c « '^ <^ « -^O ^,;^'?ff:^i^'^ ^■^' >^/> ^^ '^ s s ' <• / ^ »'o^".v. P.^ D' ""^i^^V^ '^ " ' ' a"^^ c " ^ "^ f S'' h^.' .-X- A^ > •/• ^ ♦ s, S " ' / ^ m^ c^^:^^ ^1%. ^^'-^-'^-J ^^. % ^^^> 0^ s" - V. >P i ' ^ai^-''/^: .^^* .^:'€^%', '- ^ '\^^m^ ■r * 0^ * i' . .. «^ >^' ^ "■ ■ '■■■ ^ , '^. %1 ~^ "^ '-*^ x'^^' ^^. r^'?.- ->. / c^^.-^^^^ ^!^' '"^- '-^"^ ,\ THE FIELD ENGINEER: A IHantis 33oolt of Practice IN THE URVEY, LOCATION, AND TRACK-WORK OF RAILROADS ; COXTAIXISTG A LAEGE COLLECTION Of EULES AND TABLES, ORIGINAL AND SELECTED, APPLICABLE TO BOTH THE STANDARD AND THE NARROW GAUGE. AND PREPARED WITH SPECIAL REFERENCE TO THE WANTS OF THE YOUNG 'ENGLs'EEIi. WILLIAM FINDLAYSHUNK, C.E. FIFTEENTH EDITION, EE VISED AND ENLAEGED. NEW YORK: D. VAN NOSTRAND COMPANY, 23 MUEBAY AND 27 WaKREN STRBBTS. 1903. THE LIBRARY @F CONGRESS, Two Copies Received APR f 1903 Copyright Entry CLASS CC )»evelHng and tangent screws, bring either of the cross-hairs to coincide with a well-defined object, distant from 400 to 000 feet, or as much farther as distinct vision can be had free from heat ripple. Gently rotate the telescope half-way around in the wyes. If the cross-hair selected for treatment then fails to coincide with the object, reduce the error one-half by means of the small capstan head screws at right angles to it on the telescope-barrel. Bring the spider-line again to coincide with the object by means of tbe levelling and tangent screws, and, if necessary, repeat the operation. Proceed in the same man- ner with the other cross-hair. If the error is large, bring both nearly right before undertaking their final adjustment. 2. Having thus adjusted the line of collimation upon a dis- tant point, requiring the object-tube to be drawn well in, select a point close by, which shall require it to be thrust out almost to its limit. If any error appears, correct half of it with the small screws provided for the purpose, a little forward of the diaphragm, and usually protected by a movable sleeve on the outside; correct the other half with the levelling-screws. After completing this adjustment, test the former one on a distant object, and. if necessary, repeat the operations. 3. In the transit, the small guide-ring screws used for this adjustment are covered by the bulb of the cross-bar in which the telescope is fixed, and are therefore inaccessible. The adjustment, however, is one not liable to become deranged in either instrument, and, in the transit, is of comparatively small importance. 4. The young practitioner should bear in mind that the intersection of the cross-hairs may coincide with the optical axis of the telescope, and yet be out of centre as regards the field of view. Such' eccentricity does not affect the working accuracy of the instrument, which depends upon the position TUE LEVEL. 25 Df the object-piece solely. It may be removed by manipulation of the small screws securing the inner end of the eye-piece. TO BEING THE LEVEL BUBBLE PARALLEL WITH THE TELE- SCOPE AXIS. 5. Clamp the instrument over either pair of levelling screws, and bring the bubble to the middle of its tube. Turn the tele- scope slightly on its bearings, so that the bubble-case shaii project a little on one side or the other. If the bubble sli]is, coi-rect half its movement by means of the small lateral capstan head screws at one end of the case. Return the telescope tc its first position, level up again, and repeat the operation until the erroneous movement ceases. This adjustment brings the telescope and level into the same vertical plane. G. Next, the bubble being at the middle of its tube, carefully lift the telescope out of the wyes, turn it end for end, and replace it. If the bubble settles away from the middle, bring it half-way back by means of the capstan-heads, working up and down at one end of the case. Again middle it with the levelling screws, and repeat the opei-ation until the error is corrected. rO ADJUST THE WYES ; OE, IN OTHER WORDS, TO BRING THE TELESCOPE INTO A POSITION AT RIGUT ANGLES TO THE VEETICAL AXIS OF THE INSTRUMENT. 7. Close the wyes. Unclamp. Set the telescope directly over two of the levelling screws, and with them bring the bubble to the middle of the tube. Then rotate the telescoi^o horizontally, until it stands over the same pair of screws, changed end for end. If the bubble errs, correct one-half of the deviation with the capstan head nuts at the end of the bar, and one-half with the levelling screws. Place the tele- scope over the other pair of levelling screws, liepeat the operation there; and continue the corrections, over one and the other pair of levelling screws alternately, until the bubbk' stands without varying during an entire revolution of Hk- instrument upon its vertical axis. 8. The capstan head nuts on the cross-bar should be moved by gradual stress, not by pounding. They are a I'ude deviec With so short a leverage as the length of the common adjusi- ing-pin suppUes, it is almost impossible to give them a smooth. 26 LEVELLING. manageable motion. They reproach the instrument-maker's art as unchecked hydrophobia and cancer do that of medicine, or mercenary villany that of law, and should be supplanted by better practice. 9. Having thus completed the principal adjustments in their proper order, bring the telescope and its l)ubble-case as nearly vertical in the wye bciunigs as hand and eye can make them, and by reference to a plumb-line, or the corner of a well-built house, see if the vertical hair is out of true. If so, slightly loosen two opposite screws of the diaphragm, and correct the error by turning it. Try again the adjustment of the line of collimation before pinning up the wyes. XI. LEVELLING. 1. Suppose O the starting-point; 1, 2, 3, &c., the stakes of survey; and A the initial bench-mark. Wherever convenient the elevation of A above mean tide should be ascertained. It is to be regretted that this was not done from the outset. under statute provisions requiring maps and profiles also to be filed at the several State capitals. In that case, not only would much after labor and expense by way of duplicate sur- veys have been spared, but the older Commonwealths would now have in hand materials for the preparation of physio- graphical maps, the value of which to science, to the engineer, and to the economical geologist, it were hard to measure. LEVELLING. 27 2. For the purposes of a railroad-survey, however, such determination is not needful. Any elevation may be assumed for A, taking care only that it be large enough to avoid the possibility of having minus levels, which would be inconven- ient. Zero of the datum should be below the lowest probable ground on the contemplated line. 3. Let the elevation of the initial bench-mark, A, in the figure, be taken at -f-200. Set the level at B, and suppose the rod on the BM to read 2.22. The " instrument height " then is 202.22. If the rod at sta. O reads 8.4, the elevation at that point is 202.22 — 8.4 = 193.8. The rod reading 1.9 at sta. 1, the elevation there is 202.2 — 1.9 = 200.3. If desirable to turn at sta. 2, drive a pin nearly to the ground at that stake ; sup- pose the rod on it to read 0.81. The elevation then is 202.22 — 0.81 = 201.41. Now move the instrument to C, and, sighting back to sta. 2, let the rod standing on the pin read 2.64. This makes the new " instrument heiglu" at C = 201.41, the height of sta. 2, -f 2.64 == 204.05, and the elevations at 3, 4, 5, or other points observed from C are found by deducting the readings at those points from the ascertained instrument height at the new point of observation. " 4. It thus appears how simple is the rule of levelling, namely: Find the "instrument height" by adding the "back- sight" to the elevation of the point upon which the rod stands for that purpose : from the "instrument height" thus found deduct the " foresights," severally, in order to find the eleva- tions of the points at which such foresights are taken. 5. The foregoing example woidd appear in the field-book as follows : — Sta. B. S. Inst. F.S. Eleva. Remarks. BM 200.00 B M on W. Oak. 2.22 202.22 40 ft. N. of Sta. O. .. .. 8.4 193.8 1 1.9 200.3 2 2!64 204 '.05 0.81 201.41 3 .. 3.7 200.3 4 .. 3.2 200.8 5 10.36 193.69 1 G. In levelling where great exactness is necessary, the rod at t-uruing-points should be read to thousandths, and the reading 1 is perpendicular to BC. 4'.) 50 PROPOSITIONS RELATING TO THE CIRCLE. 2. Tangents drawn to a circle from the same point are equal Thus, I B = I E. > 3. The angle DIE, at the intersection of tangents, is equal to the central angle B C E, inchuU'd bcLween radii to the tan- gent points. 4. If a chord BE connect the tangent points, the angles I BE, lEB, are equal: each of them is equal to half of the central angle BCE, or of the intersection angle DIE. 5. Any angle, BCE, at the centre, subtended by the chord BE, is double the angle BFE, at the circumference, on the same side of the chord B E. 6. Acute angles at the circumference, subtended by equal chords, are equal. 7. An acute angle, KFH, between a tangent and a chord, is called a tangential angle, and is equal to the peripheral angle LFH subtended by an equal chord; each is equal to half the central angles FCH, or HCL, subdivided by the same chords. 8. The exterior angle LHN at the circumference, between two equal chords, is called a deflection angle : it is equal to the central angle, or to twice the tangential angle, subtended by either chord. 9. If F K be made equal to F H, and H X be made equal to HL, HK is called the tangential distance, and LN the deflec- tion distance. 10. The exterior angle E HIS" at the circumference, between two unequal chords, is equal to the sum of their tangential angles, or to half the sum of their central angles. XVII. CIRCULAR CURVES ON RAILROADS. 1. The circle is divided, for convenience, into 360 equal parts, called degrees. A circle 36,000 feet in circumference would be cut by such subdivision into 360 parts, each 100 feet long, and subtending an angle of one degree at the centre; its radius would be 5,729.6 feet, usually reckoned 5,730 feet. The CIRCULAR CURVES ON RAILROADS. 51 chain 100 feet long being llie unit generally adoiDted by Ameri- can engineers for field measurements, any circular arc liaviug that radius, of 5,T;jO feet, is called a one-degree curve, for the reason that one chain is equivalent to an arc of one degree at the circumference. 2. The circumferences of circles vary directly as their radii; hence, in any circular arc struck with half that radius, or 2,865 feet, one hundred feet at the circumference would sub- tend an angle of two degrees at the centre. Such an arc is called a two-degree curve. If one-third of the primary radius of 5,730 feet, or 1,910 feel, be used, the arc is called a three- degree curve; and so on. 3. It should be borne in mind, however, that these measure- ments are supposed to be made around the arc itself, and not on lines of chords. Since field measurements with the chain are always made on the lines of the choi'ds, which are shorter between given points at the circumference than the lines of the arcs, as a bowstring is shorter than the bow, it is plain that, in advancing tov/ards the centi'e of the one-degree curve by a series of concentric circles having radii equal to one-half, one-third, &c., of the i)riniary radius, the chord 100 feet long differs more and more in length from the arc subtended by it, the bow being more and more arched in relation to the string. Thus, in the circle having a radius equal to one-twentieth of the primary radius, the chord 100 feet long subtends an angle of 20° 06', at the centre, instead of 20°, and the arc is 100.5 feet in length, instead of 100 feet. In order, therefore, that the chord of 100 feet may subtend arcs of 1°, 2°, 3°, &c., in regular succession, the radii of these successive arcs must be somewhat greater than the above method by subdivision of the primary radius would make them; though, as might be inferred from the extreme case given by way of illustration, the dif- ference is not appreciable in ordinary field practice, and radii, together with all the functions dependent on them, may usually be held to vary as the degree of curvature, or central angle per 100 feet chord, varies. TO FIND THE RADIUS OF A CURVE. XYIII. TO FIND THE RADIUS, THE APEX DISTANCE, TH] LENGTH, THE DEGREE, ETC., OF A CURVE. 1. Let D B, A L be two straight lines intersecting at D. Lay off equal distances, D A, D B; erect perpendiculars at A and B, meeting at G, and con- nect A B, D G. From the centre G, with radius G A, draw the curve A H B. The point D will be the P. I.; A and B, tangent points; D A, D B, the tan- gents, or apex distances, which denote byAD; D H, the external secant, or S; HN, the middle ord, or O. Let the long chord A B, connecting the tan- gent points, be called C, Call the deflection angle to a DBA=AGD=DGB and G A or G B, the radius, R chord of 100 feet D, as before. 3. By XVI. 3 and 4, angle DAB = iL 3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO FIND THE APEX DIST. AD. A D = R X tan. i I. Example. R= 1,910.1, I = 35° 24'. Then A D = R tan. i I =r 1,910.1 X 0.3191 = 609.5. 4. Measure from the P. I. equal distances, D M, D F, along the tangents. Measure, also, MF and D K, the distance from D to the middle point of MF. Then, by reason of similarity in the triangles M D K, D A G, MK:DK::AG:AD::R:T .•.AD=RxDK-v-MK. TO FIND THE RADIUS OF A CURVE. 53 Let M K = 190.5, D K = 60.8, R = 1910.1. Then R = 1910.1 .... 3.281056 DK= 60.8 .... 1.788904 MK= 190.5 (a.c.) '. . . 7.720105 AD= 609.6 . . . . 2.785065 5. If 100- feet chords be used, find the ap. dist. in Table XVI. corresponding to the given angle I. Divide that tabular ap. dist. by the degree of curvature corresponding to the given radius: the quotient will be the required ap. dist. Thus, Tab. A D corresponding to BS"" 24' = 1,828.7, which, divided by 3, the degree of curvature, gives 609.6, the ap. dist. sought. 6. GIVEN THE INTERSECTION ANGLE I AND AP. DIST. AD, TO FIND RADIUS R. Transposing the equation in (8), R = AD-T- tan. ^ 1= A D Xcot. i I. Example. 4D=609.6, 7=35° 24' R=ADcot. i 7=609.6 X 3.1334=1910.1. B}'- a like transposition of the equation in (4), R = ADxMK-^DK. 7. If 100-feet chords be used, find in Table XVI. the ap. dist. corresponding to the given angle I. Divide that tabular datum by the given ap. dist.; the quotient will be the degree of curvature in degrees and decimals. The radius corre- sponding to this degree of curvature may be found by (12), by Table X., or, with sufficient accuracy for ordinary practice, by dividing 5,730, the radius of a 1° curve, by it. Thus, in the foregoing example, the tabular ap. dist. cor- responding to 35° 24' is 1,828.7. Dividing by 609.6, we have 3 for the degree of curvature; and 5,730 divided "by 3 gives R= 1,910 feet. 54 TO FIND THE RADIUS OF A CURVE. %. GIVEN THE INTERSECTION ANGLE I AND CHORD A B = C, CONNECTING THE TANGENT POINTS, TO FIND RADIUS R. AG = A N-^sin. AGN ; i.e. j B = ^ C -^ sin. i I. Example. 1=35° 24', C= 1161.4. Then R = ^C^ sin. 1 1, = 580.7 -^ 0.304 = 1910.2. 9. If 100-feet cliords be used, find in Table XYI. the chord corresponding to the given angle I. Divide that chord by the given chord, for the degree of curvature in degrees and deci- mals. Determine the corresponding radius by (17), by Table X., or, for ordinary practice, by dividing 5,730 by it. Thus, in the foregoing example, the tabular chord corre- sponding to angle 35° 24' would be 3,484.2, which, divided by the given chord, 1,161.4, gives 3 for the degree of curvature, and 5,730 divided by 3 makes the radius R = 1,910 feet. 10. GIVEN THE INTERSECTION ANGLE I AND THE DEGREE OF CURVATURE OR DEFLECTION ANGLE D, WITH 100-FEET CHORDS, TO DETERMINE THE LENGTH OF THE LONG CHORD C, THE MIDDLE ORD. O, THE iLXTERNAL SECANT S, OR THE APEX DIST. A D. Take from the proper column in Table XYI., the number corresponding to the intersection angle, and divide it by the degree of curvature: the quotient will be the length required. Example. A 4P curve, I^ 50° 16'; to find the several functions above named. Table XVI. gives the designated functions of a 1° curve as follows: C 4,867.8, O 543.4, S 599.8, AD 2,688.3. Dividing by 4 the degree of curvature, we have for the corresponding functions of a 4° curve as follows: C 1,316.8, O 185.6, S 149.8, A D 673.0. BADU, DEFLECTION ANGLES, ETC. 55 11. GIVEN C, O, S, OR A D, OF ANY CURVE, AND D, THE DE- GREE OF CURVATURE, TO FIND THE INTERSECTION ANGLE, I. Multiply the given fuDction C, O, S, or AD, by the degree of curvature, D: the product will be found in the proper col- umn of Table XVI., corresponding to the required angle. Example 1. Given A D = 515, D = 5°; to find I. Then A D X D = 3,575, which corresponds in Table XVI. to 48° 34' = I. Example 3. Given C = 1,656, D = 3°; to find I. Then C X D = 4,968. which corresponds in Table XVI. to 51° 33= I. 13. GIVEN C, O, S, OR A D, OF ANY CURVE, AND THE INTER- SECTION ANGLE I, TO FIND THE DEGREE OF CURVATURE D. Take from the proper column of Table XVI. the number corresponding to the given angle I, and divide that tabular number by the length of the given part; the quotient will be D, in degrees and decimals. 1. Given A D = 587, I = 33° 36'; to find D. The A D corresponding to I in Table XVI. is 1,136.3. Then 1,136.3 -^ 587 = 1.935 = 1° 56' = D. Example 2. Given S = 64, I = 30° 25', to find D. The Ex. Sec. corresponding to I in Table XVI. is 208. Then 208 -^ 64 = 3.25 = 3° 15' = D. 13. GIVEN THE INTERSECTION ANGLE I, AND DEFLECTION ANGLE D, TO FIND THE LENGTH OF THE CURVE. Divide I by D: the quotient will be the number of chord lengths in the curve. If the degree of curvature is a whole number, the more con- venient method of effecting the division is, first, to reduce the 56 RADII, DEFLECTION ANGLES, ETC. raiiiiites, if any, in I to decimals of a degree; then divide bjf the degree of curvature. Example 1. I = 20° 40', 1) = 3°. 20° 40' is equivalent to 20.67 degrees. Dividing by :>, we liave 6.89, chord lengths for the length of the curve. If the chords, as is usual, are each 100 feet long, the length of the curve in this case will be 689 feet. If the chord lengths were 50 feet eacli, the length of the curve would be half this nunil)er of feet. 14. If the degree of curvature is fractional, the more con- venient method of effecting the division is, first, to reduce both I and D to minutes; then divide the former by the latter. Example 2, I = 30° 22', D = 2° 45'. These are equivalent, respectively, to 1,822 and 165 minutes. Dividing the former by the latter, we have 1,104 feet for the length of the curve. 15. The ingenious assistant who will attentively consider the preceding figures cannot fail to detect other obvious analo- gies which it has not been thought necessary to include in this compendium. 16. In railroad field practice it is usually sufficient to deter- mine angles to the nearest minute, and distances to the nearest foot. The nicety of seconch and tenths appears generally to be quite superfluous; the time consumed on them were better employed in pushing ahead. 17. GIVEN ANY DEFLECTION AN- GLE D, AND CHORD C, TO FIND RADIUS R. FB -^ sin. i AL B = BL ; i.e., i C-^sin. i D = B. Example. Let C=100 feet, D = 4°. Then R = ^ C -^ sin. ^ Z> = 50 ^ .0349 = 1432.7. If the chords are 100 feet long, as is usual in railroad prac- tice, radius may be found with sufficient accuracy by dividing RADII, DEFLECTION ANGLES, ETC. 57 5,700, tlio radius of a 1° curve, by the defleetiou angle, or de- fjree of curvature. Thus, in the foregoing example, 5,730 -^ 4 = 1,432.5. 18. GIVEN ANY EADIUS R, AND CHOPvD C, TO FIND THE DE- FLECTION ANGLE D. From the preceding equation and example: — SUi ^ D = i C -f- K = 50 -^ 1,432.7 = .03 '9 = sin 2° = ^ D .-. D = 4°. 19. GIVEN Any 3i.vi;ius R, and chord C, to find the de- flection DISTANCE d. First find the deflection angle by above method (18). Then, angle HAIi in the figure being made equal to D, and HA = B A, BII will be the deflection distance. Draw AK to the middle point of H B, Then cZ = HB = 2KB = 2AB X sin K A B = 2 C X sm Excmiple. Let R = 1,14G feet, C = 100 feet. By (18) D will be found = 5°. Then cZ = 2 C X sin i D = 200 X .0436 = 8.72 feet. 20. If the chords are 100 feet long, as is usual in field meas- urement, divide the constant number 10,000 , by the radius in feet: the quotient will be the deflection distance. The deflec- tion distance with radius of 10,000 feet and chord of 100 feet is one foot: this rule is based upon the principle that deflection distances, the chord length being fixed, will vary inversely as the radii. Thus, in the foregoing example, 10,000 -f- 1,140 = 8.72. 21. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN- GENTIAL ANGLE T. The angle T is equal to ^ D by construction; for mode of determining it, see preceding section (18). 58 ORDIKATES. 22. G2VEN ANY RADIUS R, AND CHORD C, TO FIND THE TAN GENTIAL DISTANCE t. First find the tangential angle, as above directed. Then, angle B AE in the figure being made equal to T, and AE == AB, BE will be the tangential distance. Draw AN to the middle point of BE. Then t =EB = 2NB = 2AB X sin N A B = 2 C X iin Example. Let R = 1,14G feet, C = 100 feet. By sect. 1, T will be found = 2° 30'. Then t = 2 C X sin i T = 200 X .0218 = 4..36 feet. 23. In ordinary railroad practice the tangential distance jnay be considered equal to half the deflection distance. I XIX. ORDIXATES. 1. GIVEN ANY RADIUS R, AND CHORD C, TO FIND THE MID- DLE ORDINATE M. ^r^ ^ E *^ ^>^ N / K N^ /\ \ \ / L In the annexed figure, H N = M, H G = R, A B = C. NG = VAG2 — AN=2 = VR''^ — iC"; HX = HG — NG, i.e., M = R — VR'^ — jC^. ORDINATE S. Example. R = 819, C=100; to find the middle ordinate, M. M = 819 — \/07076r=^2500 = 1.53. 2. Ansle 11 AN = i IIGB; IIG E = ^ AG B, .'. HAN = |AGB. IIX = AN X ian. HAN; i.e., M = i C X tan. i D; D being the central angle subtended by the chord. Example. D = 7°, C = 100; to find M, the middle ordinate. M = i C X tan. { D = 50 X 0.03055 = 1.528. 3. GIVEN THE EADIUS R, ClIOKD C, AND MIDDLE ORDINATE M, TO FIND ANY OTIIEK OllDINATE E K = M', DISTANT d FKOM N, TUE MIDDLE I'OINT OF THE CHORD. KL = NG;NK=GL;EK = EL — NG. E L = VG E-^ — N K^ = VR- — '/' ; NG (1)= Vr- — iC^. Then E K = M' = y/W^^' — VR- — i C^. 4. It is a property of ihe parabola, that ordinates vary as the products of their abscissas. Tliis property may be assigned to the circle in cases where the arc encloses a small angle. Applying it here we have — HN : EK :: AN X NB : AK X KB. Call any segments A K, K B, of the chord, a and h. Then M : M' : : i C'^ : a6, .'. M' = M X 4 a6 -^ C^, Example. M = 1.528, C = 100, a = 60, ?; = 40; to find M'. M' = 1.52S X 9600 ~ 10000 = 1.528 X 0.96 = 1.467. 5. Multiply the corresponding ordinate of a 1° curve from the annexed table by the degree of curvature: the product wili l)e the ordinate sousfht. ORDINA TES. OKDINATES OF A 1° CURVE, CHORD 100 FEET. Distances op the Oruinates from the End of'the IOO-feet Chord. Middle Feet. 50 Feet. 45 Feet. 40 Feet. O.J Feet. 30 Feet. FtTt. 20 Feet. 15 Feet. 10 Feet. 5 Lengths of the Orihnates in Feet. .218 .216 .209 .198 .183 .164 .140 .111 .078 .041 Example. What is the ordinate of a G° curve, 30 feet from the end of the IOO-feet chord? The corresponding tahiilar ordinate of a 1° curve is .183; which, multiplied by 6, gives 1.09S, the required ordhuitt^ 6. A quick way of laying otf ordinates on the ground, and one sufficiently exact for the Hold, is, after fixing the point II by means of the middle ordinate HX, to stretch' a Hue from II to A, and make the middle ordinate F O = { II X; then from F to A and F to H, making the middle ordinates = ^ F O; and so on. 7. A good track-layer will seldom require points at shorter intervals than 25 feet. TEACING CUEYES AND TUKNIISTG OBSTACLES IN THE FIELD. XX.— XXIIL TRACING CURVES AND TURNING OBSTACLES IN THE HELD. XX. TO TRACE A CUKYE ON THE GROUND WITH THE CHAIN ONLY. 1, This is best taught by an example. Example. From a point B, 18 feet in advance of A, on tangent A B, to trace a: curve of 867 feet radius to the right, with chords 66 feet long, and consuming an angle of 34° 27'. 63 64 TO TRACE A CURVE ON THE GROUND. 2. First, dividing half the unit chord, or 33 feet, by thf>' radius, 367 feet (XYIII., 18), we have 0.09-|- for the sine of tli. :angential angle, corresponding to an angle of 5° 10': the de-: flection angle, therefore, Is 10° 20'. The tangential distance! corresponding to the angle 5° 10', and chord 66 feet, is equal (XVIII., 22) to twice the chord multiplied hy the sine of half the tangential angle, = 132 X 0.04507 = 5.95 feet. The deflec- tion distance (XYIII., 19) is equal to twice the chord multi plied by the sine of half the deflection angle, = 132 X 0.09+ ^ 11.88, say 11.9 feet. 3. To find the length of the curve (XVIII., 13): Divide the total central angle by the degree of curvature. The central angle, 34° 27', is equivalent to 2067 minutes; dividing by 620, the number of minutes in the deflection angle, we have 3.33, the number of chord lengths in the curve, = 3^ chains = 220 feet. If A be a stake numbered 2, then the point of curvature, B, will be 2.18, and the point of tangent, F, will fall at 2.18 + 3.22 = stake 5.40. 4. To determine the tangential distance C P, to the first stake on the curve, either of two methods may be used: — v First, The sine of any tangential angle is equal to half the chord which limits the angle on one side divided by radius. The limiting chord B C in this instance is equal to 66 — 18 — 48 feet; half of 48, therefore, or 24 feet, divided by radius, 367 feet, gives 0.0654, the sine of 3° 45' =•- tangential angle P B C. The sine of half this angle multiplied by twice the given chord = 0.0327 X 96 = 3.14 feet, the tangential distance C P. 6. Secondly, CP may be found as fqllows, assuming that the functions of small \E angles vary directly as the angles themselves, and vice versa. Let B F be a portion of the curve. Make the tangent B E equal to the chord B F, and strike the arc E F. Draw the sub- chord B C, and strike the arc C P. Prolong B C to D. E F may be taken as the tangent tial distance due to the whole chord BF, and PC the tangen- tial distance due to the sub-chord B C. TO TRACE A CURVE ON THE GROUND. 65 Then PC : ED : : B C : BD or BF; and, by the foregomg supposition, E D : E F : : B C : B F. Combining these propor- tions, and oancelling E D, we have P C : EF : : B C'^ : BF- .'. PC = EF X (BC-^BF)2. In words, the tangential distance for a sub-chord is to that for a whole chord as the square of the sub-chord is to the square of the whole chord. The same is true of detlection dis- tances. 6. In the example we are treating, the tangential distance for the whole chord of CO feet has been found to be 5.95 feet; that f jr 48 feet, therefore, is 5.95 X 48- -f- 66- = 5.95 X 0.528 = :'•>. 14, as before. Stretch the 48 feet of chain from B to P, in prolongation of tangent A B, and ina:lc the point P ; thc:i stop aside, and stretch from B to C, making the distance PC = 3.14 feet: C will be a stake on the curve. 7. Next, run out the whole chain length from C to O in the range BC. To find CD, suppose the line jS" C T to be drawn tangent to the curve at C. Then ND may be considered the tangential distance due to the whole chord, ^= 5.95, as above determined. The angle OCN = TCB=PBC (XVI., 4); and (5) ON:ND::BC:CD.-.ON = NDxBC-v-CD, i.e.;OD = N D 4- O N = N D -f- N" D X (B C -^ C D) = 5.95 X 1 + (48 -i- 66) = 5.95X1.727 = 10.27. 8. The point N may be fixed otherwise by laying off B T = C P, and running out the chain length C N in the range C T. The point D on the curve may then be fixed by making N D equal to 5.95 feet, the tangential distance. Next run out the chain to M, in the range C D ; make M E equal to the deflection distance, 11.9 feet, and fix the point E. The points C, D, and E will be stakes 3, 4, and 5 on the curve. 9. To set the point of tangent, F, at stake 5.40, prolong the chord line D E for 40 feet to L, and suppose Y E to be drawn tangcRt to the curve at E. Then the angle LEV is equal to the tangential angle of the curve; and the sub-tangential dis- tance L V is to the wdiole tangential distance due to the 66- feet chord, as the sub-chord is to the whole chord (5); i.e., L V = 5.95 X 40 ^ 66 = 3.6 feet. By the method JUustrated in (6), the distance FV will be 66 TO TRACE A CURVE ON THE GROUND. equal to 5.95 X 40^ -^ 662 = 5.95 x 0.367 = 2.18 feet. W}*'i». the distance LF = 3.6 + 2.18 = 5.78 feet, thus obtained, and the sub-chord E F = 40 feet, the point of tangent F may be established. 10. Next, set off UE = FY = 2.18 feet, and lay out FK in prolongation of the range U F ; F K will be in the line of the terminal tangent. 11. This analysis has been somewhat minute and detailed, in order that the subject may be thoroughly understood. An instrument for measuring angles should always be used in rail- road service: it greatly simplifies and abridges the labor of tracing field-curves, and gives more exact results. But occa- sions sometimes rise, in miscellaneous practice, when strict accuracy is not required, and the chain only can be had: the young engineer should qualify against such occasions. XXI. TO TRACE A CURVE ON THE GROUND WITH TRANSIT AND 100-FEET CHAIN. 1. This, also, is best taught by an example. Let it be a general rule, in locating, to fix the intersection of tangents, and to set the tangent points, or the P. C. at least, from the P. I. There are exceptional conditions, as a steep hillside, timber or broken ground, a very long arc, unimpor- tance of exact conformity to the project, and the like, which warrant its omission; but where these conditions do not obtain or are not prohibitory, and a snug fit is desirable, time will usually be saved by fixing the P. I. It often proves serviceable as a reference point during construction: on the location, i1 gives confidence in the work and an assurance of safe progress, which are well worth a little painstaking beforehand. 2. Having established the P. I., and found the intersection angle to measure, say, 66° 45^ the first step is to find the apex distances so called, or tangent lengths IB, IF. These are each equal to R X tan. 1 1. If .a 7° 30' curve be prescribed to close the angle, R X tan. ^ I = 764 X 0.659 = 503 feet. TO TRACE A CURVE ON THE GROUND. 61 Or, referring to Ta- ble XVL, theap. dist. corresponding to 66° 45' is found by inter- polation to be 3774.6; dividing by 7.5, the rate of curvature in degrees and decimals, we have for the apex distance 503 feet, as above. 3. Before disturbing the instrument, which is presumed to stand in the range of the terminal tangent, measure I F, = 503 feet, and set the P. T. at F. Then direct the telescope to the last point fixed on the ini- tial range AB, meas- ure I B, = 503 feet, and set the P. C. at B. Move to B. 4. Suppose the P. C. to have fallen at a stake 2.50. In order to find the length of the curve, divide the intersection angle by the degree of curva- ture, having first re- duced the minutes in each to hundreths of a degree by multiplying by 10 and dividing the product by 6. Thus the intersection angle becomes 66.75°, and the degree of curva- ture 7.5° : dividing the 6S TO TRACE A CURVE ON THE GROUND. former by tlie latter, we have 890 feet for tlie length of the curve. Or, the intersection angle 60° 4o' is equivalent to 4005', and the degree of curvature 7° 30' is equivalent to 450': dividing the former by the latter, we have 890 feet for the length of the curve, as before. 5. Adding 8.90 to 2.50, the number of the P. C, the P. T. Is found to fall at stake 11.40. Let the rear chainman make a note of this, that there may be no mistake in the terminal pluf^. 6. Next, to determine the proper deflections from the line of tangent at B, bear in mind that the deflection for a whole chain is half the degree of curvature ; and that, in field-curves of more than 300 feet radius, the deflections for sub-chords, or plusses, may, without material error, beheld to vary directly as the sub-chords themselves; that is to say, the sub-deflec- tions due to 30, 60, and 80 feet, for instance, will be, to the deflection due to 100 feet, as 30, 60, and 80 are to 100. 7. Thus, in the example, 7° 30' being the degree of curva- ture, one-half of this, or 3° 45', will be the deflection due to a chord of 100 feet; and ^sjl of this, or a deflection of 1° 52^' from the line of tangent at B, will fix stake 3, 50 feet distant on the curve. 8. The following is a simple rule for finding sub-deflec- tions: — Multiply the sub-chord in feet by the rate of curvature in degrees and decimals : three-tenths of the product will be the sub-deflection in minutes. Thus, in the example, 50 X 7.5 = 375, and 375 X 0.3 = 112.5' = 1° 52i', as before. 9. Having set stake 3, stakes 4 and 5 will be fixed by succes- sive deflections of 3° 45'. In establishing stake 5, the index will read, 1° 52^' -|- 3° 45' -f 3° 45' = 9° 22f = angle C B 5. 10. Suppose. the instrument moved to 5. See that the ver- nier has not been disturbed, backsight to B, and deflect 9° 22^' right; i.e., double the index angle. The index will now read, 18° 45' = the angle I CD; and the telescope will be directed along the line C D, tangent to the curve at 5, for the reason that the angle B5C has been made equal to the angle CBS (XYI. 4). Proceed with successive deflections of 3° 45' from this tan- gent, and set stakes 6, 7, 8, and 9, at intervals of 100 feet. U. Suppose the iustrumeut moved to 9. lu fixing this TO TRACE A CURVE ON THE GROUND. 69 stake, the index will read, 18° 45' + 4 times the constant angle ;io 45/, = ISO 45' -f 150 = angle I C D + angle D 5 9, = 33° 45'. In order to place the telescope in the line D E, tangent to che curve at 9, it is now necessary to turn an angle to the right, from backsight to 5, equal to D95 = D59 = 15°; i.e., the vernier should be moved from 33° 45' to 33° 45' -f 15° = 48° 45'. The telescope will then be in tangent at 9. 12. A simple rule for tinding the hidex angle which shall place the ti;lescope in tangent at any point on the cur\ e is as follows: — From double the index angle which fixed the given point, stih- tract the index reading in tangent at the last turning-point : the remainder will he the required index angle. Thus the index angle which established stake 9 was 33° 45'. Double this angle will be 67° 30'; subtracting 18° 45', the reading in tangent at the last turning-point, we have 48° 45', the required index angle, as before. The reasons for the rule will be obvious from an examina- tion of the figure. 13. Being in tangent, then, at 9, and the index reading 48° 45', a deflection of 3° 45' will fix 10: a further deflection of 3° 45' will fix 11, and the index will stand at 48° 45' -f 7° 30' = 56° 15'. 14. To find the deflection corresponding to the sub-chord 11 F, =40 feet: by the foregoing rule (8), the degree of curva- ture, 7.5, multiplied by 40, the length of the sub-chord in feet, gives a product of 300, three-tenths of which amount to 90 minutes = 1° 30'. Adding 1° 30' to 56° 15', makes the index angle 57° 45' to fix the P. T. at 11.40. 15. Move to the P. T. at 11.40, see that the vernier has not been disturbed, and backsight to 9. By the foregoing rule (12), double the index angle, 57° 45', less the angle in tangent at 9, the last turning-point, 48° 45', = 115° 30' — 48° 45', = 66° 45', = the index angle in tangent at the P. T., = the tota] angle consumed by the curve. The work thus proves itself. 16. The preceding example would appear in the field-book as follows: — 10 TO TRACE A CURVE ON THE GROUND. U5 -* CO CO ^ bo * CO C5 H V V «v V i^. ^ ^ ^ ^ V 5 ..let . G^ "^ (M O vn o >o o to lO . • . O CO (M CO o -* CO ^ . . z o o o o rl lO 0> 'M <£> o CO o CD t^ «5 a o S S S TO TRACE A CURVE ON THE GROUND. 71 17. This mode of running curves secures a record of each step in the proceeding; so that, if any error occurs, it can readily be detected. At each turning-point, the number in the " tangent'' column must correspond with the central angle due to the length of curve to that point ; and at the P. T. that number must correspond with the total central angle. The work can thus be checked with facility during its progress, and checks itself at the end. 18. The young transitman is recommended to rule blanks after the pattern given, and exercise himself thoroughly in computing the parts, and recording the field-notes of various curves assumed at will : drawings are not necessary. 19. Another method, and in some respects a better one, is, before starting on a curve, or during its progress, to record for all its stations the deflections which would locate them if the instrument remained at the P. C. Obviously the final deflec- tion thus recorded would be half the central angle of the whole curve; and, if the instrument were placed anywhere on the curve, a B. S. to P. C, and deflection of half that central angle would locate the P. T. The same reasoning will apply to any subdivisions of the curve which may be found con- venient in field work, the deflection angles for plusses being reckoned from the P. C. and recorded as in the case of whole stations. I am indebted to Mr. Robert Burgess, C.E., for recommending this method. Following is his illustration of it as applied to the preceding example. This illustration serves also to exemplify another form for field-notes. Sta. Dep. Angle. Total Angle. Calc. Curve. Mag. Curve. Remarks. 12 O / o / o / o o / -^40 X P.T. 33 22^ 66 45 N. 15 E. N.1510E. 11 31 52i^ Apex Dis. 503 10 28 071^ ft. P. I. set. 9 X 24 22U 8 20S7}4 7 16 52 V^ 6 13 07i6 5 X 9 2214 4 3 1521^ + 50 X P.O. 7 30 R. 00 20. The transitman, at P. C. , Sta. 3 -f- 50, sets his vernier at zero, takes his B. S. and locates the stations in order to 5, where a point is given. He then moves to 5, sets vernier at 72 TO TRACE A CURVE ON THE GROUND. zero and backsights to P. C. A deflection of 9° 22^' will now place him in tangent; a deflection of 13° 07^' will locate Sta. 6, and so on, using the recorded angles. Suppose a tree to mask the site of Sta. 8. Placing a stake as near thereto as possible, to hold distance; go on deflecting to 9, the vernier reading 24° 22^', where another point is given. Moving now to 9, the transitman observes the general rule, applicable to this method, namely, to set his vernier at each new instrument point, to the angle recorded opposite the point of his proposed backsight. In this case that point is Sta. 5. He therefore sets the vernier at 9° 22i', backsights to 5, and turns to 20° 37i' to locate 8. The vernier at 24° 22^' would put him in tangent, and the suc- cessive recorded deflections complete the curve. The transit is then moved to P. T., the vernier set at 24° 22^', the recorded angle opposite last point at 9, according to the above rule, backsight taken on that point, and a final deflection to 33° 22^' turns off the tangent ahead. 21. The chief advantages of this method are the complete- ness of the record for subsequent use; its adaptation to a re- tracing of the line backward as well as forward, which please observe particularly; the little labor it imposes in mental arith- metic; and its simplicity, permitting any one who can read figures and turn an angle to relieve the transit on occasion. The writer was raised on the first method; is still partial to it, out of a certain loyal feeling to the elder generation from whom it descended to him; but it must be owned the children, in some things, have surpassed the fathers,— as needs they should, else were there no progress, — and this seems to be one of those things. TURNING OBSTACLES TO VISION IN TANGENT. 73 XXII. TURNING OBSTACLES TO VISION IN TANGENT. 1. Draw CF parallel to AB. Let lines BC, CE, FG, cut these parallels at equal inclinations. Call this angle I. Then B C = CE = FG. BE = BD + DE = 2BD. But BD = BC -COS. I, .-. BE = 2 BC cos. L EG = CF. BG = EG + BE = CF + 2BC cos. I. Example. Suppose B to be a stake 24.50 on the tangent AB, and that a deflection left of 10° be made there for 200 feet to a point C. Set transit at C, vernier reading 10° left. B S to B, and deflect 20° right. Vernier will now read 10° right, and telescope w ill be in line C E. Make C E = 200 feet. Move to E. See that vernier still reads 10° right. B S to C, and turn 10° left. Ver- nier will now read zero, and telescope will be in line E G, or tangent AB prolonged. Distance BE = 2 B C cos. 1 = 2 (200 cos. 10°) =400 X .985 = 394 feet. Then E = 24.50 + 394, = stake 28.44 on tangent A B prolonged. If a parallel line C F were run, a deflectioij of 10° right would be made at each of the points C and F. If C F were 250 feet, then B G would be = 250 + 394 = 644 feet, and the point G would fall at stake 30.94 on tangent A B prolonged. 2. If angle I = 60°, the other conditions of above method being observed, triangle BHE will be equilateral, and BE = B H = H E. If the parallel D C or DF be run, BE = BD -f DC, and BG = BD + DF. For field work see last example. 3. In turning obstacles by either of these methods, the angles should be measured with extreme niceness. Handle the instrument lightly, to avoid jarring the vernier; and, if possible, observe well-defined distant objects in the several short ranges, that the Une§ of foresight ^n4 backsight may accurately coincide, 74 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. In locating, the following method is preferable to those given above, and should always be used on long tangents. 4. Having established points A and B on the centre line, the farther apart the better within limits of distinct vision, set off the equal rectangular d i s - "] tances AE, B F, ■ ranging clear of a the obstacle. Place the transit at E or F, fix points G and H on the forward range, and, rectangularly to these points, establish others on the forward range of the centre line at C and D. The offset distances should be measured very carefully with the rod, or with a steel tape if they exceed in length the pocket rule which every engineer should have about him. XXIII. TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 1. Fix a point on tangent A B prolonged at E. Lay off at B a perpendicular of any convenient length. Move the instru- ment to D, make the angle B D A = B D E, and mark the point of intersection A. By reason of symmetry in the triangles A D B, B D E, A B = B E, and may be measured on the ground. 2. Or, fix the point E, and lay off the perpendicular BD as before. Move to D, direct the telescope to E, turn a right angle EDO, mark the point of intersection C, and measure C B. Then, by reason of simi- larity in the triangles CBD, DBE, CB:BD::BD:BE, .-. BE = BD2-i-BC. Example. Suppose B D to be 60 feet, and B C 40 feet. Then B D 2 -^ B C = 3600 ^ 40 = 90 feet = B E. 3. Or, with the instrument at D, measure the angle BDE. Then B E = B D tan. BDE. TURNING OBSTACLES TO MEASUREMENT IN TANGENT. 75 Example. B D = 120 feet, angle B D E = 54° 40'. B D tan. B D E == 120 X 1.41 = 169.2 feet = BE. 4. Or, without an instrument, lay off any convenient lines B F or C H. Mark the middle point D. Line out H Gr, parallel to AB. Mark on it the point G in range with D and E. Then GF = BE, orGH = CE. 5. Should the use of a right angle be inconvenient, turn any angle E B D = x, measure B D about equal by estimation to B E, if the ground permits, and set a point D. Move to D, and measure angle B D E =7 2/. Then the angle B E D, or z, ^ 180 — ix-\-y), and, by trigonom- etry, sin. z : sin. y : : B D : BE, . •. B E = B D sin. y -^ sin. z. Example. Let x=A4P 02', y = 71° 48', B D = 300 feet. Then z = 180° — {x -\- y) = 180° — (44° 02' + 71° 48') = 180° — 115° 50' = 64° 10'. BE = B D sin. y -^ sin. z = 300 sin. 71° 48' -f- sin. 64° 10' = 300 X .95 -^ .90 = 316.6 feet. The calculation by logarithms would be as follows: — Log. 300 2.477121 Log. sin. 71° 48' 9.977711 Sum 12.454832 Log. sin. 64° 10' 9.954274 Log. 316.6. Diff 2.500558 ' n E is invisible from B, extend the line D B towards C, until a line C E clears the obstacle. The point E must then be established by intersection of the sides CE, DE, in triangle C D E. Supposing the extension B C to have been 120 feet, the side CD will be 420 feet, the angle y 71° 48'; and, by a calculation similar to the above, the side D E, opposite angle x in the lesser triangle, identical with DE in the larger one, will be found to be 231.7 feet. The sum of the angles at the base C E of the triangle C D E = 180° —y = 180° — 71° 48' = 108° 12'. By trigonometry, two sides and the included angle being 76 TURNING OBSTACLES TO MEASUREMENT IN TANGENT. known in any plane triangle, the sum of the known sides is tc their difference as the tangent of the half sum of angles at base is to the tangent of half their difference. In triangle C D E, therefore, CD + DE or 651.7 : CD — DE or 188.3 :: tan. 108.12 -!- 2 or tan. 54° 06' : tan. 54° 06' X 188.3 -^ 651.7 = .399 = tan. 21° 45', = half the difference of the angles at the base. Log. 188.3 2.274850 Tan. 54^06' 0.140334 Sum 2.415184 Log. 651.7 2.814048 Tan. 21° 45'. Diff 9.601136 The angle at C, being evidently the lesser of the two angles at the base, is equal to the half sum of these angles decreased by their half difference, = 54° 06' — 21° 45' = 32° 21'. Set the transit, then, at C, foresight to D, deflect 32° 21' left, and fix in that range two points F and G, between which a cord may be stretched, and as nearly as can be judged on opposite sides of E. Move to D, foresight to C, deflect 71° 48' right, and establish a point E at intersection with FG. Cross to E, B S to D, and deflect the angle z = 64P 10' into the line of the tangent A B prolonged. ' SUGGESTIOISrS AS TO FIELD-WORK AND LOCATION -PROJECTS. XXIV. -XXV. SUGGESTIONS AS TO FIELD -WORK AND LOCATION -PROJECTS. XXIV. SUGGESTIONS CONCERNING FIELD-WORK. 1. The Chief Engineer, after conference with his em- ployers in regard to the character of the work contemplated, and its general route, should, before organizing field-corps, go over the ground in both directions, and, aided by the best attainable maps, qualify himself by actual observation to in- struct his assistants as to the conduct of the survey. Equipped with hand-level, pocket-compass, and in rough regions with the aneroid, he can often not only prescribe lines for examina- tion, but indicate the gradients to be tried, thus saving a vast amount of random labor and needless expense. Such thorough preliminary exploration is due both to himself and his princi- pals: it is too often omitted, or done with a perfunctory rush. In broken topography, no maps, notes, or information derived from others can supply the want of personal acquaintance with tlie ground itself. He must indispensably make that acquaint- ance, in order to project an intelligent location, — a work which should rarely be delegated; being capital service, it comes within the special function of the chief engineer, and only the necessary distribution of labor attending a great charge should relieve him from its direct performance. 2. A Field-Corps in settled regions generally consists of one senior assistant or chief of corps, one transitman, one leveller, one rodman, two chainmen, one slopeman, and two or more axemen. Tlie following notes in regard to the allotment of duties and tlie conduct of work may be acceptable. They are copied from the writer's memoranda for the guidance of his field- parties, with the addition of some detail, and practical hints here and there, to aid the inexperienced. 79 80 SUGGESTIONS CONCERNfJSrG FIELD -WORE. 3. The Senior Assistant will receive instmctions from the principal assistant in charge, or the chief engineer, and will act exclusively under his direction. He will be held responsible for the good conduct of the corps» and for the rapid, exact, and economical performance of the work. Indecent or blasphemous outcries in the field should be prohibited. The writer's various travel by land and sea has brought him acquainted with many cultivated, estimable, energetic, profane fellows, but not one in whom swearing was a grace ; nor has he ever seen an- instance where it forwarded work. Those considerate of others' pride and self-respect will generally find that a good leader makes good followers. The senior assistant is empowered to appoint and dismiss employes below the rank of rodman, and will report any inefficiency or neglect of duty in the ranks above to his chief. He will pay the authoi-ized expenses of the corps for sup- plies, repairs, transportation, and subsistence, taking duplicate vouchers. Accommodations should be sought near the work. When not thus obtainable, transportation to and from the field is to be regarded as a measure of economy for the com- pany, compensating the expense incurred by saving time and labor. He will superintend field operations in person, keeping in advance of the transit to direct and expedite the work, and establish the turning-points. On preliminary surveys, the axe should be little used ; and on alternative locations, or such as may be subject to revision, trees over four inches in diameter need rarely be felled. He should be patient with sensitive landholders. He will find exercise for that amiable virtue, also, with the field vis- itors who so often spare time from useful toil to tell him he is on the wrong line, and to show him where the right one is. Note for record the kind and quality of material to be moved, observing quarries, wells, or other indications for the purpose; the timber and rock in the country traversed, with a view to their use in construction, and the widths of passage to be pro- vided for streams, together with the character of their banks and beds. Note the names of residents in the immediate vicinity of the work on survey; and, on location, cause the property-lines to be observed and recorded also when convenient. SUGGESTIONS CONCERNING FIELD-WORK. 81 Always begin grade-lines at the summit, and work down. For such service, carry habitually a slip of profile paper, say six inches wide and two feet long. Rule the proposed grade- line on it, assume a summit cut, mark the stations, and start down. When at fault, the elevation can be spotted on the profile, which will show at a glance, without any calculation, how you stand in relation to grade. The work of each day should be compiled and recorded in the evening, that no delay may result from the loss or deface- ment of a field-book. FORM FOR SURVEY RECORD. Sta. Dis. Deflec. Course M.C. Eleva. Slope. Remarks. FORM FOR LOCATIOX RECORD. i t i IS, 6 i i i. o < > 5 u P t O i < < > Remarks. On location, check the transitman's calculation of the length of each curve and the fractional deflections. The senior assistant must be qualified to locate a line accu- rately on the ground from the project furnished him. Lateral deviations exceeding five feet on ten-degree slopes, three feet on fifteen-degree slopes, and two feet on twenty-degree slopes, will be considered errors requiring correction. Measurements to the experimental line should be made and noted frequently, in order not only to check the field-work, but that the line may by means of them be laid down on the map. The senior assistant will supply himself with drawing in- struments, colors, brushes, and the like personal furniture of an engineer. He will take care also that the stationery, field- books, instruments, and other articles of outfit suiiplied by the company, are not misused. His field equipment should always include a hand-level and a pocket-compass: to these may be 82 SUGGESTIONS CONCERNING FIELD -WORK'. added a straight, round staff, five or six feet long, steel pointed ^ it will be found exceedingly useful. If without a topographer, he should make sketches of irregu- lar ground, of streams, buildings, roads, and the like, to help in compiling the map. In hilly or wooded districts, the front chalnman carries the flag on survey, and is at the head of the line. In open, plain country^ work is greatly forwarded by detaching an axeman with flag, to accompany the senior in advance, and set turning- points for the transit. The transitman follows as rapidly as possible, and the chainmen come after, lining in their stakes by the eye from point to point. The whole force is thus kept pretty steadily in motion. On wide plains, a set of chain-pins may be used, and survey- stakes placed five hundred or a thousand feet asunder. Yery often stakes at intervals of two hundred feet are sufficient, the levels being taken every hundred. Location stakes are put in every hundred feet. 4. The Transitman will be expected to keep his instru- ment in adjustment, and to be quick and accurate in its manip- ulation. It is not needful to plant it as if for eternity. On the contrary, it should be set gently, the legs thrust but slightly into the ground, and the screws worked without straining. On long tangents it is a good plan to reverse the instrument at each new point, putting the north and soiitli ends forward alternately. Small errors in adjustment are thus balanced in some measure. Select also, in such a case, some distant object in range, when practicable, to run by. The telescope, in wind or sun, will sometimes warp a little out of line. Never omit to note both the calculated and magnetic bear- ings of the- lines on survey, and of the tangents on location. Guard against the error of reading deflections or bearings from the wrong ten mark; as, for instance, 34 instead of 26. At the beginning of a curve, let the rear chainman know the plus of the P. T. Tell the front chainman the degree of curve, and instruct him how, by multiplying 1.75 by the degree, he can find the distance of each full station from the range of the last two. A quick fellow will soon pick this up, and become wonderfully skilful in practice. Thus accomplished, he is a check on wrong deflections. In running curves, a tangential angle of fifteen degrees from one point should seldom be exceeded : twenty degrees is to be regai'ded as a maximum. SUGGESTIONS CONCERNING FIELD-WORK. 83 Carry a pocket-compass, and observe with it the magnetic bearings of streams and roads crossed. Record daily each day's run; fill out the distance column, ti'anscribe the chain-book, and, on location, record the apex distances also in the column of remarks. On survey, do not erase from the field-book the notes of abandoned lines. Simply cancel, and mark them " aban- doned," in such manner that they may still be legible. When required by the senior assistant, the transitman will aid in the making of maps. 5. The Leveller must be familiar with the adjustments of his instrument, keep it in order, and handle it rapidly. On survey, establish and mark benches at half-mile intervals, on location, four to the mile when practicable. Note the surface elevations, the depths, and the flood heights, of all considerable streams crossed. Take elevations in the beds of small streams. Six hundred feet each way should be regarded as the maxi- mum sweep of the level. Carry a hand-level, and thus save the time required to peg across narrow hollows, or over heights which can be turned with the instrument. The leveller should record his work, and make up the profile daily. 6. The Rodman will give his intermediates close by the sta- tions, observing the number of each one as a check on the chainmen, and calling it out to the leveller. He should have an eye to abrupt irregularities in the ground, and give plus elevations when necessary. He will keep note of bench-marks and turning-pegs, describ- ing the latter occasionally with reference to the nearest stake, that the levels may be taken up speedily in case of a revision of the line. When unaccompanied by an axeman, the rodman is equipped with belt and hatchet. Sometimes he is furnished also with a steel pin for turning on. The pin has a ring through the head, by which it may be hung to a spring hook in the belt. The rodman will assist the leveller at record and profiles, and transcribe the slope-book daily. If stakes of survey are set at intervals of two hundred feet give rods every hundred feet, as nearly as the midway point* can be guessed. 84 THE CURVJE -PROTRACTOR. slopes for one liundred feet on eacli side of the line at every station. 8. The Rear Ciiainman" will carry a book ia which to note the turning-points, the crossings of roads, streams, swamps, woodland, and, when convenient, i^roperty lines also. He will hand it in daily to the transitman for record. As each succes- sive chain is stretched, the rear chainman calls out the number of the stake it is stretched from: this assures the selection of the right number for the stake ahead. 9. One Axeman will be detailed to make stakes, another to mark and drive them. Additional axemen may be employed at the discretion of the senior assistant, as the work requires them. Wanton destruction of timber, fences, growing crops, or other property, should not be allowed. Axemen must be careful, in passing through the country, to do as little damage as possible. XXV. THE CURVE-PROTRACTOR, AND THE PROJECTING OF LOCATIONS. 1. The curve-protractor is simply an eight-inch, semi-circu- lar horn protractor, upon which a series of twenty-three curves, from half a degree up to eight degrees, is finely engraved, to a scale of 400 feet to an inch. After some years' use in his own practice, the contrivance was transmitted by the writer to the well-known firm of James W. Queen & Co., mathematical- instrument makers. New York and Philadelphia, by whom it is now manufactured. It greatly facilitates the projecting of lines and solution of field-problems on location. It enables the engineer, for example, by a short, graphical process and rapid inspection, to find the curve which shall close an angle between tangents, or terminate a compound curve, and pass at the same time through some fixed intermediate point, without liability to the errors, and free from the loss of time, involved in a tedious calculation. Other applications, such as the nice adjustment of line among buildings, on precipitous steeps, and the like, will suggest themselves to the experienced reader. THE CURVE -PROTRACTOR. 85 2. For office use, the writer prefers a home-made curve- protractor of mica, prepared as follows: Take a thin, clear sheet, say six by ten inches, free fron\ bubbles and cracks. Block it securely on the drawing-table with thumb-tacks, set- ting the shanks close against the edge of the sheet, but not piercing it, and the heads lapping its edge. From a centre, midway of one of the long sides and near its margin, strike the curves from 12° or less, varying outwards by half-degrees to 0°; thence by quarter-degrees to 4°; and thence by ten-minute differences to 2^°. This covers one side of the sheet, the scale being 400 feet to an inch. ?^ow release the sheet, turn it over, and on its other face strike the remaining curves, down to ten minutes, from centres on the table, in the reverse direc- tion, so that they shall cross the first series at a large angle. Space them about three-eighths of an inch asunder at the mid- dle. Use a needle-point centre for the first series, to avoid boring a large hole in the sheet! Add also, on that face, two radial lines drawn towards the corners. Score the fractional <3urves very lightly, the full figure curves a little deeper, but all of them with steadiness and delicate stress. Practise beforehand on a separate slip, for the right intensity of stroke. Engrave the numbers with a stiff steel point on the opposite side of the sheet to that upon which the corresponding curve is traced. Bring the work out by rubbing it with India ink. If preferred, the flat curves on the reverse side may be colored with carmine. Duplicate protractors will be found useful in projecting compound curves. Clip off the four corners, re- enforce the edges with a narrow ribbon of tracing-linen, folded over them and glued fast, and the article is complete. It is perfect for its use; durable, flexible, spotlessly transparent, not liable to warp or change dimensions with changes in the temperature or moisture of the air, an,d, withal, takes and pre- serves a visible line, thin as the gossamer. 3. To experienced locating engineers, the curve-protractor needs no wordy commendation. Contrasted with the incon- venient appliances of the old method, — cardboard, veneer, glass, or dividers, — its advantages will be ma-nifest. A few hints as to the manner of using it may be in place. 4. First of all, let the experimental line approximate to the probable line of location; and, upon that base, construct a contour map, with reference to which special observations shou.ld be made in the field, and the chaining done with care. 86 THE CURVE -PROTRACTOR. Extreme accuracy in the contours need not be attempted. Note the courses of streams, ravines, and ridges, the average slopes at frequent intervals, and, on irregular ground, make illustrative sketches to aid in utilizing the other notes. Prac- tice gradually teaches how to observe critical points intelli- gently, and to record them briefly. In valleys or plains, where the location indicated is made up of long tangents and easy curves, little detail is required; but on bluffy, tortuous ground, with unavoidable divides to overcome, and long reaches of maximum gradient to be fitted, the method by contours is not only the simplest and clearest way of compiling necessary information, but is an aid to the engineer in projecting the right line, which no substitute can fully replace. 5. The writer is forced by the strong constraint of experi- ence to differ on this subject with Mr. Trautwine. The dif- ference, however, is a permissible one, and implies no lack of grateful respect for that veteran engineer, whose books are our handy-books, and to whose genius we are all debtors. 6. Having made the map, with ten-foot contours, suppose, for example, that a continuous gradient five miles long is to be located. Spread the dividers to 500 feet by the scale, start at the foot of the ascent, and step up, complying with the general trend of the ground, to the summit. This needful preliminary gives about the distance you have to work on, which cannot in many cases be derived from the experimental line directly. The profile furnishes the height to be overcome ; and you are thus prepared to assume a summit cut, and determine the gradient. Having adopted one, say, of 66 feet per mile, observe that this rises five feet in 400 feet. Spread the dividers, then, to 400 feet by scale, and stand one leg on or near the summit, at a point corresponding to a five or ten unit in the elevation of the gradient. That is to say, if the grade elevation at the summit be 362, for instance, stand the leg of the dividers a little beyond or a little short of the summit, at a point where the grade elevation is 365 or 360. Thence, exer cising good judgment to conform in a general way to what the location ought to be, and to make no angular indirections which cannot be closed with the maximum curvature, step forward down the incline. Name each step mentally as it is made, 355, 350, 345, 340, &c., and spot at the same time with a pencil- point the contour or half space, directly opposite, correspond' THE CURVE -PROTRACTOR. 87 ing to it in elevation. Connect the pencil-marks with a faint dotted line. 7. Were the ground a straight, regular hillside, the steps would be made directly from contour to half-space, thence to the next contour below, and the dotted line would mark out a tangent conforming exactly to the ground surface. On devious slopes, rounding within the limit of the sharpest permissible curve, the same exact conformity could be obtained, if desired, and a grade-line laid down which should require the least possible expense in building. On irregular, winding ground, an approximation only to the dotted line can be made: it is nevertheless a guide to go by; and, the more nearly the loca- tion project approaches it, the lighter will the work of con- struction be. The dotted line, in short, is analogous to a profile ; and the engineer can prescribe his cuts and fills with reference to it, by means of curve or tangent, just as on the profile he does the same by means of grade-lines. A fairly correct map will enable him to construct a profile from the project, and to amend its errors without the trouble and ex- pense' of tentative field-work. The writer's habitual practice has been to base his preliminary estimates on a profile thus deduced from the map; and he recommends the practice to others. They will be surprised to observe the likeness between such a profile, tolerably well done, and that of the subsequent location. 8. It is a good custom, and one which cannot prudently be neglected where long reaches of maximum gradient are en- countered, to " slacken grade " on the curves. In making this adjustment, the contour map is exceedingly useful. An ap- proximate project is first required, in order to determine the curvature, and, from that, the varying gradient. The location can then be laid down on the map with satisfactory precision. Opinions differ as to the right allowance per degree of curva- ture, and no experiments seem to have been made from which to deduce an authoritative rule. Some say 0.025 per degree oer 100 feet; others, 0.05; others, variously between the two. Probably 0.05 is the safer rate. This amounts to 2.64 feet on a mile of continuous one-degree curve, and makes a nine-degree 3urve, about the curve of double resistance at ordinary passen- .,";er speeds. 9. In projecting locations, the better way generally is to strike the curves first. 88 THE CUR VE -PR O TRA CTOR. 10. The following tables may be of assistance. It was need- ful, calculating tliem at all, to calculate them right; but of course such exactness as the figures would seem to indicate is unattainable in practice. |: D d \ ! / L ::^^. i R -- -^ 11. TABLE SHOWING THE DISTANCE, D, IN FEET, AT WHICH A STRAIGHT LINE MUST PASS FROM THE NEAREST POINT OF ANY CURVE, STRUCK WITH RADIUS r, IN ORDER THAT A TERMINAL, BRANCH HAVING RADIUS K = 2 r, AND CONSUMING A GIVEN ANGLE, iC, MAY MERGE IN SAID STRAIGHT LINE. D = (R — r) X (1 — COS. ic). Angle Degree op the Main Curve. X. 2» 3' 4° 6" ■ 6° 7" 8" 9" 10» D. 1° 1.72 1.15 0.86 0.69 0.57 0.49 0.43 0.38 0.34 3° 4.01 2.67 2.00 1.60 1.34 1.15 1.00 0.89 0.80 4' 6.88 4.58 3.44 2.75 2.29 1.96 1.72 1.53 1.37 5° 10.89 7.29 5.44 4.35 3.63 3.11 2.72 2.42 2.18 6° 15.76 10.50 7.88 6.30 5.25 4.50 3.94 3.50 3.15 7° 21.49 14.32 10.74 8.59 7.16 6.14 5.37 4.77 4.30 8° 28.36 18.91 14.18 11.35 9.45 8.10 7.09 6.30 5.67 9° 35.24 23.49 17.62 14.09 11.75 10.07 8.81 7.83 7.05 10° 43.55 29.13 21.77 17.42 14.52 12.44 10.89 9.68 8.71 THE C I •/.' VE -PRO TRA C TOR. 8v^ If K = 1| r, use half the tabular distance ; if K = 3 j% use twice the tabular distance; if R = 4 r, use three times the tabular distance, and so on. 12. TABLE SHOWI]SrG THE DISTANCE, d, IN FEET, AT WHICH CURVES OF CONTKAKY FLEXURE MUST BE PLACED ASUNDER IN ORDER THAT THE CONNECTING TANGENT, T, MAY BE 300 FEET LONG. > a. D O - Degree of Curve. o 1° 2° 3° 4° 5° 6° 7" 1 8° 9' 10° o a; d. 1° 3.9 5.24 5.92 6.29 6.35 6.68 6.86 7.00 7.08 7.18 r 2° 7 84 9.43 10.38 11.20 11.70 12.20 12.55 12.80 13.06 2° 1 3" . 11.77 13.43 14.64 15.68 16.45 17.09 17.61 18.05 go 4° 15.65 17.39 18.76 19.90 20.82 21.64 22.31 4° 5° . 19.54 21.22 22.76 24.01 25.07 25.97 5° 6° 23..32 25.20 26.70 28.00 29.13 6° 7° . .. .. 27.25 29.01 30.58 31.93 ( 8° 31.05 32.82 34.41 8° 9° . .. .. .. .. 34.82 36.31 9° 10° 38.56 10. Examvples. A 7° and 4° should be 19.9 feet asunder; a 5° and 9° should be 25.07 feet asunder. As approximations, for a connecting tangent 400 feet long, take twice the tabular distance: for a connecting tangent 200 feet long, take half the tabular distance. 90 THE CURVE-PROTRACTOR. It is thought by some that the parabola is an ideal curve for railroads, and should be adopted; by others, that a spiral or parabolic " easement," so called, is sufficient by way of tran- sitional flexure from straight line to circular curve; by others, that a circular curve compounded with similar terminal curves of larger radii is to be preferred; by others, that the circular curve alone serves all conditious best. The writer holds with the last party up to curves of about 4°, and for sharper ones, in the absence of proof to the contrary, believes with the third party that circular terminal curves, not less than 200 feet long, having half the degree of the main curve, are likeliest to meet, in a fair measure, the requirements of actual service. Meanwhile the old circular curve continues to do good work. On well-regulated lines a curve is usuallj^ indicated to the traveller by the inclination of the car only; there is no jar. Some years ago one of the most intelligent, experienced, and enterprising railroad managements in this country caused a thorough practical test to be made of the second device above mentioned, with a view to -its adoption if found advisable. The engineers and superintendents who made the test reported adversely, on experimental grounds. The proposed improve- ment was not adopted. PROBLEMS m FIELD LOCATIONS. XXYI.— XXXVII. PROBLEMS IN FIELD LOCATIOISr. XXVI. HOW TO PROCEED WHEN THE P. C. IS INACCES- SIBLE. 1, Suppose, for example, a pro- jected 5° curve, beginning at stake 24.20, or B in the diagram. First Method. — At any point A, which we will assume to be stake 23. 40, set up the transit. Let it be judged that stake 27, marked D in the diagram, must fall on ac- cessible ground. Then the distance B D, around the curve is 280 feet, corresponding to an angle E B D of 7" at the circumference, or an angle of 14° at the centre. The chord of a 1° curve consuming this angle, by Table XVI., is 1,396^6 feet; that of a 5° curve, B D in the figure, is one-fifth of this, or 279.3 feet. In the triangle A B D are thus known the sides A B, B D, and the sum of the angles at A and D, which sum is equal to the angle E B D. Hence, by trigonometry, — As the sum of the sides given = 359.3 A C .... 7.444543 Is to their difference =199.3 2.299507 So is tan. h sum of angles at base =3° 30' . . . . S.78G4S0 To tan. -J their difference = 1« 8.530536 Adding half the difference to half the sum, the larger angle, A, is found to be 5° 26j' ; subtracting half the difference from half the sum, the smaller angle, D, is found to be 1° 33^''. The 93 94 ffOW TO PROCEED WHEN THE P. C. IS INACCESSIBLE. length of the side A D may he found in like manner by trigo. nometrical proportion; or, perhaps more simply, thus: — B D X nat. cos. D = D F = 279.2. B A X nat. cos. A = A F = 79.6. AF + FD = AD = 858.8. We are now prepared, from our poi]it A, to deflect the angle 5° 26^' R, and lay out the line A D to the point D on the curve. Moving the instrument to that point, and backsighting to A, a deflection of 1° 33^' R places the telescope on line DB; a fur- ther deflection of 7° places it in tangent at D, and the curve may thence be traced in both directions. 2. Second Method. — Having, as in the first method, judged that stake 27, marked D, must fall on accessible ground, and thus determined the central angle subtended by the arc B D, refer to Table XVI. for the ap. dist. of a 1° curve, corre- sponding to 14°, the given angle. It proves to be 703,5 feet. One-fifth of this, 140.7 feet, is the tangent or apex distance, BC, of a 5° curve, which may be measu'^ed on the ground. Moving the instrument to C, turning 14° R, and laying off the line C D = B C, the point D on the curve is ascertained. 3. The preceding methods are manifestly apijlicable to the ends also of curves, as well as the beginnings. A case not unfrequent in practice may be added in conclusion of the subject. Suppose a 2° curve terminating at C, in marsh or stream not measurable directly. Let C fall at stake 32.20. At any con- venient point A, say stake 29, place the transit with telescope in tangent. The arc A C, = 320 feet, includes an angle of go 24'. The ap. dist. of a 1° curve corre- sponding to this angle in Table XVI. is 320.34 feet; that of a 2° curve is therefore 1G0.2 = A B. Move to B, deflect 6° 24^ li into the range of the terminal tangent, and fix E on the opposite shore. Fix also D, and note the angle EBD. Move to E. Measure the angle DEB, and the distance D E. The tri- angle BED may then be solved. If B E is found to be 670 feet, C E = 670 — 160.2 = 509.8, and stake E = 32.20 + 509.8, — say 37.30. now To PRCCEED WHEN THE P. C. C. IS INACCESSIBLE. XXVII. HOW TO PROCEED WHEN THE P. C. C. IS INAC- CESSIBLE. 1. Suppose a 4° curve, A B, compounding at B into a 6° curve B C. FiKST Method. — Place the transit at any point A, say stalce 34. Let the pro- posed P. C. C. fall at stake 3G.25. Assume that we wish to reach C, on the second curve, by means of the straight line ADC. The arc AB, covering 225 feet of a 4° curve, subtends an angle of 9°. A D is half the chord of twice this angle. By Table XYI., the chord of 18° on a 1° curve is 1,792.7 feet. That of a 4° curve is therefore 448.2 feet, half of which = 234.1, = A D. The mid. ord. of 18° on a 1° curve, by the same table, is 70.54 feet; one-fourth of which, or 17.635, is the mid. ord. B D, corresponding to the same angle on a 4° curve. In order to find what angle on the 6° curve this mid. ord, B D, — 17.635 feet, corresponds to, multiply it by 6, and seek the prod- uct, 105.81, in Table XVI., where it is found, nearly enough for field-practice, opposite the angle 22^ 04'. The chord of that angle, on a 1° curve, is seen at the same time in the adjoining column to be 2,193.2 feet; on a 6° curve it is therefore 365.5 feet, one-half of which, = 182.75 feet, = DC, and one-half of 21° 04' = 11° 02', = the angle covered by the arc B C. Thus are found the angle at A = 9°, the angle at C = 11° 02^, and the distance AC = 224.1 + 182.75, = 406.85 feet. The angle 11° 02' corresponds to a length of 1.84 feet on the 6° curve; C, therefore, falls at stake 36.25 + 1.84 = 38.09. With these data the field-work is obvious. 2. Second Method. — Having reached the point A, and determined the arc A B = 9°, as above, find in Table XVI. the ap. dist. 450.95 feet, corresponding to the given arc, one-fourtb 96 TO SHIFT A P. G. of which = 112.7 feet, = ap. dist. for the 4° curve. Move to E, deflect 9° R; range out the line E F, made up of E B = A E ^ 112.7 feet, and BF any convenient distance, say 90 feet. Tills 90 feet is the assumed ap. dist. of some unknown angle on Ibe 6° curve. To find the angle, multiply 90 by G, and seek the product, 540, in the AD column of Table XVI., where it is found opposite 10° 46'. By moving then to F, deflecting 10° 46' R, and measuring F C = 90 feet, the point C is fixed on the second curve. 3. Should unexpected obstacles be met in carrying out either of these plans, the triangles AGC or EGF may be solved, and the point C fixed by means of the lines AG, G C. 4. The application of the foregoing methods to .-urning obstacles on simple curves needs no special instance. XXVIII. TO SHIFT A r. C. SO THAT THE CURVE SHALL TERMINATE IN A GIVEN TANGENT. 1. Suppose a 3° curve AB to have been located, containing an angle of 44° 26', and ending in tangent B E : required, that it shall end in tangent D F, parallel to B E. It is plain, from the diagram, that if the curve and its initial tangent be moved forward, like the blade of a skate, until the terminal tangent merges in D F, the P. T. will hr.ve traversed the line B D, equal and parallel to A C. If, there- fore, on the ground at B, the angle E B D, equal to the whole angle consumed by the curve, in this case 44° 26', be laid off to the right, and the distance B D to the range of the proposed terminal tangent be measured, the equal distance AC, from Ibe original to the required P. C, is thus directly ascertained. Should such direct measurement be impracticable, range out the tangent BE, and, at any convenient point, measure the distance from it square across to the proposed terminal tan- gent D F, say 56 feet. Then in the right triangle BED, mak- ing BD radius^ we have given the angle at B = 44° 2&j and TO SUBSTITUTE A CURVE OF DIFFERENT RADIUS. 97 the sine E D = 56 feel. Hence, by trigonometry, E D -^ nat sin. 440 26', or 56 -^- 0.7, = B D = 80 feet, = distance A C along the initial tangent, from the erroneous to the correct P. C. 2. This i^roblem occurs more frequently than any other in the field; and the young engineer should have it by heart, that the distance square across between terminal tangents, divided by the natural sine of the total angle turned, will give him the distance he is to advance or recede with his P. C. to make a fit. 3. Excepting on precarious rocky steeps, city streets, or like exact confines, to strike within two feet of any point desig- nated in the project, may be considered striking the mark. Astronomical nicety, whether with transit or level, in an ordi- nary railroad location, is mere waste of time. 4. The observant reader will not fail to perceive that the foregoing rule applies to systems of curves, or to compound lines also, the angle E B D being the angle included between the initial and terminal tangents, let what flexures or indirec- tions soever have been interposed; and that, if the angle re- ferred to be either 180° or 360°, adjustment by shift of P. C. is impracticable. In those cases, a change of radius becomes necessary. XXIX. TO SUBSTITUTE FOR A CURVE ALREADY LOCATED, ONE OF DIFFERENT RADIUS, BEGINNING AT THE SAME POINT, CONTAINING THE SAME ANGLE, AND ENDING IN A FIXED TERMINAL TANGENT. 1. Suppose the 4° curve AB, containing an angle of 32° 20', to have been located, and that it is required to substitute for it an- other curve A C, which shall end in a parallel tangent C F, 60 feet to the right. First M k t h o d. — Find the length of the long chord A C, = AB + B C. Referring to Table XVL, the chord of a 1° curve for 32° 20' is seen to be 3,190.8 feet; that of a 4° cui've, there- 98 TO FIND THE POINT AT WHICH TO COMPOUND. fore, = 797.7 feet, say 798 feet, = AB. To find BC, solve the triangle B D C, observing that the angle DBC = BAI = one-half of the central angle 32° 20', = 16° 10', and that D C = 60 feet. Then D C -f- nat. sin. 16° 10' = 60 -i- .278 = say 216 feet, = B C. Hence AC = AB + B C = 798 + 216 = 1,014 feet. Having thus found the length of chord A C, the radius and rate of curvature may be deduced as in X. Or, dividing the tabular chord of 32° 20' by chord A C = 1,014, the degree of the required curve is ascertained directly to he 3.15, equivalent to 3° 09'. 2. Second Method. — Find the apex distance AH, = AI -f I H. The tabular ap. dist. of 32° 20' divided by 4 gives A 1 = 415 feet. In the triangle KDC, the side DC -^ nat. sin. K = 60 -^ nat. sin. 32° 20', = 112 feet = KC = I H. Then AH = AI 4- IH = 415 + 112 = 527 feet; and the tabular ap. dist. 1,661 -^ 527 gives 3.15, equivalent to 3° 09', the degree of the required curve A C, as before. XXX. HAVING LOCATED A CURVE A B C, TO FIND THE POINT r, AT WHICH TO COMPOUND INTO ANOTHER CURVE OF GIVEN RADIUS, WHICH SHALL END IN TANGENT E F, PARALLEL TO THE TERMINAL TANGENT OF THE ORIGINAL CURVE, AND A GIVEN DISTANCE FROM IT. 1. To find B, the angle B I C must be found. Call the given distance between tangents D; the larger radius, R; the smaller one, r; the required angle, a Then, referring to the figure, observe that in the triangle I H K, I H being ra- dius, IK is the cosine a; i.e., IK -^ IH = nat. cosine a. But I H = R — r; IK = IC — KC, and KC = KF or HE + FC, =r + D; i.e.,IK = Il — r — D. Hence naU cosine (t = (R - 7- - D) -r- (R -- r) = 1 ~ D H- (R - ?•). TO SHIFT A P. C. C, 99 The same reasoning would apply if A B E were the curve first located, and a terminal curve of larger radius required to be put in. 2, We have, then, the following general rule for such cases: Divide the perpendicular distance between terminal tangents by the diffei-ence of the radii, and subtract the quotient from unityi the remainder is the natural cosine of the angle of re- treat along the located curve to the required P. C. C. Example. 3. A 3° curve on the ground, to find the P. C. C. of a 5° curve striking 27 feet to the right. Here D = 27; R — ? = 1,910 — 1,146, = 764; D ^(R - r) = 27 ^ 734, = .03534; and 1 — .03534 = .96466 = nat. cosine 15° 17'. We must go back, therefore, .509 feet on the 3° curve, to compound into the 5° curve. Had the 5° curve been located first, we must have gone back 306 feet to begin the 3° curve which should strike 27 feet to the left. In either case, time might be saved by moving directly from E to C, or the reverse, and spotting in the curve backwards. To do this, we have in the right triangle F E C, the angle E = half of 15° 17', = 7° SSi', and the side F C = 27 feet. Then E C = 27 -^ nat. sin. 7° 38^', = 203 feet; and if E were stake 54.20 on the 5° curve, B would fall at stake 54.20 — 3.06, = 51.14; and C, the P. T. of the 3° curve, at 51.14 -j- 5.09, = stake 56.23. XXXI. TO SHIFT A P. C. C, SO THAT THE TEPvMINAL BRANCH OF THE CURVE SHALL END IN A GIVEN TANGENT. First Case: the terminal branch having the shorter radius. 1. Suppose the compound curve ACN located, and that it is required to fix a new P. C. C. at B, from which the terminal branch BM shall merge in tangent ML, a given distance from NO. To fix B, the central angle B H ]M of the new terminal branch must be found, and substituted for Cl^, C^U tii§ iQUgyr r^^m lij the sUorter Qiie, r/ the 4i§' L.DfC. 100 TO SHIFT A P. a C. tance asunder of the terminal tangents, D ; the central angle, C I N, = I E K, of the located terminal hranch, b ; and the central angle, B H M, = HE F, to be substituted for it, a. In the right triangle, EIK, EK = EI cos. I E K = (R — r) COS. b. In the right triangle HFE, EF = EH cos. H E F = (R — r) COS. a. Also, F K = L O = D, since each is equal to r — K L. ThenEF = EK — FK; i.e., (R — r) cos. a=(R — r) cos.b — D. Hence nat. cosine a = nat. cosine b — [D -^ (R — r)\. Were the curve B M- located, and the curve C IST to be substi- tuted for it, — that is to say, were a given and b required, — we should have, by transposition, nat. cos. b = nat. cos. a -j- D -f- (R - r). Example. A 3°, compounding into a 5° curve at C, which consumes an angle CIN, ;= 30° 22', and ends in a- tangent, NO, which is iound, by measurement of L O, to be 34 feet too far to the left. Here, D = 34, R = 1,010, r = 1,146, b = 30° 22'; and, by the solution, nat. cos. a = nat. cos. 30° 22' — 84 -r- [1,910 — 1,146] = 0.8628 — (84 - 764). 34 764 .0445 log. 1.531479 log. 2.883093 loir. 2.G4S386 Then 0.8628 — 0.0445 with central angle I E F 0.8183 = cos. 35° 05', = angle a ; a — 6 = B IT M — CIN = B E C = the angle of retreat from the erroneous P. C. C. = 35° 05' — 30° 22' = 4° 43', equivalent to b":? feet, on the 3° curve, from C to B. 2. Second Case: the terminal branch having the longer radius. Let B K represent the terminal branch located with central angle I K O = 6, and suppose it required to determine the new arc CM, a. Call the longer radius' R, the shorter one r; the distance L N between tangents, D. In the TO SHIFT A P. C. C. 101 right triangle IKO, KO =KI cos. IKO =(R — r) cos. b. Ill the right triangle FIE, EF = EI cos. lEF = (K — r j COS. a. Also, E H = L N = D, since each is equal to R — K L. Then EF = EII + IIF = EH + KO; i.e., (R — r) cos. a={U — r) COS. 6 -j- D. Hence nat. cos. a = nat. cos. 6 + D -- (R - r). Were the curve CM located, and the curve BN to be sub- stituted for it, that is to say, were a given and b required, Ave should have, by transposition, nat. cos. b = nat. cos. a — D - (R - r). Example. A 5° compounding into a 3° curve at B, which consume? {■-n angle of 44° 20', and terminates at N, 28 feet too far to t::o left. Here D = 28, R = 1,1)10, r = 1,146, b = 44° 20; and, by the solution, nat. cos. a = nat. cos. 44° 20' + 28 H- 764. The nat. cos. 44° 20' = 0.71529 ; 28 -=- 764 = log. 1.447158 — /oj. 2.883093 = Zof/. 2.i:04G65, corresponding to the decimal 0.003C5, which, being added to nat. cos. 44° 20', gives (;.75194, the nat. cos. 41° lo . Then B K ?T — C E M = 44° 20' — 41-15' = 3° 05'= angle BIC, equivalent on a 5° curve to 62 feet, which therefore is the distance around the arc from B, the erroneous P.C.C., to C, the correct one. 3. Fro^n these formulas the follovving general rule may be drawn: Divide the distance between terminal tangents by the difference of the radii, and call the quotient Q. Find the nat- ural cosine of the terminal arc already located, and call it C. The sum or the difference of Q and C will be the natural cosine of the terminal arc to be substituted for that already located. With radii in the order R, r, should the terminal inside tangent located strike s f -a \ *^® proposed tangent; or, with radii in tlie order r, R, should the terminal tangent located strike ] • -i [ the proposed tangent, — take the Trc > of Q and C for the required cosine. ^ difference ) ^ 102 TO FIND THE POINT AT WHICH TO BEGIN A CURVE. XXXII. HAVING LOCATED A TANGENT, A B, INTERSECTING A CURVE, C D, FROM THE CONCAVE SIDE, TO FIND THE POINT E ON SAID CURVE AT WHICH TO BEGIN A CURVE OF GIVEN RADIUS WHICH SHALL MERGE IN THE LOCATED TANGENT. /' 1. Place the transit at the intersection point B. Set points at equal distances therefrom in both directions on the curve already located, by means of which the direc- tion of a tangent to that curve at B may be fixed, and the angle F B A measured. Call that angle a ; and, as shown in the figure, suppose the lo- cated curve to be prolonged in- to a terminal tangent, parallel with the newly located tan- gent A B. Complete the dia- gram. Call the larger radius R; the proposed radius, r; the central angle of the proposed curve, x. Then, obviously, the line A G = II cos. a. It is also equal to (R — r) cos. x -\- r. That is to say, R cos. a = (R — r) cos. x -}- r. Hence cos. x == {"Rcos. a — r) -i- (R — r); and x — a = angle BGE, sub- tended by the arc B E, from which the length of the arc may be deduced, and the point E ascertained. Example. DC, a 1° curve; angle a = 64° 32': to connect with a 49 curve. Here cos. x = (5,730 X 0.43 - 1,433) -5- (5,730 - 1,433) = 0.24 = cos. 76° 06'; and x — a = ll° 34', equivalent to a dis- tance from B around the 1° curve of 1,157 feet to E, the point at which to begin the 4° curve. TO LOCATE A Y. 103 XXXIII. HAVING LOCATED A TANGENT, A B, INTERSECTING A CURVE, C J), FROM THE CONVEX SIDE, TO FIND THE POINT K ON SAID CURVE AT WHICH TO BEGIN A CURVE OF GIVEN RADIUS WHICH SHALL MERGE IN THE LOCATED TANGENT. L This pr()l)lem is analo- gous to the preceding one. The preparatory steps are the same in both. Having found the angle «, however, it will be manifest to the attentive reader, that, in this case, R coH. rt = (K -j- r) COS. X -{- r. Hence cos. a: = (R cos. a — r) -^ (R + r). Example. 2. D C, a 1° curve; angle a = 64° 32': to connect with a 4° curve. Here cos. x = (5,730 X 0.43 - 1.433) ^ (5,730 + 1,433) = 0.1439 = COS. 81° 43'; and x — a = 17° il', equivalent to a distance from 13 around the 1° curve of 1,718 feet to E, the point at which to start tlie 4° curve. XXXIV. TO LOCATE A Y. B D A \ ^^^ .-^ / ^ \ \ / "^ N / \ \ c / "N ^' \ ^'"'^'x'^ \ '\.^ \c \ ^--^^ { 3 tersecting ilie tangent BA. 1. The processes of the two former problems may- be adopted. In this case the angle a vanishes, and the COS. X clearly is equal to (R-r)-^(R + r). 2. Another solution of tlie Y problem is as follows : — Draw tlie tangent E D in- Then is BD = D A, for the lea- 104 TO LOCATE A Y. son that each is equal to DE. Make GF = R + r, the diame- ter of a semicircle. Said semicircle touches tangent B A at D, its middle point; and D E being perpendicular to G F, we have by geometry GE : DE ;: DE : EF; i.e., GE X EF, orR X r, = DE2. Hence DE = B D = D A = V^R X r = R tan. i a-, and we are thus enabled to fix the points E and A. 3. In the two foregoing problems, the angle consumed by curve E A is = 180° — x. Example. BE, a 21° curve located; BA, a tangent: to complete the Y with a 6° curve, E A. By the first method, cos. x = (R — r) -^ (R + r) = (2,292 — 955) -^ (2,292 -j- 955) = 1,337 ^ 3,247 = log. 3.126131 — log. 3.511482 == 1.614649, which corresponds to log. cos. 9.614649, or to the decimal number 0.4118, indicating in either case the angle 65° 41' = x. DE = BD = DA = R tan. ix = 2,292 X 0.6455 = 1,479.4. DE may be found also by reference to Table XYL, where the ap. dist. of a 1° curve for 65° 41' is seen to be 3,698.6. Dividing this number by 2|, we have 1,479.4, as above. Or, by the second method, — D E = VR X r = \/2188860= 1,479.4. Having thus the means of fixing points E, D, and A, the curve E A can be laid down. 4. If B A is curved con- vex to the Y, construct the figure as in margin, and reason thus: — In the triangle EGF, formed by lines connect- ing the curve-centres, the sides are respectively equal to the sums of the contiguous radii : the angles may therefore be found as in Case III., Trigonometry. Lines drawn bisecting the central angles of the several TO LOCATE A Y. 105 curves will pass throiigli the points of intersection of the tan- gents to those curves severally. But lines so drawn in this case bisect also the angles of a triangle, and, demonstrably by geometry, meet in one point equidistant from the three sides of the triangle. That point, therefore, must be a com- mon P. I. for all the curves, and that equidistance the "ap. dist." length common to them all. Example. Given B A, a 3°, and B C, a 4° curve : to complete the Y with a 5° curve, C A. E F = 1,910 + 1,146 = 3,056. GF = 1,433 + 1,146 = 2,579. E G = 1,910 + 1,433 = 3,343. Then, by Case III., Trigonometry, — As EG, 3,343. . . . log. (a. c.) 6.475864 Is to E F + G F, 5,635 .... log. . . 3.750894 SoisEF— GF, 477 . . . . log. . . 2.678518 To diff. of segments of E G,"804 ...'.. 2.905276 Adding half the difference to half the sum of the segments of the base EG, we shall have the greater of them; i.e., (3,343 -f- 804) -i- 2 = 2,073.5, which is the cos. E, E F being radius. Hence 2,073.5 H- 3,056 = log. 3.316704 — log. 3.485153 = 9.831551 = cos. 47° 16' = E. By Table XVI., the ap. dist. of a 1° curve corresponding to this angle is 2,507 3 : that of a 3° curve, therefore, is 835.8 = the common ap dist. B D or D A. Multiplying the common ap. dist. by 4, we shall find opposite the product in Table XVI. the central angle of the 4° curve to be 60° 32'; multiplying it by 5, we find, in like manner, the central angle of the 5° curve to be 72° 12'. Arc B A, = 47° 16', is equivalent to 1,575 feet on the 3° curve; arc BC, = 60° 32', is equivalent to 1,513 feet on the 4° curve. Points being thus fixed at A and C, curve C A can be laid on the ground. 5. If curve B A is concave to the Y, the radii being given, construct the figure as follows: — First draw the triangle GFE, the sides of which are obvi- ously derived from the given radii. Prolong the sides E G and E F indefinitely. Bisect the exterior angles at G and F with 106 TO LOCATE A Y lilies meeting at I), and from D let fall perpendicnlars on E H. EA, and G F. Then, comparing triangles GBD, GCl), IIif angles at G are equal by construction ; the angles at B and C are right angles, the side G D common. Hence the triangles are equal in all their parts: B G = G C, and B D = D C. By like reasoning, it appears that C F = F A, and DA = DC. The point D being equidistant from the right lines E B, E A, which limit angle E, a line bisecting that angle will strike point D. 6. It may be remarked, therefore, that lines bisecting the vertical angle and the exterior angles contained between the base and the prolongation of the sides of any triangle, will meet in a point equidistant from the base and the said prolon- gations. We thus have in the figure all the conditions for fit- ness of tlie curves. It remains only to solve the triangle G F E, seeing that from its angles the required central angles can be obtained. Example. B A, a 1°, BC, a 6° curve, located: to complete the Y with an 3° curve, C A, TO LOCATE A TANGENT TO A CURVE. 107 In triangle G F E, — E F = 5,780 — 717 = 5,013. E G = 5,730 — 955 = 4,775. G F = 955 + 717 = 1,672. Then, by Case III., Trigonometry, — As E F . . . 5,013 .... log. (a. c. ) 6.299902 Is to E G + G F, 6,447 .... log. . . 3.809358 SoisEG — GF, 3,103 . . . . log. . . 3.491782 To diff. seg. of base, 3,991 . . . log. . . 3.601042 The longer segment, therefore, is 4,502; the shorter, 511- Cos. E ^ the longer segment divided by E G = 4,502 -^ 4,775 = lo[i. 3.65.3405 — 3.678973 = 9 974432 = cos. 19° 28' = angle E. Cos. GFE = the shorter segment divided byGF = 511-^ 1,672 = log. 2.708421 — log. 3.223236 = 9.485185 = cos. I'l^ 12/ = angle GFE. The central angle, B G C, of the 6° curve, is equal to 180 — F G E = the sum of the angles at E and F = 72° 12' + 19° 28' = 91° 40', making the arc B C = 1,528 feet. The arc B A, equivalent to 19° 28' of a 1° curve, = 1,947 feet. Points C and A being thus ascertained, curve AC maybe located. It will consume an angle = 180° — 72° 12' = 107° 48', equivalent, on an 8° curve, to 1,347.5 feet. XXXV. 10 LOCATE A TANGENT TO A CURVE FROM AN OUTSIDE FIXED POINT. 1. If the ground is open, and the curve can be seen from the fixed point, it may be marked by stakes or poles at short inter- vals, and the tangent laid off without more ado. 2. Suppose, however, that on cumbered ground a trial tan- gent, A B, has been run out, intersecting the curve at B: it is required then to timl the angle BAE, \\\ order that the true tangeut.AK may be laid down. 108 TO SUBSTITUTE A CURVE. Example. A B = 1,500 feet ; D H B, a 4° curve; angle F B D = 20° 13'. First, the angle FBD, between a tangent and a chord, is ^(jiial to half the central angle subtended by the same chord. Angle D C B, therefore, = 40° 26'. By Table XVI., the chord of 40° 26', for a 1° curve, = 3,960.2 feet; for a 4° curve, it is, say, 990 feet = D B; and D I = I B = 495 feet. The mid. ord. H I is, in like manner, found to be 88.25 feet. Deducting this from the radius of the 4° curve, we have I C = 1,344.4 feet. Then lC-^lA = tan. I AC; i.e., 1,344.4 H- (495 + 1,500) = 0.674 = tan. 33° 59' = angle I AC. Next, by geometry, the proposed tangent A E =\/A D X A B = V2,490 X 1,500 = 1,932.6 ; and E C -^ A E = tan. E A C = 1,432.69^ 1,932.6 = 0.7413 = «an. 36° 33' = angle E A C. Then E A C — I A C = .36° 33' —33° 59' =2° 34' = angle B A E, the angle required, which can accordingly be laid off from the fixed point A, and the tangent located. XXXVI. TO SUBSTITUTE A CURVE OF GIVEN EADIUS FOR A TANGENT CONNECTING TWO CURVES. Example. 1. A B, a 4° curve; BC = 774 feet; CD, a 6° curve: to put in the 1° curve, EF. Sketch the figure as in margni, HK being parallel and equal lo BC. Then KG = BG — BKorCH = 1,433 — 955 = 478 feet; KH -f- GK = 774 -^ 478 = 1.62 = tan. 58° 19' = angle KGH; and KH -^ ain. 58° 19' =774 4-0,851 = 909,6 ieet = GH. TO RUN A TANGENT TO TWO CURVES. 109 In the triangle GHI we have then the sides given; namely GH = say, 910 feet, HI = 5,730 — 955 = 4,775 feet, and GI = 5,730 — 1,433 = 4,297 feet: to liud the angles. Under Case 3, Trigonometry (III.), IH : IG + GH :: IG — GH:IL — LH; i.e., 4,775 : 5,207 : : 3,387 : 3,093, the differ- ence of the segments into which the base I H is divided by a perpendicular from G. Adding half the difference of the seg- ments thus found to half their sum, the longer segment, I L, is found to be 4,234 feet ; subtracting half the difference from half the sum, the shorter segment, L H, is found to be 541 feet. Then H L -i- H G = 541 ^ 910 = 0.5945 = cos. 53° 31' = angle GHI. In like manner, dividing I L by I G, we find the angle GIH to be 9° 40'. The sum of these angles = angle E G H = 63° 20', for the reason that each is equal to 180 — II G I. Finally, E G H — K G H = 63° 20' — 58° 19' 5° 01' = angle E G B, equivalent to a distance from B of 125 feet around the 4° curve to the P. C. C. at E; and GIH — EGB = 9° 49' — 5° 01' = 4° 48' = angle CHF, equivalent to a distance from C of 80 feet around the 6° curve to the P. C. C. atF. XXXVII. ^O RUN A TANGENT TO TWO CURVES ALREADY LOCATED. then be the line 1. If one curve be visible from the other, or if both be visible from some inter- mediate point, ■ mark them on the ground with stakes at short intervals. The points M or L in the range ^ of the required tangent may fixed by one or two trial settings of the transit, and put in. 110 TO RUN A TANGENT TO TWO CURVES. 2. Should obstacles prohibit this plan, measure any ccn- venient line, FG or B CD, from one to the other curve, and, completing the traverse AFGrE or ABODE, determine thence the bearings and distances asunder of the centres A and E. The right triangle A E K, in which E K = the sum of the radii, may then be solved, and the points H and I ascer- tained as in the following example : — Example. F B, a 4° curve ; G D, a 6° curve. N. S. E. W. A B, N. 20° E., 1,433 feet . . 1,346.6 490.0 BC,East, 3,570 feet . . - 3,570.0 C D, N. 34° E., 1,800 feet . . 1,492.2 1,006.2 955 feet . . 675.2 - 675.2 3,514.0 5,066.2 675.2 Then 4,391 -f- 3,514 = 1.2496 = tan. 51° 20' = bearing AE; and 4,391 -^ sin. 51° 20' = 5,624 feet = distance A E. Also, EK -f- AE = (1,433 + 955) ^ 5,624 = .sm. 25° 08' = angle EAK; and angle AEK = 90°00'— 25°08/= 64° 52'. Hence the bearing of A K or HI is N". 76°28/E., and that of A H or I E, K 13° 32^ W. Since AB bears X. 20° E., the angle HAB = 20° 00' + 13^ ^2'^::^ 33° 32', equivalent to a distance of 838 feet from B around the 4° curve to the required P. T. at H; and, since DE benrs N. 45° 00' W., the angle IE D = 45° 00' — 13° 32/ = 31° 28 . equivalent to a distance of 524 feet from D around the 6° curve to the required P. C. at I. 3. Should the curves turn in the same direction, the side EK of the triangle AEK is equal to the difference of the radii instead of their sum. In other respects, the method exemplified will apply to that case also. 4. The preceding solution may be useful as an exercise. But the problem is one of rare occurrence, and the conditions must be extraordinary which prevent a close approximation, at least, to the true line in the field. The better way in actual l)ractice, then, is to run out a trial tangent as nearly right as j.ossible. If it errs by passing outside the objective curve, ciose with a compound (XXIX.) ; if that error be inadmissible, or if it ei-rs by cutting the objective curve, measure the miss, and divide it by the length of the trial tangent. The quotient TO RUN A TANahNT TU TWO CURVES. \\\ will be the iiatural tangent of the angle of retreat or advance on the first curve recpiired to make the tangent fit. 5. A still closer ad.iustnien', would be, after determining the angle approximately as above, to find the "tangents" corre- sponding to it for the two curves in Table XYI. Subtract the sum of these tangents from the length of the trial line, if it cuts the objective curve; add the sum, if it passes outside. AVitli the number thus found, divide the measured amount of error for the tangent of the angle of retreat or advance, as the case may be. G. Suppose, for illustration, that a trial tangent, bearing by needle X. 54° 30' E., is run out from stake 24.80 of a 4° curve, intending to touch a G°, but is found to cut it. Supjwse fur- ther that the objective 0° curve was laid down and numbered in the direction of approach towards the 4° curve; that its P. C. is stake 25,10, and the magnetic bearing of its initial tan- gent S. 30° 30' W. The angle, then, between the bearing of the trial tangent and tliat of the initial tangent of the G° curve, is 24°, corresponding to a distance of 400 feet on the latter curve. At stake 25.10 + 4.0 = 29.10, therefore, a tangent to the G° curve would oe parallel to the trial tangent. Go forward on the trial tangent, accordingly, to a point oi^posite 29.10, and measure the distance square across to that plus on the G° curve. Assuming the trial tangent to be 2,500 feet long, and the amount of the miss to be 87 feet, the nat. tan. of the angle of error is 0.0348 = tan. 2°. By the method in (4), this calls for a shift of the P. T. 50 feet ahead on the 4° curve, making the new P. T. 24.80 + 0.50 = stake 25.30, and ad- vances the P. T. of the 6° curve to stake 29.43 of that numera- tion. The method in (5), applied to this case, brings the angle of error 2° 02', instead of 2°, equivalent to a deviation of 1^ feet scant in half a mile from the line corrected by the method in (4), and agreeing exactly with the correction determined by the method in (2). TRACK PROBLEMS XXXYIII.— LI. (Standard Gauge 4 Feet 8i Inches.) TRACK PROBLEMS. XXXVIII. REVERSED CURVES. The following problems will be useful in laying off turnouts, the adjustment of tracks near stations or shops, and the like; but reversed curves should never be used on the main line between stations, where they are both objectionable and unne- cessary. Ground which allows any permissible location 9t all will allow straight reaches of at least two hundred or three hundred feet between curves of contrary flexure; and in every case it is worth the small additional outlay to make such a location. XXXIX. TO CONNECT TWO PARALLEL TANGENTS BY A REVERSED CURVE HAVING EQUAL RADII. 1. The radius R, and the perpendicular distance D, between the tangents given. j^c 1 115 116 TO CONNECT TWO PARALLEL TANGENTS. Draw the tangents, radii, and curves, fixing the P. R. C. midway of D. Draw the chords G I, I E, the line B F perpendicular to G I, and the line E H in prolongation of radius C E to an intersec- tion with B H passed through centre B parallel to tangents. That I falls midway of D, follows from the necessary sym- metry of the figure ; and G I E must be a straight line, because the radii B I, I C, perpendicular to a common tangent at the same point, form a straight line, to which the chords G I, IE, are equally inclined. C H -^ C B = COS. A; but C H = 2 R — D, and C B = 2 R. .-. COS. A = (2R — D) -^2R. BH = B C sin. A = 2 R sin. A; GF = R sin. i A; GE = 4 GF. .-. GE = 4 R sin. i A, and GI or IE = 2 R sin. i A. Observe, that, in the right triangles G K E and B G F, the angles at G and B are each equal to ^ A : hence the triangles are similar. Example. K = 800 feet, D = 24 feet. To find angle A. Cos. A = (2 R — D) -^ 2 R = 1,576 ~ 1,600 = 0.985 = nat COS. 9° 56^ BH may then be found = 2 R sin. A = 1,600 X 0.1725 = 276 feet, and laid off from the P. C. at G to K, the point E being fixed by a right angle from K. Or GE may be found = 4 R sin. i A = 3,200 X 0.866 = 277.1 feet, and laid off from G to E, the point I being fixed 138.5 feet from G, and angle KGE made equal to half of A = 4° 58'. 2. The distances G K and D given, to find R. In triangle G K E, K E = D. D ^ GK = tan. i A; D -4- sin. i A = GE; and GE -f- sin. I A = 4 R. Or, having found GE, we have from the congruity of trian- gles GKE,BFG, D : GE :: ^GEorGF : R. .-. R = GE2-f-4D. TO CONNECT TWd PARALLEL TANGENTS. in Example. GK = 300 feet, D = 28 feet. D-^GK Log. 28 ... . 1.447158 Log. 300 ... . 2.477121 = Tan. ^ A . 5° 20' . . . . 8.970037 D -^ sin. i A . . . . Log. 28 ... . 1.447158 Sin. 5° 20' . . . . 8.968249 = G E . . 301.24 . . . . 2.478909 GE-^sm. iA . . .Sm. 502O' . . . .8.968249 = 4 R . . 3,241 .... 3.510660 .-. 11 = 810.2. XL. TO CONNECT TWO PARALLEL TANGENTS BY A EEYERSED CURVE HAYING UNEQUAL RADII. 1. Given the perpendicular distance, D, between two paraL lei tangents, and the unequal radii, R and r, of a reversed curve, to find the central angles. A, the chords, and the straight reach, G K, of the curve. 118 TO CONNECT TWO PARALLEL TANGENTS. Cos. A = C H ^ B C; but C H = (R + r) — D, and B C = R + r. .-. Cos. A = (R + r — D) -^ (R + r). The straight reach GK = BH=(R + r) sin. A. The sum of the chords G E = G K -h- cos. ^ A. G I = 2 R sin. i A. IE = 2 r sin. i A = GE — GI. Example. D = 28, R = 955, r = 574. Cos. A = (R 4- r — D) — (R + r) = 1,501 -^ 1,529. 1,501 . . . log. 3.176381 1,529 . . . log. 3.184407 Cos. A, 10° 59' 9.991974 GK=(R + r) sin. A. R + y, 1,529 . . . log. 3.184407 Sin. A, 10° 59' . . . log. 9.279948 GK = 291.3 2.464355 GE = GK-^ COS. 1 A. GK, 291.3 . . . log. 2.464355 Cos. i A, 5° 29i' . . . log. 9.998014 GE = 292.6 2.466341 GI = 2Rsm. iA. 2 R, 1,910 . . . log. 3.281033 Sin. -J A, 5° 29^ . . . log. 8.980916 GI = 182.8 2.261949 IE = GE — GI = 292.6 — 182.8 = 109.8. 2. The distances GK and D, and one of the unequal radii, R, given, to find the other radius, r, and the central A. REVERSED CURVE WITH UNEQUAL ANGLES. Ufi Example. G K = 422, D = 30, R == 2,292. Taw. iA = D-^-GK. D = 30 . . . log. 1.477121 GK = 422 . . . log. 2.625312 Tan. i A, 40 04' . . . , . 8.851809 .-.A = 8° 08'. GE = D-^sm. i A. D = 30 . . . log. 1.477121 Sin. i A, 4° 04' . . . log. 8.850751 GE = 423 2.626370 GI = 2Rsm. iA. 2 R = 4,584 . . . log. 3.661245 Sin. i A, 4° 04' . . . log. 8.850751 GI = 325.1 2.511996 GE — G I = 423 — 325 = 98 = IE. r = i I E -^ sin. ^ A. iIE==49 . . . log. 1.690196 -Sin. i A, 40 04' . . . log. 8.850751 r = 691 2.839445 XLI. A REVERSED CURVE HAVING UNEQUAL ANGLES. Given the angles A and B, and the length A B of a straight Jine connecting two diverging tangents, to find the radius of a leversed curve to close the angles. AI = R X tan. i A; B I = R X tan. \ B, .*. AB = R X {tan. ^ A + tan. \ B). ,*. R = A B -h {tan. ^ A + tan. i BJ. 120 REVERSED CURVE BETWEEN FIXED POINTS, Example. A = le**, B = 10°, A B = 840. AB, 840 log. 2.924279 ^ A = 8°, nat. tan. 0.14054 i B = 5°, nat tan. 0.08749 Tan. i A + tan. \B = 0.22803 . . . log. 1.357992 K = 3684 3.566287 This solution will apply also to the finding of the maximum radius for a simple curve which shall connect three tangents. XLII. A REVERSED CURVE BETWEEN FIXED POINTS. Given the angles N and K, and the length of the straight line E F connecting two divergent tangents, to find the radius of a reversed curve from E to F, connecting the tangents. 1. Denote the angle i!l C or D I F by I; the angle CEI, complement of N, by n ; and the angle D F I, complement of K, by k. Then, in triangle E C I, — E C : C I : : sin. I : sin. n. . •. E C X sin. n = C I X sin. I. REVERSED CURVE BETWEEN FIXED POINTS. 121 Also, in triangle D F I, — DF : D I : : sin. I : sin. k. .'.DFX sin. fc = D I X s£w. L Adding these equations, we have — EC X sin. » + DF x sin. & = (CI + DI) X ««w. I. But E C and DF are each equal to K; sin. n = cos. N; sin, fc = COS. K; and C I + DI = 2 R. Hence the equation becomes, — R X {cos. N + COS. K) = 2 R X sin. I. .*. sin. I = {cos. N + COS. K) -f- 2. The foregoing elegant solution is abridged from Henck. 2. Angle A = 180 — (?i + I) ; angle B = 180 — ( jk + I). To find radius, draw F H parallel, and E H perpendicular, to CD. Then E H = E F X sin. I. But EH = EG + GH; EG = RX sin. A; and GH = R X sin. B. .-. EF X sin. I = R X {sin. A + sin. B). .-. R = E F X sin. I -^ {sin. A -}- sin. B). 122 REVERSED CURVE BETWEEN FIXED FOINTS, Example, E F = 1,400, N = 30°, K = 20°. Sin, I = {cos. N + COS. K) -^ 2. N = 30°, nat, cos 0, K = 20°, nat. cos 0. 1.80572 1.80572 -^ 2 = 0.90286 = nat. sin. 64° 32^ .-.I = 64° 32'. A = 180° — {n-\-l)= 180° — (60° + 64° 32') = 55° 28'. B = 180° — (A; + I) = 180° — (70° + 64° 32') = 45° 28'. K = E F X sin. I -H {sin, A + sin. B). EF = 1,400 log. 3.146128 -ZVai. sm. I, 0.90286. . ]og. 1.955621 EG = 1,264 .3.101749 A = 55° 28' nat. sin 0.82380 B = 45° 28' na«. sm 0.71284 Sin. A + sin. B 1.53664 log. 0.186579 R = 822.6 2.915170 3. The young student should bear in mind that the addition or subtraction of the logarithms of two natural numbers gives a logarithm representing, not the sum or difference, but the product or quotient, of such numbers. When, therefore, as in the two foregoing cases, the sum or difference of two or more trigonometric functions — sines, tangents, and the like — is sought, the logarithm of the sum of the natural functions, and not the sum of their logarithms, is to be used. If, for example, sin. A X sin. B is required, the log. sin. A + log. sin. B = the logarithm of the product of the sines designated ; but, if sin. A -j- sin. B is sought, the natural sines of those angles must be added together, and the logarithm of the sum of these natural functions must be used in making logarithmic calculations. 1 TO CONNECl TWO DIVERGENT TANGENTS. 123 XLIII. TO CONNECT TWO DIVERGENT TANGENTS BY A REVERSED CURVE. 1. ADVANCING TOWARDS THE INTERSECTION OF TANGENTS. Given the angle of divergence, N, tbc i.iitial P. C. at G, the distance GH, and the radii R, r, to find the central angles A and B. GK = C G X C08. N = R cob. N. GL = GH X sm. N. GK — GL = LKorEF, CF being drawn parallel to L E. Cos. B = D F -^ D C = (r + E F) -^ (R + r). Angle GC K = 90° — N ; angle D C F = 90° — B'. Angle A = GCK — DCF= (90° — N) — (90° — B) = B — N. • lExample. N = 24° 30', G H = 854, R = 1,440, r = 1,146. GK = R COS. N. R = 1,440 . Co8. N, 24° 30' . log. 3.158362 log. 9.959023 GK = 1,310 . . . . „ 3.117385 124 TO CONNECT TWO DIVERGENT TANGENTS. GL = GH X .sm. N. GH = 854 Sin. N, 24° 30^ log. 2.931458 log. 9.617727 GL = 354 2.549185 LKorEr = GK — GL = 1,310 — 354 = 956. Cos. B = (r + EF) -^ (R + r). r + EF = 2,102 . K + r = 2,586 . Cos. B, 35° 38' . log. 3.322633 log. 3.412629 . . 9.910004 B = 35° 38'. A = B — N = 35° 38^ — 24° 30^ = 11° 08'. 2. EECEDING FEOM THE INTERSECTION OF TANGENTS. Given the angle of divergence, N, the initial P. C. at G, the distance G H, and the radii R, r, to find the central angles A and B. GK = GH X «an. N. KC = GC — GK=-R — GK. LC orEF = KC X cos. N, the line C F being drawn paral- lel to L E. Cos. B = D F H- C D = (r 4- E F) -^ (R + r). Angle A manifestly = B -)- N. TO SHIFT A P. R. C. 125 lElxam-ple. N = 18° 30', G H = 920, R = 955, r = 819. G K = G H X tan. N. GH = 920 . . . log. 2.963788 Tan. 18° 30' . . . los. 9.524520 GK = 307.8 2.488308 KG = R — GK = 955 — 307.8 = 647.2. LCorEF = KC X cos. N. K C = 647.2 . . . log. 2.811039 Cos. N, 18° 30' . . . log. 9.976957 EF = 613.8 2.787996 Cos. B = (r + EF) ^ (R + r). r + EF = 1,432.8 . . . log. 3.156185 R 4- r = 1,774 . . . log. 3.248954 Cos. B, 36° 08' 9^907231 B = 36° 08 A = B -f N = 36° 08' + 18° 30' = 54° 38^ XLIV. TO SHIFT A P. R. C. SO THAT THE TERMINAL TANGENT SHALL MERGE IN A GIVEN TANGENT PARALLEL THERETO. Given the reversed curve E F G, ending lu tangent GV: to find the angle of retreat, A, on the first branch, and the angle C of the second branch, ending in tangent PI T, parallel to GY. Measure the error T G = D, perpendicular to the terminal tangent. 126 TO SHIFT A P. R. C. In the figure, draw L K parallel to G Y, and passing through centre of first branch. Then M K = (R + r) X cos. B. NL ={R + >') X COS. C. WL = GK. KL =r + D-f GK. MK = r + GK. N L ~ M K = D. .-. (R + r) X COS. C — (R + r) X cos. B = D. .-. (R + r) X COS. C = (R + r) X cos. B + D. .-. Cos. C = [(K + r) X cos. B + D] -^ (R + r), A = (90° - C) - (90° - B) = B - C. Example. K = 1,433, r = 819, B = 34° 20^ D = 94. Cos. C = [(R + r) cos. B + D] -^ (R + r). K 4- r = 2,252 B = 34° 20', cos. log. 3.352568 log. 9.916859 (R + r) cos. B = 1,860 3.269427 Add D 94 1,954 (R + r) log. 3.290925 log. 3.352568 Cos. C, 29° 49/ 9.938357 A = B — C = 340 20' — 29° 49' = 4° 31^ CURVE THROUGH A FIXED POINT. 127 XLV. I TO PASS A CURVE THROUGH A FIXED POINT, THE ANGLE OF INTERSECTION BEING GIVEN. Given the Intersection angle, A, of two tangents, to find the radius, R, of a curve which shall pass through a point, C; the position of said point, with reference to the tangents or the point of intersection, being known. 1. By what data soever point C is located, they may be com- muted by simple processes to the form shown in the figure; namely, the ordinate BC and the distance IC to apex. Call the angle B I C a, and complete the triangle ICO. In this triangle, x = i Also, C O : I O : : sin. X ButCO = R: IO = 180 -\_a = 90°- (i A + a). sm. z. R COS. ^ A* .-.R: R COS. iA" sin. X : sm. z. Hence sin. z = COS. i A solved, and the radius ascertained The triangle ICO may then be 128 CURVE THROUGH A FIXED POINT. Exam2')le. A = 40°, B C = 32 feet, I B =- SO feet. Then BC -f- IB = 32 -=- 80 = 0.4 nat. tan. 21° 49'; I C = B C -^ nat. sin. 21° 49' = 32 ^ .372 = 86 feet. Also, a: = 90° — (i A + a) = 90° — 41° 49' = 48" 11'. and Next, sin. x, 48° 11' Divided by cos. i A, 20° = sin. z, 127° 31' log. 9.872321 los. 9.97^ log. 9.899335 Or, since the sine of any angle is equal to the sine of its sup- ])lement, the supplement in this case, 52° 29', may be taken directly from the logarithmic table, from which supplement deducting x, or 48° 11', the remainder is the angle y = 4° 18'. Finally, IC = 86 . . . log. 1.934498 Multiplied by sin. x, 48° 11' . . . log. 9.872321 = CD . . . log. 1.806819 And C D divided by sin. y, 4° 18' . . . log. 8.874938 = C O = R = say, 855 feet . . . log. 2.931881 2. In the case of a rectangular intersection, the solution is more simple. It is quite plain, from the figure, that — from which equation, R = a + 6 4- ^2ab. FROGS AND SWITCHES. 129 Example. a = 40, 6 = 80. Then R = 40 + 80 + VmOO = 200. 3. Cases of this kind are disposed of with great ease in the field by means of the curve-protractor. XLVI. FROGS AND SWITCHES. t T W o l\ \>5>r AN /I .\r^^ / V TO FIND THE RADIUS OF A TURNOUT CURVE, THE FROG ANGLES, AND THE DISTANCES FROM THE TOE OF SWITCH TO THE FROG POINTS. 1. Draw the figure as in margin, C being the centre of the turnout curve, C K parallel to main track, and O K, I E, L M, perpendicular to it. Call the angle of the frogs at O, F; that of the intermediate frog at I, 2 F'; the throw of the switch-rail for single turnout, D; its angle with main track, S; the gauge of the track, G; and radius of outer rail, R. 2. Usually the lengtli . and throw of switch-rail and the angles of the frogs at O are given. In that case, to find R, F' and U>o distances LO, LI, reason thu?:-' 130 FROGS AND SWITCHES. 3. The angle H N W, between the line of the switcli-rail pro- longed and a tangent to turnout curve at frog point O, = NOP — NIIW = F — S. The angle NOL or NLO, be- tween chord and tangent, = half the intersection angle H N W = i(F — S). The angle NOB = NOL + LOB. But NO L is seen to be = i (F — S), and NOB = F; then L OB = NOB — NOL = F — ^ (F — S) =i (F + S). The distance LO, from toe of switch to point of main frog, = LB -^ sin, LOB = (G — D) -^ sin. i (F + S). 4. Again : the angle LCY = NLO=|(F — S);LY = i LO = i (G — D) -^ sin. i (F + S). LY -^ sin. LCY = LC; i.e., [^ (G — D) -f- sin. i (F + S)] -f- sin. i (F — S) = R. 5. R may be found otherwise, as follows: — OK = OC co.s. KOC = Rcos. F; LM = LC co.s. CLM = Rcos. S; LM — OK = LB; i.e., R {cos. S — co.s. F) = (G — D). Hence R = (G — D) -^ {nat. cos. S — nat. cos. F). 6. If R be known, to find F. This equation gives nat. cos. F = nat. cos. S — I (G — D) H- R]. 1. To find the angle, 2 F', of the middle frog at L IE = I P + P E or O K; i.e., R cos. F' = i G + R co.s. F. Hence nat. cos. F' = nat. cos. F -\- (| G -4- R). 8. The angle LI Y, by similar rciisoning to that used in rela- tion to LOB, is found to be = | (F' + S). The distance L I, from toe of switch to point of middle frog, = L Y -i- sin. L I Y = (iG — D)-^sm. i(F' + S). The preceding formulas translate into the following — RULES FOn FROG.S AND SWITCHES. 9. To find the Angle of Switch-Rail with Main Track. Divide its throw, in decimals, by its length: the quotient will be the natural sine of the angle sought. 10. To find the Distance from Toe of Switch to Point of Main Frog. Subtract the throw of switch-rail from the gauge of track, both in decimals; call the remainder a. Add together the angle of switch-rail with main track and the angle of the main frog; find the natural sine of half this sum, and call it 6. Divide a by b: the quotient will be the distance sought. FROGS AND SWITCHES. 131 11. To find the Radius of Outer Bail of Turnout Curve. Subtract the throw of switch-rail from the gauge of track, both ill decimals ; call the remainder a. Subtract the natural cosine of the main frog angle from the natural cosine of the switch-rail angle; call the remainder h. Divide a by h: the quotient will be radius. 12. To find the Main Frog Angle, the Badius of the Outer Bail being known. Call the natural cosine of the switch-rail angle a. Subtract the throw of switch-rail from the gauge of track, both in deci- mals. Divide the remainder by radius; call the quotient h. Subtract b from a: the remainder will be the natural cosine of the main frog angle. 13. To find the Angle of the Middle Frog, in the Case of a Double Turnout. Call the natural cosine of the main frog angle a. Divide half the gauge of track by the radius of outer rail of turnout curve; call the quotient b. Add a and b together. Their sum is the natural cosine of half the middle frog angle. 14. To find the Distance from Toe of Switch to Point of Middle Frog. Subtract the throw of switch-rail from half the gauge of track, both in decimals ; call the remainder a. Add together the switch-rail angle and half the middle frog angle. Find the natural sine of half this sum; call said natural sine b. Divide a by 6 : the quotient will be the distance sought. 15. The use of logarithms will be found convenient in work- ing these rules. Examples. 16. Switch-rail, 18 feet; throw, 5 inches = 0.42 feet; frog angle, 5° 44'; gauge, 4.71 feet. Sin. S = 0.42 -M8 = .02334 = sin. 1° 20'. LO = (G — D) -^ sin. |(F -f S) = (4.71 — 0.42) -f- sin. 3° 32' = 4.29 -^ 0.0616 = 69.64 feet. K = (G — D) -H {nat. cos. S — nat. cos. F) = 4.29 -^ 0.00473 = 907 feet. Nat. cos. F' = nai. cos. F -|- (| G -^ K) = 0.995 -f (2.354 -H 232 FROGS AND SWITCHES. 907) = 0.99759 = cos. 3° 58^. Hence the angle of Ihe middle frog = 2 F' = 7° 57'. LI = (i G — D) 4- sin. I [W + S) = (2.354 — 0.42) -^ sin. i (3° 58f + 1° 20') = 1.934 -h 0.0463 = 41.8 feet. 17. In ordinary practice, frogs may be located with sufficient exactness by the following rules, deduced from the congruity of triangles. Great nicety in their location is not necessary. The important thing in practice is to lay the turnout curve so that the approach to the frog shall be fair and regular. How trackmen may do this without the use of instruments, in a very simple way, will be shown hereafter. Not that frogs may be set hap-hazard, and the approaches forced to fit: they ought to be nearly where they mathematically belong, and they can be thus placed by means of the rules subjoined. 18. Let N stand for the number of the frog; L the length of switch-rail in feet; F the distance from toe of outer switch-rail to point of frog in feet. Then, for standard gauge, 4 feet 8^ inches, straight switch- rail, and 5 inches throw of switch. y_ 8.6LN L 4- 0.42 N* The above may be written roundly as a rule thus: — Multiply the length of switch-rail in feet by the number of the frog, and set down the product. Multiply that product by 8^, and call the result A. Next add together the length of switch-rail in feet and two-fifths of the frog number; call the sum B. Then divide A by B, and the quotient will be the dis- tance in feet from toe of outer switch-rail to point of frog. Example. Switch-rail, 20 feet long; frog. No. 9. Length of switch-rail 20 Multiplied by frog number .... 9 Product 180 Multiplied by . . 8^ 1,530 = A. Length of switch-rail 20 Added to | frog No, 9 3.6 FROGS AND SWITCHES. 133 A ilivided by B = 1,530 divided by 23.6 = 04,8 feet, the frog distance; say, (35 feet. 19. If tlie switch-rail be curved, the formula would stand ^lus: — 8.6 LN F = L + 0.84 N Which may be made a written rule as follows: — Multiply the length of switch-rail in feet by the number of the frog, and their product by 8j; call the result A. Add together the length of switch-rail in feet and four-fifths of the trog number; call the sum B. Then divide A by B, and the quotient will be the distance from toe of outer switch-rail to point of frog in feet. 20. The foregoing rules are applicable to turnouts from curves, as well as from straight lines. 21. To find the radius of outer rail of a turnout curve from straight track. Data same as in previous rules for frogs; R 'the required radius iu feet. 8.6 L2 N2 If the switch-rail be straight, R If the switch-rail be curved, R L2 — O.nN-^ 8.6 L'^ N^ L2 — 0.68 W' 22. To find the radius of the outer rail of a turnout curve from curved track, proceed thus : — First find the radius as for a turnout from straight track by I he preceding rule; call it, as before, R. Call the radius of the :nain track R2, and the required radius of turnout curve r. Then, if the turnout be towards the concave side of main track, — R. X R ^ ~ R. + R* If the turnout be towards the convex side of main track, — Ro X R r = * . R.2 — R More explicitly, in the first case, r is equal to the product of the other radii divided by their sum ; and. In the second case, r is equal to the product of the other radii divided by thei: vlifference. 134 FROGS AND SWITCHES. 23. The angle of a frog is equal to 3,440' divided by the frog number. 24. To find the frog distances and radii for a three-foot gauge, find them by the preceding rules for standard gauge, and take five-eighths of the result, using a switch-rail reduced \n like measure. For a metre gauge, take seven- tenths of the result, using a switch-rail reduced in like measure. Or these radii and distances may be found from the appended tables for standard gauge by pro-rating as above. 25. Three frog patterns are enough for general service. They should be so proportioned, that, taken in couples, the less may fit as middle frogs on double turnouts. Numbers 5^, 7^, and lOi make an excellent suit; numbers 5, 7, and 9^- also answer very well. 26. At the terminal stations, and about the shops of busy roads, patterns necessarily multiply. The better way in such cases is to plot the situation to a large scale, and to take the required distances and angles from the drawing. TURNOUT TABLE. 135 'Si 11 ^1 II s a" ©to Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. ■Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. 32.6 138.3 21.3 19° 16' .32.2 138.5 20.9 19° 17' 31.8 138.8 20.5 19° 19' 31.2 139.4 19.9 19° 22' 30.2 140.6 19.0 19° 28' I- » H W > i 36.4 175.2 23.7 17° 08' 35.9 175.6 23.2 17"" 09' 35.4 176.0 22.7 17° 10' 34.6 176.9 22.0 17° 14' 33.5 178.9 20.9 17° 20' 1- 40.1 215.6 26.1 15° 25' 39.5 216.6 25.5 15° 28' 38.9 217.4 24.8 15° 30' 38.0 218.8 24.0 1.5° 34' 36.6 221.8 22.6 15° 41' r ^to||^ Sbbto 43.5 267.1 27.8 14° 00' 42.7 268.2 27.1 14° 02' 41.6 270.3 26.0 14° 06' 39.9 274.9 24.4 14° 14' ■^OJCO-JI 46.8 314.2 29.8 12° 56' 45.9 315.7 28.9 12° 58' 44.6 318.6 27.8 13° 02' 42.7 325.0 25.8 13° 12' i» 50.9 362.0 32.8 11° 58' 49.9 363.7 31.7 12° 00' 48.9 365.8 30.8 12° 02' 47.5 369.7 29.4 12° 06' 45.3 378.4 27.3 12° 16' 00 54.9 427.3 34.2 11° 04' 05 50 05-^ 52.6 432.6 32.8 11° 10' 50.9 438.0 31.3 11° 14' 48.4 450.2 28.9 11° 24' 58.1 485.3 37.0 10° 25' 56.8 488.4 35.6 10° 28' 55.5 492.1 34.5 10° 30' 53.7 499.2 32.8 10° 34' 50.9 515.0 30.1 10° 46' 61.8 556.2 39.3 9° 42' 60.3 560.2 37.9 9° 44' 58.8 565.0 36.5 9°47' 56.8 575.0 34.5 9° 54' 53.7 595.8 31.6 10° 04' t-lOO 65.0 624.8 41.3 9° 10' 63.4 630.0 39.7 9° 14' 61.8 636.0 38.1 9° 16' 59.5 648.0 35.9 9° 22' 66.1 675.0 32.8 9° 34' 68.7 707.0 43.3 8° 39' 66 8 713.5 41.6 8° 41' 65.0 721.0 39.9 8° 44' 62.6 737.0 37.4 8° 51' 58.8 772.0 34.0 9° 03' =8 72.0 788.5 15.2 8° 14' 70.0 796.5 43.3 8° 17' 68.0 807.0 41.4 8° 21' 65.3 826.0 38.7 8° 27' 61.3 870.0 35.1 8° 40' i^ 75.3 875.0 47.3 7° 47' 73.1 885.0 45.2 7° 50' 71.0 897.0 4.3.1 7° 54' 68.0 921.0 40.4 8° 01' 63.6 976.7 36.2 8° 15' cn 78.5 964.2 49.0 7° 27' 76.1 976.2 46.8 7° 30' 73.8 992.0 44.4 7° 34' 70.0 1,020 41.6 7° 40' 65.9 1,089 37.2 7° 55' cn 4S 82.0 1,068 50.9 7° 09' 76.4 1,083 48.3 7° 12' 76.9 1,102 45.9 7° 15' 73.4 1,137 43.0 7° 22' 68.3 1,224 38.3 7° 38' 5^ 84.8 1,157 52.9 6° 50' 82.0 1,175 49.9 6' 54' 79.4 1,196 47.6 6° 57' 75.7 1,263 43.8 7° 09' 70.3 1,342 39.2 7° 21' cn J lm< ■^*>.oooo 82.5 1.325 49.0 6° 41' 78.6 1,377 45,3 6° 49' 72.7 1,506 40.1 7° 05' 136 TURNOUT TABLE. o !z: <; "A m P3 56.3 4,117 26.5 8° 45' 2P? looo tov Spl EpI si 54.7 3,135 26.2 8° 58' 61.5 1,762 31.6 8° 03' 66.7 1,469 35.8 7° 37' 70.3 1,343 39,1 7° 21' 74.6 1,256 43.0 7° 08' ■ 53.4 2,516 25.9 9° 10' 59.9 1,549 31.1 8° 18' 64.6 1,315 35.1 7° 51' 68.2 1,215 38.3 7° 36' 72.2 1,144 41.9 7° 24' SI 25.5 9° 26' 58.1 1,351 30.4 8° 36' 62.6 1,169 34.3 8° 10' 65.8 1,090 37.3 7° 55' 69.6 1,032 40.7 7° 43' s5: 50.6 1,672 25.1 9° 42' 56.4 1,182 29.8 8° 54' 60.5 1,040 33.5 8° 29' 63.6 976.8 36.3 8° 15' 67.1 930.3 39.5 8° 03' "°: 49.0 1,370 24.6 10° 04' 54.4 1,022 29.1 9° 16' 59.1 914.7 32.5 8° 53' 61.2 865.2 35.1 8° 39' 64.4 828.3 38.1 8° 28' 47.5 1,160 24.1 10° 26' 52.6 894.7 28.4 9° 38' 56.2 811.1 31.6 9° 16' 58.8 772.1 34.1 9° 03' 61.8 742.8 36.8 8° 53' .:4 45.8 947.3 23.6 10° 52' 50.4 767.0 27.6 10° 08' 53.7 704.9 30.5 9° 46' 56.2 675.2 32.8 9° 34' 58.8 652.7 35.3 9° 24' >5 00 "-1 44.1 798.2 23.0 11° 18' 48.5 666.2 26.8 10° 36' 51.5 618.9 29.5 10° 16' 53.7 595.8 31.6 10° 04' 56.2 578.1 33.9 9° 55' "-t 42.2 656.1 22.3 11° 54' 46.1 564.3 25.8 iri4' 48.8 529.8 28.2 10° 56' 50.9 513.0 30.2 10° 44' 53.0 499.7 32.2 10° 36' t-r-l °00 40.4 5.53.9 21.7 12° 30' 44.0 487.1 24.9 11° 52' »OC0(M^ 48.4 448.3 28.9 11° 23' 50.3 438.3 30.8 11° 14' CO 38.3 451.1 20.9 13° 18' 41,5 405.7 23.8 12° 42' 43.7 387.6 25.8 12° 26' 45.3 378.4 27.3 12° 16' 47.1 371.2 29.0 12° 08' CO^ °o 36.4 375.8 20.1 14° 08' 39.3 343.7 22.7 13° 34' 41.3 330.7 24.6 13° 18' 42.7 324,0 25.9 13° 10' 44.2 318,6 27.4 13° 02' 34.4 310.2 19.3 15° 06' 36,9 288,1 21.6 14° 36' 38.7 278.9 23.2 14° 21' 39,9 274.0 24.4 14° 12' 41.2 270.3 25.7 14° 06' < 31.9 244.9 18.2 16° 30' 34.1 230.8 20.2 16° 02' 35.6 224.9 21.6 15° 48' 36.6 221.8 22.6 15° 41' 37.7 219.3 23.7 15° 35' 4 29,5 193,7 17.1 18° 06' 31.4 184.8 18.8 17° 40' 32.6 181.0 20.0 17° 28' 33,5 178,9 20.8 17° 20' 34.4 177.3 21,7 17° 15' ■i 26.9 149.6 15.8 20° 08' 28.5 144.2 17.29 19° 44' 29.5 141.9 18.25 19° 34/ 30.2 149.6 18.9 19° 28' 31.0 139.6 19.7 19° 23' Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist, Rad. outer rail. Mid. frog dist. Mid. frog angle. Main frog dist. Rad. outer rail. Mid, frog dist. Mid, frog angle. Switch-rail, L'gth. 12 ft. Ang. 4° 00'. Rad. 171.9. Switch-rail, L'eth. 16 ft. Ang. 3° 00'. Rad, 312.7. Switch-rail, L'gth. 20 ft. Ang. 2° 24'. Rad. 477.5. Switch-rail, L'gth. 24 ft. Ang. 2° 00'. Rad. 687.6, Switch-rail, L'gth, 30 ft, Ang, 1° 36', Rad. 1,074. TO LOCATE A TURNOUT, 137 XLvn. TO LOCATE A TUENOUT. \. Let the heavy parallels in the figure represent the rails of the main track. 2. Stick a pin or drive a spike at A, the toe of switch, aV, a distance from the gauge side of the main-track rail equal to the throw of the switch-rail. Lay off the distances A C and AB (if a double turnout), taken from the foregoing tables, and place the frogs C and B, or mark those points. Stretch the cord from A to B, and from B to C. Mark the middle points of those stretches at H and P. Catch the cord at H with your forefinger, and pull it outwards until your finger, at E, lines with the switch-rail, and also with the right gauge side of frog B, Stick a pin at L, half-way between H and E. Let the cord spring in against L, so that it shall stretch straight from A to L, and from L to B. Opposite the middle points, V, of those stretches, stick pins on the outside at a distance from the cord equal to one-quarter of H L. In like manner, catch the cord at P, the point midway between B and C ; stretch it to F, in line with the gauge sides of the fro^s ; and §tick a pip ^X I, half -way between J* aud Ft 138 TO LOCATE A TURNOUT. 3. Next lay off the proposed line of the near rail of the sid$ track, X D. Mark the point G on that line where the range of the proper gauge side of frog C strikes it. Measure C G. Set off G D, equal to C G, along the side-track line, and drive a pin at D. Stretch the cord from C to D. Mark the middle point of it at K, and drive a pin at N, half-way between K and G. Stretch the cord from C to N, and from N to D. Stick pins outside the middle points, M and O, of those stretches at a distance from those points, M and O, equal to one-quarter of KN. 4. These three sets of pins will fix the line of one rail of the turnout. The corresponding rail of a double turnout can be laid off from them, if required, by symmetrical measurements. 5. In the case of a single turnout, stretch the cord from the toe of switch, as above, to the point of frog, located by the foregoing tables ; catch it at the middle, and pull it outwards to a point in range with the line of the switch-rail in one direction, and the gauge side of frog in the other direction. Half-way between that point and the middle of the cord, when straight, stick a pin. Measure that half-way distance, and divide it by 4; call the quotient the "quarter-distance." Stretch the cord from the pin just set to the toe of switch in one direction, and to the point of frog in the other. Outside the middle points of these short stretches, lay off the " quarter- distance," as above found, and stick two other pins. These three pins will sufficiently mark the line of the outer rail of Ihe turnout. 6. The same methods will apply in practice to turnouts from curves. In the latter case, the distance C G is to be calculated as follows : — Multiply the distance Y D, between the nearest rails of the parallel tracks, by the number of the frog, taken from the fore- going table. Thus, on the full gauge, with a space between tracks of 7 feet and a No. 6 frog, the distance C G would be 7 X 6 = 42 feet. Lay off C G, in range of the gauge side of the frog, and stick a pin at G. Measure out GD, equal to CG, and set another pin at D, making D Y the proper distance be tween tracks. Then stretch the cord from D to C, and pro ceed to stake off the curve C N D, as above directed. CROSSINGS ON STRAIGHT LINES, 139 XLVIII. CROSSINGS ON STRAIGHT LINES. 1, Having located frogs B and C by the preceding methods, stretch the cord any convenient distance, C D, in the range of the outer gauge side of the frog C. Set off E F parallel to CD, and distant the gauge-width from it. The intersection of said parallel at F with the near rail of the side track marks the spot for point of side-track frog; the curve F G, thence to toe of switch, corresponds to A C on the main track, and may be staked out in like manner. XLIX. CROSSINGS ON CURVES. 1. Having located frogs B and C by the preceding methods, set off the width of gauge, C D, from point of frog C, and square to its outer gauge side. Stick a pin at D. 2. Next calculate the distance D E to the point of side-track frog as follows: Subtract the gauge of track from the dis- /40 CROSSINGS ON CURVES. tance, H I, between the gauge sides of the nearest rails of the main and side tracks ; multiply the remainder by the number of the frog, taken from foregoing tables. The product will be the distance from D to the point of side-trftck frog at £. ELEVATION OF THE OUTER RAIL ON CURVES. 14J 3. Suppose, for example, the gauge sides of the nearest rails of the main and side tracks are 6 feet 6 inches asunder; gauge of track, 4 feet Si inches; frog, a No. 9. Kediici ug inches to dechnals, we have then the distance between tracks 6,5 feet, less the gauge, 4.7 feet, = 1.8 feet; and 1.8 multiplied by i), the number of the frog, gives 16.2 feet for the distance D E. The proper spring will be given to rail DE on the ground; and curve E G, from frog to toe of side-track switch, will be staked off as directed in the section on turnouts. L. ELEVATION OF THE OUTER RAIL ON CURVES. 1. Great precision in this adjustment is unattainable, owiiij^- to differences in the speed of trains and to the cost of track - maintenance, if it were attempted. 2. Molesworth gives the following formula for determining the elevation of the outer rail with any gauge : — V — greatest velocity of trains in feet per second. G — gauge of railway in feet. C = length of chord whose middle ordinate will give the required elevation. Then C = i V ^ G^ A modification of this formula gives the following approximate rules : — To fix the elevation of the outer rail on the standard gauge of 4 feet ^\ inches, multiply the speed of trains in miles per hour by 5, and divide the product by 3. This will give the length of tape, C, to stretch on the gauge side of the outer rail; and the distance, c. from the middle of the tape to the gauge side of the rail, wi.l be the proper elevation. For guage of one metre, — 3.28 feet, make C equal to one and one-third times the speed of trains in miles per hour. For 3-feet gauge, make C equal to one and one-fourth time the speed of trains in miles per hour. 143 ELEVATION OF THE OUTER RAIL ON CURVES. TABLE OF ELEVATIONS OP OUTER RAIL ON CURVES. This table was formulated by the writer from Pennsylvania Rail Road practice as follows : (jV speed in miles per hour + 1) X (by the degree of curve) = elevation of outer rail expressed in 8ths of an inch. SPEED IN MILES PER HOUR. 10 20 30 40 50 60 fi VALUES IN EIGHTHS OF AN INCH. 2° 4 6 8 10 12 14 4° 8 12 16 20 24 28 6° 12 18 24 30 36 42 8° 16 24 32 40 48 10° 20 30 40 50 12° 24 36 48 14° 28 42 16° 32 48 •• •• Note.— The limit of elevation of outer rail is 6^ inches. TRACKMEN'S TABLE OF CURVES. 14< LI. ocooo-ac»tgi*^cot0h-^otooo~jcgi:^ifjcotoi-j DEGREE OP CURVE. OOMO.OI-'rfs.ollibOWQOI-itOCntOOCOaiCD^ ts4^ iUt9 COtOtOK) DEFLECTION DISTANCES IN FEET AND INCHES. 2 1-2 5 1-4 8 10 1-2 13 15 3-4 18 1-4 21 231-2 261-4 29 311-2 34 36 3-4 39 1-4 42 441-2 47 1-4 49 3-4 52 1-2 1 3 ■ I 4 '4 2 « H W |. S IS o H n '■.■''r'T"r''r'T"r"r'7"r"7-'T"r"r" "r"?^ 'r"T" oo to 00 io to 00 ifi. to 00 *>. to 00 *> 00 tt-oo it-oo *.*»*^*^(MI»COMtatOtOtOI--MI-.i-i tut £S£g|£SSS£2|||S^ 8 • COtOtOtOtOtOlCi-'Mp-'l-'f-'l-'H- t-'OiOi H-"^ OSOJ 050J 1-8 3-16 5-16 3-8 1-2 5-8 3-4 13-16 15-16 11-16 11-8 11-4 11-2 15-8 13-4 17-8 1 15-16 2 1-16 2 5-16 1 Ft. In. 463 5 328 5 266 2 232 207 7 189 8 175 6 164 1 154 10 146 6 140 134 1 129 124 1 120 116 2 112 10 109 7 106 8 104 ►S£q ff LENGTH OF CHORD IN FEET AND INCHES, WITH A MIDDLE ORDINATE EQUAL TO GAUGE OP TRACK. |.JtSOJp-'l-'©tOl-'t>OOt>0-»©tO*«.|-»M-^0© P f ^^^^^^!^^^^^«S°g-J®o«^.'^«o'i^ DEGREE OP CTJRVB. X44 trackme:n's table of curves: EXPLANATION OF THE FOREGOING TABLE. Columns 1 and 10 give the degree of curve. The use of column 2, containing the deflection distances, may be illustrated thus : Suppose stakes 4, 5, and 6 to be miss- ing from a 3-degree curve, and that stakes 2 and 3 are still standing 100 feet apart. To replace the missing stakes, pro- ceed as follows: Measure 100 feet from 3 to A, and make a mark at A exactly in range with 2 and 3. Find, in column 2 of the table, the deflection distance for a 3-degree curve, which is seen to be 5 feet 3 inches. Hold one end of the tape at A, and, stretching 5 feet 3 inches towards 4, nearly square to the range A-3, make a scratch on the ground three or four feet long, swinging the tape around A as a centre. Next lay off 100 feet from stake 3 to the scratch; where the end of that measurement strikes it, is the place for stake 4. By measuring 100 feet out to B on the range 3-4, and proceeding in like manner, stake 5 may be set ; and so on. 3. If the centre line is already staked for track at points 100 feet asunder, and the degree of curve is wanted, range out the straight line between stakes, as above, to A or B, and measure across from those marks to the neighboring location-stake. Suppose the distance B-5, for example, to be 8 feet 9 inches. Referring, then, to column 2 of the table, we find that deflec- tion distance to indicate a 5-degree curve. If the distance TRACKMEN'S TABLE OF CURVES. 145 proved to be 4 feet 4 inches, we should soon discover that that distance was about half-way between 3 feet 6 inches and 5 feet 3 inches, the nearest numbers in the table corresponding respectively to a 2-degree and a 3-degree curve, and showing the located line to be a 2^-degree curve. -1. Let A C B in the figure, which is drawn very much out of proportion in order to make the subject clear, represent the centre line of a curve. Suppose G H to be a chord 100 feet long, and G C or C H to be a chord 50 feet long. Then column 3 in the table gives the distance, C D, from the middle of the 100-feet chord to the rail, and column *4 gives the distance, E F, from the middle of the 50-feet chord to the rail, for the different degrees of curve. By the aid of these columns, pins can be set 25 feet apart on a curve where the location-stakes are 100 feet apart. Thus, for a 3-degree curve, C D is 8 inches, N ^ N and E F 2 inches. If pins were wanted at the half-way marks^ '^, their distance from the dotted short cliords would be one>= quarter of E F. It must be an uncommon case, however, that calls for stakes closer togetlier than 25 feet. 5. Columns 5, 6, and 7 give the spring of rails of different lengths for the various degrees of curve. 6. Columns 8 and 9 give figures for finding the degree of curve, by simple measurement of a straight line on the track, as follows : Suppose A C B and K I L to represent the rails of a curving track. From any point A, on the outer rail, sight icross to a point B, on the same rail, along a line just touching the inner rail at I. Measure from A to B, and seek the dis- tance in column 8 or 9, according to the gauge of track. If the distance, for example, measured 232 feet on the full gauge, then the curve would be a 4-degree curve ; if 249 feet, then it would indicate a 3i-degree curve, for the reason that the 146 TRACKMEir*S TABLE OF CURVES. measured distance falls half-way between the distances corre- sponding to a 3-degree and a 4-degree curve respectively. 7. The rate of curve can be found also very nearly by means of column 3. To do so, stretch a straight line, 100 feet long, between points on either rail ; for, though they seem very dif- ferent in the figure, the two rails of a track have practically the same curvature. Measure from the middle of the line across to the gauge side of the rail, and seek the measured distance in column 3: opposite to it, in column 1, will be found the degree of curve. 8. If, in any case, the exact figures sought are not found in the table, take out the next figure less and the next greater. Subtract one from the other, and divide the remainder by 4. Add the fourth part of the difference between them, thus determined, to the smaller number, and compare the sum with the number sought. If still too small, add another fourth part; and so on until the distance or the degree is ascertained to within a quarter part. 9. Suppose, for instance, a deflection distance measures 5 feet 7 inches. The nearest tabular numbers are 5 feet 3 inches and 7 feet. Their difference is 21 inches, one-fourth of which is 5j inches. Adding h\ inches to the smaller number, 5 feet 3 inches, gives 5 feet 85- inches, which indicates nearly enough a 3|^-degree curve. Again: if a measurement of 175 feet is sought in column 9, the track is see::> at once, witbo»at calcmat- tion, to be a 4^-degree curve. TABLES TABLES OF THE TIMES OF CULMINATION AND OF ELONGATION OF TEE POLESTAll AND OF ITS AZIMUTH AT ELONGATION. These tables are designed to facilitate the determination of a meridian line and of the magnetic declination (variation of the compass) by simple instrumental means (p. 44). For this purpose the tables are sufficiently accurate. They will also be found useful when preparing for or laying out work for a more refined determination of the astronomical azimuth and for the measures of the value of an eye-piece micrometer. 148 TABLE I. MEAN LOCAL (ASTRONOMICAL) TIM^., COUNTED FROM NOON AND FROM ZERO TO TWENTY-FOUR HOURS, OF THE CULMINATIONS AND ELONGATIONS OF POLARIS IN THE YEAR 1889. COMPUTED FOR LATITUDE 40° NORTH AND LONGITUDE 6 HOURS WEST FROxM GREENWICH. 1889. Date. E. Elong. LTppER Culm. W . Elong. Lower Culm. h. m. h. m. h. m. h. m. Jan. 1 " 15 36.2 6 31.0 5 35.7 12 25.7 11 30.4 18 29.1 17 33.8 23 37.0 Feb. ] 22 29.9 4 28.6 10 23.3 16 26.7 15 21 34.6 3 33.3 9 28.1 15 31.4 March 1 20 39.4 2 38.1 8 32.8 14 36.2 15 19 44.4 1 43.1 7 37 7 13 41.1 April 1 " 15 18 37.4 17 42.4 36.0 6 30.7 5 35.7 12 34.1 11 39.0 23 37.1 May 1 16 39.5 22 34.2 4 32.9 10 36.1 " 15 15 44.6 21 39.3 3 38.0 9 41.2 June 1 14 37.9 20 32.7 2 31.3 8 34.6 15 13 43.0 19 37.8 1 36.4 7 39.7 July 1 15 12 40.4 11 45.5 18 35.2 17 40.3 33.8 6 37.1 5 42.2 23 35.0 Aug. 1 10 39.0 16 33 8 22 28.4 4 35.7 " 15 9 44.1 15 38.9 21 33.5 3 40.8 Sept. 1 8 37.5 14 32.3 20 26.9 2 34.2 " 15 7 42.6 13 37.4 19 32.0 1 39.3 Oct. 1 15 6 39.7 5 44.7 12 34.5 11 39.5 18 29.1 17 34.1 36.4 23 37.6 Nov. 1 4 87.9 10 32.7 16 27.3 22 30.8 15 3 42.7 9 37.5 15 32.2 21 35.6 Dec. 1 2 39.7 8 34.5 14 29.2 20 32.6 15 1 44.4 7 39.2 13 34.0 19 37.3 To refer the tabular times to any year subsequent to the tabular year (1889) add 0"\33 for every year. To refer the tabular times, corrected as above, to any year in a quadrennium, observe the following rules: For the first year after a leap-year the table is correct. For the second year after a leap-year add 0™.9 to the tabular value. For the third year after a leap-year add 1™.7 to the tabular value. For leap-year and befoi'e March 1 add 2'". 6 to the tabular value. For leap-year from and after March 1 subtract 1™.2 from the tabular value. CULMINATIOXS AND ELONGATIONS OF POLARIS. 149 To refer to any calendar day other than the 1st and 15th of each mouth, subtract 3'". 94 for every day between it and the preceding tabular day, or add 3'". 94 for every day betweeu it and the succeeding tabular day. The longitude correction will amount to 0"'.16for each hour. To refer to any other than the tabular latitude, and between the limits of 25° and 50° North, add to the time of west elouga tion 0°i.l3 for every degree South of 40° and subtract from the time of west elongation 0™.18 for every degree North of 40°. Reverse these signs for corrections to times of east elongation. Observe that the year 1900 is not a leap-year, and this must be kept in view when dealing with dates from and after March 1 of that year. The 20th century begins after the ex- piration of Dec. 31, 1900. The deduced tabular times may be relied on to have no greater error than ± 0*^.3. Table II. below Lat. 24° is abridged from a table for each degree of latitude between 25° and 50° North, computed for this book by Mr. C. A. Schott, Asst. Supt. of the U. S. C. and G. Survey, with the mean declination of Polaris for each year. A closer result will be had by applying to the tabular values the following correction, which depends on the differ- ence of the mean and the apparent places of the star : For Middle of Lat. 25° Lat. 40° Lat. 50° For Middle of Lat. 25° Lat. 40° Lat. 50° Jan. Feb. March April May June - 0'.3 -0.3 - 0.1 0.0 + 0.2 + 0.3 - 0'.4 -0.3 - 0.2 0.0 + 0.2 + 0.3 -0'.4 - 0.4 -0.2 0.0 + 0.2 + 0.3 July Aug. Sept. Oct. Nov. Dec. + 0'.2 + 0.1 0.0 - 0.2 - 0.5 -0.6 + 0'.3 + 0.1 -0.1 -0.3 -0.6 -0.8 + 0'.3 + 0.2 - 0.1 - .3 -0.7 - 0.9 The deduced tabular azimuth, counted from the North, may generally be depended upon with no greater error than ± 0'.2. In making the computation the mean places of Polaris were first accurately deduced from Newcomb's Catalogue of 1098 standard clock and zodiacal stars, Washington, 1881, for five equidistant epochs. From these fundamental places those f(jr each year were readily found by interpolation. Azimuth for latitudes less than 25° was reckoned by the author from the data for that degree. 150 AZIMUTH OF POLARIS AT ELONGATION. • 03 Xi *" a XL ^5 .2 H ^ H .3 5 cc d S H ^ a g *W5 § 2 >» 2 g n o s t4 s S eg '& -^ 1 03 H ^ 53 >^ OJ ,C3 QJ H -d s t; *^ H fl d rt O S S t^ p s § d -1 ;_i H CS 3 3 bO d c3 W +-' A H i y-i O d w 1 1? ^ o ■^ tJ s n-t S^ s O 03 O 03 ^5 J ° ^«'2?SS^^g5g^^^?§^^:5^^S o «»^oeoo5i-iTt'QOooT-iOi-iTtoooojT)'so«o S;;?3?5S^SS5JS5S^S??^^^i55 05 1 osift«cwcoiOi-i-ic-'>aC0»O00C000i0i0CDO«ii0<.-0»(NJ0 i l-ii-»3i»00l-l?0r-l050SO-«'!=>05T--iiO SS^'^$^Sg55S^^=;jg^53S^^5^S i i i-(£-?oc:«50s»-ia5i:>ioOT-i«5c^?j-0:JOX)OCOTl<«3^050COO(N05 SS^SS?:j^^^7^5;S?i§^^^5SL5 1 WX)00— iC5C^ID'rjQ0l-00O>O00T) 2546.6 3.405961 6-594039 .491 3-93 1.96 ! 20 2455 -7 3-390175 6.609825 -509 4.07 2.04 li 25 2371.0 3-374932 6.625068 .527 4.22 2.ir i ! 30 2292.0 3.360215 6.639785 .545 4-36 2.18 1 35 2218. I 3-345982 6.654018 •564 4-51 2.25 1 40 2148.8 3.332196 . 6.667804 .582 4-65 2.33 45 2083.7 3-318835 6.681165 .600 4.80 2.40 1 so 2022.4 3-305867 6.694133 618 4.94 2.47 1 55 1964.6 3-293274 6.706726 636 5-09 2.54 3 1910.1 3.281056 6.718944 655 5-23 2.62 1 i 5 l8q8.5 3.269163 6.730837 673 5.38 2.69 10 1809.6 3.257584 6.742416 691 5-53 2.76 ! IS 1763-2 3.246301 6.753699 709 5-67 2.84 20 1719.1 3-235301 6.764699 727 5-82 2.91 1 25 1677.2 3-224585 6.775415 745 5.96 2.98 30 1637-3 3.214129 6.785871 764 6.11 3.05 35 1599.2 3.203902 6.796098 782 6.25 3-13 40 1562.9 3.193931 6.806069 800 6.40 3-20 45 1528.2 3.184180 6.815819 818 6.54 3-27 50 1495.0 3. 1 74641 6.825359 836 6.69 3-34 55 1463.2 3-165303 6.834607 855 6.83 3-4^ 4 1432.7 3-156155 6.843845 873 6.98 3-49 5 1403.5 3.147212 6.852788 891 7.12 3-56 10 1375-4 3.138429 6.861571 909 7.27 3-63 15 1348.4 3.129819 6.870181 927 7.42 3-71 20 1322.5 3-121395 6.878605 945 7-56 3-78 25 1297.6 3.113141 6.886859 964 7.71 3-85 30 1273.6 3-105033 6.894967 982 7-85 3-93 35 1250.4 3.097048 6.902952 I, 00 8.00 4.00 40 1228.1 3-089233 6.910767 I. 02 8.14 4.07 RADII JJfD THEIR LOGARITHMS. 157 Degree Logarithm Arithmetical Middle Deflec- Tangen- of Radius. of Comple- v_/rQmatej Chord tion Dis- tial Dis- 1 Curve. o / 4 45 Radius. ment. 100 Feet. tance. tance. j 1206.6 3.081563 6.918437 1.04 8.29 4.14 i 50 1185.8 3.074011 6.925989 1.05 8-43 22 55 1165.7 3.066587 6.933413 1.07 8.58 29 1 5 1146.3 3.059299 6.940701 1.09 8.72 36 1 5 1127.5 3.052117 6.947883 I. II 8.87 43 \ TO 1109.3 3.045050 6.954950 1-13 9.01 51 i 15 1091.7 3.038103 6.961897 1-15 9.T6 58 I 20 1074.7 3.031287 6.968713 1. 16 9-30 65 ! 25 1058.2 3.024568 6.975432 1. 18 9-45 72 ; 30 1042.1 3.017910 6.982090 1.20 9.60 80 1 35 1026.6 5.011401 6.988399 1 .22 9-74 87 i 40 1011.5 3.004967 6.995033 1.24 9-89 94 1 45 996.9 2.998652 7.001348 1-25 10. 02 1 50 982.6 2.992377 . 7.007623 1.27 10.2 09 ! 55 968.8 2.986234 7.013766 1.29 10.3 16 G 955-4 2.980185 7.019815 1. 31 10.5 23 5 942.3 2.974189 7.025811 1-33 10.6 31 10 929.6 7.031704 1-35 10.8 38 ^5 , 917.2 2.962464 7-037536 1.36 10.9 45 20 905.1 2.956697 7-043303 1-38 11.0 52 25 893.4 2.951046 7.048954 1.40 11.2 60 ! 30 882.0 2.945469 7-054531 1.42 11-3 67 : 35 870.8 2.939918 7.060082 1-44 II-5 74 1 40 859.9 2.934448 7-065552 1-45 11.6 81 1 45 849-3 2.929061 7.070939 1-47 11.8 5 89 \ 50 839.0 2.923762 7.076238 1-49 II. 9 5 96 ! 55 828.9 2.918502 7.082498 1-51 12. 1 6 03 1 7 819.0 2.913284 7.086716 1-53 12.2 6 10 ! 5 809.4 2.908163 7.091837 1-55 12.3 6 18 1 10 800.0 2.903090 7.096910 1-56 12.5 6 25 { 15 790.8 2.898067 7-101933 1.58 12.6 6 32 I 20 78T.8 2.893096 7.106904 1.60 12.8 6 39 ! 25 773-1 2.888236 7.111764 1.62 12.9 6 47 ! 30 764-5 2.883377 7. I 16623 1.64 131 6 54 1 35 756.1 2.878579 7.121421 1.65 13.2 6 61 1 40 747-9 2.873844 7.126156 1.67 13.4 6 68 1 45 739-9 2.869173 7.130827 1.69 ■13-5 6 76 1 50 732.0 2.864511 7-135489 1-71 13-7 6 83 55 724-3 2.859918 7.140082 1-73 13.8 6 90 8 716.8 2.855398 7.144602 1-75 14.0 6 98 5 709.4 2.850891 7.149109 1.76 14.1 05 ! 10 702.2 2.846461 7-153539 1.78 14.2 12 1 15 695.1 2.842047 7-157953 1.80 14-4 19 i 20 688.2 2.837715 7.162285 1.82 14-5 27 25 681.3 2-833338 7.166662 1.84 14.7 34 30 674.7 2.829111 7.170889 1.85 14.8 41 1 35 668.1 2.824841 7-175159 1.87 15.0 48 i 40 661.7 2.820661 7-179339 1.89 15-1 56 1 45 655-4 2.816506 7.183494 1.91 15.3 63 50 649-3 2.812445 7-187555 1-93 15-4 70 55 643.2 2.808346 7.181654 1-95 15-5 '' 1 9 637-3 2.804344 7-195656 1.96 15-7 1 85 5 631.4 2.800305 7.199695 1.98 15-8 92 10 625.7 2.796366 7.203634 2.00 16.0 7 99 15 620.1 2-. 792462 7.207538 2.02 16.1 8.06 II 1 158 RADII AND THEIR LOGARITHMS. Degree Logarithm Arithmetical Middle Deflec- Tangen- tial Dis- of Radius. of Comple- Ordinate, tion Dis- Curve. Radius. ment. Chord 100 Feet. tance. tance. 9 20 614.6 2.788593 7.211407 2.04 16.3 ■ 1 8.14 25 60.^. I 2.784689 7.215311 2.06 16.4 8.21 j 30 603.8 2.780893 7.219107 2.07 16.6 8.28 1 35 598.6 2.777137 7.222863 2.09 16.7 8.35 1 40 593-4 2.773348 7.226652 2.11 16.8 8.43 45 588.4 2.769673 7.230327 2.13 17.0 8.50 50 583.4 2.765966 7.234134 2.15 17. 1 8.57 55 578.5 2.762303 7.237697 2.16 17-3 8.64 10 573-7 2.758685 7.241315 2.18 17.4 8.72 10 564.3 2.751510 7.248490 2.22 17.7 8.86 20 555.2 2.744449 7.255551 2.26 18.0 9.00 30 546.4 2-7375" 7.262489 2.29 18.3 9.15 40 537-9 2.730702 7.269298 2-33 18.6 9.30 50 529.7 2.724030 7.275970 2.36 18.9 9.44 11 521.7 2.717421 7.282579 2.40 19.2 9-58 10 513.9 2.710879 7.289121 2.44 19.5 9-73 20 506.4 2.704494 7.295506 2.47 19.7 9-87 30 499.1 2.698188 7.301812 2.51 20.0 10. 40 492.0 2.691965 7.308035 2.55 20.3 10.2 50 485.1 2.685831 7.314169 2.58 20.6 10.3 18 478.3 2.679700 7.320300 2.62 20.9 10.4 10 471.8 2.673758 7.326242 2.66 21.2 10.6 20 465.5 2.667920 7.332080 2.69 ^^'l 10.7 30 459.3 2 . 662096 2.656386 7-337904 2.73 21.8 10.9 40 453.3 7.343614 2.77 22.1 II. 50 447.4 2.650696 7.349304 2.80 22.4 11.2 13 441.7 2.645127 7.354873 2.84 22.6 II-3 10 436.1 2.639586 7.360414 2.88 22.9 11-5 20 430.7 2.634175 7-365825 2.91 23.2 II. 6 30 425.4 2.628797 7.. 371 203 2.95 23.5 11.7 .40 420.2 2.623456 7-376544 2.98 23.8 II. 9 50 415.2 2.618257 7.381743 3.02 24.1 12.0 14 410.3 2.613102 7.386898 3.06 24.4 12.2 10 405.5 2.607991 7.392009 3-09 24.7 12.3 20 400.8 2.602928 7.397072 3.13 25.0 12.5 30 396.2 2.597914 7.402086 3.t7 25.2 12.6 40 391.7 2.592954 7.407046 3.20 25.5 12.8 50 387-3 2.588047 7.411953 3-24 25-8 12.9 1 15 - 383-1 2.583312 7.416688 ■ 3-28 26.1 13.0 10 378-9 2-578525 7.421475 3-3^ 26.4 13.2 20 374-8 2.573800 7 . 426200 3.35 26.7 13-3 30 370.8 2.569140 7.430860 3-39 27.0 13.5 40 366.9 2.564548 7.435452 3.42 27.3 13.6 50 363-0 2-559907 7.440093 3.46 27-5 13.8 16 359.3 2-555457 7.444543 3.50 27.8 13.9 10 355-6 2.550962 7.449038 3.53 28.1 14. 1 20 352.0 2.546543 7-453457 3-57 28.4 14.2 30 348.4 2.542078 7.457922 3.61 28.7 14-3 40 345.0 2.537819 7.462181 3-^^ 29.0 14.5 50 341.6 2.533518 7.466482 3.68 29.3 14.6 17 338.3 2.529302 7.470698 3.72 29.6 14.8 10 335.0 2.52504s 7-474955 3.75 29.9 14.9 RADTI AN-D THEIR LOGARITHMS. 159 Deflec- Tangen- Degree Logarithm Arithmetical Middle of Radius. of Comple- Ordinate, tion Dis- tial Dis- i Curve, Radius. ment. Chord 100 Feet. tance. tance. o / j 17 20 33^-8 2.520876 7.479124 3-79 30.1 15-1 30 328.7 2.516800 7.483200 3-82 30-4 15-2 40 325-6 2.512684 7.487316 3-86 30.7 ^5-4 50 322.6 2 . 508664 7-491336 3-90 31.0 15-5 18 319.6 2.504607 7-495393 3-93 3T.3 T5-6 10 316.7 2.500648 7-499352 3-97 31.6 15.8 20 313-9 2.496791 7.503209 4.01 31-9 15-9 30 311-1 2.492900 7.507100 4.04 32.1 16. 1 40 308.3 2.488974 7.5T1026 4.08 32-4 16.2 50 305-6 2-485153 7.514847 4.12 32-7 16.4 19 302.9 2.481299 7.518701 4-15 33-0 T6.5 10 300.3 2-477555 7-522445 4.19 33-3 16.6 ■20 297.8 2.473925 7-526075 4-23 33-6 16.8 30 295.2 2.470116 7.529884 4.26 33-9 16.9 40 292.8 2.466571 7-533429 4-30 34-2 17. 1 50 290.3 2.462847 7-537153 4-34 34-4 17.2 30 287.9 2.459242 7-540758 4-37 34-7 17.4 1_ . __l TABLE X.— (See p. 160.) FOR USE WITH A 20-METRE CHAIN. Engineers accustomed to thinking their degree of curvature with reference to the 100-ft. chain may find it convenient to remember that the degree of curvature, if a 20-metre chain be used, is approximately tico-thirds of the foregoing. Thus a 3° metric curve would be about equivalent to a ^r^'' curve laid out with the 100-ft. chain. A 20-metre chain = 65.618 feet ; a 100-ft. chain = 1.524 chains of 20 metres each, one foot being equal to 0.3048 of a metre, and a metre equal to 3.2809 feet. If a metric curve is to be retraced with a 100-ft. chain, the exact degree of curvature should be ascertained with reference to the radius in feet, as set forth in Art. XVIII. It is convenient to mark stakes with the even numbers, 2, 4, 6, etc., when using the 20-metre chain, distance being thus recorded in tens of metres. 160 METRIC CURVE TABLE. Degree Radius Loga- 1 Arithme- Mi J.Ord. Deflec- Ta ngen- of in rithm of ticalCom- C hord tion Dis- tia 1 Dis- Curve. Metres. Radius. plement. 20 ]^ Metres. tance, tc ince. lO 6875.50 3-837304 6.162696 0076 .0582 0291 20 3437-75 3-536274 6 463726 0144 .1164 .0582 30 2291.84 3-360184 6 639816 0218 .1745 0873 40 1718.88 3-235246 6 764754 0290 .2327 1 1 64 50 I375-II 3-138338 6 861663 0363 .2909 1454 1 1145-93 3.059158 6 940842 0437 -3491 1745 10 982.23 2.992213 7 007787 0509 -4072 2036 20 859-46 2.934226 065774 0582 ■4654 2327 30 763-97 2.883076 1 1 6924 0655 .5236 2618 40 687.57 2.837317 162683 0727 .5818 .2909 50 625,07 2.795929 204071 0800 -6399 3200 2 572.99 2.758147 241853 0873 .6981 3490 10 528.92 2 • 723390 276610 0945 -7563 3781 20 491.14 2.691205 308795 1018 -8144 4072 30 458.40 2 661245 338755 1091 .8726 4363 40 429.76 2.633226 366774 1:64 .9308 • 4654 50 404.48 2.606897 393103 1237 .9889 4945 3 382.02 2.582086 417914 1309 1.047 5235 10 361.91 2.558601 440399 1382 1 .105 5526 20 343-82 2-536331 463669 1454 1. 163 .5817 30 327-46 2-515158 484842 1527 1.222 6108 40 312.58 2.4H961 505039 1600 1.280 6398 50 298.99 2 475657 524343 1673 1.338 6689 4 286.54 2.^57185 542815 1746 1.396 6980 lO 275.08 2-4.39459 560541 i8i8 1-454 7^71 20 204 51 2.422442 577558 1891 1.512 7561 30 254-71 2 . 406046 59J954 1964 1-570 7852 40 245.62 2.390264 609736 2036 1 . 629 8143 50 237 16 2.375041 624959 2109 1.687 bH 5 229.26 2.360328 639672 j 2182 1-745 8726 20 214.94 2.332317 667683 1 2328 1. 861 9308 40 202 . 30 2.305996 694004 2473 1-977 9889 6 191.07 2.280193 719807 2619 2.093 I 047 20 i8r.o3 2-257751 742249 2764 2.210 I 105 40 171 98 2 235478 764522 2910 2.326 I 163 7 163.80 2.214314 7S5686 3055 2.442 I 222 30 156.37 2.194153 805847 3201 2-558 1 280 40 149.58 2.174874 825,26 3347 2.674 I 338 8 143-36 2.156428 843572 ! 3492 2.790 I 396 20 137-63 2.138713 861287 363S 2 906 I 454 40 132.35 2 121724 878276 3783 3 . 022 I 512 9 127-45 2. 1053 to 894660 3929 3.138 I 570 20 122. 9 t 2.089587 910413 4075 3-254 1 ^o^ 40 118.68 2.074378 925622 4220 3-370 I 687 10 114.74 2 059715 940285 4366 3486 I 745 33 109.29 2.038580 961420 4585 3.660 I 832 11 c 104.33 2.018409 981591 4803 3834 919 30 99-81 I. 999174 8 000826 5022 4.008 2 006 12 95-67 1.980776 8 019224 5241 4.181 2 093 30 91.86 I. 963126 8 036S74 5460 4 355 2 i8x 13 88.34 I. 946157 8 053843 5679 4528 268 30 85.08 1.920828 8 070172 5897 4.701 2 355 14 82.06 1.914132 8.085868 6117 4-875 2 442 TABLE XI. SQUARES, CUBES, ETC., OF NUMBERS FROM 1 TO 1042. Note. — If N be taken to represent any number in any column of this table, then the algebraic significance of the re- maining numbers, on the same line, in terms of N, will be as given in the following synopsis : N N2 N^ i^N fN 1 N |/N N VN^ fN fN 1 4/N 1 fN Vn fN^ N 'fN fN N2 W N6 N fN^ 1 N2 1 X3 IS-3. K6 N9 i/W N 1 N 1 N=^ 1 N^ »4 ^1 N TABLE SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS .. Squaws. Cubes. Square Roots. Cube Roots. Reciprocals. 1 1 1 10000000 1-ooooono 100000000 2 4 8 1-4142136 12599210 500000000 3 9 27 1-7320508 1-4422496 .333333333 4 16 64 2-0000000 1-5874011 250000000 5 25 125 2-2360680 1-7099759 200000000 6 36 216 2-44948;t7 1-8 1712. (6 166666667 7 49 343 2-6457513 1-9129312 142857143 8 64 512 2-8284271 2-OOOOUOO 125000000 9 81 729 3-0000000 2-0800837 111111111 10 100 1000 3-1622777 2-1544347 100000000 il 121 1331 3-3166248 2-2239801 090909091 12 144 1728 3-4641016 2-2894288 083333333 13 169 2197 3-6055513 2-3513347 076923077 14 196 2744 3-7416574 2-4101422 071428571 15 225 3375 3-8729833 2-4602121 066666667 IG 256 4096 4-0000000 2-5198421 062500000 17 289 4913 4-1231056 2-5712816 058823529 05.55o5556 18 324 5832 4-2423407 2-02j7414 J9 3f)l 6859 4-3588989 2-6684016 052631579 20 400 8000 4-4721360 2-7144177 050000000 21 411 9261 4-5825757 2-7589-243 047619048 22 484 10648 4'6904158 2-8020393 045454545 23 529 12167 4-7958315 2-8438670 (^3478261 24 570 13824 4-8989795 2-8844991 041666667 25 625 15625 5-0000000 2-9240177 040000000 26 676 17576 5-0990195 29624960 038461538 27 729 19683 5-196152.4 3-0000000 037037037 28 784 21952 5-2915026 30365889 035714236 29 841 24389 5-385 1G48 3-07-23168 034482759 30 O'M) 27000' 5-4772256 3-1072325 033333333 31 9;ii 29791 5-56:7044 3-14138U6 032258065 32 1024 32708 5-6568542 3-1748021 031250000 33 lum 35937 5-744>>l)26 3-2075343 0303(»;<030 34 1156 39304 5-8309.:49 3-2396118 029411765 35 1225 42875 5-91 1)0798 3-2710663 0285714-29 3(i 1296 46656 6-0000000 3-3019272 027777778 37 1369 50653 6-0827625 3-3322218 027027027 38 1444 54872 6-1644140 3-3619754 026315789 39 1521 59319 6-2449980 3-3912114 025641026 40 1000 64000 6-3245553 3-4199519 025000000 41 1681 68921 6-4031242 3-4482172 024390244 42 1764 74088 6-4807407 3-4760266 023809524 43 1849 70507 6-5574385 3-5U33981 023255814 44 1936 85184 6-6332496 3-5303483 022727273 45 2025 91125 6-7082039 3-556a933 02222-2-222 46 2116 97336 6-7823300 3-5830479 021739130 47 2209 103823 6H55t)546 3-6088261 021276600 48 2304 110592 6-9282032 3-634-2411 020833333 49 2401 117649 7-0000000 3-6593057 020408163 $0 2500 mm 7-0710678 3-68403H 02000WOO T63 SQUARES, CUBES, ETC., OF NUMBERS. 163 No. Squarea, Cubes. Square Roots. Cube RooU. Reciprocali. 51 2601 132651 7-1414284 3-7084298 •019607843 .•58 2704 140608 7-2111026 3-7325111 -019230769 53 2809 148877 7-2801099 3-7.562858 •018867925 54 2916 157464 7-3484692 3-7797631 -018518519 5a 3025 160375 7-4] 6 1985 3-8029525 -018181818 56 3136 175616 7 4833148 3-8258624 •0178.57143 57 3249 185193 7-5498.344 3-8485011 •017.543860 58 3364 195112 7-6157731 3-87087(i6 •017241.379 59 3481 205379 7-6811457 3-8929905 •016949153 61» 3600 216000 7-7459667 3-9148670 •016666667 61 3721 226981 7-8102497 3-93o4;i72 -01639.3443 6i 3^44 238328 7-87401)79 3-9578^15 -016129032 6:< 3969 250047 7-9372539 3-9790.571 •01587.3016 64 4096 262144 8-0000000 4-00000 JO -01562.5000 65 4225 274625 8-0622577 4-02J7256 -01.5384615 66 4356 287496 8-1240384 404J-2401 -015151515 67 44H9 300763 8-1853.128 4-0615480 •014925373 68 4624 314432 8-2462113 4-0816551 -01470.5882 69 4761 328509 8-3066239 4-101.5661 •014492754 70 4900 343U00 8-3666003 4-121-2853 •01428.5714 7J 5041 357911 8-4261408 4- 1408 J 78 •014084517 72 5 J 84 373248 8-4852814 4-160](,76 -013888889 73 5329 389017 8-544()():{7 4-1793390 •013698630 74 5476 405224 8-6023253 4-19d3j64 •013513514 75 5625 421875 8-6602.J40 4-2171633 •013333333 76 5776 438976 8-7177979 4-2358236 •0131.57895 77 5929 456533 8-7749644 4-2543210 -012987013 78 6084 474552 8-8317609 4-2726586 -01-2820513 79 6-241 493039 8-8881944 4-2908404 -01-26.58228 80 6400 512000 8-9442719 4-3088695 •012500000 81 6561 531441 90000000 4-3-267487 •012345679 82 6724 551 368 90553851 4-3444815 012195122 83 6889 571787 9-11043.36 4-3620707 •012048193 84 7056 592704 9-1651514 4-3795191 •011904762 85 7225 614125 9-219.5445 4-3968296 011764706 86 7396 636056 9-2736 185 4-4140049 •011627907 87 7569 658503 9-3273791 4-4310476 -011494253 88 7;44 681472 9-3808315 4-4479602 •011363636 89 7921 704969 9-4339811 4-4647451 •011235955 90 8100 729000 94868330 4-4814047 ■011111111 91 8281 753571 9-5393920 4-4979414 •010989011 92 8464 778688 9-5916630 45143574 -010869565 93 8649 804357 9-6436508 4-5306549 0107.52688 94 8836 830584 9-6953597 4-5468359 •010638298 95 9025 857375 9-7467943 4-5629026 -010.526316 96 9216 884736 9-7979590 4-5788570 -010416667 97 9409 912673 9-8488578 4-5947009 •010309278 98 9604 941192 9-8994949 4-6104363 •010204082 99 9801 970299 9-9498744 4-6-260650 -010101010 100 10000 1000000 100000(100 4-641.5888 010000000 lOJ 10201 1030301 100498756 4-6570095 •009900990 102 10404 1061208 10-0995049 4-6723287 -009803922 103 10609 1092727 10-1488916 4-6875482 -009708738 104 10816 1124864 101980390 4-7026694 009615385 105 11025 1157625 10-2469508 4-7176940 •009523810 106 11236 1191016 10-2956301 4-7326235 -009433962 107 11449 1225043 l>/-3440804 4-7474594 •009345794 108 11064 1259712 10-3923048 4-7622032 •009259259 109 llddl 1295029 10-4403065 4-7768562 009174312 110 12J00 1331000 10-4880885 4-7914199 •009090909 111 12321 1367631 10-5356.538 4-8058995 •009009009 m I'^tl }mw |g-58300§5} 4-820^845 •U0892857J 164 SQUARES, CUBES, ETC., OF NUMBERS, No. Squares. Cube*. Square Roots. Cube Roots. Reciprocal* 113 12769 1442897 10-6301458 4-8345881 •008849.558 114 12996 1481.544 10-6770783 4-8488076 •008771930 115 1.3225 1.520875 10-7238053 4-8629442 •0086956.52 116 13456 1560896 10-7703296 4-8769990 •008620690 117 1.3689 1601613 10-8166538 4-8909732 •008547009 118 13924 1643032 10-8627805 4-9048681 •008474576 119 14161 16851.59 10-9087121 4-9186847 •008403361 120 14400 1728000 10-9544512 4-9324242 •008333333 121 14641 1771561 11-0000000 4-9460874 •008264463 122 14834 1815848 11-0453610 4-9596757 -008196721 123 15129 1860867 n -090.5365 4-9731898 -0081,30081 124 1.5376 1906624 11-13.5.5287 4-9866310 -008064516 125 15625 1953125 11-1803399 5-0000000 -008000000 126 15876 2000376 11-2249722 .50132979 -007936508 127 16129 2048383 11-2694277 5-0265257 •007874016 128 16384 20971.52 11-3137085 5-0396842 •007812500 129 16641 2146689 11-3.578167 50527743 -007751938 130 16900 2197000 11-4017.543 50657970 -007692303 131 17J61 2248091 11-445.5231 50787531 •007633588 132 17424 2299968 11-4891253 5-0916434 •007575758 133 17689 23.52637 11-5325626 5- J 044687 -007518797 134 179.56 2406104 11 -57.58369 51172299 •007462687 135 18225 2460375 11-6189500 5-1299278 •007407407 136 18496 251.5456 11-6619038 .5-1425632 •007.3.52941 137 18769 2.571353 11-7046999 5-1551.367 •007299270 138 19044 2828072 11-7473401 5-1676493 ■007246377 139 19321 2685619 11-7898261 5-1801015 •007194245 140 19600 2744000 11-8321596 5-1924941 •007142857 141 19881 280322] 11-8743421 5-2048279 •007092199 142 20164 2863288 11-91637.53 .5-2171034 •007042254 143 20449 2924207 ]p<)58-2607 5-2293215 •006993007 144 20736 2985984 ]2-(t0O00OO 5-2414828 •006944444 145 21025 3048G25 1C;04 15046 5-2535879 -00(5896552 146 21316 311213(5 1208304()0 5-2()56374 ■006849315 147 21609 3176523 12-1243.557 5-2776321 •006802721 148 21904 3241792 12-16.55251 5-2895725 •((06756757 149 22201 3307949 12-2065556 5-.30 14.592 •00671 1409 150 22500 3375000 12-2474487 5-3132928 -00666()6(i7 151 22801 3442951 12-2882057 5-3250740 -006622517 152 23104 3511808 12-3288280 5-3368033 -006.578947 153 23409 3581577 12-3693169 5-3484812 •006535948 154 23716 3652264 12-4096736 5-3601084 •006493506 155 24025 3723875 12-4498996 5 37168.54 •006451613 156 24336 3796-1 ]fi 12-4809960 5-3832126 •006410256 157 24649 3869893 12-5299641 5-3946907 •006369427 158 24964 3944312 12 5698051 5-4061202 •006329114 159 25281 4019G79 12-6095202 5-4175015 •006-289308 160 25G00 4096000 12 6491108 5-4288352 -0062.50000 161 25921 4173281 12-6885775 5-4401218 •00(5211180 162 26244 4251.528 12-7279221 5-4513618 •006172840 163 26569 4330747 12-7671453 5-4625.556 •006134969 164 26896 4410944 12-8062485 5-4737037 •006097561 165 27225 4492125 12-84.52326 5-4848066 •0060(50606 166 27556 4574296 12-8840987 5-4958647 •006024096 167 27889 4657463 12-9228480 5-5068784 •005988024 168 28224 4741632 129614814 5-5178484 •005952.381 169 28561 4826809 13-0000000 5-5287748 •005917160 170 28900 4913000 13-0384048 5-5396583 •005882353 171 29241 5000211 13 0766968 5-5504991 •005847953 172 29584 5088448 13 1148770 5-5612978 •005813953 173 29929 5177717 13-1529464 5-5720546 •005780347 174 30276 5268024 13-1909060 5-5827702 •005747126 SQUARES, CUBES, ETC, OF NUMBERS. 165 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals, 175 .30625 5359375 13-2287566 5-5934447 005714286 176 30976 5451776 13-2664992 5-6040787 •00.5681818 177 31329 5545233 13-.3041347 .5-6146724 •005649718 178 31684 5639752 13 3416641 5-62.52263 •005617978 179 32041 5735339 13-3790882 5 63.57408 •005586592 180 32400 5832000 13-4164079 5-6462162 •005555556 181 32761 5929741 13-4536240 5-6566528 •005.524862 182 33124 6028568 13-4907376 .5-6670511 •005491.505 183 33489 6128487 13-5277493 .5-6774114 •00546448 1 184 33856 6229504 l3-564(i600 5-6877340 •005434783 185 34225 6331625 13-6014705 5-6980192 •00540.5405 186 34596 6434856 13-6381817 5-7082675 •005376344 187 34969 6539203 13-6747943 5-7184791 •005347594 188 35344 6644672 ]3-7!13U92 5-7286.543 •005319149 189 35721 6751269 13-7477271 5-7387936 •005291005 UK) 36100 6859000 13-7840488 5-7488971 •00.5263158 191 36481 6967871 13-8202750 5-7589652 •005235602 192 36864 7077888 13-8564065 5-7 -89982 •00.5208333 193 37249 7189017 13-8924440 5-7789966 •005181347 194 37636 73013&4 1:^-9283883 5-7889604 •005154639 195 38025 741 4875 ] 3964-2400 5-7988900 •005128205 196 38416 7529536 14-00001)00 5-8087857 •005102041 197 38809 7645373 14-0356688 5-8186479 -005076142 196 39204 7762392 14071^:173 5-8284767 •005050.505 199 39601 7880599 1410G7360 5-8382725 •005025126 200 40000 8000000 14-14213.-).'> 5 84803.55 •005000000 201 40401 8120601 14- 1774469 5-8.577660 •004975124 202 40804 8242408 14-2i207O4 5-8674643 . -004950495 203 41209 8365427 14-24T8(;68 .5-8771307 •004926108 204 41616 8489664 14-28-2H569 5-8867653 •004901961 205 42025 8615125 14-3178-21 1 5-8963685 •004878049 206 42436 8741816 14-3527001 59059406 •00485436'.) 207 42849 8869743 14.3874946 .5-91.54817 •004830918 208 43264 8998912 14-4222051 5-9249921 -004807692 209 43681 9129329 14-4568.323 5-9344721 •004784689 210 44100 9261000 14-4913767 5-9439220 -004761905 211 44521 9393931 14-5258390 .5-9.5.33418 •C04739336 212 44944 9528128 14-5602198 5-9627320 •004716981 213 45369 9063597 14-5945195 5-9720926 •004r;94836 214 45796 9800344 14-6287388 . 5-9814240 •004672897 215 46225 9938375 14-6628783 5-9907264 •004651 163 216 46056 10077696 14-6909385 GOOOOOOO -004629630 217 47089 10218313 147309199 6-0092450 -004608295 218 47524 10360232 14-7648231 6-0184617 •004.587156 219 47961 10503459 14-7986486 6-0276502 •004566210 220 48400 10648000 14-8323970 6-0368107 ■004.5454.55 221 48841 10793861 14-8660687 60459435 •004524887 222 49284 10941048 14-8996644 60550489 •004501505 223 49729 11089567 14-9331845 6-0641270 •004484305 2'J4 50176 11239424 14-9006-295 6-0731779 •0044()4286 OOj 50625 11390625 15-.KJ00000 6-0822020 -004444444 226 51076 11543176 15-033-2964 6-0911994 -004424779 227 51529 11697083 15-0665192 6-1001702 -004405286 228 51984 1 1852352 15 0996689 6-1091 147 -004385965 229 .52441 12008989 15 1327460 6-1180332 -004366812 230 52900 12167000 15- 1657509 6-12692.57 •004347826 231 53.161 12326391 15-1986842 6-1357924 •004329004 232 53824 12487168 15-2315462 6-1446337 •004310345 233 54289 12649337 15-2643-375 6-1534495 •004291845 234 547.56 12812904 15-2970585 6-1622401 •004273504 235 55225 12977875 15-3297097 61710058 •0042.5.5319 236 55696 13144256 15-3622915 6-1797466 -00423V288 166 SQUARES, CUBES, ETC., OF NmfBERS. Na Squares. Cubes. Square Roots. Cube Roots. Reciproca!*. 237 56169 13312053 15-3948043 6-1884G28 •004219409 238 56644 13481272 15-4272486 6-1971544 •004-201681 239 57121 13651919 15-4596248 6-20.58218 •004184100 240 57600 13824000 15-49] 9334 6-2144650 •004166667 24) 58081 13997521 15-5241747 6-2230843 •004 14:4378 212 585G4 14172488 15-5563492 6-2316797 •004 13223 J 243 59049 14348907 15-5884573 6-2402515 -004]1.>2-2G 244 59536 14526784 15-6204994 6-2487998 -004098361 245 60025 14706125 15-65-24758 6-2573248 -004081633 246 60516 14886936 15-6843871 6-26.58266 •004065041 247 61009 15069223 157162336 6-2743054 -004048583 248 61504 15252992 15-7480157 6-2827G13 -004(«2-2.")8 249 62001 15438249 15-7797338 6-2911946 -004016064 250 6-2500 15625000 15-8113883 6-29960.53 •0O4U00000 251 63001 15813251 15-8429795 6-3079935 •003984064 252 63504 16003008 15-8745079 6-3163536 -003968254 253 64009 16194-277 15-9059737 6-3247035 -003952569 254 64516 163870(>4 15-9373775 6-3330256 -003937008 255 65025 16581375 15-9687194 6-3413257 •003921589 256 65536 16777216 16-0000000 6-3496042 -00390G250 257 68049 lGiJ74593 lG-03 12195 6-3.57861 1 •00:i8!H051 258 665G4 17173512 1G0G23784 6-3660968 •0038759G9 259 67081 17373979 16-0934769 6.37431 J 1 -003861004 260 67600 17576000 IG- 1245 155 6-3825043 -003846154 261 68121 17779581 16-1554944 6-39067G5 -003831418 262 68644 17984728 16- 1864 141 6-3988279 •00381()794 263 69169 181 91447 16 2172747 6-4()G9585 •003802281 264 69696 18399744 10-2480768 6-4150687 •003787879 265 70225 18609625 16-2788206 6-4231583 •003773585 266 70756 18821096 16-3095064 6-4319276 •003759398 267 71289 19034163 16-3401346 6-4399767 •003745318 268 71824 19248832 16-3707055 6-4473057 -003731343 269 72361 19465109 16-4012195 6-4553148 •003717472 270 72900 19683000 16-4316767 6-4633041 •003703704 271 73441 19902511 16-4620776 6-4712736 •003690037 272 73984 2U123648 16-4924225 6-479-2236 •003676471 273 74529 20346417 16-5227116 6-4871541 •003663004 274 75076 20570824 16-5529454 6-4950653 •003649635 275 75625 20796875 16-5831240 6-5029572 -00363(5364 276 76176 21024576 16-6132477 6-5108300 •003623188 277 76729 21253933 16-6433170 6-5186839 •003610108 278 77284 21484952 16-673332) 6-5265189 •003597122 279 77841 21717639 16-7032931 6-5343351 •00.3.584229 280 78400 2195-20U0 16-7332005 6-54213-26 •003.571429 281 78961 22188041 16-7630546 6-5499] 16 -003.5.58719 282 79524 2-2425768 16-7928556 6-5576722 •00354(5099 283 80089 226G5187 16-8226038 6-5654144 -003533569 284 80656 22906304 16-8522995 6-5731385 -003.522127 285 81225 23149125 16-8819430 6-5808443 •00350H772 286 81796 23393G56 16-9115345 6-588.5323 •003496503 287 82369 23639903 16-9410743 6-5962023 •003484321 288 82944 23887872 16-9705627 6-6038545 •003472222 289 83521 24137569 17-0000000 6-6114890 •00346020H 290 84100 24389000 17-0293864 6-6191060 •003448276 291 84681 24642171 17-0587221 6-62(57054 -0034.3643) 292 85264 24897088 17-0880075 6-6342874 -003424f55S 293 85849 25153757 171172428 6-6418522 •0034129'!:) 294 86436 25412184 17-1464282 6-6493998 •003401 3^51 295 87025 25672375 17-1755640 6-65()9302 •003.38,K{| 296 87616 25934336 17-2046505 6-6644437 •00.3378378 297 88209 26198073 17-2336879 6-6719403 •003367003 298 88804 26463592 17-2626765 6-6794200 -0033o5705 SQUARES, CUBES. ETC., OF NUMBERS. 107 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 239 8M01 2G730899 17-291GJ65 6-6868831 •003344482 300 9;)noo 27000000 1 7-3205081 6-6943295 •00333:i;i33 301 90001 27270901 17-3493516 6-7017593 •003322259 30'i 91204 27543608 17-3781472 6-7091729 •0033112.58 303 91809 27818127 17-4068952 6-7165700 •003301330 304 924 J 6 28094464 17-4355958 6-7239508 •003289474 305 G3;;25 28372025 17-4042492 6-7313155 •003278G89 306 93G36 28652616 17-4928557 6-7386641 •0032G7974 307 94249 28934443 17-5214155 6-74599G7 •0032.57329 338 94804 29218112 17-5439288 6-7533134 •003240753 309 93481 29503G29 17-5783958 6-760G143 •00323G24G 310 9G100 29791000 17-0068169 6-76789J5 •0032258.;G 311 9G721 3008C23J 17-6351921 6-7751 G90 •003215434 312 97344 30371328 17-6035217 6-7824229 •003205128 3J3 97969 30G64297 17-G9180G0 6-7896613 •003194888 3J4 98596 30959144 17-7200451 6 7968844 •003184713 3J5 99225 31255875 17-7482393 6-8040921 •003174;;^3 3J6 99856 31554496 17-7763883 6-8112847 •0031G4.:57 317 100489 31855013 17-8044938 6-8184620 •003154574 318 101124 32157432 17-8325545 6-825G242 •003144G.54 319 1017G1 32461759 17-0605711 6-8:]27714 •003134796 320 102400 32768000 17-8885438 6-8399037 •003125.;30 321 103041 33076161 17-9164729 6-84702 J 3 •003115205 322 103684 33386248 17-9443584 6-85412-10 •0031055;)J 323 104329 33698267 17-9722008 G-8G12120 ■0O3O95.)75 324 104976 34012224 18-00000:;0 6-81382855 •00308G420 325 105625 34328125 18-02775G4 6-87534 W •00307G923 326 106276 34645976 18-0554701 6-8823888 0030G7485 327 106929 349G5783 18-0831413 6-8894188 •003058104 328 107584 35287552 18-1107703 6-8964345 •003048780 329 108241 35611289 181383571 6-9034359 •003039514 330 108900 35937000 18-1659021 6-9104232 •003030303 331 109561 36264691 18-1934054 6-9173964 •003021148 332 110224 36594368 18-2208G72 G-9243556 •003012048 333 110889 36926037 •18-2482376 6-9313008 •003003003 334 111556 37259704 18-2756G69 6-9382321 •002994012 335 112225 37595375 18-3030052 6-9451496 •002985075 336 112896 37933056 18-3303028 6-D520533 •002976190 337 113569 38272753 18-3575598 6-95Hr)434 •002967359 338 114244 38614472 18-3847763 6-9058198 •0029.58580 339 114921 38958219 18-4119526 6-9720826 ■002949853 340 115600 39304000 18-4390889 6-9795321 •002941176 341 116281 39G51821 18 4661853 6-9863681 -002932551 342 1169G4 40001688 18-4932420 6-9931906 •002923977 343 117649 40353607 1^^5202592 7-0000000 -002915452 344 118336 40707584 18-5472370 7-00G7962 -002906977 345 119025 41063625 18-5741756 7-0135791 -002898551 346 119716 41421736 18-6010752 7-0203490 •002890173 347 120409 4178 J 923 18-6279360 7-0271058 •002881844 348 121104 42144192 18-6547581 7-0338497 •002873563 349 121801 42508549 18-6815417 7-U4058015 -00286.53.30 350 122500 42875000 18-7082869 7-0472987 •002857143 351 123201 43243551 18-7349940 7-0540041 •002849003 352 123904 43614208 18-7616630 7-0606967 •002840909 353 124609 43986977 18-7882942 7-0673767 •002832861 354 125316 44361864 18-8148877 7-0740440 ■002824859 355 126025 44738875 18-8414437 7-0806988 •002816901 356 126736 45118016 18-8679623 70873411 •002808989 357 127449 45499293 18-8944436 7-0939709 •002801120 358 128164 45882712 18-9208879 7-1005885 •002793296 359 128881 46268279 18-947W53 7-1071937 •0027855J5 360 129«(H> 46656000 18-9736660 71137866 i •002777778 168 SQUARES, CCBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 361 130321 47045881 190000000 7-1203674 002770083 362 131044 47437928 190262976 7-1269360 002762431 363 131769 47832147 190.525589 71334925 002754821 364 132496 48228r)44 190787840 71400370 002747253 365 133225 4862712.5 191049732 7 1465695 002739720 366 133956 49027896 191311265 7-15.30901 00273224) 367 134689 49430863 191572441 71.59.5988 0027247(t6 368 135424 49836032 191833261 7-1660957 00271731)1 369 13G161 50243409 192093727 7 1725809 002710(W7 370 136900 50653000 19 23.53841 7-1790.544 002702703 371 137041 5 10648 U 192613603 7-18.55162 002695418 372 138384 51478848 19 2873015 7-1919663 002688172 373 139129 51895117 193132079 7-1984050 002680965 374 139876 52313624 193390796 7 2048322 002673797 375 140625 52734375 W3649167 7-2112479 002666667 376 141376 53157376 193907194 7-2176522 002659574 377 142129 53582633 19 4164878 722404.50 00265252J 378 142884 540101.32 1944229,91 7-2304268 002645503 379 143641 54439939 194679223 7 23679/2 002638521 380 144400 M872000 194935887 7 2431565 002631579 3«1 145181 55306341 19 5192213 72495045 002024672 382 145924 55742968 19 5448203 7 2558415 002617801 383 146689 5G181887 195703858 72021675 OOQGlOlKJo 384 147456 56G23104 19 5959179 72684824 002G041G7 385 148225 57066625 19 6214 169 7-2747864 002597403 386 148996 57512456 19 6468827 7-2810794 002590674 387 149769 57960603 19 6723156 7-2873617 002583979 388 150544 58411072 19 6977156 7-2936330 002577320 389 151321 58863869 197230829 7 2998936 00257(;()94 390 152100 59319000 197484177 7-3061436 002564103 31M 152881 59776471 19 7737199 7-312.3828 002557.545 W'.Vl 15361)4 60236288 19 7989899 7-318()lI4 00255 IOC J 393 154449 60698457 19 8242276 7-32482;)5 002544.529 394 155236 61162984 19 8494332 7 3310369 0!)2.538;)7l 395 156025 61629875 19 8746069 7-3372339 002.531646 396 156816 62099136 19-8997487 7-3434205 002525253 397 157609 62570773 19-9248588 7-3495966 002518892 398 158404 63044792 19 9499373 7-3557624 002512563 L99 159201 63521199 19 9749844 7-3619178 002.506266 400 160000 64000000 200000000 7-3(i8t)630 002.500000 401 160801 64481201 20 0249844 7-3741979 002493766 402 161604 64964808 20-0499377 7-3803227 002487.562 403 162409 65450827 20 0748599 7-3864373 002481390 404 163216 65939264 20 0997512 7-3925418 002475^248 405 164025 66430125 20 1246118 7 3986363 002469136 406 164836 66923416 20-1494417 7-41)47206 00241)3054 407 1(35649 67419143 20 1742410 74J079.50 002457002 408 1664()4 67917312 20-1990099 7-4168595 002450980 409 167281 68417929 20 2237484 74229142 002444988 410 168100 68921000 20 2484.567 7-4289589 002439024 411 168921 69426531 20 2731.349 7 4349938 1102433090 412 1(59744 ti; (934528 20 2!>77831 7 4410189 002427184 413 170569 70444997 203224014 7 4470342 ()024213(,8 414 171396 70957944 20 3469899 74530399 0024 J 54,59 415 172225 71473375 20-3715488 7 4590359 002409639 416 173056 71991296 20-.3960781 7-4650223 002403846 417 173889 72511713 20-4205779 7 4709991 002398()82 418 174724 73034632 20-4450483 7-4769664 002392344 419 175561 73560059 20-4694895 7-4829242 002386635 420 176400 74088000 20-4939015 7-4888724 002380952 421 177241 74618461 20-5182845 7-4948113 00237.5297 422 178084 75151448 20v42ti386 7-3007406 0023696G8 SQUARES, CUBES, ETC., OF NUMBERS. 169 N Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 423 178929 75686967 20 1669038 75066607 -002.364006 424 179776 76225024 20-5912003 7-512.5715 -002358491 4'i5 180625 76765625 20-615,1281 7-51847.30 002352941 4'J6 181476 77308776 20-0397674 7.1243652 •002347418 4'27 182329 778.14483 20-6639783 7-5302482 •002341920 4-J8 18^184 784027.12 20-6881609 7-5361221 ■002330449 429 184041 78953589 20-71231.12 7-5419867 002.331002 4; 255025 128787825 22 4722051 7-9633743 •001980198 503 253033 129554216 2249444.38 7-9686271 •001976285 507 257049 130323843 22 5168805 7-9738731 •001972387 503 258064 131098512 22 5388553 7-9791122 -001968504 53.) 259081 131872229 22 5810283 7-9843444 •001964637 51) 230100 132651000 22-5331796 7-989^837 •001960785 511 261121 133432831 22-6053091 7-9947883 •001956947 512 232144 134217728 22-6274170 8-00C0030 •001953125 513 233169 135005697 22-6495033 8-00520:3 •001949318 514 284196 135798744 22-6715381 80104032 •001945525 515 265225 136590875 22 0938114 8 0155943 •001941748 5r, 266256 137383035 22 7150331 80207794 •001937984 517 267289 138188413 22 737834 1 8 0259574 •001934236 5 IS 233-24 133991832 22-7598131 8 0511287 •0019.30502 5i;) 239331 139798359 22-7815715 80362935 •001926782 520 270400 149308000 22-8035335 80414515 ■001923077 521 271411 141420761 228254244 8-0466030 ■001919386 522 272484 142238648 22-8473193 80517479 •091915709 523 273529 143055667 228691933 8-0568832 001912046 524 274576 1438/7824 228910-183 80620130 •001908397 525 275625 144703125 22 9123785 8-0671432 -001904762 523 276676 145531576 22 9",43399 8 0722323 •001901141 527 2VV729 143133183 229564836 8 0773743 -001897533 523 278784 M7 197932 22 9732536 80824303 •001893939 52J 279841 143335333 23 0000000 80875734 -001890359 53) 23.9JD 148877001 230217289 80928723 •001886792 5! I 281931 149721231 23 0434372 80977539 -001883239 5:j 283324 153588783 23C651252 8 1023333 •001879699 5:3 2.84089 151419437 23 0807928 8-1079128 •001876173 531 285153 152273334 23-1084409 8-1129803 •001872859 5.15 283225 153133.375 23 1303870 8-1180414 •0018691.'-.0 535 287298 153393358 23 15187.38 8-1230932 •001865672 537 238339 154854153 23-1732305 81281447 -001362197 533 289444 155723872 23 134^273 8-1331870 -0fll8.'')873fi 539 290521 155590819 23 2133735 8-1-S2230 ■001855238 54'J 291600 1 "7484000 23 2379331 8 1432529 -001851852 541 292681 1:8340421 23 2594087 8 1482765 -001848429 542 293764 159220088 23 2338335 8- 1532939 •00184.W18 543 294849 160103007 23 3023604 81.5H3051 •001841621 544 235933 160989184 23-n238376 8 1333102 -0018382.35 545 297025 161878625 23 .3452351 8- 1683002 -00]H34H(1'-J iM6 2981 13 182771336 233666429 8-1733020 mmm SQCAIIFS, CUBES, ETC., OF NUMBERS. 171 Sc.uares. Square Roots. Cube Roots, Reciprocals. 299-209 300304 301401 302500 303601 304704 305809 306916 308025 309 J 36 310249 311364 312481 313600 314721 315844 316969 318096 319225 320356 321489 322624 323761 324900 326041 327184 328329 329476 330625 331776 332929 334084 335241 336400 337561 338724 339889 341056 342225 343396 344569 345744 346921 348100 349281 350464 351649 352836 354025 355216 356409 357604 35880J 360U0O 36J201 3(32404 363609 364816 366025 367236 368449' 369664 163667323 184566592 16546914!> 166375000 ]672.-^4J51 168196608 169112377 170031464 170933875 171879616 172808693 J 73741 IJ 2 174676879 175616000 176558481 177504328 178453547 179406144 180362125 181321496 182284263 183250432 184220009 185193000 186169411 187149248 188132517 189119224 190109375 191102976 192100033 193100552 194104539 195112000 196122941 197137368 198155287 199176704 200201625 201230056 202262003 203297472 204336469 205379000 206425071 207474688 208527857 209584584 210644875 211708736 212776173 213847192 214921799 216000000 217081801 218167208 2h)256227 2-2():i4dr64 2.; 1445 J 25 22-2545016 223648543 224755712 23-3880311 23-4093998 23-4307490 23-452()788 23-4733892 23-4946802 23-5159520 23-5372046 23-5584380 23-5796522 23-6008474 23-6220236 23-6431808 23-6643191 23-6854386 23-7065392 23-7276210 23-7486842 23-7697286 23-7907545 23-8117618 23-8327506 23-8537209 23-8746728 23-8956063 23-9165215 23-9374184 23-9582971 23-9791576 24-0000000 24-0208243 240416306 24-0624188 24-0831891 24-1039416 24-1246762 24-1453929 24-1660919 24-1867732 24-2074369 24-2280829 24-2487113 24-2693222 24-2899156 24-3104996 24-3310501 24-3515913 24-3721152 24-3926218 24-4131112 24-4335834 24-4540385 24-4744765 24-4948974 24-5153013 24-5356883 24-5560583 24-5764115 24-5967478 24-6170673 24-6373700 24-6576560 8-1782888 8-1832695 8-1882441 8-1932127 8-1981753 8-2031319 8-2080825 8-2130271 8-2179657 8-2-228985 8-2278254 8-2327463 8-2376614 8-2425706 8-2474740 8-2523715 8-2572633 8-2621492 8-2670294 8-2719039 8-2767726 8-2816355 8-2864928 8-2913444 8-2961903 8-3010304 8-3058651 8-3106941 8-3155175 8-3203353 8-3251475 8-3299542 8-3347553 8-3395509 8-3443410 8-3491256 8-3539047 8-3586784 8-3634466 8-3682095 8-3729668 8-3777188 8-3824653 8-3872085 8-3919423 8-3966729 8-4013981 8-4061 ISO 8-4108326 8-4155419 8-4202460 8-4249448 8-4296383 8-4343287 8-4390098 8-4436877 8-4483605 8-4530281 8-4576906 8-4623479 8-4670001 8-4716471 •001828154 •001824818 •001821494 •001818182 •001814832 •001811594 •001808318 -001805054 •001801802 •001798561 •001795332 •001792115 •001788909 •001785714 •001782531 •00 < 77935 J •001776199 •001773050 •001769912 •001766784 •001763668 •001760533 •001757469 •001754386 -001751313 •001748252 001745201 •00J7421f.0 •001739130 •00173G111 •001733102 •001730104 •001727116 •001724138 •001721170 •001718213 •001715266 •0017123-29 •00 1709402 •001706485 •001703578 •001700680 •001697793 •001694915 •001692047 •001689189 •001686341 •001683502 •0016806 r2 •001577852 •001675U12 •001672-241 •001669449 •001666667 •0016638J4 •001661130 •001658375 •00 165562 J •00165-2893 •001650165 •001647446 •001644737 172 SQUARES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciproeali). 609 370881 225866529 24-6779254 8-4762892 •001642036 610 .372100 226981000 24-6981781 8-4809261 •001639344 611 373321 228099131 24-7184142 8-4855579 •001636661 612 374544 229220928 24'7386338 8-4901848 •001633987 613 375769 230346397 24-7588368 8-4948065 •001631321 614 376996 231475544 24-7790234 8-4994233 •001628664 615 378225 232608375 24-7991935 8-5040350 •001626016 616 379456 233744896 24-8193473 8-5086417 •001623377 617 380689 234885113 24-8394847 8-5132435 •001620746 618 381924 236029032 24-8596058 8-5178403 •001618123 619 383161 237176659 24-8797106 8-5224331 •001615509 m) 384400 238328000 24-8997992 8-5270189 •001612903 621 385641 239483061 24-9198716 8-5316009 •001610306 6-22 38(3884 . 240641848 24-9399278 8-5361780 •001607717 6-2:{ 388129 241804367 24-9599679 8-5407501 •001605136 624 38:!37() 242970624 24-9799920 8-5453173 •001602564 62.1 390C)25 244140625 25-0000000 8-5498797 •001600000 626 391876 245314376 25-0199920 8-5544372 •001597444 627 393129 246491883 25-0399681 8-5589899 •001594896 628 394384 247673152 25-0599282 , 8-5635377 •001.592357 62;l 395641 2488.58189 25-0798724 8-5680807 •001589825 6;iO 39ti900 250047000 25-0998008 8-5726189 •001587302 6:u 398161 251239591 25-1197134 8-5771523 •001584786 632 399424 252435968 25-1396102 8-5816809 •001582278 633 400689 253636137 25-1594913 8-5862047 •001579779 634 401956 254840104 25 1793566 8-5907238 •001577287 635 403225 256047875 251992063 8-5952380 •001574803 636 404496 257259456 25-2190404 8-5997476 •001572327 637 405769 258474853 25-2388589 8-6042525 •001569859 638 407044 259694072 25 2586619 8-6087526 •001567398 631) 408321 2609J7119 25-2784493 8-6132480 •001564945 640 401)(i00 262144000 25-2982213 8-6177388 •001562500 641 410881 263374721 25-3179778 8-6222248 •001560062 642 412164 264609288 25-3377189 8-6267063 •001557632 643 413449 265847707 25-3574447 8-6311830 •001555210 644 414736 267089984 25-3771551 8-6356551 •001552795 645 416125 258336125 25-3968502 8-6401226 •001550388 646 417316 26958R136 25-4165302 8-6445855 •001547988 647 418609 270840023 25-4361947 8-6490437 •001545595 64y 419904 272097792 25-4558441 8-6534974 •001543210 64399U4 001259446 SQi^AA'ES, (JUBES, ETC., OF NUMBERS. 175 Sinmres. Cubes. Square Roots. Cube Roots. Reciprocals. g:j2u->.) 502459875 28- 1957444 92637973 -001257862 633Ul(i 5.,4:J5.<33() 28 2134720 9-2676798 -001256281 635209 506261573 28-2311884 92715592 -001254705 636804 508169592 28-2488938 9-2754352 001 253 J 33 638-101 510082399 28 2665881 92793081 00J251364 640000 512000000 28 2842712 9-2831777 -001250000 6416U1 513922401 28 3019434 9-2870444 001248439 643204 515849008 28-3196045 92909072 001246883 644809 517781627 28 3372.546 9-2947671 •001245330 646416 519718464 28 3548938 9-2986239 -001243781 648025 521660125 28-3725219 93024775 001242230 (i49636 523606()16 28 3901391 9-3063278 001240095 651249 525557943 284077454 93101750 0012391.57 652864 5275)4112 284253408 9-3140190 -0012376-24 654481 529475129 284429253 9-3178599 ■00123G094 650100 531441000 28-4694980 9-3216975 -001234308 65772J 533411731 28-4780017 9-3255320 -001233U46 G59344 535387328 28 49.50137 9-3293634 ■001231527 060969 537307797 285131.549 9-3331916 -001230012 662596 539353144 285300852 9-3370167 -0012-28501 064225 541343375 28-5482048 9-3408336 -001226994 665850 543338496 28-50.57137 9-.3446.575 -001225490 6o7489 545338513 28 5832119 9-3484731 001-2-23990 009124 547343432 286000993 9-352-2857 -001222494 070761 549353259 28-6181760 9-3560952 -001221001 672400 551368000 28 6356421 93599016 -001219512 674041 553387661 286530976 9-3637049 •001218027 675684 555412248 286705424 9-3675051 -001216545 677329 557441767 28 6879760 9 3713022 •001215007 678976 559476224 28-7054002 9 3750963 -001213592 680625 501515625 287228132 9-3788873 001-212121 682276 563559970 287402157 9-3826752 001210654 683929 565609283 28-7576077 9-3864600 -001209190 685584 567663552 28-7749891 9-3902419 -001207729 687241 509722789 28-7923601 93940206 -001206273 688900 571787000 28-8097206 9-3977964 -001204819 690561 573850191 28-8270706 9-4015691 -001203369 692224 575930308 28-8444102 9-4053387 ■001201923 693889 578009537 28-8617394 9-4091054 -001200480 695556 580093704 28-8790582 9-4128690 ■001199041 097225 582182875 28-8963666 9-4100297 ■001197005 098896 584277050 289136046 9-4203873 001190172 700569 586376253 289309523 9-4241420 • 001194743 702244 588480472 28 9482297 9-4278936 ■001193317 703921 590589719 28-9054907 9-4316423 ■001191895 705600 592704000 28 9827535 9-4353880 ■001190476 707231 594823321 290000000 9-4391307 ■001189061 708964 596947083 290172363 9-4428704 ■001187648 710049 599077107 29 0.344623 9-4466072 ■001186240 712336 00121 1584 29 0510781 9-4.503410 ■001184834 714025 603351125 290088837 9-4540719 ■001183432 715716 005495736 29 0860791 9-4577999 •001182033 717409 607045423 291032644 9-4615249 •001180638 719104 009800192 29-1204396 9-4052470 •001179-245 7208fl 011900049 291370046 9-4089661 •001177856 722500 014125000 291.547595 9-4726824 -001176471 724201 616295051 291719043 94763957 •001175088 725904 618470208 29 1890390 9 4801061 •001173709 727609 620650477 29 2001037 94838136 •001172333 729316 622835864 29-2232784 9-4875182 •001170960 731025 625026375 29-2403830 9 4912200 -001169591 732736 627222016 29-2574777 9-4949188 001168224 176 SQUARES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Koote. Reciprocals. "ssT 734449 629422793 29-2745023 9-4986147 •001106801 858 736164 631(128712 29-2916370 9-5023078 •001165501 859 737881 633839779 29 .3087018 9-5059980 •001164144 860 739600 636056000 29-3257566 9-.5096854 •001162791 861 741321 638277381 29-3428015 9-5133699 •001161440 862 743044 640503928 29-3.598365 9-5170515 •001160093 863 744769 642735647 29-3768616 9-5207303 •001158749 864 746496 644972.544 29-3938769 9.5244063 •OOJ 1.57407 865 748225 64721 4t)25 29-5 108823 9-52.80794 •001 156069 866 749956 64946189(1 29-4278779 9-5317497 •001154734 867 751689 6517143(53 2944486.37 9-5354172 -001153403 868 753424 653972032 29-4618397 9-5390818 •001152074 869 755161 656234909 29-4788059 9-5427437 -001150748 870 756900 658503000 29-4957624 9-5464027 •001149425 871 758641 660776311 29-5127091 9.5.500.589 •001148106 872 760384 663054848 29-5296461 9 5537123 •001146789 873 762129 665338617 29-54(55734 9-5573630 •001145475 874 763876 667627024 29-5634910 9-5610108 •001144165 875 765625 669921875 29-5803989 9-5646559 •001142857 876 767376 672221376 29-5972972 9-5682782 •OOJ 14 1553 877 769129 67452GJ33 29-6141858 9.5719377 •001140251 878 770884 676836152 29-6310648 9 5755745 •001138952 879 77264J 679151439 296479.342 9-5792085 •0011.37656 880 774400 681472000 29 6647939 9-5828397 •001136364 881 7761(i] 683797841 29 6816442 95864682 -001135074 882 777924 686128968 29-6984848 9-5900939 •091133787 883 779689 688465387 29-7153159 9-5937169 •001132503 884 781456 690807104 29-7.321375 9-5973373 •0011.31222 885 783225 693154125 29-7489496 9-6009548 •001129944 886 7849'J6 695506456 29-7657521 9-6045696 -001128668 887 786769 697864103 29-7825452 9-6081817 001127396 888 788544 700227072 29-7993289 9 6117911 -001126126 889 790321 7025953(i9 29-8161030 9-6153977 •001124859 890 792100 704969000 29-8328678 9-6190017 •001123596 891 793881 707347971 29-8496231 9-6226030 -001122334 892 7956G4 709732288 29-8663690 9-6262016 •001121076 893 797449 712121957 29-8831056 9 6297975 •001119821 894 799236 714516984 29-8998328 9-6333907 •001118568 895 801025 716917375 29-9165506 9-6369812 •001117818 896 802816 719323136 29-9332591 9-6405690 -001116071 897 804609 721734273 29-9499583 90441542 •001114827 898 806404 724150792 29-9666481 9-6477367 •001113586 899 808201 726572699 29-9833287 9-6513166 •001112347 900 810000 729000000 30-0000000 9-6548938 •OOllllUl 901 811801 731432701 30-0166621 9-6584684 •OOJ 109878 902 813604 733870808 30-0333148 9-6620403 •001108647 903 8154U9 736314327 300499584 9-6656096 •001107420 904 817216 738763264 30-0665928 9-6691762 •001106195 905 819025 741217625 30-0832179 9-6727403 •001104972 906 820836 743677416 30-0998339 9-6763017 •001103753 907 822649 746142643 301164407 9-6798604 •001102.536 908 824464 748613312 30-1330383 9-6834166 001101322 909 826281 751089429 301496269 9-686(^701 •001 1001 10 910 828100 753.571000 301662063 9-690.5211 •00109890C 911 829921 756058031 30-1827765 9-6940694 •001097695 912 831744 758550528 301993377 9-6976151 •001096491 913 833569 761048497 30-2158899 9-7011583 •00109.5290 914 835396 763.551944 30-2324329 9-7046989 •OOJ 094092 915 837225 76()060875 30-2489669 9-7082369 00 J 092896 916 839056 768575296 30-2654919 9-7117723 -001091703 917 840889 771095213 30-2820079 9-7153051 -00 10! 105 13 9J8 842724 773620632 30-2985148 9-7188354 •001089325 SQUARES, CUBES, ETC., OF NUMBERS. m No. Squares, Cubes. Square Roots. Oube Hoots, Reciprocals. 919 844561 776151559 30-3150128 9-7223631 001088139 9-JO 846400 778688000 30-3315018 9-7258883 001086957 921 848241 781229961 30-3479818 9-7294109 00108.5776 922 850084 783777448 30-3644529 9-7329309 001084599, 923 851929 786330467 30-3809151 9-7364484 001083423 924 853776 788889024 30-3973683 9-7399634 001082-251 925 855625 791453125 30-4J3,sj-27 9-7434758 001081081 92G 857476 79402277G 30-430-2481 9-7469857 001079914 927 859329 796597983 30-4466747 9-7504930 001078749 928 861184 799178752 30-4630924 9-7539979 001077586 929 863041 801765089 30-4795013 9-7575002 001070426 930 864900 804357000 30-4959014 97610001 001075-209 931 866761 806954491 30-5122926 9-7644974 001074114 932 868624 809557568 30-5286750 9-7679922 00 107-2901 933 870489 812166237 30-5450487 9-7714845 001071811 934 872356 814780504 30-5614136 9-7749743 001070064 935 874225 817400375 30-5777697 9-7784616 001069519 936 876096 820025856 30-5941171 9-7819466 001068376 937 877969 822656953 30-6104557 9-7854288 001067236 938 879844 825293672 30-6267857 9-7889087 001066098 939 881721 827936J19 30-6431069 9-7923861 001064963 940 883600 830584tJcJ0 30-6594 194 9-7958611 00 106383 J 941 885481 833237621 30-6757233 9-7993336 001002G99 942 887364 835896888 30-6920185 9-8028036 001001571 943 889249 838561807 30-7083051 9-8002711 00]G0o445 944 891136 841232384 30-7245830 9-8097302 001659322 945 893025 843908625 30-7408523 9-8131989 001(i.)8-2Jl 940 894916 846590536 30-7571130 9-81065J1 001057082 947 896869 849278123 30-7733651 9-8-201109 001055966 948 89i^7u4 851971392 30-7896086 9-8-235723 001054852 949 900601 854670349 30-8058436 98270252 001053741 950 902500 857375000 30-8220700 9-8304757 00105-2032 951 904401 860085351 30-8382879 9-8339238 001051525 952 906304 862801408 30-8544972 9-8373695 00105042J 953 908209 865523177 30-8706981 9-8408127 001(/4>JJl8 954 910116 868250664 30-8868904 9-8442536 001('48218 955 9l2u25 870983875 30-9030743 9-8470920 001047 120 956 913936 873722816 30-9192497 9-8511280 001046O-25 957 915849 876467493 3^-9354166 9-8545617 001044Ly.>2 958 917764 879217912 30-9515751 9-857'Ji)29 001043841 959 91'j681 881974079 30-9677-251 9-8014218 001042753 96U 921600 884736000 30-9838068 9-8648483 001041007 961 923521 887503681 31-0000000 9-8682724 001040583 962 925444 890277128 310101248 9-8716941 001039501 963 927369 893056347 31-03-2-2413 9-8751135 001038422 964 929296 895841344 31-0483494 9-8785305 001037344 965 931225 898632125 310644491 9-8819451 001030-209 966 933156 901428696 31-0805405 9-8853574 001035197 967 935089 9042310G3 310966236 9-8887673 001034120 968 937024 907039232 3 1 1126984 9-8921749 001033053 969 938961 909853209 3r 1287648 9-8955801 001031992 970 940900 912673000 311448230 9-8989830 001030928 971 942841 915498611 3l 1608729 9-9023835 001029866 972 944784 918330048 31 1769145 9-9057817 001028807 973 946729 921167317 3] 1929479 9-9091776 001027749 974 948676 924010424 31-2089731 9-91257)2 001026694 975 950625 926859375 31 2249900 9-91596-24 001025641 976 952576 929714176 31 2409987 9-9193513 001024590 977 954529 932574833 31-2569992 9-9227379 0010-23541 978 9564H4 935441352 312729915 9-9261222 001022495 979 958441 938313739 31-2889757 9-9-295042 001021450 980 960400 941192000 31 3049517 9-9328839 001020408 178 SQUARES, CUBES, ETC., OF NUMBERS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocala. 981 962361 944076141 31-3209195 9-9362613 001019168 982 964324 946966168 31-3338792 9-9396.363 •00101833ft 983 966289 949862087 31-3.528308 9-!M3(t(l!l2 -0111017294 «84 968256 9527t)3!(l)4 31-3687743 9-9463797 •001016260 985 970225 955671625 31-3847097 9-9497479 •001015228 986 972196 958585256 31.4006369 9-9.531138 •001014199 987 974169 961504803 31-416.5561 9-9564775 •001013171 988 976144 964430272 31-4.324673 9-9.598389 •001012146 989 978121 967361669 31-4483704 9-9631981 •001011122 990 980100 970299000 31-46426.54 9-966.5549 •001010101 991 982081 973242271 31-4801525 9-9699095 •001009082 992 984064 976191488 31-4960315 9-9732(U9 •001008065 993 986049 979146657 31-5119025 9-9766120 •001007049 994 988036 982107784 31-.i2776.55 9-9799.599 •001006036 995 990025 985074875 31.54.36206 9-9833055 001005025 996 992016 988047936 31 -.5,594677 9-9866488 •001004016 997 994009 991026973 31 -.57.53068 9-9899900 •001003009 998 996004 994011992 31-.591]380 9-99.33289 •001002004 999 998001 997002999 31-6069613 9-9966656 •001001001 1000 1000000 1000000000 31-6227766 100000000 •001000000 1001 1000201 1003003001 31-638.5840 10-0033222 •0009990010 1002 1004004 1006012008 31-6.543836 10-0066822 •0009980040 1003 1006009 1009027027 31.67017.52 10-0099899 •0009970090 1004 1008016 1012048064 31-6859.590 10-01.331.55 •0009960159 1005 1010025 1015075125 31-7017349 100166.389 •0009950249 1006 1012036 1018108216 31-717.5030 100199601 •0009940358 1007 1014049 1021147343 31-73.32633 100232791 •0009930487 1008 t016064 1024192512 31-74901.57 10-02t>5958 •0009920635 1009 1018081 1027243729 31-7647603 10-0299104 •0009910803 1010 1020100 1030301000 31-7804972 10-0332228 -0009900990 1011 1022121 1033364331 31-7962262 100365330 -0009891197 1012 1024144 1036433728 31-8119474 10- 03984 10 •0009881423 1013 1026169 1039509197 31-8276609 10-0431469 •0009871668 10J4 1028196 1042590744 31-8433686 100464,506 •0009861933 10J5 1030225 1045678375 318.590646 100497.521 •0009852217 lOlG 1032256 1048772096 31-8747.549 10-0530514 •0009842.520 1017 1034289 1^51871913 31-8904374 100563485 0009832842 1018 1036324 1054977832 31-9061123 10-0.596435 0009823183 1019 1038361 10580898.59 319217794 10-0629364 0009813543 1020 1040400 1061208000 31-9374.388 l0-066i!271 0009803922 1021 1042441 1064332261 31-95.30906 1006951.56 0009794319 1022 1044484 1067462648 31-9687347 10-0728020 0009784736 1023 1046529 1070599167 31-9843712 10-07608()3 0009775171 A024 1048576 1073741824 32-0000000 100793684 0009765625 1025 1050625 1076890625 320156212 10-0826484 0009756098 1026 1052676 1080045576 320312348 10-0859262 0009746589 1027 1054729 1083206683 320468407 10-0892019 0009737098 1028 1056784 1086373952 32-0624391 10-0924755 0009727626 1029 1058841 1089547389 32-0780298 10-09.57469 -0009718173 1030 10(50900 1092727000 32 0936131 10-0990163 •0009708738 1031 1062961 1095912791 32-1091887 101022835 •0009699321 1032 1065024 1099104768 32 1247568 10-1055487 -0009689922 1033 1067089 1 102302937 321403173 10-1088117 -0009680542 1034 1069156 1105507304 3215.58704 101120726 -0009671180 1035 1071225 1108717875 32-17141.59 10-11.53314 -0009661836 1036 1073296 1111934656 32- 1869539 10-1185882 1)009652510 1037 1075369 11151.57653 32-2024844 10-1218428 0009643202 1038 1077444 1118386872 32-2180074 1012,50953 0009633911 1039 1079521 1121622319 32 2335229 10-1283457 -0009624639 1040 1081600 1124864000 32-2490310 10-1315941 -0009615385 1041 1083681 1128111921 32-2645316 10-1348403 -0009606148 1042 1085764 1131366088 32-2800248 10-1380845 •0009596929 TABLE XII. LOGAEITHMS OF NUMBERS FROM 1 TO 10000. TABLE, CONTAnrUTB THE LOGAEITHMS OE NUMBERS FROM 1 TO 10,000. NUMBERS FROM 1 TO 100 AND THEIR LOGARITHMS, WITH THEIR INDICES. Ka Logarithm. No. Logarithm. No. Logarithm. No. Logarithm. No. Logarithm. 1 0-ootiuuo 21 1-322219 41 1-612784 61 1-785330 81 1-908495 2 0-30 lUoO 22 1-342423 42 1-623249 02 1-792392 82 1-913814 3 0-477r21 23 1-301728 43 1-633468 63 1-799.341 83 1-919078 4 0-G020G0 24 1-380211 44 1-643453 64 1-806180 84 1-924279 5 0-698;)70 25 1-397940 45 1-653213 65 1-812913 85 1-D29419 6 0-778151 20 1-414973 46 1 •602758 66 1819.544 86 1-934498 7 0-845098 27 I-4313G4 47 1-672098 67 1-826075 87 1-039519 8 0-9030Ua 28 1-447158 48 1-681241 68 1-832509 88 1-944483 9 0-954243 29 1-402398 49 1-690190 69 1-838849 89 1-949390 10 1-0001)00 30 1-477121 50 1-098970 70 1-845098 90 1-954243 11 1-041393 31 1-491302 51 1-707570 71 1-851258 91 1-959041 12 1-079J81 32 1-505150 52 1-716003 72 1-857332 92 1-963788 13 1- 113943 33 1-518514 53 1-724270 73 1-8G3323 93 1-968483 14 1 14G128 34 1-53 J 479 54 1-732394 74 1-669232 94 1-973128 15 1-170091 35 1-544008 55 r 740303 75 1.875001 95 1-977724 Hi 1-204120 3(5 1-55G303 50 1-748188 76 1-880814 96 1-982271 17 1-23U449 37 1-568202 57 l-7.-)5875 77 1-88G491 97 1-986772 18 1-255273 38 1-579784 58 1-763428 78 1-892095 98 1-991226 19 1-278754 39 1-591065 59 1-770852 79 1-897627 99 1-995G35 20 1-301030 40 1-602060 60 1-778151 80 1-903090 100 2-000000 Note. — In the following pare of the Table, the characterisrics are omitted, as they can be very easily supplied. Thus, the chaiacteristic of the logarithm of every integer number, consisting only of one number, isO; of two figures, 1; of three figures, 2; of four figui-es, 3; being always a uuit less than the number of flgiu-es contained in the integei- number. 180 LOGARITHMS OF NUMBERS. 181 Na 1 3 3 4 1 5 6 1 7 8 9 (Diff l()0!(iOOOOO 0004341000868 001301 001734 002166 002598 003029 003461 003891 432 1 4321 4751 5181 5609 6038 6466 6894 7321 7748 8174 42-1 2 8(i00 9026 9451 9876 01030U 010724 011147 011570 011993 012415 424 3 012837 013259 013680 014100 452] 4940! 5360 5779 6197 (5()I6!420 4 7033 7451 7868 8284 8700 9116 9532 9947 02U3(il 020775 416 5 021189 021603 022016 022428 022841 023252 («36ti4 024075 448(5 489li;412 () 5306 5715 6125 6533 6942 7350 7757 HI (54 8571 8978; 408 7 9384 9789 030195 030600 031004 031408 031812 03221(5 032619 033021,404 8 033424 033826 4227 4(i28 5029 5430 5830 62:50 (56211 702H!40(» 9 7426 7825 8223 8620 9017 9414' 9811 04(J207 040602 040998:397 110 041393 041787 042182 042576 042969 043362 043755 044148 044540 044932 393 1 53231 5714 6105 6495 68851 7275! 7661 8053 8442 H83:i MW 2 9218 9606 9993 050380 050766 051153 051538 051924 0523(i'hf 052(594 3Hii 3 053078 10534(53 053846 4230 4613 4996 5378 57(5(1, > 42 ()524 3s:{ 4| 6905 7286 7666 8046 8426 8805 9185 9563 9942 06032(1 379 510600981061075 061452 061829 062206 062582 062y.l8 063;j33 063709 4083 37(; U; 4458 4832 5206 5580 5953 i 6326! 6(599 7071 7443 7815 373 7 818!)! 8557 8928 9298 9668 070038 070407 07077(5 071145 071514 370 8 071882 072250 072617 072985 073352 3718 4085 4451 4816 5182 3ii(i 9 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 3(53 120 07918J 079543!079904 0802(56 080626 080987 081547 081707 082067 082426 360 1 082785! 083 144 '083503 3861 4219, 4576 4934 5291 5647 6004 357 2 6360 67161 7071 7426 .7781 8136 8490 8845 9198 9552 355 3 9905 (190258 090611 090963 091315 091667 092018 092370 092721 093071 352 4 093422 37721 4122 4471 4820 5169 5518 58(56 6215 65(52 349 5 0910 7257 7604 7951 8298 8644 8990 9335 9681 1000261346 6 100371 100715 101059 101403 101747 102091 102434 102777 103119 3462! 343 7 3804 4146 4487 4828 51(59 5510 5851 6191 6531 6871 I341 8 7210 7549 7888 »>27 8565 8903 9241 9579 9916 1102531338 9 110590 110926 111263 111599 111934 112270 112605 112940 113275 3609,335 130 ] 13943 114277 114611 114944 11 5278 115611 115943 11(5276 116608 116940 333 1 7271 7603 7934 8265 8595 8926 9256 9586 9915 120245 1 330 2 120574 120903 121231 121560 121888 122216 122544 122871 123198 35251328 3 3852 4178 4504 4830 5156 5481 5806 6131 6456 67811325 4 7105 7429 7753 8076 8399 8722 9045 9368 9690 1300121323 5 130334 130655 130977 131298 131619 131939 132260 132580 132900 32191321 6 3539 3858 4177 4496 4814 5133 5451 5769 6086 64031318 7 6721 7037 7354 7671 7987 8303 8618 8934 9249 95641316 8 9879 1140194 140508 140822 141136 141450 141763 142076 142389 142702; 314 9 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818311 140 146128 146438 146748 J 47058 147367 147676 147985 148294 148603 148911 309 1 9219 9527 9835 150142 150449 150756 151063 151370 15167(5 1519821307 2 152288 152594 152900 3205 3510 3815 4120 4424 4728 5032 305 3 5336 5640 5943 6246 6549 6852 7154 7457 7759 80611303 4! 83621 8664 ftj65 9266 9567 9868 1601(i8 1604(59 160769 161068 |301 5| 161368! 161667 161967 16226(5 162:564 162863 31(51 3460 3758 4055 1 299 6 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 '297 7 7317 7613 7908 8203 8497 8792 9086 9380 9674 99681295 8 170262 170555 170848 17114! 171434 171726 172019 172311 172(503 172895 293 9 3186 ^78 3769 4060 4351 4641 4932 5222 5512 5802 291 150 ! 176091 176381 176670 176959 177248 177536 177825 178113 178401 1786891289 11 8077 9264 9552 9839 180126 180413 180699 180986 181272 181558 287 2 181844 182129 182415 182700 2985 3270 3555 3839 4123 4407 285 3i 4691 49751 5259 5542 5825 6108 6391 6674 6956 7239 283 4| 752 J 1 78031 8084 836(5 8647 8928 9209 9490 9771 190051 281 5 190332 1906121 190892 191171 191451 191730 192010 192289 192567 2846 279 6 3125 3403! 3681 3959 4237 4514 4792 5069 5346 5623 278 7| 59UU' 6176! 6453 6729 7005 7281 7556 7832 8107 8382 276 8 86571 8932| 9206 9481 9755 200029 200303 200577 200850 201124 274 920139712016701201943 202216 202488 2761 3033 3305 3577 3848,272 No. I I 1 I 3 I I 4 I 5 I I 7 I 8 i 9 lOiff 182 LOGARITHMS OF NUMBERS. Nal 1 1 1 a I 3 1 4: 1 5 1 6 1 7 1 8 1 9 lD.fl: 160 204120 |204391| 204663 204934 205204 205475 205746 206016 206286 206556 271 1 6826 7096 7365 7634 7904 8173 8441 8710 8979 9247 269 2 9515 9783 210051 210310 210586 210853 211121 211388 211654 211921 267 3 21218;:^ 212454 2720 2986 3252 3518 3783 4049 4314 4579 266 4 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 5 7484 7747 8010 8273 8530 8798 9060 9323 9585 9846 262 6 220108 220370 220631 220892 221153 221414 221675 221936 222196 222456 261 7 2716 2976 3336 3490 3755 4015 4274 4533 4792 5051 259 8 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 9 7887 8144 8400 8657 8913 9170 9426 9682 9938 230193 '256 170 230449 230704 230960 231215 231470 231724 231979 232234 232488 232742 255 1 2996 3250 3504 3757 4011 4264 4517 4770 5023 5276 253 2 5528 5781 6033 6285 6537 6789 7041 7292 7544 7795 252 3 8046 8297 8548 8799 9049 9299 9550 9800 240050 240300 250 4 240549 240799 241048 24 J 297 241546 241795 242044 242293 2541 2790 249 5 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 6 5513 5759 6006 6252 6499 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8709 8954 9198 9443 9687 9932 250176 245 8 250420 250664 250908 251151 251395 251638 251881 252125 252368 2610 243 9 2853 3096 3338 3580 3822 4064 4300 4548 4790 5031 242 180 255273 255514 255755 255996 256237 256477 256718 256958 257198 257439 241 1 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 2 260071 260310 260548 260787 261025 261263 261501 261739 261976 262214 238 3 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 5 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 6 9513 9746 9980 270213 270446 270679 270912 271144 271377 271609 233 7 271842 272074 272306 2538 2770 3001 3233 3464 3696 3927 232 8 4158 4389 4620 4850 5081 5311 5542 5772 6002 6232 230 9 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 229 190 278754 278982 279211 279439 279667 279895 280123 280351 280578 280806 228 1 281033 281261 281488 281715 281942 282169 2396 2622 2849 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 3 5557 5782 6007 6-232 6456 6681 6905 7130 7354 7578 225 4 7802 8026 8249 8473 8696 8920 9143 9306 9539 9812 223 5 290035 29U257 290480 290702 290923 291147 291369 291591 291813 292034 222 6 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 221 7 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 8 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 9 8853 9071 9289 9507 9725 9943 300161 300378 300595 300813 218 m 301030 301247 301464 301681 301898 302114 302331 302547 302764 302980 217 1 3196 3412 3628 3844 40&9 4275 4491 4706 4921 5136 216 2 5351 5566 5781 5996 6211 6425 6639 6854 7068 ■ 7282 215 3 7496 7710 7924 8137 8351 8564 8778 8991 9204 9417 213 4 9630 9843 310056 310268 310481 310693 310906 311118 311330 311542 212 5 311754 311966 2177 2389 2600 2812 3023 3234 3445 3656 211 6 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 320146 320354 320562 320769 320977 321184 321391 321598 321805 322012 207 210 322219 322426 322633 322839 323046 323252 323458 323665 323871 324077 206 1 4282 4488 46:)4 4899 5105 5310 5516 5721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 8380 8583 8787 8i*91 9194 9398 9601 9805 330008 330211 203 4 330414 330617 330819 331022 331225 331427 331630 331832 2034 2236 202 5 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 6 4454 4055 4850 5057 5257 5458 5658 5859 6059 6260 201 7 6460 6660 6860 7060 7260 7459 7659 7858 8058 8257 200 8 8456 8656 8855 9054 9253 9451 9650 9849 340047 340246 199 9 340444 340642 340841 341039 341237 341435 341632 341830 2028 2225 198 No. 1 1 1 » a 1 3 J 4 1 5 1 6 1 'i I 8 1 9 'Dili 1 LOGARITHMS OF NUMBERS. isa No. 1 1 1 1 3 1 3 1 4 1 5 • 6 1 7 1 8 9 Ditt 220 342423 342620 342817 343014, J43212 ;:43409 343606 •J43802 343999 344196 i97 1 4392 4589 4785 4981 51781 5374 5570 5766 5962 6157 196 o 6353 6549 6744 6939 ! 7i:r)i 7330 7525 7720 7915 8110 195 3 8305 8500 8694 8889 9083 9278 9472 966(5 9860 350054 194 4 350248 350442 350636 350829 551023 351216 351410 351603 351796 1989 193 5 2183 2375 2568 2761 2954 3147 3339 3532 37-24 3916 193 6 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 192 7 6026 6217 6408 6599 6790 1)981 7172 7363 7554 7744 191 8 7935 8125 8316 8506 8696 i'.Am 9076 926() 9456 9646 190 9 9835 360025 360215 360404 360593 360783 360972 361161 361350 361539 189 230 361728 361917 362105 362294 362482 362671 362859 363048 36;»e36 363424 188 1 3612 38UI) 3988 4176 4363 4551 4739 4926 5 J 13 5301 188 2 5488 5675 5802 6049 6236 6423 6610 6706 6983 7169 187 3 7356 7542 7729 7915 8101 82^7 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 370143 370328 370513 376()98 370883 185 5 371068 371253 371437 371622 371806 1991 2175 2360 2544 2728 184 G 21)12 30^6 3280 341)4 3647 3831 4-015 4198 4382 4565 184 7 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8634 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 380030 181 24;) 380211 380392 380573 380754 380934 381115 381296 381476 381656 381837 181 1 2017 2197 2377 2557 2737 2917 :i097 3277 3456 3636 18C 2 3815 3995 4174 4353 4533 4712 4891 5070 5249 5428 179 3 5606 5785 5964 6142 6321 6499 6677 6856 7034 7212 178 4 7390 7568 7746 71.^^3 8101 8279 8456 8634 8811 8989 178 5 9166 9343 9520 96981 9875 390051 390228 390405 390582 390759 177 6 390935 391112 391288 391464 391641 1817 1993 2169 2345 2521 176 7 2()97 2873 3048 3224 3400 3575 3751 3926 411)1 4277 176 8 4452 4627 4802 4977 5152 5320 5501 5676 5850 6025 175 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 77(56 174 250 397940 398114 398287 398461 398634 398808 398981 399154 399328 399501 173 1 9674 9847 400020 400192 400365 400538 400711 400883 401056 401228 173 2 401401 401573 1745 1917 2089 2261 2433 2605 2777 2949 172 3 3121 3292 3464 3635 3807 3978 4149 4320 4492 4663 171 4 4834 5005 5176 5346 5517 5688 5858 6029 6199 6370 171 5 6540 6710 6881 7051 7221 7391 ■7561 7731 7901 8070 170 6 8240 8410 8579 8749 8918 9087 9257 9426 9595 9764 169 9933 410102 410271 410440 410609 410777 410046 411114 4.11283 411451 169 8 411620 1788 19.--6 2124 2293 2461 2629 2706 2.-)l4 3132 168 9 3300 3467 3635 3803 3970 4137 43',(5 4472 463!) 4806 167 26,) 414973 415140 415307 415474 415641 415808 415974 416141 4163()f^ 41(5474 167 J 6641 6807 6973 7139 7306 7472 7t)38 7804 :!»70 8135 166 2 8301 8467 8633 8798 8964 9129 9295 946u '.)(i25 9791 165 3 9956 420121 4202S6 420451 420616 420781 420945 421110 421275 421439 165 4 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5208 5371 5534 5697 5860 6023 61W6 ))349 163 ' 6511 6674 6836 6999 7161 7324 7486 7648 7811 7973 162 8 8135 8297 8459 8621 8783 8944 9106 9268 942J 9591 162 9 9752 9914 430075 430236 430398 430559 430720 430881 431042 431203 161 270 431364 431525 431685 431846 432007 432167 432328 4324HH;432li4!» 432H09 161 1 29691 3i:i0 32J,) 3450 3610 3770 3930 4iV.«); 4241) 4409 160 2 45691 4729 4888 5048 5207 5367 5526 5685 5844, 6004 159 3 61631 6322 6481 6640 6799 6957 7116 7275 7433 75921 159 4 7751 7909 8067 8226 8384 8542 8701 8859 9.17 9175 158 5 9333 9491 9648 9806 9964 440122 440279 440437 440594 440752 158 6 440909 441066 441224 441381 441538 1695 1652 2009 2166 2323 157 7 248^ 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 £ 404i j 4201 4357 4513 4669 4825 49^1 5137 5293 5449 156 S 5604 1 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 No. |011|31314:|5|6|7|8|9 Die 184 LOGARITHMS OF NUMBERS. No. 1 3 3 ^ 5 G 7 8 9 D,ff, 280 447158 447313 447468 447623 447778 447933 448088 448242 448397 448552 155 1 870G 8861 9015 9170 93-24 9478 9033 9787 3341 450095 154 2 4.:j:4D 453403 450557 453711 450805 451010 451172 451320 451479 1033 154 3 J78.'i 1940 2033 2247 2400 2553 2706 2859 3012 3165 153 4 3 J 18 3471 3o24 3777 3930 4032 4235 4387 4540 4632 l.':3 5 4845 4997 5150 5302 5454 5;i0{; 575(1 5310 6062 6214 152 6 onoa 0518 6070 6821 6373 7125 7276 7428 7579 7731 152 7 7c^82 80r,3 8104 8336 8487 8038 8783 8940 9091 9-242 151 8 9332 9543 on. 14 9845 9935 460146 46023() 460447 460597 460748 151 9 460898 461048 4GU38 401348 461499 1649 1793 1948 2098 2248 150 290 402338 462548 402337 402>S47 462997 463140 463296 4(53445 403594 463744 150 i; 38a3 4042 4131 4340 4430 4633 4788 4936 5085 5234 149 2 5383 5532 5i)«0 5.-3 5377 612.; 6274 6423 6571 6719 149 3 6868 7.116 7164 7312 7400 7608 7756 7904 8052 8200 148 4 8347 8405 8043 8790 8938 9085 9233 9380 9527 9675 148 5 9822 9963 470116 470263 4704 Iv) 470557 470704 470851 470998 471145 147 6 471292 471438 1585 1732 1S78 2025 2171 2318 2464 2010 146 7 2756 2903 3043 3195 334] 3487 ?633 3779 3925 4071 146 8 4216 4362 4508 4J53 4793 4344 5090 5235 5381 5526 146 9 5671 5816 5962 6107 6252 6397 6542 6687 6832 6376 145 300 477121 477266 477411 477555 477700 477844 477989 478133 478278 478422 145 1 8560 48000f 8711 8855 8999 9143 9287 9431 9575 9719 9863 144 2 480151 480294 480438 480582 480725 480869 481012 481156 481299 144 3 1443 1586 1729 1872 2016 2159 2302 2445 2588 2731 143 4 2874 3016 3159 3302 3445 3587 3730 3872 4015 4157 143 5 4300 4442 4585 4727 4869 5011 5153 5295 5437 5579 142 6 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 142 7 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 8 8551 8692 8833 8974 9114 9255 9396 9537 9677 9813 141 9 9958 490099 490239 490380 490520 490661 490801 490941 491081 491222 140 310 491362 491502 491642 491782 491922 492062 492201 492341 492481 492621 140 1 2760 2900 3040 3179 3319 3458 3597 3737 3876 4015 139 o 4155 4294 4433 4572 4711 4850 4989 5128 5267 5406 139 3 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 139 4 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 5 8311 8448 8586 8724 8862 8999 9137 9275 9412 9550 138 6 9687 9824 9962 500099 500236 500374 500511 500648 500785 500922 137 7 501059 501196 501333 1470 1()07 1744 1880 2017 2154 2291 137 8 2427 2564 2700 2837 2973 3109 3246 3382 3518 3655 136 9 3791 3927 4063 4199 4335 4471 4607 4743 4878 5014 136 320 505150 50528() 505421 505557 505693 505828 505964 506099 506234 506370 136 1 6505 6640 6776 6911 7046 7181 7316 7451 7586 7721 135 2 7856 7991 8126 8260 8395 8530 8664 8799 8934 9068 135 3 9203 9337 9471 9606 9740 9874 510009 510143 510277 510411 134 4 510545 510679 510813 510947 511081 511215 1349 1482 1616 1750 134 5 1883 21)17 2151 2284 2418 2551 2684 2818 2951 3084 133 6 3218 3351 3484 3617 3750 3883 4016 4149 4282 4415 133 7 4548 4681 4813 4946 5079 5211 5344 5476 5609 5741 133 8 5874 60!)6 6133 6271 6403 6535 6668 6800 6932 7064 132 9 7196 7328 7460 7592 7724 7855 7987 8119 8251 8382 132 330 518514 518646 518777 518909 519040 519171 519303 519434 519566 519697 131 1 9828 9959 520030 520221 520353 520484 520615 520745 520876 521007 131 2 521 KW 521269 1430 1530 1661 1792 1922 2053 2183 2314 13J 3 2444 2575 2705 2835 2966 3090 3226 3356 3486 3616 130 4 374(i 3876 40J6 4136 4266 4396 4526 4656 4785 4915 130 5 5045 5174 53()4 5434 5563 5693 5822 5951 6081 6210 129 6 6339 6463 653H 6727 6856 6985 7114 7243 7372 7501 129 7 7630 7759 7888 8016 8145 8274 8402 8531 8660 8788 129 8 8917 9045 9174 9302 9430 9559 9687 9815 9943 530072 128 9 530200 530320 530456 530584 530712 530840 530968 531096 531223 1351 128 Wd.|0I119I3|4:|$ r I 8 I 9 iDifl LOGARITHMS OF NUMBERS. 185 3 I I 7 I 8 / 9 iDiffi 340 5J1-17Q 531607 531734 5318621 531990 532117 532245 532372 532500 532627 128 1 2754 2882 3009 313()! 3264 3391 3518 3645 3772 3899 127 'J 4026 4153 4280 4407 4534 4661 4787 4914 5041 5167 127 3 5294 5421 5547 5674 5800 5927 6053 6180 6306 6432 126 4 6558 6685 6811 6937 7063 7189 7315 7441 7567 7693 126 5 7819 7945 8071 8197 8322 8448 8574 8699 8825 "951 126 6 9076 9202 9327 9452 9578 9703 9829 9954 540079 540204 125 7 540329 540455 540580 540705 540830 540955 541080 541205 1330 1454 125 8 1579 1704 1829 1953 2078 2203 2327 2452 2576 2701 125 9 2825 2950 3074 3199 3323 3447 3571 3696 3820 3944 124 350 544068 544192 544316 544140 544564 544688 544812 544936 545060 545183 124 1 5307 5431 5555 5678 5802 5925 6049 6172 6296 6419 124 2 6543 6666 6789 6913 7036 7159 7282 7405 7529 7652 123 3 7775 7898 8021 8144 8267 8389 8512 8635 8758 8881 123 4 9003 9126 9249 9371 9494 9616 9739 9861 9984 550106 123 5 550228 550351 550473 550595 550717 550840 550962 551084 551206 1328 122 6 1450 1572 1694 1816 1938 2060 2181 2303 2425 2547 122 7 2668 2790 2911 3033 3155 3276 3398 3519 3640 3762 121 8 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 121 9 5094 5215 5336 5457 5578 5699 5820 5940 6061 6182 121 360 556303 556423 556544 556664 556785 556905 557026 557146 557267 557387 120 1 7507 7627 7748 7868 7988 8108 8228 8349 8469 8589 120 2 8709 8829 8948 9068 9188 9308 9428 9548 9667 9787 120 3 9907 560026 560146 560265 560385 560504 560624 560743 560863 560982 119 4 561101 1221 1340 1459 1578 1698 1817 1936 2055 2174 119 5 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 119 6 3481 3600 3718 3837 3955 4074 4192 4311 4429 4548 119 7 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 118 8 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 9 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 568202 568319 568436 568554 568671 568788 568905 569023 569140 569257 117 1 9374 9491 9608 9725 9842 9959 570076 570193 570309 570426 117 2 570543 570660 570776 570893 571010 571126 1243 1359 1476 1592 117 3 1709 1825 1942 20.58 2174 2291 2407 2523 2639 2755 116 4 2872 2988 3104 3220 3336 3452 3568 3684 3800 3915 116 5 4031 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 6 51G8 5303 5419 5534 5650 5765 5880 5996 6111 6226 115 7 6341 6457 6572 6687 6802 6917 7032 7147 7262 7377 115 8 7492 7607 7722 7836 7951 8066 8181 8295 8410 8525 115 9 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 579784 579898 580012 580126 580241 580355 580469 580583 580697 580811 114 1 580925 581039 1153 1267 1381 1495 1608 1722 1836 1950 114 2 2063 2177 2291 2404 2518 2631 2745 2858 2972 3085 114 3 3199 3312 3426 3539 3652 3765 3879 3992 4105 4218 113 4 4331 4444 4557 4670 4783 4896 5009 5122 5235 5348 113 5 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 113 6 6587 6700 6812 6925 7037 7141) 7262 7374 7486 7599 112 7711 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 8 8832 8944 9050 9167 9279 9391 9503 9615 9726 9838 112 9 9950 590061 590173 590284 590396 590507 590619 590730 590842 590953 112 390 591065 591176 591287 591399 591510 591621 591732 591843 591955 592066 111 1 2177 2288 23a9 2510 2621 2732 2843 2954 3064 3175 111 3286 3397 35U« 3618 3729 3840 3950 4061 4171 4282 111 3 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 110 4 5496 5006 5717 5827 5937 6047 6157 6267 6377 6487 110 5 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 110 6 7695 7805 7914 8024 8134 8243 8353 8462 8572 8681 110 7 8791 89C0 9009 9119 9228 9337 9446 9556 9665 9774 109 8 9883 9992 600101 600210 600319 600428 600537 600646 600755 600864 109 9 600973 601082 1191 1299 1408 1517 1625 1734 1843 1951 109 W.|0|l|a|3|4:|5|6|7| I 9 ID* 186 LOGARITHMS OF NUMBERS. No. I I- 1 I I 3 I 4 I I 6 I 7 I 8 I 9 iDifS 400 602060 602169 602277 602386 602494 502603 602711 602819] 602928 603038 108 1 3144 3253 3361 3469 3577 3638 3794 39021 4010 4118 108 2 4226 4334 4442 4550 4658 4766 4874 4982 5089 5197 108 n 5305 5413 5521 5r.28 5736 5844 5951 6059 6166 6274 108 4 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 107 5 7455 7562 7669 7777 7884 7991 8098 8205 8312 8419 107 6 8526 8633 8740 8847 8954 9061 91G7 9274 9381 9488 107 7 9594 9701 9808 9914 610021 610128 610234 610341 6104-7 G1053-; 107 8 610660 6107G7 610873 610970 1086 1192 1298 1405 1511 1617 106 9 1723 1829 1933 2042 2148 2254 2360 2466 2572 2678 106 410 612784 612890 612996 613102 613207 613313 613419 613525 613630 613736 106 1 3842 3947 4053 4159 4264 4370 4475 4581 4688 4792 106 4897 5003 5108 5213 5319 5424 5529 5G34 5740 5843 105 3 5950 6055 6160 6265 6370 6476 6581 6686 0790 6895 105 4 7000 7105 7210 7315 7429 7525 7829 7734 7839 7943 105 5 8048 8153 8257 8362 8466 8571 8676 8780 8884 8989 105 6 9093 9198 9302 9406 9511 9615 9719 9824 9928 620032 104 7 620136 620240 620344 620448 620552 62065S 620700 623864 0209G8 1072 104 8 117G 1280 1384 1488 1592 1695 1709 1903 2007 2110 104 9 2214 2318 2421 2525 2628 2732 2835 2939 3342 314G 104 420 023249 623353 62345G 623559 623663 623766 623869 G23973 624076 624179 103 1 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 6032 6133 6238 103 3 6340 6442 6546 6648 6751 6853 6956 7058 716! 7263 103 4 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 102 5 8389 8491 8593 8695 8797 8900 9092 9104 920G 9308 102 6 9410 9512 9613 9715 9817 9919 630021 G30123 630224 630323 102 7 G30428 630530 630631 630733 630835 630936 1038 1139 1241 1342 102 8 ^-14 1545 1647 1748 1849 1951 2352 2133 2235 2336 131 9 2437 2559 2660 2761 2862 2963 30. ;4 3165 3266 33G7 101 430 G33468 633569 633G70 633771 633872 633973 634074 634175 634276 634376 101 1 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 101 2 5484 5584 5685 5785 5886 5986 6087 6187 6287 6388 100 3 6488 0588 G688 6789 6889 6989 70S9 7189 7290 7390 100 4 7490 7590 7690 7793 7890 7990 8090 8190 8290 83H9 100 5 8489 853J 81389 8789 8838 8988 90,88 9188 9287 9337 100 9486 9386 9686 9785 9885 9984 640084 640183 640283 040382 99 7 640481 640581 640680 640779 640879 640978 1077 1177 1276 1375 99 8 1474 1573 1672 1771 1871 1970 2069 2168 2267 2306 99 9 2465 2563 2662 2761 2860 2959 3058 3156 3255 3354 99 440 643453 643551 643650 643749 643847 643946 644044 644143 644242 644340 98 1 4433 4537 4636 4734 4832 4931 5029 5127 5226 5324 .98 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 98 3 6404 C5!J2 6600 6698 6796 6894 6992 7089 7187 7285 98 4 7383 7481 7579 7676 7774 7872 7969 8067 8165 8362 98 5 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 6 9335 9432 9530 9627 9724 9821 9919 650016 650113 650210 97 7 650308 650405 650502 650599 650696 G50793 650890 0987 1084 1181 97 8 1278 1375 1472 1509 1666 1762 1859 1956 2053 2150 97 9 2246 2343 2440 2536 2633 2730 2876 2923 3019 3110 97 150 653213 653309 653405 653502 653598 053695 653791 653888 653984 654080 96 1 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 96 2 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 96 3 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 4 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 m 5 8011 8107 8202 8298 8393 8488 8584 «679 8774 8870 93 e 8965 906( 9155 9230 9346 9441 9536 9631 9726 9821 95 7 9916 650011 630101 C03231 C6029G 663331 6G3486 6G3581 660676 060771 95 h 660865 096U 1055 1150 1245 1339 1434 1529 1623 1718 95 s J813 1907 2002 2396 2191 2286 2380 2475 25C9 2663 95 Wo. 10 111^131415161718 |9|D.a / <)(r. U.' I TILVS OF NUMBERS. 187 NoJ 1 Q .1 4 5 6 7 470 1 2 3 4 5 6 7 8 9 480 1 2 3 4 5 C 7 8 9 490 1 2 3 I I '^ I 4 5 I 6 r I 8 9 I Diff 002758 3701 4042 558J 0518 7453 8380 9317 670240 1173 072098 3021 3942 4801 5778 6094 7007 8518 9428 680330 081241 2145 3047 3947 4845 574 6036 7529 8420 9309 690196 1081 1905 284- 372' 4005 5482 6356 7229 81U1 098970 9838 700704 1508 249i 4330 5167 9924 700790 1054 2517 3377 4230 5094 5949 080j 707055 8500 9355 710202 1048 1892 2734 3575 4414 5251 700011 0877 1741 2603 34()3 4322 5179 0035 0888 707740 8591 9440 71028 li32 1970 2818 3i;59 4497 5335 3983 4924 58(52 0799 7733 8005 9590 ()70524 1451 672375 3297 42J8 5137 6053 09C8 7881 8791 9700 680607 681513 24 K) 3317 4217 5114 6010 6904 7790 8C8 9575 690402 134' 2230 3111 3991 4808 5744 01 ;i 7491 8302 099231 700098 09()3 1827 2089 3549 4408 5205 0120 0974 670017 670710 15431 1636 4078 5018 5950 6892 7826 8759 9089 4172 5112 0050 0980 7920 8852 9782 G72407 3390 4310 5228 0145 7059 7972 8882 9791 080098 681003 2500 3407 4307 5204 0100 6994 788tj 8770 9004 090550 1435 2318 3199 4078 4950 5832 t)700 7578 8449 09931 700184 1050 1913 2775 3035 4494 5350 0200 7059 ro7820 8()70 9524 r 10371 1217 2000 2902 3742 4581 5418 707911 8701 9009 710456 1301 2144 2980 3820 4605 5502 1 426() 4300 5206 5299 0143 6237 7079 7173 80131 8100 8945' 9038 98751 9j67 070802 070895 1728 i 1821 672560 3482 4402 5320 6236 7151 8063 8973 9882 680789 681693 2596 3497 4396 5294 0189 7083 7975 8805 9753 09U()39| 1524 2400 3287 4100 ."044 0793 7665 8535 099404 700271 1130 19J9 28ol 3721 4579 5430 0291 7144 707996 8846 9t;94 710540 1385 2229 3070 3910 4749 5586 672652' 35741 4494 54121 6328 7242 8154 9064 9973 680879 681784 2086 3587 4480 5383 0279 7172 8004 8953 9841 69072?^ 1012 2494 3375 42.'>4 Ot;t)7 ()880 7752 8022 099491 00358 1222 20H0 2947 3ci07 4005 5522 037() 7229 r03081 8931 9779 r 10025 1470 2313 3154 3994 4833 5669 672744 3006 4586 5503 6419 7333 8245 9155 680003 0970 4454 5.393 0331 7206 8199 913]! 070000; 070 153! 0988 108!) 1913 2o05 93 4548 94 5487, 94 6424 j 94 7300 91 8293 93 9'224i 93 53 j 93 180 ! 93 3077 4570 5473 0308 7201 8153 9042 9930 090810 1700 2583 3403 4312 .">219 0094 0968 7839 8709 099578 00444 1309 2172 3033 ;)d93 4751 5007 0402 7315 r08l06 9015 9803 ^0710 i.-.:.4l 2:i.;| 3238 40781 49161 5753 672836 3758 4077 5595 6511 7424 8336 9246 680154 1000 681904 2807 3707 4G00 5563 0458 7351 8242 9131 690019 690905 1789 2071 3551 4430 530' 6182 7055 7926 8796 699604 700531 1395 2258 ,3119 3979 483 5093 0547 7400 708251 9100 9948 710794 1039 2481 3323 4 J ('2 5000 5836 672929 3850 4709 5687 6002 7510 8427 9337 080245 1151 682055 2957 3857 4750 5652 0547 7440 8331 9220 69010' 690993 1877 2759 3039 4517 5394 6209 7142 8014 888:^ 099751 70001 1482 2344 3205 4005 4922 5778 6632 7485 708336 9185 710033 0879 1723 2566 3407 4246 I 50»4 I 5920 No.| O 1 .1 1 ^ 1 3 1 4: 1 5 I 6 I 7 I 84 84 I 9 \om 188 LOGARITHMS OF NIUIBERS. No. I O I 1 I 3 ( 1*15 1 I r I 8 f I DiS. 520 716003 716087 716170 716254 716337 716421 716504 716588 716671 716754 83 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 83 2 7671 7754 7837 7i>20 8003 8080 81(59 8253 8336 8419 83 3 8502 8585 8668 8751 88.14 8917 9000 90«3 9165 9248 83 4 9331 9414 9497 9580 9()03 9745 9828 9911 9994 720077 83 5 720159 720242 720325 720407 720490 720573 720655 720738 720821 0903 83 6 0986 1068 1151 1233 1316 1398 1481 1563 1046 1728 82 7 1811 1893 1975 2058 2140 Q-)00 2305 2387 2469 2552 82 8 2034 2716 2798 2881 2963 3045 3127 320:) 3291 3374 82 9 3456 3538 3620 3702 3784 3800 3948 4030 4112 4194 82 530 724276 724358 721440 724522 724604 724085 724767 724849 724931 725013 82 1 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 82 2 5912 5993 6075 6156 6238 6320 6401 6483 6564 6(546 82 3 6727 6809 6890 6972 7053 7134 7216 7297 7379 7460 81 .4 7541 7023 7704 7785 7860 7948 8029 8110 8191 8273 81 5 8354 8435 8516 8597 8678 8759 8841 8922 9003 9084 81 6 9165 9240 9327 9408 9489 9570 9651 9732 9813 9893 81 7 9974 730055 730136 730217 730298 730378 730459 730540 730621 730702 81 8 730782 0803 0944 1024 1105 1186 1266 1347 1428 1508 81 9 158D 1869 1750 1830 1911 1991 2072 2152 2233 2313 81 540 732394 732474 732555 732635 732715 732796 732376 732956 733037 733117 80 1 3197 3278 3358 3433 3518 3598 3679 3759 3839 3919 80 2 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 3 4800 4880 4960 5040 5120 5200 5279 5359 5439 5519 80 4 5599 5679 5759 5;S3^ 5918 5998 (5078 6157 6237 (5317 80 5 6397 6476 0556 6035 6715 6795 0874 6954 7034 7113 80 6 7193 7272 7352 743! 7511 7590 7670 7749 7829 7908 79 7 7987 8067 8140 822') 8305 8384 8403 8543 8622 8701 79 8 8781 8860 8939 9!)!H 9097 9177 9256 9335 9414 9493 79 9 9572 9651 9731 9810 9889 9968 740047 740126 740205 740284 79 550 740363 740442 740521 740000 740078 740757 740836 740915 740994 741073 79 1 1152 1230 1309 1388 1467 1540 1624 1703 1782 1860 79 o 1939 2018 2096 2175 2254 2332 2411 2489 2568 2047 79 3 2725 2804 2382 29(U 3039 3118 3196 3275 3353 3431 78 4 3510 3588 3607 3745 3823 3902 3980 4058 4136 4215 78 5 4293 4371 4449 4528 4600 4684 4762 4840 4919 4997 78 6 5075 5153 5231 5309 5387 5405 5543 5621 5699 5777 78 7 5855 5933 6011 6089 6167 6245 0323 6401 6479 6556 78 8 6634 6712 6790 68()8 6945 7023 7101 7179 7256 7334 78 9 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 5G0 748188 748266 748343 748421 748498 748576 748(553 748731 748808 748885 77 ] 8963 9040 9118 9195 9272 9350 9427 9504 9582 9059 77 2 9736 9814 9891 9968 750045 750123 750200 750277 750354 750431 77 3 750508 750586 750663 750740 0817 0894 0971 1048 1125 1202 77 4 1279 1356 1433 • 1510 1587 1604 1741 1818 1895 1972 77 5 2048 2125 2202 2279 2350 2433 2509 2586 2663 2740 77 6 2816 2893 2970 3047 3123 3200 3277 3353 3430 3500 77 7 3583 3060 3736 3813 3889 39156 4042 4119 4195 4272 77 8 434S 4425 4501 4578 4654 4730 4807 4883 4900 503(5 76 9 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 no 755875 755951 750027 750103 756180 756256 756332 750408 756484 75(5560 76 I 6636 6712 6788 6804 6940 7016 7092 7168 7244 7320 76 2 7396 7472 7548 7624 7700 7775 7851 7v>27 8003 8079 76 3 815S 8230 830(5 8382 8458 8533 8009 8685 8761 8836 76 4 8912 8988 9063 9139 9214 9290 9366 9441 9517 9592 76 5 9668 9743 9819 9894 9970 760045 760121 760196 760272 760347 75 6 760422 760498 760573 7011049 7(50724 0799 0875 0950 1025 1101 75 7 1176 1251 132(i 14()J 1477 1552 1()27 1702 1778 1853 75 8 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 75 9 2679 2754 2829 2904 2978 3053 3128 3203 3278 3353 75 No.|0|l|S|3|4:|5|6|7|8|9|Diff LOGARITHMS OF NUMBERS. 189 No. i 1 1 1 3 1 3 i * 1 5 1 6 7 8 9 iDiff 580 763428 763503 763578 763653 763727 763802 763877 763952 764027 764101 75 1 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 75 o 4923 4998 5072 5147 5221 5296 5370 5445 5520 5594 75 3 5669 5743 5818 5892 5966 6041 6115 6190 6264 6338 74 4 6413 (>487 6562 6G36 6710 6785 GS59 0933 7007 7082 74 5 7156 7230 7304 7379 7453 7527 7601 7075 7749 7823 74 f) 7898 7972 8046 8120 8194 8268 8342 8410 8490 8564 74 7 8638 8712 87o6 88G0 8934 9008 9082 9156 9230 9303 74 8 9377 9451 9525 9599 9G73 9746 9820 9894 9968 770042 74 9 770115 770189 770263 770336 770410 770484 770557 770631 770705 0778 74 590 770852 770926 770999 771073 771146 771220 771293 771307 771440 771514 74 1 1587 1661 1734 1808 1881 1955 2028 2102 2175 2248 73 2 2322 2395 24G8 2542 2615 2688 270x,M 2835 2908 2981 73 3 3055 3128 3201 3274 3348 3421 3494; 3567 3040 3713 73 4 3786 3860 3933 4006 4079 4152 4225 4293 4371 4444 . 73 5 4517 4590 46iJ3 4736 4809 4882 4955 5G28 5100 5173 73 6 5246 5319 5392 5465 5538 5G10 5683 5756 5829 5902 73 7 5974 6047 6123 6193 62G5 6338 6411 6483 6556 6629 73 8 6701 6774 G84G 6919 6992 7084 7137 7209 7282 7354 73 9 7427 7499 7572 7644 7717 7789 78G2 7934 8006 8079 72 600 778151 778224 778290 778368 778441 778513 778585 778658 778730 778802 72 1 8874 8947 9019 9091 9163 9236 9308 9380 9452 9524 72 2 9596 9069 9741 9813 9885 9957 780029 783101 783173 780245 72 3 780317 780389 780461 780533 780605 780877 0749 0821 0833 0965 72 4 1037 1109 1181 1253 1324 1396 1468 1540 1612 1684 72 5 1755 1827 1839 1971 2042 2114 21 8G 2258 2329 2401 72 6 2-173 2544 2G1G 2688 2759 2831 2902 2974 304G 3117 72 7 3189 3260 3332 3403 3475 3546 3GI8 3689 37G1 3832 71 8 3904 3975 4046 4118 4189 4261 4333 4403 4475 4546 71 9 4617 4C89 4760 4831 4932 4974 5045 5116 5187 5259 71 610 785330 785401 785472 785543 785615 785686 785757 785828 785899 785970 71 1 6;)41 6112 6183 6254 6325 6393 C467 6538 6609 6680 71 2 0751 0822 6893 6964 7035 7103 7177 7248 7319 7390 71 3 74G3 7331 7602 7673 7744 7815 7885 7956 8027 8098 71 4 8168 8233 8310 8381 8451 8522 8593 8G63 8734 8804 71 5 8375 8346 9016 9U87 9157 9228 9239 9369 9440 9510 71 6 9581 9G51 9700 9792 9863 9933 790004 790074 790144 790215 70 7 790235 790356 790426 790496 790567 790637 0707 0778 0848 0918 70 8 0988 1059 1129 1193 1269 13-10 1410 1480 1550 1620 70 9 1631 1761 1831 1901 1971 2341 2111 2181 2252 2322 70 620 792393 792462 792532 792602 792672 792742 792812 792882 792952 793022 70 1 3092 3162 3231 330] 3371 3-141 3511 3581 3651 3721 70 2 3793 3360 3930 4000 4070 4139 4209 4279 4349 4418 70 3 4488 4558 4627 4697 4767 4836 4996 4976 5045 5115 70 4 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 5 5883 5949 6019 6088 6158 6227 6297 6366 6436 6505 69 6 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 7 7268 7337 74G0 7475 7545 7614 7G83 7752 7821 7890 69 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 9 8651 8720 8789 8858 8927 8930 90G5 9134 9203 9272 69 330 799341 799409 799478 799547 799616 793685 799754 799823 799892 799961 69 1 800029 800098 800167 800236 800305 800373 800442 800511 800580 800648 69 2 0717 0786 0854 0923 09G2 ICGI 1123 1198 1266 1335 69 3 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 4 2089 2158 2226 2295 2363 2433 2500 2563 2637 2705 68 5 2774 2842 2910 2979 3047 31JG 3184 3252 3321 3389 68 G 3457 3525 3594 3GG2 3730 3793 3867 1 3935 4003 4071 68 7 4139 4208 4276 4344 4412 4480 45481 4616 4685 4753 68 8 4821 4889 4957 5025 5093 5161 52391 5297 5365 5433 68 & 5501 5569 5637 5705 5773 58411 59081 5976 6044 6112 68 No. I |lia|3|4:|5|6|7|8|9|D^ 190 LOGARITHMS OF NUMBERS. N* 1 1 1 3 3 4 1 5 1 6 7 1 8 9 I Did t)4(l 8()(;iHn 806^^l8:8063]6iS06384 806451 806519 806587 806655 806-'23 8067901 68 74671 68 J 0858 f.!)26 6994^ 7061 7129 7197 72(54 7332 7400 o 7535 7(503 7670 7738 7806 7873 7941 8008 8076 8143 68 3 8211 8279 8346 8414 8481 8549 861(5 8(584 8751 8818 67 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 ."5 9560 9627 9HH4 9762 9829 9896 f)l)(54 810031 810098 8101(5S 67 6 810233 810300 810367 810434 810501 810569 810(536 0703 0770 0837 67 7 0904 0971 1039 110(5 1173 1240 1307 1374 1441 1508 (57 8 J 575 1642 1709 1776 1«43 19 "0 l;/77 2044 2111 2178 67 9 2245 2312 2379 2445 2512 2579 2(54(5 2713 2780 2847 67 650 8] 29 13 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 i 3581 3648 3714 3781 3848 3914 39«1 4048 4114 4181 67 2 4J48 4314 4381 4447 4514 4581 4647 i-lW 4780 4847 67 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 66 4 5578 5644 5711 5777 5843 5910 5976 6042 6109 6175 66 5 6241 6308 6374 GMO 6503 6573 6G39 6705 6771 G838 6(5 6 0904 0970 7036 7102 7169 7235 7301 7367 7433 7499 66 7 75G5 7631 7698 7764 7830 7896 79G2 80;:8 8094 8160 66 8 8226 8292 83.^8 8^:^ 8490 8556 8622 8688 8754 8820 66 9 8885 8951 9917 9083 9149 9215 9281 9346 9412 9478 66 660 819544 8I9G10 8IC676 819741 819807 819873 819939 820004 820070 820136 06 1 820201 820267 820333 820399 820464 820530 820595 0661 6727 0792 66 2 0858 09-34 09!^9 105,3 1120 1186 1251 1317 1382 1443 66 3 1514 1579 1645 1710 1775 1841 1906 1972 2337 2103 05 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 65 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 6 3474 3539 3i5i:.-, 3i570 3735 3800 3335 393,) 3996 4061 65 7 4126 4191 425'5 4321 4386 4451 4516 4581 4646 4711 65 8 4776 4841 49!)G 4971 503G 5101 51G6 3231 5296 5361 65 9 5426 5491 555t5 5621 5686 5751 5815 5880 5945 6010 65 670 826075 826140 826204 826269 826334 820399 826464 826528 826593 826658 65 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 65 2 7369 7434 7499 75G3 7G28 7692 7757 7821 7886 7951 65 3 8015 8080 8144 8209 8273 8338 8-102 8467 8531 8595 64 4 8660 8724 8789 8853 8918 8982 9346 9111 9175 9239 64 5 9304 ur.np 9:'^2 9497 9561 9625 9GJ0 9754 9818 9882 64 6 9947 83001! 830075 830139 830204 83026a 83u332 83039(5 830460 830525 64 7 830589 o(i:.3 (:717 C781 0845 0909 6973 1037 1102 1166 64 8 1230 1294 1358 1422 1480 1550 1614 1678 1742 1806 64 9 1870 19;!4 1998 2062 2126 2189 2253 2317 2381 2445 64 680 832509 83257:; 8;!2;^57 832700 832764 832828 832892 832956 833920 833083 64 J 3147 3211 3:75 3338 3402 3466 3530 3593 3657 3721 64 2 3764 3848 3i)12 3975 4039 4103 4166 4230 4294 4357 64 3 4421 4484 4548 4611 4675 4739 4802 4866 4929 4993 64 4 5056 512U 5183 5247 5310 5373 5437 5500 5564 5627 63 5 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 6 6324 6337 0451 6514 6577 6641 6704 6767 6830 6894 63 7 6957 7020 7083 7146 7210 7273 733(5 7399 7462 7525 63 8 7588 7652 7715 7841 7904 7967 8030 8093 8156 63 9 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 838849 S38912 838975 839038 839101 839164 839227 839289 839352 839415 63 1 9478 9541 9604 9(567 9729 9792 9855 9918 9981 840043 63 2 8401)6 84016!) 840232 840294 840357 840420 840482 840545 840(508 0671 63 3 0733 0793 0859 0921 0984 1046 1109 1172 1234 1297 63 4 1359 1422 1485 1547 1610 1672 1735 1797 1860 1922 63 5 1985 2047 2110 2172 2235 2297 2360 2422 2484 2.547 62 6 2609 2(572 2734 2796 2859 2921 2983 3046 3108 3170 62 7 3233 3295 3357 3420 3482 3544 3606 3(569 3731 3793 62 8 3855 3918 3980 4042 4104 4166 4229 4201 4353 4415 62 9 4477 4539 4601 4664 4726 4788 4850 4912 4974 5036 62 No. I O I 1 I 3 I 4 I 5 j 6 I 7 I 8 I 9 |D>£ LOGARITHMS OF NUlfBERS. 19] No. I I 3 I * I 8 I 9 I Diff. "700 845098 845160 845222 845284 845346 84.5408 845470 845532 845594 845656 62 ] 57J8 5780 5842 5904 5966 6028 6090 6151 6213 6275 62 2 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 3 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 4 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 5 8189 8251 8312 8374 8435 8497 8559 8620 8682 8743 62 6 8805 8866 8928 8980 9051 9112 9174 9235 9297 9358 61 '/ 9419 9481 9.542 9(504 9665 9720 9788 9849 9911 9972 61 8 850033 850005 850156 850217 850279 850340 850401 850462 850524 850585 61 9 064!) 0707 j 076^) ()83(» 0891 0952 1014 1075 1136 1197 61 710 851258 851320 851381 851442 851503 851564 851625 851686 851747 851809 61 1 1870 1931 1992 2053 2114 2175 2236 2297 2358 2419 61 2 2480 2541 2602 2663 0704 2785 2846 2907 2968 3029 61 3 3090 3150 3211 3272 3333 3394 3455 3516 3577 3637 61 4 3698 3759 3820 3881 3941 4002 4063 4124 4185 4245 61 5 4306 4367 4428 4488 4549 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 ^156 5216 5277 5337 5398 5459 61 7 5519 5580 5640 5701 5761 5822 5882 5943 6003 6064 61 8 611M 6185 6245 6306 6366 6427 6487 6548 6608 6668 60 9 6729 6789 6850 6910 6970 7031 7091 7152 7212 7272 60 "20 857332 857393 8574 j3 857513 357574 857534 857694 857755 857815 857875 60 1 7935 7995 8056 8116 8176 8236 8297 8357 8417 8477 60 2 8537 8597 8657 8718 8778 8838 8898 8958 9018 9078 60 3 9138 9198 9258 9318 •9379 9439 9499 9559 9619 9679 60 4 9739 97U9 860398 9859 9918 9978 860038 860098 860158 860218 860278 60 5 860338 800158 860518 860578 0637 0697 0757 0817 0877 60 6 0937 0996 1056 1116 1176 1236 1295 1355 1415 1475 60 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 60 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 60 9 2728 2787 2847 2906 2966 3025 3085 3144 3204 3263 60 730 863323 863382 663442 863501 863561 863620 863680 863739 863799 863858 59 i 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 59 2 4511 4570 4630 4r89 4748 4808 4867 4926 4985 5045 59 3 5104 5163 5222 5282 5341 54(10 5459 5519 5578 5637 59 4 5696 5755 5814 5874 5933 5992 6051 6J10 6169 6228 59 5 6287 6346 6405 6465 6524 6583 6642 6701 6760 6819 59 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 7467 7526 7585 7644 7703 7762 7821 7^80 7939 7998 59 8 8056 8115 8174 8233 8292 835U 8409 8468 8527 8586 59 9 8&44 8703 8762 8821 8879 8938 8997 9056 9114 9173 59 740 809232 8'J9290 869349 869408 869486 869525 869584 869642 869701 869760 59 1 9818 9877 9935 9994 870053 870111 870170 870228 870287 870345 59 •2 870404 8704O2 870521 370579 0638 0696 0755 0813 0872 0930 58 3 0989 IU47 1106 1164 1223 1281 1339 1398 1456 1515 58 4 1573 1631 1690 1748 1806 1865 li)23 1981 2040 2098 58 5 2lo6 2215 2273 2331 2389 2448 2506 2564 2622 2681 58 6 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 58 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 58 8 3902 3960 4018 4076 4134 4192 4250 4308 4366 4424 58 9 4482 4540 4598 4656 4714 477g 4830 4888 4945 5003 58 750 875061 875: 19 875177 875235 875293 875351 875409 875466 875524 875582 58 1 5640 5698 5756 5813 5871 5929 5987 6045 6102 6160 58 2 6218 6276 6333 6391 6449 6507 6564 6622 6680 6737 58 3 6795 6853 6010 6968 7026 7083 7141 7199 7256 7314 58 4 7371 7429 7487 7544 7602 7659 7717 7832 7889 58 5 7947 8004 8062 8119 8177 8234 8292 8349 8407 8464 57 6 8522 8579 8637 8694 8752 8809 8866 8924 8981 9039 57 7 9090 9153 9211 9268 9325 9383 9440 9497 9555 9612 57 8 9S09 9726 9784 9841 9898 9956 380013 880070 880127 880185] 57 9; 880-242 88C299I 8803561 8804131 880471 88052tf 0585 0642 06991 07561 57 N*lO|l|3|3|4:j5j I 7 I 8 i 9 |Di£ 192 LOGARITHMS OF NlUfBEBS, No 1 3 3 4 5 7 8 9 Dift 7fX) 880814 H8087] 880..28| 880985 881042 881099 881156 881213,881271 8813-28 ~;>7 1 1385 1442 149it 1556 1613 1670 17-27 1784 1841 1898 5' 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 3 2525 2581 2()38 2r>95 2752 28i)9 2866 2923 2980 3037 57 4 3093 3150 3207 3264 3321 3377 3434 3491 3548 3605 57 5 36(31 3718 3775 3832 3888 3945 4002 4059 4115 4172 57 «) 4-229 4285 4342 4399 4455 4512 4569 4625 4(i82 4739 57 7 47;).') 4852 4909 4965 5j22 5078 5135 5192 5243 5305 57 8 53()1 5418 5474 5531 558/ 5644 5700 5757 53i;{ 5870 57 9 5i)2t) 5983 6039 6096 6152 6209 6265 6321 6378 6434 56 770 886491 888547 886604 886tu;o 8S6716 886773 886829 886885 886942 886998 56 1 7054 71J1 7167 7223 7280 7331) 7392 7449 7505 7561 56 2 7617 7674 7730 778(i 7842 7898 7955 8011 8067 8123 56 3 8179 8236 8292 8348 8404 84()0 8516 8573 8629 8685 56 4 8741 8797 8853 8999 8965 9021 9077 9134 9190 9246 56 5 9302 9358 9414 9470 9526 9582 9633 9694 9750 9806 56 6 9862 9918 9974 890030 890086 890141 890197 890253 890309 890365 56 7 890421 890477 890533 0589 0(i45 0700 0756 0812 0868 0921 56 S 0980 1035 1091 1147 12:)3 1259 1314 1370 1426 1482 56 9 1537 1593 1649 1705 176U 181.; 1872 1928 1983 2039 56 rso 892095 892150 892296 892262 892317 89237:-: 892429 892484 892540 892595 56 1 2651 2707 2762 2818 2873 2J2J 29.-<5 3040 3096 3151 56 2 3207 3262 3318 3373 3429 3484 354;) 3595 3651 3706 56 3 3762 3dlV 3873 3928 3984 4039 40J1 4150 4205 4261 55 4 4316 4371 4427 4482 4533 4593 4648 4704 4759 4814 55 5 4870 4925 4980 5;i;j6 5091 5146 5201 5257 5312 5367 55 6 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 55 7 5975 6U30 6085 6140 6195 6251 6306 6361 6416 6471 55 8 6526 658.1 6636 6692 6747 6802 6857 6912 6967 7022 55 9 7077 7132 7187 7212 7297 7352 7407 7462 7517 7572 55 790 897627 897682 897737 897792 897847 897902 897957 898012 898067 898122 55 1 8176 8231 8286 8341 8396 8451 850() 8561 8()15 8670 55 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 55 3 9273 9328 9383 9437 9492 9547 9602 965() 9711 9766 55 4 9821 9875 9930 9985 900039 900094 900149 900203 900258 900312 55 J 900367 900422 90U476 900531 0586 0640 0695 0749 0804 0859 55 6 0913 0968 1022 1077 1131 118;; 1240 1295 1349 1404 55 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 54 8 2003 2057 2112 2166 2221 2-275 2329 2384 2438 2492 54 9 2547 2601 2655 2710 2764 2818 2873 2927 2981 3036 54 800 903090 9J3I44 903199 903253 903307 903361 903416 903470 903524 903578 54 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 54 2 4174 4229 4283 4337 4391 4445 44 9; > 4553 4607 4661 54 3 4716 477.J 4824 4878 4932 4986 5040 5094 5148 5202 54 4 5256 5310 53(M 5418 5472 5526 5530 5634 5688 5742 54 5 5796 5850 5904 5958 6012 6066 6119 6173 0-227 6231 54 6 6335 6389 6443 6497 6551 6(i04 6658 6712 6766 6820 54 7 6874 6927 6931 7035 7089 7143 7196 7250 7304 7358 54 8 7411 74G5 7519 7573 7621) 7680 7734 7787 7841 7895 51 9 7949 8002 8056 8110 8103 8217 8270 8324 8378 8431 54 810 908485 908539 908592 908646 908699 908753 908807 908860 908914 908967 54 1 9021 9i)74 9128 9181 9235 9289 9342 9396 9449 9503 54 2 955() yijio 9663 971 ;; 9770 9823 9877 9930 9984 910037 53 3 910U91 910144 910197 910251 910304 910358 910411 910464 910518 0571 53 4 0624 0673 0731 0784 0838 0891 0944 0998 1051 1104 53 5 1158 1211 12(i4 1317 1371 1424 1477 1530 1584 1637 53 6 1690 1743 1797 1850 1903 1956 2009 2963 2116 21G9 53 7 2222 2275 2328 2381 2435 2488 2541 2594 2647 2700 53 8 2753 2806 2859 2913 2966 3019 3072 31-25 3178 3231 53 9 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 3|4|5|6|7j8j9JDiff %1 I-OGAIUTIUIS OF XUMBERS. 193 Na I I 1 3 I 3 I 4 7 I 8 9 I 820 913814 913867 913920 913973 914026 914079 914l32i914184 914237 914290 53 1 4343 4396 4449 4502 4555 4608 4660 4713 4766 4819 53 2 4872 4925 4977 5030 5083 5136 5189 5241 5294 5347 53 3 5400 5453 5505 5558 5611 5664 5716 5769 5822 5875 53 4 5927 59S0 6033 6085 6138 6)91 6243 6296) 6349 6401 53 5 6454 6507 6559 6612 6664 6717 6770 6822 6875 6927 53 6 . 6980 7033 7085 7338 7190 7243 7295 7348 7400 7453 53 7 7506 75.58 7611 7663 7716 7768 7820 7873 7925 7978 52 8 8030 8083 8135 8188 8240 8293 8345 8397 8450 8502 52 9 8555 8607 8t;59 8712 87(14 8816 8869 8921 8973 9026 52 830 J9 19078 919130 919183 919235 919287 919340 919392 919444 919496 913549 52 1 9601 9653 97(16 9758 9810 9862 9914 9967 920019 920071 52 2 920123 920176 92U228 92l»-if'0 920332 520384 9204361920489 054 1 0593 52 3 0645 0697 074'.l 08U1 0853 0906 09581 1010 1062 1114 52 4 1166 1218 1270 1322 1374 1426 1478 1530 1582 1634 52 5 1686 1738 1790 1842 1894 1946 1998 2050 2102 2154 52 6 2206 2258 2310 2362 2414 2466 2516 2570 2622 2674 52 7 2725 2777 2829; 2881 2933 2985 3037 3089 3140 3192 52 8 3244 3296 3348 3399 3451 3503 3555 3607 3658 3710 52 9 3762 3814 381)5 3917 3969 4021 4072 4124 4176 4228 52 840 924279 924331 924383 924434 924486:924538 924589 92464! 924693 924744 52 1 4796 4848 4899 4951 5003 5(S54 5106 5157 5209 5261 52 2 5312 5364 5415 5467 5518 5570 1 5621 5673 5725 5776 52 3 5828 5879 5931 5982 6034 6085; 6137 6188 6240 6291 51 4 6342 6394 6445 6497 6548 6600 6651 6702 6754 6805 51 5 6857 6908 6959 7011 7062 7114 7165 7216 7268 7319 51 6 7370 7422 7473 7524 7576 7627 7678 7730 7781 7832 51 7 7883 7935 7986 8037 8088 8140 8191 1 8242 8293 8345 51 8 8396 8447 8498 8549 8601 8052 87031 8754 8805 8857 51 9 8908 8959 9010 9061 0112 9163 9215 9266 9317 9368 51 850 929419 929470 929521 929572 929623^929674 929725 929776 929827 929879 51 1 9930 9981 930032 930083 930134 930185 930236 930287 930338 930389 51 2 930440 930491 0542 0592 0643 j 0694 0745 0796 0847 0898 51 3 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 51 4 145S 1509 1560 1610 1661 1712 1763 1814 1805 1915 51 5 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 51 6 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 51 7 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 51 8 3487 3538 3589 3639 3690 3740 3791 3841 3892 3943 51 9 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 51 860 934498 934549 934599 934650 934700 934751 934801 934852 934902 934953 50 1 5003 5054 5104 5154 5205! 5255 5306 5356 5406 5457 50 2 5507 5558 5608 5658 57091 5759 5809 5860 5910 5960 50 3 6011 0061 6111 6162 62l2i 6262 6313 6363 6413 6463 50 4 6514 6564 0614 6665 6715 6765 6815 6865 6916 6966 50 5 7016 7066 7117 7217 7267 7317 7367 7418 7468 50 6 7518 7568 7618 7668 7718 7769 7819 7869 7919 7969 50 7 8019 8069 8119 8169 8219! 8269 8320 8370 8420 8470 50 8 8520 8570 8620 8670 87201 8770 8820 8870 8920 8970 50 9 9020 9070 9120 9170 92201 9270 9320 9369 9419 9469 50 870 939519 939569 939619 939669 939719*939769 839819 939869 939918 939968 50 1 940018 940068 9401 18 94016S 940218 940267 940317 940307 940417 940467 50 2 0516 0566 06J6 0666 0716 0765 0815 0865 0915 0964 50 3 10141 1064 1114 1163 1213 1263 1313 1362 1412 1462 50 4 1511 1561 1611 1660 1710 1760 1809 1S59 1909 1958 50 5 2008 2058 2107 2157 2207' 225() 2306 2355 2405 2455 50 6 2504 2554 2603 2653 2702 2752 280 1 2851 2901 2950 50 7 3000 3049 3009 3148 3198 3247 3297 3346 3396 3445 49 8 3495 3544 3593 3643 3692 3742 379! 3841 3890 3939 49 9i 39d9 4038 4088 4137 4186 4236 4285 4335 4384 4433 49 Np, I I I j J§ I 3 1 4 I 5 i 7 I 8 9 'D^ 194 LOGARITHMS OF NUMBERS. Na I O I 1 I 3 * I I 7 I 8 I 9 iDiff 880 1 2 3 4 5 6 7 8 9 890 1 2 3 4 5 6 7 8 9 900 1 2 3 4 5 910 1 2 3 4 5 6 7 8 9 9*20 930 944483 944532 944581 944631 944680 944729 944779 944828 4976 5025 5074 5124 5173 5222 5272 5321 5469 5518 5567 5616 5665 5715 5764 5813 5961 6010 6059 6108 6157 6207 6256 6305 6452 6501 6551 6600 6649 6698 0747 6796 6943 6992 7041 7090 7140 7189 7238 7287 7434 7483 7532 7581 7630 7679 7728 7777 7924 7973 8022 8070 8119 8168 8217 8266 8413 8462 8511 8560 8609 8657 8706 8755 8902 8951 8999 9048 9097 9146 9195 9244 949390 949439 949488 949536 949585 949634 949083 949731 9878 9926 9975 950024 950073 950121 950170 950219 950365 950414 950462 0511 0560 0608 0657 0706 0851 0900 0941, 0997 1046 1095 1143 1192 1338 1386 1435 1483 1532 1580 1629 1(577 1823 1872 1920 1969 2017 2066 2114 2163 2308 2356 2405 2453 2502 2550 2599 2647 2792 2841 2889 2938 2986 3(J34 3083 3131 3276 3325 3373 3421 3470 3518 3566 3615 3760 3808 3856 3905 3953 4001 4049 4098 954243 954291 954339 954387 954435 954484 954532 954580 4725 4773 4821 4869 4918 496(3 5014 5062 5207 5255 5303 5351 5399 5447 5495 5543 5688 5736 5784 5832 5880 5928 5976 6024 6168 6216 6265 6313 6361 6409 6457 6505 6C49 6697 6745 6793 6840 6888 6936 6984 7128 7176 7224 72-2 7^2.y 73oe 7416 7464 7607 7655 7703 7751 7799 7847 7894 7942 8086 8134 8181 8229 8277 8325 8373 8421 8564 8612 8659 8707 8755 8803 8850 8898 959041 959089 959137 959185 959232 95J280 959328 959375 9518 9566 9614 9661 9-.rJ 9757 9lii3 ubuJ UWj 0756 U«(M 0946 0:1114 1U41 1089 1136 1184 123 J 1279 1421 1469 1516 1563 Kill 1658 170() 1753 1895 1943 1990 2038 2085 2132 2180 2227 2369 '2417 2464 2511 2559 2606 2653 2701 2843 289U 2937 2985 3032 3U79 3126 3174 3316 3363 3410 3457 3504 3552 3599 364() 963788 963835 963882 963929 963977 964024 964071 964118 426U 4307 4354 4401 4448 4495 4542 459.* 4731 4778 4825 4872 4919 49()6 5013 506) 5202 5249 5296 5343 5390 5437 5484 5531 5672 5719 5766 5813 5860 5907 5954 (iOOi 6142 6189 6236 6283 6329 6376 6423 (3470 6611 6658 6705 6752 6799 6845 6892 6939 7080 7127 7173 7220 72; ,7 73J4 7361 7408 7548 7595 7642 7688 7735 7782 7H29 7875 8U1() 8002 8109 8156 8203 8249 8296 8343 968483 968530 968576 968623 968670 968716 968763 968810 8950 8996 9043 9090 9136 9183 9229 9276 9416 9463 9509 955t) 9()02 9649 9695 9742 9882 9928 9975 970021 970068 970114 970161 970207 970347 970393 970440 048() 05.J3 '0579 0(')2() 0(572 0812 0858 0904 0951 09ii7 1044 1090 1137 1276 1322 1369 14J5 1461 1508 1554 1601 1740 1786 1832 1879 1925 1971 2018 20(;4 2203 2249 2295 2342 2388 2434 2481 2527 2666 2712 2758 28(M 2851 2897 2943 2989 944877 1944927 5370 58(32 6354 6845 7336 7826 8315 8804 9292 949780 950267 0754 1240 1726 2211 2696 3180 3363 4146 954628 5J10 5592 6072 6553 7032 7512 7990 8468 8946 959423 9900 96i'376 08." 1] 1326 1801 2275 2748 3221 3f);13 964165 4(i37 5108 5578 6048 6517 6986 7454 7922 8390 968856 9323 9789 970254 0719 1183 1647 2110 2573 3035 5419 5912 6403 6894 7385 7875 8364 8853 9341 949829 950316 0803 1289 1775 2260 2744 3228 3711 4194 954677 5158 5640 6120 6601 7080 7559 8038 8516 8994 959471 9947 960423 0899 1374 1848 2322 2795 3268 3741 964212 4684 5155 5625 6095 ()5(J4 7033 7501 79()9 843(3 968903 93(>9 9835 970300 0765 1229 2157 21) 19 3082' |0|l|»i3)4:|d|617|8|9|Oiff LOGARITHMS OF XUJIBERS. I 3 I 3 I 4 I 5 6 19-1 I 9 iDtfl 940 973128 973174 973220 973266 973313 973359 973405 97345! 973497 973543 46 1 3590 3636 3()82 3728 3774 3820 3866 3913 3959 4005 46 2 4051 4097 4143 4189 4235 4281 4327 4374 4420 4466 46 3 ' 4512 4558 4004 4650 4696 4742 4788 4834 4880 4926 46 4 4972 5018 5(J04 5110 515e 5202 5248 5294 5340 5386 46 5 543i> 5478 5524 5570 5616 5662 5707 5753 5799 5845 46 6 5891 5937 5983 0029 6075 6121 0107 6212 6258 ()304 46 7 6350 6396 6442 6488 6533 6579 6625 6671 0717 6763 46 8 6808 6H54 6900 6946 6992 7037 7083 7129 7175 7220 40 y 7260 7312 7358 7403 7449 7495 7541 7586 7032 7678 46 gsu 977724 977769 977815 977801 977900 977952 977998 978043 978089 978 J 35 46 1 8181 8226 8272 8317 8363 8409 8454 8500 8540 8591 46 2 8637 8683 8728 8774 8819 88()5 8911 8J50 9002 9047 46 3 9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 4 9548 9594 9039 9085 9730 9770 9821 9867 9912 9958 46 5 980003 980049 980094 980140 980185 980231 980270 980322 980307 980412 45 fi 0458 0503 0549 0594 (JG40 0085 0730 0770 0821 0867 45 7 0912 0957 1003 1048 1093 1139 1184 1229 1275 1320 45 8 1300 1411 1450 1501 1547 1592 1637 I6s:i 1728 1773 45 9 1819 1804 1909 1954 2000 2045 2090 2135 2181 2226 45 960 982271 982310 982302 982407 982452 982497 982543 982588 982()33 982678 45 1 2723 2709 2814 2859 2904 2949 29!:4i 3040 3085 3130 45 2 3175 3220 3205 3310 3350 3401 34;0| 3491 3536 3581 45 3 3{>20 3(i71 3716 3702 3807 3852 38j71 3942 2M1 4032 45 4 4077 4122 4107 4212 4257 4302 4347 4392 4437 4482 45 5 4527 4572 4017 4002 4707 4752 4797 4842 4887 4932 45 G 4977 5022 5007 5112 5157 5202 5247 5292 5337 5382 45 7 5420 5471 5510 5501 5000 5051 5090 5741 5780 5830 45 8 5875 592(» 5905 0010 0055 6100 6144 6189 6234 6279 45 9| 0324 6:509 0413 6458 6503 6548 0593 6637 6682 6727 45 970 980772 980817 980801 986906 986951 986996 987040 987085 987130 987175 45 1 7219 7204 7309 7353 7398 7443 7488 7532 7577 7622 45 2 7000 7711 7750 7800 7845 7890 7934 7979 8024 8068 45 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 45 41 8559 8G04 8048 8693 8737 8782 8826 8871 8910 8960 45 5| 9005 9049 9094 9138 9183 9227 9272 9310 9361 9405 45 945U 9494 9539 9583 9028 9672 9717 9701 9806 9850 44 7 9895 9939 9983 990028 990072 990117 990161 990200 990250 990294 44 8 990339 990;i83 990428 0472 0510 0501 0005! OboO 0094 0738 44 9 0783 0c^27 0871 0916 0900 1004 1049 1093 1137 1182 44 980 99122G 991270 991315 991359 991403 991448 991492 991536 991580 991625 44 1 1009 IT 13 1758 1802 1840 189U 1935 1979 2023 2067 44 % 2111 2J50 2200 2244 2288 2333 2377 2421 2465 2509 44 3| 2554 2598 2042 2086 2730 2774 2819 2803 2907 2951 44 4 '2995 3(J39 3083 3127 3172 3210 3300 3304 3348 3392 44 5 3436 3480 3524 3568 3613 3657 3701 3745 3789 383:i 44 6 3877 3921 3965 4009 4053 4097 414J 4185 4229 4273 44 7 4317 4301 4405 4449 4493 4537 4581 4025 4009 4713 44 8 4757 48U1 4845 4889 4933 4977 5021 5005 5108 5152 44 9j 5190 5240 52t4 5328 5372 5416 5400 5504 5547 5591 44 990 995635 995(;79 995723 995767 995811 (,95854 995898 995942 995980 990030 44 1 6074 0J17 0101 6205 0249 6293 0337 0380 6424 041 KM 44 2 6512 0555 6599 6643 6087 6731 0774 0818 0862 09U(i 44 3 6949 6993 7037 7080 7124 7108 72 J 2 7255 7299 7343 44 4 7386 7430 7474 7517 7561 7005 7048 1 7692 7736 7779 44 5 7823 7807 7910 7954 7998 8041 8085 8129 8i72 8210 44 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 805-_> 44 7 8695 8739 8:82 8820 8869 8913 8950 9000 9043 9087 44 8 9131 9174 9218 920J 9305 934H 9392 9435 9479 9522 44 _9 9565 9009 9652 9696 9739 9783 9826 9870 9913 9957 4J No. I O I 1 I a I 3 5 I 6 j ? I 8 » 9 IDjS TABLE XIII. LOGARITHMIC SINES, C@SINES, TANGENTS, AND COTANGENTS. N. B. — The minutes in the left-hand column of each page, increasing downwards, belong to the degrees at the top ; and those increasing upwards, in the right-hand column, helone to the degrees below. In using the differences for one second, in columns D, the two right-hand figures should be marked off as decimals. Thus the difference for log, sin. 1° 12' 5" would be 99.83 X 5 = 499.1, additive to the mantissa .321027 treated as an integer, and the difference for log. cos. 8° 30' 50" would be 0.30 X 50 = 16.0 subtractive from the mantissa .995203 treated us an integer. The differences in columns D range opposite the upper one of the two functions to which they respectively apply. The first column D refers to Sines, the second to Cosines, the third to both Tangents and Cotangents. V>' ICS (0]ic,LcrcfO LOGAEITHMIC ZJNES, COSIXKS. ETC. M. Sine D. Cosine ! D. 1 Tan?. 0-000000 D. Cotansr. Inttniie. 60 (1 Inf. Ne;,'. 1 0-000000 I (i-4«>3726 501717 000000 00 6-463726 501717 13-536274 59 2 704756 293485 000000 00 764756 293485 235^44 58 :i 940847 208231 000000 00 940847 208231 059153 .57 4 7-()65786 J615I7 000000 00 7-0(55786 161517 12-934214 .56 ,") 162696 131968 000000 00 Iti2696 131969 837304 55 () 241877 1 1 1578 9-999999 01 241878 111578 758122 54 7 308824 96(i53 999999 01 308825 99653 691175 .53 8 366816 85254 999999 01 366817 85254 &33I83 52 9 417968 76263 999i)99 01 417970 762(53 582030 51 10 463725 68988 999998 01 463727 68988 536273 .50 > 11 7505] 18 62981 9-999998 01 7-505120 6298; i2-494H80 49. ' 12 542906 57936 999997 01 542909 57938 457091 48 13 577668 53641 999997 01 577672 53642 4-2-2:i->8 47 14 609853 49938 999996 01 609857 49939 390143 46 15 639816 A^' !4 999996 01 639820 46715 3(;()180 45 If) 667845 -)3H8J 999995 01 6(57819 43882 332151 44 17 694173 41372 999995 01 694179 41373 305821 43 18 718997 39135 999994 0! 7!9(M.'3 39136 280it97 42 19 742477 37127 999993 01, 742484 37128 257516 41 '^0 7(M754 35315 999993 01 764761 35317 235239 40 21 7 785943 33672 9-999992 01 7-785951 33673 12-214049 39 22 806146 32175 999991 01 806155 32176 193845 38 23 825451 30805 999990 01 825460 30806 174540 37 24 843934 29547 9999P9 02 843944 29549 156056 36 25 861662 28388 999988 02 861674 28390 138326 35 2() 878695 27317 999988 02 878708 27318 121292 .34 '}? 895085 2()323 999987 (12 895099 26325 104901 33 2t; 910879 25399 999986 02 910894 25401 0891(i6 32 29 926J19 24538 999985 02 926 Kll 24540 0738(5(5 31 30 940842 23733 999983 02 940858 23735 059142 30 31 7-955082 22980 9-999982 02 7-955100 22981 12044900 29 32 96H87() 22273 9999!^ 1 02 968889 22275 031111 28 33 982233 21(108 999980 02 982253 21610 017747 27 34 995198 2ilit8l 999979 02 995219 20983 004781 26 35 8-U07787 20390 999977 02 8-',|(l78(19 2(1392 11 ■992191 25 3C. 020021 19H3i 999!t76 02 (12(1045 19833 979955 24 37 (>3»'.)I9 19302 999975 02 o;U945 19305 9(58055 23 3H 043501 I8r'0l 999973 02 043527 18803 956473 22 39 05478 1 18:^25 999972 02 054809 18327 945191 21 40 065776 17872 999971 02 065806 17874 934194 20 41 8076500 17441 9-999969 02 8-07(553 1 17444 11-923469 19 42 086965 17031 999968 02 086997 17034 913003 18 43 097183 16639 999966 02 097217 1(5(542 902783 17 44 1(17167 16265 999964 03 107202 162(58 892797 ?6 45 116926 J 5908 999963 03 1 1(5963 15910 883037 J5 4G 126471 15566 999961 03 12(5510 15568 873490 14 47 135HI0 15238 999959 0.-. 135H5I 15241 8(54149 13 48 144953 14924 999958 (.3 144996 14927 855004 12 49 15391)7 1-1622 999956 03 153952 14627 846048 11 50 1626.'J| 1433;; 999954 03 162727 . 14336 837273 10 51 8l7i^2H() 14054 9-999952 03 8- 171328 14057 11-828(572 9 52 179713 13786 999950 03 179763 13790 820237 8 53 187985 13529 999948 03 188036 13532 811964 7 54 I'M;|fi2 13280 999946 03 196156 13284 803844 6 55 2iM07(» 13041 999944 03 204126 13044 795874 5 5() 211895 12810 999942 04 211953 12?3I0(( 2332 998243 19 954^^5(5 2351 045144 51 in 954499 2325 998232 19 956267 2344 043733 53 11 8-95")894 2317 9-998220 19 8-957674 2337 11042326 49 12 957284 2310 998209 19 959075 2;}29 040925 48 i:j 958(570 2302 998197 19 960473 2323 039527 47 14 9601152 2295 998186 19 961 S66 2314 038 K54 46 15 961429 2288 998174 19 963255 2307 036745 45 Ifi 9(>2-'i)l 2280 998163 19 9(54639 2300 0353(51 44 17 964170 2273 998151 19 966019 2293 033981 43 18 9(55534 2266 998139 20 967394 2286 032606 42 Vi 906893 2259 998128 20 968766 2279 031234 41 2(1 968249 2252 998116 20 970133 2271 029867 40 2) 81)69600 2244 9-998104 20 8-971496 2265 11028504 39 22 970947 2238 998092 20 972855 2257 027145 38 2:» 972289 2-231 998080 20 974209 2251 025791 37 ^ 973628 2224 998068 20 975560 2244 024440 36 2.-. 974962 2217 998056 20 976906 2237 023094 35 2 989374 2144 997922 21 991451 2165 008549 24 37 990660 2138 997910 21 992750 2158 007250 23 :<8 991943 2131 997897 21 994045 2152 005955 22 39 993222 2125 997885 21 995337 2146 00461)3 2I 4U 994497 2119 997872 21 9966-24 2140 003376 20 41 8-995768 2112 9-997860 21 8-997908 2134 11-002092 19 42 997036 2106 997847 21 999188 2127 000812 18 43 998299 2100 997835 21 9-000465 2121 10-999535 17 44 999560 2094 997822 21 001738 2115 998262 16 45 9-000816 2087 997809 21 003007 2109 996993 15 46 002069 2082 997797 21 004272 2103 995728 14 47 003318 2076 ttc)77«4 21 005.^34 2097 994466 13 48 004563 2070 997771 21 006792 2091 993208 12 49 005805 2064 997758 21 008047 2085 991953 11 50 007041 2058 997745 21 009298 2080 990702 10 51 9-008278 2052 9-997732 21 9 -01 0546 2074 10-989454 9 52 009510 2046 997719 21 01J7i»0 2068 988210 8 53 010737 2040 997706 21 013031 2062 986969 7 54 011962 2034 997693 22 014-268 2056 985732 6 55 013182 2029 997680 22 015502 2051 984498 5 56 014400 20-23 997667 22 016732 2045 983268 4 57 015613 2017 997654 22 017959 2040 982041 3 58 016824 •2012 997641 22 019183 2033 980817 2 59 018031 2006 997628 22 020403 2028 979597 1 60_ 019235 2000 997614 22 021620 2023 978380 Cnnine Sine 84 Degre 1 Cotang. es. ' 1 Tang. 1 M. 204 (6 Degrees.) LOGARITILMIC iSINES, COSINES, ETC. H. 1 Sine 1 D. 1 Cosine I D. 1 Tang. 1 D. Coia.i^. 9-019235 2000 9997614 22 9-021620 • 2023 10-978380 ; 60 1 020435 1995 997601 22 0-2-2834 2017 977 KiO :>9 2 0'21ti:i2 1989 9i 17588 22 024044 2011 975956 58 3 02-2H^25 1984 997574 22 0-25251 2006 974749 57 4 <)'24(»I6 1978 997561 22 026455 2.000 973545 56 a 0-25203 1973 997547 22 027 (i55 1995 972345 55 6 ()26:W6 1967 997534 23 028852 1990 971148 54 7 027567 1962 997520 23 030046 1985 969954 53 J i<->i744 1957 997507 23 031-237 1979 9()8763 52 9 (i-Jj Sme Degre Cotang. es. I Tang. I k LOGARITHMIC SINES, CO SIXES, ETC. (7 Degrees.; M 1 Sine 1 ^• 1 Cosine 1 D. 1 Tan?. 1 D. 1 Cotan^ 1 ~0 9-085894 1713 9-996751 26 9-089144 1738 10-910856 60 1 ((H(iil-22 1709 <)96735 26 090187 1734 909813 59 2 087947 1704 996720 26 091228 1730 908772 58 3 088970 1700 99()704 26 092266 1727 907734 .57 4 089990 1696 996688 26 093302 1722 'J()(S(!98 56 5 091(108 1()92 99667:5 26 094336 1719 9()5()64 55 6 092024 ](i88 996657 26 095367 1715 904633 .54 7 0930S7 lii84 99()(»41 26 096395 1711 903605 53 8 094047 Jfi80 99()625 96 097422 1707 902578 52 9 0950.')() l(i76 996610 26 098446 1703 901554 51 10 09(i0()2 1(573 996594 26 099468 1699 900532 50 11 9-097(inr) KiCS 9-99657S o- 9-J0(!487 1695 10-899513 49 1-2 O'jSOOli ]i;(i-) !)!)(;.">( 12 07 101504 1691 898496 48 13 0990(55 ]ii(ii i (ill ;.-,-( 6 27 102519 1687 897481 47 14 1000()2 Hi. -.7 !t!l(i.");<0 27 103532 1684 896468 46 15 10105(> i(;:)3 H'.Ki.'iU "7 104542 I68ii 895458 45 16 102048 l(i49 it! MM! (8 27 105550 1676 894450 44 17 103037 l(i45 9!l(;-182 27 J 06556 1672 893444 43 ■18 104025 ](i41 9!Mi4(i5 27 I07o59 i(;69 2 195925 1323 994580 33 201345 135() 798655 5" 3 196719 1321 994560 34 202159 1354 797841 57 4 197511 1318 994540 34 202971 1352 797029 5fi 5 198302 1310 994519 34 203782 1349 79(:218 55 6 199091 1313 994499 34 204592 1347 795408 54 7 199879 1311 994479 34 205400 1345 794(;00 53 8 200(H)f) 1308 994459 34 206207 1342 793793 52 9 201451 1306 994438 34 2070 IS 1340 792987 51 10 202234 1304 994418 34 207817 1338 792183 5«:' ]l 9-203017 1301 9-994397 34 9-208619 1335 10-791381 49 li 203797 1299 994377 34 209420 1333 790580 48 i:i 204577 1296 994357 34 210220 1331 789780 47 14 205354 1294 99433() 34 211018 1328 788982 46 15 20G131 J 292 994316 34 211815 1326 788185 45 l(i 206906 1289 994295 34 212611 1324 787389 44 17 207679 1287 994274 35 213405 1321 786595 43 18 2*ia452 1285 994254 35 214198 1319 ■; 85802 42 19 209222 1282 994233 35 214989 1317 785011 41 20 209992 1280 994212 35 215780 1315 784220 40 21 9-210760 1278 9-994191 35 9-216568 1312 10-783432 39 22 211526 1275 994171 35 217356 1310 782644 38 23 212291 1273 994150 35 218142 1308 781858 37 24 213055 1271 994129 35 218926 1305 781074 36 25 213818 1268 994108 35 219710 1303 780290 35 2(5 214579 1266 994087 35 220492 1301 779508 34 27 215338 1264 994066 35 221272 1299 778728 33 28 216097 1261 99^045 35 222052 1297 777948 32 29 216854 1259 994024 35 222830 1294 777170 31 30 217609 1257 994003 35 223606 1292 776394 30 31 9-218363 1255 9-993981 35 9-224382 1290 10-775618 29 32 219116 1253 993960 35 225156 1288 774844 28 33 219868 1250 993939 35 225929 1286 774071 27 34 220618 1248 993918 35 2-26700 1284 773300 26 35 221367 1246 993896 36 227471 1281 772529 25 30 222115 1244 993875 36 228239 1279 771761 24 37 2-?.-?S61 1242 9D3854 36 229007 1277 770993 23 38 223606 1239 993832 36 229773 1275 770227 22 39 224349 1237 993811 36 230539 1273 769461 21 40 225092 1235 993789 36 231302 1271 768698 20 41 9-225833 1233 9-993768 36 9-232065 1269 10-707935 19 42 22G573 1231 993746 36 232826 lt:67 767174 18 43 227311 1228 993725 ;;(> 233586 l-i65 766414 17 44 228048 1226 993703 36 2::4;;45 1262 765655 16 45 228784 1224 993681 36 2;-r>i;;3 1260 764897 15 46 229518 2020 993660 3ii 2;:5t59 1258 764141 11 47 230252 1220 993638 36 2::6(;i4 1256 763386 13 48 230984 1218 993616 36 2:;73G8 1254 762632 12 49 231714 12i() 993594 !i7 2:;8i2u 1252 761889 11 50 232444 1>J4 993572 37 238872 l-:50 761128 10 51 9-233172 1212 9-993550 37 '.)-2n:ir,22 1248 10-760378 9 52 233899 ■ 1209 9935-28 37 24u::7i 124(i 759629 8 53 234625 1207 993506 37 241118 m4- 758882 7 54 235349 1205 993484 37 241865 1242 758135 e 55 236073 1203 993462 37 2426W 1240 757390 5 5{) 236795 1201 993440 37 243354 1238 756646 4 57 237515 1199 993418 37 244097 1236 755903 3 5>) 23e2;!5 1197 993396 37 244839 1234 755161 2 59 238953 1195 993374 37 ^5579 1232 754421 1 i GU 239670 1193 993351 37 246319 1230 753681 Cosine ! 1 Sme 1 80 Degre Cotaii^. 3S. Tang. 1 M. 208 (10 D<'grees.) LOGARITHMIC SINES, COSINES, ETC. M. 1 Sine i D. 1 Cosine I D. f Tan^. 1 D. 1 Cotang. o' 9-239670 1193 9-993351 37 9-246319 1230 10-75G681 60 1 240386 1191 993329 37 247057 1228 752943 59 2 241101 1189 993307 37 247794 1-226 752206 58 3 241814 1187 993285 37 248530 1224 751470 57 4 242526 1185 993262 37 249264 1222 750736 56 5 243237 1183 993240 37 249998 ]2-2d 750002 55 6 243947 J 181 993217 38 250730 1218 749270 54 7 244656 1179 993195 38 251461 1217 748539 53 8 245363 1177 993172 38 252191 1215 747809 52 9 246069 1175 993149 38 252920 1213 747080 51 10 246775. 1173 993127 38 253648 1211 746352 50 11 9-247478 1171 9-993104 38 9-254374 1209 10-745G26 49 12 248181 IIGU 993081 38 255100 1207 744J00 48 13 248883 1107 993059 38 255824 1205 744176 47 14 249583 1165 993036 38 256547 1203 743453 46 15 230282 1163 993013 38 257269 1201 742731 45 16 250980 1161 992990 38 257990 1200 742010 44 17 251677 1159 992967 38 258710 1198 741290 43 18 252373 1158 992944 38 259429 1196 740571 42 19 253067 1156 992921 38 260146 1194 739854 41 20 253761 1154 992898 38 260863 1192 739137 40 21 9-254453 1152 9-992875 38 9-261578 1190 10-738422 39 22 255144 1150 99-2852 38 262292 1189 737708 38 23 255834 1148 992829 39 263005 1187 736995 37 24 256523 1146 992806 39 2G3717 1185 73G283 36 25 257211 1144 992783 39 2G4428 1183 735572 35 26 257898 1142 992759 39 2G5138 1181 734862 34 27 258583 1141 992736 39 265847 1179 734153 33 28 259268 1139 992713 39 26o555 1178 733445 32 29 259951 1137 992690 39 2672G1 117G 732739 31 30 260633 1135 992606 39 2G7967 1174 732033 30 31 9-261314 1133 9-992643 39 9-268671 1172 10-7313-29 29 32 261994 1131 992619 39 2G9375 1170 730625 28 33 262673 1130 992596 39 270077 11G9 729923 27 34 203351 1128 992572 39 270779 1167 729221 26 35 2G4027 1126 992549 39 271479 11G5 728521 25 36 264703 1124 992525 39 272178 1164 727822 24 37 265377 1122 992501 39 272876 1162 727124 23 38 2G6051 1120 992478 40 273573 IIGO 726427 S 39 266723 1119 992454 40 274269 1158 725731 21 40 267395 1117 992430 40 274964 1157 725036 20 41 9-268065 1J15 9-992406 40 9-275G58 1155 10-724342 19 42 268734 1113 992382 40 276351 1153 723649 18 43 2G9402 1111 992359 40 277043 1151 7t>2957 17 44 270069 1110 992335 40 277734 1150 722266 16 45 270735 1108 992311 40 278424 1148 721576 15 46 271400 1106 992287 40 279113 1147 720887 14 47 272064 1105 9922G3 40 279801 1145 720199 13 48 272726 1103 992239 40 280488 1143 719512 12 49 273388 1101 992214 40 281174 1141 718826 11 50 274049 1099 992190 40 281858 1140 718142 10 51 9-274708 1098 9-992166 40 9-282542 1138 10-717458 9 52 275367 1096 992142 40 283225 1136 716775 8 53 276024 1094 992117 283907 1135 716093 7 54 276681 1092 992093 284588 1133 715412 6 55 277337 1091 992069 2852G8 1131 714732 5 56 277991 1089 992044 285947 1130 714053 4 57 278644 1087 992020 286G24 1128 713376 58 279297 1086 991996 287301 1126 712699 "i 59 279948 1084 991971 287977 1125 712023 1 60 280599 1082 991947 288652 1123 711348 -9 1 Cosine 1 Sine 79 1 Degree Cotang. 1 1 Tang. 1 K. LOGARITHMIC 8INE8, COSINES, ETC. C^l Degrees.) 201' .M. 1 Sins D. Cosine D. Tang. D. I Cotang. 9-280599 1082 9-991947 41 9-288652 1123 10-711348 60 1 281248 1081 991922 41 289326 1122 710674 59 2 281897 1079 991897 41 289999 1120 710001 58 3 282544 1077 991873 41 290671 1118 709329 57 4 283190 1076 991848 41 291342 1117 708658 56 5 283836 1074 991823 41 292013 1115 707987 55 fi 2«4480 1072 991799 41 292682 1114 707318 54 - 2-S5I24 107 1 991774 42 293350 1112 706650 53 8 285766 1069 991749 42 294017 1111 705983 52 9 286408 1067 991724 42 294684 1109 70.5316 51 10 287048 1066 991099 42 295349 1107 ^04651 50 11 9-287687 1064 9-991674 42 9-296013 1106 10-703987 49 12 288326 1063 991649 42 296677 1104 703323 48 13 288964 1061 991624 - 42 297339 1103 702661 47 14 289600 1059 991599 42 298001 1101 701999 46 15 290236 1058 991574 42 298662 1100 7013.38 45 10 290870 1056 991549 42 299322 1098 700678 44 17 291504 1054 991524 42 299980 1096 700020 43 le 292137 1053 991498 42 300638 1095 699362 42 19 292768 1051 991473 42 301295 1093 698705 41 20 293399 1050 991448 42 30I95I 1092 698049 40 21 9-294029 1048 9-991422 42 9-302607 1090 10-697393 39 22 294658 1046 991397 42 303261 1089 696739 38 23 295286 1045 991372 43 303914 1087 696086 37 ai 295913 1043 991346 43 304567 1086 695433 36 25 296539 1042 991321 43 303218 1084 694782 35 26 297164 1040 9D12J5 43 305869 1083 694131 34 27 297788 1039 991270 43 306519 1081 693481 33 28 298412 1037 991244 43 3071 08 1080 69'2832 32 29 299034 1036 991218 43 307815 1078 692185 31 30 2i)9655 1034 991193 43 308463 1077 691537 30 31 9-300276 1032 9-991167 43 C-30Dlfir) 1075 10-690891 29 32 300895 1031 991141 43 309754 1074 690246 28 33 301514 1029 991115 43 310398 1973 689602 27 34 302132 1028 991090 43 311042 1071 688958 26 35 302748 1028 9910G4 43 311685 1070 688315 25 36 303364 1025 991038 43 312327 1068 687673 24 37 303979 1023 991012 43 312967 1067 687033 23 38 304593 1022 990986 43 313608 1065 686392 22 39 305207 1020 990960 43 314247 1064 685753 21 40 305819 1019 990934 44 314885 1062 685115 20 41 9-306430 1017 9-990908 44 9 315523 1061 10-684477 19 42 307041 1016 990882 44 3161.59 1060 683841 18 43 307650 1014 990855 44 316795 1058 683205 17 44 308259 1013 990829 44 317430 1057 682570 16 45 308867 1011 900803 44 318064 1055 681936 15 46 n;)9474 1010 990777 44 218697 1054 681303 14 47 310080 1008 9907.:0 44 319.329 10.33 680671 13 48 310685 1007 990724 44 319961 1051 680039 12 49 311289 1005 990697 44 320592 1050 679408 11 50 311893 1004 99(1671 44 321222 1048 678778 10 5J 9-212495 1003 9-990644 44 9-321851 1047 10-678149 9 52 31309/ 1001 9906J8 44 322479 1045 G77.521 8 53 313698 1000 990.591 44 3-231(i6 1044 676894 54 314297 998 990565 44 323733 1043 67G267 6 55 214897 997 99J538 44 ■.vl-2-.S 1041 675642 5 5«i 31.5495 996 990511 45 ^"24983 1040 675017 4 57 SI 6092 994 990485 45 3-25C^7 1039 674393 3 58 3Jf)6«9 993 990458 45 320231 1037 6727G9 2 59 317284 991 990431 45 326853 1036 673147 1 60 317879 990 990404 45 327475 1035 672525 Cosine I I Sine 1 78 Degre* Cotang. )8. ' Tang. 1 M. 210 (12 Ttri, rccs.) LOGARITHMIC SINES, COSLYES. ETC. M. Sine D. Cosine D. 1 Tang. 1 D. ColUMg. u" 9-317879 990 9-990404 45 9-327474 1035 10-672^26 60 1 318473 988 990378 45 328095 1033 671L05 59 2 319066 987 990351 45 328715 1032 671285 58 3 319658 986 990324 45 329334 1030 67066G 57 4 320249 984 990297 45 329953 1029 670047 56 5 320840 983 990270 45 330570 1028 669430 55 6 321430 982 990243 45 331187 1026 668813 54 I 7 322019 980 990215 45 331803 1025 668197 53 8 322607 979 990188 45 332418 1024 667582 52 9 323194 977 990161 45 333033 1023 666967 51 10 323780 976 990134 45 333646 1021 666354 50 11 9-3243()6 975 9-990107 40 9-334259 1020 10-605741 49 12 32495* ) 973 990079 40 334871 1019 665129 48 13 325534 972 990052 46 335482 1017 604518 47 J4 326117 970 990U25 40 336093 1016 6639b'7 40 J5 326700 969 989997 46 336702 1015 663298 45 1() 32728J 968 989970 46 33731] 1013 662689 44 17 3278(i2 96G 989942 40 337919 1012 6C2C81 43 18 328442 965 989915 46 338527 1011 661473 42 19 329021 964 989887 46 339133 1010 660867 41 20 329599 962 9S98G0 46 339739 1008 660261 40 21 9-330176 961 9-989832 46 9-340344 1007 [0-659656 39 22 330753 960 989804 40 340948 1006 659052 38 23 331329 958 989777 46 341552 1004 658448 37 24 331903 957 989749 47 342155 1003 657845 36 25 332478 956 989721 47 342757 1002 657243 35 20 333051 954 989693 47 343358 1000 656642 34 27 333624 953 989G65 47 343958 999 656042 33 28 334195 952 989037 47 344558 998 655442 32 29 334706 950 989609 47 345157 997 654843 31 30 335337 949 989582 47 345755 996 654245 30 31 9-335906 948 9-989553 47 9-346353 994 10-653647 29 32 336475 946 989525 47 346949 9:)3 653051 28 33 337043 945 989497 47 347545 992 652455 27 34 337610 944 9894Ci9 47 348141 991 651859 26 35 338176 943 989441 47 348735 990 651265 25 36 338742 941 989413 47 349329 988 650671 24 37 339306 940 989384 47 349922 987 650078 23 j 38 339871 939 989356 47 350514 986 649486 22 39 340434 937 98'.>328 47 35J106 985 648894 21 40 340996 936 989300 47 351697 983 648303 20 41 9-341558 935 9-989271 47 9-352287 982 10 647713 19 42 342119 934 989243 47 352876 981 647124 18 43 342(579 932 989214 47 353465 980 64(5535 17 44 343239 931 989186 47 354053 979 645947 16 45 343797 930 989157 47 354640 977 645360 15 46 344355 929 989128 48 355227 976 644773 14 47 344912 927 989100 48 355813 975 644187 13 48 345469 926 989071 43 356398 974 643602 12 49 346024 925 989042 48 356982 973 043018 U 50 346579 924 989014 48 357566 971 642434 10 51 9-347134 922 9-98ffii85 988956 48 9-358149 970 10-641851 9 52 347687 921 48 358731 969 641260 8 53 34824(1 920 988927 48 359313 968 (540(587 7 54 348792 919 988898 48 359893 967 (i4()I07 6 55 349343 917 988869 48 360474 966 639520 5 36 349893 916 988840 48 361053 965 (538947 4 57 350443 915 988811 49 361632 9(i3 63P3C.8 3 58 350992 914 988782 49 362210 962 637790 2 59 351540 913 988753 49 362787 961 637213 1 60 352088 911 988724 49 363364 960 636636 Cosine 1 Sine T 7 Degrt Cotang. »es. Tang. ; JVL LOQABITHMIG SINES, COSINES, ETC. (13 Degrees.) 211 M. 1 Sme 1 D. Cosine 1 D. Tan J. D. Cotang. 9-352088 911 9-988724 49 9-363364 960 -^0-636636 60 1 352635 910 988695 49 363940 959 636060 59 2 353181 909 988666 49 364515 958 635485 58 3 353726 908 988636 49 365090 957 634910 57 4 354271 907 988607 49 365664 955 634336 56 5 354815 905 988578 49 360237 954 633763 55 6 355358 904 988548 49 366810 953 633190 54 7 355901 903 988519 49 367382 952 632618 53 8 356443 902 988489 49 367953 951 632047 52 9 356984 901 988460 49 368524 950 631476 51 10 357524 899 988430 49 309094 949 630906 50 11 9-358064 898 9-988401 49 9-369663 948 10-630337 49 12 358603 897 988371 49 370232 946 629768 48 13 359J41 896 988342 49 370799 945 629201 47 14 35U()78 895 988312 50 371367 944 628633 46 15 360215 893 988282 50 371933 943 028067 45 16 360752 892 988252 50 372499 942 027501 44 17 361287 891 988223 50 373064 941 620936 43 18 361822 890 988193 50 373629 940 626371 42 19 362356 889 988163 50 374193 939 625807 41 20 3!)2889 888 988133 50 374756 938 625244 40 21 9-363422 887 9-988103 50 9-375319 937 10-624681 39 22 3()3954 885 988073 50 375881 935 624119 38 23 304485 884 988043 50 376442 934 623558 37 24 365016 883 988013 50 377003 933 622997 36 25 3(;5546 882 987983 50 377563 932 62-2437 35 26 366075 881 987953 50 378122 931 621878 34 27 366604 880 987922 50 378081 930 621319 33 28 367131 879 987892 50 379239 929 020761 32 29 3(;7n59 877 987862 50 379797 928 620203 31 30 368185 876 987832 51 380354 927 619646 30 31 9-368711 875 9-987801 51 9-380910 926 10-619090 29 32 369236 874 987771 51 381466 925 618534 28 33 369761 873 987740 51 382020 924 617980 27 34 370285 872 987710 51 382575 923 617425 26 35 370808 871 987679 51 383129 922 010871 25 36 371330 870 987649 51 383682 921 616318 24 37 371852 869 987618 51 384234 920 615766 23 38 372373 867 987588 51 384786 9J9 615214 22 39 372894 866 987557 51 385337 918 614663 21 40 373414 865 987526 51 385888 917 014112 20 41 9-373933 804 9-987496 51 9-3864:^8 915 10-013502 19 42 374452 863 987465 51 380987 914 013013 18 43 374970 862 987434 51 387536 913 012404 17 44 375487 861 987403 52 388084 912 611916 16 45 376003 860 987372 52 388631 911 61 1309 15 46 376519 859 987341 52 389178 910 010822 14 47 377035 858 987310 52 389724 909 610-276 13 48 377549 857 987279 52 390270 908 609730 12 49 378063 856 987248 52 •3908 15 907 609185 11 50 378577 854 987217 52 391360 906 608640 10 51 9-379089 853 9-987186 52 9-391903 905 10-608097 9 52 379601 852 987155 52 392447 904 607553 8 53 380113 851 987124 52 392989 903 607011 7 54 380624 850 987092 52 393531 902 606409 6 55 381134 849 987061 52 394073 901 005927 5 56 381643 848 987030 52 394614 900 605386 4 57 382152 847 986y98 52 395154 899 604846 3 58 382661 846 986967 52 395094 898 604306 2 59 383108 845 986936 52 396233 897 603767 I 60 383675 844 • 986904 52 396771 896 603229 I I Cotaug. 76 Degrees. Taug. I 212 (14 Degrees.) LOGARITHMIC SINES, COSINES, BTC I Tang. I D. I Cotang. | 9-383675 844 9-986904 52 9-396771 896 10603229 60 I 384182 843 986873 53 397309 896 602691 59 2 384087 842 986841 53 397346 895 002154 58 3 385 192 841 986809 53 398383 894 601617 57 4 385G97 840 986778 53 398919 SD3 601(,81 56 5 380231 ■^::9 980740 53 ;;;;j-^55 892 600545 55 G 38G704 b;>8 986714 53 399990 891 600010 54 7 387207 837 986683 53 400524 890 599476 53 .8 387709 836 986651 53 401058 889 598942 52 9 388210 835 980619 53 401591 888 598409 51 10 388711 834 986587 53 402124 887 597876 50 11 9-38921 1 833 9-986555 53 9-402050 886 10-597344 49 12 389711 832 986523 53 403187 885 596813 48 13 390210 831 986491 53 403718 884 596282 47 14 390708 830 980459 53 404249 883 595751 46 15 391206 828 980427 53 404778 882 595222 45 16 391703 827 980395 53 405308 881 594692 44 17 392199 826 986303 54 405836 880 594164 43 18 392095 825 980331 54 406364 879 593636 42 19 393191 824 980299 54 406892 878 503108 41 20 393085 823 986206 54 407419 877 592581 40 21 9-394179 822 9-986234 54 9-407945 876 10-592055 39 22 394 (i73 821 986202 54 408471 875 591529 38 23 395106 820 986109 54 408997 874 591003 37 24 395058 819 980137 54 409521 874 590479 36 25 390150 818 986104 54 410045 873 589955 35 26 390041 817 980072 54 410509 872 589431 34 27 397132 817 986039 54 411092 871 588908 33 28 397021 816 986007 54 4U015 870 588385 32 29 398111 815 985974 54 412137 869 587863 31 30 398000 814 985942 54 412658 868 587342 30 31 C-3990a8 813 9-985909 55 9-413179 867 10-586821 29 32 399575 812 985876 55 413699 806 586301 28 33 400062 811 985843 55 414219 805 585781 27 34 400549 810 985811 55 414738 864 585262 26 35 401035 809 985778 55 415257 804 584743 25 30 401520 808 985745 55 415775 803 584225 24 37 402005 807 985712 55 416293 802 583707 23 38 402489 806 985679 55 416810 8()1 583190 22 39 402972 805 985646 55 417320 860 582674 21 40 403455 804 985613 55 417842 859 582158 20 41 {)-403938 803 9-985580 55 9-418358 858 10-581642 19 42 404420 802 985547 55 418873 857 581127 18 43 404901 801 985514 55 419387 856 580613 17 44 405382 800 985480 55 419901 855 580099 16 45 405862 799 985447 55 420415 855 579585 15 46 406341 798 985414 56 420927 854 579073 14 47 406820 797 985380 56 421440 853 578560 13 48 407299 796 985347 56 421952 852 578048 12 49 407777 795 985314 56 422-463 851 577537 11 50 408254- 794 985280 56 422974 850 577026 10 51 9-408731 794 9-985247 56 9-423484 849 10-576516 9 52 409207 793 985213 50 423993 848 576007 8 53 409(«2 792 985180 50 424503 848 575497 7 54 410157 791 985146 56 425011 847 574989 6 55 410632 790 985113 56 425519 846 574481 5 56 411106 789 985079 56 426027 845 573973 4 57 411579 788 985045 56 426534 844 573466 3 58 412052 787 985011 56 427041 843 572959 2 59 412524 786 984978 56 427547 1 843 572453 1 60 412996 785 984944 56 428052 1 842 571948 Coiin» 1 Sine i 75 1 Degre Cotang. I es. Taag. 1 VL LOGARITHMIC SIXES, CO SIXES, ETC. (To Ix-grees.) 213 M. ( D. I TaMR. 9-412996 1 785 9-9R4944 57 9-428052 842 10-571948 ' 1 4i:!4fi7 784 9^4910 57 428557 841 571443 2 4i3y;;8 1 783 984876 57 429062 840 570938 3 414408 783 984842 57 429566 839 570434 4 414878 782 98-1808 57 430070 838 509930 5 415347 781 984774 57 430573 838 569427 6 415815 780 984740 57 431075 837 5C8925 7 416283 779 984706 57 431577 836 568423 8 410751 778 984672 ! 57 432079 835 507921 9 4J7217 777 984637 57 432580 834 567420 10 ■ 417G84 776 984603 57 433080 833 566920 11 9-418150 775 9-984569 57 9-433580 832 iO-5G0420 12 4J86I5 774 984535 57 434080 832 505920 13 419079 773 984500 57 434579 831 505421 ^ 14 410544 773 984466 57 435078 830 564922 15 420007 772 984432 58 435576 829 564424 16 420470 771 98-1397 58 436073 828 563927 17 420933 770 984363 58 436570 828 563430 18 421395 769 984328 58 437007 827 562933 A ^9 42J857 708 984294 58 437563 826 562437 I 20 4223J8 707 984259 58 438059 825 561941 21 9-422778 767 9-984224 58 9-438554 824 10-561446 \_ 22 423238 766 984190 58 439048 823 560952 [ 23 423697 7C5 984155 58 439543 823 560457 [ 24 424156 764 984120 58 440036 822 559964 r 35 424G15 763 984085 58 440529 821 559471 ; 26 425073 762 984050 58 441022 820 558978 ' 27 425530 761 984015 58 441514 819 558486 2 28 425987 700 983981 58 442006 819 557994 ; 29 420443 760 983946 58 442497 818 557503 3 30 426899 759 98391 1 58 442988 817 557012 3 31 9-427354 758 9-983875 58 9-443479 816 10-556521 2 32 427809 757 983840 59 443908 816 556032 33 428263 756 983805 59 444458 815 555542 c 34 428717 755 983770 59 444947 814 555053 2 35 429170 754 983735 59 445435 813 554565 t_ 36 429023 753 983700 59 445923 812 554077 2 37 430075 752 983664 59 446411 812 553580 38 430527 752 983629 59 446898 811 553103 ; 39 430978 751 983594 59 447334 810 552010 .; 40 431429 750 983558 59 447870 809 552130 2 41 9-431879 749 9-983523 59 9-448356 809 10-551644 1 42 432329 749 983487 59 448841 808 551159 1 43 432778 748 983452 59 44932G 807 550674 I 44 433226 747 983416 59 449810 806 550190 1 45 433G75 746 983381 59 450294 806 549706 1 4G 434 J 22 745 983345 59 450777 805 549223 1 47 434.169 744 983309 59 •451260 804 548740 1 4f 435016 744 983273 60 451743 803 548257 1 4t 435462 743 983238 60 452225 802 547775 1 5(, 435908 742 983202 60 452706 802 547294 1 51 9-436353 741 9-983l(!6 60 9-453187 801 10-546813 52 436798 740 983130 60 453668 800 546332 53 437242 740 983094 60 454148 799 545852 54 4376H6 739 983058 60 454628 799 545372 55 438129 738 983022 60 455107 798 544893 56 438572 737 982986 60 455586 797 544414 57 439014 736 982950 60 456064 796 543936 58 439456 736 982914 60 456542 796 543458 59 439897 735 982878 60 457019 795 542981 60 440338 734 982842 60 457496 794 542504 1 Sine I I Cotang. I \ Tan§ 74 Degrees. 214 (10 DogifcK.; LOUARirilMIC SINES, COSIXES, ETC. M. 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' 1A LOGARITHMIC SINES, COSINES, ETC. (17 Degrees.) 215 M. / I D. I I D. 1 Tang I Colang. I f)-465935 688 9-98()596 64 9-485339 755 10-514661 60 J 466348 688 980558 64 485791 752 514209 59 2 466761 687 980519 65 486242 751 513758 58 3 467173 686 980480 65 48(i693 751 513307 57 4 467585 085 98(t442 65 487143 750 512857 56 5 467996 685 980403 65 487593 749 512407 55 6 468407 684 980364 65 488043 749 511957 54 7 468817 683 980325 65 488402 748 511508 53 8 469227 683 980286 65 488941 747 511059 52 9 469637 (i82 980247 65 489390 747 510610 51 10 470046 681 960208 65 489838 746 510162 50 ]1 9-470455 680 9-980169 65 9-4CC286 746 10-509714 49 12 470863 680 980130 65 490733 745 509267 48 i:j 471271 679 980091 65 491180 744 508820 47 ]4 471679 678 980052 65 491627 744 508373 46 J5 472086 078 980012 65 492073 743 507927 45 16 472492 677 979973 65 492519 743 507481 44 17 472898 67G 979934 66 492965 742 507035 43 18 473304 676 979895 66 493410 741 506590 42 19 473710 675 979855 66 493854 740 506146 41 20 474115 674 979816 66 494299 740 505701 40 21 9-474519 674 9-979776 66 9-494743 740 10-505257 39 22 474923 (■)73 979737 66 495186 739 504814 38 23 475:i27 072 979697 66 495C30 728 504370 37 24 475730 672 979658 66 496073 737 503927 36 25 476133 671 979618 66 496515 737 503485 35 26 47C53C 670 979579 66 496957 736 503043 34 27 476938 669 979539 66 497399 736 502601 33 28 477340 669 979499 66 497841 735 502159 32 29 477741 668 979459 66 498282 734 501718 31 30 478142 667 979420 66 498722 734 501278 30 31 9-478542 067 9-979380 66 9-4C91G3 733 10-500837 29 32 478942 ()G6 979340 66 4G9CC3 733 500397 28 33 479342 G65 979300 67 500042 732 499958 27 34 479741 6{i5 979260 67 50C481 731 499519 26 35 480140 664 979220 67 506920 731 499080 25 36 480539 663 979180 67 601359 730 498641 24 37 48(;30306 583 369694 53 8 594251 493 963596 90 ()30656 583 369344 52 <) 594547 492 963542 90 631005 582 368995 51 i!; 594842 492 96;i488 90 631355 582 368645 50 Ji 9-595137 491 9-90:i434 90 9-631704 582 10-368296 49 1-2 595432 491 963379 90 632053 581 3r.7947 48 iA 595727 491 963325 90 632401 581 3(i7599 47 14 596021 490 903271 90 632750 581 367250 46 15 596315 490 963217 90 633098 580 366902 45 iK 596C09 489 963163 90 633447 580 366553 44 17 596903 489 963108 91 633795 580 366205 43 18 597196 489 963054 91 634143 579 365857 42 19 597490 488 962999 91 634490 579 365510 41 20 597783 488 962945 91 634838 579 365162 40 21 9-59P075 487 9-962890 91 9-635185 578 10-364815 39 22 598368 487 962836 91 635532 578 364468 38 23 598660 487 962781 91 635879 578 364121 37 24 598952 486 962727 91 636226 577 363774 36 25 599244 486 962672 91 636572 577 363428 35 2C 599536 485 962617 91 636919 577 363081 34 27 599827 485 962562 91 637265 577 362735 33 28 600118 485 9C2508 91 637611 576 362389 32 29 600409 484 962453 91 637956 576 362044 31 30 600700 484 962398 92 638302 576 361698 30 31 9-000990 484 9-962343 92 9-638647 575 10-361353 29 32 601280 483 962288 92 638992 575 361008 28 33 601570 483 962233 92 639337 575 360663 27 34 691860 482 962178 92 639682 574 360318 26 35 602 I5U 482 962123 92 640027 574 359973 25 36 602439 482 962067 92 640371 574 359629 24 37 602728 481 962012 92 640716 573 359284 23 38 603017 481 961957 92 641060 573 358940 22 39 603305 481 961902 92 641404 573 358596 21 40 603594 480 961846 92 641747 572 358253 20 41 9-603882 480 9-961791 92 9-642091 572 10 357909 19 42 6G4170 479 961735 92 642434 572 357566 18 43 604457 479 961680 92 642777 572 357223 17 44 604745 479 901624 93 643120 571 356880 16 45 605032 478 961569 93 643463 571 356537 15 4G 005319 478 961513 93 643806 571 356194 14 47 605606 478 961458 93 644148 570 355852 13 48 605892 477 961402 93 644490 570 355510 12 49 606 J 79 4 ;~ 961340 93 644832 570 355168 11 50 606465 476 961290 93 645174 569 354826 10 51 9-606751 476 9-961235 93 9-645516 569 10-354484 9 52 607036 476 961179 93 645857 569 354143 8 53 607322 475 961123 93 646199 5(39 353801 7 54 607607 475 961067 93 646540 568 353460 6 55 607892 474 961011 93 646881 568 353119 5 56 608177 474 960955 93 647222 568 352778 4 57 608461 474 960899 93 647562. 567 352438 3 58 608745 473 960843 94 647903 567 352097 2 59 609029 473 960786 94 648243 567 351757 1 60 609313 473 960730 94 648583 566 351417 Coane | Sine I Cotang-. I Tang. I Bl 66 Degrees. 222 (24 I)i-giec's.) LOGARITHMIC SIXES, COSINES, ETC. Sine UW>93]y J3()yo97 609f<80 610IH4 610447 6 J 0729 fi]J(ll2 61J:i94 611576 61lri58 612140 9-612421 612-1(2 612983 6] 3264 6 J 3545 613825 61411)5 6)4385 6I4H65 614944 9-615223 615502 615781 616338 617172 617450 617727 9-618004 618281 618558 618834 619110 619386 619662 619938 6202 J 3 620488 9-620763 621038 62)3)3 621587 621861 622135 622409 622682 622!»56 623229 9-623502 623774 624047 6243)9 62459) 624863 625 J 35 62540H 625677 D. Cosine | D. 1 Tang. 1 D. Cotang. 473 9-960730 94 9-648583 566 10 351417 "60 472 960674 94 648923 566 351077 5S 472 960618 94 649263 566 350737 58 472 960.itil 94 649602 566 350398 57 471 960505 94 649942 565 350058 56 471 960448 94 6.50281 565 349719 5i 40 960392 94 6.50620 5hi5 349380 54 4 1) 96o:«5 94 650959 564 349041 53 4 ) 960279 94 651-297 51 ;4 348703 52 4'k) 960-222 94 651636 564 348364 5 4()9 960 i 65 94 651974 563 348026 5t 4r>9 9-960109 95 9-6.5-2312 563 10-347688 4L 4nrt 900002 95 65-2650 563 347350 48 4r>8 959995 95 652988 563 3470)2 4" 4h7 959938 95 653326 562 34t)674 4t 4ti7 959882 95 653663 562 346337 4i 4h7 959825 95 654000 562 346000 4-f 4(i6 959768 95 654337 561 345663 4: 466 9597 1 1 95 654674 561 345326 4; 466 959654 95 6.V)0ll 561 344989 4 465 959596 95 655348 561 344652 4( 465 9-959539 95 9-655684 560 10-344316 3i 465 959482 95 656020 560 343980 3t 464 959425 95 656356 560 343644 3- 464 959368 95 656692 559 343308 3t 464 959310 9t; 657028 559 342972 3. 463 959253 96 657364 559 342636 3^ 463 959195 96 657699 559 342301 3: 462 959138 96 658034 558 341966 3; 462 959081 96 658369 558 341631 3 462 959023 96 658704 558 341296 3( 461 9-958965 96 9-659039 558 10-340961 2i 461 958908 WJ 659373 5.57 340627 2i 461 958850 96 659708 557 340292 2- 460 958792 96 660042 557 339958 2( 460 958734 96 660376 557 339624 2. 460 958677 96 660710 556 339290 2^ 459 958619 96 661043 556 338957 2: 459 958561 96 66 J 377 556 338623 2S 459 958503 97 661710 555 338290 2 458 958445 97 662043 555 337957 2( 458 9-958387 97 9-662376 5.55 10-337624 457 958329 97 662709 554 337291 457 9.58271 97 663042 554 336958 457 958213 97 663375 5.54 336625 456 958)54 97 663707 554 336293 456 95H096 97 6640.S9 553 335961 456 958038 97 664371 553 335629 455 957979 97 664703 5.)3 335297 455 957921 97 66.5035 553 334965 455 957863 97 665366 552 334634 454 9-957804 97 9-66.5697 5.52 10-334.303 454 957746 98 666029 552 333S7] a:a 957687 98 666360 551 333640 453 957628 98 666691 551 333309 453 957570 98 667021 551 332979 453 9575)1 98 6673.'»2 551 332648 452 957452 98 667(i82 550 332318 452 957393 98 668013 550 331987 452 957335 98 668343 550 3316.57 451 957276 98 668672 550 331328 1 Cosine I i Sme I Coiang. I Tang. 65 Degrees. KUiARITTTMIC STXES, COSIKES, ETC. (2:. Doki-pps.) 22S Sine I I Cosine | .D. | Tang. | 9-625948 451 9-957276 98 9-668673 550 |0-3:(I327 60 J 526219 451 957217 98 669002 549 330998 59 2 626490 451 957158 98 669332 549 330668 58 3 626760 450 957099 98 669661 549 330339 57 4 627030 450 957040 98 669991 548 330009 .56 5 6273^)0 450 956981 98 670320 548' 329680 55 6 6275 70 449 956921 99 670649 548 329351 54 7 627M0 449 956862 99 670977 548 329023 53 8 628 J 09 449 956803 99 671306 547 328694 .52 9 628378 448 956744 99 671634 547 328366 .51 10 628647 448 956684 99 671963 547 328037 50 11 "> 628916 447 9-956625 99 9-672291 547 10-327709 49 12 629 H5 447 956566 99 ' 672619 546 327X81 48 13 629453 447 956506 99 672947 546 327(153 47 14 62^721 446 956447 99 673274 546 326726 46 15 629989 446 956387 99 673602 546 326:^98 45 16 630257 446 95H327 99 673929 545 326(171 ■^4 17 63II.V24 446 956263 99 674257 545 325- 43 A 1 iA SHU-; 92 445 956208 100 M', 4584 545 325JI6 4 . J9 6311(59 445 956148 J 00 674910 544 3251 "HO 41 20 63 J 326 445 956089 J 00 675237 544 324163 40 21 9-631593 444 9-956029 100 9-675564 544 10-324436 39 22 631859 444 955969 100 675890 544 324110 3H 23 632125 444 955909 100 676216 543 323784 37 24 632392 443 955849 100 676543 543 323457 36 2o 6326.58 443 955789 100 676869 543 323131 35 2« R32923 443 955729 100 677194 543 322806 34 27 633189 442 955669 100 677520 542 322480 33 28 633454 442 955609 100 677846 542 3221.54 32 2!» 533719 442 955548 100 678171 542 32IH29 31 30 633984 441 955488 100 678496 542 321504 30 31 9-634249 441 9-955428 101 9-678821 541 10-321179 29 32 6:}4514 440 955368 101 679146 541 32(»8.54 28 33 634778 440 955307 101 679471 541 320529 27 34 635042 440 955247 101 679795 541 3202(t5 26 35 6:;5306 439 955186 101 680120 540 319880 25 36 635570 439 9551-26 101 680444 540 319.5.56 24 37 fi35H34 439 955065 101 680768 540 319232 23 38 636097 438 955005 101 681092 540 318908 22 39 636360 438 954944 101 681416 539 318.584 21 40 636623 438 954883 101 681740 539 318260 20 41 9-636886 437 9-954823 101 9-682063 539 10-3179.37 19 42 637148 437 954762 101 682387 539 317613 \S 43 637411 437 954701 101 682710 538 317290 17 44 637673 437 954640 101 683033 538 316967 16 45 637935 436 954579 101 683356 538 316644 15 46 638197 436 954518 102 683679 538 316321 14 47 638458 436 954457 102 684001 537 31.5999 13 48 6387-20 435 954396 102 684324 537 315676 12 49 638981 435 954335 102 684646 537 3153.54 11 50 639242 435 954274 102 684968 537 315032 10 51 9-630503 434 9-954213 102 9-685290 536 10-314710 9 52 639764 434 954152 102 685612 536 314388 8 53 640024 434 954090 102 6859:M 536 314066 7 54 640284 433 954(t29 1(12 686255 536 31.3745 6 55 640544 433 9539«i8 102 686577 535 313423 5 5fi 640804 433 953906 102 686898 535 313102 4 57 64illt)4 432 953845 102 687219 535 312781 3 58 641324 432 953783 102 687540 535 312460 2 59 641584 432 953722 103 687861 534 312139 1 60 641842 431 953660 103 688182 534 311818 \ Sine I I Cotaiig. I I TAag. J U. 224 (20 Deg rces.) LOGARITHMIC SIA'^FS, COS/^'FS, ETC. M. Sine , D. Cosine D. Tang. D. 1 Cotang. 1 9-641842 431 9-953660 103 9-688182 534 10-311818 60 6 1.21 01 431 953599 103 688502 534 311498 59 2 w^^3r*o 421 953537 103 088823 534 311177 58 3 642618 430 953475 103 689143 533 310857 57 4 642877 430 953413 103 •689463 533 310537 56 5 643135 430 953352 103 689783 533 310217 55 6 643393 430 953290 103 690103 533 309897 54 7 64365'-) 429 953228 103 690423 533 309577 53 8 6439U8 429 9531CG 103 690742 532 309258 52 9 644105 429 953104 103 691 002 532 308938 51 10 6444':3 428 953042 1C3 691281 532 308GI9 50 11 9 644C80 428 9-952980 104 9-691700 531 10-308300 49 12 64493G 428 952918 104 092019 531 307i)81 48 13 64519;] 427 952855 104 092338 531 307002 47 14 645450 427 952793 104 092056 531 307344 46 15 64570G 427 952731 104 692975 531 307025 45 16 6439C2 426 952G69 104 693293 530 300707 44 17 64G218 426 9526C6 104 693012 530 30G388 43 18 646474 426 952544 104 093930 530 306070 42 19 646729 425 952481 104 G94248 530 305752 41 20 64G984 425 952419 104 094566 529 305434 40 21 9-G47240 425 o-o:;2356 104 9-C94883 529 10-305117 39 22 647494 424 952294 104. 695201 529 304799 38 23 647749 424 952231 104 695518 529 304482 37 24 C480G4 4C4 9521C8 105 G05C3G 529 304164 36 25 648258 424 9521C6 105 CDG153 528 303847 35 26 648512 423 952043 105 030470 528 303530 34 27 6487CG 423 951980 105 69G787 528 303213 33 28 649020 423 951917 105 697103 528 302897 32 29 649274 422 951854 105 697420 527 302580 31 30 649527 422 (,51791 105 69773G 527 3022G4 30 31 9-649781 422 <;-951728 1C5 9-698053 527 10-301947 29 32 650034 •i' i 951GG5 105 698309 527 301031 28 33 650287 421 951G02 105 698085 526 301315 07 34 650539 421 95]5;:9 105 699001 520 300999 26 35 C50792 421 951476 105 6993 IG 526 300684 25 36 G51044 420 951412 105 699032 526 300368 24 37 651297 420 951349 106 099947 526 300053 23 38 651549 420 951286 100 700203 525 299737 22 39 G51800 419 951222 106 700578 525 299422 21 40 652052 419 951159 106 700893 525 299107 20 41 9-652304 419 9-951096 106 9-701208 524 10-298792 19 42 652555 418 951032 106 701523 524 298477 18 43 G52806 418 950968 106 701837 524 298163 17 44 653057 418 950905 106 702152 524 297848 16 45 653308 418 950841 106 702406 524 297534 15 46 053558 417 950778 lOG 702780 523 297220 14 47 653808 417 950714 106 7G3U95 523 296905 13 48 654059 4.7 950650 106 703409 523 29()591 12 49 654309 410 950586 106 703723 523 290277 11 50 654558 416 950522 107 704036 522 295964 10 51 654808 416 9-950458 107 9-704350 522 10-295650 9 52 655058 416 950394 107 704()63 522 295337 S 53 655307 415 950330 107 704977 522 295023 7 54 (J55."5G 415 950266 107 705290 522 294710 6 55 6558(15 415 950202 107 705603 521 294397 5 56 65(5054 414 950138 107 705910 521 294084 4 57 656302 414 950074 107 706228 521 293772 3 58 656551 414 950010 107 706541 521 293459 it 59 656799 413 949945 107 70fi854 521 293146 1 60 657047 413 949881 1(»7 707166 520 292834 1 C«une 1 1 Sine 63 1 Degre Cotang. Ta..g. 1 M. LOGARirmilC SIXES, COSTXES, ETC. (27 Degrees.) 225 M. Sine D. 1 Cosine 1 D. Tang. D. Coiaiig. 1 9-657047 413 9-949881 107 9-707166 520 10-292834 60 1 657295 413 949816 107 707478 520 292522 59 2 657542 412 94y7o2 107 707790 5ii0 292210 58 3 657790 412 949688 108 708102 520 291898 57 4 658037 412 949623 108 708414 519 291586 56 5 658284 412 949558 108 708726 519 291274 55 6 658531 411 949494 108 709037 519 290963 54 7 658778 411 949429 108 709349 519 290651 53 8 659025 411 94t»364 108 709660 519 290340 52 !> 659271 410 949300 108 709971 518 290029 51 ](• 659517 4J0 949235 108 710282 518 289718 50 11 9-659763 410 9-949170 108 9-710593 518 10-289407 49 12 66U009 409 949 1 05 108 710904 5,8 289096 48 13 660255 4(t9 949040 108 711215 518 288785 47 14 6605(11 409 948975 108 711525 5J7 288475 46 15 630746 409 948910 108 711836 517 288164 45 lo 6609<»1 408 948845 108 712148 517 28V854 44 17 661236 408 948780 109 712456 517 287544 43 ]• 661481 4J8 948715 109 712766 516 287234 42 19 661726 407 948650 109 713076 516 286924 41 2U 661970 407 948584 109 713386 516 286614 40 21 9-662214 407 9-948519 109 9-713696 516 1(1-286304 39 22 662459 407 948454 109 714005 5l() 285995 38 23 662703 406 948388 109 714314 515 285686 37 24 662946 406 948323 109 714624 515 . 285376 36 25 683190 406 948257 109 714933 515 285067 35 26 663433 405 948192 109 715242 515 284758 34 27 663677 405 948126 109 715551 511 284449 33 28 663920 405 948060 109 715S60 514 284140 32 29 664163 405 947995 110 716168 514 283832 31 30 664406 404 941929 110 716477 514 983523 30 31 9-664648 404 9-947863 110 9-716785 514 ](l-283215 29 32 664891 4(14 947797 110 717093 513 282907 28 33 665133 403 947731 110 717401 513 282599 27 34 665375 403 947665 110 717709 513 282291 26 35 665617 403 947600 110 718017 513 281983 25 36 665859 402 947533 110 718325 513 281675 24 37 666100 402 947467 110 718633 512 281367 23 38 666342 402 947401 110 718940 512 281060 22 39 666583 402 947335 110 719248 512 280752 21 40 666824 401 947269 110 719555 512 280445 20 41 9-367065 401 9-94/203 110 9-719862 512 10-280138 19 42 667305 401 947136 111 720169 511 279831 18 43 667546 401 947070 111 720476 511 279524 17 44 667786 400 947004 111 720783 51i 279217 16 45 668027 400 946937 111 721089 511 278911 15 46 668267 400 946871 111 721396 511 278604 14 47 668506 399 946804 111 721702 510 278298 13 48 668746 399 946738 111 722009 510 277991 12 49 668986 399 946671 111 722315 510 277685 11 50 669225 399 946604 111 722621 510 277379 10 51 9-669464 398 9-946538 111 9-722927 510 10277073 9 52 669703 398 946471 111 723232 509 2767(58 8 53 669942 398 946404 111 723538 509 276462 7 54 670 181 397 946337 111 723844 509 276156 6 55 670419 397 946270 112 724149 509 275851 5 56 670658 397 946203 112 724454 509 275546 4 57 67(1896 397 946136 112 724759 508 275241 3 58 671134 396 946069 112 725065 508 274935 2 50 671372 396 9460(12 112 725369 508 274631 1 6U 671609 396 945935 112 725674 508 274326 I Coeine | I I Coiang, I 62 Degrees. Tanj. I M 226 (28 Degrees.) LOOAHITHMTC SINES, COSINES, ETC. I Cosine | I Tang. I Cotang. I Cofine I I Cotang. ^ Pegr«eii. 9-671609 396 9-945935 112 9-725K74 508 10-274326 60 1 671847 395 345868 l.i 7-.i597y 508 274021 59 2 672084 395 945800 li2 726284 507 273716 58 3 672321 395 945733 112 726588 507 273412 57 4 672558 395 945666 112 726892 507 273108 56 5 672795 394 945598 112 727197 507 272803 55 6 673032 394 945531 112 727501 507 272499 54 ■ 7 673268 394 945464 113 727805 506 272195 53 8 673505 394 945396 113 728109 .506 27l«91 52 9 673741 393 945328 113 728412 .506 271,588 51 10 673977 393 945261 113 728716 506 271284 511 11 9-674213 393 9-945193 113 9-729020 506 10-270980 49 12 674448 392 945125 113 7293-23 505 270677 48 13 6746H4 392 945058 113 729626 505 270374 47 14 674919 392 944990 113 729929 505 270071 46 15 675155 392 944922 113 730233 505 269767 45 16 675390 391 944854 113 730535 505 269465 44 17 675624 391 944786 113 730838 504 269162 43 18 675859 391 944718 113 731141 504 26rtH59 42 39 676094 391 944650 113 731444 504 26^556 41 20 676328 390 944582 114 731746 504 268254 40 SI 9-676562 390 9-944514 114 9-732048 504 10-2679.52 3f 22 676796 390 944446 114 732351 503 267649 38 23 677030 390 944377 114 732653 503 267.347 37 24 677264 389 944309 114 732955 503 267045 36 2.'i 677498 389 944241 114 733257 503 266743 35 2f) 677731 389 944172 114 733558 503 266442 34 27 677964 388 944104 114 733860 502 266140 33 28 678197 388 944036 214 734162 502 265838 32 29 678430 388 ^43967 ;i4 734463 502 265537 31 30 678663 388 943899 .14 734764 502 265236 30 31 9-678895 387 P 943830 114 9-735066 502 10-264934 29 32 679128 387 943761 114 735367 502 264633 28 33 679360 387 943693 115 735668 501 2643.32 27 34 679592 387 943624 115 735969 501 264031 26 35 679824 386 043555 115 736269 501 263731 25 36 680056 386 9434H6 115 736570 501 263430 24 37 680288 386 94.3417 115 736871 501 263129 23 38 6805 J 9 385 943.348 115 737171 500 262829 22 39 680750 385 943279 115 737471 • 500 26'i529 21 40 680982 385 943210 115 737771 500 262229 20 41 9-681213 385 9-943141 115 9-738071 500 10-261929 19 42 681443 384 943072 115 438371 500 261629 18 43 681674 384 943003 115 738671 499 261.329 17 44 681905 384 942934 115 738971 499 26)()'J9 16 45 68213.5 384 942864 115 739271 499 260729 15 46 682365 383 942795 1J6 739570 499 260430 14 47 68ii595 383 942726 116 739870 499 2601.30 13 48 6H2825 383 942656 116 740169 499 259831 12 49 683055 383 942587 116 740468 498 259532 11 50 683284 382 942517 116 740767 498 259233 10 51 9-683514 382 9-942448 116 9-741066 498 10-2.58934 9 52 683743 382 942378 116 741365 498 2.58635 8 53 683972 382 942308 116 741664 498 258336 7 54 684201 381 942239 316 741962 497 2.58038 6 55 684430 381 942169 116 742261 497 257739 5 56 684658 381 942099 116 742559 497 2.57441 4 57 684887 380 942029 116 7428.58 497 2.57142 3 58 685115 380 941959 116 7431.56 497 256H44 2 59 685343 380 941889 117 743454 497 2.56.546 1 60 685571 380 941819 117 743752 496 256248 I Tftog. I M. LOGARITHMIC SIXES, CO SIXES, ETC. (20 Degrees.) 227 M. I i! I Cosine | D. | Tang. I Cotang. 9-68.5571 380 9-941819 117 9-7437.52 496 10-256248 685799 379 941749 117 744050 496 255950 68(j()27 379 941679 117 744348 496 255652 686^254 379 941609 117 744645 496 255355 68tt485> 379 941539 117 744943 496 255057 68b7U9 378 94141)9 117 745240 496 254760 686936 378 941398 117 745538 495 254462 687J63 378 941328 117 745835 495 254165 687389 378 941258 117 746132 495 253868 687616 377 941187 117 746429 495 253571 6M7843 377 941117 117 746726 495 253274 9-688069 377 9-941046 118 9-747023 494 10-252977 6W295 377 940975 118 747319 494 2.r2C81 ^Kyil 376 940905 118 747616 494 252384 6887-17 376 940834 118 747913 494 252087 68^^972 3T6 940763 118 748209 494 251791 689198 376 910693 118 748505 493 251495 689423 375 940622 118 748801 493 251199 689648 375 940551 118 749097 493 250903 689873 375 940480 118 749393 493 250607 690098 375 94041)9 1J8 749689 493 250311 9-690323 374 9-9-10338 118 9-749985 493 10-2500I5 690548 374 94U267 118 750281 492 249719 690772 374 940196 118 750576 492 249424 690996 374 940125 119 750872 492 2491-28 69J-2-20 373 940054 119 751167 492 248833 691444 373 939982 119 751462 492 248538 691668 373 939911 119 751757 492 248243 691892 373 939840 119 752052 491 247948 692115 372 939768 119 752347 491 247653 692339 372 939697 119 752642 491 247358 9-692562 372 9-939625 119 9-752937 491 10-247063 692785 371 939554 119 753231 491 246769 6930(J8 371 939482 119 753526 491 246474 693231 371 939410 119 753820 490 246180 693453 371 939339 119 754115 490 245885 693676 370 939267 120 754409 490 245.591 693898 370 939195 120 754703 490 24.5297 694120 370 939123 120 754997 490 245003 694342 370 939052 120 755291 490 244709 694564 369 938980 120 755585 489 244415 9-69.1786 369 9-938908 i20 9-755878 489 10-244122 695007 369 938836 J 20 756172 489 243828 69.5229 369 938763 120 756465 489 243535 695450 368 938691 190 756759 489 243241 695671 368 938619 1-^-u 757052 489 242948 695892 368 938.547 120 757345 488 242655 696113 368 938475 120 757638 488 242362 696334 367 938402 121 757931 488 242069 696554 367 938330 121 7.58224 488 241776 696775 367 938258 121 758517 488 241483 9-696995 367 9-938185 121 9-758810 488 10-241190 697215 366 938113 121 759102 487 240898 697435 366 938040 121 759395 487 240605 697654 366 937967 121 759'>87 487 240313 697874 366 937895 121 759979 487 240021 698094 365 9378-22 121 760272 4S7 239728 698313 365 937749 121 760564 487 239436 698532 365 937078 121 760856 486 239144 698751 365 93-004 121 761148 486 2388.52 698970 364 937J31 121 761439 486 23856] Q??!RP I [ Sm I I Cotang. §9 Pegre§|, ?»o?. I ¥^ 228 (30 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine D. Cosine D. 1 Tang. D. Cotang. 9-698970 364 9937531 121 9-761439 486 10-238561 60 1 G99189 364 937458 122 761731 486 238269 59 2 699407 364 937385 122 762023 486 237977 58 3 699626 364 937312 122 762314 486 237686 57 4 699844 363 937238 122 762606 485 237394 56 5 700062 363 937165 122 762897 485 237103 55 6 700280 363 937092 122 763188 485 236812 54 7 700498 363 937019 122 763479 485 236521 53 8 700716 363 936946 122 763770 485 236230 52 9 700933 362 936872 122 764061 485 235939 51 30 701151 362 936799 122 764352 484 235632 468 203368 57 4 725017 336 928104 132 796til3 468 203087 56 5 725219 336 928025 132 797194 468 202806 55 6 725420 335 927946 132 797475 468 202525 54 7 725622 335 927867 132 797755 468 202245 53 8 725823 335 927787 132 798036 467 201964 52 9 726024 335 927708 132 798316 467 201684 51 10 726225 335 927629 132 798596 467 201404 5(f 11 9-726426 334 9-927549 132 9-798877 467 10-201123 49 12 726026 334 927470 133 799157 467 200843 48 13 726827 334 927390 133 799437 467 200563 47 14 727027 334 927310 133 799717 467 200283 46 15 727228 334 927231 133 799997 466 200003 45 16 727428 333 927151 133 800277 466 199723 44 17 727628 333 927071 133 800557 466 199443 43 18 727828 333 926991 133 800836 466 199164 42 19 728C27 333 926911 133 801116 466 198884 41 20 728227 333 926831 133 801396 4()6 198604 40 21 9-728427 332 9-926751 133 9-801675 466 10-198325 39 23 728626 332 926671 133 801955 466 198045 38 23 728825 332 926591 133 802234 465 197766 37 24 729024 332 920511 134 802513 465 197487 36 35 729223 331 926431 134 802792 465 197208 35 26 729422 331 926351 134 803072 465 196928 34 27 729621 331 926270 134 803351 465 196649 33 28 729820 331 926190 134 803630 465 196370 32 29 730018 330 926110 134 803908 465 196092 31 30 730216 330 926029 134 804187 465 195813 30 31 9-730415 330 9-925949 134 9-804466 464 10-195534 29 32 730613 330 925868 134 804745 464 195255 28 33 730811 330 925788 134 805023 464 194977 27 34 731009 329 925707 134 . 805302 464 194698 96 35 731200 329 925626 134 805580 464 194420 25 3G 731404 329 925545 135 805859 464 194141 24 37 731602 329 925465 135 806 137 464 193863 23 38 7317'jy 329 925384 135 8C6415 463 193585 22 29 731096 328 925303 135 806693 463 193307 21 40 732193 328 925222 135 806971 463 193029 20 41 9-732300 328 9-925141 135 9-807249 463 10-192751 19 42 732587 328 925060 135 807527 463 192473 18 43 7:^2784 328 924979 135 807805 4G3 192105 17 44 732980 327 924897 135 808083 463 191C17 18 45 733177 327 924816 135 808361 463 191639 15 4(5 733373 327 924735 136 808638 4G2 1913C2 14 47 733539 327 924G54 136 808916 462 191C84 13 48 7337G5 327 924572 136 - 809193 462 190807 12 49 733961 326 924491 136 809471 462 190529 11 50 734157 326 924409 136 809748 462 190252 iO 51 9-734353 326 9-924328 136 9-810025 462 10-189975 9 52 73^1549 32G 92-^2^6 136 810302 4G2 183698 b 53 734744 325 924164 136 813580 462 189429 7 54 734939 325 924083 136 810857 462 180143 6 55 735135 325 924001 136 811134 461 188866 5 56 735330 325 923919 136 811410 461 188590 4 57 735525 325 923837 136 811687 461 188313 2 58 735719 324 923755 137 811964 461 188036 2 59 735914 324 923673 137 812241 461 187759 1 fiO 736109 324 923591 137 812517 461 187483 ) Cosine 1 1 Sine i 1 >7Degre Cotang. es. 1 fang. M. ^OGAA'/TI/J/IC SnVFS, COSINES, ETC. (33 Degrees.) 231 Sine D. Cosine D. Tang. 1 D. Cotanff. 1 !i-736109 324 9-923591 137 9-812517 461 10-187482 00 7:56303 324 923509 137 812794 461 1872G6 59 7304 98 324 923427 137 813070 461 1869.30 ^^ 736692 323 923345 137 813347 460 186653 57 736886 323 923263 137 813623 460 186377 56 737080 323 923181 137 813899 460 180101 55 737274 323 923098 137 814175 460 185825 54 737467 323 923016 137 814452 460 185548 53 737661 322 922933 137 814728 460 185272 52 737855 322 922851 137 815004 460 184996 51 738048 322 922768 138 815279 460 184721 5C 9-738241 322 9-922686 138 9-815555 459 10 184445 49 738434 322 922(i03 138 815831 459 1841()9 48 738627 321 922520 138 816107 459 183893 47 738820 321 922438 138 816382 459 183618 46 739013 321 922355 138 810658 459 183342 45 739206 321 922272 138 816933 459 183067 44 739398 321 922189 138 817209 459 182791 43 739590 320 922106 138 817484 459 182516 42 739783 320 922023 138 817759 459 182241 41 739975 320 921940 138 818035 458 181965 40 9-740167 320 9-921857 139 9-818310 458 10-181690 39 740359 320 921774 139 818585 458 181415 38 740550 319 921691 139 818860 458 181140 37 740742 319 921607 139 819135 458 180865 30 740934 319 921524 139 819410 458 180590 35 741J25 319 921441 139 819684 458 180316 34 741316 319 921357 139 819959 458 180041 33 7415C8 318 921274 139 820234 458 179766 :^ 741699 318 921 190 139 820508 457 J 79492 3; 74J889 318 921107 139 820783 457 179217 3U 9-742080 318 9-921023 139 9-821057 457 10-178943 29 742271 318 920939 140 821332 457 178668 28 742462 317 92C856 140 821606 457 178394 27 742652 317 920772 140 821880 457 178120 26 742842 317 92C088 140 822154 457 177846 25 743033 317 92CGC4 140 822429 457 177571 24 743223 317 92C520 140 822703 457 177297 23 743413 310 92C436 140 822977 456 177023 22 743602 316 920352 140 823250 456 176750 21 743792 316 920268 140 823524 456 170476 20 9-743982 316 9-920184 140 9-823798 456 10-176202 19 744171 316 920099 140 824072 456 175928 18 744361 315 92U015 140 824345 456 175655 17 744550 315 919931 141 824019 456 17i381 16 744739 315 919846 141 824893 456 175107 15 744928 315 919762 141 825160 456 174834 14 745117 315 919677 141 825439 455 174561 13 745306 314 919593 141 825713 455 174287 12 745494 314 919508 141 825986 455 174014 11 745683 314 919434 141 826259 455 173741 IC 9-745871 314 .9-919339 141 9-826532 455 10-173468 9 746059 314 919254 141 826805 455 173195 8 746248 313 919169 141 827078 455 172922 7 746436 313 919085 141 827351 455 172649 6 746624 313 919000 141 827624 455 172376 5 746812 313 918915 142 82'/89V 454 172103 4 746999 313 918830 142 828170 454 171830 3 747187 312 918745 142 828442 454 171558 2 747374 312 918659 142 828715 454 171285 1 747562 312 918574 142 828987 454 171013 * 1 CcMine ( I J Cotiuag. I I 1»*>B- lift 232 (34 Degrees.) LOGARITHMIC RINES, COSIMIJS, ETC. M. 1 Sine i D- 1 Cosine I D. Tar-. 1 D. Colang. 9-747562 312 9-918574 142 9-828987 454 10-171013 ~60 1 7-1774J 3J2 918489 142 829260 454 170740 59 2 747936 312 918404 142 829532 454 170468 58 3 748123 311 918318 142 829805 454 170195 57 4 7483 10 311 918233 142 83tj077 454 169923 56 5 748497 311 918147 142 830349 453 169651 55 6 748^83 311 i* 18062 J 42 83(J621 453 169379 54 7 748870 311 917976 143 830893 453 169107 53 8 74905G 310 917891 143 831165 453 168835 52 9 749243 310 917805 143 831437 453 168563 51 10 749423 310 917719 143 831709 453 J68291 50 11 9-749615 310 9-917034 143 9-831981 453 10-168019 49 J2 749801 310 917548 143 832253 453 167747 48 13 749987 309 917462 143 832525 453 167475 47 14 750172 309 917376 143 832796 453 167204 46 15 750358 309 9 J 7290 143 833068 452 J 66932 45 .10 750543 309 917204 143 833339 452 166661 44 17 750729 309 917118 144 833611 452 166389 43 18 750914 308 917032 144 833882 452 166118 42 19 751099 308 916946 144 834154 452 165846 41 20 751284 308 916859 144 834425 452 165575 40 21 9-751469 308 9-916773 144 9-834696 452 10-165304 39 22 751G54 308 916687 144 834967 452 165033 38 23 751839 308 91G600 144 835238 452 l{i4762 37 24 752U23 307 910514 144 835509 452 1C4491 36 25 752208 307 916427 144 835780 451 1M2-20 35 26 752392 307 916341 144 830051 451 163949 34 27 752576 307 91o254 144 836322 451 103678 33 28 752760 307 916167 145 836593 451 163407 32 29 752944 306 916081 145 836864 451 163136 31 30 753128 306 915994 145 837134 451 162866 30 31 9-753312 306 9-915907 145 9-837405 451 10-162595 29 32 7u3495 306 915820 145 837075 451 162325 28 33 753679 306 915733 145 837946 451 162054 27 34 753862 305 915646 145 838216 451 1G1784 26 35 754046 305 915559 145 838487 450 161513 25 36 754229 305 915472 145 838757 450 161243 24 37 754412 305 915385 145 839027 450 1G0973 23 38 754595 3U5 915297 J45 839297 450 160703 22 39 754778 304 915210 145 839568 450 160432 21 40 754960 304 915123 146 839838 450 160162 20 41 9-755143 304 9-915035 146 9-840108 450 10159892 19 42 755326 304 914948 146 840378 450 159622 18 43 755508 304 914860 146 840647 450 159353 17 44 755690 304 914773 146 840917 449 159083 16 45 755872 303 914685 146 841187 449 158813 15 46 756054 303 914598 146 841457 449 158543 14 47 756236 303 914510 146 841726 449 158274 13 48 756418 303 914422 146 841996 449 158004 12 49 756600 303 914334 146 842266 449 J 57734 11 50 756782 302 914246 147 842535 449 157465 10 51 9-756963 302 9-914158 147 9-842805 449 10157195 9 52 757144 302 914070 147 843074 449 156926 8 53 757326 302 9 J 3982 147 843343 449 156657 7 54 757507 302 913894 147 843612 449 156388 6 55 757688 301 9 J 3806 147 843882 448 156 J 18 5 56 757869 301 913718 147 844151 448 155849 4 57 758050 301 913630 147 844420 448 155580 3 58 758230 301 913541 147 844689 448 155311 2 5tt 758411 301 913453 147 844958 448 -155042 1 60 758591 301 913365 147 845227 448 154773 Cosine | \ Cotong. I I Tang. I 55 Degrees. LOGARITHMIC SIXES, COSIXES, ETC. (35 Degrees.) 23S I Cosine I ThM^T I Colanjf. 9-758591 301 9-913365 147 9.845227 448 10-154773 60 1 758772 300 913276 147 W4.")496 448 154.504 59 75895-2 300 913187 148 b45764 448 1 54:^30 58 3 753132 300 913099 148 S46033 448 153967 57 4 75D312 300 913010 14ti 6-,u.jo2 448 153ujri 50 5 759452 300 912922 148 84()570 447 153430 55 6 759G72 2yy 912833 148 846639 447 153161 54 7 759852 2j9 912744 148 847107 447 152893 53 8 76C031 299 912G55 148 847.376 447 152G24 52 9 7C0211 299 9125GG 148 S47G-;4 447 15235G 51 10 7G03D0 299 912477 143 847913 447 152387 50 II 9-7G05G9 298 9-912388 148 9.8^8181 447 10151819 49 12 7G0748 2;,8 912-299 149 Si'6-WJ 447 151551 4H 13 7GG'J27 298 912210 149 C-:87JI7 447 15I2S3 47 It 7C110G 298 912121 149 84Si}oG 447 151014 40 1.-. 7Gi-235 298 912031 149 84D254 447 15074G 45 16 7Gin;i ■2J3 91 1042 149 84L522 447 150478 44 17 7G1G42 297 911853 149 8^0790 446 150210 43 18 7G182I 297 9117G3 149 850358 446 149942 42 lii 7GlJ9il 297 91 1074 149 S5G325 446 149075 41 •20 7G2177 297 911584 149 85G593 446 149407 40 21 97G235G 297 9-911495 149 9850:^61 446 10-149139 30 22 7G2534 296 911405 149 851129 446 148871 38 23 7G2712 29G 911315 150 851396 44G 148G04 37 24 7G2889 29G 9J122G 150 8.:jCG4 44G 14S:^3G 30 2.) 7(i30G7 296 9Jlio6 150 851^31 446 14S.G9 35 •2tj 7t)3245 2.'iG 9iliMG 150 852199 446 147801 34 ■J 7 703422 2;>ti 9JU956 150 852466 446 147534 33 .'■s 7l)3filiO 2j.') 9W866 15U 8.52733 445 147267 32 29 7(i3777 2J5 910776 150 85;i001 445 146999 31 30 7.13954 295 916G86 15 J 853268 445 146732 30 31 9-7G4131 295 9-910596 150 9-853535 445 10-146465 29 3-2 7G4308 295 9i;}5GG 15 J 8..3t)J2 445 146198 28 33 7G4-I85 294 910415 150 8540v39 445 145931 27 34 7G4(iG2 294 910325 151 8543.J6 445 145664 26 35 75()15 294 910144 151 854870 445 145130 24 37 765191 294 910054 151 855137 445 144863 23 38 7G5367 294 909963 151 8554U4 445 144.596 22 39 705544 293 909873 151 8.55671 444 1443-29 21 4G 765720 293 909782 151 8559.i8 444 144062 20 41 9-7G5896 293 9-909691 151 9-85G2V.4 444 10-14379G 19 42 766072 293 909601 151 850471 444 1435-29 18 43 766247 293 909510 151 85G737 444 1432G3 17 44 766423 293 909419 151 8570;)4 444 142996 6 45 766598 2i)2 909328 152 S57270 444 142730 5 46 766774 2:12 909237 152 857537 444 1424 G3 4 47 76(i949 2l)2 909 J 46 152 8578C3 444 14-2 J i)7 13 48 767124 2i)2 909055 152 858069 444 14193! 12 49 767300 292 908964 152 858336 444 141664 U 50 767475 291 90P873 152 858G02 443 141398 10 51 9-7G7649 291 9-908781 152 9-858868 443 10-141132 9 52 767824 291 90P690 152 8.59134 443 140866 8 53 767999 291 908599 152 859400 443 140600 7 54 768173 291 903507 152 859666 443 140334 6 55 768348 290 908416 153 859932 443 140068 5 56 768522 290 908324 153 860198 443 139802 4 57 768697 290 908233 153 860464 443 139536 3 58 768871 290 908141 153 860730 443 139270 2 59 769045 290 908049 153 860995 443 139005 1 60 769219 290 907958 153 861261 443 138739 I C(Mine I I Sine I Cotang. I I Tang. I M. 54 Degrees. 234 (36 Degr( LVC Mi I Til MIC SIX/.S, CO, SIXES, ETC M. Sine 1 D. 9-769219 290 1 769393 289 2 769566 289 3 769740 289 4 7699J3 289 5 770087 289 G 770260 288 7 770433 '288 8 770606 2BS 770779 288 i;) 770952 288 11 9 77-125 288 i-i 771298 287 13 771470 287 J4 771643 287 ]"> 771815 287 Ifi 771987 287 ]7 772159 287 16 772331 •286 19 772503 2,^6 20 772675 'J8i) 21 9-772847 236 22 773018 286 23 773190 28G 24 773361 285 25 773533 285 m 773704 285 27 773875 285 28 774046 285 2;J 774217 285 30 774388 284 31 9-774558 284 32 774729 284 33 774899 284 34 775070 284 35 775240 284 36 775410 283 37 775580 283 38 775750 283 39 775920 283 40 776090 283 41 9-77625C 283 42 776429 282 43 776598 282 44 776768 282 45 77(i937 282 46 777106 282 47 777275 281 48 777444 281 49 777613 281 50 777781 281 51 9-777950 281 52 778J 19 281 53 778287 280 54 778455 280 55 778624 280 56 778792 280 57 7789«0 280 58 779128 280 59 779295 279 60 779463 279 Cosine D. 1 Tang. D. Cotang. 9~9~U7958 153 9-861261 443 10-138739 907866 153 861527 443 138473 907774 153 861792 442 138208 907682 253 862058 442 137942 907590 J 53 862323 44j» 137677 9o7498 153 8625«9 442 137411 907406 153 8t)2854 442 137146 907314 154 86;;] 19 442 136881 907222 154 863385 442 136615 907129 154 863650 442 136350 907037 154 863915 442 136(»85 9-906945 154 9-864180 442 10-135820 906852 154 864445 442 135555 906760 154 864710 442 135290 906667 154 864975 441 135025 906575 154 865240 441 134700 906482 154 865505 441 134495 906389 155 865770 441 134230 906296 155 866035 441 133965 906204 155 866300 441 133700 906111 155 866564 441 133436 9-906018 155 9-866829 441 10-13317; 905925 155 867094 441 132906 905832 155 867358 441 '32642 905739 155 867623 441 132377 905645 155 8D7S87 441 132113 905552 155 86815:^ 440 131848 905459 155 868416 440 131584 905366 156 86»d60 440 131320 905272 156 868945 440 131055 905179 156 86D209 440 130791 9-905085 156 9-869473 440 10-130527 904992 156 869737 440 ]3026:{ 904898 156 870001 440 J 29999 904804 156 870265 440 129735 904711 156 870o29 440 129471 904617 156 87G793 440 129207 904523 156 871057 440 128943 904429 157 871321 440 128679 904335 357 871585 440 128415 90424] 157 871849 439 128151 0-904147 157 9-872112 439 10-127888 904053 157 872376 439 127624 903959 157 872640 439 127360 903864 157 872903 439 127097 903770 157 873167 439 126833 903676 157 873430 439 126570 903581 157 873694 439 12»)306 903487 157 873957 439 126043 903392 158 874220 439 125780 903298 158 874484 439 125516 9-903203 158 9-874747 439 10-125253 903108 158 875010 439 124990 90:'(I14 158 875273 438 124727 9i;29l9 158 875536 438 124464 902824 158 875800 438 124200 902729 158 876063 438 123937 902634 158 876326 438 123674 902539 159 876589 438 123411 9112444 159 876851 438 12:n49 902349 159 877114 438 122886 I Cosine I Sine I I Cotang. ©3 Pegrwg, Tws- LOGARITHMIC SINES, COSINES, ETC. (37 Degrees.) 235 M. I Sine flf79463 779631 779798 779966 780133 7H03U0 780467 78U634 7808U1 780968 781J34 9-781301 781468 781634 781800 781966 781; 132 782298 782464 782(;30 782'/ 96 9-782961 783127 783292 783458 783623 783788 783953 784118 784282 784447 9-784612 784776 784941 785105 785269 785433 785597 785761 785925 786089 9-786252 786416 786579 786742 786906 7^57069 7H7232 787395 787557 787720 9-787883 788i;45 7H8208 7HH:rO 7«H532 78>'«i94 788856 789018 789180 789342 279 279 279 279 279 278 278 278 278 278 278 277 277 277 277 277 277 276 276 276 276 276 276 275 275 275 275 273 273 273 273 273 272 272 272 272 272 272 271 271 271 271 271 271 271 2-; I) 270 270 270 270 270 Cosine 9-902349 902253 902158 90-io63 9(>1967 90 J 872 901776 901681 90 J 585 901490 901394 9-901298 901202 901106 901010 900914 900818 900722 900626 900529 900433 9-900337 900240 900144 900047 899951 899854 899757 899660 8995(>4 899467 9-899370 899273 899176 899078 898981 898884 8D8787 898680 818592 898494 9-898397 898299 898202 898104 898006 897908 897810 897712 897614 897516 9-897418 897320 8H722-2 )^:t7l-23 897(K25 89H926 896828 896729 89663 1 896532 1 D. Tanj. D. Cotang. 159 9-877114 438 10-122886 60 159 877377 438 122623 59 159 877640 438 122360 58 159 877903 438 122097 57 159 878 J 65 438 121835 56 159 878428 438 121572 55 159 878691 438 121309 .54 159 878953 437 121047 53 159 879216 437 120784 52 159 879478 437 120522 51 160 879741 437 120259 50 160 9-880003 437 10-119997 49 160 880265 437 1 J 9735 48 160 880528 437 119472 47 160 880790 437 119210 46 160 881052 437 118948 45 160 88J314 437 118686 44 160 881573 437 118424 43 160 881839 437 118161 42 160 882101 4:n 117899 41 161 882363 436 117637 40 161 9-882625 436 10117375 39 161 882887 436 117113 38 iril 883148 436 116852 37 1151 883410 436 116590 36 161 883672 436 116328 35 161 883934 436 116066 34 161 884 J 96 436 115804 33 161 884457 436 115543 32 161 884719 436 115281 31 162 884980 436 115020 30 162 9-885242 436 10-114758 29 162 885503 436 114497 28 162 885765 436 1 J 4235 27 162 886026 436 113974 26 162 886288 436 1137i2 25 162 886549 435 113451 24 162 886810 435 113190 23 102 887072 435 112928 22 162 887333 435 112667 21 163 887594 435 112406 20 163 9-P87855 435 10-112145 19 163 888116 435 111884 18 163 888377 435 111623 17 163 888639 435 111361 16 163 888900 435 111100 15 163 889160 435 110840 14 163 889422 435 110579 13 1()3 889(;82 435 110318 12 163 889943 435 110057 11 163 890204 434 109796 10 164 9-890465 434 10-109535 9 164 890725 434 109275 8 164 890986 434 109014 7 164 891247 434 108753 6 164 891507 434 108493 5 164 891768 434 108232 4 164 892028 434 107972 3 164 892289 434 107711 2 164 892549 434 107451 1 164 892810 434 107190 I ^vm,% I I Sine I Cotang, I I fSPf. P? Pegr??8, 236 (38 Degrees.) LOGARITHMIC SINES, COSINES, ETC. M. Sine -D. 1 Cosine D. Tang. D. Cotang. 9-789342 269 9-896532 164 9-892810 434 10-107190 60 ] 789504 269 896433 165 893070 434 106930 59 2 789665 269 896335 165 893331 434 106669 58 3 789827 269 896236 165 893591 434 106409 57 4 789988 269 896137 165 893851 434 106149 56 5 790149 269 896038 165 8941 1 1 434 105889 55 6 790310 268 895939 165 894371 434 105629 54 7 790471 268 895840 J65 894632 433 105368 53 8 790632 268 895741 165 894892 433 105108 52 9 790', \fi 268 89;>641 165 8951,52 433 104848 51 10 790954 268 895542 165 895412 433 104588 50 II 9-791115 268 9-895443 166 9-895672 433 10-104328 49 1-2 791-275 i;3()2 431 093698 8 53 797777 261 891217 170 906.560 431 093440 7 54 797934 261 891115 170 9068 19 431 093181 6 55 798091 261 89 1013 170 907077 431 092923 5 56 79H247 261 89(1911 170 907.3.36 431 092664 4 57 79M403 260 890809 no 907.594 431 092406 3 58 79H5H() 260 890707 170 907852 431 092148 2 59 798716 260 890605 ■ 170 908111 430 091889 . 1 89 798872 260 890503 170 908369 430 091631 I Cosine | I Cotang. I Tang. 61 Degrees. LOGARITHMIC SINES, COSINES, ETC. (39 Degrees. ) M. S.ne D. Cosine n. T.n-. D. Cot.n:.?. ; 9-798872 260 9-89U503 170 9-9083G9 430 10-091031 60 1 799U28 260 890400 171 908628 430 091372 59 2 7i)9184 260 890298 171 90888G 430 091 1 14 58 3 799339 259 890195 171 909144 430 09035G 57 4 799495 259 890093 171 909402 430 090598 50 5 799651 259 889990 171 909060 430 090340 55 (') 7D980G 259 889888 171 909918 430 090082 54 7 799962 259 889785 171 910177 4.10 C89823 53 8 8:)01 17 259 889682 171 910435 430 0895G5 52 » 800272 258 889579 171 910693 430 089307 51 10 800427 258 889477 171 910951 430 089049 50 11 9-800582 258 9-889374 172 9-911209 430 10-088791 49 12 800737 258 889271 172 911467 430 088533 48 13 800892 258 889168 172 911724 430 088276 47 14 801047 258 889064 172 911982 430 088018 46 \:^ 801201 258 888961 172 912240 430 0877GO 45 IG 801356 257 888858 172 912498 430 087502 44 17 801511 257 888755 172 91-2756 430 087244 43 l^ 801065 257 888651 172 913014 429 086980 42 19 801819 257 888548 172 913271 429 086729 41 2U 801973 257 888444 173 913529 423 080471 40 21 9-802 J 28 257 9-888341 173 9-913787 429 10-086213 39 22 802-282 250 888^7 173 914044 429 085956 38 23 802436 256 888134 173 914302 429 085098 37 24 802589 256 888030 173 9145G0 429 085440 36 25 8;;2743 256 887926 173 914817 429 085183 35 26 802897 256 887822 173 915075 429 084925 34 27 803050 256 887718 173 915332 429 084668 33 28 803204 256 887614 173 915590 429 084410 32 29 803357 255 887510 173 915847 429 084153 31 30 803511 255 887400 174 916104 429 083896 30 31 9-803664 255 9-887302 174 9-916362 429 10-083638 29 32 803817 255 887198 174 916619 429 083381 28 33 803970 255 887093 174 916877 429 083123 27 34 804123 255 886989 174 917134 429 C82866 26 35 804276 254 886885 174 917391 429 082609 25 36 804428 254 886780 174 917648 429 082352 24 37 804581 254 886676 174 917905 429 082095 23 38 804734 254 886571 174 918163 428 081837 22 39 804886 254 886466 174 918420 428 081580 21 49 805039 254 886362 175 918677 428 081323 20 41 9-805191 254 9-886257 175 9-918934 428 10-081066 19 42 805343 253 886152 175 919191 428 0808)9 18 43 805495 253 886047 175 919448 428 080552 17 44 805647 253. 885942 175 919705 428 080295 16 45 805799 253 885837 175 919902 428 080038 15 46 805951 253 885732 175 920219 428 079781 14 47 806103 253 885627 175 920476 428 079524 13 48 806254 253 885522 175 920733 428 079207 13 49 806406 252 885416 175 920990 428 079010 11 5U 806557 252 885311 176 921247 428 078753 10 51 9-806709 252 9-885205 176 9-921.503 428 10-078497 9 52 806860 252 885100 176 921700 428 078-240 8 53 807011 252 884994 176 922017 428 077983 7 54 807163 252 884889 176 922274 428 077726 6 55 807314 252 884783 176 922530 428 077470 5 56 807465 251 884677 176 922787 428 077213 4 57 807015 251 884572 176 923044 428 076956 3 58 807766 251 884466 176 923300 428 076700 2 59 807917 251 884360 176 923557 427 876443 1 60 808067 251 884254 177 923813 427 076187 I Sine I ^ Couuag. \ I Tang. I M. ^J>«^re» 238 (40 Deg recs.) LOGARITHMIC SIKES, COSiyFS, ETC. M. 1 Sine 1 D. Cosine 1 D. 1 Tang. 1 D. 1 Cotan?. 1 9-«08(l67 251 9-884254 177 9-923813 427 W076 187^7 60 1 808218 251 884148 177 924070 427 075y;«) 59 2 808368 251 884042 177 924327 427 07.5673 58 3 808519 250 883936 177 924583 427 075417 57 4 8U8669 250 883829 177 924840 427 075160 56 5 808819 250 883723 177 92.5096 427 ' 074904 55 6 808969 250 883617 177 925352 427 074648 54 809119 250 883510 177 925609 427 074391 53 H 809269 250 883404 177 925865 427 074135 52 it 809419 249 883297 178 926122 427 073878 51 10 809569 249 883191 178 9-J6378 427 073622 50 11 9-809718 249 9-883084 178 9-926634 427 10-073366 49 J -2 809868 249 882977 .178 926890 427 073110 48 13 8)0017 249 882871 178 927147 427 072853 47 14 810167 249 882764 178 927403 427 072597 46 15 810316 '248 882657 178 927659 427 072341 45 16 8)0465 248 882550 178 927915 427 072085 44 IT 810614 248 882443 178 928171 427 071829 43 J8 810763 248 882336 179 928427 427 071573 42 J9 810912 248 882229 179 928683 427 071317 41 20 81 J 061 248 882121 179 928940 427 071060 40 21 9-81 1210 248 9-882014 179 9-929196 427 10-070804 39 22 811358 247 881907 179 929452 427 070548 38 23 811507 247 881799 179 929708 427 070292 37 24 811655 247 861692 179 929964 426 070036 36 25 811804 247 881584 179 930220 426 069780 35 26 811952 247 881477 179 930475 426 069525 34 27 812100 247 881369 179 930731 426 069269 33 58 8 12248 247 881201 180 930987 426 069013 32 29 812396 246 881153 180 931243 426 008757 31 30 812544 246 881046 180 931499 426 068501 30 31^ 9-812692 246 9-880938 180 9-931755 426 10-068245 29 32 812840 246 880830 180 932010 426 007990 28 33 812988 246 880722 180 932266 426 0G7734 27 34 813135 246 880613 180 932522 426 0G7478 26 35 813283 246 880505 180 932V'/8 426 GG7222 25 36 813430 245 880397 180 933033 426 0C69G7 24 37 813578 245 880289 181 933289 426 066711 23 38 813725 245 880180 181 933545 426 066455 22 39 813872 245 880072 181 933800 426 066200 21 40 814019 245 879963 181 934056 426 065944 20 41 9-814166 245 9-879855 181 9-934311 426 10-065689 19 42 814313 245 879746 18/ 9345()7 426 065433 18 43 814460 244 879637 181 934823 426 065177 17 44 814607 244 879529 181 935078 426 064922 16 45 814753 244 879420 181 935333 426 0646r)7 15 46 814900 244 879311 181 935589 426 064411 14 47 815046 244 879202 182 935844 426 064156 13 48 815193 244 879093 182 936100 426 0f;3900 12 49 815339 244 878984 182 936355 426 0(53645 11 50 815485 243 878875 182 936610 426 063390 10 51 9-8i.v;3i 243 9-8787(16 182 9-936866 425 10-0631.34 9 52 815778 243 878656 ]H2 937121 425 062879 8 53 815924 243 878547 182 937376 425 062624 7 54 8lti069 243 878438 182 9376.32 425 062.3<)8 6 55 816215 243 878328 182 937887 425 062)13 5 56 816361 243 878219 183 938142 425 0618.58 4 57 8:r,5(»7 242 878109 183 938398 425 001602 3 58 816652 242 877999 183 938653 425 061347 2 59 816798 242 877890 183 938908 425 061092 1 60 816943 242 877780 183 939163 425 060837 I Coauie i Sina ) I Cotang. J Tang. 1 .^Df^gnies. LOGARITHMIC iSIXES, fV SINES, ETC. ^41 Degrees.) 23? M. Sine D. Cosine 1). Tang. D. Cotang. 9-816943 242 9 877780 183 9-939163 425 10-060837 60 1 817088 242 877ti70 183 939418 425 060582 50 2 817233 242 W77.3f;0 183 939673 425 060327 58 3 817379 242 8774.50 183 939928 425 060072 57 4 817524 241 877;{40 183 940Jf33 425 059817 56 5 817668 241 877230 184 940438 425 059562 53 6 817813 241 877120 184 940094 4-.;5 039306 54 7 817958 241 877010 184 940949 423 059051 33 8 818103 241 876899 1^<4 941204 425 058796 32 9 81.-^*47 241 876789 184 941438 425 058542 31 10 81831)2 241 876678 184 941714 425 058286 50 11 9-81H536 240 9 876368 184 9-94 19G8 425 100.')8032 49 12 818681 240 87()457 184 942223 . 425 037777 48 i:j 818825 240 871)347 184 942478 425 037.322 47 14 818969 240 87ti236 185 942733 425 057267 46 J 5 819113 240 876125 185 i 942988 425 ■ 0.37(112 45 16 819257 240 876014 185 943243 425 05(:7.-,7 44 17 8194.11 240 875904 185 943498 425 05(_.3l/2 43 18 819.145 239 873793 183 9437 .")2 425 056248 42 19 8iyt;89 239 873682 185 944(107 425 055993 41 20 81ii832 239 875571 185 944262 425 055738 40 21 9-819976 239 9-875439 185 9-944317 425 10-05.5483 39 22 82012(1 239 873348 185 944771 424 053229 38 23 82021 >3 239 87.->237 185 943026 424 054974 37 '>4 821)406 239 873126 186 945281 424 034719 36 ^5 82()5.")0 238 873014 186 945335 424 054465 35 % 8206!)3 238 874903 186 945790 424 054210 34 27 820836 238 874791 18() 946045 424 053955 33 28 820979 238 874680 186 94(5299 424 0.33701 32 29 821122 238 874568 186 94(]554 424 05344(j 31 30 821265 238 874436 186 946808 424 053192 3C 31 9-821407 238 9-874344 186 9-947063 424 10052937 m 132 821530 238 874232 187 947318 424 052682 28 33 8210U3 237 874121 187 947572 424 052428 27 ."■1 821835 237 874009 187 947826 424 052174 2f 33 821977 237 873896 187 948(181 424 051919 25 36 822120 237 873784 187 948.136 424 051664 24 37 82-22:32 237 873G72 187 948390 424 051410 23 38 82241)4 237 8733C0 187 948844 424 051156 22 39 822546 237 87.3448 187 949099 424 050901 21 40 822688 236 873335 187 949333 424 050647 20 41 9-822830 236 9-873223 187 9-949607 424 10.050.393 19 42 822972 236 873110 188 949862 424 050138 18 43 823114 236 872998 188 950116 424 049884 17 44 823255 236 872885 1P8 950370 424 049630 16 45 823397 236 872772 IHR 950625 424 049375 15 46 823339 2.36 872639 188 950879 424 049121 14 47 823HS0 235 872.^47 188 951133 424 048867 13 48 823Hi>l 235 87LM-4 188 931388 424 048612 12 49 8239(i3 235 8723.21 188 95ir'»2 424 048338 11 50 824104 235 872208 188 95189<) 424 048104 10 51 9-824245 235 9-872093 189 9-9.32150 424 10-047850 9 52 8243H6 235 871981 189 9.32405 424 047595 a 53 824.V27 235 871H68 189 952659 424 047341 7 54 82461 « 234 8717.35 189 9.52913 424 047087 6 55 824808 234 871641 189 9.33167 423 046833 5 56 824949 234 871528 189 9.33421 423 046579 4 57 823090 234 871414 189 9.-)3675 423 046325 3 58 82.3230 2.34 871301 189 933929 423 046071 (4li JJogiccs.) LOiJAniTinilC SINES, COSINES, ETC. M. Sine 1 D. Cos.ne D. Ta„g. 1 D. 1 Cotang. 1 9-825511 234 9-871073 190 9-954437 423 10-045563 60 I 823(i51 233 870960 190 954691 423 0453(t9 59 2 825791 233 870846 190 954945 423 045055 58 3 825^31 233 870732 190 955200 423 044800 57 4 82607 1 233 870618 190 955454 423 044546 56 5 8262 11 233 870504 190 955707 423 044293 55 6 826351 233 870390 190 955961 423 044039 54 7 826491 233 870276 190 956215 423 043785 53 8 826631 233 870161 190 956469 423 043531 52 9 826770 232 870047 191 956723 423 043277 51 10 826910 232 869933 191 956977 423 043023 50 11 9-827049 232 9-869818 191 9-957231 423 10-042769 49 V2 827189 232 869704 191 957485 423 042515 48 i:t 827328 232 869589 191 957739 423 042261 47 14 827467 232 869474 191 957993 423 042(»07 46 15 827606 232 8693C0 191 958246 423 041754 45 16 827745 232 869245 191 958500 423 04 1500 44 17 ■ 827884 231 869:30 191 958754 423 041246 43 18 828023 231 869015 192 959008 423 04(<992 42 19 828162 231 868900 192 9592(52 423 040738 41 20 828301 231 868785 192 959516 423 040484 40 2] 9-828439 231 9-86P670 192 9-959769 423 10-040231 39 22 828578 231 8()8555 192 960023 423 039977 ;i8 23 828716 231 8()8440 192 900277 423 039723 37 24 828855 230 8()8324 192 900531 423 039469 36 25 828993 230 8(i8209 192 960784 423 039216 35 26 829131 230 868093 192 961038 423 02C962 34 27 829269 230 867978 193 961291 423 Gr07C9 33 28 829407 230 86:8i;2 193 901545 423 occ-:c5 52 29 829545 230 867747 193 901799 423 C38201 31 30 829683 230 867631 193 962U52 423 037C48 30 31 9-829821 229 9-867515 193 9-962306 423 10-C37C94 29 32 829959 229 867;!:i9 193 9C25C0 423 037440 28 33 830097 229 8(J7283 193 9C2813 423 037187 27 34 830234 229 867167 193 963067 423 036933 26 35 830372 229 867051 193 903320 423 036680 25 3t) 830509 229 86(3935 194 963574 423 036426 24 37 830646 229 866819 194 903827 423 036173 23 38 830784 229 866703 194 9G4081 423 035919 22 39 830921 228 8<)(5586 194 964335 423 035665 21 40 831058 228 866470 194 964588 422 035412 20 41 9-831195 228 9-866353 194 9-964842 422 10035158 19 42 831332 228 806237 194 965095 422 034905 18 43 831469 228 866120 194 965349 422 034651 17 44 831606 228 866004 195 965602 422 034398 16 45 831742 228 865887 195 965855 422 034145 16 46 831879 228 8P5770 195 966109 422 033891 14 47 832015 227 865653 195 9()63()2 422 033638 13 48 832152 227 80.':536 195 966616 422 033384 12 49 832288 227 865419 195 966869 422 033131 11 50 832425 227 865302 195 967123 422 032877 10 51 9-832561 227 9'865185 195 9-967376 422 10032624 9 52 832697 227 865068 195 967629 422 032371 8 53 832833 227 864950 195 967883 422 032117 7 54 832969 226 864833 196 968136 422 03IIS64 6 55 833105 226 864716 196 968389 422 031611 5 56 833241 226 864598 196 968643 422 031357 4 57 833377 226 864481 196 968896 422 031104 3 58 833512 226 864363 196 969149 422 030851 2 59 833648 226 864245 196 969403 422 030597 1 60 833783 226 864127 196 969656 422 030344 Cotang. ^ Tang. 47 Degrees. LOGARITHMIC SINES, COSINES, ETC. (43 Degrees.) 241 I D. I Cosii I Cotaii^, 9-833783 226 9-864127 196 9-9G9656 4-.-2 10-030344 1 60 1 8339 J 9 225 864010 196 9G9909 422 030091 59 2 834054 225 863892 197 970 162 422 029838 58 3 834189 225 863774 197 970416 422 029584 57 4 834325 225 863656 197 970669 422 029331 56 5 8344G0 225 863538 197 970922 422 029078 55 6 834595 2-25 863419 197 971175 422 028825 54 7 834730 225 863301 197 971429 422 028571 53 & 8348G5 225 863183 197 971682 422 028318 52 9 834999 224 8630G4 197 971935 422 028(.C5 51 10 835134 2-24 8G2946 198 972 188 422 027812 50 11 9-835269 224 9-862827 198 9-972441 422 10-027559 49 12 835403 224 862709 iJ8 972694 422 027306 48 13 835538 224 862590 198 972948 422 027052 47 14 835672 224 86247] 198 973201 4-22 G26799 15 835807 224 862353 198 973454 422 C2G54G 45 IG 835941 224 862234 198 973707 422 026293 44 17 &3G075 223 8C2115 198 973960 422 02G040 43 18 83G209 223 8G199G 198 974213 4-22 025787 4 -J 19 836343 223 861877 198 9744G6 422 C25534 41 20 83G477 223 8G1758 199 974719 422 025281 40 21 9-836611 223 9-861C38 199 9-974973 4-22 10-025027 39 22 836745 223 8GI5J9 199 97522G 422 02477-1 38 23 836878 223 8G1400 1D9 975479 422 (;:4.:2i 24 837012 222 861280 199 975732 422 024268 36 25 837146 222 8GIIG1 1J9 9751.85 422 G24015 35 2G S37279 222 8G1U41 199 97G238 422 0237C2 34 27 837412 222 860922 199 97G491 422 0235G9 33 28 837546 ooo 860802 199 97G744 422 023256 .'52 29 837679 222 860682 200 97G997 422 023003 31 30 837812 222 8;;o562 200 977250 422 C22750 30 31 9-837945 222 9-860442 200 9-977503 422 10-02-2497 29 32 838078 221 8G0322 200 977756 422 022244 28 33 838211 221 860202 200 978009 422 02 J 991 27 34 838344 221 8G0082 200 9782G2 422 021738 2() 35 838477 221 8599G2 200 978515 422 021485 25 3G 838G10 221 859842 200 97t<7G8 422 021232 24 37 838742 221 859721 201 !y7'JU21 422 02U979 23 38 838875 221 859601 201 t)'7' NAT. C OSINE. NATURAL SINES. 24ri / 8° 9° 10° 11° 12° 13° 14° 1-:° / 1391731 156 4345 173 6482 190 8090 •207 9117 2-24 9511 241 9219 258 8190 CO i 1 4G12 721S 93-lG 191 0945 •208 1902 225 2345 2 1:2 2041 250 LXI 5'J ., 7492 157 0091 174 2211 3801 4807 5179 4863 3810 58 3 140 0372 29Gn 5075 6656 7652 8013 7G85 eeio 57 4 3252 583t 7939 9510 •209 0497 226 0846 •243 0507 9428 56 5 G132 8705 175 0803 •192 2365 3341 3680 3329 260 2-237 55 6 9012 158 1581 3G67 5220 6186 6513 6150 5045 b\ 7 1411892 4453 6531 8074 9030 934G 8971 7853 53 8 ' 4772 7325 9395 •193 0928 •210 1874 227 2179 2441792 261,0662 52 9| 7651 159 0197 176 2258 3782 4718 5012 4G13 3469 51 10 142 0531 3069 6121 6636 7561 7844 7438 6277 50 11 341U 5940 7984 9490 •211 0405 228 0677 245 0254 90S; 4:) 12 6289 8812 177 0847 •194 2344 3248 3509 3074 '262189^ 48 13 j 91Gb •IGO 1GS3 3710 5197 6091 0341 5891 469; 17 14 1 143 2047 4555 6573 8050 8934 9172 8713 750 40 15 i 492u 7426 9435 •195 0903 •2121777 2-29 2004 246 1533 ■203 031: 15 1(5 i 7805 •161 0297 178 2298 3756 4619 4835 4352 311' 44 17 ■•144 0684 3167 5160 6609 7462 7666 7171 592; 43 IS 3562 6038 8022 9iCl •213 0304 230 0497 9990 S73-; 42 19 6440 8909 •179 0884 •196 2314 3146 332S 247 2809 2G4153i 41 20 9319 •162 1779 3746 5166 5988 6159 50'27 4342 40 21 •145 2197 4650 C607 8018 8829 8989 8445 7147 39 22 5075 7520 9469 •197 0870 •2141671 231 1819 248 1263 9952 38 23 7953 •163 0390 •180 2330 3722 4512 4649 4081 265 2757 37 24 •146 0830 3260 5191 6573 7353 7479 6899 5561 36 25 3708 6129 8052 9425 •215 0194 232 0309 9716 8366 35 26 6585 8999 -181 0913 •198 2276, 30351 3138 •249 2533 2661170 34 27 9463 •1641868 3774 5127 5876 5967 5350 3973 33 28 •147 2340 4738 6635 7978 8716 8796 8167 6777 32 2'J 5217 7607 9495 •199 0829 •2161556 •2331625 ■250 0984 9581 31 30 8094 •165 0476 ■182 2355 3679 4396 4454 3800 -267 2384 30 31 •148 0971 3345 5215 6530 7236 7282 6616 5187 29 32 3848 6214 8075 9380 •217 0076 •234 0110 9432 7989 28 33 6724 9082 •183 0935 •200 2230 2915 2938 •251 2248 •268 0792 27 34 9601 •1661951 3795 5080 5754 5706 5063 3594 20 35 •149 2477 4819 6654 7930 8593 8594 7879 6396 25 36 5353 7687 9514 •201 0779 •218 1432 •235 1421 •252 0694 ■ 9198 24 37 8230 •167 0556 ■184 2373 3629 4271 4248 3508 •2^9 2000 23 38 •150 not 3423 5232 6478 7110 7075 63-23 4801 22 J9 3981 6291 8091 9327 9948 9902 9137 7602 21 to 6857 9159 •185 0949 •202 2176 •219 2786 •236-2729 •253 1952 ■270 0403 20 si ' 973:^ •168 2026 3808 5024 5624 5555 4766 3204 19 12 1 1512608 4894 6666 7S73 8462 8381 7579 6004 18 43 1 5484 7761 9524 •203 0721 •2201300 •237 1207 •2540393 8805 17 44 I 8359 •169 0628 •186 2382 3569 4137 4033 3206 •271 1605 16 45 1521234 3495 5240 6418 6974 6859 6019 4404 15 46 410'j G362 8098 9265 9811 9684 8S32 7204 14 47 6984 9-228 •187 0956 •204 2113 •221 2648 •238 2510 •255 1645 •2720003 13 48 985S •170 2095 3813 4961 5485 5335 4158 2802 12 49 ■153 2732 4961 6670 7808 8321 8159 7270 5601 11 50 5607 7828 9528 •205 065S •2221158 •239 0984 ■256 0082 8400 10 51 848--; •171 0694 •188 2385 3502! 3994 3808 2894 •2731198 9 52 •1541351 3560 5241 6349 6830 6633 5705 3997 8 53 423C 6425 8095 91951 9666 9457 8517 6794 7 54 710-1 9291 •189 0954 •206 2042 -223 2501 •240 2280 •257 1328 9592 6 55 9975 •172 215P 3811 4888 5337 5104 4139 •274 2390 5 56 •155 285] 5022 6667 77341 8172 79-27 6950 5187 4 57 572e 7887 9523 -207 0580 -224 1007 •241 0751 976( 79S4 3 58 8595 •173 0752 ■190 237C 3426, 3842 3574 -258 -2570 •275 0781 2 59 •156 1471 3:;i7 523^ t 6272 6676 G39C 5381 3577 1 60 434^ ) 648i 809( ) 9117 9511 9219 8190 6374 / 81° 80° 79° 1 78° 77° 76° 75° 74° / ; NAT. COSINE 246 NATURAL SINES. 16° 17^^ 1 18° 19° 20° 21° 22° 1 23° ! / 275 C374 -292 37171 309 0170 325 5682 342 0201 858 3679 374 6066 •390 7311' 60 9170 64991 2936 8432 2935 6395 8763 9989 59 276 1965 9280 5702 326 1182 5668 9110 3751459.3912666 58 47C1 293 2061 8468 3932 8400 359 1825 4156 5343; 67 755C 4842 310 1-234 6681 343 1133 4540 6852 8019' 56 277 0352 7623 3999 9430 3805 7254 9547 •392 0695! 55 3147 294 0403 6764 327 2179 6597 9968 376 2243, 3371] 54 5941 3183 95«9 4928 9329 360 2682 4938 6047 53 873e 5963 311 2294 7676 3442060 5395 7632 8722! 62 278 1530 8743 5058 328 0424 4791 8108 377 0327 3931397' 61 4324 295 1522 0-, 0*^^22 3172 7521 •361 0821 3021 4071 50 7118 4302 312 058(3 5919 345 0252 3534 5714 6745 49 9911 7081 3349 8666 2982 62461 8408! 9419 48 279 2704 9859 6112 329 1413 5712 8958 378 1101 394 2093 47 5497 296 2638 8875 4160 8441 •362 1669 3794 4766 46 8290 5416 •313 1638 6906 346 1171 4380 6486 7439 45 2801083 8194 4400 9653 3900 7091 9178 395 0111. 44 3875 297 0971 7163 330 2398 6628 9802 •379 1870 2783 43 6667 3749 9925 5144 9357 •363 2512 4562 5455 42 9459 6526 •314 2686 7889 347 2085 5222 7253 8127 41 281 2251 9303 5448 331 0634 4812 7932 9944 •396 0798 40 5042 •298 2079 8209 3379 7540 ■364 0641 380 2634 3468 39 7833 4856 •315 0969 6123 •348 0267 3351 5324 6139 38 282 0624 7632 3730 8867 2994 6059 8014 8809 37 3415 •299 0408 6490 •3321611 5720 8768 •381 0704 •397 1479 36 6205 3184 9250 4355 8447 •365 1476 3393 4148 36 8995 5959 •316 2010 7098 •349 1173 4184 6082 6818 34 2831785 8734 4770 9841 3898 6891 8770 9486 33 4575 •3001509 7529 •333 2584 6624 9599 •3821459 ■398 2155 32 7364 4284 •317 0288 5326 9349 •366 2306 4147 4823 31 2840153 7058 3047 8069 •350 2074 5012 6834 7491 30 2942 9832 5805 •3340810 4798 7719 9522 .399 0158 29 5731 •301 2606 8563 3552 7523 ■367 0425 •383 2209 2825 28 8520 5380 •318 1321 6293 ■351 0246 3130 4895 5492 27 285 1308 8153 4079 9034 2970 5836 7582 8158 26 4096 •302 0926 6836 •3351775 5693 8541 •384 0268 •400 0825 25 6884 3699 9593 4516 8416 ■368 1246 2953 3490 24 9671 6471 •319 2350 7256 •3521139 3950 5639 6IS6 23 •2862458 9244 5106 9996 3862 6654 8324 8821 22 524e ■303 2016 7863 •336 2735 6584 9358 •385 1008 •401 1486 21 8032 4788 •320 0619 5475 9306 ■369 2061 3693 4160 20 •287 0819 7559 3374 8214 •353 2027 4765' 6377 6814 19 3605 •304 0331 6130 •337 0953 4748 7468 9060, 9478 18 6391 3102 8885 3691 7469 •370 0170 •386 1744 •402 2141 17 9177 5872 •321 1640 6429 •3540190 2872 4427 *" 4804 16 •2881963 8643 4395 9167 2910 5574 7110 7467 16 4748 •305 1413 7149 •338 1905 5630 8276 9792 •403 0129 14 7533 4183 9903 4642 8350 •371 0977 •387 2474 *" 2791 13 •289 0318 6953 •3222657 7379 •355 1070 3678 6156 6453 12 3103 9723 5411 •339 0116 3789 6379 7837 8114 11 5887 •306 2492 8164 2852 6508 9079 •388 0518 •404 0775 10 8671 5261 •3230917 5589 9226 •3721780 3199 3436 9 •2901455 8030 3670 8325 •3561944 4479 5880 6096 8 4239 •307 0798 6422 •340 1060 4662 7179 8560 8756 7 7022 3566 9174 3796 7380 9878 •389 1240 •405 1416 6 9805 6334 •3241926 6531 •357 0097 •373 2577 3919 4075 5 •291 2588 9102 4678 9265 2814 5275 6598 6734 4 5371 ■308 1869 7429 •341 2000 5531 7973 9277 9393 3 815S 463e •325 0180 4734 8248 •374 0671 •390 1955 •406 2051 2 •292 093E 740S 2931 7468 •358 0964 3369 4633 4709 1 3717 •309 017C 5682 •342 0201 367S 6066 7311 7366 73° 72° 71° 70° ' NAT. < 69° 30SINE. 68° 1 67° 66° / NATURAL SINES. 247 24° •406 7366 •407 0024 2881 5337 7993 ■408 0649 3305 5960 8615 ■409 1269 3923 65 ■ 9230 •410 1883 4536 7189 9841 ■411 2492 5144 7795 •412 0445 3096 5745 8395 •413 1044 3693 6342 8990 •414 1638 4285 6932 9579 •415 2226 48' 7517 •416 0163 2803 5453 8097 •417 0741 3385 6028 8671 •418 1313 3956 659 9233 •419 1880 4521 7161 9801 •420 2441 5080 7719 •421 0358 2996 5634 8272 •422 0909 3546 6183 65 25° I 422 6183 8819 423 1455 4090 6725 9360 424 1994 4628 7262 9895 425 2528 5161 7793 426 0425 3056 5687 8318 427 0949 3579 6208 8838 428 1467 4095 6723 9351 429 1979 46061 7233 9859 430 2485 5111 7736 431 0361 2986 5610 8234 432 0857 3481 6103 8726 433 1348 3970 6591 9212 434 183: 4453 7072 9692 435 2311 4930 7548 430 0166 2784 6401 8018 437 0634 3251 5866 8482 •438 1097 3711 64° 26° 27° 28° 1 •438 3711 •453 9905 ^469 4716 • 6326 •454 2497 7284 • 8940 50881 9852 •4391553 7679 -470 2419 4166 •455 0269 4986 6779 2859 7553 • 9392 5449 •4710119 440 2004 8038 2685 4615 •456 0627 5250 7227 3216 7815 • 9838 5804 •472 0380 441 2448 8392 2944 5059 •457 0979 5508 7668 356G 8071 • 442 0278 6153 •473 0634 2887 8739 3197 5496 •458 1325 5759 8104 3910 8321 • •443 0712 6496 •474 0882 3319 9080 3443 5927 •459 1665 6004 8534 4248 8564 •444 1140 6832 •475 1124 3746 9415 3683 6352 •460 1998 6242 8957 4580 8801 •445 1562 7162 ■4761359 4167 9744 3917 6771 ■461 2325 6474 9375 4906 9031 •4461978 7486 •477 1588 4581 •462 0066 4144 7184 2646 6700 9786 5225 9255 •447 2388 7804 •478 1810 4990 •463 0382 4364 7591 2960 6919 •448 0192 5538 9472 2792 8115 •479 2026 5392 ■464 0692 4579 7992 3269 7131 •449 0591 5845 9683 3190 8420 •480 2235 5789 ■465 0996 4786 8387 3571 7337 •450 0984 6145 9888 3582 8719 •481 2438 6179 •466 1293 4987 8775 3866 7537 •4511372 6439 •482 0086 3967 9012 2634 6563 •467 1584 5182 9158 4156 7730 •4521753 6727 •483 0277 4347 9298 2824 C941 ■468 1869 5370 9535 4439 7916 •453 2128 7009 •484 0462 4721 9578 3007 7313 •469 2147 5552 9905 4716 8096 63° 62° 61° 29° ! 484 8096 ■ 485 06401 3184J 57-271 8-2701' •486 08121 33541 5895! 8436 •487 0977 3517 6057 8597 •488 1136 3674 6-212 8750 •489 1288 3825 6361 8897 •490 1433 3968 6503 9038 •491 1572 4105 6638 9171 •492 1704 4236 0767 9298 •493 18-29 4359 6889 9419 •494 1948 4470 7005 95321 •495 2060 4587 7113 9639 •496 2165 4690 7215 9740 •497 2264 4787 7310 9833 ■498 2355 4877 7399 9920 •499 2441 4961 7481 ■500 0000 60° 30° 31° 500 0000 •515 0381 2519 2874 5037 5367 7556 7859 •501 0073 •516 0351 2591 2842 5107 5333 7624 7824 •502 0140 •517 03U 2655 2804 5170 5293 7685 7782 503 0199 ■518 0-270 2713 2758 5227 5246 7740 7733 ■504 0252 519 0219 •2765 2705 5-276 5191 7788 7676 •505 0298 ■520 0161 2809 2646 5319 5130 78-28 7613 ■506 0338 521 0096 2846 2579 5355 5061 7863 7543 ■507 0370 •522 0024 2877 2505 5384 4986 7890 7466 •508 0396 9945 2901 •523 2424 5406 4903 7910 7381 ■509 0414 9859 2918 ■524 2336 5421 4813 79-24 7290 •510 042C 9766 292S ■525 2-241 54-20 4717 793i- 7191 •511 0431 9665 2331 •526 2139 5431 4613 7930 7085 ■512 04-29 9558 2927 •527 ^2030 5425 4502 7923 697.:' ■513 04-20 9443 2916 •528 1914 5413 4383 7908 685S •514 0404 9322 2899 •529 1790 5393 4258 7887 6726 •515 0381 9193 59° 58° NAT. COSINE. 248 NATURAL SINES. 32° 33° 34° 35° 36° 37° 38° 39° 1 f 629 9193 544 6390 559 1929 573 5764 587 7853 601 8150 615 6615 ^629 3204 60 530 1659 8830 4340 8147 588 0206 e02.0473 8907 i 5464 59 ^ 4125 545 1269 6751 574 0529 2558 2795 GIG 1198: 7724 58 i G591 37071 ylGi: 2911 4910 5117 3489 9983 57 ; 9057 0145 560 157:^ 5292 7262 7439 5780 •630 2242 56 ■■ 5311521 8583 39S1 1 5651 7661 9671 ■723 1681 7705 9712 7241719 3724 5729 7734 9738 725 1741 3744 5746 774 9748 7261748 3748 5747 7745 9743 727 1740 47° 731 3537 5521 7503 9486 782 1467 3449 5429 7409 938^ .7331367 51 3345 5322 7299 9-275 734 1250 3225 5199 7173 914C 735 1118 5061 7032 9002 736 0971 2940 4908 6875 8842 •737 0808 2773 4738 6703 8666 •738 0629 2592 4553 6515 84' •739 0435 3736 5732 7728 97-22 728 1716 3710 5703 7695 2394 4353 6311 8268 740 0225 2181 413 6092 8040 ■7291677/7410000 3668 i 1953 10 50571 390r 9 7646 5857 8 90351 7808 7 730 1 023 9758 6 3610-7421708 5 5597 365F 4 7583 5606 3 9568 7554 2 •731 15531 9502 1 35371 743 1448 43° 1 42° / NAT. COSINE 50 JVA run A L S INES 48° I 49° •7-t3 1448 ' -754 709G 3394 ' 9004 5340 ' -755 0911 728:) 9229 i 7411173! 3115 5058 6999 8941 •745 0881 2821 4760 669 J 8636 •746 0574 2510 4446 0382 8317 •747 0251 2184 4117 6049 7981 9912 •748 1842 3772 5701 7629 . 9557 •74£ 1484 3411 5337 7202 9187 •7501111 3034 I 4957 I 687 U i 8800! •751 0721 2641 ; 4561 6480 8398 •752 0316 2233 4149 6065 V980 9894 •753 1808 3721 5634 7546 9457 •75U3G8 3278 5187 7096 41° 281 X 4724 C630 8535 •756 0439 2342 424'J 6148 8050 9951 •757 1851 3751 5050 7548 9446 ■7581343 3240 5136 7031 8926 •759 0820 2713 4606 6498 8389 •760 02S0 2170 4060 594;) 7837 •7011011 3197 53t)3 7 268 9152 •762 1U36 ■7641714 3590 5465 7340 9214 ■765 1087 2960 4S32 6704 8574 ■766 0444 40° 50° 766 0444 • 2314 4183 ; 6051 7918, 9785 • ■767 1652 I 3517 5382 i 7246: 9110, ■768 0973 • 2835 4697 t 6558! 8418 1 •769 0278 • 2137 ' 3996: 5853: 7710 I 9567 ! •770 1423 1 • 3278 5132 I 6986 i 8840 1 •771 0692 I ■ 2544 i 4395 6246 8095 i 9945 1 121794; 3642 I 5189 I 7336; 9182 I r31027 ■ 2872 2919 4716 4802 6559 6683 8402 8561 •774 0244 63 0445 2086 2325 392o 4-204 5767 6082 7606 7960 9445 9838 •775 1-283 3121 4J57 6791 80-29 ■776 0464 2298 ■ 4132 6965 7797 9629 •777 1460 39° 51° 52° 53° 777 1460 •788 0108 •798 6355 3290 1898 8105 5120 3688 9855 6949 5477 •799 1604 8777 7266 3352 •778 0604 9054 5100 2431 •789 0841 6847 4258 2627 8593 6084 4413 •800 0338 7909 6198 2083 9733 7983 3827 •779 1557 9767 5571 3380 •790 1550 7314 5202 3333 9050 7024 5115 •801 0797 8845 6896 2538 •780 0665 8676 4278 2485 •791 0456 6018 4304 2235 7756 6^23 4014 9495 7940 5792 •802 1232 9757 7569 2969 •781 1574 9345 4705 3390 •792 11-21 6440 5205 2896 8175 7019 4671 9909 8833 6445 •8031642 •782 0646 8218 3375 2459 9990 5107 4270 •793 1762 6838 6082 3533 8569 7892 5304 •8040299 9702 7074 2028 •7831511 8843 3756 3320 •794 0611 5484 51-27 2379 7211 6935 4146 8938 8741 5913 •805 0664 •784 0547 7678 2389 2352 9444 4113 4157 ■795 1208 5837 t901 2972 7560 7764 4735 9-283 9566 6497 •806 1005 •785 1368 8259 2726 3169 •796 0020 4U6 4970 1780 6106 6770 3540 7885 8569 5-299 9603 •786 0367 7058 •807 1321 2165 8815 3038 3963 •797 0572 4754 5759 2329 0470 7555 4084 8185 9350 5833 9899 •787 114G 7594 •8081612 2339 9347 3325 4732 •7981100 5037 6524 •,^853 6749 8316 4604 8460 •788 010S 6355 •809 0170 38° 37° 36° 54° 809 0170 1879 3588 5296 7004 8710 810 041C 21-22 3826 5530 7234! 8936 I 811 0638 2339 4040 5740 7439 9137 812 08135 2532 4229 5925 7620 9314 813 1008 2701 4393 6084 7775 9466 8141155 2844 4532 6220 7906 9593 815 1-27 2963 4647 6330 8013 9695 8161376 3056 4736 6416 8094 977: 817 1449 3125 4801 6476 8151 9S'24 •SIS 1497 3169 4S41 i--.v: 818-J 9852 -819 1520 35° NAT. COSINK. I,\iTURAL SINES. 251 60 65° 56° 57° 58° 59° 60° 61° 819 1520 •8-29 0376 •838 6706 •848 0481 •857 1673 •866 0254 •S74 6197 3189 2002 8290 2022 3171 1708 7607 4856 3628 9873 3562 4068 3161 9016 6523 5252 •839 1455 510-! 6164 4014 •875 0425 8189 6877 3037 0641 76C0 60G6 1832 9854 8500 4618 8179 9155 7517 3239 •820 1519 •830 0123 6199 9717 •858 0649 89C7 4&45 3183 1745 7778 •849 1254 2143 •867 0417 6051 4840 3366 9357 •2790 3635 1866 7455 6509 4987 •840 0936 4325 5127 3314 8859 8170 6607 2513 5860 6619 4762 ■876 0263 9832 8220 4090 7394 8109 6209 1665 •821 1492 9845 5066 8927 9599 7C55 3067 3152 •831 14C3 7241 •850 0459 •859 108S 9100 4468 4811 3080 8816 1991 2576 •868 05 44 5SG8 6469 4696 ■8410390 3522 4064 1988 7268 8127 6312 1963 5053 5551 3431 8660 9784 7927 3536 65S2 7037 4874 •877 0004 •8221440 9541 5108 8111 85'23 6315 1402 3096 •8321155 6679 9639 •800 0007 7750 2858 4751 2768 8249 •851 1167 1491 9196 4254 6405 4380 9319 2693 2975 •869 0630 5049 8659 5991 •842 1388 4219 4457 2074 . 7043 9712 7602 2956 5745 5939 3512 8437 •8231364 9212 45^24 7269 7420 4949 9830 3015 ■833 0822 6091 8793 8901 6380 •878 1222 4666 2430 7657 •852 0316 •8610380 ■^821 2613 6316 4038 9222 1839 1859 9256 4004 7965 5640 ■843 0787 3360 3337 ■870 0691 5394 9614 7252 2351 4881 4815 2124 6783 •824 1262 8858 3914 6402 6292 3557 8171 2909 •834 0463 5477 7921 7768 4989 9559 4556 2068 7039 9440 9243 6420 •879 0946 6202 3672 8600 •853 0958 •802 0717 7851 2332 7847 5275 •844 0161 2475 2191 9281 3717 9491 6877 1720 399 J 3' < ; ■■^710710 5102 •825 1135 8479 3273 55uo 5i;;-, 2138 6486 2778 •835 0080 4838 7u23 oeos 3566 7869 4420 1680 6395 8538 8079 4993 9251 6062 3279 7952 •8540051 9549 6419 •880 0633 7703 4878 9508 1564 ■8631019 7844 2014 9343 6476 •845 1004 3077 2488 9269 3394 •826 0983 8074 2618 4588 3956 •872 0693 4774 2622 9670 4172 6099 5423 2116 6152 4260 •836 1266 5726 7609 6889 3538 7530 5897 2862 7278 9119 8355 4960 8907 7534 4456 8830 •855 0627 9820 6381 •881 0284 9170 6050 •846 0381 2135 •8641^284 7801 1660 •827 0806 7643 1932 3643 2748 9221 3035 2440 9^236 3481 5149 4211 •873 0640 4409 4074 •837 08-27 5030 6G55 5673 2058 5782 5708 2418 6579 8160 7134 3475 7165 7340 4009 81-26 9664 8595 4891 8527 8972 5598 9673 •856 lies ■S05 0065 6307 9898 •828 0603 7187 ■847 r219 2671 1514 77^22 ■882 1269 •2234 8775 2765 4173 2973 9137 2638 3864 •838 03a3 4309 5674 4430 •874 0550 4007 5493 1950 5853 7175 6887 1963 5376 7121 3530 7397 8675 7344 3375 C743 8749 5121 8939 •857 0174 8799 4786 8110 •829 0376 6706 •848 0481 1673 •866 0254 6197 9476 34° 33° 82° 31° 30° 29° 28° NAT. COSINK. 252 NA TUBAL SINES. 62° 63° 64° 65° 66° 67° 68° •882 947(3 .891 00G5 •898 7940 •906 3078 •913 5455 •920 5049 •927 1839 •883 0841 1385 9215 4307 6637 6185 2928 2206 2705 •899 0489 5535 7819 7320 4016 3569 4024 1763 6762 9001 8455 5104 4933 5342 3035 7989 •9140181 9589 6191 6295 6059 4307 9215 1361 .921 0722 7277 7656 7975 5578 •907 0440 2540 1854 8363 9017 9291 6848 1665 3718 2986 9447 •8840377 •892 0606 8117 2S88 4[:.: 4116 •928 0531 1736 1920 9386 4111 ec7- 5246 1614 3095 3234 •900 0654 5333 7247 6375 2696 4453 4546 1921 6554 8422 7504 3778 5810 5858 3188 7775 9597 8632 4858 7166 7169 4453 8995 ■915 0770 9758 5938 8522 8480 5718 •908 0214 1943 •922 0884 7017 9876 9789 6982 1432 3115 2010 8096 •885 1230 ■893 1098 8246 2649 4286 3134 9173 2584 2406 9508 3866 5456 4258 •9290250 3936 3714 •9010770 5082 6626 6381 1326 5288 5021 2031 6297 7795 6503 2401 6639 6326 3292 7511 8963 7624 3475 7989 7632 4551 8725 •916 0130 8745 4549 9339 8936 5810 9938 1297 9865 5C22 •886 0688 •894 0240 7068 ■9091150 2462 •923 0984 6694 2036" . 1542 8325 2361 3627 2102 7765 3383 2844 9582 3572 4791 3220 8835 4730 4146 ■902 0838 4781 5955 4336 9905 6075 5446 2092 5990 7118 6452 ■930 0974 7420 6746 3347 7199 8279 6567 2042 8765 8045 4600 8406 9440 7682 3109 •887 0108 9344 5853 9613 •917 0601 8795 4176 1451 •895 0641 7105 •910 0819 1760 9908 5241 2793 1938 8356 2024 2919 •9241020 6306 4134 3234 9606 3228 4077 2131 7370 5475 4529 ■903 0856 4432 6234 3242 8434 6815 5824 2105 5635 6391 4351 9496 8154 7118 3353 6837 7546 5460 •931 0558 9492 8411 4600 8038 8701 6568 1619 •888 0830 9703 5847 9238 9855 7676 2679 2166 .896 0994 7093 •911 0438 •918 1009 8782 3739 3503 2285 8338 1637 2161 9888 4797 4838 3575 9582 2835 3313 •925 0993 5855 6172 4864 •9040825 4033 4464 2097 6912 7506 6153 2068 5229 5614 3201 7969 8839 7440 3310 6425 6763 4303 9024 •889 0171 8727 4551 7620 7912 5405 •932 0079 1503 ■897 0014 5792 8815 9060 6506 1133 2834 1299 7032 •912 0008 ■919 0207 7606 2186 4164 2584 8271 1201 1353 8706 3-238 5493 3868 9509 2393 2499 9805 4290 6822 5151 •905 0746 3584 3644 •926 0902 5340 8149 6433 1983 4775 4788 2000 6390 9476 7715 3219 5965 5931 3096 7439 •890 0803 8990 4454 7154 7073 4192 8488 2128 ■898 0276 5088 8342 8215 5286 9535 3453 1555 6922 9529 ■ 9356 6380 •933 0582 4777 2834 8154 ■913 0716 •920 0496 7474 1628 6100 4112 9386 1902 1635 8566 2673 7423 5389 ■906 0618 3087 2774 9658 3718 8744 6665 1848 4271 3912 •927 0748 4761 •891 0065 7940 3078 5455 5049 1839 5804 27° 26° 25° ■>\o 23° 22° 21° NAT. COSINE. NATURAL SINES. 25?, 69° 70° 933 5804 •939 6926 6846 7921 7888 8914 8928 9907 9968 •940 0899 gai 1007 1891 1 2045 2881 ; 3082 3871 4119 4860; 5154 5848 6189 6835 7223 7822 8257 8808 9289 9793 •935 0321 •941 0777 1352 1760 1 2382 2743' 3412 3724 4440 4705 5468 5686 6495 6665 7521 7644 8547 8621 9571 9598 •936 0595 •942 0575 1618 1550 2641 2525 3662 3498 4683 4471 5703 5444 6722 6415 7740 7386 8758 8355 9774 9324 •937 0790 •943 0293 1806 1260 2820 2227 3833 3192 4846 4157 5858 5122 6869 6085 7880 7048 8889 8010 9898 8971 •938 0906 9931 1913 •944 0890 2920 1849 3925 2807 4930 3764 5934 4720 7940 6630 8942 7584 9943 8537 0943 9489 1942 •945 0441 2340 1391 3938 2341 4935 3290 5931 4238 6926 5186 0° 19° 71° 72° 945 5186 •951 0565 6132 1464 7078 2361 8023 3258 89(58 4154 9911 5050 94G0854 5944 1735 6838 2736 7731 3677 8623 4616 9514 5555 •952 0404 6493 1294 7430 2183 8366 3071 9301 3958 917 0236 4844 1170 5730 2103 6615 3035 7499 3966 S3S2 4897 9264 5827 •953 0146 6756 1027 7684 1907 8612 2786 9538 3664 •948 04tU 4542 1389 5418 2313 6294 3237 7170 4159 8044 5081 8917 6002 9790 6922 •954 0662 7842 1533 8760 2403 9678 3273 •949 0595 4141 1511 5009 2426 5876 3341 6743 4255 7608 5168 8473 6080 9336 6991 •955 0199 7902 1062 8812 1923 9721 2784 •950 0629 36JG 1536 4502 2443 5361 3348 6218 4253 7074 5157 7930 6061 8785 6963 9639 7865 8766 •956 0492 1345 73° 74° •956 3048 •961 2017 3898 3418 4747 4219 5595 5019 0443 5818 7290 6616 8136 7413 8981 8210 9825 9005 957 0669 9800 1512 •902 0594 2354 1387 3195 2180 4035 2972 4875 3762 5714 4552 6552 5342 7389 6130 8225 6917 9060 7704 9895 8490 958 0729 9275 1562 •363 0060 2394 0843 3226 1626 4056 2408 4886 3189 5715 3969 6543 4748 7371 5527 8197 6305 9023 7081 9848 7858 •959 0672 8633 1496 9407 2318 •964 0181 3140 0954 3961 1726 4781 2497 5600 3268 0418 4037 7236 4806 8053 5574 8869 6341 9684 7108 •960 0499 7873 1312 8638 2125 9402 2937 •965 0165 3V48 0927 •951 0565 18° 2197 3048 17° 4558 5368 6177 6984 7792 8598 9403 •961 0208 1012 1815 2617 16° 2449 3209 3968 4726 5484 6240 6996 7751 8505 9258 15° 75° 965 9258 966 0011 0762 1513 2263 3012 3761 4508 5265 6001 6746 7490 8234 8977 9718 967 0459 1200 1939 2678 3415 4152 4888 5624 6358 7092 7825 8557 9288 968 0018 0748 1476 2204 2931 3658 4383 5108 5832 6555 7277 7998 8719 9438 •9690157 0875 1593 2309 3025 3740 4453 5167 5879 6591 7301 8011 8720 9428 •970 0135 0842 1548 2253 2957 14° NAT. COSINE. 254 NATURAL SINES. 21 22 23 24 25 I 26 I 27 1 28 29 76° 77° 78° 79° 80° 1 81° 82° / •970 2957 •974 3-^01 •978 1476 •981 6272 •9848 078 •9876 883 •9902 681 60 36o0 4355 2080 6826 582 •9877 338 •9903 085 59 4363 5008 2684 7380 •9849 086 792: 489 58 5065 5660 3287 7933 589 •9878 -245 891 57 5766 6311 3889 8485 •9850 091 697 •9904 293 1 56 6466 6962 4490 9037 593 •9879 148 694 ' 55 7165 7612 5090 9587 •9851 093 599 •0905 095 ! 54 7863 8261 5689 •982 0137 593 •9880 048 494 53 8561 8909 6288 068G •9852 092 497 893 52 9258 9556 6886 1234 590 945 •9906 290 51 9953 •975 0203 7483 1781 •9853 087 •9881 392 687 60 •9710649 0849 8079 2327 583 838 •9907 083 49 1343 1494 8674 2873 ■9854 079 •9882 284 478 48 2036 2138 9268 3417 574 728 873 47 2729 2781 9862 3961 •9855 (36S •9883 172 •9908 266 46 3421 3423 •979 0455 4504 501 615 659 45 4112 4065 1047 5046 •9856 053 •9884 057 •9909 051 44 4802 4706 1638 5587 544 498 442 43 5491 6345 2228 6128 •9857 035 939 832 42 6180 59S5 2818 6668 524 •9885 378 •9910 221 41 6867 6623 3406 7206 •9858 013 817 610 40 7554 7260 3994 7744 501 •9886 255 997 39 8240 7897 4581 8282 988 692 •9911 384 38 8926 8533 5167 8818 •9859 475 •9887 128 770 37 9610 9168 5752 9353 960 564 •9912 155 36 •972 0294 9802 3337 9888 •9860 445 998 540 35 0976 •976 0435 6921 •983 0422 929 •9888 432 923 34 1658 1067 7504 0955 •9861 412 865 •9913 306 33 2339 1699 8086 1487 894 •9889 297 688 32 3020 2330 8667 2019 •9862 375 728 •9914 069 31 3699 2960 9247 2549 856 •9890 159 4i9 30 4378 3589 9827 3079 •9863 336 588 828 29 5056 4217 •980 0405 3608 815 •9891017 •9915 206 28 5733 4845 0983 4136 •9864 293 445 584 27 6409 5472 1560 4663 770 872 961 26 7084 6098 2136 5189 •9865 246 •■9892 208 •9916 337 25 7759 6723 2712 5715 722 723 712 24 8432 7347 3286 6239 •9866196 •9893 148 •9917 086' 23 9105 7970 3860 6763 670 572 459 22 9777 8593 4433 7286 •9867 143 994 832 21 •973 0449 9215 5005 7808 615 •9894 41G •9918 204 20 1119 9830 6576 8330 •9868 087 838 574 19 1789 •977 045G 6147 8850 557 •9895 258 944 18 2458 1075 6716 9370 •9869 027 677 •9919 314 17 3125 1693 7285 9889 496 •9896 096 682 16 3793 2311 7853 •984 0407 9C4 514 •9920 049 15 4458 2928 8420 0924 •9870 431 931 416 14 5124 3544 89SG 1441 897 •9S97 347 782 13 5789 4159 9552 1956 •9871 363 762 •9921 147 12 6453 4773 •981 0116 2471 827 •9898177 511 11 7116 5386 0680 2985 ■•9872 291 590 874 10 7778 6999 1243 3498 754 •9899 003 •9922 237 9 8439 6611 I 1805 4010 •9873 216 415 599 8 9100 7222 2306 4521 678 826 959 7 9760 7832 i 2927 5032 •9874138 •9900 237 •9923 319 6 •9740419 8441 ! 3486 5542 598 646 679 5 1077 9050 4045 6050 •9875 057 •9901 055 •9924 037 4 1734 9658 4603 6558 514 462 394 3 2390 •978 0265 5160 7066 972 869 751 2 3046 0871 i 5716 7572 •9876428 •9902 275 •9925 107 1 3701 1476 I 6272 8078 883. 681 462 13° 12° 1 11° Ni 10° VT. C08IN 9° E. 8° 70 / NATURAL SIKES. 25ri 83^ 84° 85° 8G° 87° 88° 89° / ■9925 402 '9945 219 •9961 947 •9975 641 •eS86 295 9993 CC8 ■99'c-8 477 60 816 523 •9962 200 843 447 9994 009 5i7 59 •9926 16'J 825 452 •9976 045 598 110 577 58 521 •9946127 704 245 748 209 625 57 ■ 873 428 954 445 898 308 €73 56 •9927 22i 729 ■9963 204 645 •9987 046 405 720 55 573 •9947 028 453 S43 194 502 7 CO 54 922 327 701 •9977 040 340 698 812 53 •9928 271 625 948 237 486 €93 856 52 618 921 •9964195 433 631 788 900 51 965 •9948 217 440 627 775 881 942 50 •9929 310 513 685 821 919 974 964 49 656 807 929 ■9978 015 •9988 061 •9995 066 •9999 026 48 999 •9949 101 •9965 172 207 203 157 065 47 •9930 342 393 414 399 344 247 105 46 685 685 655 589 484 236 143 45 •9931 026 976 895 779 623 424 l&l 44 367 •9950 260 •9966135 9L8 761 512 218 43 706 556 374 •9979 15C £99 599 254 42 •9932 045 SW 612 343 •9989 C35 184 289 41 384 •9951 132 849 530 171 770 323 40 721 419 •9967 085 716 306 854 357 39 •9933 057 705 321 900 440 937 389 38 393 990 555 •9980 084 573 •9996 020 421 37 728 -9952 274 789 2l7 706 101 452 36 •9934 062 557 •9968 022 450 837 182 482 35 395 840 254 631 968 262 511 34 727 •9953 122 485 811 •£990 068 341 5S9 33 •9935 058 403 715 991 227 419 567 32 389 683 945 •9981 170 355 497 593 31 719 962 •9969173 348 482 573 619 30 •9936 047 •9954240 401 525 6(..9 649 644 29 375 517 628 701 734 724 6C8 28 703 794 854 877 859 798 692 27 •9937 029 •9955 070 •9970 080 •9982 052 983 871 714 2C 355 345 304 225 •9991 106 943 736 25 679 620 528 398 228 •9997 015 756 24 •9938 003 893 750 570 350 086 776 2\, 326 ■9956165 972 742 470 156 795 2-2 648 437 •9971193 912 590 224 813 21 969 708 413 •9983 082 709 292 831 2C •9939 290 978 CSS 250 827 360 847 IS 610 1 ^9957 247 851 US 944 426 863 18 928 ; 515 •9972 0G9 i85 •9992 0(0 492 878 17 ■9940 246 783 286 751 176 656 892 le 5C5 •9958 049 502 917 290 620 905 \l 880 315 717 •9984 OSl 404 683 917 14 •9941 1 95 580 931 245 517 745 928 ic 510 844 •9973145 408 629 807 939 li 823 •9959 107 357 570 740 867 949 n •9942 136 370 5C9 731 851 927 958 i( 448 631 780 891 960 986 966 < 7C0 892 990 ■9985 050 •9993 069 •9998 044 973 J •9943 070 •9960 152 •9974199 209 177 101 979 ' 379 411 408 367 284 157 985 6f8 C69 615 524 390 213 9S9 993 926 822 680 495 267 993 ' •9944 303 •9961183 •9975 028 835 600 321 996 009 438 233 989 704 374 998 914 693 437 •9986143 806 426 1-0000 000 •9945 219 947 641 295 908 477 000 6° I 5° 1 4° 3° AT. COSI 2° NB. 1° 0« ' 256 NATURAL TANGENTS. 0° 1° •000 0000 •017 4551- 2909 7460- 5818 •018 0370 8727 3280 •001 1636 6190 4544 9100 7453 •019 2010 •002 0362 4920 3271 7830 6180 •020 0740 9089 3650 •003 1998 6560 4907 9470 781(i •021 2380 •004 0725 5291 3634 8201 6542 •0221111 9451 4021 •005 2360 6932 5260 9842 8178 023. 2753 •006 1087 5663 399( 8574 6905 •02H484I 9814 4395 •007 2723 7305 5632 •025 0216 8541 3127 •008 1450 6038 4360 8948 7260 •026 1859 •009 017? 4770 3087 7681 5996 •027 05921 8905 3503 •010 1814 6414 4724 9325 7C3G •028 2236 •Oil 0542 5148 3451 8059 63C1 •029 0970 9270 3882 •012 2179 6793 5088 9705 799? •030 2616 •013 0907 5528 3817 8439 072r ■031 1351 9636 4263 •014 2545 7174 5454 •032 0086 8364 2998; •015 127S 5910| 4183 88221 7093 •033 17341 •016 0002 4646! 2912 7558 582] •034 0471! 8731 3383' •017 1641 6295 '• 4551 9208 j 89° 88° 1 2° I 034 9208-05 035 21-20 5033' 7945' 030 0858: 37711 6683 95961 037 2500, 5422 8335 038 1-248| 4161' 7074 9988, 039 2901' 5814 8728 040 lG4i; 4555 7409' 041 0383 3296 6-210 9124 042 2038 4952' 7866: 043 07811 36951 4078 6995 9912 •053 2829 5746 8663 •054 1581 4498 7416 ■055 0333 3251 6169 908' •056 2005 4923 7841 -057 0759 3678 0596 9515 •058 2431 5352 8271 -059 1190 4109 7029 9948 •060 2867 5787 6609 061 1626 9524: 4546 044 2438! 7466 5353 062 0386 8268: 3306 045 1183; 6226 4097! 9147 7012-063 2067 9927 1 4988 046 2842 7908 5757: 8673 ■047 1588 4503: 7419 ■048 0334' 3250; 6166' 9082 0491997' 4913 7829 ■050 0740, 3662 6578 9495 051 2411 5328, 8-244 0521161 4078 87° I -064 0829 3750 0671 9592 -065 2513 5435 8356 -0661-278 4199 7121 -067 0043 2965 5887 8809 -068 1732 4654 7577 -069 0499 3422 6345; 9268 069 9268 070 2191 5115 8038 ■071 0:)61 3885 6S09 072 2657 5581 8505 ■073 1430 4354 7279 ■074 0203 3128 6053 8979 ■075 1904 4829 7755 ■076 0680 3606 087 4887 7818 088 0749 31)81 6612 9544 089 '2476 5408 8341 090 1273 4200 7138 091 0071 3004 5938 8871 092 1804 4738 7672 093 0606 3540 6474 9409 6532 -094 2344 9458 5278 077 2384 8213 5311 -095 1148 8237 4084 078 1164 7019 40901 9955 7017; 9944 -079 2871! 5798! 87261 -080 1653 1 4581 7509 •081 0437 3365 6293 9221 •082 2150 5078 8007 •083 0936 3865 6794 9723 •084 2653 5583: 8512| ■085 1442 4372 73021 086 0233' 31631 6094 9025' 087 1956i 4887 85° I •096 2890 5826 8763 -097 1 4635 7572 -098 0509 3446 6383 9320 -099 2257 5194 8133 •100 1071 4009 694 9886 •101 2824 5763 8702 •1021041 4580 75-20 ■103 0460 3399 6340 9280 104 2220 5161 8101 105 1042 84° 6° 70 105 1042 •122 7846 3983 -1-23 0798 6925 3752 9866 6705 106 2808 9658 5750 -1-24 2612 8692 5560 107 1634 8520 4576 125 1474 7519 4429 108 0462 7384 3405 -120 0339 6348 3294 9-291 6249 109 2234 9205 5178 -127 2161 8122 5117 -110 1060 8073 4010 -128 1030 6955 3986 9899 6943 -111 2844 9900 5789 ■129 2858 8734 5815 -1121680 8773 4625 •130 1731 7571 4690 -113 0517 7648 3463 •131 0607 6410 3566 9356 6525 -114 2303 9484 5250 '132 2444 8197 5404, -115 1144 8364 4092 -133 1324 7039 4285 9987 7246 -116 2936 -134 0207 5884 3168 8832 6129 ■117 1781 9091 4730 -135 2053 7679 5015 -118 0628 797r 3578 -136 0940 6528 3903 9478 G80C 119 2428 9830 5378 •137 2T93 8329 120 1279 5757 8721 4-230 7182 1210133 3085 6036 89S8 12219411 4893 7846; 83° •138 1685 4650 7615 139 0580 3545 C510 9476 140 2442 5408 82*^ NAT. COZAN. NA run A L TANGENTS. 257 ' 1 80 9° 1 10° 11° 12° 13° 1 U° 1 15° 1 / -140 5408 158 38441.176 3270 194 3803 •212 556«i -23 1868-2 249 3280-267 9492 60 1 1 S:i75 6826! 6269 6822 8606 -231 1746 6370-268 2610 59 2 -141 134- 98JJJ 9209 9841 -213 1647 4811 9460] 57 28 58 3 1 4oJ< 153 2791-177 2269 195 2861 4688 7876 250 2551! 8847 57 4 1 727^ 5774 5270 5881 7730-232 0341 5642, -269 1967 56 5 -U'iO-itJ 8757 8270 8901 214 0772 4007 87341 5087 55 6 3211 11.01740-1781271 196 1922 3814- 7073 251 1826| 8-207 54 7 617'.. 47241 4-273 4*4:3 6857-233 01401 4919-270 1328 53 8 9147 77081 7274 7964 99001 3207 8012 4449 52 9 U3 2115 161 0692'-179 0276 197 0981: 215 2944 6274 25211061 . 7571 51 10 50S4 3677 3-279 400? 5988 9342 4200 -271 0694 50 11 8053 6662 0281 7031 9032 234 2410 7^294 ' 3817 49 12 •14H0J-: 96 i7 92.^ •198 0053 ■210 2077 5479 253 0383 6940 48 13 1 3a J 1 1G2 2632 -ISO -2287 3076 5122 8548 3484 -272 0064 47 14 1 6901 5ol8 5291 6100 8167 235 1617 (;-5S0 3188 46 15 1 9J31 8603 8295 9124 -217 1213 4687 9676, 6313 45 Id .•145 2;>Jl -ir;3 15901-181 1-299 •199 2148 4259 7758 254 2773, 9438 44 17 5872 4576 4303 5172 7306 236 08-29 5870-273 2564 43 18 88 ii 7563 7308 8197 -218 0353 3900 89G8 5690 42 19 14e 1813 •164 0550! 182 0313 -200 1222 3400 6971 -255 -2066; 8817 41 20 4784 3537 3319 4248 6448 .237 0044 5165-2741945 40 21 7756 6525 63-24 7-274 9496 3116 8264 5072 39 22 •147 0727 9513 9330 -201 0300 ■219 2544 6189 •253 1363 8201 38 23 36J9 -105 2501 183 2337 3327 5593 9'262 4163 -275 1330 37 24 0G72 5489 5343 6354 864.3 -238 2336 7564 4459 36 25 9644 8478 8350 9381 •220 1692 5410 •257 0664 7589 35 26 •148 2617 •106 1467 •184 1358 -202 2409 4742 84S5 37 66 -276 0719 34 27 5590 4456 4365 5437 7793 •2391560 6868 3850 33 is 8563 7446 7373 8465 •2-21 0844 4635 9970 6981 32 29 •149 1536 •107 0436-185 0382 •203 1494 3S95 7711 •258 3073 ■277 0113 31 30 4510 3420 3390 45-23 6947 -240 0788 6176 3245 3C 31 7484 6417 .6399 7552 9999 3864 9280 6378 29 32 •150 0458 9407 9409 -204 0582 •222 8051 6942 •259 2384 9512 28 33 3433 •168 23381-186 2418 3612 6104 -241 0019 5488 •278 2646 27 34 6408 5390 54-28 6643 9157 3097 8593 5780 26 35 9383 8381 8439 9674 -223-2211 6176 •260 1699 8915 25 36 •151 2358 •169 1373 ^•187 1449 •205 2705 5265 9255 4805 -279 2050 24 37 5333 4366! 4460 5737 8319 -242 2334 7911 5186 23 38 8303 7358 7471 8769 -2241374 5414 •261 1018 8322 22 39 •152 1285 •170 0351 •188 0483 •2061801 4429 8494 4126 -280 1459 21 40 4262 3344 3495 4834 7485 -2431575 7234 4597 20 41 7238 6338 6507 7867 •225 0541 4656 •262 0342 7735 19 42 •153 0215 9331 9520 •207 0900 3597 7737 3451 -281 0873 18 43 3192 •171 23-25i-189 2533 3934 6654 -244 0819 6560 4012 17 44 6170 5320 5546 6988 9711 3902 9670 7152 16 45 9147 8314 8559 •208 OOOC •226 -2769 6984 •263 2780 •282 0292 15 46 154 2125 •172 1309 -1901573 3035 5827 -245 0068 5891 3432 14 47 510C 43041 4587 607.: 8885 3151 9002 6573 13 48 808L 7300 7 60-: 9103 •227 1944 6236 •264 2114 9715 12 49 •1551061 •173 0296 -191 0617 •209 214£ 5003 9320 5226 •283 2857 11 50 404C 3292 363L 5181 8063 -246 2405 8339 5999 10 51 7019 6288 664? 821? •22s 1123 5491 •265 14521 9143 9 52 9398 9285 9664 ••210 125£ 4184 85771 4566-2842286 8 53 •156 237? •174 2282 •192 268C 429C 7244 -24- 1663: 7680 5430 7 54 595S 5279 569f 7331 •229 030fi 4750,-266 0794 8575 6 55 C33: 8^277 871C •211 036' 3367 7837i 3909|-2851720 5 56 ■157 191C •175 1275 -1931731 340- 6429 -24« 09-25' 7025 4866 4 57 490C 4273 4745 644^ 949-2 4013-267 0141' 8012 3 58 7881 7272 776( ) 94Sf •230 -2555 710-2 3-257:-2861159 2 59 •158 086C •176 0-271 -194 078-1 \ -2X2 25-2. 5618 •249 0191 6374 4306 1 60 384^ [ 3270 3Sor 5 656f 8682 328C 9492 7454 / 81° 80° 79° 78° 77° 76° 75° 74° / NAT. COTAN. 258 NATURAL TANGENTS. / 1 16° 17° 18° 19° 20° 21° 22° 23° / 286 7454 305 7307 324 9197 344 327G 3G3 9702 383 8640 404 0262 •424 4748 60 1 287 0602 306 0488 325 2413 6530 364 2997 384 1978 3646 8182 59 2 3761 3670 5630 9785 6291 5317 7031 ■425 161C 58 3 6900 6852 8848 345 3040 9588 8656 4050417 5051 57 4 288 0050 307 0034 326 206t, 6296 365 2885 385 1996 3804 8487 56 5 3201 3218 5284 9553 6182 5337 7191 •4261924 55 6 6352 6402 8504 346 2810 M80 8679 406 0579 5361 54 7 9503 9586 327 1724 6068 366 2779 386 2021 3968 8800 63 8 289 2655 308 2771 4944 9327 6079 5364 7358 •427 2239 52 9 5808 5957 8165 347 2586 9379 8708 •407 0748 5680 61 10 8961 9143 328 1387 5846 367 2680 387 206G 4139 9121 50 11 290 2114 30£ 2330 4610 9107 5981 5398 7531 •428 25G3 49 12 5269 5517 7833 348 2368 9284 8744 •408 0924 6005 48 13 8423 8705 329 1050 5630 368 258-, 388 2'J9] 4318 9449 47 14 291 1578 310 1893 4281 8893 589( 543^! 7713 •429 2894 46 15 4734 5083 7505 349 21 5C 919L 87 s; •409 1108 6339 45 16 7890 8272 330 0731 5420 369 2500 ■389 213C 4504 9785 44 17 292 1047 311 1462 3957 8685 580t 5486 7901 •430 3232 43 18 4205 4653 7184 350 1950 9112 8837 ■410 1299 6680 42 19 7363 7845 ■3310411 521G ■370 2420 ■390 2189 4697 •431 0129 41 20 •293 0521 •312 1036 3039 8483 5728 5541 8097 3579 40 21 3080 4229 68L& 351 1750 9036 8894 •411 1497 7030 39 22 6839 7422 •332 0097 5018 •371 234! ■391 2247 4898 •432 0481 38 2? 9999 ■313 0616 3327 8287 565t 5602 8300 3933 37 1'4 •294 3160 3810 6557 3521556 8967 8957 •4121703 7386 36 25 6321 7005 9788 4826 •372 227 S •392 2313 5106 •433 0840 35 26 9483 •314 0200 ■333 3020 80De 559:; 5070 8510 4295 34 27 •295 2645 3396 6252 .3„136S ^^^ 4640 8903 9027 •413 1915 7751 33 28 5808 6593 9485 .373 2217 ■393 2386 5321 .434 1208 32 29 «971 9790 •334 2719 7912 5532 5745 8728 4665 31 30 •296 2135 •315 2988 5953 •354118! 8847 9105 •414 2136 8124 30 31 5299 6186 9188 41;:.. ■C74 2lL.. ■394 24C5 5544 •435 1583 29 32 8464 93S5 •335 2424 7734 547: 5S-:'i 8C53 5043 28 33 •297 1630 •316 2585 5660 •3551010 87 i.-. 91 1 9 •415 2CG3 8504 27 34 479G 5785 889C 42SL ■375 211i ■395 25C2 5774 •436 196C 26, 35 7962 8986 •336 2134 7502 54:JC 591C 91SC 54-29 25 36 ■298 1129 •317 2187 5372 ■356 0840 8753 9280 •410 2598 8893 24 37 4297 5389 8610 411S •37G 2073 •3962C4C 6012 •437 2357 23 38 7465 8591 •337 1850 7397 5394 6on 9426 68-23 22 39 •299 0634 ■318 1794 509G ■357 0fi7C 871G 9376. ■417 2841 9289 21 40 3803 4998 8330 39oC ■377 2038 ■337 274C 6257 •438 2756 20 41 6973 8202 •338i571 7237 5361 6114 9C73 6224 19 42 •300 0144 •319 1407 4S1L ■358 OOlf 8C85 9483 -418 3091 9693 18 43 3315 4C13 805; 3801 ■378 2010 •398 2853 6509 •439 3163 17 44 6486 7819 •339 129'J 70S3 5335 6224 9928 6634 16 45 9658 •320 1025 4543 •359 0367 8G61 959i •419 0348 •440 0105 15 46 •301 2831 4232 7787 3GG1 ■3791988 •399 23Cf C7C9 3578 14 47 6004 7440 •340 1032 693C 5315 634] ■.420 CI 90 7051 13 48 9178 •321 0649 427 £ •360 0222 8C44 97K 3C1L ■441 052C 12 49 •302 2352 385S 7524 350!: •380 1973 •400 3089 703C 4001 11 50 5527 7067 .341 0771 f705 5302 6465 •421 04CU 7477 10 51 8703 •322 0278 4C19 ■3C1 oos': sens 9841 3885 4420954 9 52 •3031879 3489 72C7 ■38119G4 .401 3218 7311 4432 8 53 5055 C70C •342 05K crc( 529C C59f •422 0738 7910 7 54 8232 991;^ 37 To 9940 8629 9974 4165 •4431390 6 5o •3041410 •323 3125 701 r •362 324f ■3821962 •402 3354 7594 4871 5 56 4588 633? •343 026C 6531 529C 6734 •4231022 8352 4 57 7767 9552 3518 9S23 8631 •403 0115 4452 •4441834 3 58 •305 094r •324 276f 6770 •363 3115 •3831967 3496 788^4 5318 2 59 4126 5981 •3440023 640S 5303 6879 •4241 31 C 8802 1 60 7307 9197 3276 9702 8640 •404 0262 474S •445 2287 / . 73° 72° 71« 70° WAT. ( 69° ?07An 68' 07« §6° / NATURAL TAA'GE2^TS. 250 24° I 445 2287 1 • 57731 9260- 446 27471 6236: 9726- 447 3216: 6708; 448 0200' 3693 j 7187! 449 0682 • 4178 1 7675 4501173I' 4672: 817 1! 451 1G72;' 5173! 807 6 1' 452 2179 50831 9188 453 2694 1 62011 97091 454 3218' 6728i 455 0238' 3750 7263 456 07761 4290 7806 457 13221 4839 1 8357 i' 458 1877 1 53971 8918,' 459 2439' 5962' 9486 400 3011 6537; 4010063: 3591! 7119 462 0649 4179 7710 ' 4631243 4776 8310 4641845: 5382 465 2457; 5996 9536 466 30771 65' 25'' I ■466 3077 6618 467 0161 3705 7250 •468 0796 4342 7890 469 14391 4988J 8539, 470 2090' 5643, 9196 •471 2751 6306 9863: '472 3420 6978 '473 0538' 4098' 7659, •474 1222 4785 8319 •475 1914' 5481, 9J4S •476 2616 6185, 9755' •477 3326' 689d •478 0472' 4046, 7621 ■479 1197 4774 8352 •480 1932J 5512, 9093 •481 2675 6258, 9842 •482 3427 7014 •483 0601 4189 7778 •434 1368 4959 8552| •485 21451 6739 93341 •486 2931 6528' •487 0126, 3726 73261 64« 26* ■487 7326 488 092 4530 8133 •489 173 5343 61661 9775| 4913386'^. •492 0610 4224' 7838 •493 1454'' 5071 1 8689! •494 2308!' 5928 9549 •495 3171 6794 •496 0418, 4043' 76691 •49712971 4925 8554 •498 2185 6816 9449 •499 3082 6717 •500 0352 3989 7627 •501 12661 49061 S547| ■502 2189! 5832! 9476: •503 31211 6768i ■504 0415: 40631 7713 ■505 1363 5015, 8668' •506 2322 • 5977 9633 ■ 507 3290 6948 508 0607 • 4267 7929 509 1591 ■ 5254 63« I 27° 28° 509 5254 •531 7094 • 8919 •532 0826 510 2585 4559 • 6252 8293 9919 •533 2029 511 3588 5765 • 7259 9503 5120930 •534 3242 4602 6981 • 8275 •535 0723 513 1950 4465 ■ 5625 8208 9302 •536 1953 514 2980 5699 • 6658 9446 515 0338 •537 3194 • 4019 6943 7702 •538 0694 5161385 4145 • 50C9 8198 8755 •539 1952 517 2441 5707- 6129 9464 9818 •540 3221- 518 3508 6JS0 7199 •541 0740 519 0891 4501 • 4584 8263 8278 •542 2027 • 520 1974 5791 5671 9557 9368 •543 33-24 • 521 3067 7092 6767 -544 0862 522 0468 4C32 - 4170 8404 7874 -545 -2177 • 523 1578 5951 5284 9727 8990 •546 3503 • 524 2398 7281 6407 ■547 1060 • 525 0117 4840 3829 8621 7541 •548 2404 •5261255 6188 4969 9973 8685 •549 3759 •527 2402 7547 6120 •550 1335 9839 5125 •528 3560 8910 7281 •551 2708 •529 1004 6502 4727 •552 0297 8452 4093 •530 2178 7890 5906 -553 1688 9634 5488 •531 3364 9288 7094 -554 3091 62^ 6P 29° 30° 31° f 554 3091-577 3503 •600 8606 60 6894 738-i •601 2566 59 555 0698 ^578 1262 65^27 58 4504 5144 •602 0490 57 8:-.ll 9027 4464 56 556 2119-579 2912 8419 55 6929 0797 -G03 2386 54 9739 -580 Ot)»4 6364 53 -557 35511 457 o -604 03'23 52 7364 8402 4294 51 -558 1179-5812353 8266 50 4d94 02^5 -605 2240 49 8811 -682 0139 6215 48 •559 2629 4034 -606 0192 47 6449 7930 4170 46 •560 0269 •5831S2S 8149 45 4091] 572b -607 2130 44 7914! 9627 6112 43 •561 1738 •584 3528 •608 0095 42 55641 743J, 4080 41 9391h5S5 1335 8067 40 •562 3219! 5241 ■609 2054 39 7048| 9L48 6043 38 •563 0J79/5S6 305u ■610 0034 37 4710 6965 4026 36 S543;5S7 0870 8019 35 •5612:378 4788 ■611 2014 34 0J13 8702 6011 33 -565 0J50:-588 2616 •612 0008 •32 3888 6533 4007 31 7728 •589 0450 8008 30 •566 1568 4369 •613 2010 "29 5410 8289 6013 28 9254 •590 2211 •6140018 27 -567 3J98 6134 4024 26 6944 -591 0058 8032 25 -568 0791 3984 •615 2041 24 4639 7910 6052 23 8488-5921839 ■616 0064 22 •569 2339 5768 4077 21 6191 9699 8092 20 •570 0045! •593 3632 ■617 2108 19 3899| 7565 7755 •5941501 6126 18 ■618 0145 17 •5711612 5437 4166 16 54711 9375 8188 15 93311 •595 3314 •619 ■2-21 1 14 •572 3192 7255 6^236 13 7054 •5961196 •620 0263 12 •573 0918 6140 4291 11 4783 9084 8320 10 8649 •597 3030 •G21 2351 ^ •574 2516 6978 6.'383 3 6385 •598 0926 •622 0417 7 •575 0-255 4877 4452 6 41261 882S 8488 5 7999j •599 2781 •623 25-27 4 •576 1873 6735 6566 3 57 48, •000 0691 •624 0607 2 9625 4648 4650 1 ■577 3503 8606 8694 60° ^d^ 08« / VAT' OOTAW, 260 NATURAL TANGENTS. 32° 624 8694 625 2739 6786 626 0834 4884 8935 627 2 7042 628 1098 5155 9214 629 3274 7336 •630 1399 5464 9530 ■631 3598 7667 ■6321 5810 9883 •633 3959 8035 •634 2113 619 .635 0274 435 8441 •636 2527 6614 •637 0703 4793 7073 •639 1169 5267 9366 •640 3467 7569 •641 1673 5779 9886 •642 3994 SI •643 -2216 6329 .6440444 4560 8678 •645 2797 6918 •646 1041 5165 9290 •647 3417 7546 •648 1676 5808 9941 •649 4076 57° 33° 649 4076 8212 650 2350 6490 651 0631 4774 8918 652 3064 7211 ■653 13601 5511 ■ 9663 ■654 3817 7972 •6G5.2129 6287 •656 0447 4609 8772 •657 293' 7103 •6581271 5441 9612 •C59 3785 7960 •GGO 2136 6313 •6C1 0492 4673 8856 •662 3040 7225 •663 1413 5601 9792 •664 3984 8178 •665 2373 6570 •666 0769 4969 9171 •667 3374 7580 •6681786 5995 •669 0205 4417 8630 •670 2845 7061 •671 1280 5500 9721 •672 3944 8169 •673 2396 6624 •674 0854 5085 56° 34° 674 5085 9318 675 3553 7790 676 2028 6268 677 0509 4752 8997 678 3243 7492 679 1741 5993 680 0246 4501 8758 ■681 3016 7276 ■682 153' 5801 ■683 0066 4333 8601 ■684 2871 7143 •6851416 •686 4247 8528 •687 2810 7093 ■6881379 5666 9955 •689 424 8538 •690 2S32 7128 •691 1425 5725 692 0026 4328 8633 .693 2939 7247 •6941557 586S ■695 0181 449( 881f .696 3131 7451 •697 1773 6097 •698 0422 4749 9078 •699 3409 7741 •700 2075 55° 35° 36° ' 37° ! 700 2075 726 5425 •753 5541 • 6411 1.871 •754 0102 701 0749 727 4318 4666 • 5089 8767 9232 9430 •728 3218 •755 3799 702 3773 7671 8369 8118 •729 2125 •756 2941 703 2464 6582 7514 6813 •730 1041 •757 2090 704 1163 5501 6668 5515 9963 •758 1248 9869 •731 4428 5829 705 4224 S8'D4 •759 0413 8581 ■732 33G2 4999 7062940 7832 9587 7301 •733 2305 •7C0 4177 707 1664 6777 8769 6028 •7341253 •TGI 3363 708 0396 5730 7959 4763 •735 0210 •762 2557 9133 4691 7157 709 3504 9174 •7631759 787? •736 3G60 6363 710 2253 8147 •764 0969 6630 •737 2636 5577 711 1009 7127 •7C5 01S8 5393 ■738 1G20 4800 9772 6115 9414 712 4157 739 OCll 7GG4a31 8543 5110 SG49 713 2931 9G11 •7G7 3270 7320 ■740 4113 7893 714171? 8618 •768 2-517 610C ■741 3124 7144 715 0501 7633 •7691773 4S9S ■742 2143 6404 9297 6655 ■770 1037 716 369? •7431170 5672 8100 5686 •771 0309 •717 2505 •7410204 4948 6911 4724 9589 •7181319 9246 •772 4233 5729 •745 3770 8878 •719 0141 8296 ■773 3526 4554 •746 2824 8176 8970 7354 ■774 2827 •720 3387 •747 1886 7481 78or 6420 •775 2137 •721 2227 ■748 0956 6795 665C 5494 •776 1455 7221075 •749 0033 6118 550-.' 4575 •777 0782 9930 9119 5448 •723 4361 750 3665 •778 0117 8793 8-212 4788 •724 3227 ■751 2762 9460 7663 7314 •779 4135 •725 2101 ■7521867 8812 6540 6423 ■780 3492 •726 0982 •753 0981 8173 5425 ■ 5541 •781 2856 54° 53° 52° 38° 39° r 781 2856 809 7840 60 7542 810 2658 59 782 2229 7478 58 6919 811 2300 57 •7831611 7124 56 6305 ■812 1951 55 ■784 1002 6780 54 5700 ■813 1611 53 ■785 0400 6444 52 5103 •814 1280 51 9808 6118 50 ■786 4515 •815 0958 49 9224 5801 48 ■787 3935 816 0646 47 8049 5493 46 ■788 3364 •817 0343 45 8082 5195 44 •789 2802 •818 0049 43 7524 4905 42 •790 2248 9764 41 6975 •819 4625 40 ■791 1703 9488 39 6434 ■820 4354 38 ■7921167 9222 37 5902 •821 4093 36 •793 0640 8965 35 5379 •8-22 3840 34 •794 0121 8718 33 4865 .823 3597 32 9611 8479 31 ■795 4359 ■824 3364 30 9110 8251 29 ■796 3862 ■825 3140 .28 8617 8031 27 ■797 3374 ■826 2925 26 8134 7821 2.^ ■798 2895 •827 2719 •24 7659 7620 23 •799 2425 ■828 2523 22 7193 7429 21 ■8001963 ■829 2337 20 6736 7247 19 •8011511 ■830 2160 18 6288 7075 17 •802 1067 •831 1992 16 5849 6912 15 ■803 0632 •832 18«4 14 5418 6759 13 ■804 0-206 •833 1686 12 4997 6615 11 9790 •8341547 10 •805 4584 6481 9 9382 ■835 1418 8 ■806 4181 6357 7 8983 ■836 1-298 6 •807 3787 6-242 5 8593 ■837 1188 4 •808 3401 6136 3 8-212 ■838 1087 2 •809 3025 6041 1 7840 ■839 0996 51° 50° / NAT. COTAN. NATURAL TANGENTS. 261 ,' 40° 41° 42° 43° 44° 45° 4G° 47° / 3 839 0990 869 2867 •900 4040 932 5151 •965 6888 1^00 00000 1^03 55303 V-Q- 23687 60 1 595o 7970 9309 933 C591 •96G2511 05819 61333 29943 59 840 0915 870 3087 •9014580 6034 8137 11642 67367 36203168 3 5878 8200 9854 934 1479 •967 3767 17469 73404 42467157 4 841 0844 871 3316 •902 5131 6928 9399 23298 79445 48734 56 5 5812 8435 •903 0411 935 2380 •968 5035 29131 85489 55006 65 6 842 0782 872 355G 5! 95 7834 •969 0074 34968 91538 61282164 7 5755 8r8C •904 007'.: 936 3292 6316 40807 97589 67561153 8 •8430730 873 3S0G 6207 8753 •9701962 46051 1^04 03645 73845162 9 6708 8935 •905 1557 937 4216 7610 5-2497 09704 80132 51 10 •844068S 8744067 6851 9383 •971 3262 58348 15767 86423 60 11 5C70 9201 •90G2147 938 5153 8917 64201 21833 92718 49 12 •8450C55 875 433S 7440 939 0625 •972 4575 70058 27904 99018 48 13 5643 947? •907 2748 6101 •973 0-230 75918 33977 V08 05321 47 14 •846 0C33 8764020 8053 940 1579 5901 81782 40055 11628,46 15 5C25 9765 •903 3300 7061 •9741569 87649 46136 1793945 16 •847 0C20 877 4012 8G71 941 2545 7240 93520 52221 24'254!44 17 5G17 878 0062 •909 3984 8033 •975 2914 99394 58310 30573143 18 •848 0C17 5215 9300 942 3523 8591 1-01 05272 64402 36896142 19 5G19 879 0370 •910 4619 9017 •976 4-272 11153 70498 43223 41 20 •849 0C24 5528 9940 •943 4513 9956 17038 76598 49554 40 21 5031 880 0:8? •911 5265 •944 C013 •977 5643 22925 82702 55889 J39 22 •850 0640 5C52 •912 0592 5516 •978 1333 28817 88809 62228 [38 23 5C53 881 1017 5922 ■945 1021 7027 34712 94920 6857137 24 •851 0CC7 61 86 •9131255 6530 •979 2724 40610 r05 01034 74918136 25 5C84 •882 1357 6591 •946 2042 8424 46512 07153 812'J9i36 26 •852 0704 6531 •914 1929 7556 •980 4127 5-2418 13275 87624 34 27 572L •8831707 7270 •947 3074 9833 58326 19401 9.-.984I33 28 •853 075: 6886 •915 2015 8595 •981 5543 64239 25531 1-09 00347132 23 5777 •884 2068 7962 •948 4119 •9821256 70155 31664 06714 31 30 •8540807 7253 •916 3312 9646 6973 76074 37801 13085 30 31 5S39 •885 2440 8665 •949 5176 •983 2692 81997 43942 19460 29 32 •855 0873 7030 •917 4020 •950 0709 8415 87923 50087 26840 28 33 5910 •8862822 9379 6245 •9844141 93853 56235 32223 27 34 •856 0950 8017 •918 4740 •951 1784 9871 99786 623F-8 38610 26 35 5992 •887 3215 •919 0104 7326 •985 5603 1-02 05723 68.544 45002 25 36 •857 1037 8415 5471 •952 2871 •986 1339 11664 74704 51397 24 37 6084 •888 3619 •920 0841 8420 7079 17C08 80867 57797 23 3S •858 1133 8825 6214 •953 3971 •987 2821 23555 87035 64201 22 39 6185 •889 4033 •921 1590 9526 8567 29506 93206 70609 21 40 •859 1240 9244 6969 •954 5083 •988 4316 35461 99381 77020 20 41 6297 •890 415? •922 2350 •955 0644 •989 0069 41419 1^06 05560 83436 19 42 •860 1357 9675 7734 6208 5825 ■ 47381 11742 89857 18 43 6419 •891 4894 •923 3122 •9561774 •990 1584 53346 17929 96281 17 44 •861 1484 •892 0116 8512 7344 7346 59315 24119 MO 02709 16 45 6551 5341 •924 3905 •957 2917 •991 3112 65287 30313 09141 15 46 •8621621 •893 0569 9301 8494 8881 71263 36511 15578 14 47 6694 5799 •925 4700 •958 4073 •992 4054 77243 42713 22019 13 48 •8631768 •8941032 •926 0102 9655 •993 0429 83226 48918 28463 12 49 6846 6268 5506 •959 5241 6-208 89212 55128 3491211 50 •864 1926 •895 1506 •927 0914 •960 0829 •9941991 95203 61341 41365 |l0 51 7009 6747 6324 6421 7777 1-03 01196 67558 47823 9 52 •865 2094 •8961991 •9281738 •961 2016 •995 3566 07194 73779 54284 8 53 7181 7238 7154 7614 9358 13195 80004 60750 7 54 •866 2277 •897 2487 •929 2573 •962 3215 •996 5154 19199 86233 67219 6 55 736b 7739 7996 8819 •997 0953 25208 9246f^ 73693 5 56 •867 246G •898 2994 •930 3421 •963 4427 6756 31220 98702 801711 4 57 7558 8251 8849 •964 0037 •998 2562 37235 1^07 04943 86663 3 58 •868 2659 •899 3512 •931 4280 5651 8371 43254 11187 93140 2 59 7762 8775 9714 •965 1268 .999 4184 49277 17435 996301 1 60 •869 2867 •900 4040 •932 5151 6888 l^OOOOOOO 55303 23687 Ml 061251 / 49° 48° 47° 46° 45° 44° 43° 42° ' SAT, COZUb 202 NATURAL TANGENTS. f 4»-.' 49° 50° 51° 52° 53° 54° / Ill 06125 115 03684 119 17536 1-23 48972 1-27 99416 1-32 7044<, 1-37 63819 60 1 12,624 10445 24579 56319 1-28 07094 78483 72242 59 19127 17210 31626 63672 14776 86524 80672 58 3 25635 23979 38679 71030 2-2465 94571 89108 57 4 32 U6 30754 45736 78393 30160 1-33 026:^4 97551 56 5 386(32 37532 52799 85762 37860 10684 1-38 06001 55 f ■ 45182 44316 59866 93136 45566 18750 14458 54 7 5i70() 51104 66938 1-24 00515 53-277 26822 22922 53 8 582:J5 57896 74015 07900 60995 34900 31392 52 9 W7G8 64693 81097 15-290 08718 42984 39869 51 10 71305 71495 88184 22685 76447 51075 48353 50 11 77840 78301 95276 30080 84182 59172 56844 49 12 84391 85112 1-20 0-2373 37492 91922 67276 65342 48 13 90941 91927 09475 44903 99669 75386 73847 47 14 97495 98747 16581 52320 129 07421 83502 82358 46 15 1-12 04053 1-16 05571 23693 59742 15179 91624 90876 4£ 16 10616 1-2400 30810 67169 22943 99753 99401 44 17 17183 19384 37932 74602 30713 1-34 07888 1-3907934 43 18 23754 26073 45058 82040 38488 16029 16473 42 19 30329 32916 52190 89484 46270 24177 25019 41 20 36909 39763 59327 96933 54057 3-2331 33571 40 21 43493 46615 6f>468 1-25 04388 61850 40492 4-2131 39 22 50081 53472 73615 11848 69649 48658 50698 38 23 5G674 60334 80767 19313 77454 56832 59272 37 24 63271 67200 87924 26784 85265 65011 67852 36 25 69872 74071 95085 34260 93081 73198 76440 35 26 7(478 80347 1-21 02252 41742 1-30 00904 81330 85034 31 27 83088 87827 094-24 49229 08733 895S9 93636 33 28 £9702 94712 16601 56721 16567 97794 1-40 02245 32 29 P0321 1-17 01601 23783 64219 24407 1-35 06006 10860 31 30 1-1302944 08496 30970 71723 32254 14224 19483 30 31 09571 15395 38162 79-232 40106 22449 •28113 29 32 16203 2 2238 45359 86747 47964 30680 36749 28 33 22839 23207 52562 94267 55828 38918 45393 27 34 29479 36120 59769 1-26 01792 63699 47162 54044 26 35 3612 i 43038 66982 09323 71575 55413 6'2702 ,25 36 42773 49960 74199 1CS60 79457 63670 71367 24 37 49427 56.SS8 81422 24402 87345 71934 80039 23 38 56085 6382) 88650 31950 95-239 80204 88718 22 39 62747 70756 95883 39503 1-3103140 88481 97406 21 40 69414 7769S 1-2203121 47062 11046 96764 1-41 06098 20 41 76086 84644 10364 54626 18958 1-36 05054 14799 19 42 82761 91595 17613 62196 26876 13350 23506 IS 43 89441 98551 24866 69772 34801 21653 32221 17 44 96126 ■1-18 05512 32125 77353 42731 29963 40943 16 45 1-U02815 1-2477 39389 84940 50668 38-279 49673 15 46 09508 19447 46658 92532 58610 46602 5840C. 14 47 16206 26422 53932 1-27 00130 66559 54931 67153 13 48 22908 33402 61211 07733 74513 63267 75904 12 49 29615 40387 68496 15342 82474 71610 84662 11 60 363^1 47376 75786 22957 90441 79959 03427 10 61 430 U 54370 83081 30578 98414 883i5 r42 02200 9 52 49762 C1369 90381 38204 1-32 06393 96678 10979 8 53 56486 68373 97687 45835 14379 1-37 05047 19760 7 54 63215 75382 1-2304997 53473 22370 13423 28561 6 55 69949 82395 12313 61116 30368 •21800 37362 5 5o 76687 89414 19634 68765 38371 30195 46171 4 87 83429 96437 26961 76419 40381 38591 54988 3 58 90176 1-19 03465 34292 84079 54397 4G994 63811 2 59 96928 10498 41629 91745 62420 55403 7'2642 1 60 1 -la 03684 17536 48972 99416 70448 63819 81480 / 410 40° 39° 38° AT. COTi 37° 36° 36° / NATUBAL TANGENTS. 26:3 / 55° 56° 57° 58° 59° 60° 61° / 1-42 81480 L-4S 25610 l-:3 9S650 1-60 03345 1-66 4-2795 1-73 20508 1-80 40478 CO 1 90320 34916 1-54 08-160 13709 537 06 32149 52860 59 2 99178 44231 18280 24082 C4748 43803 C:)25G 58 3 1-43 08039 53554 2S10S 34465 75741 55468 77664 57 4 1G90C 62SS4 37940 44858 S6744 07144 00080 56 5 257 SI 72223 47792 55260 97758 78833 ' 1-61 U2521 55 6 34604 81570 57647 65672 1-67 08782 90533 14969 54 7 43554 90925 07510 76094 19818 1-7402245 27430 53 8 52451 L-49 00288 77383 80525 30804 13969 39904 52 y 61356 09659 87264 96966 41921 25705 52391 51 10 70268 19039 97155 1-61 07417 52988 37453 64892 50 11 79187 28426 1-55 07054 17878 64067 49213 77405 49 12 88114 37822 16963 28349 •^5150 60984 89932 48 13 97049 47225 26880 38829 86250 72768 1-82 02473 47 14 L-44 05991 56637 36806 49320 973C7 84564 15026 40 15 14940 6605S 46741 59820 1-68 08489 96371 27593 45 16 23897 75486 56685 70330 19621 1-75 08191 40173 44 17 3286:: 84923 06639 80850 307C5 20023 52767 43 18 4183i 943G7 76601 91380 41919 31860 65374 42 19 50814 1-50 03821 86572 1-62 01920 53085 43722 77994 41 20 59801 13282 96552 12469 64261 55590 90628 40 21 68796 22751 1-56 06542 23029 75449 67470 1-83 03275 39 22 77798 3-2229 16540 33599 86647 79362 15936 38 23 86808 417ie 26548 44178 97856 91267 28610 37 U 95825 51210 36564 54768 1-69 09077 1-7603183 41297 36 lb 1-45 04850 60713 46590 65368 20308 15112 53999 35 26 13883 702-24 56625 75977 31550 27053 66713 34 27 22923 79743 66669 86597 42804 39007 79442 3D 28 31971 89271 76722 97227 54069 50972 92184 32 29 41027 98807 86784 1-63 07867 65344 62950 1-8404940 31 30 50090 1-51 08352 96856 18517 76631 74940 17709 30 31 59161 17905 1-57 06936 29177 87929 86943 30492 29 32 68240 27466 17026 39847 99238 98958 43289 28 33 77326 37036 27128 50528 1-70 10559 1-77 10985 56099 27 34 86420 46614 37234 61218 21890 23024 68923 26 35 95522 56201 47352 71919 33233 35076 81761 25 36 1-46 04632 65796 57479 82630 44587 47141 94613 24 37 13749 75400 67615 93351 55953 59218 1-85 07479 23 38 22874 85012 77760 1-64 04082 67329 71307 20358 23 39 32007 94632 87915 14824 78717 83409 33252 21 40 41147 1-52 04261 98079 25576 90116 95524 46159 20 41 50296 13899 1-58 08253 36338 1-71 01527 1-78 07651 59080 19 42 59452 23545 18436 47111 12949 19790 72015 18 43 68616 33200 28628 57893 24382 31943 84965 17 44 77788 42863 38830 68687 35827 44107 97928 16 45 86967 52535 49041 79490 47283 56285 1-8610905 1& 46 96155 62215 59261 90304 58751 68475 1 23896 14 47 1-47 05350 71904 69491 1-65 01128 70230 80678 36902 13 48 14553 81602 79731 11963 81720 92893 49921 12 49 23764 91308 89979 22808 93222 1-79 05121 62955 11 50 32983 1-53 01023 1-59 00238 33663 1-72 04736 17362 76003 10 51 42210 10746 10505 44529 16261 29616 89065 9 52 51445 20479 20783 55405 27797 41883 1-87 02141 8 53 606S8 30219 31070 66292 39346 54162 15231 T 54 69938 39969 41366 77189 50905 66454 28336 6 55 79197 49727 51672 88097 62477 78759 41455 5 56 88463 59494 61987 99016 74060 91077 54588 4 57 97738 6927C 72312 1-66 09945 85654 1-80 03408 67736 3 58 1-48 07021 79054 82647 20884 97260 15751 80898 2 59 16311 8884S 9-2991 31834 1-73 08878 28108 94074 1 60 25610 9865C 1-60 0334£ \ 42795 20508 40478 1-88 07266 / 340 33° 32° 31° f 30° 29° 28° ' J iAX. COT iN '^64 NATURAL TANGENTS. ' 1 62° 63° 64° 65° 66° 67° 68° / 1-88 07265 1-96 -26105 ^•05 03038 2-1445069 2-24 60368 2-35 58524 2-47 50869 60 1 20470 41)227 18185 61366 77962 77590 71612 59 2 33C0G 54314 3334U 77b83 955S0 96683 92386 58 3 46924 68518 48531 94021 2-25 13221 2-3615801 2-48 13190 57 4 00172 82688 63732 2-15 1037S 30885 34946 ^4023 56 5 73436 96874 78950 •26757 48572 54118 54887 55 6 86713 1-97 11077 94187 43156 60283 73310 75781 64 7 1-89 00006 25296 2-06 09442 59575 84016 92540 90706 53 8 13313 39531 24716 76015 2-20 01773 2-37 11791 2-49 17660 52 9 26635 53782 40008 92476 19554 31068 38645 51 10 39971 68050 55318 2-16 08958 37357 50372 59661 60 11 53322 82334 70646 25460 55184 69703 80707 49 12 66688 96635 85994 41983 73035 89060 2-50 01784 48 13 80068 1-98 10952 2-07 01359 58527 90909 2 38 08444 22891 47 14 934C4 25286 16743 75091 2-27 0S807 27855 44029 46 15 1-90 06874 39636 32146 91677 26729 47293 65198 45 16 20299 54003 47567 2-17 08283 44674 66758 86398 44 17 33738 68387 63007 24911 62643 86250 2-5107629 43 18 47193 82787 78465 41559 80636 2-39 05769 28890 42 19 60G63 97204 93942 58229 9S653 25316 50183 41 20 74147 1-99 11637 2-0809438 74920 2-28 16693 44889 71507 40 21 87647 26087 24953 91631 34758 64490 92863 39 22 1-91 01162 40554 40487 2-18 08364 52846 84118 2-52 14249 38 23 14691 55038 56039 25119 70959 2-40 03774 35667 i 37 24 28236 69539 71610 41894 89096 23457 57117 1 36 25 41795 84056 87200 , 58691 -2-29 07257 43168 78598 ! 35 26 55370 98590 2-09 02809 75510 25442 62906 2-53 00111 ' 34 27 68960 2-00 13142 18437 92349 43651 82672 21655 ' 33 28 82565 27710 34085 2-19 09210 61885 2-41 02465 43231 i 32 29 96186 42-295 49751 26093 80143 22286 64839 31 30 1-92 09821 56897 65436 42997 98425 42136 86479 i 30 31 23472 71516 81140 59923 2-30 16732 62013 2-54 08151 29 32 37138 86153 96864 76871 35064 81918 29855 ' 28 33 50819 2-01 00806 210 12607 93840 53420 2-4201851 51591 i 27 34 64516 15477 441 5C 2-20 10831 71801 21812 73359' 26 35 78228 30164 27843 90206 41801 95160; 25 36 91956 44869 59951 44878 2-31 08637 61819 2-55 16992 I ^ 37 1-93 05699 59592 75771 61934 27092 81864 38858' 23 38 19457 74331 91611 79012 45571 2-43 01938 607561 22 39 33231 89088 2-11 07470 96112 64076 22041 82686 21 40 47020 2-02 03862 23848'2-21 13234 82606 42172 2-56 04649! 20 41 60825 18654 3924e 30379 2-32 01160 62331 26645! 19 42 74545 33462 55164 47545 19740 82519 486741 J8 43 88481 48289 71101 64733 38345 2-44 02736 70735 ' 17 44 1-9402333 63133 87057 81944 56975 22982 92830' 16 45 16200 77994 2-12 03034' 99177 75630 43256 2-57 14957 i 15 46 30083 92873 19030 2-22 16432 9431 1 63559 37118 • 14 47 43981 2-03 07769 35046 33709 2-3313017 83891 59312 1 J3 48 57896 22683 51082 51009 31748 2 45 04252 81539 12 49 7182e 37615 671371 68331 50505 24642 2-58 03S00 11 50 85772 52565 83-213 85676 69287 45061 26094 10 51 9973C 67532 993082-23 03043 88095 65510 48421 9 52 1-9513711 82517 2-1315423 20433 2-34 06928 85987 70782 8 53 27704 97519 31559 37845 25787 2.46 06494 9?177 , 7 54 4171^ 2-04 12540 47714 55280 44672 27030 2-59 15606 6 55 5573( 27578 63890 72738 63582 47596 38068 5 56 6978( ) 42534 80085 90218 8-2519 68191 60564 4 57 8883' 57708 963012-24 07721 2-35 01481 88816 83095 3 58 9791( ) 7280C 2-1412537 25247 20469 -:-47 09470 2-60 05659 2 59 1-961200 ) 8791( 28793 42796 ■ 39483 30155 28258 1 60 2610512-05 0303J 45069, 60368 58524 50869 508911 • 27° 1 26° 25° i 24° 23° 22° 21° 1 / VAX. «OTAJS. NATURAL TANGENTS. f 69° 70° 71° 2-60 50891 2-74 74774 ■2-90 42109 3- 1 73558 99661 69576 3- 2 96259 2-75 24588 97089 3 2-61 18995 49554 2-91 24649 4 41766 74561 5225C 5 64671 99608 79909 3- 6 87411 2-76 24685 2-9207010 7 2-62 10286 49822 35358 8 33196 74990 63152 3- 9 56141 2-77 00199 90995 10 79121 25-US ■2-9318885 11 2-6302136 50738 46822 3- 12 25186 76069 74807 13 48271 2-78 01440 2-94 02840 14 71392 26853 30921 3- 15 94549 52307 16 2-64 17741 77802 87227 17 40969 2-79 03339 2-95 15463 3- 18 64232 28917 43727 19 87531 54537 72050 20 2-65 10867 80198 •2-96 00422 21 34238 -2-80 05901 28842 3- 22 57&15 31646 57312 23 81089 57433 85831 24 2-66 04569 83-263 2-97 14399 :> 25 28085 2-81 09134 43016 26 51638 35048 71G83 27 75227 61004 2-98 00400 3- 2,-> 98853 S7003 -291 G7 2.1 2-67 22516 2-8213045 57983 30 46215 39129 86850 ■->- :51 69951 65256 2-99 15766 ?>l 93725 91426 44734 '■'>■) 2-68 17535 2-83 17639 73751 3- \n 413S3 43S96 3-00 02820 3'. 652G7 70196 31939 36 89190 96539 61109 3- 37 2-69 13149 2-84 229-26 90330 38 37147 49356 3-01 10603 33 61181 75831 48923 3- 40 85254 2-85 02349 78301 41 2-70 093&4 28911 3-02 07728 42 33513 55517 37207 3- 43 57699 8216S 06737 44 81923 2-86 08863 96320 45 2-71 06186 35602 3-03 25954 3- 46 30487 62386 55641 47 54826 89215 85381 48 79204 ■2-8716088 3-04 15173 3- 49 2-72 036-20 43007 45018 50 28076 69970 74915 51 52569 96979 3-05 04866 3- 52 77102 2-88 24033 34870 53 2-73 01674 51132 &49-28 oi 26284 78277 95038 3- 55 50934 2-89 05467 3-06 25-203 56 756-23 32704 55421 57 2-74 00352 5;)9St^ 85694 3- 53 25120 87314 3-07 16020 59 499-27 2-90 146SS 46400 6C 74774 42109 76835 3- / 2a° 19° 18° 72° 07 76835 08 07325 37869 68468 99122 09 29831 60596 91416 10 22291 53223 84210 11 15254 46353 77509 12 08722 71317 13 02701 34141 65639 97194 14 28807 60478 92207 15 23994 55S40 87744 16 19706 51728 73° 3-27 08526 42588 76715 3-28 10907 45164 79487 3-29 1J^7G 82851 3-30 17438 52091 86811 3-31 21598 56452 91373 3-32 26362 61419 96543 3-33 31736 66997 3-34 02326 37724 73191 3-35 08728 44333 80008 3-3615753 51568 87453 3-37 23408 1715948 48147 I 804063 18 12724 45102 3 77540 19 10039 42598 3' 75217 20 07897 59434 95531 38 31699 67938 39 04249 40631 77085 40 13612 50210 86882 40638 3-4123626 73440 60443 21 0G304! 97333 39228 3-42 34297 722151 71334 22 05233 3-43 08446 38373] 45631 7154G, 82891 23 04780 3-44 20226 380781 57035 74° 3-48 74144 3-49 12470 50874 • 89356 3-50 27916 66555 3-51 05273 44070 82946 3-52 21902 i-53 00054 39251 78528 1-541788G 57325 96840 ;-55 36449 76133 ;-56 15900 55749 95681 ;-57 3569G 75794 1-58 15975 56241 96590 ;-59 37024 77543 .-60 18146 58835 714381 24 04860 3- 383461 71895 3- 25 055081 391841 72924 3- 26 067281 40596' 74529 3- 27 085261 17° 95120 I 45 32679 1 3 70315 I 46 08026 3 45813 83676 ,3' 47 21616 59632 1 97726 3' 48 35896 i 74144 3- 16° I 61 40469 81415 62 22447 63566 6304771 46064 87444 64 28911 70467 65 12111 53844 95C65 66 37575 79575 67 21665 63845 68 06115 48475 90927 69 33469 76104 70 18830 61648 71 04558 47561 90658 72 33847 77131 73 20508 15° 76° 3-73 20508 63980 3-7407546 51207 94963 3-75 38815 82763 3-76 26807 70947 3-77 15185 59519 3-78 03951 48481 93109 3-79 37835 82661 3-SO 27585 72609 3-81 17733 62957 3-82 08281 53707 99233 3-83 44861 90591 3-84 36424 82358 3-85 28396 745S7 3-86 20782 67131 3-87 13584 60142 3-88 06805 53574 3-89 00448 47429 94516 3-90 41710 89011 3-91 36420 83937 3-92 31563 79297 -93 27141 75094 •94 23157 71331 3-95 19615 68011 5 16518 65137 3-97 13868 62712 3-98 11669 60739 3-99 09924 59223 4-00 08636 58165 4-01 07809 14° NAT. COTAIT. 266 NATURAL TANGENTS. 76° 4-01 07803 57570 4-02 074-10 57440 4-03 07550 57779 4-(>4 081-25 585J0 4-05 09174 59877 1-06 10700 61643 4-07 12707 4-08 15199 66627 4-0918178 69852 4-10 21649 73560 4-11 2561 i 777S4 4-12 30079 82493 4-13 3504G 87719 4-1440519 93446 4-15 46501 99685 4-16 52998 4-17 06440 60011 4-18 13713 67540 4-19 21510 75606 4-20 29835 84190 4-21 38690 Q3318 4-22 48080 4-23 02977 58003 4-24 13177 684S2 4-25 23923 79501 4-26 35218 91072 27 47066 4-28 03199 59472 4-29 15885 72440 4-30 29130 85974 4-31 42955 4-32 00079 57347 4-3314759 13° 77° 78° 79° 80° 1 81° I 82° 1 / 4-33 14759 k 1-70 46301 5-1 445540 5-6 712818 6 31375151 --1 153697! 60 72310 t-71 13086 525557 809440 2566011 304190 59 i-34 30018 81256 005813 906394 370126 455308 58 878t6 ^ t-72 49012 686311 3-7 003663 496092 607056 57 4-35 45C61 4-7316354 7C7051 101256 610502 759437 56 4-36 04 J03 .85083 848035 199173' 737359 91^456 55 62293 - 4-74 53401 9232C4 2974161 858G65 7-206G116 54 4-37 20731 4-75 21907 5-2 010738 3959881 980422 220422 53 79317 93603 092459 494889 0-4 102G33| 375378 52 4-38 38054 - 4-76 53490 174428 594122 2253011 5309871 51 93940 4-77 28568 256047 C93CS8, 348428 687255 50 4-39 55977 97837 339110 793588 472017 844184 49 4-40 15164 i-78 67300 421836 893825! 596070 7-3 001780 48 74504 4-79 36957 504809 994400 720591 160047 i 47 4-41 33996 4-80 00808 588035 5-8 095315 845581 318989: 46 93641 70854 671517 190572 971043 478610' 45 4-42 53439 4-81 47096 755255 298172 6-5 096981 638916 44 4-43 13392 4-8217536 839251 400117 223396 799909 43 73500 88174 923505 502410 350293 961595 42 4-44 33702 4-83 59010 5-3 008018 605051 477672 7-4123978 41 94181 4-8430045 092793 708042 605538 287064 40 4-45 54V5G 4-85 01282 177830 811386 733892 450855 39 4-4615489 72719 263131 915084 862739 6153571 38 76379 4-86 44359 348696 5-9 019138 992080 780570' 37 4-47 37428 4-87 10201 434527 123550 6-6121919 94G514' 36 9S636 88248 520020 228322 25-2258 7-5113178 35 4-48 60004 4-88 60499 600993 333455 438952 383100 280571 34 4-49 21532 4-89 32956 693030 514449 448699 33 83221 4-90 05620 780538 544815 646307 617567 32 4-50 45072 78491 867718 051045 778677 787179 31 4-51 07085 4-91 51570 955172 757644 911562 957541 30 692C1 4-92 24859 5-4 042901 864014 6-7 044966 7-6128657 29 4-52 31601 98358 130906 971957 178891 300533 28 94105 4-93 72068 219188 0-0 079676 313341 473174 27 4-53 56773 4-94 45990 307750 187772 448318 646584 26 4-54 19608 4-95 20125 39G592 296247 583826 820769 25 82008 94474 485715 405103 719867 995735 24 4-55 45776 4-90 69037 575121 514343 856446 7-7 171486 23 4-50 09111 4-97 43817 664812 623967 993565 348028 22 72615 4-98 18813 7547S8 733979 6-8 131227 525366 21 4-57 36287 94027 845052 844381 269437 703506 20 4-58 00129 4-99 69459 935004 955174 408190 882453 19 64141 5-0045111 5-5 026446 6-1 066360 547508 7-8 062212 18 4-59 28325 5-01 20984 117579 177943 G87378 ' 242790 17 92680 97078 209005 289923 827807 424191 16 4-60 57207 5-02 73395 300724 402303 968799 606423 15 4-61 21908 5-03 49935 392740 515085 6-9 110359 789489 14 86783 5-04 26700 485052 628272 252489 973396 13 4-62 51832 5-05 03693 577003 741S65 395192 7-9 158151 12 4-63 17050 80907 670574 855867 538473 343758 11 82457 5-06 58352 763786 970279 682335 530224 10 4-64 48034 5-07 3'o025 857302 6-2 085106 826781 717555 9 4-0513788 5-08 13928 951121 200347 971806 905756 8 79721 92061 5-6 045247 316007 7-0 117441 8-0 094835 7 4-66 45832 5-09 70420 139680 432086 263662 284796 6 4-67 121-24 5-10 490-24 23-4421 548588 410482 475647 5 78595 5-11 27855 329474 665515 557905 667394 4 4-68 45248 5-12 00921 . 4-i4838 782868 705934 860042 3 4-09 12083 862-24 520516 900651 854573 8-1 053599 2 79100 5-13 65703 610509 6-3 018866 7-1 003826 248071 1 4-70 46301 5-14 45540 712818 137515 153697 443464 Jgo W Vi 10° VT. OOXAt 99 Eft go r / NATURAL TANGENTS. f 83° 84° ! 85° 1 86° 1 87° 1 88° 1 89° / 3-1 443464 3-5143645 11.430052 |l4-300e66!l9-081137 |28-636253| 57-289962 60 1 639786 410613 468474 3606961 187930 877089 58-201174 59 2 837041 679068 i 507154 421230 295922 29-122006 69-265872 58 3 5-2 035239 949022 546093 482273 405133 371106 60-305820 57 4 234384 9-6 220486 585294 543833 515584 024499 61-382905 56 6 434485 493475 6'24761 605916 027296 f 82299 62-499154 55 c;;5547 768000 664495 6685-29 740-291 30-144(19 63-650741 54 ■" 8:>7579 Q-7 044075 704500 731079 854591 411680 64-858008 53 8 S-3 (1405SC 3-21713 744779 795372 97U219 6^3307 00-105473 62 9 244577 600927 785333 859010 20-087 19 J 959928 67-401854 61 10 449558 381732 826167 924417 205553 31-241577 68-750087 60 11 65553C 9-8 164140 867282 989784 325308 528392 70-153346 49 12 862519 448166 908682 15-055723 446486 820516 71-615070 48 13 S-4 070515 733823 950370 122242 569115 3-2-11 8099 73-138991 47 14 279531 9-9 0-21125 992349 189.349 693220 421295 74-729165 46 16 489573 310088 12.034622 2.57052 818S2S 730265 76-390009 45 IG 700051 600724 077192 325358 945966 33-045173 78-126342 44 17 912772 893050 120062 • 394276 21-074664 366194 79-943430 43 IS 8-5 125943 10018708 16323G 463814 204949 693509 81-847041 42 19 340172 048283 206716 533981 336851 34-027303 83-843507 41 20 5554C8 078031 250505 604784 470401 367771 85-939791 40 21 771838 1079.54 294609 07 6233 605630 715115 88-143572 39 22 989290 138054 339028 748337 742569 35-069546 90-463336 38 2;-. 8-6 207833 108332 383768 821105 881251 431282 92-908487 37 24 427475 198789 428831 894545 22-021710 800553 95-489475 36 2o 648223 229428 474221 968667 163980 36-177596 98-217943 35 20 870088 2r0249 519942 16-04.T4S2 30SC97 562059 101-10690 34 27 8-7 093G77 291255 .:^65997 118998 454096 950001 104-17094 33 28 31719: .•:22U7 01-2390 195225- 602015 37-357892 107-42048 32 29 542461 353S27 659125 272174 751892 76r613 110-89206 31 30 76SS74 385397 706205 349855 903766 CS.-1PS459 114-58865 30 31 990446 417158 753034 428-279 23-057077 617738 118-54018 29 32 8-8 225186 449112 801417 507456 21306r 39-056771 122-77396 28 33 455103 481261 849557 587396 371777 505895 127-32134 27 34 686206 513607 898058 C68112 532052 905400 132-21851 26 35 918505 646151 946924 749614 694537 40-435837 137-50745 25 30 8-9 152009 578895 9961 CO 831915 859277 917412 143-23712 24 37 386726 611841 13-045769 915025 24-026320 41-410588 149-40.-,u-j 23 38 6226C* 644992 095757 998957 195714 915790 156-2.5908 22 39 859843 678343 146127 17-083724 367509 42-4S3464 163-70019 21 40 9-0 098261 711913 196883 169337 .'^41758 964077 171-88540 20 41 337933 745687 248031 255809 718512 43-508122 180-9:^220 19 42 578867 779073 299574 343155 897826 44-006113 190-98419 18 43 821074 813872 351518 431385 25-079757 638596 202-21875 17 44 9-1 064564 848288 403867 520516 264361 45-22ri 41 214-85762 16 45 309348 882921 456625 610559 451700 8-2f'3-l 229-18166 15 46 55543G 917775 509799 701529 641832 40-44SS<-2 246-55198 14 47 802838 952850 563391 793442 834823 47-085343 264-44080 13 48 9-2 051564 988150 617409 886310 26-0307:^6 739501 286-47773 12 49 301627 11-0-23676 671856 980150 229638 48-412084 312-52137 11 60 553035 059431 726738 18-074977 431600 49-10?881 343-77371 10 61 805802 095416 782060 17OS07 636690 315726 381-97099 9 62 9-3 059936 131635 837827 267654 844984 .50-548500 429-71757 8 53 315450 168089 894045 365537 27-056557 51-303157 491-10600 7 54 672355 •204780 950719 464471 271486 52-080673 572-95721 6 55 830663 ^41712 14-007856 664473 489853 882109 687-54887 5 56 9-4 090384 278885 065469 665562 711740 53-708587 859-43630 4 67 351531 316304 123536 767754 937233 54-561300 1145-9153 3 58 614116 353970 182092 871068 28-166422 55-441517 1718-87.32 2 59 878149 391885 241134 975523 399397 56-350590 3437-7467 1 60 9.5 143645 430052 300666 19-081137 636253 57-289962 Infinite. / 6° §0 4° 9 3° AT. go7^ 2® P O*' / SLOPES, FOR TOPOGRAPHY. TABLE XV SLOPES, FOR TOPOGRAPHY. 1 1 Degrees. Vertical Rise in loo Horizontal. Horizontal Distance to a Rise of 10. Degrees. Vertical Rise in TOO Horizontal. Horizontal Distance to a Rise of 10. I 1-75 572.9 19 34.43 29.0 2 1 3-49 286.4 20 36.40 27.5 1 3 5-24 190.8 21 38.40 26.0 4 6-99 143.0 22 40.40 24.7 5 8.75 "4.3 23 42.45 23.5 6 10.51 95.1 24 44-52 22.4 7 12.28 81.4 25 46-63 21.4 8 14.05 71.2 26 48.77 20.5 9 15.83 63.1- 27 50.95 19.6 lO 17-63 56.7 28 53.17 18.8 II 19.44 51.4 29 55.43 18.0 1 I. 21.25 47.0 30 57.73 17.3 13 23.09 43.3 35 70.02 14.. 2 14 24-93 40.1 40 83.91 11.9 15 26.79 37.3 45 100.00 10. i6 28.67 34.9 50 119.17 8.4 17 30.57 32.7 55 142.81 7.0 i8 32.49 30.7 60 173.20 5.7 Note.— See page 52, Art. XVHI., for examples in the use of Table XV. TABLE XVI. CHORDS, MIDDLE ORDINATES, EXTERNAL SE- CANTS, AND APEX DISTANCES OF A ONE- DEGREE CURVE. The angles of the table are the intersection angles, I, equal to the total central angle inchided between the tangent points. To find the corresponding func- tion for any other curve, divide the tabular number by the de- gree of curvature. The unit chord is assumed to be one hundred feet long. By using radius of 5,730 feet, the chord column of the table can be made serviceable for plotting. To use the table for curves having chords of 20 metres each, divide the several tabular functions, that is to say, tbe Chord, Ordinate, Ex-sec, and Apex Distance, by 5 times the proposed metric degree of curve, for the proper values of said functions in metric measure. Thus, for a 2" metric curve, chords 20 metres each, the divisor would be 2 X 5 = !0; for a 2° 30' curve, 12.5, and so on; taking care to reduce minutes to deci- mals of a degree before making the multiplication for a divisor in each case. Also, if the length in metres of any proposed Chord, Ordi- nate, Ex-sec or A. D. for a given angle be known, the tabular function corresponding to that angle divided by 5 times the known chord, ordinate, etc., will ascertain the degree of curvature for chords of 20 metres each, using the foregoing precaution as to decimals. Example 1.-4° curve, intersection angle 48°, 20-metre chords, . '. apex dist. = 2551.1 -7-5x4 = 127.55 metres. Example 2. — Apex distance 200 metres, angle 58'' 20', .*. de- gree of metric curve = 3198 -f- 200 X 5 = 3^^^ degrees = 3° 12', chords being 20 metres each. 270 FUNCTIONS OF A ONE-DEGREE CURVE. 2 o n n S S 8 t-. m t^ ro t^ ^S^^ 8 1^ ro 8 t^mo r^roo t->ro 8 ro O t^ m 8 c \o ro OvO VU en Qso m HI M „ 8 < a^O^O^ O ON 0^ u a rOOO •* 30 » t^ t^oo o rovo Ov N \D o o m -lO in -^ro en inoo o m moo m^ OO M -^ t^ N 1.>1 ro ro (<) <1 ■* m ir, LOVO vo O vO Cv t^oooooo O O r » o CO 0000 Tl- t-- ^ N OO oo r^ t^oo a\ cr,\0 CT. N vO n in M \o rr, •n in Tj- ro ■*vo inoo m\c> oo N (.M ro m ro ro (^ ■* in in lO^ kO VO VO t^ t^ f^ t^OO 00 00 o o o O O O O O o O O O s Q 8 8 8 8 8 8 8 O rONO 00 OS o rovo o ro^O X 8 rovo o m^o n rovD o rO^O n mvo n rovD s r<-)^ o mvo n rovO mvn OS " n- m vo vo t^ t^oooooo Cy. On ON ON :^ N TtvO 00 o M ^vOCO O ^vOOO o (M Tt vO 00 N T^^O00 N ^NDOO T)- 'I- •<1 ^ ' O N -^nO 00 O -*VO 00 O N -^nO ( Tj- -.1- -.^ in in Q < ONo"roONo"roONS' roONo"roOND roONO roOvo'rnONo'foONO coOno roO H m invo CO o w m invo 00 w ro invo 00 w rn invo 00 m ro inso 00 O.OOO O.OOO O.OOI 0.002 0.004 0.006 0.009 0.012 000000000 666666666 6066600 o'odcodd t^ On D in ON •* ON \ N N m 'I- -* inNO t^ I 00000000000 0000000006666666666600 O ■'J-oo , ON O H M N N 000000000 i d 8 6 mvo 5 mvo m f-, rONO mNO fONO rOND mso rONO rONO mo O a CJ fONO rONO ^^^ -^0 mNO mNO^ ^NO mvO^ mg mvo 3 5 -^NO 00^ j^NOOO s s ^^^ ^ ^z-^^ % % ?^°^ a ^ ^^^^s FUNCTIONS OF A ONE DEGREE CUEVE. 1 1 N T^^oo M ^\o 00 N -+VO 00 !N ^^ 00 M -^'O 00 w 'i-vr CO MMr,H(NOcin(Nmmmmm-r'*-'j- 1- tc i/^ lo m u, ir,v; 0- t^ Ti- M 00 lA fi t-- 1- t^ 'S- 0\0 m a\vo mo t-^ ^ m o^d mo ^^'^0 t-~M-Moo mw t^-TTO i-.TfM t^-^>H r.,mo t^^O t-^O 1^ ^0 t^ ■*• w roiovooo O M mu-)vooo o w m iomd oo O " ro in^ oo m ro lo^o oo O in vn in u-> in mvo •o^oyD'Ovo t>t^t^t^r>- t^oo 0003000000 osoooooo „„„M«MMMM«„N CO 1^8 Jg^HHS^^SSSi^^^cgi H^g^S^HE,'^^! M N n 0) (N (N (NOiNWf)f)(NNP)NiN(NiNN(NCjmf)mmmmmmm 2 Q -)-oo rooo moo vD in ^^ -too M M in N n^^ -* m m Pi n n -^vO ooomt-^fivowvomot^'*-" mob (Nti^tvJtN i^D i^Pi r^NOT moo ^fOMnO^ n r^mo> N (N U vO ON in N OnCNVO ON^O OnN^O OClvO OPivo 0>C1^0 ON^ 0^ fi^O o\ g; S'^ g^ ::^ mmmmmmmmmmmmmmmmmmmmmmmmm N Ti-^ 00 N ^^CO N ThvOOO N ^\C CO i-o>t^>nmcj(NPiNMO) ^i-vo 00 ^ -^ a o Q 00 oOnOnOO M H M ci M mmm^- s^sls^l NNNn(N(NP)(NPlP)(N(NO(NO(N0)NCSNC>)NPlNM 2 N -4-vOOO N ^^00 N ^\O0O N TJ-^00 P) Tj-^OOO O N ^vO 00 O 272 FUNCTIONS OF A ONE-DEGREE CURVE. -| h 2 « ^NOOO ^j ;^NOO0 Cj ^NOCg 0,f^?;^~ ^^VOOO « ;J;nO00 < u ■^ lOvo NO t^oo ONONO>-.00 m »0 10 10 in 10 10 lANO vOvOvOnOnOnOnOnononOvO t^c^t-^t^t^t^t^r^t^t^^. Q Q ■* lANO NO t>.oo 00ON0HP)(Nro-*>A lANO t>«oo ONOOMwro-^iri xovo t-«00 vn in lAiiAno in in invo NOvONONdNONONONdNOvONO t^t^t^t^t^t^t^r^t^t^t^ § 5 §8 S SlcS-a JS^ S^Si.c^'S :^o^|:rS>c?^S,c?^ 5^cg ir=?^cS ?5^,S JT^cg ■u-ininininininioinininininininininininininininininininininmin N -^NO 00 N -^NO 00 N T)-NO 00 P) ^ND 00 PI rhNO CO r> M -^NO 00 H M M M M N 01 P) PI « rorocororo-?J.^^<^5-^5^°5-S>S>S^'S>v8 Q T^ w^ o^-o r<-) 1- 00 in ro (^ lo O PI m m t^oo P) m in t^oo in in in in in in\o vo ^ ^o 'O ^ rocncommcnmrnromrocn P) m m t~«oo N m in f^oo d N m in t^ Oi O M w P) p) fnmm^- On M m T^ m\D r^oo On m m ■* mo t^oo o> d Q \or^o>o«P)rn-m^-)T^TJ--^Lnm in\0 vo ^ c^ r^ t^oo oo oo cy. o On 1 P) -^^00 P) -I-VDOO PI „ „ M M w D D S-^'S ^^^'^'^^%Z%%^^Z'^%S I 1 z O P) -^vOOO O M T^MD0O M ^^'S ^ ??, ^^"^ 5- ^ ^^°^ 2> S. Ji^-S^vS < ^^^^%^^S:^^S^ SoSJ^SNg^P^S-f^^S^K^-^?:^'?^? 8S ?S'^'?2 " ^i:'^" PI rn in t-^oo PI m m r>.oo o pi m in t>.oo O rncnrnrnrncnrorocncnrnfotTic^CTjcnrof^rn MMMMMMMMMMI-I^O vo in ■* rn P) M ooo t-xvo in ooo\Oi-P)m-*Tf u-ivo r^oo mOwOmOmONNPI ON o M PI rr, -:^ mo r~-oo On - PI m ^ mo tx t^ t^03 oooooooooooooooooo 00 OnOnOnOnOnOnOnOnOnOnO O O O O o d ooonQ h. n mm-* mvo t-oo ON moo PI t^i-o o m^Tj-m mmPI PI M M O O OnOnOnOnOnOnOnOnOvOnOn ON M w m 'i- mo o f».oo on o « pi m -^ mo tv t^oo oooooooooooooooooo 00 OnOnOnOnOnOnOnOnOnOnOnO O O O O Q s 1 i z" « -"l-vO 00 O (N -*vO 00 Pi ^^~ PJ^^O<» ^^NOOO O p. JJNOOO^ 274 FUNCTIONS OF A ONE-DEGREE CURVE. N T^^o 00 o Tf-^-^Tt-^Tj-^-^Tj- t^ t-~ Ir^oo c 100 oo>o>oooo ro^O a. C-) o a\ cj vn 0> N ^ O M VO O (N lO On M lO n\ N in CTi N m n\ u-)inininu-)iniouoio ionO vO C/J S.S<2'S^?J,°^^cS^^^ r-^MCO -"l-MOO ION ONt-^'l-N OnvO ■* M On-O Tfo OO o. M ro T^^o 00 OA M ro inso oo o w ro lo x' ^^■^^ !;rfrs-?r?r^^=S OOOOOOCO 0^0^0^0^0^0^0 >- m >-. n ooo M Tj-oo w -a-oo H Tj-oo M ^00 m -*oo w rj-oo m ^ r~, t- -s- r-, ONOOOi-iHMONNrOrOro-^i-^-^ioin lonO nO no c^ i^ t^oO 00 oo On On On OHHWI-IWHMl-l,-.MWHMHWMMMHMMMMMMM«e-WM O .-*0 r^-^woo -^moo ini OnnO ro O O ro O r^ H ro -"J-nO 00 On i O O O O O O ' 'l-NO 00 ON M w ro -^nO oo on m ro 10\0 00 I rorororororo■!)-Tf■T^Tl-T^l lOioioioiotomiOLoioioi rooo rooo rooo rooo rooo On -^ O lO O vO iOnO vD ^ no . Q OS o oo CT« O N ro loS £»* 00 rooo rooo rooo ^ ON M (M Tj- lO t-^00 ON-^Ov-^O lOH t^rooN'i-ONO Noo 1-1 ro t)-no ooOnm pi -"j-mt^ONO PI ro N N N (N N N CN) 0) 0) (N O ooooooooi-'wHMCMNNfiwfO mronror^roromroro^oro^ororOTt•■^■^•.t■■^■.^■^■^•£) u^u^u^u^u^^Tl-.*rorO(NHHlHHHWMHHHH CO N -*u-)t-^a>H roint^ONH romt^ONH romc^ONH mu-)^^o^M foiot^ vd t^ t^ t^ t^ r^ t^OO 00O000COONaNt>ONON6660dHHH.^H(Nc!(N(N c^rnrocnrOrorororOrO(^rorororororOTt-->i--^-^-<^-^-^-^-.^-O0O N lOOO nvO On rONO OnC. t^oo OO oo on On On rommrot^rnrornrorororomi^rorororororomrororot^rnroro O M ^nO 00 O O N Tj-o 00 O O O O C vO NO NO vD V I O (N -^mi-^ONO N -"j-ini-^ONO (N Tt-Lnr^ONM n -i-vo t^ on m n 1 M M M M ii M « N o o N PJ ror O (M CJ 01 OJ rororor^m'.*-'-*"'-*-T}"--i-ioiOi/^iOio ionO nO no nO nO f^ m ro CO ro CO m I . ro t^ r'lNO Ov N moo N inoo H.^tvO-*t^Oro o X U On N D N ■.M ■<^r>.M -nj-t^M ^t^o -0 Tj-t^o -.j-t^o ■^^l->.o ■«^t^ M M CN| IN C) r»imrO-.t-.t.^c^t^r^r^r^t>..t.~t-^t.^t^r^t^t.~t-^f^t^r^t^c^t^t^t.%t^ t^oo oo oo oo .^vo 0> M ^vO 00 y mo 00 rOO O P) moo m M-t-~0 Tht^H TCb-M rt-OO M T)-o 00 M ro m t^ N ^vO ON M ro moo pi m t>- on p) .*o on w roo oo ro OnonOnO 6 6 C) M M H H M (N (N pj pj rororororo^T5-4-4-mmin mo o Tj--*.. o Pi -^^O On m roo oo ro m t^ j ^ONONONONO^OOO^J^J^I^^^wcj^C^C^C^rorom 1 1 o X ON P) moo M Th r^ O roo on pi moo w r*- t-. o roo On P4 moo w Th r^ o roo On mONP) mONPi mONPo moo p) moo w m>oo P) irioo w m,oo m moo w moo w ■* OnOnO O " ^- " P» P^ P< rororOTl-^T(-mm mvO O O f- t^ t^oo oo oo On On i s i° ! < o o>oo t.^ t.^0 t^oo O PI Tt-mr^ONO P^ T^lrlt^ON0 Pi -i- (N PI roforofororo.^T^T^.^.^.^mmm Tj-O t^oo O m t^ On " Thv , m 'ir^'^ o o o T)-0 00 O P) .* ro m r^ O pi tj- 00 00 On ON On S ! lOO o o o o ■^O p.- On O P) t-~ t.^00 OO 00 00 ON Q o On N moo M moo H ^ t-.Q roo On P) O ON PI m 00 M ■>*• r-O ro t>~ roo ON o o On roo ON roo ON roo On roo On PI O r-h N O ON N vn rTN (N O ON N O On PI in ON ON o m m mo O O 30 00 ON ON U z o P) O § 278 FUNCTIONS OF A ONE-DEGREE CURVE. 1 N ^nO 00 (N ^vo 00 (N Tt-^O 00 (N -.J-^D OO W ^vD 00 P) ^vO 00 „„„„„CH(NN(NiNroMmrorOTt-Tf^Ti-Tt-inin in 2^ mvo < in invO \0 vO tv. t-^ i^co oooo OnOnOnQ C h m pj rorh inyD no t-^oo on "-i N roo t^n-MOO inw onnO ro o t^ rt- n onno mO t^rf^O) inpj onno ^ m oo in NO 00 ON H ro -4-^0 OO ON M rn ino co d w ro m t^oo 6 pi rn m t^oo 6 pi 4- in i-^ in in \D'^ NOnC/nOnOnO c-..tv.t^l-^ t^co oococoooco OnOnOnOnOnOnO o Id C/} ■^ONinOMO n '^ PI t^ rnoo mONinM i-^pioo ^^j-Qno ro onno pi on-o p) on in pi NO CO M T)-vo ON M '^NO O H ^j- ^o Onpi -^t^ONpi mt-^O P< moo O rONO 03 W ^ rorOTt-T]-.I^T)-lnlnln in^o no no no t^ t^ t-^ t-^co ooooonO-OnOnOOOOmm nOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnOnO t^t^f-t^t-^r^ I O Q 1 Tl-oo rooo rooo PI t^ PI t^ P) t-^ roco TfON^O u-nhno m t^roONinONO PJOO -^ c^ H .^vo ON H ^No ON H T^NO On m ^no Oncj Tf-t-^ONp) T^t^ONP^ inc--o N in N mrororo-. M T)-cO P) no •^CO PJ no rt- On rOOD P4 NO m, in T^ 1 t^ ON PI ^ t-. On M Tf-'O CO M ro inco o roini~,o Pi int^ONPi T^^^ONPl .*-r-,ON in inNO NO NO NO t^ t^ t^ t^od oococooNONONONoddoOMMMMpiNcJpi i inmininininininininininininininin inNO nononononononononononono 1 On PJ inoo I-; 'f f;- rONO On pj inCO ►^ -f t;- rONO ON P) inoo ►^ ^ t;~ rONO On 1 S) S)NO NO^NONONONDNONO vcTno^nO^nonOnO NO NO NO NO NO S no'nO^noS S S no'no' „HMMMMMMHMHMMMMM„MMMMHMMMMMHMMM j 1 C< -^nO CO N n-NO OO PJ -^nO 00 M .^hNO OO O PJ -+nO OO O PJ -shNO 00 I FUNCTIONS OF A ONE-DEGREE CURVE. 279 1H Z S « rhvOOO ^. 2-^00 O Cj ^vocg O JJ^^vOC^ O ^^vOOO O C^;J;vOeO O Q »0 t^ O N r^invooo o m n ^f u-|^ OO N Th\0 00 P4 ■* 00 lOroO f^■<^woo uiN t^-*-MOO irimO t^M-M Oivo ro oo i/i M 0>vO •'t- Hiisil^ §; S; S S; ^ iiii?^l^f S^ § 111 § § §1 1 1 vo ON S tn S^'o fo^D O N inoo rovo" On (N iot^ r? ? t^ rO'O On N lOOO fO t^ t-OO OOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 0&-0QO0 CO 00 00 Q O Q OONO fOHOOvO Tj-H ONt^iorOH OCOVO ior»lH Ooo t^iDTj-N H ON t^NO in lOOO w •♦-O ON N m t-^ rONO On fJ ■* t^ rOvO ON m ■* t>. rONO On H ■>!- t^ O 0000 OnOnOnOnQ " w >- *~ CN M N nc-omm-*->*--*ioinm rnvo no vo t^ t- t^ c^ t^ c^ t^oo oocooooooooooooooooooooooooocooooooooooooooooooo d o 5 moo H fONO On N moo ponO on O in t^ cOnO On 0) -^ t^ rONO On m ij- t^ l^cl^l^ 8n n Sn s^ s; §N 1 s; s s s H 1 1 S sHi §^ ^HI^In 1 2 ^ O c* ^NOOO « ;^NDoo c; ^ND^ %^'^'%%%%T%'%^^^%%^ 00 1 1 1 N ^NO 00 O N ^NO 00 O N ThNO oo N ^VO 00 N t(-nO 00 N 'l-NO 00 O < ^^Z^^^'^^^ Snn^S^^^^^cS^nS ?J>??:^^g§ sas ?l^I?S 0NON0N0NON0>0N0NaN0N0N0N0N0N0N0N0NO>ON0NON0N0N0NON0NON0N0NO>0N J a N ONNO roo r-^MOo inN ONt^^w ONt^-*f) ON^^T^ OOi-ii-iMi-ip)NNpqrom(T)m-<^Tf--*mmm mvo no no vo t^ t^ t>.00 00 oo Q r^ O fovo ON N m (^ fovo on n moo h ■* t>. o mvo On h -^ t>. o fO\o o ci in | r£gi»f^^i'ii'i:"3i^tKHsi„-i=Hs'ii' N -l-NO 00 N ■♦no 00 M tJ-vO 00 N t)-vO 00 O N •♦nO 00 N •♦nO OO O „ „ „ „ M ^ -"J-MOO iDMO (>.■*« OnvO r0 ro h oo M ro in f^oo O N ^iot^O\w N Thvo t^ o H ro t)-^o oo O h ro lovo oo O N ro vOvDvO^^ t^t^t-^r^t^ r^oo ooooooooco C!\ a^ o^ C!\ W t^ t^oo coco onOvOnQ O M H M IN N M rororo-*-*-^-*ir)io invo ^D vo t^ O^O^O^O^q^O^O^O^OOOOOOOOOOOOOOOOOOOOOOO UTO >0 t^ t^oo OnOnOOOOOO On . H ■* t^ O rr)\0 ON P) lOOO N lONO NO NO t^ t-^ t^oo 0000 OnOnOnOnO O O h h h (M P) P) roror<-)(T),j-Tj.^ir) OnOnOnOnOnOnOnOnOnOnOnOnOnOnOOOOOOOOOOOOOOOOO lOCO O rONO ON H T»- t^ O PI inoo O p^nO on h tI- t^ o n lOOO h ponO On h ■* t^ 00 M irioo M -^co M -^00 1-1 -^t-^H -o mv O P) ■'(-NO 00 P) ^NO 00 P) T)-NO CO N -^ND OO N -4-NO CO PI ""(-NO 00 < ^MCO inPi t^'J-HOONO roc r-.Tl-H onno poOoo iapj o^c^'I-hoono roO O N rOint^ONO N ^int^ONM pg T^ND t^ on « ro 'S-nO oo On m M iriNO 00 P) o ?o*?o'o'o'o'o'o"o"o'o'o'o^o^o^ o"o "o OnOnOnOnOnOnOnOnOnOnO i pono On w inoo w -!^ t^ o T)- OOOOOO OnOnOnO O O 1 irjNO NO NO NO o onoo t~~NO NO in 1 O POnO On POnO On On On On Q Q On On On On O D lOOO H -^ t^ O rONO ON I rONO ONP4NO ONPINO ONPI inoNi OOOOOOOOO rn m CT) -^ '^ -^ irt \ t^ O rovo On inoo H moo tv r^oo 00 00 o o o o o FUNCTIONS OF A ONE-DEGREE CURVE. 281 ■ct>6 o o woo rovo O m ["> I M M M^ pi rnror^- inoo N vo o mvo O in invo vo vo t-. t^oo in m mvo vo vo r^ i • t^ H -^ r^ H -"j-oo H -*oo H inoo . t^oo OOOOO^O^O^OOOwHM inoo N moo n m C3^ w (N N romrOTj-Tj-Tt-m •"1-vO Ovw r}-t^o>p) Tj-t^o O m moo O m^o i (-00 H ■a>000ww r^ O tn t^ O rOvO O rOMD on fOvO O N ^D Ov I O M -*vO00 O O N -^^ 00 O *- m rocom(r)roro^^-^T^T^mmmmm mNO no no no i N O ON NvO ON NVO 0vO ■* w CTi p tM p IT) t-^ C^O IN _-?-vO _t>j5N M ro -3-^ 00 O H rfi to t>CO O N Tj-irjt^osM CJ t^ r^ r^ f^ f^ t^oo < in OS rovO -^00 IN ir)0\rof-.M ino r<-)\o O -"l-OO N \0 O -"t-OO M in O (^ I^ N inosNvo O mt^o M-co H inoo N \o o m t^ w inoo n in ct> m t>. o -^oo -^\0 O w 'JfO t> M -*^ OS H TfvO OS HI -^^O OS H TfsO OS M "^vO OS w -^vO OS O roso O cn\0 OS roso osoiso osN inos(N inoo C) inoo w moo m T^oo « ■■^ r~ oqoqoo qsc^ososqoqwMM_nNNrornrn-!j-^^inin mso so so r^ t^ i^ O (N -*sO 00 O M ^sO 00 O IN -^SO 00 O (N ^sO 00 O P) -^sO 00 O (N -^sO 00 O 1-1 I-. M M H p) 0) 0) 0) c) romrotr)m->f-^- rr, en <^ (^ l-^CO 00 00 m n ro m m ? 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" "' "' ^ ^ 'I- ^ 286 FUNCTIONS OF A ONE-DEGRKE CURVE. O Pi ^\0 00 O N -^O 00 M ■^'O 00 O P) -^^ 00 O P) Tj-^O 00 O N -^VO 00 O H »- H H 1-1 p4 N PI pj pj mromrofO'*-^i-'*--<*-->j-iom>oin mvo rOM ONf-iO'J-M 000>O \r, r<^ n 0\ t^vO tJ-n OoomD irirOM o t^vO -^ P) O 00 (-» O O PI "i-vo 00 1- romt^oO P> -^mD oo O ►^ roir)f^cj\0 Pi -^^O 00 O w a>0>OOOOOMHMMwi-ip)NPj(NPirororororor<^-00'-ii-iNP)r^. ro->*-iA invo vD t^ r^oo 00 o O O O w w P)PlPiP)P)PlPIP)P)P4NP)PiP)PIP)P)P»P)P)P)piP)NNNPjP)P -i- On ■* 0^ rooo POOO VO VD \0 >^ (-^00 00 O O O tv. P) t^ PI VO -!i- T^ in lovo vo lO O •>*■ 0\ ■* O -^ . t^OO 00 00 On 0\ 6 ■^vo 00 O PI "«i-vD i^o»w mlo^^o^H■fo mvo oo o N -^vo ■ ON-'*o»^ONrt-ON-*ONThONTfo mo mo mo mo m«vo h^o m OMHP)Ncoco^-*m m^ vO t^ t^oo a- O O m h p) n cocOT^■.*-m m\o COCOCOCOCOCOCOfOCOCOrOCOCOCOCOCOCOCOrf•-P)P)P)PlP)P)NPlP)'PiP)P)DP4PlPlP)NP)PlPlPlPlP)P)P)(NP)P)P) O m 0> -^oo cooo p) ~ N Pt Q PI PI M m O •<^ 0> -"i-oo cot^. N t^pivo m mo mO "^0 mvo ^ t^ t^oo 0000 ooo O M M n p) coco-^-^in m\D PJ PJ PI PI P) PI PI PJ PI PI cococococococococococococo PlPlP)PlPlP)PIPIPlPlP)PIP)P»INP)PgWP)PIP)P4Pl O PJ -"it-NO 00 o p) ■^^vo OO o PI m t^ o> I CONO O covO On P) M pomt^ONii fomt^ON cocococococococoi o PI -"i-NO 00 o w -^No 00 o O O H M H MPIP)P«NP»WMNOP)PlPiPlC4MPlPl icorocOfOcorocococorofOcocofofOfOfO FUNCTIONS OF A ONE-DEGREE CURVE. 287 r 1 1 ! i 1 1 N ">1"0 00 O M -+vO 00 P) ■^"O 00 O W -*^ CO O N -^-^O OO N -^^vO 00 < 1 t>.ir)rON 000 t^mroN Ooo t^inmn Ooo t^in-^N h o t^vo ^ ro m ooO vooo N -^lOt^ONM (r> iri\0 00 o -*0 t^ o >-< m in t>.oo O P) ■^'O oo c>. m >- ►- " ►- '1 M M C) CJ c< o rororor^r<^o^■. r-^0 mM t^woo -^j-omo^ n i^roo> r-< 1^ t^ t>.oo 00 00 oo .P) t^p) i-Npj rs.p) t^pi t^p4 t^pi f-.p4 r^P) r^p) i^pj i^pjoo ■* MNP»NP4P)P)PlP)P»PiPOvOOOOMi-'>-iPJ-T^-*•T^r^-^-<^^,^T^r^TJ-lnlnlnlnlnlnlnlnlnlnlnlnlnu^ N -"J-vOOO O P» - t^\o ■<^p^ ON(^in--*0 loOvo M t^N t^ fooo mon-*0 inovo H t^c»oo mo\^ O «nti M e) p» fi ■* •* m mvo >o t^ t>.oo oo o>o>0 h « n pi co m -^ -^ tn mvo t^ t^oo ^p)pjp.ONw Pi "^vo 00 P) foiot^ONM fo -£> vo vo t-^ t^ 1^00 oooooio>a>OOOHiHi-iP4NP)rr)fOfnro-M ro I- w (N e) N N N rocnrofOfOm->i-->i--.MOMnH t^MOMONOO ■^0>0 (NOO -^OvO (N N N fO T^ ■vO t^ r^OO OOO^O^OMM(^^N<^•4■•>i- >nv6 VO t^ t^OO OOO WMHMiHMMHiHHMl-ll-ii-cNWflPINNNMWOONWNNPirO Q t^ OOO ro o> •* o> >o O »o I . rooo r-1 On •* OMO O vO W fO irivo 00 On i N •<*-vO fv On O N CO triNO 00 O (T) -^vO tv On I COf^COOfOfOCOfOfOCOfO< NO VO NO NO NO NO NO NO NO NO NO NO t^ !>• I rororocDOrororocoroi O N -"I-nOOO O •^nO 00 O N -!^NO 00 O N -^nO 00 O ■*NO 00 O « -^NO 00 cororororO'^-<*-'^-^-^ioioiom \r,\i O N ■*nO00 O rororocO'r>ir)io u^no OOnO iDMN O ONt^tr)'»NO ■«l-mH OOO tvu->-. t^OO OOOOOOOOOOOnOnOnONOnOOOOOOmwhw 000000000000000000000000000000000000000000 OnOnOnOnONOnOnOvONOn 0\'<^OnO tJ t^oONinONO NOO roONiOHNO 0)00 -^ O inw t^roONiOM r>.ro ■^ iDNO NO (^ r~.oo ooo^OOMMP^c^f^^■. t^OO 00 OnOnO O h h n fOfO'J--*in idno 00000000000000000000000000000000000000 OnOnOnOnOnOnOnOnOnOnONOn NC00 t^vTiTj-rON m OOO t^vO \r, -^ ^ n o OnOO t^ xr, . M N ■^\6 OO 6 N •ivO t^O>M cou^t^o^M ro tj-vo oO 6 ( OOOOOOOOOOOOOOf — ■ r^ (T) OO N 00 ■* M t^ M OnO N OMrt h ( 00 C3^O^0 " w N roro-^M" ir)vO vO t^oo 00 0> O I ro fo fO fo ro (^Onu^OvO « r^MO-^OvO H moo -^ O lAH t^NOO Tj-OMTIMVO 00 Oi O O O . t^oo 00 O O O I fO CO CO CO CO CO < I CO CO CO CO CO CO I N C) CO CO ■* 1 CO CO CO CO CO CO CO ( invO 00 O O N CO ■«J-<0 t^OO O I- N -^ lOvO 00 O O N CO -^vO t^OO O >-< N ■* tO r^ O co\0 O covD 0> 01 lAoo M irioo >- •*■ c^ o co t^ O covO O CO CO CO Tf -"^ -"l- U-) 1 in\o vo vo vo t^ i 00 00 00 o o> a> I OO 00 00 OO 00 OO < CO CO CO CO CO CO ( ■*-'0 t^ O M CO lO t>. O I o o o o o o o t". CO OVO N 00 •* w t^ CO CO ■<}- -"l- m^O vO t^OO 00 (cococococococococococococococococococococococococo Q o N c-> cooo ■* o lO vO M t^ PJOO CO 0-* lO H\0 N t^ cooo ■* o inO\0 M t^ M M CO CO ■* -S- irivO vo t^ t^OOOO OOO „ M N N CO CO T^ Tl- iri m^ t~~ t^oo no O N N Ci W IN C4 COCO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO S Q CO Tj-vO t^ O N COIOVOOO O M N ^ in t^oo w CO -^vO t^ OO N CO O X „ ^ t^ O COVO O COVO Of) lOOO K lOOO ^ T(- c^ ■* c>. O covo ^ H vO O CJ in irjvO vO vO u a o N ■<*-vO00 O N -<*-vO00 O P) -^vOOO O N ■^VOOO O N ■<^\ooo n N -J-VOOO O S 290 FUNCTIONS OF A ONE-DEGREE CURVE. O « -^^ 00 O N -^VO 00 (SOWN 0\ H ro m t^ CT» I O O09 t~»vO lO ■* 'i- CO N O OOO t^vO lOvO *0 vO vO ^ N CH W N M 1 « CTi^ CO 0>\0 CO m ro CO ro r<^ fo I ICOCOCOCOCOCOCOCOI lCOCO->J--.l-Tj--^Tt-T4-- 0> -* in w vo 100 -"J-O^ OOO -"i-OMD . r^OO 00 On o O _ . t^ r^ t^ t^oo 00 icococococococororococococococococococococococococococo •^^ t^oo o O »A^ t>.00 M N CO ■<*- 1 oooooooooo -* C^ O CO t^ O co^ T^ "^ IT) m lo^o vo vo oooooooo OOOOOOOOOOmhm O N ■*vO 00 O N -^VOOO O M ■<*->JD I •>l-vO 00 O PI -^VO 00 O M -^^ 00 O W O OnOO t^vO in CO N O OnOO t^vO O ON t->NO l-O 00 O .aNOvOOOOOMMHMHHNNC^NNCOCOCOCOCOTj--^ OOmmmmhwmhmmmhmihi-iwmmmmhmm 0(NNOCJOP)(NOWPiC)OMNNC)N(N(NNNNC)00 00 On CO On (N* cj P) O I cocococococococo I 00 On M P) CO Tj-NO tvOO O O CO t^ O COND On C>) in On U-INO NO NO NO t~~ t^ r ' inNO t^ On O ■^nO 00 O N ■•t-NO 00 o FUNCTIONS OF A ONE-DEQREE CURVE. 291 ! 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OnO " w 1-1 H H P) P^ P) P) PI mmrororom^^TfThTHninioiD iHPiP)NP)p)P)P)P)Pipqp4P)NP4Pipqp)P)NP)P)Npqp)P)P)Piri- invo VD t^oo OOOOOwPlNrOiJ-ir) invO t^ t^oo d O Q s VD NCO r^0vO PJOO ^OvO PJOO ^0\O PlOO tCOvO P)00 ^OvO PiOO '^O t^ Owi-iPirorOTf'*- xn^o vO t^ r^oo oso>0 w P< P) rorOTi-u-) lovO O t^oo 00 oooooococococooooooooocooooooooo 0\ (j\ o\ o\ 0^ 0^ 0^ 0^ 0^ 0\ 0^ 0^ 0^ 0^ 0^ d a: o a O CJi 1-1 PI fO -"l- lA^ 00 H N ro Tj- lovo t^OO On M P» ■* "^^ t^OO CTi M ^ 2 ^^ 2^S J?^ f^^^:^5:^S,??^S?^^vg^?l-lC'i^.So^c!5-aSS;8 7; P) ■♦^80 N ■j--oo 00 00 t^ t-^'O vOvO inmm-*-^-^r>-i roint^Ow ro Th^ oo d N -"l-^O oo O N 'i-^ 00 O w -^^ o6 O N t^ t-^ t^ r~.oo ooooooooooONOOOOOOOt-iMwi-MpjNMDNroro mrororofomroc-orOfnrorororo-J--*-*-"^-l---»--^^ WNNNMNtM*--<)--^-*--J--»J--*■■*■^l-•*u^mlr)l^lU1lOu^u-)ln m>0 r-» tvOO OvOOMNrOro-* irivo vO t^oo a>OOHNroro-0 t^oo inoo M -!t- t^ Q ■* t^ O rovo On n uioo w -* t-^ ■* r^ O ro^ C^ N looo m ■* t-^ 0000 OOOO O O w w H M PI P4 w mro(-ri,j..^-^u-)vnui invo vo vo t^ t-^ f^ r(^romrofo-^j-■^^■^^-•*■-J-■*.^-.f•*-:^-^ ■*.■* T^->J-■^■^J-■*•-*•■^T)-■*■*■^^•<^- 1*•■*■*■0 00 o 1-1 M HI M M PI PI P) PI P) rororororO-^-^->i--*-*mioir)io ui\0 PI P) P) PI rorocororO'-a--lrlLnu^ mvo vD ^ \0 vO t-> t-v MPip.T)-M 0\^ rOHOO u-ino t-~u-)P) Ot^Tj-P) Ot^-*M ONVO -.roq^ P) ONiriwoO -^0 t-^ro OvO PIOO triM t^-^OvD ro on-O -> " N PI ri^T)-T)- invO vO t^ t^OO OO^O m ■-'PI mrn-*T)- lO^O M«MM«rq(NPi(NMN(NP)P)Pip)WPIPiPiPirorOf^rororor^r»iroro d w PI P) ro ^ u^vO t-^ r--oo o M n (N ro '*- mvo t^ t-^co oOwPiPirOTj-io o o o o o o Tj- -^ Tf -*j- -"^ Tf V V -'i- W- W- W- V W- W* -^ -^ V V V 'OnOiO>0>0 O O r- ■- -1 ►- M M N D M o forororO(-o->J-Thr}--«-Tj-u-) PINP)PlMP)P)P)NPICaNNP)NNP4NP)PlP)P)r4P)PlMP)P)P)(NP) X 1^ rOOOOvO -*P) OoovO -<1-P1 GOOvO -"l-PJ OOO^o -^N OOOvO 'i-P) QoovO tK^ OOONOiOwPlrOrOT*- ino C-N t^OO ONOf->-p)ro-«-iO u-)\D r-00 Oi ON M M M M M C4 N Pt PI p< P4 PI p) PJ p) P) P) f0^or<^r^o^rof<^^or<^roromro^-<^Tt■ Q o N ONiOPJOO inpJOO lAM t^roo t^roo t^roO t^-*MOO lOP) O^ ro t^ -^ lO lovo P^ r-«oo OnOnO ■"! w PI rom-OnOnCT> a i u t^ t^oo oOONONOOwMpgpifTiro^^ir) invo vO tv t^oo OOOnOnQOmmPI ON PI ^oo M T^oo « 1^ 1^ mo ON P) inoo m ^ t^ o pomo on pj vn on P) iooo m NO r-N t^ f^oo 0000 OnonOnO O O w H i-i D M (N r^ro^om■<^-T^T^u-)u-l invO miOLOu-)u-)iJ^u-i LOin u~,\0 nOnOvOnOnOnOnOnOnOnOOnOvOnOnOnOnOnOnOvO p. ^NOCO O 2 JNOOO g ^vocg J^^vOOO O « ^NOOO.O p. ;5^vO00^ ■ 1 « ^VOOO O 2 2"^^ PJ ^0<^ 0^« ^^WO^ 5^0 00.0^ C^^NO 00 o Q PJINwhwhOOOOnOnOn OnoO 0° 00 00 t^ t^- t~. t-.NO nOnOnOvO ininiOlAin pJ 4-vO oo 6 c^ ^NO oo ON- roint^ONM r^int^o-" roiAt^ONM roint^ONM r<^ror»^m■^l-Tl-T^-^1i-T^u^lOu^lr) idno vDvDnOnO t^t^t-^t^ r^oo oo oo oo oo On PlP)PtP)P)P)P)P)P)P)NPg,g>g>C^n^?,D^;;:,;;:,S^j:;,;;:,n^l;!, Q O Q lOMOo ThM ^>.T^o t-^foo t-^rooNO i^Ono co onno co onno p» onvo p) ONin n lONO NO t^oo 0OONOO^-lP^P^ror^^ vanO no t^ t~-oo OnOnQ •- " PJ rOrOTj-io in in LO LO LO in LO\D nOnOnOnOnOnDnOnOnOnOvOnOnOnOnO t^r^f^-t^t^t^f^ts. Q o a ooosONOOMWP)r. f-- m t^ rONO On Pi inoo " ^i- t^ o (^vo on pi no On n inoo M ':^ t^ ponO 0> r^oo 0000 ononOnonq o o 1-1 M M « N (N Pi fomm-^-^-^ioin iono no no no 55 PJ^NOOOO PJ 2-^<» CJ j^NO^ p^ PJ^ M;nO c« ^ ^NO 00 p. ;*vo 00 ^ 294 FUNCTIONS OF A ONE-DEGREE CURVE. z" n O O M N (N (-0 mhmmmmOMOJ.NNNi lOioiOLoinioiOLOLOLotOLoi cs 04 0) N IN 04 M rommroror^iroromro LOLOiOLommioiouiioioioiOLOinmio [-^ t^OO OOOO 0^0^0^0 O O O r^ i-^co 00 00 00 00 00 ( ■*^ 00 O D -^^ 00 O CM ThvD I 10 U-> in IT) lOVi M(NM04(NNNtM M rou^t^O>H romt^ONw roxot^OM iri iri m IT) m^ vo^vo^o t^t^c^r^ t^oo < iTi ir^ iTi vr^ iTi LTi \j~i w) \j~i iTi u^ iTi iry in w^ \D \o \o VO rOM o^^~^o■*o^ ooo^o mroH a> t^vo ■*« Ooo t^uiroN Ooo t^^iorON O p) ro ro ■* lOvo r~.oo ooONO'-CMNro-'i- m\0 t^ t^oo On O w N N ro -^l- in^ r~ -<)- t^ Tt- HOO Tl- H 00 m N 00 in N ON NO 0) OnnO ro no fOO p~ ■* MOO m N ON mvO \0 t^OO CO On 8 8 ^ ^^§8^8^ « ^ H ro ro Tj- w M (N W ro ro ro Tf -* in m'ONONO t^ t^ tvOOOO ON ON ON „ „ N C^ " ro ro w "^ f^ roND ON CJ inoo w -^ r^ O roNO on n NO NO NO t^ t^ t^ t^OO OOOOOnOnOnOOOOw nOnOnOnOnOnOnOnOnOOnOnOnO r^f^C^t^t^ •*nO 00 O 0) -^nO 1 O 0) 'J-nO 00 O P) -"J-nO 00 O •^\0 00 O N -^nD ( rorororo-^- O I- ro i m ro ro ro ■* ■* ■ r^ t^ t^ t^ r^ t~» I r^OO 00 Ov o^ O a> I- ro m t^ o ■*vO 00 O C4 ■* W IN N W C) O O t>»vD m "T i-o N O^ O OOO t^vO 00 O^ 6 M IN OOCTvO'-'MN'^'l- invO t^OO On O O " D m I*- lAvO I^C M tH (N 01 N CI i- mvo t-- t^oo o O in in in\o \0'0^n£i\onOnononO'Ono no ^o t^t^c^t^t^t^t^t^t^c^t^c^t^t^ minininininminininLninininininininininininminininininininm ^ mo o^ CN inoo w -^ t^ O m\0 o tN moo w -^ t^ o fo\D O I z; ^nro^o-*••^■^^lr)ln in\o \o ^ \o t^ t^ t^oo oo oo o on o^ o . ' X O^ONC3NO^ONO^O^C>O^C^»O^O^O^O^ONO^G^C^ONONO^C7^0^0 U-) u-> lo in IT) m O N '^^O 00 O t^vO in m w O Onoo t^ m ■* m pj w ooo t^vo m ininminininmin invo \dno\ono\osdndonovonovo'OnOvovOnO\0'Ovovo Q o 00 in N o\vo m t-^ ^ M a^^ mo t^rhMoo mo o t^-*Moo mo ovo ^ n vO t^oo ooONOMMPjmrOTt- inMD vo t^oo ooONOwMomro-^in m^o r^oo mmmmro•*T^T^^Tl--*•T^T^T^T)-T^^T>-T^lnlnlnlnlnlnlnlnlnlnlnln ininininininininininininininininininininininininininininininin d N N N PI N rorommrommrfTj--i-r*-^^,f-*-»nininininin inso vo vo vO O 5 00*00 00 CO t»o? 00 XI OT co"oo"co"oo'oo 00 00 S'S' OO CO 00 lOvO ^O VO ^ VO t-^ t>~ I t^ tvOO 00 00 OO N (N D .00 CT> O o o" o'o'l ^VO 00 O O -^VO (.. . , . 100000 0>0>0>CT'00 O (_ 1000000000000000 O^C^O^O^O^O^O^O^O^O^ t-^vO lO -^ -* ro N N O O On OCO t^ t^vo <0 ro m ■<*• lovD (N ro T^ tnvO VO vD vO vO VO vO vO VO vO vO vo vO vo vO vO vO vO vO vO VO vO vO _ _ _ _ _ f4 ro ■* tnvD t^oo t^OO OOOOOOOOOOOOOOOOOOOO OvCT>OvO>ONOO>OSOva>l - ---- - - - -vovOvj- ---- ■ I vO VO VO VO VD Ooo loroooo loroooo loroooo inroooo mr^Ooo inrOHOO iritriooovo N IN' lOOO H -^ I t^ t^ t^oo t uniommioiommi O N -^vo 00 o O P) -^VO 00 O N -!j-vO 00 O N ■ ■>*-vD 00 O N -^vo ( t-»00 00Ov00HNNrO->J--ii- \DvO VO t^OO 00Ov0H(NNro-4-m yrwo t^oO On ■*vo 00 O romt-^ONH romt>.o\H roiAt>.ONH ti-vo oo 6 I VOVD VO VO vO ' •^"^"^•^-^"^iriirimiomi tr) M- lOVO t^OO 00 On I ro -^ invO t^OO On ov O N ON t>. ■<*■ H OvvO rOMOO inroo t^inw Ovr>-*N O ov O H ! «^00 00 O lo lo in 1 ro Tj- in lovo r^ t--.co a\ o t^ t^ r^ tN> t-» I . t-. t^ t^vO vOvOvOvOvOvOvOvOvOvOvOvOvOVOvOvOvOvOvO in 00 M -:}■ r^ O rovo Ov P) >ooo m ■* _ _ VO t-^ «^ t^OO OOOOCO OvOvOvO O uOOOOOOOOOOOOOOOOOOOOOOww inininininininioinininininininininininininioininin rr^^D 0\ CN inoo H -<*■ f^ o rovo PJ M p) rorom-^-^Tj-ini •*-\O00 O N Tf^OO O Vi f^ -^ \r, t^oo O O O w M ■<*• "^^ Tt-vO OO O N -^^O ( o o o o o On 0> C3> OOO 00000000 t^t^t^t^l ON d t^ i^ (^ t^ 1 ro ■>*- lOvO t^oo On O HI r^ t^VO NO VO NO NO VO NO NO On O H w r<1 Tj- lONO t^oo ON . .- .- .- in ID IT) t^ t^ t^ ■^ M OnnO tJ- n O t^ in ro I CI O 00 NO ■<*- I tvOO 00 ON O NO t-^00 OOONOMNNrO-^in lOND r-^OO On On O to lO lO l/) tONO NO NO NO NO NO NO NO NO NO NO VD NO " NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO NO t^NO NO in m ■ in in lo tn m lo 1 M -*• t-~ O N N N m ONOO 00 t-^ r~.NO NO inin-*-*mrO(N n m I inco iH T^ t^ o fONO On o moo m 'j- t^ o ifOcO'^-^'^minin inNO no no t^ t^ t^oo iinininininininininininininininin ■^NO 00 O « -^NO ( « ^NOCO O PJ ;*NO<« O CJ ^NOCg CJ^M^NOOO^O « :5:nO00 « 5;nO<» O < inNO 00 On w N m Tj- lANO t^oo On w N <^ mNO t^oo ON o M N n- inNO i~-oo ON w n moo (N -^no oo P) -^no Onm mmt^ONM ro moo Pi -^nO oo o m i X* On onoo 00 00 c^ t^ t^NO Nommm^-^poforowNPiwMWHOGOOOON 8ooooo'oo'oo'2l:;«!^MM'2£r'2 2'SSS^^J?'g^'SS^N' t^t^r-.t^t^r^t^t^i>.t>.c^t^c^ir^r-.t^t^t^r-.r^t^t^t^t^i^t^t^t^t->c^c>. 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M 'i-vO 00 t-l O 0) lOt^O N "3t>.0 NVOOO Th t^ O N >J^oo VO \0 vO VO M O O O O I I O H P) ■* uTO t^ 0> O H ro Tj- invO < f-> t-> (^ t^ t^ tv. t^ t^OO 00 00 00 00 OO ( OOOOOOOOOOOOOO ■.vo m Tf m N h o CTioo r^vo m -^ M (M n o o>oo t^vo tn -^ m 000000 o^O^C^O^O^O^C^C^O»C^O»O^C^C^O^O^C^C^O^O^O^O^O^CTlO^O^O^O^ • lOrOM OM^iOfOH OtxiOrOH 0>t>.tnrOH OOOVO •<*-Cl OOOvO "<*-N OOOvO O N -"l-vO 00 O M 't-vO < ■*t^O M-i-nO C0t>.0 mt^O O N »0 t^ Ov N -^-O CTi OOOOOOOOOO OCTiOlO ui t^ O w •<*- t~. Ol 1 rorocororofororococococoi t^ O N ■<*■ t^ OM ■»^^0 On w rOvO 00 m ro moD O roiOt^O N lOt^O M lOt^O ON O H C» "^ invO C^ ON O t^ Ol O •"• P) ■* lAvO OOOOOOOOOOOOO rj- rnio t^ 0> O H N OOOOOOOOOOOOO f^ 0» w OOO t^vO o> t>.vo lo T^ I O ONOO t^vO IT) •<*■ <*> ■<»- \n lovo t^oo ON O w N ro fO -"h lovo r^oo ct> O h w M ro , r>. t^ p~ t^ r>. t^ p~ I ONOnO^OnOnOnOiOnO* T 00 00 00 00 CO Q 0>t^uT.fM OOOVO >orow O. J^\0 ■<♦-« 000 t~,iArOM OnOO v3 -* o o o> r^ «a O u N u^OO M •* t^ On N u^OO M rOvO On N lOCO O rovo ON N •»!- t^ O rOvO On m ■<*- t^ VOVOVONONONONONOVOVONOVONONONOVOVOVOVOVOVOVOVOVONOVOVOVCIVOVOVO FUNCTIONS OF A ONE-DEGREE CURVE. 303 "»^ « •i O N VvO 00 O N ^^ CO N -^^ 00 O N -rvD 00 O N "-l-vO 00 O N M-vO 00 O H « M « M M N N N (N (DCOCOrOrri-^Tf-^^-^lOlOmiO invfi 1 < vO ^CO N vD 'i-OO IN M3 T^00 NvO ^Oror^M inov Thoo N ^O m O IN ior-~ow -^i-^cTiM T)-\o O M rovO 00 « ro irioo Or<^lOt-~OMlAt^ONr^, OOO^OO O M w w w N N N N r«^mn^^-)T^-^■>^T^u-ll0^o invo vD \D 1 oorororomrororoi^fOroi^rorororororororororororofnrororornroro 1 u [I] in W lAOO M T^f^« "^t^O rot^O -1-t^M T)-oo M inoo n in o M \D O rovo O rovo i « (N 4-ui^oo ON c3 rn 4-^ t^oo (D - (N Tj-mooo (> 6 IN ro 4-\0 t^ OMD -i 1 Q O Q i oo t^-O inio^mp) N M O ooo t--vO vOm^roroNnOO O^oo oo t^vo vO i M P) fO -J- irivO t^OO On " '-' N 1^ ^ mvO t^OO On >- (N CO -J- ■*■ lOvO t^OO ON O^O^O^O^O^O^a^O^O^O^O^O^a^O^O t^ fONO oo •- ■'^ r-. 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H NO '*■ t^ H inoo 01 vo ON ro t>. -*oo H inoNNvo O rot>«H uiovf^t^n inov .^c^c^-c^cS'^cg'oNSNSSf^g^S S ?S-| S-8^2 S ^2-^^^ 8 ^ | d O Q • voio-*-*roo)oiHOO o>oo 00 r-vo vOlnTJ--<^roo^o^MOO onoo oo r-* t^ (■* ScgcScS.tnrooco mrOHOONO rOM VONONONONONONONONONOVONONONONONONONONONONONONONOVONOVONONONONO O M '!^NO00 0) ^VOOO O 0) ■'J-nOOO O 01 -*-NO00 O M ""l-NOOO N Tl-NOCO o HHHMH0icici(NC 0) r- 0) oo fooo <^oo Th o\--* oui w ^'^ R'R'R'R'S 22"2 2 • rooo N fs. 01 vo H in '^^ 0^ ■<^ O ^00 rooo rooo oi t^oi t^:: t--oi i-^oi (^ 'OOM^H^^^^'P.H ^ ^^H^^^^^^HH HI; 1 ^ ^ mmOOOOnOO 003 00 00 CO t^ t^ t^ tvvO vOvO »ninin>n->i--*-.i-rorrirorri invO t>.00 oo CT> O M 01 ro ■* invo t^oo cy> O " 01 ro ^ in\0 t^oo o O ►- 01 ro ■* vOvO^o^^o t^t^r^t^i-^t>.t-~t^r^ t^oo oooooocooooooooooo octvOOO ooooooooooooooooooooooooooooooo 0\\0 mO t^-*MC0vO roo t^-- -!f >-i Tt- t^ O ro inoo w rovo o oi -"i- r-^ o oi moo O rovo On w ^J- t^ o 01 inoo O roo in in'O vo vO \0 t~. t^ r^ t^oo ooooovONONONOOOOi-''-'i-'-'Oioioir»-iror-> VO VO *0 ^ ^ VO VO vO vO SO "O vO OvO^'O'O'OO'OvO^^^OvOOvO'OOvO^O O N -^VO 00 O M ^vO 00 O 0) ThvOOO O 01 - M OS coroc«oiMMHOOONO> o\oo 00 b> t^ r^\o vo miom^^rororooi oi « « o a \0 t^oo Oi H 01 ro ■* ■<*• invo t-^oo Oi ►-i oi ro ■* ino t^oo on m n ro ^ lo roromrOTj--*-.l--^-^-.*-Tj-^Tj-^^minininininininin invO vo vD vo vO vo ooooooooooooooooooooooooooooooo p ro i^ in 01 ON^D roMOOinoi Oni^'^moo inroo t^-*M onvO ro O t-, m 01 O o X rovo 00 M ^ 01 in r^ O ro moo h ^md On oi in r^ O ro moo m -!j-\o on oi t)- mmminininmmm invo vovdnovonononovondnonononovonOvdvOnonono VOVONOVONOVOVOVOVOVOVOVOVOVOVOVONOVOVOOVOVOVOVOVOVONOVOVOVOVO 2 01 -^nO 00 01 -"J-nO oo 01 -1-vD 00 01 •^\0 CO O 01 rj-vo 00 01 tJ-xO 03 O H M H M M 01 01 01 c^ 01 rororororOTj-^^n-^mmmm mvo 806 FUNCTIONS OF A ONE-DEGREE CURVE. O N •'1-VOOO O N •<*-^O00 O N -"J-^OO N "^VOOO N -^^ 00 O W -^VO 00 < OvO M l>.rOOMnH t>NOO -^OVO MOO -i- OO MOO T)-0^O NOO -<^0vO MOO N. 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N -"l-^O 00 (N ->1-vO 00 M 'i-'O 00 O N ■<*-\0 00 N ^VO 00 (N tJ-vO 00 « < 00 lOroO t^iAN Ot^rt-H OMD mOOO lOroOOO lOroOOO iDroOOO inroO OnvO CO O t^ "^ I w ro Lovo 00 O M ro in t^oo O L. . . . . 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N VC H in 0^00 CO t^ H vo . inocn 00 PJ t^ M vo M r^ 00 PI PI CO 1 1 T^ ^ -+ ^ 'I 2 00 Y1 cs P) P) (N eg cs (N P) PJ P) N p) PI PI Q* CO 1-^-001^ rj- H CO 10 N CO .nP, c^ >i PI OVO r-l 00 CO vo C-. Cvo CO (^ S ,1- ^ r^ - t-1 -r ~ r~. C^ - P4 5^ h^rr, - S "5 - 8^ M ro ^ \0 00 ON M ro -J- 00 -L COOOCOOOCOOOCOOOOOOO f^^ 000 000000 0\ C7^ 0^ 000 OOC^OC3^C^OOO^O^C3^C^ OC3^ t> 0^ cr 000 n PI , -f-vn 00 O Pi -fO 00 o pj -^vo 00 o fO'O o PI 1000 H inco CO -r ^ invo 'O r^oo 0000000- 10 o rovo ^00 CO -r M I-^ -:_•- o VD " PI CO CO -+ ir> I -)' o 10^ vo vC ^ ^O >0 vc 'O vr> ' O O O M . 1 in 10 10^ vo _ _ _ oc- 000000000 -5h03 N O O -+00 10 ^ O " t-^ PI t^ O O O H i-' PI M CO t-^ " \D mo r-.OO -r c- 10 o \o n CO -1- Tt- ir-AO ^ in 10 10 Li-> iTj ui in "^O-l-OTt-O-^O^OinO^ 00 n r^ >o o -1-00 cot-^wvo -*ocoi^pivo o mo cooo N \o m in o ->*-i -] -i-O l~-00 O >-' CO -hvo r^•--">-' i-^P) -t^i-^t^t^t-^t^t^t-^t-^i-^ f^o- cocooooooocncoooooooooooooooi PIPIPIPIP)(NPIP)PIPIPIPIP)P)P)P1PICIP1PIPIPIPIP)PIP) M OVO CO ' -' -ho CO ( rOOVO CO O t-^ -vo 00 O P) in PI 0\0 CO t~« -* 1 O PI CO 10 t~^ o O 1 vo vo vo \o r^ OVO CO 6 pi -i- TABLE XVII. RISE PER MILE OF VARIOUS GRADES. 332 TABLE XVII.-RISE PER MILE OF VARIOUS CEADEr Rise Feet per Rise Feet per ^ Mile. c ise Feet per Rise Feet p r Cent. Mile. Cent. er ent. Mile. Cent. Miie. .01 .528 .61 32.208 1 21 63.888 1.81 95-568 .02 1.056 .62 32.736 I 22 64.416 1.82 96.096 .03 1.584 .63 33.264 1 23 64.944 1.83 96.624 .04 2. 112 .64 33.792 I 24 65.472 1.84 97.152 •05 2.640 • 65 34.320 I 25 66.000 1.85 97.680 .06 3.168 .66 34.848 I 26 66.528 1.86 98.208 .07 3.696 .67 35.376 I 27 67.056 1.87 98.736 .08 4.224 .68 35.904 I 28 67.584 1.88 99.264 .09 4.752 .69 36.432 I 29 68.112 1.89 99.792 .10 5.280 .70 36.960 I 30 68.640 1,90 100.320 .11 5.808 •71 37.488 I 31 69.168 1.91 100.848 .12 6.336 .72 38.016 1 32 69.696 1.92 101.376 •13 6.864 •73 38.544 I 33 70.224 1-93 101.904 .14 7-392 •74 39.072 I 34 70.752 1.94 102.432 •15 7.920 •75 39.600 I 35 71.280 1^95 102.960 .16 8.448 .76 40.128 I 36 71.808 1.96 103.488 -.11 8.976 •77 40.656 I 37 72.336 1.97 104.016 9.504 •78 41.184 I 38 72.864 1.98 104.544 .19 10.032 'P 41.712 I 39 73.392 1.99 105.072 .20 10.560 .80 42.240 I 40 73.920 2.00 105.600 .21 11.088 .81 42.768 I 41 74.448 2.10 110.880 .22 II. 616 .82 43.296 1 42 74.976 2.20 116. 160 .23 12.144 •83 43.824 I 43 75.504 2.30 121, 440 .24 12.672 .84 44.352 , I 44 76.032 2.40 126.720 •25 13.200 •85 44.880 I 45 76.560 2.50 132.000 .26 13.728 .86 45.408 I 46 77.088 2.60 137.280 .27 14.256 .87 45.936 1 47 77.616 2.70 142.560 .28 14.784 .88 46.464 I 48 78.144 2.80 147.840 .29 15.312 .89 46.992 I 49 78.672 2.90 153.120 .30 15.840 .90 47.520 I 50 79.200 3.00 158.400 •31 16.368 .91 48.048 1 51 79.728 3.10 163.680 .32 16.896 .92 48.576 I 52 80.256 3.20 168.960 •33 17.424 •93 49.104 I 53 80.784 3.3ci 174.240 •34 17.952 .94 49.632 I 54 81.312 3.40 179.520 •35 18.480 •95 50.160 1 55 81.840 3.50 184.800 •36 19.008 .96 50.688 I 56 82.368 3.60 190.080 •37 19.536 •97 51.216 I 57 82.896 3.70 195.360 .38 20.064 .98 51.744 1 58 83.424 3.80 200.640 •39 20.592 •99 52.272 I 59 83.952 3.90 205.920 .40 21.120 1. 00 52.800 I 60 84.480 4.00 211.200 .41 21.648 1. 01 53.328 I 61 85.008 4.10 216.480 .42 22.176 1.02 53.856 I 62 85.536 4.20 221.760 •43 22.704 1.03 54.384 1 63 86.064 4-30 227.040 •44 23.232 1.04 54.912 I 64 86.592 4.40 232.320 •45 23.760 1.05 55.440 I 65 87.120 4.50 237.600 .46 24.288 1.06 55.968 I 66 87.648 4.60 242.880 ■''I 24.816 1.07 56.496 I 67 88.176 4.70 248.160 .48 25.344 1.08 57.024 I 68 88.704 4.80 253.440 •49 25.872 1.09 57.552 I 69 89.232 4.90 258.720 •50 26.400 1. 10 58.080 I 70 89.760 5.00 264.000 •51 26.928 I. II 58.608 I 71 90.288 5.10 269.280 •52 27.456 1. 12 59.136 I 72 90.816 5- 20 274.560 •53 27.984 1.13 59.664 I 73 91.344 5.30 279.840 •54 38.512 1. 14 60.192 I 74 91.872 5.40 285.120 1 •55 29.040 1.15 60.720 I. 75 92.400 5.5c 290.400 .56 29.568 1. 16 61.248 I. 76 92.928 5.60 295.680 •57 30.096 1. 17 61.776 I. 77 93.456 5.70 300.960 .58 30.624 1. 18 62.304 I. 78 93.984 5.80 306.240 •59 31.152 1. 19 62.832 1. 79 94.512 5.90 311.520 ) .6c 31.680 1.20 63.360 I. 80 95.040 6.00 316.800 ADDENDA. ADDEISTDA TABLE OF FEET, INCHES AND RECIPROCALS OF VARIOUS TRACK GAUGES. 6' 72" .013389 Metre 39.375" .0254 5' 60" .01667 3' 6" 42" .0238 4' 9" 57" .01754 3' 4" 40" .0250 4' 81" 56i" .01769 3'0 36" .0278 TABLE MINUTES OF A DEGREE EXPRESSED IN DECIMALS. 1 .0167 16 .2667 31 .5167 46 .7667 2 .0333 17 .2833 32 .5333 47 .7833 3 .0500 18 .3000 33 .5500 48 .8000 4 .0667 19 .3167 34 .5667 49 .8167 5 .0833 20 3333 35 .5833 50 8333 6 .1000 21 .3500 36 .6000 •51 .8500 7 .1167 ! 22 .3667 37 .6167 52 .8667 8 .1333 1 23 .3883 38 .6333 53 .8833 9 .1500 j 24 .4000 39 .6500 54 .9000 10 .1667 I 25 .4167 40 .6667 55 .9167 11 .1833 26 .4333 41 .6833 56 .9333 12 .2000 27 .4500 42 .7000 57 .9540 13 .2167 1 28 .4667 43 .7167 58 .9667 14 .2333 29 .4833 44 .7333 59 .9833 15 .2500 30 .5000 45 .7500 60 1.0000 386 ADDENDA. CONDENSED TABLE OF RADII INCLUDING SHORT CHORDS. Degree OP 100' Chord, 50' Chord. 25' Chord. Curve. r 5729.66 5729.60 2° 2864.93 2864.82 3^ 1910.08 1909.91 4° •1432.69 1432.47 5° ■ 1146.28 1146.01 6° 955.37 955.04 7° 819.02 818.64 8° 716.78 716.34 9° 637.28 636.78 10° 573.69 573.14 ir 521.67 521.07 12° 478.34 477.68 13° 441.68 440.97 14° 410.28 409.51 15° 383.06 382.25 16° 359.26 ■ 358.17 17° 338.27 337.11 18° 319.62 818.46 19° 302.94 30164 20° 287.94 286.57 21° 274.37 272.93 22° 262.04 260.54 23° 250.79 249.22 24° 240.49 238.84 25° 235.65 229.30 26° 222.27 220.49 27° 214.18 212.30 28° 206.68 204.76 29° 199.70 197.70 30° 193.60 190.79 ADDENDA. 337 To express gradients per cent, (page 332), in angular meas- ure, multiply the rate per cent, by 34.3; the product will be given in minutes of a degree. EXAMPLE. Gradient PER CENT. Minutes. GUADIENT PER CENT. Minutes. .20 6.86 2.50 85.75 .40 13.72 3.00 102.90 .60 20.58 3.50 120.005 .80 27.44 4. 137.20 1. 4.30 4.50 154.35 1.20 41.16 5. 171.5 1.40 48.02 6. 205.8 1.60 54.88 1.80 61.74 2. 68.60 If the gradient per cent, be multiplied by 57. 14, the result will be expressed in hundredths of a degree. SOUND. At freezing temperature, 32 degrees Fahrenheit, in calm air, the velocity of sound may be assumed 1100 feet per second. For lower temperatures subtract, and for higher add, a half foot per degree. The intensity of sound varies inversely as the square of the distance. The velocity varies directly as the temperature. It is nearly four times as great in water as in air; and in wood ten to sixteen times as great. AMERICAN AND FRENCH EQUIVALENTS. LINEAR MEASURE. 1 inch = 2.54 centimetres ; 1 centimetre = .394 inches. 1 foot = .3048 metres: 1 metre = 3.2809 feet. 1 yard = 3 feet = .9144 metres; 1 metre — 1.0936 yards 1 rod = 16.5 feet = 5.029 metres; 1 metre = 0.2 rods. 338 ADDENDA. 1 surveyor's chain = 66 feet = 4 rods = 20.117 metres; 1 metre = .05 chains. 1 kilometre = .6214 miles = 3281 feet. 1 statute mile = 5280 feet = 80 rods = 1.6093 kilometres. AMERICAN AND FRENCH EQUIVALENTS. SQUARE MEASURE. 1 square inch = 6.4515 square centimetres. 1 square centimetre = 0.1550 square inches, 1 square foot = 0.929 square metres. 1 square metre = 1.19659 square yards. 1 square acre = 43560 square feet = 4840 square yards. 1 square hectare = 2.4711 acres = 11960 square yards. 1 acre = 0.4047 hectares. 1 square kilometre = .3861 square miles. 1 square mile = 2.5899 square kilometres. 1 square rod = 272.25 square feet = .00259 hectares. AMERICAN AND FRENCH EQUIVALENTS. CUBIC MEASURE. 1 cubic inch = 16.383 cubic centimetres. 1 cubic centimetre = .0610 cubic inches. 1 cubic foot = 28.316 cubic decimetres. 1 cubic decimetre = ,0353 cubic feet. 1 cubic yard — .7645 cubic metres. 1 cubic metre = 1.308 cubic yards. ^ INDEX. PAGE Abbreviations explained ix Acres, roods, and perches in square feet, Table VI 152 Adjustment and use of instruments 23 Angles of frogs, to find 129 index, to find , 69 intersection, to find 55 plane 12 to read on verniers 43 tangential and deflection 50 of switch-rails 130 Apex distance of curves, to find 52 Arc, functions of, to find 13 Arithmetical complement 6 Axemen, duties of 84 Azimuths of North Star, Table 11 150 Barometer, levelling by 29 Bench-marks, proper intervals for • 83 Bubble, to adjust on level . 25 to adjust or transit 40 Chain, to lay out curves with 63 Chainman, duties of 42 Chief engineer, duties of 79 Chords, to calculate 54, 58 Table XVI 269 Circle, propositions concerning 49 Circular arcs to radius of 1, Table VT 152 Complement of an angle 12 arithmetical 6 Compoimd curves. See Curves. Contour maps, utility of 85 Correction for curvature and refraction in levelling 28 Cosines defined 12 Crossings, plain rules for laying off 139 Cross-hairs, to adjust 24, 26, 40 eccentricity of 24 to put in new 44 340 INDEX. PAGE Cross-sectioning. See Slope stakes. Cubes and cube roots of numbers, Table XI 161 Curves, circular, on railroad defined 51 to find radius, length, degree, apex distance, chord, mid- ordinate, and external secant 53, 56 form for field notes 70 Curves, how to lay out on the ground, — with the chain only 63 with tiansit and chain 66 hints as to field-work 82 protractor for 84 slackening grade on 87 terminal 88 Cui-ves, simple, location of, — how to proceed when the P. C. is inaccessible 93 to shift the P. C. in order to strike a fixed tangent 96 to change radius from same P. C. in order to strike a fixed tan- gent 97 to triangulate on 94 to pass through a fixed point . 127, 128 Curve«, compound, — how to proceed when the P. C. C. is inaccessible 95 to compound a curve in order to strike a fixed tangent .... 98 to shift a P. C. C. in order to strike a fixed tangent 99 summary of rules for 101 to compound into a tangent intersecting main curve on concave side 102 to compound into a tangent intersecting main curve on convex siae 103 Curves, reversed, — parallel tangents, radii equal 115 parallel tangents, radii unequal 117 angles unequal, radii equal 119 angles unequal, tangent points fixed, radii equal . . . . . . 120 divergent tangents, radii equal, advancing towards intersection . 123 receding from intersection 124 to shift a P. R. C. in order to strike a fixed tangent 125 Curves, miscellaneous, — elevation of outer rail 141, 142 degree of, to find by calculation 52, 55 to find on ground 145, 146 to connect curves of contrary flexure by short tangents ... 89 to locate a Y from a tangent 103 from a convex curve 104 from a concave curve 106 to locate a tangent to a curve from a fixed point 108 to two curves already located 109 to substitute a curve for a tangent connecting two curves ... 109 terminal curves 88 INDEX, 341 PAGE Curves, miscellaneous — continued. trackmen's table of curves and spring of rails 143 vertical curves, to calculate 36 to project 39 Datum in levelling 27 Decimals of an acre per 100 feet for various widths, Table IV. ... 151 Deflection angles and distances explained 50 to find 57, 64, 68 short rule for sub-deflections 68 limit in field-practice 82 Degree of curve, to calculate 52, 55 to find on ground 145, 146 Deviations from project admissible on location 81 Distances, tangential and deflection, defined 50 table of 155 of frogs from toe of switch 130, 132 tables of . 135, 136 Elevation of outer rail on curves 141 table of 142 Excavation and embankment, to stake out 30 External secants, to fiud 54 of a r curve. Table XVI 269 Extreme elongations of North Star, Table 1 148 Feet in decimals of a mile. Table VII 153 Field-work, suggestions concerning 79, 85 Field-book, form of, for level 27 for transit 70 for slope stakes 33, 34, 35 Frogs and switches 129 rules for angles and distances 130 table of, switch-rails straight 135 switch-rails curved 136 plain rules for locating, switch-rails straight 132 switch-rails curved 133 on narrow gauges 134 patterns for 134 Functions, trigonometrical, defined 12 logarithmic, of arcs, to find 14 General propositions in trigonometry 15 as to circles 49 Grade, to slacken on curves 87 rise per mile. Table XVII 332 Grade lines, how to project on map 86 how to trs.ce in field 81 Heights, to find by barometer and thermometer 29 342 INDE3L PAGE Inches in decimals of a foot, Table VIII 153 Index angles, to determine 69 Instruments, adjustment and use of 23 Intersection angles of tangents, to find 55 desirable to fix on ground 66 Level, to adjust 24 Leveller, duties of 83 Levelling, art of 26 by barometer and thermometer 29 correction for curvature and refraction 28 form for field-book 27 rules for exact work 27 rules for survey and location 28 suggestions concerning 83 Location, problems in field 94 admissible errors on ground 81 form of record for 81 projects, hints concerning . . . , , . . . 84 of terminal curves 88 of a Y 103, 104, 106 Logarithms explained 3 multiplication by 5 division by 6 of numbers, to find 4 Table XII 179 roots and powers by 7 Logarithmic sines, tangents, &c., to find 13 table of, XIII 197 Maps, contour, utility of 85 notes for 82, 83 not sufficient for intelligent projects 79 Meridian, to establish 44 by equal shadows 45 by North Star 45 times of passage of North Star, Table I l^^ Multiplication by logarithms 5 Natural sines, tangents, &c., defined ' 12 Table XIV •. 243, 256 Needle, magnetic, to adjust 41 to re-magnetize 44 hints as to management 44 bearings should always be noted 82 North Star, to establish meridian by 45 times of meridian passage, Table 1 148 extreme elongations of, Table 1 14S azimuths and natural tangents, Table II ' . . . . 150 INDEX. 343 PAGB Obstacles in the field to vision 73 to measnrement 74 Ordinates of circular curves, to find 58, 59 of parabola, to find 36 of a r curve, Table X 155 Parabola, ordinates of . 36, 59 Plane trigonometry 12 Powers and roots of numbers by logarithms .......... 7 Propositions, general, in trigonometry * 15 Protractor for curves described 84 how to make 85 Rails, table of spring for trackmen 143 Radius of a curve, how to find . . . . . . . . . . . . . 52, 54, 56 of a turnout curve 129 plain rule for, on curves 133 for narrow gauges ' .' . . . . . . . 134 Radii and their logarithms. Table IX. . ...".'.'.'. . . . . 155 Records, forms for ......"........ 81 Refraction and curvature, correction for 28 !^eversed curves. See Curves. Rise per mile of various grades, Table XVII. 332 Rod, levelling . 28 how to read 42 Rodman, duties of . 83 Roods and perches in decimals of an acre. Table tn 151 Roots and powers of numbers by logarithms ......... 7 Senior assistant, duties of 80 equipment for 81 Sines defined . , . 12 Shadows, to fix true north by 44 Slopes for topography, Table XV 268 Slopeman, duties of 84 Slope stakes, to set 30 for earth excavation 31 for embankment . . .... . 33 for hillsides and rock 85 field record of work 84 Spring of rails, table for trackmen 143 Squares, cubes, and roots of numbers. Table XI. 161 Supplement of an angle 12 Survey, form for record 81 to facilitate 82 Switch-rails, angles of 130 tables of 135,138 Tangent, or apex distance of curve, to find 52, 54 344 INDEX. PAGE Tangent of a 1° curve, Table XVI 269 to curve from a fixed point, how to locate 108 to two cui'ves on the ground, how to locate 109 Tangential angles and distances explained 50 how to find 57, 58, 64 Thermometer, levelling by 29 Track problems 115 Trackmen's plain rules for finding frog distances 132, 133 tables of turnouts 135,136 plain rules for laying off turnouts with tape-measure and pins . 137 , .crossings on straight lines and on curves ....... 139 elevation of outer rail . 142 instructions how to put in missing stakes on curves with tape- - ? measure 144 table of curves and spring of rails .......... .143 explanation of the trackmen's tables , . 144 how to find the degree of a curve 145, 146 Transit, adjustment of 40 cross-hairs . . 24 Transitman, duties of 82 Triangles, solution of, — two angles and a side given 16 two sides and an angle given 17 three sides given 18 Triangles, right-angled, solution of - 19 Trigonometry, plane 12 general propositions 15 Turnouts. See Trackmen. Vernier explained ,..-.....,.- 42 on transit 43 Versed sines defined 12 to calculate 54, 58 of a r curve. Table XVi . 269 Vertical curves, to calculate 36 to project 39 Tables: Ordinates of a 1° curve • 60 For locating terminal curves 88 Tangents between curves of contrary flexure ....... 89 Turnouts, switch-rails straight . 135 switch-rails curved 136 Elevation of outer rail on curves 142 Curves and spring of ra?ls . . . » 143 I. Culminations and elongations of North Star 148 INDEX. 345 PAGE n. Azimuths of North Star, and their natural tangents „ , , 150 in. Roods and perches in decimal parts of an acre 151 IV. Decimals of an acre in one chain length of 100 feet, and of various widths 151 V. Acres, roods, and perches in square feet , 152 VI. Circular arcs to radius of 1 152 VII. Feet in decimals of a mile 153 VIII. Inches reduced to decimal parts of a foot 153 IX. Radii and their logarithms, middle ordinates, and deflection distances 155 X. Metric-curve table 159 XI. Squares, cubes, roots, and reciprocals of numbers, from 1 to 1,042 , 161 XII. Logarithms of numbers from 1 to 10,000 179 XIII. Logarithmic sines, cosines, tangents, and cotangents ... 197 XIV. Natural sines and cosines 243 Natural tangents and cotangents 256 XV. Slopes for topography 268 XVI. Functions of a 1° curve 269 ivVn. Rise per mile of various grades , 388 ADDENDA. Table of feet, inches and reciprocals of various track gauges .... 335 Table of minutes of a degree expressed in decimals 335 Condensed table of radii including short chords . 336 To express gradients per cent, in angular measure 337 Sound , 337 American and French equivalents— Linear measure 337 American and French equivalents— Square measure 338 American and French equivalents— Cubic measure 338 APR 1 1903 l( L 1 A-^ •%'^ / .^' ^.0 N -^ ./ : ^.\. "■ .On : '^^.^^'^ .^^IC^"' ■%<- V 1 R . ••/' > ^^ ^^ ,0^ ^- .% *