98272 20 50 .1880 .1914 1.0181 5.320 5.226 .98218 10 11 .1908 .1944 1.0187 5.241 5.145 .98163 79 10 .1937 .1974 1.0193 5.164 5.066 .98107 50 20 .1965 .2004 1.0199 5.089 4.989 .98050 40 30 .1994 .2035 1.0205 5.016 4.915 .97992 30 40 .2022 .2065 1.0211 4.945 4.843 .97934 20 50 .2051 .2095 1.0217 4.877 4.773 .97875 10 12 .2079 .2126 1.0223 4.810 4.705 .97815 78 10 .2108 .2156 1.0230 4.745 4.638 .97754 50 20 .2136 .2186 1.0236 4.682 4.574 .97692 40 30 .2164 .2217 1.0243 4.620 4.511 .97630 30 40 .2193 .2247 1.0249 4.560 4.449 .97566 20 50 .2221 .2278 1.0256 4.502 4.390 .97502 10 13 .2250 .2309 1.0263 4.445 4.331 .97437 77 10 .2278 .2339 1.0270 4.390 4.275 .97371 50 20 .2306 .2370 1.0277 4.336 4.219 .97304 40 30 .2334 .2401 1.0284 4.284 4.165 .97237 30 40 .2363 .2432 1.0291 4.232 4.113 .97169 20 50 .2391 .2462 1.0299 4.182 4.061 .97100 10 14 .2419 .2493 1.0306 4.133 4.011 .97030 76 10 .2447 .2524 1.0314 4.086 3.962 .96959 53 20 .2476 .2555 1.0321 4.039 3.914 .96887 40 30 .2504 .2586 1.0329 3.994 3.867 .96815 30 40 .2532 .2617 1.0337 3.949 3.821 .96742 20 50 .2560 .2648 1.0345 3.906 3.776 .96667 10 15 .2588 .2679 1.0353 3.864 3.732 .96593 75 10 .2616 .2711 1.0361 3.822 3.689 .96517 50 20 .2644 .2742 1.0369 3.782 3.647 .96440 40 30 .2672 .2773 1.0377 3.742 3.606 .96363 30 40 .2700 .2805 1.0386 3.703 3.566 .96285 20 50 .2728 .2836 1.0394 3.665 3.526 .96206 10 74° Cosin. Cotg. Cosec. Sec. Tan. Sin. Angle Cosin. Cotg. Cosec. Sec. Tan. Sin. Angle
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Natural Trigonometrical Functions Angle Sin. Tan. Sec. Cosec. Cotg. Cosin. 16 .2756 .2867 1.0403 3.628 3.487 .96126 74 10 .2784 .2899 1.0412 3.592 3.450 .96046 50 20 .2812 .2931 1.0423 3.556 3.412 .95964 40 30 .2840 .2962 1.0429 3.521 3.376 .95882 30 40 .2868 .2994 1.0438 3.487 3.340 .95799 20 50 .2896 .3026 1.0448 3.453 3.305 .95715 10 17 .2924 .3057 1.0457 3.420 3.271 .95630 73 10 .2952 .3089 1.0466 3.388 3.237 .95545 50 20 .2979 .3121 1.0476 3.357 3.204 .95459 40 30 .3007 .3153 1.0485 3.326 3.172 .95372 30 40 .3035 .3185 1.0495 3.295 3.140 .95284 20 50 .3062 .3217 1.0505 3.265 3.108 .95195 10 18 .3090 .3249 1.0515 3.236 3.078 .95106 72 10 .3118 .3281 1.0525 3.207 3.048 .95015 50 20 .3145 .3314 1.0535 3.179 3.018 .94924 40 30 .3173 .3346 1.0545 3.152 2.989 .94832 30 40 .3201 .3378 1.0555 3.124 2.960 .94740 20 50 .3228 .3411 1.0566 3.098 2.932 .94646 10 19 .3256 .3443 1.0576 3.072 2.904 .94552 71 10 .3283 .3476 1.0587 3.046 2.877 .94457 50 20 .3311 .3508 1.0598 3.020 2.850 .94361 40 30 .3338 .3541 1.0608 2.996 2.824 .94264 30 40 .3365 .3574 1.0619 2.971 2.798 .94167 20 50 .3393 .3607 1.0631 2.947 2.773 .94068 10 20 .3420 .3640 1.0642 2.924 2.747 .93969 70 10 .3448 .3673 1.0653 2.900 2.723 .93869 50 20 .3475 .3706 1.0665 2.878 2.699 .93769 40 30 .3502 .3739 1.0676 2.856 2.675 .93667 30 40 .3529 .3772 1.0688 2.833 2.651 .93565 20 50 .3557 .3805 1.0700 2.811 2.628 .93462 10 21 .3584 .3839 1.0711 2.790 2.605 .93358 69 10 .3611 .3872 1.0723 2.769 2.583 .93253 50 20 .3638 .3906 1.0736 2.749 2.560 .93148 40 30 .3665 .3939 1.0748 2.729 2.539 .93042 30 40 .3692 .3973 1.0760 2.708 2.517 .92935 20 50 .3719 .4006 1.0773 2.689 2.496 .92827 10 22 .3746 .4040 1.0785 2.670 2.475 .92718 68 10 .3773 .4074 1.0798 2.650 2.455 .92609 50 20 .3800 .4108 1.0811 2.632 2.434 .92499 40 30 .3827 .4142 1.0824 2.613 2.414 .92388 30 40 .3854 .4176 1.0837 2.595 2.394 .92276 20 50 .3881 .4210 1.0850 2.577 2.375 .92164 10 23 .3907 .4245 1.0864 2.559 2.356 .92050 67 10 .3934 .4279 1.0877 2.542 2.337 .91936 50 20 .3961 .4314 1.0891 2.525 2.318 .91822 40 30 .3987 .4348 1.0904 2.508 2.300 .91706 30 40 .4014 .4383 1.0918 2.491 2.282 .91590 20 50 .4041 .4417 1.0932 2.475 2.264 .91472 10 66 Angle Sin. Tan. Sec. Cosec. Cotg. Cosin. 24 .4067 .4452 1.0946 2.459 2.246 .91355 68 10 .4094 .4487 1.0961 2.443 2.229 .91236 50 20 .4120 .4522 1.0975 2.427 2.211 .91116 40 30 .4147 .4557 1.0989 2.411 2.194 .90996 30 40 .4173 .4592 1.1004 2.396 2.177 .90875 20 50 .4200 .4628 1.1019 2.381 2.161 .90753 10 25 .4226 .4663 1.1034 2.366 2.145 .90631 65 10 .4253 .4699 1.1049 2.351 2.128 .90507 50 20 .4279 .4734 1.1064 2.337 2.112 .90383 40 30 .4305 .4770 1.1079 2.323 2.097 .90259 30 40 .4331 .4806 1.1095 2.309 2.081 .90133 26 50 .4358 .4841 1.1110 2.295 2.066 .90007 10 26 .4384 .4877 1.1126 2.281 2.050 .89879 64 10 .4410 .4913 1.1142 2.268 2.035 .89752 50 20 .4436 .4950 1.1158 2.254 2.020 .89623 40 30 .4462 .4986 1.1174 2.241 2.006 .89493 30 40 .4488 .5022 1.1190 2.228 1.991 .89363 20 50 .4514 .5059 1.1207 2.215 1.977 .89232 10 27 .4540 .5095 1.1223 2.203 1.963 .89101 63 10 .4566 .5132 1.1240 2.190 1.949 .88968 50 20 .4592 .5169 1.1257 2.178 1.935 .88835 40 30 .4617 .5206 1.1274 2.166 1.921 .88701 30 40 .4643 .5243 1.1291 2.154 1.907 .88566 20 50 .4669 .5280 1.1308 2.142 1.894 .88431 10 28 .4695 .5317 1.1326 2.130 1.881 .88295 62 10 .4720 .5354 1.1343 2.119 1.868 .88158 50 20 .4746 .5392 1.1361 2.107 1.855 .88020 40 30 .4772 .5430 1.1379 2.096 1.842 .87882 30 40 .4797 .5467 1.1397 2.085 1.829 .87743 20 50 .4823 .5505 1.1415 2.073 1.816 .87603 10 29 .4848 .5543 1.1434 2.063 1.804 .87462 61 10 .4874 .5581 1.1452 2.052 1.792 .87321 50 20 .4899 .5619 1.1471 2.041 1.780 .87178 40 30 .4924 .5658 1.1490 2.031 1.767 .87036 30 40 .4950 .5696 1.1509 2.020 1.756 .86892 20 50 .4975 .5735 1.1528 2.010 1.744 .86748 10 30 .5000 .5774 1.1547 2.000 1.732 .86603 60 10 .5025 .5812 1.1566 1.990 1.720 .86457 50 20 .5050 .5851 1.1586 1.980 1.709 .86310 40 30 .5075 .5890 1.1606 1.970 1.698 .86163 30 40 .5100 .5930 1.1626 1.961 1.686 .86015 20 50 .5125 .5969 1.1646 1.951 1.675 .85866 10 31 .5150 .6009 1.1666 1.942 1.664 .85717 59 10 .5175 .6048 1.1687 1.932 1.653 .85567 50 20 .5200 .6088 1.1707 1.923 1.643 .85416 40 30 .5225 .6128 1.1728 1.914 1.632 .85264 30 40 .5250 .6168 1.1749 1.905 1.621 .85112 20 50 .5275 .6208 1.1770 1.896 1.611 .84959 10 58
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"TRIGONOMETRIC FORMULAE" Right Triangle Oblique Triangles Solution of Right Triangles For Angle A. sin = a/c, cos = b/c, tan = a/b, cot = b/a, sec = c/b, cosec = c/a Given Required a, b A, B, c tan A = a/b = cot B, c = √(a² + b²) = a√(1 + b²/a²) a, c A, B, b sin A = a/c = cos B, b = √((c+a)(c-a)) = c√(1 - a²/c²) A, a B, b, c B=90°-A, b=a cot A, c=a/sin A. A, b B, a, c B=90°-A, a=b tan A, c=b/cos A. A, c B, a, b B=90°-A, a=c sin A, b=c cos A, Solution of Oblique Triangles Given Required A, B, a b, c, C b = (a sin B)/sin A, C = 180°-(A+B), c = (a sin C)/sin A A, a, b B, c, C sin B = (b sin A)/a, C=180°-(A+B), c=(a sin C)/sin A a, b, C A, B, c A+B=180°-C, tan ½(A-B)=((a-b)tan ½(A+B))/(a+b), c=(a sin C)/sin A a, b, c A, B, C s=(a+b+c)/2, sin ½A=√(((s-b)(s-c))/(bc)), sin ½B=√(((s-a)(s-c))/(ac)), C=180°-(A+B) a, b, c Area s=(a+b+c)/2, area = √(s(s-a)(s-b)(s-c)) A, b, c Area area = (bc sin A)/2 A, B, C, a Area area = (a² sin B sin C)/(2 sin A) REDUCTION TO HORIZONTAL Slope distance Vert. Angle Rise Horizontal distance When the rise is known, the horizontal distance is approximately:—the slope dist- ace less the square of the rise divided by twice the slope distance. Thus: rise=14 ft. slope distance=302.6 ft. Horizontal distance=302.6-(14×14)/(2×302.6)=302.6-0.32=302.28 ft. Horizontal distance=Slope distance multiplied by the cosine of the vertical angle. Thus: slope distance=319.4 ft. Vert. angle=5° 10'. From Table, Page IX. cos 5° 10'= .9959. Horizontal distance=319.4×.9959=318.09 ft. Horizontal distance also=Slope distance minus slope distance times (1-cosine of vertical angle). With the same figures as in the preceding example, the follow- ing result is obtained. Cosine 5° 10'= .9959. 1-.9959=.0041. 319.4×.0041=1.31. 319.4-1.31=318.09 ft. MADE IN U. S. A.
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GENERAL OUTCROP SECTION OF THE BLACK HILLS AREA FORMATION SECTION THICKNESS IN FEET DESCRIPTION QUATERNARY SANDS AND GRAVELS 0-50 Sand, gravel, and boulders PLIOCENE OGALLALA GROUP 0-200 Light colored sands and silts MIOCENE ARIKAREE GROUP 0-50 Light colored clays and silts White bed at base OLIGOCENE WHITE RIVER GROUP Brule Formation Chadron Formation 0-600 Light colored clays with sandstone channel fillings and local limestone lenses PALEOCENE FORT UNION FORMATION TONGUE RIVER MEMBER 0-425 Light colored clays and sands, with coal beds further north CANNONBALL MEMBER 1-225 Green MARINE shales and yellow sandstones; latter often as concretions LUDLOW MEMBER 0-350 So[?] gray clays and sandstones with beds of lignite. CRETACEOUS HELL CREEK FORMATION (Lance Formation) 425 Smoky colored soft brown shale and gray Sandstone, with thin lignite lenses in the upper part Lower half more sandy Many loglike concretions and thin lenses of iron carbonate FOX HILLS SANDSTONE 25-200 Grayish white to yellow sandstone Calc. PIERRE SHALE 1200-2000 Principal horizon of limestone lenses giving tepee buttes Dark gray shale containing scattered concretions. Widely scattered limestone masses, giving small tepee buttes. Black fissile shale with concretions. NIOBRA Formation 100-225 Impure chalk and calcareous shale CARLILE FORMATION Turner Sand Zone Wall Creek Sands 400-750 Light-gray shale with numerous large concretions and sandy layers Dark gray shale GREENHORN FORMATION (25-30) (200-350) Impure slabby limestone Weathers buff Dark gray calcareous shale, with thin Orman Lake limestone at base BELLE FOURCHE SHALE 300-450 Gray shale with scattered limestone concretions Clay sour bentonite at base MOWRY SHALE 150-250 Light-gray siliceous shale Fish scales and thin layers of bentonite NEWCASTLE SANDSTONE 20-60 Brown to light yellow and white sandstone SKULL CREEK SHALE 170-270 Dark gray to black shale. FALL RIVER [OAKOTA (?) S.S.] FUSON SHALE 10-200 Massive to slabby sandstone. MINNEWASTE LIMESTONE 10-188 Gray to purple shale Thin sandstones LAKOTA SANDSTONE 0-25 Massive gray limestone 25-485 Coarse, hard, crossbedded sandstone mostly buff to gray Conglomerate locally and coal at base MORRISON FORMATION 0-220 Green to maroon shale Thin sandstone. UNKPAPA SANDSTONE 0-225 Massive fine-grained sandstone. SUNDANCE FORMATION 250-450 Greenish-gray shale, thin limestone lenses. Glaucanitic sandstone; red xs near middle GYPSUM SPRING FORMATION 0-45 Red siltstone, gypsum, and limestone NUGGET (?) FORMATION 0-60 Massive crossbedded salmon sandstone TRIASSIC ? SPEARFISH FORMATION 300-700 Red sandy shale, soft red sandstone and siltstone with gypsum and thin limestone layers Gypsum locally near the base. PERMIAN MINNEKAHTA LIMESTONE 30-50 Massive gray, laminated limestone OPECHE FORMATION 50-135 Red shale and sandstone PENNSYLVANIAN MINNELUSA SANDSTONE 300-850 Yellow to red cross-bedded sandstone, limestone, and anhydrite locally at top Interbedded sandstone, limestone, dolomite, shale, and anhydrite Red shale with interbedded limestone and sandstone at base MISSISSIPPIAN PAHASAPA (MADISON) LIMESTONE 300-630 Massive light-colored limestone Dolomitic in part ENGLEWOOD LIMESTONE 30-60 Pink to buff limestone Shale locally at base ORDOVICIAN WHITEWOOD (RED RIVER) FORMATION 0-60 Buff dolomite and limestone WINNIPEG FORMATION 0-100 Green shale with siltstone at top and white sandstone at base CAMBRIAN DEADWOOD FORMATION 10-400 Massive to buff sandstone Greenish glauconitic shale, fraggy dolomite and flat-pebble limestone conglomerate Sandstone, with conglomerate locally at the base PRE-CAMBRIAN METAMORPHIC and IGNEOUS ROCKS Schist, slate, quartzite, and arkose grit Intruded by diorite, metamorphosed to amphibolite, and Sr granite and pegmatite