■mim i§m ill, iiiiiiiiiiniiil ]m\\ i| Hi liiiiii ( RAILROAD CONSTRUCTION. THEORY AND PRACTICE. A TEXT-BOOK FOB THE USE OF STUDENTS IN COLLEGES AND TECHNICAL SCHOOLS. BY / / WALTER LOEIXG WEBB, C.E., Associate Membey American Society of Civil Engineers: Assistant Professor of Ciuii Engineering in the University of Pennsylvania ; etc. FIRST EDITION. FIRST THOUSAND. KEW YORK: JOHN WTLEY & SONS. London : CITAPNfAN of the rod. h is then on the 165- FiG. 6. foot contour, and the horizontal distance ah added to the liori- zontal distance ac gives the position of that contour from tlie center. The contours on the lower side are found similarly. The first rod reading will he 8.3, giving the 155-foot contour. Plot the results in a note-book which is ruled in cpiarter-inch squares, using a scale of 100 feet per inch in both directions. 12 BAILROAD CONSTRUCTION. § 13. Plot tlie work up the page ; then when looking ahead along the line, the work is properly oriented. When a contonr crosses the survey line, the place of crossing may be similarly deter- mined. If the ground flattens out so tliat five-foot contours are very far apart, the absolute elevations of points at even fifty- foot distances from the center should be determined. The method is exceedingly rapid. Whatever error or inaccuracy occurs is confined in its effect to the oue station where it occurs. The work being thus plotted in the field, unusually irregular topography may be plotted with greater certainty and no great error can occur without detection. It would even be possible by this method to detect a gross error that might have been made by the level party. 13. Stadia method. This method is best adapted to fairly open country where a "shot" to any desired point may be taken without clearing. The haclcbone survey line is the same as in the previous method except that each course is limited to the practicable length of a stadia sight. The distance between stations should be checked by foresight and backsight — also the vertical angle. Azimuths should be checked by the needle. Considering the vital importance of leveling on a railroad survey it might be considered desirable to run a line of levels over the stadia stations in order that the leveling may be as precise as possible ; but when it is considered that a preliminary survey is a somewhat hasty survey of a route that onay be abandoned, and that the errors of leveling by the stadia method (which are com- pensating) may be so minimized that no proposed route would be abandoned on account of such small error, and that the effect of such an error may be easily neutralized by a slight change in the location, it may be seen that excessive care in the leveling of the preliminary survey is hardly justifiable. Since the students taking this work are assumed to be familiar with the methods of stadia topographical surveys, this j^art of the subject will not be further elal)orated. 14. " First " and " second " preliminary surveys. Some engi- neers advocate two preliminary surveys. When this is done, §15. RAILROAD SURVEYS. 13 the first is a very rapid survey, made perhaps witli a compass, and is only a better grade of reconnoissance. Its aim is to rapidly develop the facts which will decide for or against any proposed route, so that if a route is found to be unfavorable another more or less modified route may be adopted without having wasted considerable time in the survey of useless details. By this time the student should have grasped the fundamental idea that both the reconnoissance and preliminary surveys are not surveys of li)ies but of areas \ that their aim is to survey only those topographical features which would have a deter- mininof influence on anv railroad line which mii^ht be constructed through that particular territory, and that the vahie of a locating engineer is largely measured by his ability to recognize those 'determining influences with the least amount of work from his -surveying corps. Frequently too little time is spent on the comparative study of ]u'eliminary lines. A line will be hastily decided on after very little study ; it will then be surveyed with minute detail and estimates carefully worked up, and the claims of any other suggested route will then be handicapped, if not disregarded, owing to an unwillingness to discredit and throw away a large amount of detailed surveying. The cost of two or three extra preliminary surveys {at critical 2>oints and not over the whole line) is utterly insignificant compared with the j^rob- able improvement in the "operating value" of a line located after such a comparative study of preliminary lines. LOCATION SURVEYS. 15. "Paper location." When the preliminary survey has been plotted to a scale of 200 feet per inch and the contours drawn in, a study may be made for the location survey. Disre- garding for the present the effect on location of transition curves, the alignment may be said to consist of straight lines (or " tan- gents ") and circular curves. The " paper location '* therefore consists in plotting on the preliminary map a succession of straight lines which are tangent to the circular curves connect- 14 RAILROAD CONSTRUCTION. § 15. ing tliem. Tlie determining points should first be considered. Such points are the termini of the road, the lowest practicable point over a summit, a river-crossing, etc. So far as is possi- ble, having due regard to other considerations, the road should be a ''surface" road, i.e., the cut and fill should be made as small as possible. The maximum permissible grade must also have been determined and duly considered. The method of location differs radically according as the lines joining the deter- mining points have a very low grade or have a grade that ap- proaches the maximum permissible. With very low natural grades it is only necessary to strike a proper balance between the requirements for easy alignment and the avoidance of exces- sive earthwork. When the grade between two determined points approaches the maximum, a study of the location may be begun by finding a strictly surface line which will connect those points with a line at the given grade. For example, suppose the required grade is 1.6^ and that the contours are drawn at 5-foot intervals. It will require 312 feet of 1.6^ grade to rise 5 feet. Set a pair of dividers at 312 feet and step off this in- terval on successive contours. This line will in general be very irregular, but in an easy country it may lie fairly close to the proper location line, and even in difficult country such a surface line will assist greatly in selecting a suitable location. When the larger part of the Kne will evidently consist of tangents, the tan- gents should be first located and should then be connected by suitable curves. When the curves predominate, as they gener- ally will in mountainous country, and particularly when the line is purposely lengthened in order to reduce the grade, the curves should be plotted first and the tangents may then be drawn connecting them. Considering the ease with which such lines may be drawn on the preliminary map, it is frequently advisable, after making such a paper location, to begin all over, draw a new line over some specially difficult section and compare re- sults. Profiles of such lines may be readily drawn by noting their intersection with each contour crossed. Drawing on each profile the required grade line will furnish an approximate idea of the § 16. BAILROAD SURVEYS, 15 coTTiparative amount of earthwork required. After deciding on the paper location, the length of each tangent, the central angle (see § 21), and the radius of each curve sliould be measured as accurately as possible. Since a slight error made in such meas- urements, taken from a map with a scale of 200 feet per inch, would by accumulation cause serious discrepancies between the plotted location and the location as afterward surveyed in the field, frequent tie lines and angles should be determined between the ])lotted location line and the preliminary line, and the loca- tion should be altered, as may prove necessary, by changing the length of a tangent or changing the central angle or radius of a curve, so that the agreement of the check-points will be suffi- ciently close. The errors of an inaccurate preliminary survey may thus be easily neutralized (see § 33). When the pre- liminary line has been properly run, its "backbone" line will lie very near the location line and will probably cross it at fre- quent intervals, thus rendering it easy to obtain short and nu- merous tie lines. 16. Surveying methods. A transit should be used for align- ment, and only precise work is allowable. The transit stations should be centered with tacks and should be tied to witness- stakes, which should be located outside of the range of the earth- work, so that they will neither be dug up nor covered up. All original property lines lying within the limits of the right of way should be surveyed with reference to the location line, so that the right-of-way agent may have a proper basis for settlement. "When the property lines do not extend far outside of the re- quired right of way they are frequently surveyed completely. The leveler usually reads the target to the nearest thousandth of a foot on turning-points and bench-marks, but reads to the nearest tenth of a foot for the elevation of the ground at stations. Considering that yif-oir ^^ ^ ^^^^ ^^^^ ^^^ angular value of only 7 seconds at a distance of 300 feet, and that one division of a level-bubble is usually about 30 seconds, it may be seen that it is a useless refinement to read to thousandths unless corre- sponding care is taken in the use of the level. The leveler 16 RAILROAD CONSTRUCTION. § 17. should also locate liis bench-marks outside of the range of earthwork. A knob of rock protruding from the ground affords an excellent mark. A large nail, driven in the roots of a tree, which is not to be disturbed, is also a good mark. These marks should be clearly described in the note-book. The leveler should obtain the elevation of the ground at all station-points ; also at all sudden breaks in the profile line, determining also the distance of these breaks from the previous even station. This will in- clude the position and elevation of all streams, and even dry gullies, which are crossed. Measurements should preferably be made with a steel tape, care being taken on steep ground to insure horizontal measure- ments. Stakes are set each 100 feet, and also at the beginning and end of all curves. Transit-points (sometimes called " plugs " or "hubs") should be driven flush with the ground, and a " witness- stake," having tlie "number" of the station, should be set three feet to the right. For example, tlie witness-stake might have on one side " 137 + 69.92," and on the other side " P C 4° K," which would signify that the transit hub is 69.92 feet beyond station 137, or 13769.92 feet from the beginning of the line, and also that it is the "point of curve" of a " 4°- curve ' ' which turns to the right. Alignment. The alignment is evidently a part of the loca- tion survey, but, on account of the magnitude and importance of the subject, it will be treated in a separate chapter. 17. Form of Notes. Although the Form of ]N"otes cannot be thoroughly understood until after curves are studied, it is nere introduced as being the most convenient place. The right-hand page should have a sketch showing all roads, streams, and property lines crossed with the bearings of those lines. This should be drawn to a scale of 100 feet per inch — the quarter- inch squares which are usually ruled in note-books giving con- venient 2 5 -foot spaces. This sketch will always be more or less distorted on curves, since the center line is always shown as straight regardless of curves. The station points ("Sta." in first column, left-hand page) should be placed opposite to their §17. RAILROAD SURVKYS. 17 sketched positions, which means that even stations will be recorded on every fourth line. This allows three intermediate lines for substations, which is ordinarily more than sufficient. The notes should read up the page, so that the sketch will be properly oriented when looking ahead along the line. The other columns on the left-hand page will be self-explanatory when the subject of curves is understood. If the ' ' calculated bearings ' ' are based on azimuthal observations, their agreement (or constant diiference) with the needle readings will form a valuable check oh the curve calculations and the instrumental work. FORM OF NOTES. [Left-hand papre.] [Right-hand page.] Sta. 54 53 0+72.2 52 51 O 50 49 48 0-1-32 47 46 Align- ment Vernier P.T. P.O. 9° 11' 7 57 6 15 4 33 2 51 1 09 0° Tang. Defl. 18° 22' Calc. Bearing. N 54° 48' E N 36° 26' E NeedU N 6S° 15' 1 N 14° 0' ]■ CHAPTEK II. ALIGNMENT. In this chapter the alignment of the center line only of a pair of rails is considered. When a railroad is crossing a sum- mit in the grade line, altliough the horizontal projection of the alignment may be straight, the vertical projection will consist of two sloping lines joined by a cnrve. When a curve is on a grade, the center line is really a spiral, a curve of double curva- ture, although its horizontal projection is a circle. The center line therefore consists of straight lines and curves of single and double curvature. The simplest method of treating them is to consider their horizontal and vertical projections separately. In treating simple, compound, and transition curves, only the horizontal projections of those curves will be considered. SIMPLE CURVES. 18. Designation of curves. A curve may be designated either by its radius or by the angle subtended by a chord of unit length. Such an angle is known as the ' ' degree of curve ' ' and is indicated by D. Since the curves that are practically used have very long radii, it is gener- ally impracticable to make any use of the actual center, and the curve is located without reference to it. If AB in Fig. 7 represents a unit chord ((7) of a curve of radius i?, then by the above defini- 18 §19. ALIGNMENT. 19 tion the angle AOB equals D, Then AO sin ^D = iAB ^ iO. (1) .-. B = or, bj inversion, sin ^J) = sin il) C_ 2E (2) The unit chord is variously taken throughout the world as 100 feet, (S^ feet, and 20 meters. In the United States 100 feet is invariably used as the unit chord length, and throughout this work it will be so considered. Table I has been computed on this basis. It gives the radius, with its logarithm, of all curves from a 0° 01' curve up to a 10° curve, varying by single minutes. The sharper curves, which are seldom used, are given with larger intervals. An approximate value of i? may be readily found from the following simple rule, which should be memorized : B = 5730 IT' Although such values are not mathematically correct, since jR does not strictly vary inversely as D, yet the resulting value is within a tentli of one per cent for all commonly used values of ^, and is suf- ficiently close for many purposes, as will be shown later. 19. Length of a sub-chord. Since it is impracticable to measure along a curved arc, curves are always measured by laying off 100-foot chord lengths. Tliis means that the actual arc is always a little longer than the chord. It also means that a suhchord (a chord shorter than the unit length) will be a little longer than the ratio of the angles subtended would call for. The truth of this may be seen without calcu- FiG. 8. 20 RAILROAD CONSTRUCTION. § 20. lation by noting that two equal subcliords, each subtending the angle j-T), will evidently be slightly longer than 50 feet each. If c be the length of a subchord subtending the angle d, then, as in Eq. (2), sm 2,ct — Q~o) or, by inversion, c=^2B sin ^d (3) d The no?ninal length of a subchord = 100—. For example,, a nominal subchord of 40 feet will subtend an angle of -^-^q of D° ; its true length will be slightly more than 40 feet, and may be computed by Eq. 3. The difference between the nominal and true lengths is maximum when the subchord is about 57 feet long, but with the low degrees of curvature ordinarily used the difference may be neglected. "With a 10° curve and a nominal chord length of 60 feet, the true length is 60.049 feet. Very sharp curves should be laid off with 50-foot or even 25- foot chords (nominal length). In such cases especially the true lengths of these subcliords should be computed and used instead of the nominal lengths. 20. Length of a curve. The length of a curve is always indicated by the quotient of 100/^ -^ D. If the quotient of z/ -=- Z^ is a whole number, the length as thus indicated is the true length — measured in 100-foot choi'd lengths. If it is an odd number or if the curve begins and ends with a subchord (even though A -^ D \& a whole number), theoretical accuracy requires that the true subchord lengths shall be used, although the difference may prove insignificant. The length of the arc (or the mean length of the two rails) is therefore always in excess of the length as given above. Ordinarily the amount of this excess is of no practical importance. It simply adds an insignificant amount to the length of rail required. Examjple. Required the nominal and true lengths of a 3° 45' curve having a central angle of 17° 25'. First reduce §22. ALIGNMENT. 21 the degrees and minutes to decimals of a degree. (100 X 1T° 25') -h 3° ^5' = 17^:1.007 -=- 3.75 = U^AU. The curve has four 100-foot chords and a nominal chord of Gir.-l-ll:. The true chord should be 61:. 451. The actual arc is 17M:1G7 X 7t IbO' X E = 461:. 527. The excess is therefore 46-1.527 - 464.451 = 0.076 foot. 21. Elements of a curve. Considering the line as running from A toward I>^ tlie beginning of tlie curve, at A^ is called i\\Q point of curve {PC). The other end of the curve^ at ^, is' called the point of tangency (PT). The intersection of the tangents is called the vertex {V). The angle made bj the tangents at T", which equals the angle made by the radii to the extremities of the curve, is called the central angle [A). A T^and B T", the two equal tangents from tlie vertex to the PC and PT^ are called the tangent distances {T). The chord AB is called the long chord (LC). The intercept HG from the middle of the long chord to the middle of the arc is called the middle ordinate (M). That part of the secant G V from the middle of the arc to the vertex is called the external distance (E), From the figure it is very easy to derive the follow^ing frequently used relations : Fig. 9. T= R tan ^ J LC M 2^ sin i J E = (4) (5) R vers ^z/ (6) R exsec ^A (7) 22. Relation between T, E, and A. Join A and G in Fig. 9. The angle VAG = iA, since it is measured by one half of the 22 RAILROAD CONSTRUCTION. § 23. arc AG between the secant and tangent. AGO z= 90° —\A. AY: VG:\miAGV:dn YAG) miAGY=^ QinAGO — cosiz?; T'.Ew cos \A : sin \A ; T=Eq,oI\A (8) The same relation may be obtained by dividing Eq. 4 by Eq. 7, since tan a -^ exsec a — cot \a. 23. Elements of a 1^ curve. From Eqs. 1 to 8 it is seen that the elements of a curve vary directly as R. It is also seen to be very nearly true that B, varies inversely as D. If the ele- ments of a 1° curve for various central angles are calculated and tabulated, the elements of a curve of Z^° curvature may be approximately found by dividing by D the corresponding elements of a 1° curve having the same central angle. For small central angles and low degrees of curvature the errors involved by the approximation are insignificant, and even for larger angles the errors are so small \h2Xf0r many jpurposes they may be disre- garded. In Table II is given the value of the tangent distances, external distances, and long chords for a I'' curve for various central angles. The student should familiarize himself with the degree of approximation involved by solving a large number of cases under various conditions by the exact and approximate methods, in order that he may know when the approximate method is sufficiently exact for the intended purpose. The approximate method also gives a ready check on the exact method. 24. Exercises, [a) "What is the tangent distance of a 4° 20' curve having a central angle of 18° 24' ? {h) Given a 3° 30' curve and a central angle of 16° 20', how far will the curve pass from the vertex ? [Use Eq. 7.] (c) An 18° curve is to be laid off using 25-foot (nominal) chord lengths. What is the true length of the subchords ? §25. ALIGNMENT. 23 {(l) Given two tangents making a central angle of 15° 2^'. It is desired to connect these tangents by a curve which shall j^ass 16.2 feet from their intersection. How far down the tangent will the curve begin and what will be its radius ? (Use Eq. S and then use Eq. -i inverted.) 25. Curve location by deflections. The ano-le between a secant and a tangent (or between two secants intersecting on an arc) is measured by one half of the intercepted arc. Beginning at the PC {A in Fig. 10), if the first chord is to be a full cliord we may deflect an angle VAa [= jP), and the point «, which is 100 feet from ^•1, is a point on the curve. For the next station, ^, deflect an additional angle hAa {— ^D) and, with one end of the tape at a, swing the other end until the 100-foot point is on the line Ah. The points is then on the curve. If the final chord cjB is a subchord, its additional deflection {^a) is something less than 4-7>. The last deflection {BA Y) is of course ^//. It is particularly inqwrtant, when a curve begins or ends with a subchord and the defiections are odd quantities, that the last additional defiection should be care- fully coni})uted and added to the previous deflection, to check the mathematical work by the agreement of this last conqnited deflection with -g-z/. Example. Given a 3° 2-1' curve having a central angle of 18° 22' and beginning at sta. -IT -\- 32, to conq^ute the deflections. The nominal length of curve is 18° 22'- 3° 24' =18.367 — 3.40 = 5.402 stations or 540.2 feet. The curve therefore ends at sta. 52 + 72.2. The deflection for sta. 48 is y^o X K^^ ^^0 = 0.68 X 1°.T = 1°.156 = r 09' nearly. For each additional 100 feet it is 1° 42' additional. The final additional deflection for the final subchord of 72.2 feet is Fig. 10. ^^ X K3° 24') = 1°. 2274 100 ^^ ^ 1° 14' nearly. 24 RAILROAD CONSTRUCTION. §26. The defections are P. C . . . . Sta. 47 + 32 0° 48 0° + 1° 09' = 1° 09' 49 1° 09'+ 1° 42' = 2" 51' 50 2° 51'+ 1° 42' r=4° 33' 51 4° 33'+ 1° 42' ==G° 15' 52 0° 15'+ 1° 42' = 7° 57' P. T 52 + 72.2 . . . /7° 57' + 1° 14^ = 9° 11' As a check 9° 11' = i(18° 22') = |^. (See the Form of Notes in § 17.) 26. Instrumental work. It is generally impracticable to locate more than 500 to 600 feet of a curve from one station. Obstructions will sometimes require that the transit be moved up every 200 or 300 feet. There are two methods of setting off the angles when the transit has been moved up from the PC. (a) The transit may be sighted at the previous transit station with a reading on the plates equal to the deflection angle from that station to the station occupied, but with the angle set oif on the other side of 0°, so that when the telescope is turned to 0° it will sight along the tangent at the station occupied. Plunging the telescope, the forward stations may be set off by deflecting the proper deflections from the tangent at the station occupied. This is a very common method and, when the degree of curva- ture is an even number of degrees and when the transit is only set at even stations, there is but little objection to it. But the degree of curvature is sometimes an odd quantity, and the exi- gencies of difiicult location frequently require that substations be occupied as transit stations. Method {a) will then require the recalculation of all deflections for each new station occupied. The mathematical work is largely increased and the probability of error is very greatly increased and not so easily detected. Method {!)) is just as simple as method (a) even for the most simple cases, and for the more difiicult cases just referred to the superiority is very great. §26. ALIGNMENT. 26 (b) Calculate the deflection for each station and substation throughout the curve as though the whole curve were to be lo- cated from the PC. The computations may thus be completed and checked (as above) before beginning the instrumental work. If it unexpectedly becomes necessary to introduce a substation at any point, its deflection from the P(7may be readily inter- polated. The stations actually set from the PC are located as usual. Rule. When the transit is set on any forward station, backsight to ANY previous station with the plates set at the deflec- tion angle for the station sighted at. Plunge the telescope and sight at any forward station with the deflection angle originally computed for that station. AVlien the plates read the deflection angle for the station occupied, the telescope is sighting along the tangent at that station — which is the method of getting the for- ward tangent when occupying the PT. Even though the sta- tion occupied is an unexpected substation, when the instrument is properly oriented at that station, the angle reading for any station, ^^ forward or back, is that originally computed for it from the P(7. In diflicult work, where there are obstructions, a valuable check on the accuracy may be found by sighting back- ward at any visible station and noting whether its deflection agrees with that originally com- puted. As a numerical illustration, assume a -t° curve, with 28° curvature, with stations 0, 2, 4, and T occupied. After setting stations 1 and 2, set up the transit at sta. 2 and backsight to sta. with the deflection for sta. 0, which is 0°. The reading on sta. 1 is 2° ; when the reading is 4° the telescope is tangent to the curve, and when sighting at 3 and 4 the deflections will be 6° and 8°. Occupy 4 ; sight to 2 with a reading of 4°. is 8° the telescope is tangent to the curve and, by plunging the telescope, 5, 6, and 7 may be located with the originally com- FlG. 11. When the reading 26 RAILROAD CONSTRUCTION. 27 puted deflections of 10°, 12°, and 14°. When occupying 7 ?, backsight may be taken to any visible station with the plates read ing the deflection for that station ; then when the plates read 14° the telescope will point along the forward tangent. The location of curves by deflection angles is the normal method. A few other methods, to be described, should be con- sidered as exceptional. 27. Curve location by two transits. A curve might be located more or less on a swamp where accurate chaining would be ex- ceedingly difficult if not impossible. The long chord AB may be determined by triangulation or otherwise, and the elements of Fig. 13. Fig. 13. the curve computed, including (possibly) subchords at each end. The deflection from A and B to each point may be computed. A rodman may then be sent (by whatever means) to locate long stakes at points determined by the simultaneous sightings of the two transits. 28. Curve location by tangential offsets. When a curve is very flat and no transit is at hand the following method may be § 29. ALIGNMENT. 27 used : Produoe the back tangent as far forward as necessary. Compute the ordinates Oa\ Oh\ Oc\ etc., and the abscissae a' a., h'h, c'c. etc. If Oa is a full station (100 feet), then Oa' = Oa' =^ 100 cosiD, also = R' sin D; ] 01/ = Oa'Ara'h' = 100 cos *j9+ 100 cos |/>, also — li sin 'ID ; L p^ Oc' = Oa +a'h' -\-h'c' =^100{cos>il) +C06 U) + co^^I))j also — Ic sin 3D ; etc. a' a = 100 sin ^Z>, also = /*' vers 7>; ^ h"b = a:' a + h''h =100 sin ii> + 100 sin :] />, | also = J^vers2D; I mq) c'c' = Jj'h + c'c = 100(siniZ) + sin|Z> + sinfZ>), also — I^versoD] etc. The functions Ji>, fi^, etc., may be more conveniently used without logarithms, by adding the several national trigonometrical functions and pointing off two decimal places. It may also be noted that oV (for example) is one half of the long chord for four stations; also that h'h is the middle ordinate for four stations. If the engineer is provided with tables giving the long chords and middle ordinates for various degrees of curvature, these quantities may be taken (perhaps by interpolation) from such tables. If the curve begins or ends at a substation, the angles and terms will be correspondingly altered. The modifications may be readily deduced on the same principles as above, and should be worked out as an exercise by the student. 29. Curve location by middle ordinates. Take first the simpler case when the curve begins at an even station. If we consider (in Fig. 14) the curve produced back to ^, the chord za = 2 X 100 cos iD, A'a = 100 cos iZ>, and A' A = am = .iit = 100 sin ^D. Set off A A' perpendicular to the tangent and A'a parallel to the tangent. ^1^1' = aa' = hh' = cc\ etc. — 100 sin \D. Set ofi aa' per^^endicular to a' A. Produce Aa' 28 BAILROAD COISSTRUCTION. §30. until a'h =-■ A'a^ thus determining h. Succeeding points of the curve may thus be determined indefinitely. Suppose the curve begins with a subchord. As before ra = Am' = c' cos \d' ^ and rA = am! = c' sin \d' . Also sz ■=■ An' = c" cos \d'\ and sA = zn' = c" sin \d" . in which Fm. 14. Fig. 15. (d' -\- d") = D. The points ^ and a being determined on the ground, aa' may be computed and set off as before and the curve continued in full stations. A subchord at the end of the curve may be located by a similar process. 30. Curve location by offsets from the long chord. (Fig. 16.) Consider at once the general case in which the curve commences with a subchord (curvature, d'), contains with one or more full cliords (curvature of each, D)^ and ends with a subchord with curvature d" . The numerical work consists in computing first AB^ then the various abscissae and ordinates. AB=2B sin ^Zi. Aa' = Aa! = c' cos ^{J — d'); Ah' = Aa' + a'b' = d cos \{A-d')-^' 00 cos \{A - 2d' - D) ; Ac' = Aa' + a'b' + b'c' = c' cos i(z/ - d') + 100 cos {{/I - 2d' - D) + 100cosi(z/-2(f"-Z>); also -AB-Bc' =z2Bs\n^J- c" co%^{A - d"). Kii) §32. ALIGNMENT. 29 a'a=: a'a = c' sin |(z/ — d'); ") bb = a'a + mb= c sin 1{J - d')-\- 100 siu \{^ - 2d' - V); j c'c = bb - nb = c' sin 1{J - d')-{- 100 siu |(z/ - M - D) )■ (12) -lOOsin i(J-2(r'-Z>); \ also =c" s\u\{J — d"). J The above formulae are considerably simplified when the enrve begins and ends at even stations. When the curve is very long a regular law becomes very apparent in the formation of all terms between the first and last. There are too few terms in the above equations to show the law. 31. Use and value of the above methods. The chief value of the above methods lies in the possibility of doing the work without a transit. The same principles are sometimes employed, even when a transit is used, when obstacles pre- vent the nse of the normal method (see § 32, c). If the terminal tangents have already been ac- curately determined, these methods are useful to locate points of the curve when rigid accuracy is not essential. Track foremen frequently use such methods to lay out unimportant sidings, especially when the engineer and his transit are not at hand. Location by tangential offsets (or by offsets from the long chord) is to be preferred when the curve is flat (i.e., has a small central angle ^) and there is no obstruction along the tangent, or long chord. Location by middle ordinates may be employed regard- less of the leno;th of the curve, and in cases when both the Fig. 16, tangents and the long chord are obstructed. The above methods are but samples of a large number of similar methods which have been devised. The choice of the particular method to be adopted nmst be determined by the local con- ditions. 32. Obstacles to location. In this section will be given only a few of the principles involved in this class of problems, with illustrations. The engineer must decide in each case, which is 30 JRAILROAD CONSTRUCTION. 32. tlie best metliod to use, and it is frequently advisable to devise a special solution for some particular case. a. When the vertex is inaccessible. As shown in § 26, it is not absolutely essential that the vertex of a curve should be located on the ground. But it is xery evident that the angle between the terminal tangents is determined with far less prob- able error if it is measured by a single measurement at tlie ver- tex rather than as the result of numerous angle measurements along the curve, involving several positions of the transit and comparatively short sights. Sometimes the location of the tangents is already determined on the ground (as by hi and am, Fig 17), and it is required to join the tangents by a curve of given radius. Method. Measure ab and the angles Vba and ha V. A is the sum of these angles. The distances h Fand a Y are computable from the above data. Given A and R, the tan- FiG. 18. gent distances are computable, and then Bb and aA are found by subtracting h V and a V from the tangent distances. The curve may then be run from A, and the work may be checked by noting whether the curve as run ends at £ — previously lo- cated from h. b. When the point of curve (or point of tangency) is inacces- sible. At some distance {As, Fig. IS) an unobstructed line pn § 33. ALIGNMENT. 31 may be run parallel with A V. nv = jpy = As =z Ji vers a. vers (Y = As ^ U. ns =j)s = It sin a. At ?/, which is at a distance jps back from the computed posi- tion of ^i, make an oifset sA to 7>>. Ilun i)ii parallel to the tano-ent. A tansjent to the curve at n makes an anoxic of « with np. From n the curve is run in as usual. . If the point of tangencj is obstructed, a similar process, somewhat reversed, may be used. /5 is that portion of A still to be laid off when m is reached, tin =^ tl =^ It sin /?. iriz = tB = lx =1 B vers ft. c. When the central part of the curve is obstructed. a is the central angle between two points of the curve between which a chord may be run. a may equal any angle, but it is prefer- able that a should be a multiple of Z^, the degree of curve, and that the points vi and n should be on even stations. m)i = 2/t sin iot, A point s may be located by an oifset hs from the chord tnn by a ^^"""T^-^.^ similar method to that outlined in § 30. / tjS^^^^-,^ The device of introducin£i^ the dotted / / \'^o^ curve ???./? havino; the same radius of cur- / / \i'v^ vature as the other, although neither / / \\ necessary nor advisable in the case shown / / ^' in Fig. 19, is sometimes the best method a5^_, — - — """" of surveying around an obstacle. The oifset from any point on the dotted curve to the corresponding point on the true ^^^^' ^^• curve is twice the "ordinate to the long chord," as computed in § 30. 33. Modifications of location. The following methods may be used in allowing for the discrepancies between the " paper location " based on a more or less rough preliminary survey and the more accurate instrumental location. (See § 15.) They are also frequently used in locating new parallel tracks and modify- ing old tracks. 32 RAILROAD CONSTRUCTION. 33. a. To move the forward tangent parallel to itself a distance a?, the point of curve (^1) remaining fixed. (Fig. 20.) VV = -T V'h X sinAFF' sin ^ • • (13) AV = A V+ VV\ The triano^le BmB' is isosceles and Bin = B'm. R' - R^ O'O^mB^ B'r X vers BinB vers A .-. R'^R-\- X vers A . . . . (U) The solution is very similar in case the tangent is moved in- ward to V^'B^\ IS'ote that this method necessarily changes the Z^r-^ ^-2^^^^ ^^4-> /t^^ ^^^^^ ^^^^^-3: ^\\ /i / / r" \A /i/j/j \ 0' 6 6" V' V V" Fig. 20. Fig. 21. radius. If the radius is not to be changed, the point of curve must be altered as follows : b. To move the forward tangent parallel to itself a distance x, the radius being unchanged. (Fig. 21.) In this case the whole §33. ALIGNMENT. 33 curve is moved bodily a distance 00' = A A' = W = BB', and moved parallel to the first tangent A V. BB' = -. B'n X sin iiBB' sin A = AA\ (15) c. To change the direction of the forward tangent at the point of tangency. (Fig. 22.) This problem involves a change («-) in the central angle and also requires a new radius. An error in the determination of the central angle furnishes an occasion for its use. ^, J, a, A Vy and B Fare known. /}' = /} — a. Bs = B vers A. Bs = R' vers A' , vers A E' = E vers {A — ^) * As = E ^m A. A's z= E' sin A' , (16) .-. AA' = A's — As = E' sin A' — E sin A. . (17) The above solutions are given to illustrate a large class of problems which are constantly arising. All of the ordinary Fig. 22. Fig. 23. problems can be solved by the application of elementary ge- ometry and trigonometry. 34 RAILROAD CONSTRUCTION. % 34. 34. Limitations in location. It may be required to run a curve that shall join two given tangents and also pass through a given point. The point (P, Fig. 23) is assumed to be determined by its distance ( VP) from the vertex and by the angle A VP = p. It is required to determine the radius {E) and the tangent distance (^ F). A is known. PVG = i(180° - J) - /5 = 90° - (iJ + ^). PP' =2VP sin PVG ^2 VP cos (iJ + /?). PSV = U. .', SP = YP^""^ sin hA' AS = VSP X SP' = VSP{SF + PP')' = )/ FP4^r FP-g^ + 2 FP cos (iz^ + ^) sm t^L sm -^n __ Yp i/ sin' /g 2 sin yg cos (jz/ + jS) sin' \A sin ^-z/ sni \/i AV^AS^SY =_-^rsin (i// +/i) + |/sin' /3 + 2 sin ^ sin i^/ cos (i^ +/i)]. (18) sin ^i^ P:= ^FcOtiZl. In the special case in which P is on the median line F, y5 = 90° - iJ, and (Jz^ + /?)== 90°. Eq. (IS) then reduces to VP AV= ^-T^(l + cos i J) = FP cot iJ, sm 2 '^ as mi^ht have been immediately derived from Eq. (8). § 35. ALIGNMENT. 35 111 case the point P is given by the offset PK and by tlie distance YK^ the triangle PKV \\\a^\ be readily solved, giving the distance YP and the angle /?, and the remainder of the solution will be as above. 35. Determination of the curvature of existing track, (a) Vs'niy a trcuisit. Set up the transit at any point in the center of the track. Measure in each direction 100 feet to points also in the center of the track. Sight on one point with the plates at 0°. Plunge the telescope and sight at the other point. The angle between the chords" equals the degree of curvature. (b) Using a tape and string. Stretch a string (say 50 feet long) between two points on the inside of the head of the outer rail. Measure the ordinate [x) between the middle of the string and the head of the rail. Then _, chord' . ^ ^ ^ = — g^ (very nearly) (19) For, in Fig. 24, since the triangles AGE and ADC are similar, AO : AE : : AP : PJC or P = lAP" ~ x. When, as is usual, the arc is very short compared with the radius, AD = -J^-^? very nearly. Making this sul)stitution we have Eq. (19). With a chord of 50 feet and a 10° curve, the resulting difference in x is .0025 of an inch — far within the possible accuracy of such a method. The above method gives the radius of the inner head ^^^- ~^- of the outer rail. It should be diminished by %g for the radius of the center of the track. With easy curvature, however, this will not affect the result by more than one or two tenths of one per cent. The inversion of this formula gives the required middle or- dinate for a rail on a given curve. For example, the middle ordinate of a 30-foot rail, bent for a 6° curve, is a; = 900 -^ (8 X 955) = .118 foot = 1.1 inches. 36 BAILBOAD CONSTRUCTION. § 36. Another much used rule is to require the foreman to have a string, knotted at the centre, of such length that the middle or- dinate, measured in inches, equals the degree of curve. To- find that length, substitute (in eq. (19)) 5730 -^- D for H and 2> -=- 12 for X. Solving for chords we obtain chord =61.8 feet. The rule is not theoretically exact, but, considering the uncertain stretching of the string, the error is insignificant. In fact, the distance usually given is 62 feet, which is close enough for all purposes for which such a method should be used. 36. Problems. A systematic method of setting down the solution of a problem simplifies the work. Logarithms should always be used, and all the work should be so set down that a revision of the work to find a supposed error may be readily done. The value of such systematic work M^ill become more apparent as the problems become more complicated. The twa solutions given below will illustrate such work. a. Given a 3° curve beginning at Sta. 27+60 and running to Sta. 32 -J- 45. Compute the ordinates and offsets used in locating the curve by tangential offsets. h. With the same data as above, compute the distances to locate the curve by offsets from the long chord. c. Assume that in Fig. 17 ab is measured as 217.6 feet,, the angle ah F= 17° 42', and the angle haV = 21° 14'. Join the tangents by a 4° 30' curve. Determine hB and aA, d. Assume that in a case similar to Fig. 18 it was noted that a distance {As) equal to 12 feet would clear the building. Assume that A = 38° 20' and that !> = 4° 40'. Eequired tiie value of a and the position of n. Solution : YQY^ a z=z As -^ R ^5=12 log = 1.07918 R (for 4° 40' curve) log = 3.08923 0^= 8° 01 ' log vers a = 7.98994 m = B sin a log sin a = 9.14445 log R = 3.08923 71^ = 171.27 log = 2.2336^ § 37. ALIGNMENT. 37 e. Assume that the forward tangent of a 3° 2(V curve havinir a central anii^le of 1G° 50' must be moved 3.62 feet inward^ witliout altering the P. C. Required the cliange in radius. f. Given two tangents making an angle of 36° 18'. It is required to pass a curve through a point 93.2 feet from the vertex, the line from the vertex to the point making an angle of 42° 21' with the tangent. Required the radius and tangent distance. Solution: Applying eq. (18), we have 2 /? = 42° 21' ij = 18° 09' (ij + /?) = 60° 30' .20667 logsinV = 9.656S8 .45382 2| 9.81987 .66049 9.90993 .81271 nat sin 60° 30'= .8703 log =z 0.30103 log sin = 9.82844 log sin = 9.49346 log cos ^^ 9.69234 9.31527 1.6836 log= 0.22610 YP= 93.2 ]og= 1.9694i 2. 19551 log sin iz/ =: 9.49346 tang. dist. AY = 503.56 log= 2.70205 log cot iJ = 10.48437 7?= 1536.1 3.18642 1) =-- 3° 44' COMPOUND CURVES. 37. Nature and use. Compound curves are formed by a succession of two or more simple curves of different curvature. The curves must have a common tangent at the ])oint of com- pound curvature {P.C.C.). In mountainous regions there is frequently a necessity for compound curves having several changes of curvature. Such curves may be located separately as a succession of simple curves, *but a combination of two 38 RAILROAD CONSTRUCTION. 3a simple curves has special properties wliicli are worth investigat- ing and utilizing. In the following demonstrations H^ always represents the longer radius and B^ the shorter^ no matter which succeeds the other. T^ is the tangent adjacent to the curve of shorter radius (A*,), and is invariably the shorter tan- o-ent. A^ is the central angle of the curve of radius i?j , but it may be greater or less than z/,. 38. Mutual relations of the parts of a compound curve havings two branches. In .Fig. 25, ^6^ and CB are the two branches of Fig 25. the compound curve having radii of B^ and B^ and central ano-les of z/, and z/,. Produce the arc AC to n so that Ao{ii = A. The chord Cn produced must intersect B. The line ^16', parallel to CO^ , will intersect BO^ so that Bs = sn = (9^(9j = 7?^ — B^. Draw Am perpendicular to O^n. It will be parallel to lik. Br = S7i vers Bs7i = (7?, — B,) vers A^ ; m7i = AO^ vers A0^7i = R^ vers A ; Ak =^AV sin A Vh = T, sin A ; Ale = J nil = mn -\- nli = 7/171 -f- Br. ,'. T^ sin A^B, vers A-\-{B, — B,) vers A^. . (20) § 38. ALIGNMENT. 39 Similarly it may be shown that 7; sin A ^ R^ vers A — {B, — B,) vers A,. . (21) The mutual relations of the elements of compound curves may be solved by these two equations. For example, assume the tangents as fixed {/} therefore known) and that a curve of given radius Ii^ shall start from a given point at a distance T^ from the vertex, and that the curve shall continue through a given angle ^,. Required the other parts of the curve. From Eq. (20) we have T, sin J — i?i vers A i?, - 7?, = vers ^2 T, sin /I- B^ vers /I „•. It., = li -] J-. -rr . . . (22) * ' vers (// — -^j ^ ^ T^ may then be obtained from Eq. (21). As another problem, given the location of the two tangents, with the two tangent distances (thereby locating the PC and PT)^ and the central angle of each curve ; required the two radii. Solving Eq. (20) for R^ , we have T. sin A — R„ vers J„ vers A — vers A^ Similarly from Eq. (21) we may derive „ _ J!, sin A — R^ (vers A — vers z/,) vers ^1 Equating these, reducing, and solving for R^ , we have T, sin A vers z/, — T^ sin A (vers A — vers z^,) '''^ ~ vers A^ vers ^, — (vers A —vers ^,)(vers A — vers A^' ^^ Althouorh the various elements mav be chosen as above with considerable freedom, there are limitations. For example, in Eq. (22), since R^ is always greater than R^ , the term to be added to R^ must be essentially positive — i.e., T^ sin A must be 40 RAILROAD CONSTRUCTION. §«^9. VGrs ^ greater than 7?, vers /I. This means that 7T > IL —. — ;r* or that T^ > ^^ tan |-z/, or that T, is greater than the corre- sponding tangent on a simple curve. Similarly it may be shown that T^ is less than i?, tan ^^ or less than the correspond- ing tangent on a simj^le curve. E^evertheless T^ is always greater than T^, In the limiting case when /^ = -^i , T, = T and z/^ — ^1- 39. Modifications of location. Some of these modifications may be solved by the methods used for simple curves. For example : a. It is desired to move the tangent VB, Fig. 26, parallel to itself to V^B\ Run a new curve from the P. C. O. which shall reach the new tangent at B\ where the chord of the old curve Fig. 26. Fig. 27. intersects the new tangent. The solution is almost identical with that in § 33, c^. b. Assume that it is desired to change the forward tangent (as above) but to retain the same radius. In Fig. 27 (7?2 — ^i) cos J, = O^n ; (7?, - B,) cos /},' = 0:n'. X — O^n — O^n' = {R^ — P,)(cos z/^ — cos z//). X cos /^/ = cos Z/^ — n __ p ' . . . (24) §39. ALIGNMENT. 41 The P. C. C. is moved hackward along the sliarper curve an angular distance of ^/ — ^^ — z^i — /^/. In case the tangent is moved inward rather than outward, the solution will apply bj transposing A^ and ^/. Then we will have cos A' = cos ^. + ' p _ p - . . . (25) Tlie P. C. C. is then moved for- ward. c. Assume the same case as (b) ex- cept that the larger radius comes first and that the tangent adjacent to the smaller radius is moved. In Fisr. 28 (i?2 — R,) cos J, = 0,71 ; (i?, - B,) cos zf/ = 0;n', '' "n^-^ Fig. 28. X = 0/n' - 0^71 = (P, - i?,)(cos z// _ cos ^,). cos /I/ = COS A, + p^^_^ > . . . (26) The P. C. C. is moved forward along the easier curve an angular distance of ^/ —. /i^ = /l^ — A^, In case the tangent is moved inward, transpose as before and we have cos Al = cos A^ — X p.-pr . . (27) The P. C. C. is moved hachioard. d. Assume that the radius of one curve is to be altered witli- out changing either tangent. Assume conditions as in Fig. 29. For the diagrannnatic solution assume that P, is to be in- 42 RAILROAD CONSTRUCTION. 39. creased by O^S, Then, since ^/ must pass through 0, and ex- tend beyond 0, a distance 0,S, the locus of the new center must lie on the arc drawn about 0, as center and with OS as radius. The locus of 6*/ is also given by a line O^'j? parallel to ^ F and at a distance of ^/ (equal to S ... P. CO.) from it. The new center is therefore at the intersection 0^\ An arc with radius i?/ will therefore be tangent at Jj' and tangent to the old curve ^^r^- chiced at new P.C.C. Draw O^n perpendicular to O^B. With 0^ as center draw the arc 0,m, and with 0^ as center draw the arc 0{in' . mB = m' B' = B,. .-. mn = m'n' = \s/U Fig. 29. (i?/ - B,) vers J/ = {B, - B,) vers /J,. •. vers /i: = ^, ^ vers J, {It^ — It,) . . (28) 0,71= (B,-B,) sin/},', 0,n'={B:-B,)sm z//. BB' = 0,n'-0,n - {B:-B,) sin zJ/- {B~B) sin z^,. (29) This problem may be further modified by assuming that the radius of the curve is decreased rather than increased, or that the smaller radius follows the larger. The solution is similar and is suggested as a profitable exercise. It might also be assumed that, instead of making a given change in the radius ^,, a given change BB' is to be made, z?/ and ^/ are required. EHminate B^' from Eqs. 28 and 29 and solve the resulting equation for z^/. Then determine B^' by a suitable inversion of either Eq. 28 or 29. § 41. ALIGNMENT. 43 As in §§ 32 and 33, the above problems are but a few, although perhaps the most common, of the problems tlie engineer may meet with in compound curves. All of tlie ordinary problems may be solved by these and simihir methods. 40. Problems, a. Assume that the two tangents of a com- pound curve are to be 348 feet and 624 feet, and that ^, = 22° 16' and ^, = 28° 20'. Kequired the radii. [Ans. ^,:= 326.92; 7?, = 1574.85.] h. A line crosses a valley by a compound curve which is first a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is afterward decided that the last tangent should be 6 feet farther up the hill. What are the required changes ? {Note. The second tangent is evidently moved outward. The solution corresponds to that in the first part of § 39, c. The P. C. C. is moved forward 16.39 feet. If it is desired to know how far the P. T. is moved in the direction of the tangent (i.e., the projection of JSJ3\ Fig. 28, on V B), it may be found by observing that it is equal to 7in' = (7?2 — ^,)(sin A, — sinz//). In this case it equals 0.65 foot, which is very small because A^ is nearly 90°. The value of z/^ (^^° ^^0 is not used, since the solution is independent of the value of A^. The student should learn to recognize which quantities are mutually related and therefore essential to a solu- tion, and which are independent and non-essential.] TRANSITION CURVES. 41. Superelevation of the outer rail on curves. When a mass is moved in a circular path it requires a centripetal force to keep it moving in that path. By the principles of mechanics we know that this force equals Gv' -r- gli^ in which G is the weight, V the velocity in feet per second, g the acceleration of gravity in feet per second in a second, and R the radius of curvature. If the two rails of a curved track were laid on a level (trans- versely), this centripetal force could only be furnished by the 44 RAILROAD CONSTRUCTION. 41. pressure of the wlieel-flanges against tlie rails. As tliis is very objectionable, the outer rail is elevated so that the reaction of the rails against the wheels shall contain a horizontal component equal to the re- cpiired centripetal force. In Fig. 30, if oh represents the reaction, oc will repre- sent the weight G, and ao will represent ___^_-l--— -"[^ the required centripetal force. From similar triangles we may write S7i : sm : : ao\oc. Call ^=32.17. Call B = 5730 -^ Z), which is sufficiently accurate for this purpose (see § 19). Call v= 5280 F-^ 3600, in whieh T^is the velocity in miles per hour, mn is the distance between rail centers, which, for an SO-lb. rail and standard gauge, is 4.916 feet, sin is slightly less than this. As an average value we may call it 4.900, which is its exact value when the superelevation is 4i inches. Calling sn = gR G 32.17 X 3600' X 5730 .0000572 F'Z> (30) It should be noticed that, according to this formula, the required superelevation varies as the sqttare of the velocity, which means that a change of velocity of only 10,^ would call for a change of superelevation of 21^. Since the velocities of trains over any road are extremely variable, it is impossible to adopt any superelevation which will fit all velocities even approximately. The above fact also shows why any over- refinement in the calculations is useless and why the above approximations, which are really small, are amply justifiable. For example, the above formula contains the approximation that i? = 5730 -f- Z^. In the extreme case of a 10° curve the error involved would be about Ifo, A change of about i of I'fo in §42. ALIGNMENT. 46 the velocity, or say from 40 to 40.2 miles per hour, would mean as much. The error in e due to tlie assumed constant value of s?n is never more than a very small fraction of Ifc. Tlie rail-laying is not done closer than this. The following tabular form is based on Eq. 30 : SUPERELEVATION OF THE OUTER RAIL (IN FEET) FOR VARIOUS VELOCI- TIES AND DEGREES OF CURVATURE. Velocity ill Miles per Hour. Degree of Curve. 1° o& 3° .15 .27 .43 4° .20 .37 5° 6° 7° 8° 9» 10° 30 40 50 60 .05 .09 .14 .20 .10 .18 .29 .41 .26 .46 .31 .36 .41 .46 1.51 i .55 .86 .64 .73 .82 1 .57 .82 .71 1 .62 42. Practical rules for superelevation. A much used rule for superelevation is to ''elevate one half an inch for each degree of curvature." The rule is rational in that e in Eq. 30 varies directly as I), The above rule therefore agrees with Eq. 30 when Fis about 27 miles per hour. However applica- ble the rule may have been in the days of low velocities, the elevation thus computed is too small now. Another (and better) rule is to "elevate for the speed of the fastest trains." This rule is further justified by the fact that a four-wheeled truck, having two parallel axles, will always tend to run to the outer rail and will require considerable flange pressure to guide it along the curve. The effect of an excess of superelevation on the slower trains will only be to relieve this flange pressure somewhat. This rule is coupled with the limita- tioirthat the elevation should never exceed a limit of six inches —sometimes eight inches. This limitation implies that locomo- tive engineers must reduce the speed of fast trains around sharp curves until the speed does not exceed that for which the actual superelevation used is suitable. The heavy line in tlie tabular form (§ 41) shows the six-inch limitation. 46 BAILTtOAD CONSTRUCTION, §48. Some roads furnish their track foremen with a Hst of the superelevations to be used on each curve in their sections. This method has the advantage that each location may be separately studied, and the proper velocity, as affected by local conditions {e.g.^ proximity to a stopping-place for all trains), may be determined and applied. Another method is to allow the foremen to determine the superelevation for each curve by a simple measurement taken at the curve. The rule is developed as follows : By an inversion of Eq. 19 we have X = chord' -^ 8B (31) Putting X equal to ^ in Eq. 30 and solving for ^'chord,^^ we have chord' = .0000572 V'DSE = 2.621 F\ chord = 1.62 F. . (32> To apply the rule, assume that 50 miles per hour is fixed as the velocity from which the superelevation is to be computed. Then 1.62 F== 1,62 X 50 = 81 feet, which is the distance given to the trackmen. Stretch a tape (or even a string) with a length of 81 feet between two points on the inside head of the outer rail or the outer head of the inner rail. The ordinate at the middle point then equals the superelevation. The values of this chord length for varying velocities are given in the accompanying tabular form. Velocity in miles per hour Chord length in feet •20 32.4 25 30 40.5 48.6 35 56.7 40 64.8 45 72.9 50 81.0 55 89.1 60 97.2 43. Transition from level to inclined track. On curves the track is inclined transversely; on tangents it is level. The transition from one condition to the other must be made gradu- § 45. ALIGNMENT. 47 ally. If there is no transition curve, there must be either in- clined track on the tangent or insufficiently inclined track on the curve or both. Sometimes the full superelevation is continued through the total length of the curve and the " run- oft" " (having a length of 100 to 200 feet) is located entirely on the tangents at each end. In other practice it is located partly on the tangent and partly on the curve. Whatever the method, the superelevation is correct at only one point of the run-olf. At all other points it is too great or too small. This (and other causes) produces objectionable lurches and resistances when entering and leaving curves. The object of transition curves is to obviate these resistances. 44. Fundamental principle of transition curves. If a curve has variable curvature, beginning at the tangent with a curve of infinite radius, and the curvature gradually sharpens uiitil it equals the curvature of the required simple curve and there becomes tangent to it, the superelevation of such a transition curve may begin at zero at the tangent, gradually increase to the required superelevation for the simple curve, and yet have at every point the superelevation required by the curvature at that point. Since in Eq. (30) e is directly proportional to />, the required curve must be one in which the degree of curve increases directly as the distance along the curve. The mathe- matical development of such a curve is quite complicated. It has, however, been developed, and tables have been computed for its use, by Prof. C. L. Crandall. The following method has the advantage of great simplicity, while its agreement w^itli the true transition curve is as close as need be, as will be shown. 45. Multiform compound curves. If the transition curve commences with a very flat curve and at regular even chord lengths compounds into a curve of sharper curvature until the desired curvature is reached, the increase in curvature at each chord point being uniform, it is plain that such a curve is a close approximation to the true spiral, especially since the rails as laid will gradually change their curvature rather than main- tain a uniform curvature throughout each chord loTi£rtli and 48 RAILROAD CONSTRUCTION. § 46. then abruptly change the curvature at the chord points. Such a curve, as actually laid^ will be a much closer approximation to the true curve than the multiform compound curve by which it is set out. There will actually be a gradual increase in curvature which increases directly as the length of the curve. 46. Required length of spiral. The required length of spiral evidently depends on the amount of superelevation to be gained, and also depends somewhat on the speed. If the spiral is laid off in 25-foot chord lengths, with the first chord subtend- ing a 1° curve, the second a 2° curve, etc., the fifth chord will subtend a 5° curve, and the increase from this last chord to a 6° curve is the same as the uniform increase of curvature between the chords. The same spiral extended would run on to a 12° curve in (12 - 1)25 = 275 feet. The last chord of a spiral should have a smaller degree of curvature than the simple curve to which it is joined. If the curves are very sharp, such as are used in street work and even in suburban trolley work, an increase in degree of curvature of 1° per 25 feet will not be sufticiently rapid, as such a rate would require too long curves. 2°, 10°, or even 20° increase per 25 feet may be necessary, but then the chords should be reduced to 5 feet. Such a rapid rate of increase is justified by the necessary reduction in speed. On the other hand, very high speed will make a lower rate of increase desirable, and therefore a spiral whose degree of curva- ture increases only 0° 30' per 25 feet may be used. Such a spiral would require a length of 375 feet to run on to an 8° curve, which is inconveniently long, but it might be used to run on to a 4° curve, where its length would be only 175 feet. Three spirals have been developed in Table lY, each with chords of 25 feet, the i*ate of increase in the degree of curvature being 0° 30', 1° and 2° per chord. One of these will be suitable for any curvature found on ordinary steam-railroads. 47. To find the ordinates of a l°-per-25-feet spiral. Since the first chord subtends a 1° curve, its central angle is 0° 15' and the angle aQY (Fig. 31) is 7' 30". The tangent at a makes an angle of 15' with VQ. The angle between the chord ha and §48. ALIGNMENT. 49 die tangent at a is J(30') ^ 15', and the angle hah"= 4(30') + 15' = 30'. Similarly the angle chc" = i(45') + 3U' + 15' ^ OT' 3U'' = 1° 07' 30", and the angle dcd" is 2° 0'. The ordinate aa = 25 sin 7' 30", and Qa = 25 cos 7' 30". Qh' = Qa! + aV ^ Qa: 4- aV = 25 (cos 7' 30" + cos 30'). hb' = W + ^^/' = 25 (sin 7' 30" + sin 30'). Similarly the ordinates of c, d, etc., may be obtahied. Fig. 31. FrG. 32. 48. To find the deflections from any point of the spiral. aQV=7' 30". Tan hQ V =^ hh' -^ QV ; tan cQV = cc' -^ Qc ; etc. Thus we are enabled to find the deflection angles from the tangent at Q to any point of the spiral. The tangent to the curve at c (Fig. 32) makes an angle of 1° 30' with Q V, or cm F = 1° 30'. Qcm = cm V - cQm. The 50 RAILROAD COI^STRUCTION. 48. value of cQm is known from previous work. The deflection from c to Q then becomes known. acm = cmV — cap = ciiiV — caq — qap, caq is the deflec- tion angle to c from the tangent at a and will have been previously computed numerically, qaj? = 15'. acm therefore becomes known. hcm^iofW = 22' 30"; den = ioi 60' = 30'. ecn = ecd"— ncd'\ ncd" = cmV^ tan ecd" = {ee'— d"d')-r- c'e\ all of which are known from the previous work. By this method the deflections from the tangent at any Fig. 33. point of the curve to any other point are determinable. These values are compiled in Table lY. The corresponding values of these angles when the increase in the degree of curvature per chord length is 30', and when it is 2°, are also given in Table lY. § 49. ALIGNMHNT. 51 49. Connection of spiral with circular curve and with tangent. See Fig. 3o.''' Let A V and I^ V be the tangents to be cunnected bj a D° curve, having a suitable spiral at each end. If no spirals were to be used, the problem would be solved as in simple curves giving the curve AMB. Introducing the spiral has the effect of throwing the curve away from the vertex a distance MM' and reducing the central angle of the D^ curve by 20. Continuing the curve beyond Z and Z' \o A' and B\ we will have AA' = BB' = MM'. ZK — the x ordinate and is therefore known. Call MM' — m. A'N =^ x — R vers 0. Then n.^^^, 4 4, ^'^ X — R vers ,^^^ m = MM' = AA' = —: == — , . . . (33) cos ^A cos "l"^ ^ ^ iTJL = AA ' sin l^ = (x — R vers 0) tan i^. VQ = QK- KN-\-NA + AV — ij — R sin + (,^; — 7? vers 0) tan \A ^ R tan \A — y — R sin + a; tan 4-^ + i^ cos tan \A, . (34) When A 'N has already been computed, it may be more con- venient to write VQ=zy^R (tan ^A - sin 0) + A'N tan ^A. (35) V2r ^ VM+MM' = i? exsec i^ A —. r^p. . . (oo) ^ ' cos ^A cos ^^ ^ ^ = ?/ — i? sin + (a? — ^ ^^ers 0) tan ^A. (37) Example. To join two tangents making an angle of 34° :2(>' by a 5° 40' curve and suitable spirals. Use l°-per-25-feet * The student should at once appreciate the fact of the necessary distor- tion of the figure. The distance MM' in Fig. 33 is perhaps 100 times its real proportional value. 52 RAILROAD CONSTRUCTION. §50. spirals with five chords. Then = 3° 45^ x = 2.999, ^J = 17° 10', and y = 124.942. (Eq. 33) B vers 312.471 AQ = 59.042 AF 3.00497 7.33063 2.166 0.33560 X = 2.999 A'N = 0.833 cos ^A 9.92064 9.98021 771 = = Mir = AA' =^ 0.872 9.94043 (Eq. 36) R 3.00497 exsec ^A 8.66863 V3f = : 47.164 1.67365 on = 0.872 35) y = = 124.942 V2r = nat. 48.036 = .30891 (Eq. tan Jz/ - nat. sin = = .06540 .24351 R 9.3865! 3.00497 246.314 [See above^ A'N- 2.39148 9.92064 tan ^A 9.48984 VQ = 0.257 Aj}^ 9.41048 = 371.513 (Eq. 37) R tan \A 3.00497 9.48984 2.49481 50. Field-work. When the spiral is designed during the original location, tlie tangent distance VQ should be computed and the point Q located. It is hardly necessary to locate all of the points of the spiral until the track is to be laid. The extremities should be located, and as there will usually be one and perhaps two full station points on the spiral, these should § 51. ALIGNMENT. 53 also be located. Z may be located by setting off QK — y and KZ ^= a', or else by the tabular deflection for Z from Q and the distance ZQ, which is the long chord. Setting up the instru- ment at Z and sighting back at Q with the proper deflection, the tangent at Z may be found and the circular curve located as usual, its central angle being ^ — 20. A snnilar operation will locate Q' from Z' . To locate points on the spiral. Set up at Q, with the plates reading 0" when the telescope sights along VQ. Set oft" from Q the deflections given in Table IV for the instrument at Q, usino" a chord length of 25 feet, the process being like the method for simple curves except that the deflections are irregu- lar. If a full station-point occurs within the spiral, interpolate between the deflections for the adjacent spiral- points. For ex- ample, a spiral begins at Sta. 5G + 15. Sta. 57 comes 10 feet beyond the third spiral point. The deflection for the third point is 35' 0"; for the fourth it is 56' 15". |f of the difference (21' 15") is 8' 30" ; the deflection for Sta. 57 is therefore 43' 30". This method is not theoretically accurate, but the error is small. Arriving at z, the forward alignment may be obtained by sight- ing back at Q (or at any other point) with the given deflection for that point from the station occupied. Then when the plates read 0° the telescope will be tangent to tlie spiral and to the succeeding curve. All rear points should be checked from z. If it is necessary to occupy an intermediate station, use the de- flections given for that station, orienting as just explained for 5, checking the back points and locating all forward points up to z if possible. After the center curve has been located and z' is reached, the other spiral must be located but in reverse order, i.e., the sharp curvature of the spiral is at z' and the curvature decreases toward Q'- 51. To replace a simple curve by a curve with spirals. This may be done by the method of § 49, but it involves shifting the whole track a distance m^ which in the given example equals 0.87 foot. Besides this the track is appreciably shortened. 54 RAILROAD CONSTRUCTION. §51. which would require rail-cutting. But the track may be kept at j^racticallj the same length and the lateral deviation from the old track may be made very small by slightly sharpening tlie curvature of the old track, moving the new curve so that it is wholly or partially outside of the old curve, the remainder of it witli the spirals being inside of the old curve. It is found by experience that a decrease in radius of from Ifo to 5^ will answer o Fig. 34. the purpose. The larger the central angle the less the change. The solution is as indicated in Fig. 34. O'V = (9'ir sec 1^ = ^ ' cos sec ^^ -\- X sec ^^, m = MM' = 2fV-M'V = Eex8eci^ -{O'V - B') = B exsec J^ — B' cos sec ^^ — x sec J^ + B'. (38) AQ = QK- KN'-\-.¥V- YA — y—B' sin + (^' cos + x) tan ^^ — B tan ^^ =y—B' sin

, = 8°; Z^^ =r 4° ; ^, = 36" and 4, = 32°. Use l°-per-25-feet spirals ; 0^ = 7° 0' ; 02 = 1° 30'. Assume that the sharper curve is sliarpened from 8° 0' to 8° 12'. §53. [Eq. 38] ALIGNMENT. 169.209 Ri' = 699.326 868.535 857.970 9.429 [Eq. 43] 215.974 nat. cos (p = .99966 nat. cos /1 2 — .84805 i?,' = 1424.54 [4°1'22'] [Eq. 39] 2/, = 174.722 85.226 504.302 679.024 600.461 515.235 600.461 ^Q, = 78.563 59 exsec 36° 2.85538 9.37303 2.22842 11.' cos 01 sec z/i 2.84408 9.99075 0.09204 2.93347 3*1 sec A I 0.88241 0.09204 0.97445 m\ 867.399 = 1.136 217.700 1.726 ^•2 867. 0. = 0, 1 399 963 .763 .726 vers 32° m, = 1.136 cos 32° 3.15615 9.18175 2.33785 0.05538 9.92842 9.98380. 2.33440 .15161 9.1807S 3.15307 sin 01 2.84468 9.08589 1.93057 cos 01 tan \AIA, = 36°] 2.84468 9.99675 9.86126 R, = 716.779 X, = 7.628 2 . 70269 709.151 tan \A 2.85074 9.86126 2.7120a 60 [Eq. 39] RAILROAD CONSTRUCTION. 2/a = 74.994 889.843 964.837 932.060 37 290 894.770 932.060 AQ-.^ 32.777 For the length of the old track we have 53. R,' sin 02 3.15367 8.41792 1.57159 R,' cos 0a tan iz/(J2 = 32°) 3.15367 9.99985 9.79579 2.94931 i?a = 1432.69 CCa = 0.76 1431.93 tan iz/ 3.15592 9.79579 2.95171 100 100 D, = "0 - = 450. ^2 i>2 = 100 -° = 800. AQ, = 78.563 AQ^ = 32.777 1361.340 For the length of the new track we have : 100^i^'^ = 100-J?l= 353.659 i>/ 8°.20 100^i^l^ = 10oS.= 758.140 A 4°. 023 Spiral on 8° 12' curve 175.000 " " 4° 01' 22" " 75. Length of new track = 1361.799 " old " = 1361.340 Excess in length of new track = 0.459 feet. § 55. ALIGNMENT. 61 Since the new track is slightly longer than the old, it shows that the new track runs too far outside tlie old track at the P.C.C. On the other hand the offset 7n is only 1.186. The maximum amount by which the new track comes inside of the old track at two points, presumably not far from Z' and Z, is vei-y dithcult to determine exactly. Since it is desirable that the maxinnini offsets (inside and outside) should be made as nearly equal as possible, this feature should not be sacrificed to an effort to make the two lines of precisely equal length so that the rails need not be cut. . Therefore, if it is found that the offsets inside the old track are nearly equal to m (1.136), the above figures should stand. Otherwise vi may be diminished (and the above excess in length of track diminished) by increasing R^ very slightly and making the necessary consequent changes. VERTICAL CURVES. 54. Necessity for their use. AYhenever there is a change in the rate of grade, it is necessary to eliminate the angle that Avould be formed at the point of change and to connect the two grades by a curve. This is especially necessary at a sag be- tween two grades, since the shock caused by abruptly forcing an upward motion to a rapidly moving heavy train is very severe both to the track and to the rolling stock. 55. Required length. Theoretically the length should de- pend on the change in the rate of grade, the greater change requiring a longer curve. The importance of this was greater in the days when link couplers were in universal use and the ^' slack " in a long train was very great. Under such circum- stances, when a train was moving down a heavy grade the cars would crowd ahead against the engine. Reaching the sag, the engine would begin to pull out, rapidly taking out the slack. Six inches of slack on each car would amount to several feet on a long train, and the resulting jerk on the couplers, especially those near the rear of the train, has frequently resulted in 62 RAILROAD CONSTRUCTION. § oQ. broken couplers or even derailments. A vertical curve will practically eliminate tins danger if the curve is made long enough, but the rapidly increasing adoption of close spring couplers and air-brakes, even for freight trains, is obviating the necessity for such very long curves. Two hundred feet may be considered sufficiently long for all ordinary changes of grade. Four hundred feet would probably suffice for the greatest change ever found in practice. 56. Form of curve. In Fig. 36 assume that A and C, equi- FiG. 36. distant from B^ are the extremities of the vertical curve. Bi- sect AG ^i e\ draw Be and bisect it at h. Bisect AB and BC at h and I. The line kl will pass through h. A parabola may be drawn with its vertex at h which will be tangent to AB and BC 2ii A and B. It may readily be shown from the proper- ties of a parabola that if an ordinate be drawn at any point (as at n) we will have sn : eh (or JiB) ; : Am : Ae , , Aori" or sn = eri—j~Y (44) Since the elevation of any point along AB or BO is readily determinable, the elevation of any point on the curve may be computed by adding the correction sn. 57. Numerical example. Assume that B is located at Sta. 16 + 20; that the curve is to be 200 feet long; that the grade of AB is — 0.8^, and of B0-{- 1.2^; also that the elevation of B above the datum plane is 162.6. Then the elevation of the various points is as follows: A, 163.4; 0, 163.8; 6, § 57. ALIGNMENT. 63 J(163.4 + 163.8) = 163.6; A, |-(168.0 + 1G2.G) = 103.1. Tlien eh=^ 0.5. The elevations of the points on the curve are: Sta. 15 + 20, {A) 1G3.4 '' 16 , 163.-1- (.80 X 0.8) + (.SO' X 0.5) = 163.08 " 17 , 162.6 + (.80 X 1.2) + (.20' X 0.5) = 163.58 u 17_|_20, {C) 163.8 A theoretical inaccuracy in tlie above method lies in the fact that eh and all parallel lines are not truly vertical. In the above case the variation from the vertical is 0° 07^, while tlie effect of this variation on the elevations in this case (as in the most extreme cases) is absolutely inappreciable. The grades in the figure are necessarily very greatly exaggerated, which increases the apparent inaccuracy. CHAPTER III, EARTHWORK. FORM OF EXCAVATIONS AN^D EMBANKMENTS. 58. Usual form of cross- section in cut or fill. The normal form of cross-section in cut is as sliown^^in Fig. 37, in which e . . . g represents the natural surface of the ground, no matter how irregular; ab represents the position and width of the re- e \ quired roadbed ; ac and hd represent the ' ' side slopes ' ' which begin at a and h and which intersect the natural surface at such d Fig. 38. points {g and d^ as will be determined by the required slope angle (^). 64 g 00. EARTHWORK. 65 The normal section in iill is as shown in Fig. 38. The points c and d are likewise determined by the intersection of the re- quired side slopes with the natural surface. In case the required roadbed {ah in Fig. 39) intersects the natural surface, both cut Fig. 39. and fill are required, and the points c and d are determined as before. Is^'ote that ^ and /?' are not necessarily equal. Their j)roper values will be discussed later. 59. Terminal pyramids and wedges. Fig. 40 illustrates the general form of cross-sections when there is a transition from cut to fill, a . . . g represents the grade line of the road which passes from cut to fill at d. sdt represents the surface profile. A cross-section taken at the point where either side of the road- bed first cuts the surface (the point m in this case) will usually be triano-ular if the ground is regular. A similar cross-section should be taken at o^ where the other side of the roadbed cuts the surface. In general the earthwork of cut and fill terminates in two pyramids. In Fig. -10 the pyramid vertices are at n and h^ and the bases are Ihrn and opq. The roadbed is generally wider in cut than in fill, and therefore the section Ihm and the altitude In are generally greater than tlie section oj^q and the altitude ^A:'. When the line of intersection of the roadbed and natural surface {nodkm) becomes perpendicular to the axis of the roadbed {ag) the pyramids become wedges whose bases are the nearest convenient cross- sections. 60. Slopes, a. Cuttings. The required slopes for cuttings vary from perpendicular cuts, which may be used in hard rock which will not disintegrate by exposure, to a slopeof perhaps 66 RAILROAD CONSTRUCTION. 60. 4 horizontal to 1 vertical in a soft material like quicksand or in a clayey soil winch flows easily when saturated. For earthy materials a slope of 1 : 1 is the maximum allowable, and even this should only be used for firm material not easily affected by Fig. 40. saturation. A slope of IJ horizontal to 1 vertical is a safer slope for average earthwork. It is a frequent blunder that slopes in cuts are made too steep, and it results in excessive work in clearing out from the ditches the material that slides down, at a much higher cost per yard than it would have cost to take it out at first, to say nothing of the danger of accidents from possible landslides. b. Embankments. The slopes of an embankment vary from 1 : 1 to 1.5 : 1 . A rock fill will stand at 1:1, and if some care is taken to form the larger pieces on the outside into a rough dry wall, a much steeper slope can be allowed. This method is sometimes a necessity in steep side-hill work. Earthwork em- bankments generally require a slope of 1 J to 1. If made steeper at first, it generally results in the edges giving way, re- quiring repairs until the ultimate slope is nearly or quite IJ- : 1. The difficulty of incorporating the added material with the old embankment and preventing its sliding off frequently makes these repairs disproportionately costly. § 62. EARTIIWOIiK. 67 61. Compound sections. AVlien the cut consists partly of earth and partly of rock, a compound cross-section must be Fig. 41. made. If borings have been made so that tlie contour of the rock surface is accurately known, then the true cross-section may be determined. The rock and earth should be calculated sepa- rately, and this will require an accurate knowledge of where the rock "runs out" — a difficult matter when it must be deter- mined by boring. During construction the center part of the earth cut would be taken out first and the cut widened until a sufficient width of rock surface had been exposed so that the rock cut would have its proper width and side slopes. Then the earth slopes could be cut down at the proper angle. A " berni " of about three feet is usually left on the edges of the rock cut as a margin of safety against a possible sliding of the earth sloj^es. After the work is done, the amount of excavation that has been made is readily computable, but accurate preliminary estimates are difficult. The area of the cross- section of earth in the figure must be determined by a method similar to that developed for borrow-pits (see § 89). 62. Width of roadbed. Owing to the large and often dis- proportionate addition to volume of cut or fill caused by the ad- dition of even one foot to the width of roadbed, there is a natural tendency to reduce the width until embankments become unsafe and cuts are too narrow for proper drainage. The cost of maintenance of roadbed is so largely dependent on the drain- age of the roadbed that there is true economy in making an 68 RAILROAD CONSTRUCTION, §63.. ample allowance for it. The practice of some of the leading railroads of the country in this respect is given in the following table, in which are also given some data belonging more properlj to the subject of superstructure. WIDTH OF ROADBED FOR SINGLE AND DOUBLE TRACK-SLOPE RATIOS- DISTANCES BETWEEN TRACK CENTERS. Road. A., T. & Santa Fe. ... Chi., Burl. & Quincy Chi., Mil. & St. Paul. C, C, C. &St. Louis Illinois Central. . — Erie ... Lehigh Valley L. S. & Michigan So. Louisville & Nashv. . Michigan Central. . . N. Y. N. H. &H.... Norfolk & Western... Pennsylvania -j Union Pacific Single Track. Cut. j 28' earth 1 2-i' rock 14 + (2 X.5) * 18 + (:i X 6) 20 + (2 X 4) 32.5 20' 81^" 14 + (2 X3.5) 13 + (2X4.5) 21' 2" earth 16' rock 19' 2" light traffic 27' 2" heavy " 14 -i- (2 X 3.5) Fill. 20 16 20 to 24 20 18 20' 81^" 16 16 17' 2' 19' 2" 19' 2" 16 Double Track. Cut. 28 + (2 X 5) 31 + (2 X 6) 33 + (2X4) 33' 81^" 27 + (2X3.5) 33 + (8X7.25; 33 + (2X2.5) 30 34' 2" earth 29' rock 31' 4"+(2 X 4) Fill. 30 33 to 37 33 33' 81^" 30 32 33 30 30' 2' 31' 4" Slope Ratios. Cut. 1 : Va 1.5 1.5 1.5 1.5 1.5 1 : 1.5 1 : 1.5 1.5 1 5 1.5: 1 1 : 1 Fill. 1.5: 1 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 I 1.5: 1 1.5 : 1 J? c .2£E-i 14' 13' 13' 13' 13' 13' 13' 12' 13' 13' 12' 2" * (2 X 5) signifies two ditches each 5 feet wide: the following cases should be interpreted similarly. It may be noted from the above table that the average width for an eartliworh cut, single track, is about 24.7 feet, with a minimum of 19 feet 2 inches. The widths of fills, single track, averasre over 18 feet, with numerous minimums of 16 feet. The widths for double track may be found by adding the distance- between track centers, which is usually 13 feet. 63. Form of subgrade. The stability of the roadbed depends largely on preventing the ballast and subsoil from becoming saturated with water. The ballast must be porous so that it will not retain water, and the subsoil must be so constructed that it will readily drain off the rain-water that soaks through the ballast. This is accomplished by giving the subsoil a curved form, convex § 64. EARTHWORK. 69 upward, or a surface made up of two or three planes, the two outer phines liaviiig a slope of about 1 : 21 (sometimes more and sometimes less, de23ending on the soil) and the middle plane, if three are used, being level. When a circular form is used, a crownini^ of 6 inches in a total width of 17 or 18 feet is iren- erally used. Occasionally the subgrade is made level, especially in rock-cuts, but if the subsoil is previously compressed by rolling, as required on the X. Y. C. cV: H. K. E. E., or if the subsoil is drained by tile drains laid underneath the ditches, the necessity for slopes is not so great. Eock cuts are generally required to be excavated to one foot below subgrade and then tilled up again to subgrade with the same material, if it is suit- able. 64. Ditches. ' ' The stability of the track depends upon the strength and permanence of the roadbed and structures upon wliich it rests ; whatever will protect them from damage or pre- vent premature decay should be carefully observed. The worst enemy is water, and the further it can be kept away from the track, or the sooner it can be diverted from it, the better the track will be protected. Cold is damaging only by reason of the water which it freezes ; therefore the first and most impor- tant provision for good track is drainage." (Eules of the Eoad Department, Illinois Central E. E.) The form of ditch generally prescribed has a flat bottom V2" to 24'' wide and with sides having a minimum slo])e, except in rock- work, of 1:1, more generally 1.5 : 1 and sometimes 2:1. Sometimes the ditches are made Y-shaped, which is objection- able unless the slopes are low. The best form is evidently that which will cause the greatest flow for a given slope, and this will evidently be the form in which the ratio of area to wetted perhneter is the largest. The semicircle ful- fills this condition better than any other form, but the nearly vertical sides would be difiicult to maintain. (See Fig. 42.) A ditch, Fig. 42. with a flat bottom and such slopes as tlie soil requires, which approximates to the circular form will therefore be the best. 70 RAILROAD COJSSTRUCTION. % Qb. When the flow will probably be large and at times rapid it will be advisable to pave the ditches with stone, especidly if the soil is easily washed away. Six-inch tile drains, placed 2^ under the ditches, are prescribed on some roads. (See Fig. 43.) l^o better method could be devised to insure a dry subsoil. The ditches through cuts should be led off at the end of the cut so that the adjacent embankment will not be injured. Wherever there is danger that the drainage from the land above a cut will drain down into the cut, a ditch should be made near the edge of the cut to intercept this drainage, and this ditch should be continued, and paved if necessary, to a point where the outflow will be harmless. I^eglect of these simple and inexpensive precautions frequently causes the soil to be loosened on the shoulders of the slopes during the progress of a heavy rain, and results in a landslide which will cost more to repair than the ditches which would have prevented it for all time. Ditches should be formed along the bases of embankments ; they facilitate the drainage of water from the embankment, and may prevent a costly slip and disintegration of the embankment. 65. Effect of sodding the slopes, etc. Engineers are unani- mously in favor of rounding off the shoulders and toes of embankments and slopes, sodding the slopes, paving the ditches, and providing tile drains for subsurface drainage, all to be put in during original construction. (See Fig. 43.) Some of the highest grade specifications call for the removal of the top layer of vegetable soil from cuts and from under proposed fills to some convenient place, from which it may be afterwards spread on the slopes, thus facilitating the formation of sod from grass- seed. But while engineers favor these measures and their economic value may be readily demonstrated, it is generally impossible to obtain the authorization of such specifications from railroad directors and 2)i*C)moters. The addition to the original cost of the roadbed is considerable, but is by no means as great as the capitalized value of the extra cost of mainte- nance resulting from the usual practice. Fig. 43 is a copy of ^e5. EAHTUWORK. 71 PROPOSED SECTION OF ROADBED IN EXCAVATION. CUSTOMARY SECTION OF ROADBED ON EMBANKMENT. GRAVELn PROPOSED SECTION OF ROADBED ON EMBANKMENT. GRAVEL, Fig. 43.— " VVhittemoke on Railway Excavation and Embankments,' Trans. Am. Soc. C. E., Sept. 1894 72 RAILROAD CONSTRUCTION. § 66. designs * presented at a convention of the American Society of Civil Engineers by Mr. D. J. Wliittemore, Past President of the Society and Chief Engineer of the Chi., Mil. tfe St. Paul R.R. The " customary sections " represent what is, with some variations of detail, the practice of many railroads. The '' pro- posed sections" elicited unanimous approval. They should be adopted when not prohibited by financial considerations. EAETHWOKK SURVEYS. 66. Relation of actual volume to the numerical result. It should be realized at the outset that the accuracy of the result of computations of the volume of any given mass of earthwork has but little relation to the accuracy of the mere numerical work. The process of obtaining the volume consists of two distinct parts. In the first place it is assumed that the volume of the earthwork may be represented by a more or less com- plicated geometrical form, and then, secondly, the volume of such a geometrical form is computed. A desire for simplicity (or a frank willingness to accept approximate results) will often cause the cross-section men to assume that the volume may be represented by a very simple geometrical form which is really only a very rough approximation to the true volume. In such a case, it is only a waste of time to compute the volume with minute numerical accuracy. One of the first lessons to be learned is that economy of time and effort requires that the accuracy of the numerical work should be kept proportional to the accuracy of the cross-sectioning work, and also that the accuracy of both should be proportional to the use to be made of the results. The subject is discussed further in § 94. 67. Prismoids. To compute the volume of earthwork, it is necessary to assume that it has some geometric form whose vol- ume is readily determinable. The general method is to consider * Trans. Am. Soc. Civil Eng., Sept. 1894. g 08. EAnrilWORK. 73 the volume as consisting of a series of jprisinoids, which are solids having parallel plane ends and bounded by surfaces which may be formed by lines moving continuously along the edges of the bases. These surfaces may also be considered as the sur- faces generated by lines moving along the edges joining the cor- responding points of the bases, these edges being the directrices, and the lines being always parallel to either base, which is a plane director. The surfaces thus developed may or may not be planes. The volume of such a prismoid is readily determi- nable (as explained in g 70 et seq.), while its definition is so very general that it may be applied to very rough ground. The " two ])lane ends " are sections perpendicular to the axis of the road. The roadbed and side slopes (also plane) form three of the side surfaces. The only approximation lies in the degree of accuracy with which the plane (or warped) surfaces coincide with the actual surface of the ground between these two sections. This accuracy will depend (a) on the number of points whicli are taken in each cross-section and the accuracy with which the lines joining these points coincide with the actual cross-sections ; (h) on the skill shown in selecting places for the cross-sections so that the warped surfaces shall coincide as nearly as possible with the surface of the ground. In fairly smooth country, cross- sections every 100 feet, placed at the even stations, are suf- ficiently accurate, and such a method simplifies the computations greatly ; but in rough country cross-sections must be inter- polated as the surface demands. As will be exj^lained later, carelessness or lack of judgment in cross-sectioning will introduce errors of such magnitude that all refinements in the computations are utterly wasted. 68. Cross-sectioning. The process of cross-sectioning con- sists in determining at any place the intersection by a vertical plane of the prism of earth lying between the roadbed, the side slopes, and the natural surface. The intersection with the road- bed and side slopes gives three straight lines. The intersection with the natural surface is in general an irregular line. On smooth regular ground or when approximate results are accc])t- 74 BAILED AD CONSTRUCTION. 68. able this line is assumed to be strais^ht. Accordins: to the irreo^- ularitj of the ground and the accuracy desired more and more " intermediate points " are taken. The distance {d in Fig. 44) of the roadbed below (or above) the natural surface at the center is known or determined from Fig. 44. the profile or by the computed establishment of the grade line. The distances out from the center of all ' ' breaks ' ' are determined with a tape. To determine the elevations for a cut, set up a level at any convenient point so that the line of sight is higher than any point of the cross-section, and take a rod reading on the center point. This rod reading added to d gives the height of the instrument (H. I.) above the roadbed. Subtracting from H. I. tlie rod reading at any ^' break " gives the height of that point above the roadbed (A,, hi, h^, etc.). This is true for all cases in excavation. For fill, the rod reading at center minus d equals the II. I., which may be positive or negative. When negative, add to the " H. I." the rod readings of the inter- mediate points to get their depths below " grade " ; when posi- tive, subtract the " H. I." from the rod readings. The heights or depths of these intermediate points above or below grade need only be taken to the nearest tenth of a foot, and the distances out from the center will frequenth^ be sufii- 69. EAHTUWORK. 75 ciently exact when taken to tlie nearest foot. Tlie roughness of the surface of farming land or woodland generally renders use- less any attempt to compute the volume with any greater accu- racy than these ligures would im])ly unless the form of the ridges and hollows is especially well delined. The position of the slope- stake points is considered in the next section. Additional dis- cussion regarding cross-sectioning is found in § 82. 69. Position of slope-stakes. The slope-stakes are set at the intersection of the required side slopes with the natural surface, which depends on the center cut or till ((/). The distance of 1>^ y^- >- _ ; rp >! 1 SI y : 1 . 1 Fig. 45. the slope-stake from the ceiiter for the lower side is ,r = \lj + s{d -|- y) ; for the up-hill side it is x = JZ* + s{d — y'). s is the " slope ratio " for the side slopes, the ratio of horizontal to vertical. In the above equation both x and y are unknown. Therefore some position must be found by trial which will sat- isfy the equation. As a preliminary, the value of x for the point a =z ^h -[- sd, which is the value of x for level cross- sections. In the case of fills on sloping ground the value of x on the doiim-Mll side is greater than this; on the up-liill side it is less. The difference in distance is s times the difference of elevation. Take a numerical case corresponding with Fig. 45. The rod reading on - ^'.)7 J- The volume of a section of infinitesimal lengtli will be AJx, and the total volume of the prismoia will be * 7'' X^ x'y * students unfamiliar with the Integral Calculus may take for granted the fundamental formula, th^t fdx = a, that fxdx = l^, and that fx^dx = ix- also that in integrating between the limits of I and (zero) the value of the integral may be found by simply substituting I for x after integration. '^'8 RAILROAD CONSTRUCTION. § 70. = ^[4^.^. + iU^ + K) + ^\{K + K) + iKh;] = - [J, + 4^,„ + ^J, (45) in which ^, , ^^ , and A,,, are the areas respectively of the two bases and of the middle section. IS'ote that A^ is not the mean of ^land ^., , although it does not necessarily differ very greatly from it. The above proof is absolntely independent of the values, ab- solute or relative, of 5^ , Z>, , h, or h^. For example, h^ may be zero and the second base reduces to a line and the prismoid be- comes wedge-shaped ; or l^ and h^ may both vanish, the second base becoming a point and the prismoid reduces to a pyramid Since every prismoid (as defined in § 67) may be reduced to a combination of triangular prismoids, wedges, and pyramids, and since the formula is true for any one of them individually, it is true for all collectively ; therefore it may be stated that ^ The volume of a prismoid equals one sixth of the perpendic- tdar distance hetween the hases multiplied hy the sum of the areas of the two hases plies four times the area of the middle section. While it is always possible to compute the volume of anv prismoid by tlie above method, it becomes an extremely compli- cated and tedious operation to compute the true value of tlie middle section if the end sections are complicated in form. It * The student should note that the derivation of equation (45) does not com- plete the proof, but that the statements in the following paragraph are logi- cally necessary for a general proof. § 72. EAHTUWOKK. 79 therefore becomes a simpler operation to compute volumes by ap- proximate formult>3 and apply, if necessary, a correction. The most common methods are as follows : 71. Averaging end areas. The volume of the triangular prismoid (Fig. -iB), computed by averaging end areas, is —[:J^i A, -[-J />,/',]. Subtracting this from the true volume (as given in the equation above, Eq. (45) ), we obtain tlie correction Y2^iJ\-h:){K-h,)-]. .... (46) Thi« shows that if either the A's or 5's are equal, the correc- tion vanishes ; it also shows that if the bases are roughly similar and h varies roughly with h (which usioally occurs, as will be seen later), the correction will be negative^ which means that the method of averaging end areas usually gives too large results. 72. Middle areas. Sometimes the middle area is computed and the volume is assumed to be equal to the length times the middle area. This will equal - X ' ' X - ' T" ' . Subtract- ing this from the true volume, we obtain the correction 24(^1 - ^.)(/^ - /O (^7) As l)efore, the form of the correction shows that if either the A's or ^'s are equal, the correction vanishes; also under the usual conditions, as before, the correction is positive and only one-half as large as by averaging end areas. Ordinarily the labor involved in the above method is no less than that of applying the exact prismoidal formula. 73. Two-level ground. When aj)j)roxir)iate computations of earthwork are sufficiently exact the field-work may be materi- ally reduced by observing simply the center cut (or fill) and the 80 RAILROAD CONSTRUCTION. §73. natural slope «, measured with a clinometer. The area of such a section (see Fig. 48) equals N % t£^ -J _-_(-r^ ^\,^ ^ ! 1 ^-^^^^^^^7 \d' \/ \ / . ^^k^^^;==^^^^^^^ r Fig. 47. Fig. 48. i{a-\-d){Xi-\-x,)- ab 2* But from which Similarly, Substituting, soi tan /3 = a -{- d ~\- Xj tan or, __ <^ + ^ ' tan /J — tan a a -\- d Xj~, — Area = {a -\- d) tan /? -[- tan a tan /? ^5 tan^ j3 — tan^ o' 2 * * . (48) The values a^ tan /?, tan" /? are constant for all sections, so that it requires but little work to find the area of any section. As this method of cross-sectioning implies considerable approxi- mation, it is generally a useless refinement to attempt to com- pute the volume with any greater accuracy than that obtained by averaging end areas. It may be noted that it may be easily proved that the correction to be applied is of the same form as that found in § 71 and equals ^[(«/+ <) - W + <')][(^"+ «) - {d'+ a)], § 74. EARTHWORK. 81 which reduces to ^ . M T/ , 7 N t^D /^ / , 7//V tan ^ Tr „/ „-. I b|L tun- /:^— tan* a tan'/^— tau'a J^ i When cZ" = d' the correction vanislies. This shows that when the center heights are equal there is no correction — regardless of the slope. If the slope is uniform throughout, the form of the correction is simplified and is invariably nega- tive. Under the usual conditions the correction is negative^ i.e., the method generally gives too large results. 74. Level sections. AVhen the country is very level or when only approximate preliminary results are required, it is some- times assumed that the cross-sections are level. The method of level sections is capable of easy and rapid convputation. The area may be written as {a + dp-^- (50) /7777mm77m777mm/J/777777777mm7777777g77mm7m Fig. 49. 1 This also follows from Eq. (48) when « = and tan /? = -. 5 here represents the " slope ratio," 2. 6., the ratio of the hori- zontal projection of the slope to the vertical. A table is very readily formed giving the area in square feet of a section of given depth and for any given width of roadbed and ratio of side-slopes. The area may also be readily determined (as illus- trated in the following example) without the use of such a table; a table of squares will facilitate the work. Assuming S2 RAILROAD CONSTRUCTION. § 75, t-he cross-sections at equal distances ^(= Z) apart, tlie total ap- proximate volume for any distance will be ^[^, + 2(^ + ^,+ ...^,^_j + .4j. . . (51) The prismoidal correction may be directly derived from Eq. (46) as :^\2{a + d')s - 2{a + d")s][{a + d") -{a-\- d')], which reduces to -^-^{d'-d-'r or -^l{d'-d")\ . (52) This may also be derived from Eq. (49), since a = tan a = 0^ and tan ^ = 2a ~ h. This correction is always negative, showing that the method of averaging end areas when the sections are level, always gives too large results. The prismoidal correction for any one prismoid is therefore a con- stant times the square of a difference. The squares are always positive whether the differences are positive or negative. The correction therefore becomes -I2^a^^'^'~'^"y- • • • . . (53) 75. Numerical example : level sections. Given the following center heights for the same number of consecutive stations 100 feet apart; width of roadbed 18 feet; slope IJ to 1. The products in the fifth column may be obtained very readily and with sufficient accuracy by the use of the slide-rule described in § 79. The products should be considered as {a -f d){a -f 6?) -f- -. In this problem s = 1^,- = .6667, ^ s To apply the rule to the first case above, place 6667 on scale jB over 89 on scale A, then opposite 89 on scale B will be found §76 EAHTUWOPdv. 83 118.8 on scale A. The position of the decimal point will be evident from an approximate mental solution of the prob- lem. 1 sta. Center HeiKlit. a + d (a + dV (a + d)2s 17 18 19 20 21 22 2.9 4.7 6.8 11.7 4.2 1.6 8.9 10.7 12.8 17.7 10.2 7.6 79.21 114.49 163.84 313.29 104.04 57.76 118.81 171.741 245.76 . 469.93 f 156.06 J 86.64 Areas. X2=^ 118.81 r 343.48 491.52 939.86 1^312.12 86.64 d' ~ d" ' 1 (d' ~ d")" 1.8 3.24 2.1 4.41 i 4.9 24.01 i 7.5 56.25 2.6 6.76 2 6_XJ8 2 = 54 1752.43x100 2292.43 10 X 54 = 540 1752.43 94.67 2X2 Corr. = — = 3245 cub. yards = approx. vol. 100 X 18 X 94.67 = — 91 cub. yds. 12X6X27 3245 — 91 = 3154 cub. yds. = exact volume. The above demonstration of the correction to be applied to the approximate volume, found by averaging end areas, is intro- duced mainly to give an idea of the amount of that correction. Absolutely level sections are practically unknown, and the error involved in assuming any given sections as truly level will ordinarily be greater than the computed correction. If greater accuracy is required, more points should be obtained in the cross-sectioning, which will generally show that tlie sections are not truly level. 76. Equivalent sections. When sections are very irregular the following method may be used, especially if great accuracy is not required. The sections are plotted to scale and then a uniform slope line is obtained by stretching a thread so that the undulations are averaged and an equivalent section is ol)tained. The center depth (d) and the slope angle {oi) of tliis line can be obtained from the drawing, but it is more convenient to measure the distances {xi and x^ from the center. The area 84 RAILROAD CONSTRUCTION. % 76 may then be obtained, independent of the center depth as follows : Let s — the slope ratio of the side slopes = cot fi =. — . (See Fig. 48.) Then the . JL /mS; "T~ "^r I / \ /■ T I I OjO (64) The true volume, according to the prismoidal formula, of a length of the road measured in this way will be I roo/xj ah . . foe/ + x," x^ + xj!' 1 ah\ . xl'xj!' ab-^ 6 'I ^r ah (xl -\- xl' x^ + V 1 ^^^\ , ocl'x^' If computed by averaging end areas, the approximate volume will be (j Xl Xy OjO Xl Xj, O'O Subtracting this result from the true volume, we obtain as the correction Correction = w-(.^/' — ocl){xJ — Xr'). . . (55) This shows that if the side distances to either the rip:ht or left are equal at adjacent stations the correction is zero, and also that if the difference is small the correction is also small and very probably within the limit of accuracy obtainable by that method of cross-sectioning. In fact, as has already been shown in the latter part of § 75, it will usually be a useless refinement to compute the prismoidal correction when the method of cross-sectioning is as rough and approximate as this method generally is. 77. EARTHWORK. S^ 77. Equivalent level sections. These sloping "two-level" sections are sometimes transformed into " level sections of equal area," a»d tli^ volume computed by the method of level sections (§ 74). But the true volume of a prismoid with sloping ends does not agree with tliat of a prismoid with equivalent bases and level ends except under special conditions, and wdien this method is used a correction nmst be applied if accuracy is desired, although, as intimated before, the assumption that the sections have uni- form slopes will frequently introduce greater inaccuracies than that of this method of computation. The following demonstra- tion is therefore given to show the scope and limitations of the errors involved in this much used method. In Fig. 50, let d^ be the center height which gives an Fig. 50. equivalent level section. The area will equal {a + 6?,)V — — , A QC X (to 1) which must equal the area given in § 76, -^ -r, 5 = .-r-. s "A '2(1 .-. (a-\- d,ys = vLitiU' I'^r or a -\- d^ = Vxix^ (.56) To obtain d^ directly from notes, given in terras of d and n', 86 RAILROAD CONSTRUCTION. §77. we may substitute the values of x^ and x^ given in § 73, which gives tan ^ _ a-\- d ' ^ ' '^ r tan /^ — tan a \\ -^ s tan^ «: The true volume of the equivalent section may be repre- sented by ![(« + ^^.')' + <'-^+ "-^) + (« + '^/o^; . From this there should be subtracted the volume of the ^' grade prism " under the roadbed to obtain the volume of the cut that would be actually excavated, but in the following com- parison, as well as in other similar comparisons elsewhere made, the volume of the grade prism invariably cancels out, and so for the sake of simplicity it will be disregarded. This expression for volume may be transposed to f'^ f — 1 The true volume of the prismoid with sloj^ing ends is (see 76^ [^+ 'ry,''ry, '' rp " rv. "\ OS = ^{ V^/^'- Vx/'x/y. (58) This shows that ^'equivalent level sections" do not in general give the true volume, there being an exception when ^ 78. EAUTUWOliK. 87 Xi^J' = x['Xr . This condition is fiillilled when the slo])e is uniforu], i.e., when a' = tx" . When this is nearly so the error is evidently not large. On the other hand, if the slopes are in- clined in opposite directions the error may be very considerable, particularly if the angles of slope are also large. 78. Three-level sections. The next method of cross-section- \\\^ in the order of complexity, and therefore in the order of wmiiijnmih Fig. 51. accuracy, is the method of three-level sections. The area of the section is i{a + d){Wr + w^) — -— , which may be written ah . \{(i -\- d)w — —^ ^ m which vj — \0r-\- v\. If the volume is computed by averaging end areas, it will equal I -\(a^d')io' -al-\-{a + cl'yD'' - ab\ . . (59) If we divide by 27 to reduce to cubic yards, we have, when I = 100, Yol (,...,,) = U{a + djw' - ^ah + ^a + d'^v" - ^al. For the next section Yol. (//*•• ///) — Tl ^(a + d'yo''- l\ah + f «(« + d'")io'" - ^ah. S8 RAILROAD CONSTRUCTWI^. § 78.. For a partial station length compute as usual and multiply - , leno^th in feet ^, . . , , result by — ^^.^ . ilie prismoidal correction mav be ^ 100 '^ obtained by applying Eq. (46) to each side in turn. For the left side we have r^[(« + d') — {a-\- d'^)']{w{^ — Wi)^ which equals ^id' - d"){iv{' - io{). For the right side we have, similarly, l-{d' - d"){wj' - <). The total correction therefore equals = i^{d' - d")iw" - to'). Reduced to cubic yards, and with I = 100, Pris. Corr. = ||-(^' - d'yw''-w'). . . (60) When this result is compared with that given in Eq. (55) there is an apparent inconsistency. If two-level ground is con- sidered as but a special case of three-level ground, it would seem as if the same laws should apply. If, in Eq. (55), x/ = Xr'\ and x/^ is different from x/, the equation reduces to zero ; but in this case d' would also be different from d" ; and since x/ -f- x/ would = io\ and xC + x^' = w" in Eq. (60), V3" — %o' would not equal zero and the correction would be some finite quantity and not zero. The explanation lies in the difference in the form and volume of the prismoids, according to the method of the §78. EARTHWORK. 89 formation of tlie warped surfaces. If the surface is supposed to be generated bj the locus of a Hue moving parallel to the ends as plane directors and along two straight lines lying in the side- slopes, then ,T;""^- will ecpial ^(a?/ + a?/'), and it',.""'^- will ecpial J(x/ -|- a?/'), but the protile of the center line will not be straight and t^"""*- will not equal \{(l' + d"). On the other hand, if the surfaces be generated by tioo lines moving parallel to the ends as plane directors and along a straight center Hne and straight side lines lying in the slopes, a warped surface will be generated each side of the center line, which will have uni- form slopes on each side of the center at the two ends and no- where else. This shows that when the upper surface of earth- work is warped (as it generally is), two-level ground should not be considered as a special case of three-level ground. This dis- cussion, however, is only valuable to explain an apparent incon- sistency and error. The method of two-level ground should only be nsed when such refinements as are here discussed are of no importance as affecting the accuracy. The following example is given to illustrate the method of three-level sections. S, m 17 18 +40 19 20 •J. C O a + d w Yards. d' - d" -5.5 -2.6 +4.3 +2.7 w"—w' +11.7 + 8.7 -13.4 -15.1 at ^6 -20 - 3 -11 -13 14.7 18.6 23.1 17.9 8.4 ^(■^rv .* > c 5 3 +4 +4 +5 +3 3A' 8.1F 10.7^ 6.4F Z.IF 10.6F 0.8F 7.3 12.8 15.4 11.1 8.4 31.1 42.8 51.5 38.1 23.0 210 507 734 392 179 595 448 602 449 +1 +3 +6 +2 +1 2-,'. 9 1.5.SF 8.2 3.4F 30.7 •20.2F 37.3 14.0F 12.1 4.8F 14.2 2AF 28.0 5.8 F 10.1 0.2F 15.7 7.3 Roadbed, 14' wide in fill. Approx. Vol, =2094 Slope IJ^ to 1. Pris. corr. = 47 ■47 ^ = iT~ = ~5~ = 4. ( ; 25^ 27 ah = 61. True Vol. =2047 (disregarding curv. corr).* * For the Derivation of the curvation correction, see § 93. +16 90 RAILROAD CONSTRUCTION. § 79. In the first column of yards 210 = ff(« + d)iv = f 5 X 7.3 X 31.1 ; 507, 734, etc., are found similarly ; 595 = 210 - 61 + 507 - 61; 448 = yVo(507 - 61 + 734 - 61); 602 = -j-Vo('^34 - 61 + 392 - 61) ; 449 = 392 - 61 + 179 - 61. For the prismoidal correction, - 20 = l{{cr - fr'){io" - w') = |f(2.6 - 8.1)(42.8 - 31.1) = fK-5-5)(+ii.7). For the next Hne, - 3 = -/^o_[|5(_ 2.8)(+ 8.7)], and similarly for the rest. The "7^" in the columns of center heights, as well as in the columns of " right " and '^ left," are inserted to indicate Jill for all those pooints. Cut would be indicated by " (7." 79. Computation of products. The quantities ~{a-\-d)w 25 and —ah represent in each case the product of two variable terms and a constant. These products are sometimes obtained from tables which are calculated for all ordinary ranges of the variable terms as arguments. A similar table computed for 25 — (d^ — d"){io" —- lo') wiir assist similarly in computing the ol prismoidal correction. Prof. Charles L. Crandall, of Cornell University, is believed to be the first to prepare such a set of tables, which were first published in 1886 in "Tables for the Computation of Railway and Other Earthwork." Another § 79. EARTHWORK. 91 easy method of obtaining tliese products is by the nse of a slido rule. A slide-rule has been designed by the autlior to accom- pany this volume. It is designed particularly for this special work, although it may be utilized for many other purposes for which slide-rules are valuable. To illustrate its use, suppose {ic + d) ^ 28.2, and 2^ = 62.4; then 25, , -,, 28.2 X 62.4 ~—{a -f- a)iv = . 27^ ^ 1.08 Set 108 (which, being a constant of frequent use, is specially marked) on the sliding scale {B) opposite 282 on the other scale (A), and then opposite 624 on scale J] will be found 1629 on scale A, the 162 being read directly and the 9 read by estima- tion. Although strict rules may be followed for pointing off the final result, it only requires a very simple mental calculation to know that the result must be 1629 rather than 162.9 or 16290. For products less than 1000 cubic yards the result may be read directly from the scale; for products between 1000 and 5000 the result may be read directly to the nearest 10 yards, and the tenths of a division estimated. Between 5000 and 10,000 yards the result may be read directly to the nearest 20 yards, and tlie fraction estimated; but prisms of such volume will never be found as simple triangular prisms — at least, an as- sumption that any mass of ground was as regular as this would probably involve more error than would occur from faulty esti- mation of fractional parts. Facilities for reading as high as 10,000 cubic yards would not have been put on the scale ex- cept for tlie necessity of finding such products as |7(^^^-1 X 9.5), for example. This product would be read off from the same part of the rule as f |(91 X 95). In the first case the ])roduct (80.0) could be read directly to the nearest .2 of a cubic yard, which is unnecessarily accurate. In the other case, the jirod- uct (8004) could only be obtained by estimating -^^^ of a division. The computation for the prismoidal correction may be made 92 MAILROAD CONSTBUCriON, § 80. similarly except that the divisor is 3.21 instead of 1.08. For example, |f(o.5 X 11. T) = ^liAliiJ. get the 324 on scale B (also specially marked like 108) opjDOsite 55 on scale A^ and proceed as before. 80. Five-level sections. Sometimes the elevations over each edge of the roadbed are observed when cross-sectioning. These are distinctively termed " five level sections." If the center, the slope-stakes, and one intermediate point on each side [not necessarily over the edge of the roadbed) are observed, it i-s termed an " irregular section." The field-work of cross-section- ing five-level sections is no less than for irregular sections with one intermediate point; the computations, although capable of peculiar treatment on account of the location of the intermediate point, are no easier, and in some respects more laborious; the cross-sections obtained will not in general represent the actual cross-sections as truly as when there is perfect freedom in locat- ing the intermediate point ; as it is generally inadvisable or un- necessary to employ five-level sections throughout the length of a road, the change from one method to another adds a possible element of inaccuracy and loses the advaniage of uniformity of method, particularly in the notes and form of computations. On these accounts the method will not be further developed, except to note that this case, as well as any other, may be solved by dividing the whole prismoid into triangular prismoids, computing the volume by averaging end areas, and computing the prismoidal correction by adding the comjDuted corrections for each elementary triangular prismoid. 81. Irregular sections. In cross-sectioning irregular sec- tions, the distance from the center and the elevation above ''grade" of every "break" in the cross-section must be observed. The area of the irregular section may be obtained by computing the area of the trapezoids ^fi've^ in Fig. 44) and subtracting the two external triangles. For Fig. 44 the area would be §81. hi + h EARTHWORK. 93 {^i - yi) + h + d d +jr , jr + ^' -y/+ ^r + \yr — 2r) + -v+i,, _,,_».(,_ I) -|.(,_ I). ^rh ^. Fig. 44. Expanding this and collecting terms, of which many will cancel, we obtain Area = - ^Xiki + yi(d - hi) + x^ + yAjr - K) Jr2M-h)+\(hi + K)\. . . (61) An examination of this formula will show a perfect regu- larity in its formation which will enable one to write out a similar formula for any section, no matter how irregular or how many points there are, without any of the preliminary work. The formula may be expressed in words as follows : Area equals one-half the sum of products obtained as folloics : the distance to each slope- stake times the height above grade of the point next inside the slope-stake y the distance to each intermediate point in turn times the height of the point just inside minus the height of the point just outside / finally^ one-half the width of the roadbed times the sum of the slope-stake heights. 94 BAILROAD CONSTRUCTION. 82. If one of the sides is perfectly regular from center to slope- stake, it is easy to show that the rule holds literally good. The ' ' point next inside the slope-stake ' ' in this case is the center; the intermediate terms for that side vanish. The last term must always be used. The rule holds good for three -level sections, in which case there are three terms, which may be reduced to two. Since these two terms are both variable quan- tities for each cross-section, the special method, given in § 78, in which one term (-^J is a constant for all sections, is pref- erable. In the general method, each intermediate "break" adds another term. 82. Volume of an irregular prismoid. If there is a break at one cross-section which is not represented at the next, the ridge (or hollow) implied by that break is supposed to ' ' vanish ' ' at the next section. In fact, the volume will not be correctly Fig. 52. represented unless a cross- section is taken at the point where the ridge or hollow "vanishes" or "runs out." To obtain the true prism oidal correction it is necessary to observe on the ground the place where a break in an adjacent section, which is not represented in the section being taken, runs out. For example, in Fig. 52, the break on the left of section A" . at a g S3. EARTHWORK. 95 distance of y/'from the center, is observed to run out in section A' at a distance of yi from the center. The vohime of the prismoid, computed by the prismoidal fornmkx as in § 70, will involve the midsection, to obtain the dimension of which will require a hiborious computation. A simpler process is to compute the volume by averaging end areas as in § 81 and apply a prismoidal correction. To do this write out an expression for each end area similar to that given in Eq. 61. The sum of these areas times -r gives the approximate volume. As before, - . - . - - 1 . -1 1 11 length in feet lor partial station lengths, multiply the result by — ^7;?; • There will be no constant subtractive term, f f«^, as in § 78. The true prismoidal correction may be computed, as in § 83, or the following approximate method may be used : Consider tlie irregular section to be three-level ground for the purpose of computing the correction only. This has the advantage of less labor in computation than the use of the true prismoidal correc- tion, and although the error involved may be considerable in individual sections, the error is as likely to be positive as nega- tive, and in the long run the error will not be large and generally will be much less than would result by the neglect of any prismoidal correction. 83. True prismoidal correction for irregular prismoids. As intimated in § 82, each cross-section should be assumed to have the same number of sides as the adjacent cross-section when computing the prismoidal correction. This being done, it per- mits the division of the whole prismoid into elementary triangu- lar prismoids, the dimensions of the bases of which being given in each case by a vertical distance above grade line and l)y the horizontal distance between two adjacent breaks. Tlie summa- tion of the prismoidal corrections for each of the elementary triangular prismoids will give the true prismoidal correction. Assuming for an example the cross-section of Fig. 44, witli a cross-section of the same number of sides, and witli dimensions 96 RAILROAD CONSTRUCTION. §83. similarly indicated, for the other end, the prismoidal correction becomes (see Eq. 46) l4 (^^/- V )[{xi" - yi") - {xi - 2/01 + {ki' - ki")[{xi" - yi") - (xi' - yn] + (ki' - ki"){yr - y{) + {d' - cV){y{' - yi') + {d! - d"){Zr" - Zr') + {>' - >'0(2r" - Zr!) + OV -jr")\_{yr" - Zr") - (^r' - s/)l + {kr' - kr")[{yr" - Zr") - {yr' - 2r')l + (A;/-V')[(^V''-2/r'')--(3V'-2/r')l + (Ar'-/^r'')[(2;r''-yr'')-CV-yr')] Expanding this and collecting terms, of which many will cancel, we obtain Pris. Corr. = -^^xf -x{\k{ - h") + {yi" - yi')[{d' - hi') - {d" - Jh")] + iXr" - Xr'){kr' - kr") + {yr" - yr')[{jr' - hr) - {jr" - h/')] + (Zr" -er')[{d' -kr')-{d" -kr")]] (62) By comparing this equation with Eq. 61 a remarkable coincidence in the law of formation may be seen, which enables this formula to be written by mere inspection and to be applied numerically with a minim'um of labor from the computations for end areas, as will be shown (§ 84) by a numerical example. For each term in Eq. 61, as, for example, yXjr — /^r)? there is a correction term in Eq. 62 of the form ivr" - 2//)[ov' - V) - ijr" - wn ■ Each one of these terms (^z/', 1/r\ (J/ — h/), and (J/' — /?/') ) has been previously used in finding the end areas and has its place in the computation sheet. The summation of the products of these differences times a constant gives the total true pris- moidal correction in cubic yards for the whole prismoid considered. The constant is the same as that computed in § 78, i.e., ||^. §84. EAHTHWORK. 97 84. Numerical example ; irregular sections ; volume^ with true prismoidal correction. Sta. 19 18 17 + 43 16 \ cut Ceiiter-s <>r fill. 0.6c 2.3c ,6c 10.3c 6.8c Left. 3.6c 1^174 4 .2c 1573 8.2c 3l73 ]2.2c 27.3 8.9c 3274 ^8.2/ \6.0/ 6^c sTi l(h2c TtTI" |12.3c\ \2270/ 3^2 c y.2 8.0c 6.1 12.6c 8.2 7.6c 12.0 Right. 0.1c 472 /1^9cV V3.6/ |5^c| \8.0/ 6.2c 775" 3.2c 0.4c 9:6" 1.2c U) 8 4.2c 1573 8.4c 2176 2.6c 12.9 Koadbed IS feet wide in cut; slope 1|^ to 1. The figures in the bracket \-^^-7v j mean that it was noted in the field that the break, indicated at Sta. 17 as being 17.4 to tlie left, ran out at Sta. 10 + 42 at 22.0 to the left. By inter- polation between 8.2 and 27.3 the height of this ])oint is ur' u tfa 100 RAILROAD CONSTRUCTION. 87. poses of the correction only\ the error for tlie different sections is sometimes positive and sometimes negative, and in tins case Sections. 6 s 3 O > 0) 3 U H Approx. vol. by averaging end areas. Difference or true pris. corr. Approx. pris. corr. on basis of three-level ground. Error. Approx. vol., computed from center and side heights onhj. Error. 16 16 + 42 16 + 42. ..17 17 18 18 19 373 581 174 378 584 528 177 1667 - 5 - 3 - 16 - 3 - 6 - 6 - 17 - 1 - 1 - 3 - 1 + 2 - 3 396 577 463 147 + 23 - 4 - 49 - 27 - 57 1640 - 27 - 30 15S3 amounts to only 3 yards in 1640 — less than \ of 1^. If the prismoidal correction had been neglected, the error would have been 27 yards — nearly li. The approximate results are here too large for each section — as is usually the case. If points between the center and slope stakes are omitted and the volume computed as if the ground were three-level ground, the error is quite large in individual sections, but the errors are both posi- tive and negative and therefore compensating. 87. Cross-sectioning irregular sections. The prismoids con- sidered have straight lines joining corresponding points in the two cross-sections. The center line must be straight between tAvo cross-sections. If a ridge or valley is found lying diago- nally across the roadbed, a cross-section m^icst be interpolated at the lowest (or highest) point of the profile. Therefore a ' ' break ' * at any section cannot be said to run out at the other section on the opposite side of the center. It must run out on the same side of the center or possibly at the center. Yery frequently complicated cross-sectioning may be avoided by computing the volume, by some special method, of a mound or hollow when the ground is comparatively regular except for the irregularity referred to. 88. Side-hill work. When the natural slope cuts the roadbed there is a necessity for both cut and fill at the same cross-section. When this occurs the cross-sections of both cut and fill are often so nearly triangular that they may be considered as such without 88. EARTUWORK, 101 great error, and the volumes inaj be computed separately as triangular prismoids without adopting tlie more elaborate form of computation so necessary for complicated irregular sections. AVhen the ground is too irregular for this the best plan is to follow the uniform system. In computing the cut, as in Fig. 53, Fia. 53. the left side would be as usual ; there would be a small center cut and an ordinate of zero at a short distance to the right of the center. Then, ignoring the fill ^ and applying Eq. 61 strictly, we have two terms for the left side, one for the right, and the term involving \h^ which will be ^lii in this case, since li^ = 0, and the equation becomes Area = \\xihi -f- yifl — ^i) + x^d + ^Jfhj']. The area for fill may also be computed by a strict application Fig. 54. of Eq. 61, but for Fig. 54 all distances for the left side are zero and tlie elevation for the first point out is zero, d also must be 102 BAILROAD CONSTRUCTION. § 89. considered as zero. Following the rule, § 81, litexallj, the equation becomes Area(Fin) = ^[x^h + yr{o — K) + z,.{o — h) + iHo + A,-)]? which reduces to {Note that Xy^ Kj etc., have different significations and values in this and in the preceding paragraphs.) The "terminal pyramids ' ' illustrated in Fig. 40 are instances of side-hill work for very short distances. Since side-hill work always implies hoth cut and fill at the same cross-section, whenever either the cut or fill disappears and the earthwork becomes wholly cut or wholly fill, that point marks the end of the ' ' side-hill work, ' ' and a cross-section should be taken at this point. 89. Borrow-pits. The cross-sections of borrow-pits will vary not only on account of the undulations of the surface of the 'JillHlllllllllllUIIIHNIIIillllllllllWlllWllinillllllllliilllllUIIIWIUIIWIIili' Fig. 55. ground, but also on the sides, according to whether they are made by widening a convenient cut (as illustrated in Fig. 55) or simply by digging a pit. The sides should always be prop- erly sloped and the cutting made cleanly, so as to avoid un- sightly roughness. If the slope ratio on the right-hand side (Fig. 55) is ,9, the area of the triangle is ^sm^. The area of the section is 2^119 -\-{g-\-h)v-\-{h-\-j)x-^{j-\-li')y-\-{k-\-m)z — smj''\. If all the horizontal measurements were referred to one side as an origin, a formula similar to Eq. 61 could readily be devel- oped, but little or no advantage would be gained on account of any simplicity of computation. Since the exact volume of the €arth borrowed is frequently necessary, the prismoidal correc- § 90. EARTHWORK. 103 tiou sliould be computed ; and since such a section as Yir ~ 3 xi + X, - 3"^''^" ^^)- (^^) 2 "^ 2 The side toward x^. being considered positive in the above demonstration, if x^, > Xi^ e would be negative, i.e., the center of gravity would be on the left side. Therefore, for three-level ground, the correction for curvature (see Eq. 64) may be written Correction = ^[A'(x/ — cc/) + A"{x/' — x/')]. Since the approximate volume of the prismoid is ^{A + A') = I A' + \a" = r + F", in which V^ and V represent the number of cubic yards corresponding to the area at each station, we may write Corr. in cuh. yds. = ^^ V\xj '- .t/)+ F"(-^/"- ^'")]- (^^<^> 106 RAILROAD CONSTRUCTION. § 91. It should be noted that the value of €, derived in Eq. 65, is the eccentricity of the whole area including the triangle under the roadbed. The eccentricity of the true area is greater than this and equals true area -f- i^t^^ true area e X : -^— = e,. The required quantity {A'e of Eq. 64) equals true area X ^n which equals {true area -\- ^ah) X e. Since the value of e is very simple, while the value of e^ would, in general, be a complex quantity, it is easier to use the simple value of Eq. Q^ and add ^ah to the area. Therefore, in the case of three-level ground the subtractive term ^^ah (§ 78) should not be subtracted in computing this correction. For irregular ground, when com- puted by the method given in §§ 81 and 82, which does not involve the grade triangle, a term f |<^5 must be added at every station when computing the quantities V and V" for Eq. QQ. It should be noted that the factor 1 ~- Sic, which is constant for the length of the curve, may be computed with all necessary accuracy and without resorting to tables by remember- ing that deojree of curve Since it is useless to attempt the computation of railroad earthwork closer than the nearest cubic yard, it will frequently be possible to write out all curvature corrections by a simple mental process upon a mere inspection of the computation sheet. Eq. QQ shows that the correction for each station is of the form — P — —. 3i? is generally a large quantity — for a 6° curve Sic it is 2865. {xi — x^) is generally small. It may frequently be seen by inspection that the product Y{Xi — x,) is roughly twice or three times 3^, or perhaps less than half of 3^, so that the corrective term for that station may be written 2, 3, or cul)ic yards, the fraction being disregarded. For much larger absolute §92. EARIHWORK. 1()7 amounts the correction must be computed with a correspondingly closer percentage of accuracy. The algebraic sign of the curvature correction is best deter- mined by noting that the center of gravity of the cross-section is on the riglit or left side of tlie center according as x,. is greater or less than xi^ and that the correction is positive if the center of gravity is on the outside of the cur\'e, and iiegative if on the inside. It is frequently found that Xi is uniformly greater (or uni- formly less) than x,. throughout the length of the curve. Then the curvature correction for each station is uniformly positive or negative. But in irregular ground the center of gravity is apt to be irregularly on the outside or on the inside of the curve, and the curvature correction will be correspondingly positive or negative. If the curve is to the right, the correction will be positive or negative according as {xi — x,.) is positive or negative ; if the curve is to the left, the correction ^\'ill be positive or nega- tive according as {Xy — xi) is positive or negative. Therefore when computing curves to the 7'ight use the form {xi — x,>^ in Eqs. 66 and Q^ ; wdien computing curves to the left use the form {x^ — x^ in these equations ; the algebraic sign of the correction will then be strictly in accordance with the results thus obtained. 92. Center of gravity of side-hill sections. In computing the Fi(.. 57. correction for side-hill work the cross section would be treated as trianirular unless the error involved would evidentlv be too 108 RAILROAD CONSTRUCTION. 92, great to be disregarded. Tlie center of gravity of the triangle lies on the line joining the vertex with the middle of the base and at ^ of the length of this line from the base. It is therefore equal to the distance from the center to the foot of this line plus ^ of its horizontal projection. Therefore e = 2 ~ 2 \2 + ^' + 1 r Xl 2 ~2 \2+^' •JCy Xl t-O'V - 4 ~ 2 +^ h Xl 3 3 Xy 3 -^ + {xi-x,)j. 12+6 n (67) Bj the same process as that used in § 91 the correction equation may be written Corr. in cub. yds. = .IPf^I + (^/ - a-/)) + ^"^2'^^'^'" ~ ""'"^ (68) It should be noted that since the grade triangle is not used in this computation the volume of the grade prism is 7wt involved in computing the quantities V and V" . The eccentricities of cross-sections in side-hill work are never zero, and are frequently quite large. The total volume is generally quite small. It follows that the correction for curvature is generally a vastly larger proportion of the total volume than in ordinary three-level or irregular sections. If the triangle is wholly to one side of the center, Eq. 67 can still be used. For example, to compute the eccentricity of the triangle of fill, Fig. 57, denote the two distances to the slope-stakes by y,. and — yi (note the minus sign). Applying Eq. 67 literally (noting that -^ must here be considered as nega- tive in order to make the notation consistent) we obtain 1 e = 3 - ^+(- 2/^- Vr)]' § 94. EABTIlWOliK. 109 which reduces to in n ^ = -^[j + yi + f/'j (^>'^) As the algebraic signs tend to create confusion in tliese forinula3, it is more simple to remember that for a triangle lying on hoth sides of the center e is always numerically c(|ual lYh n to ^ - -{- {xi'^ x^) , and for a triangle entirely on one side, e is -j- the numerical su/n of the two dis- numerically equal to — 3 L2 tances out]. The algebraic sign of e is readily determinable as in § 91. 93. Example of curvature correction. Assume that the fill in § 78 occurred on a 6° curve to the ri(/ht. ^-jj = . The quantities 210, 507, etc., represent the quantities V\ V'\ etc., since they include in each case the 61 cubic yards due to the grade prism. Then V(xi — Xr) 210(22.9 - 8.2) _ 3101.7 _ 3i? - 2865 2865 The sign is plus since the center of gravity of the cross-sec- tion is on the left side of the center and the road curves to the right, thus making the true volume larger. For Sta. 18 the correction, computed similarly, is -f- 3, and the correction for the whole section is 1 -f- 3 = 4. For Sta. 18 -|- 40 the cor- rection is computed as 6 yards. Therefore, for the 40 feet, the correction is y\%-(3 -[- 6) = 3.6, which is called 4. Computing the others similarly we obtain a total correction of -\- 16 cubic yards. 94. Accuracy of earthwork computations. The preceding methods give the precise vohc7ne (except where approximations are distinctly admitted) of the prismoids which are sujij^osed to represent the volume of the earthwork. To appreciate the accuracy necessary in cross-sectioning to obtain a given accuracy 110 JiAIfAlO AD CONSTRUCTION. §94. in volume, consider that a fifteen-foot length of the cross-section, which is assumed to be straight, really sags 0.1 foot, so that the cross-section is in error by a triangle 15 feet wide and 0.1 foot high. This sag 0.1 foot high would hardly be detected by the eye, but in a length of 100 feet in each direction it would make an error of volume of 1.4 cubic yards in each of the two pris- moids, assuming that the sections at the other ends were perfect. If the cross-sections at both ends of a prismoid were in error by this same amount, the volume of that prismoid would be in error by 2.8 cubic yards if the errors of area were both plus or both minus. If one were plus and one minus, the errors would neutralize each other, and it is the compensating character of these errors which permits any confidence in the results as obtained by the usual methods of cross-sectioning. It demon- strates the utter futility of attempting any closer accuracy than the nearest cubic yard. It will thus be seen that if an error really exists at any cross-section it involves the prismoids on hoth sides of the section, even though all the other cross-sections are perfect. As a further illustration, suppose that cross-sec- tions were taken by the method of slope angle and center depth (§ 73), and that a cross-section, assumed as uniform, sags 0.4 foot in a width of 20 feet. Assume an equal error (of same sign) at the other end of a 100-foot section. The error of volume for that one prismoid is 38 cubic yards. The computations further assume that the warped surface, passing through the end sections, coincides with the surface of the ground. Suppose that the cross-sectioning had been done with mathematical perfection ; and, to assume a simple case, suppose a sag of 0.5 foot between the sections, which causes an error equal to the volume of a pyramid having a base of 20 feet (in each cross-section) times 100 feet (between the cross-sections) and a height of 0.5 foot. The volume of this pyramid is i(20 X 100) X 0.5 z= 333 cub. ft. = 12 cub. yds. And yet this sag or hump of 6 inches would generally be utterly un- noticed, or at least disregarded. When the ground is very rough and broken it is sometimes §96. eartuwohk. Ill practically impossible, even with tVeqiient oross-sections, to locate warped surfaces which will closely coincides with all the sudden irregularities of the ground. In such cases the compu- tations are necessarily more or less approximate and dependence must be placed on the compensating character of the errors. 95. Approximate computations from profiles. As a means of comparing the relative amounts of earthwork on two or more proposed routes which have been surveyed by preliminary surveys, it will usually be sufficiently accurate to com})are the areas of cutting (assuming that the cut and till are approximately balanced) as shown by the several profiles. The errors involved may be large in individual cases and for certain small sections, but fortunately the errors (in comparing two lines) will be largely compensated. The errors are nmch larger on side-hill work than when the cross-sections are comparatively level. The errors become large when the depth of cut or fill is very great. If the lines compared have the same general character as to the slope of the cross-sections, the proportion of side-liill work, and the average depth of cut or iill, the error involved in considering their relative volumes of cutting to be as the relative areas of cutting on the profiles (obtained perhaps by a planim- eter) will probably be small. If the volume in each case is computed by assuming the sections as level^ with a depth espial to the center cut, the error involved will depend only on the amount of side-hill work and the degree of the slope. if these features are about the same on the two lines compared, the error involved is still less. FORMATION OF EMUANKMKNTS. 96. Shrinkage of earthwork. The evidence on this subject as to the amount of shrinkage is very conflicting, a fact which is probably due to the following causes : 1. The various kinds of earthy material act very differently as respects shrinkage. There has been l)ut little uniforniity in the classification of earths in the tests and experiments tliat have been made. 112 RAILROAD CONSTRUCTION. § 96. 2. Yery much depends on tlie method of forming an em- bankment (as will be shown later). Different reports have been based on different methods — often without mention of the method. 3. An embankment requires considerable time to shrink to its final volume, and therefore much depends on the time elapsed between construction and the measurement of what is supposed to be the settled volume. P. J. Fljnn quotes some experiments {Eng. News^ May 1, 1886) made in India in which pits were dug, having volumes of 400 to 600 cubic feet. The material, when piled into an em- bankment, measured largely in excess of the original measure- ment — as is the universal experience. The pits were refilled with the same material. As the rains, very heavy in India, settled the material in the pits, more was added to keep the pits full. Even after the rainy season was over, there was in every case material in excess. This would seem to indicate a per- manent expansion^ although it is possible that the observations were not continued for a sufficient time to determine the final settled volume. On the contrary, notes made by Mr. Elwood Morris many years ago on the behavior of embankments of several thousand cubic yards, formed in layers by carts and scrapers, one winter intervening between commencement and completion, showed in each case a permanent contraction averaging about 10^. All authorities agree that rockwork expands permanently when formed into an embankment, but the percentages of expansion given by different authorities differ even more than with earth — varying from 8 to 90^. Of course this very large ranore in the coefficient is due to differences in the character of the rock. The softer the rock and the closer its similarity to earth, the less will be its expansion. On account of the conflict- ing statements made, and particularly on account of the influence of methods of work, but little confidence can be felt in any given coefficient, especially when given to a fraction of a per § 97. . EARTHWORK. 113 cent, but the consensus of American practice seems to avera"-e about as follows : Permanent contraction of earth about 10^ " expansion of rock 40 to GO^ These values for rock should be materially reduced, according to judgment, when the rock is soft and liable to disintegrate. The hardest rocks, loosely piled, may occasionally give even higher results. The following is given by several authors as the permanent contraction of several grades of earth : Gravel or sand about 8^ Clay " lOfo Loam " 12^ Loose vegetable surface soil '^ 15^ It may be noticed from the above table that the harder and cleaner the material the less is the contraction. Perfectly clean gravel or sand would not probably change volume appreciably. The above coefficients of shrinkage and expansion may be used to form the following convenient table. Material. To make 1000 cubic yards of embankment will require 1000 cubic yards measured in excavation will make Gravel or sand Clav 1087 cubic yards 1111 " 1136 " 1176 " 714 " 625 " measured in excavation 920 cubic yards 900 " 880 " 8.50 " 1400 " 1600 " of embaukmeut. Loam Loose vegetable soil Rock, larg(,' pieces *' small " 97. Allowance for shrinkage. On account of the initial expansion and subsequent contraction of earth, it becomes necessary to form embankments higher than their required ultimate form in order to allow for the subsequent shrinkage. As the shrinkage appears to be all vertical (practically), the embankment must be formed as shown in Fi<2:. .58. Tlic effect 114 RAILROAD CONSTRUCTION. 97. of shrinkage should not be confounded with that of sUpping of the sides, which is especially apt to occur if the embankment is subjected to heavy rains very soon after being formed, and also when the embankments are originally steep. It is often difficult Fig. 58. to form an embankment at a slope of 1 : 1 which will not slip more or less before it hardens. Very high embankments shrink a greater percentage than lower ones. Various rules giving the relation between shrink- age and height have been suggested, but they vary as badly as the suggested coefficients of contraction, probably for the same causes. As the fact is unquestionable, however, the extra height of the embankment must be varied somewhat as in Fig. 59, which represents a longitudinal section of an embankment. Fig. 59. As considerable time generally elapses between the completion of the embankment and the actual riinnincr of trains, the o^rade ad will generally be nearly flattened down to its ultimate form before traffic commences, but such grades are occasionally objec- tionable if added to what is already a ruling grade. With some kinds of soil the time required for complete settlement may be as much as two or three years, but, even in such cases, it is § 98. EARTHWORK. 115 probable that one-half of the settlement will take place during the first six months. The engineer should therefore require the contractor to make all fills about 8 to 15^ (accordin*^ to the material) higher than the profiles call for, in order that subsequent shrinkage may not reduce it to less than the re- quired volume. 98. Methods of forming embankments. When the method is not otherwise ol>jectionable, a high embankment can be formed very cheaply (assuming that carts or wheelbarrows are used) by dumjiing over the. end and building to the full height (or even higher, to allow for shrinkage) as the embankment proceeds. This allows more time for shrinkage, saves nearly all the cost of spreading (see Item 4, § 111), and reduces the cost of roadways (Item 5). Of course this method is especially applicable when the material comes from a place as liigh as or higher than grade, so that no up-hill hauling is required. Another method is to spread it in layers two or three feet thick (see Fig. GO), which are made concave upwards to avoid limiHUUiiimmmimiiimmuiiniv Fig. 60. possible sliding on each other. Spreading in layers has the advantage of partially ramming each layer, so that the subse- quent shrinkage is very small. Sometimes small trendies are dug along the lines of the toes of the embankment. This will frequently prevent the sliding of a large mass of the embank- ment, which will then require extensive and costly repairs, to say nothing of possible accidents if the sliding occurs after the road is in operation. Incidentally these trenches will be of value in draining the subsoil. AVhen circumstances require an embankment on a hillside, it is advisable to cut out "steps" to prevent a possible sliding of the whole embankment. Merely 116 RAILROAD CONSTRUCTION. § 99. ploughing the side-hill will often be a cheajDer and sufficiently effective method. Fig. 61. Occasionally the formation of a very high and lono- embank- ment may be most easily and cheaply accomplished by building a trestle to grade and opening the road. Earth can then be procured where most convenient, perhaps several miles away, loaded on cars with a steam-shovel, hauled by the trainload, and dumped from the cars with a patent unloader. On such a large scale, the cost per yard would be very much less than by ordi- nary methods — enough less sometimes to more than pay for the temporary trestle, besides allowing the road to be opened for traffic very much earlier, which is often a matter of prime financial importance. It may also obviate the necessity for extensive borrow-pits in the immediate neighborhood of the heavy fill and also utilize material which would otherwise be wasted. COMPUTATION OF HAUL. 99. Nature of subject. As will be shown later when analyz- ing the cost of earthwork, the most variable item of cost is that depending on the distance hauled. As it is manifestly imprac- ticable to calculate the exact distance to which every individual cartload of earth has been moved, it becomes necessary to devise a means which will give at least an equivalent of the haulao-e of all the earth moved. Evidently the average haul for any mass of earth moved is equal to the distance from the center of 2:rav- ity of the excavation to the center of gravity of the embank- §100. EARTHWORK. 117 merit formed by the excavated material. As a rongli approxi- mation the center of gravity of a cut (or fill) may sometimes be considered to coincide with the center of gravity of that part of tlie profile representing it, but the error is frequently very large. The center of gravity may be determined by various methods, but the method of the " mass diagram " accomplishes the same ultimate purpose (the determination of the haul) with all-suffi- cient accuracy and also furnishes other valuable information. 100. Mass diagram. In Fig. 62 let A' R . . . G' represent a profile and grade line drawn to the usual scales. Assume A' Fig. 63.— Mass Diagram. to be a point past which no earthwork will be hauled. Above every station point in the profile draw an ordinate which will represent the algebraic sum of the cubic yards of cut and fill (calling cut + and fill — ) from the point A' to the point considered. In doing this shrinkage must be allowed for by considering how much embankment would actually be made by so many cubic yards of excavation of such material. For example, it will be found that 1000 cubic yards of sand or gravel, measured in place (see § 97), will make about 920 cubic yards of embankment; therefore all cuttings in sand or gravel should be discounted in about this proportion. Excavations in rock should be increased in the proper ratio. In short, all ex- cavations should be valued according to the amount of settled embankment that could be made from them. The computations may be made systematically as shown in the tabular form. Place 118 RAILROAD CONSTRUCTION. 101. in the first column a list of the stations; in the second column, the number of cubic yards of cut or fill between each station and the preceding station ; in the third and fourth columns, the kind of material and the proper shrinkage factor; in the fifth column, a repetition of the quantities in cubic yards, except that the excavations are diminished (or increased, in the case of rock) to the number of cubic yards of settled embankment which may be made from tliem. In the sixth column, place the algehraic sum of the quantities in the fifth column (calling cuts + and fills — ) from the starting-point to the station considered. These algebraic sums at each station will be the ordinates, drawn to some scale, of the mass curve. The scale to be used will depend somewhat on whether the work is heavy or light, but for ordi- nary cases a scale of 5000 cubic yards per inch may be used. Drawing these ordinates to scale, a curve A^ B^ . . . G may be obtained by joining the extremities of the ordinates. Sta. Yards{-* + Material. Shrinkage factor. Yards, reduced for shrinkage. Ordinate in mass curve. 46 +70 47 48 + 60 49 50 51 32 + 30 53 + 70 54 + 42 55 56 57 + 175 + 1788 + 2341 + 2198 + 1292 - 693 -2414 - 2526 -2243 - 1954 -2006 -2077 -1828 - 710 + 462 + 195 + 1792 + 614 - 143 - 906 -1985 -1721 - 112 + 177 + 180 - 52 - 71 + 276 + 1242 + 1302 Clayey soil — 10 per ceut -10 -10 + 175 + 1613 + 553 - 143 - 906 -1985 -1721 - 112 + 283 + 289 - 52 - 71 + 249 + 1118 + 1172 Hard rock 1 < (« +60 per cent +60 " Clayey soil i i It <( 1 < — 10 percent -10 " -10 101. Properties of the mass curve. 1. The curve will be rising while over cuts and falling while over fills. 2. A tangent to the curve will be horizontal (as at B^ 7), E^ F^ and G) when passing from cut to fill or from fill to cut. § 101. EARTHWORK. 119 3. Wlieii the curve is helow the " zero line " it shows that material must be drawn backward (to the left) ; and vice versa^ when the curve is above the zero line it shows that material must be drawn ybri^arc? (to the right). 4. When the curve crosses the zero line (as at A and 6') it shows (in this instance) that the cut between A' and B' will just provide the materinl required for the fill between^' and 6^', and that no material should be hauled past C\ or, in general, past any intersection of the mass curve and the zero line. 5. If any horizontal line be drawn (as ah)^ it indicates that the cut and fill between a' and b' will just balance. 6. When the center of gravity of a given volume of material is to be moved a given distance, it makes no difference (at least theoretically) how far each individual load may be hauled or how any individual load may be disposed of. The summation of the products of each load times the distance hauled will be a constant, whatever the method, and will equal the total volume times the movement of the center of gravity. The a/verage haul, which is the movement of the center of gravity, will therefore equal the summation of these products divided by the total volume. If we draw two horizontal ])ar- allel lines at an infinitesimal distance dx a})art, as at «/>, the small increment of cut dx at a' will fill the corresponding incre- ment of fill at b\ and this material must be hauled the distance ab. Therefore the product of ab and dx, which is the product of distance times volume, is represented by the area of the infinitesimal rectangle at ab, and the total area ABC represents the summation of volume times distance for all the earth move- ment between A' and C . This sunnnation of ])roducts divided by the total volume gives the average haul. 7. The horizontal line, tangent at E and cutting the curve at e,f, and g, shows that the cut and fill between e' and E' will just balance, and that a possible method of hauling (whether desirable or not) would be to 'M>orrow" earth for the fill between C and e' , use the material between D' and K' foi- the 1^0 RAILROAD CONSTRUCTION. § 101. fill between e and I)\ and similarly balance cut and fill between E' andy^ and also between y^ and g' . 8. Similarly the horizontal line hldw, may be drawn cuttino- the curve, which will show another possiUe method of hauling. According to this plan, the fill between C and h' would be made by borrowing ; the cut and fill between h' and h' would balance; also that between Jc' and V and between V and m' . Since the area ehDkE represents the measure of haul for tlie earth between e and E\ and the other areas measure the corre- sponding hauls similarly, it is evident that the sum of the areas eliDhE and ElFmf^ which is the measure of haul of all the material between e' and/', is largely in excess of the sum of the areas IWk^ hEl^ and IFm^ plus the somewhat uncertain measures of haul due to borrowing material for e'h' and wastino- the material between m and/'. Therefore to make the meas- ure of haul a minimum a line should be drawn which will make the sum of the areas between it and the mass curve a minimum. Of course this is not necessarily the cheapest plan, as it implies more or less borrowing and wasting of material, which may cost more than the amount saved in haul. The comparison of the two methods is quite simple, however. Since the amount of fill between e and li' is represented by the differ- ence of the ordinates at e and A, and similarly for m' and/', it follows that the amount to be borrowed between e' and li will exactly equal the amount wasted between in and /'. By the first of the above methods the haul is excessive, but is definitely known from the mass diagram, and all of the material is util- ized ; by the second method the haul is reduced to about one- half, but there is a known quantity in cubic yards wasted at one place and the same quantity borrowed at another. The leno>th of haul necessary for the borrowed material would need to be ascertained ; also the haul necessary to w^aste the other material at a place where it would be unobjectionable. Frequently this is best done by widening an embankment beyond its necessary width. The computation of the relative cost of the above methods will be discussed later (§ 116). § 102. EARTHWORK. 121 9. Suppose that it were deemed best, after drawing the mass curve, to introduce a trestle between s' and v' . tlms savin (»• an amount in fill equal to tv. If such Lad been tlie original desi --■'(a) -J (&) c posts and complicated joints. Ftg 72. (d) Split caps and sills. These are described in ^ 129. Their advantages apply with even greater "force to framed trestles. (e) Dowels and drift-bolts. These joints facilitate cheap and rapid construction, but renewals and repairs are very difficult, it being almost impossible to extract a drift-bolt which has been driven its full length without splitting open the pieces contain- ing it. Notwithstanding this objection they are extensively used, especially for temporary work which is not expected to be used long enough to need repairs. 137. Multiple-story construction. Single-story framed trestle bents are used for heights up to 18 or 20 feet and excejitionally up to 30 feet. For greater heights some such construction as is illustrated in a skeleton design in Fig. 73 is used. By using split sills between each story and separate vertical and batter posts in each story, any piece may readily be removed and 164 RAILROAD CONSTRUCTION. 138. renewed if necessary. The height of these stories varies, in different designs, from 15 to 25 and even 30 feet. In some designs tlie structure of each storj is independent of the stories above and below. This greatly facilitates both the original con- struction and subsequent repairs. In other designs the verticals and batter- posts are made continuous through two consecutive stories. The structure is somewhat stiffer, but is much more diffi- cult to repair. Since the bents of any trestle are usually of variable height and those Fig. 73. heights are not always an even multiple of the uniform height desired for the stories, it becomes necessary to make the upper stories of unifoi-m height and let Fig. 74. the odd amount go to the lowest story, as shown in Figs. 73 and 74. 138. Span. The shorter the span the greater the number of trestle bents; the longer the span the greater the required strength of the stringers supporting the floor. Economy de- mands the adoption of a span that shall make the sum of these requirements a minimum. The liigher the trestle the greater the cost of each bent, and the greater the span that would be justifiable. Nearly all trestles have bents of variable height, but the advantage of employing uniform standard sizes is so great that many roads use the same span and sizes of timber not only for the panels of any given trestle, but also for all trestles §139 TRESTLES. 165 regardless of height. The spans generally used vary from 10 to 16 feet. The Norfolk and Western E. K. uses a span of 12' Q" for all single-story trestles, and a span of 25' for all multiple-story trestles. The stringers are the same in both cases, but wheu the span is 25 feet, knee- braces are run Fig. 75. from the sill of the lirst story below to near the middle of each set of stringers. These knee-braces are connected at the top by a ^'straining-beam" on which the stringers rest, thus support- ing the stringer in the center and virtually reducing the span about one-half. 139. Foundations, (a) Piles. Piles are frequently used as a foundation, as in Fig. 76, particularly in soft ground, and also for temporary structures. These foundations are cheap, quickly con- structed, and are particularly valuable when it is financially necessary to open (^Va ^fAJ SILL tne road tor tratlic as soon as possible ^^m^^M^m^mm^^ and with the least expenditure of ''"^ ''■"' U U money; but there is the disadvantage Fig. 76. of inevitable decay within a few years unless the piles are chemi- cally treated, as will be discussed later. Chemical treatment, however, increases the cost so that such a foundation would often cost more tlian a foundation of stone. A pile should be driven under each post as shown in Fig. 76. 166 RAILROAD CONSTRUCTION. §140. (b) Mud-sills. Fig. 77 n w/ n illustrates the use of mud-sills as built bj the Louisville and Nash- ville E. K. Eight blocks 12" X 12" X 6' are used under each bent. When the ground is very soft, two additional timbers (12" X 12" X length of bent- sill), as shown by the dotted lines, are placed underneath. Fig. 77. The number required evidently de- pends on the nature of the ground. (c) Stone foundations. Stone foundations are the best and the most expensive. For very high trestles the JS'orfolk and Western R.R. employs foundations as shown in Fig. 78, the * SILL 0!^ TRESTLE r L - -- -- -- -- -- -- :: - -i -J 1 SILL 1 i- -- - -- -- -- -- -- -J < ^13 > " — 8 — ^ , -^ 13 > Fig. 78. walls being 4 feet thick. When the height of the trestle is 72 feet or less (the plans requiring for 72' in height a foundation- wall 39' 6' long) the foundation is made continuous. The sill of the trestle should rest on several short lengths of 3" X 12'' plank, laid transverse to the sill on top of the wall. 140. Longitudinal bracing. This is required to give the structure longitudinal stiifness and also to reduce the columnar length of the posts. This bracing generally consists of hori- zontal " waling-strips " and diagonal braces. Sometimes the braces are placed wholly on the outside posts unless the trestle is very high. For single-story trestles the P. R. R. employs the "laced "' system, i.e., a line of posts joining the cap of one bent with the sill of the next, and the sill of that bent with the cap of the next. Some plans employ braces forming an X in alternate panels. Connecting these braces in the center more than doubles their columnar strength. Diagonal braces, when bolted to posts, should be fastened to them as near the ends of 143. TI^ESTLES. 1G7 the posts as possible. The sizes employed vary largely, depend- ing on the clear length and on whether they are expected to act by tension or compression. 3" X 12" planks are often used when the design would require tensile strength only, and S" X S" posts are often used when compression may be expected. 141. Lateral bracing. Several of the more recent designs of trestles employ diagonal lateral bracing between the caps of adjacent bents. It adds greatly to the stiffness of the trestle and better maintains its alignment. 6" X 6" posts, formincr an X and connected at the center, will answer the purpose. 142. Abutments. When suitable stone for masonry is at hand and a suitable subsoil for a foundation is obtainable without too much excavation, a masonry abutment will be the best. Such an abutment would probably be used when masonry footings for trestle bents were employed (§ 139, c). Another method is to construct a "crib" of 10" x 12" timber, laid horizontally, drift-bolted together, securely braced and embedded into the ground. Except for temporary con- struction such a method is generally objectionable on account of rapid decay. Another method, used most commonly for pile trestles, and for framed trestles having pile foundations (§ 139, a\ is to use a pile bent at such a place that the natural surface on the up-hill side is not far below the cap, and the thrust of the material, filled in to bring the surface to grade, is insig- nificant. 3" X 12" planks are placed Fig. 79. behind the piles, cap, and stringers to retain the filled material. FLOOR SYSTEMS. 143. stringers. The general practice is to use two, three, and even four stringers under each rail. Sometimes a strin, iz; O O J <3 <1 fa ^ K o SB 03 cc W c 2 M 5 0) o o o o o o O O '?» o o o o o • o o iO o • o o o o oo o o o o o o o o oo o o o o oo o o 3 b o «s o o o o • o o o o o o o . o o o o o o o . o o o o o o o • o O o lO O lO o • o o o IC O 00 t^- • CO O CD • oooooooo •oooooooo loooooooo ■oooooooo ■ oo»ciooo»co • t- CO -^ -Tt^ CO O CO o C a; =« 'r ^ '/I H oooooooo oooooooo Ot-OJT-iOOOQOi> oooooooo oooooo»co QOi>coaoxaoc-GO 2-3 u - oooo O O lO o O C^ CO CO o o o •coo CVJ Oi Ci o o o o o o o lt o o i.t o ' CQ 1-1 Oi (M 07 W oooo • o o o o o o o o o o o o oooo • o o o o o o o o o o o o C5 t- O CO • GO oo GO O O CO 00 GO X) O OD 00 fe M (« a c t^ c3 o <; W) ^ d ^ 5d oooo OOOO rf 1— I CO O ooo o o 01 CQ CQ o o> CQ o o o o O CQ ooo O O «D CQ OO • • -o oooooooooooooooo oooooooooooooooo Oi>CQCQO0i0500OO00C0C00005t- a §154. TRESTLES. 179 On page 177 there are quoted the vahies taken from the U. S. Government reports on the strength of timber, the tests probably beino- the most thorough and reliable that were ever made. On page 178 are given the "average safe allowable work- in o- unit stresses in pounds per square inch," as recommended b^^the committee on '' Strength of Bridge and Trestle Timbers," the work being done under the auspices of the Association of Eailway Superintendents of Bridges and Buildings. The report was presented at their fifth annual convention, held in New Orleans, in October, 1895. 154. Loading. As shown in § 138, the span of trestles is always small, is generally 14 feet, and is never greater than IS' except when supported by knee-braces. The greatest load that will ever come on any one span will be the concentrated loading of the drivers of a consolidation locomotive. With spans of 14 feet or less it is impossible for even the four pairs of drivers to be on the same span at once. The weight of the rails, ties, and guard-rails should be added to obtain the total load on the string- ers, and the weight of these, plus the weight of the stringers, should be added to obtain the pressure on the caps or corbels. This dead load is almost insignificant compared with the live load and may be included with it. The weight of rails, ties, etc., may be estimated at 200 pounds per foot To obtain the weight on the caps the weight of the stringers must be added, which depends on the design and on the weight per cubic foot of the wood employed. But as the weight of the stringers is comparatively small, a considerable percentage of variation in wei^'-ht will have but an insignificant effect on the result. Disreo-ardinfi- all refinements as to actual dimensions, the ordi- nary maximum loading for standard gauge railroads may be taken as that due to four pairs of driving-axles, spaced 5' 0" apart and giving a pressure of 25,000 pounds per axle. This should be increased to 40,000 pounds per axle (same spacing) for the heaviest trafiic. On the basis of 25,000 pounds per axle the following results have been computed : 180 RAILBOAD CONSTRUCTION'. 155. STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OF 25,000 POUNDS, SPACED 5' 0" APART. Span in feet. Max. mom.— ft. lbs. Max. shear. Max load on one cap. 10 12 14 16 18 65 000 103 600 142 400 181 400 220 600 38 500 45 000 49 600 54 725 60100 52100 62 700 74 200 85 700 97 900 Although the dead load does not vary in proportion to the live load, jet, considering the very small influence of the dead load, there will be no appreciable error in assuming the corre- sponding values, for a load of 40,000 lbs. per axle, to be |o ^f those given in the above tabulation. 155. Factors of safety. — The most valuable result of the gov- ernment tests is the knowledge that under given moisture condi- tions the strength of various species of sound timber is not the variable uncertain quantity it was once supposed to be, but that its strength can be relied on to a comparatively close percentage. This confidence in values permits the employment of lower fac- tors of safety than have heretofore been permissible. Stresses, which when excessive would result in immediate destruction, such as cross- breaking and columnar stresses, should be allowed a higher factor of safety — say 6 or 8 for green timber. Other stresses, such as crushing across the grain and shearing along the neutral axis, which will be apparent to inspection before it is dangerous, may be allowed lower factors — say 3 to 5. 156. Design of stringers. — The strength of rectangular beams of equal width varies as the square of the depth ; therefore deep beams are the strongest. On the other hand, when any cross- sectional dimension of timber much exceeds 12" the cost is much higher per M., B.M., audit is correspondingly difi^cult to obtain thoroughly sound sticks, free from wind-shakes, etc. Wind-shakes especially affect the shearing strength. Also, if the required transverse strength is obtained by using high nar- row stringers, the area of pressure between the stringers and the § 156. TRESTLES, 181 cap niaj become so small as to induce crushing across the grain. This is a very common defect in trestle design. As already in- dicated in § 138, the span should vary roughly with the average hei^'ht of the trestle, the longer spans being employed when the trestle bents are very high, although it is usual to employ the same span throughout any one trestle. To illustrate, if we select a span of l-i feet, the load on one cap will be 74,200 lbs. If the stringers and cap are made of long-leaf yellow pine, which require the closely determined value of 1180 lbs. per square inch to produce a crushing amounting to ^fo of the height on timber with 12^ moisture, we may use 200 lbs. per square inch as a safe pressure even for green tim- ber; this will require 371 square inches of surface. If the cap is 12^' wide, this will require a width of 31 inches, or say 2 stringers under each rail, each 8 inches wide. For rectangular beams Moment = ^R'hh\ Using for 11' the safe value 1575 lbs. per square inch, we have 142400 X 12 = i X 1575 X 32 X h\ from which h = 15^'. 9. If desired, the width may be increased to 9" and the depth correspondingly reduced, which will give similarly h = 14''. 8, or say 15". This show^s that two beams, 9'' X 15'', under each rail will stand the transverse bending and have more than enough area for crushing. The shear per square inch will equal 3 total shear 3 49600 ^^^., . ^ -. — = - zrz = 138 lbs. per so. inch, 2 cross section 2 4 X 9 X lo ^ ^ which is a safe value, although it should preferably be less. Hence the above combination of dimensions will answer. The deflection should be computed to see if it exceeds the 182 RAILROAD CONSTRUCTION, % 157. somewhat arbitrary standard of gi^ of the span. The deflection for tiniforni loading is A = Z^WE in which I = length in inches ; IF = total load, assumed as uniform ; E = modulus of elasticity, given as 2,070,000 lbs. per sq. in. for long-leaf pine, 12^ dry, and assumed to be 1,200,000 for green timber. Then _ 5 X 72800 X 168 - ~ 32 X 36 X 15" X 1200000 ~ ^^X168''=0".84, so that the calculated deflection is well within the limit. Of course the loading is not strictly uniform, but even with a lib- eral allowance the deflection is still safe. For the heaviest practice (40000 lbs. per axle) these stringer dimensions must be correspondingly increased. 157. Design of posts. Four posts are generally used for single-track work. The inner posts are usually braced by the cross-braces, so that their columnar strength is largely increased ; but as they are apt to get more than their share of work, the ad- vantage is compensated and they should be treated as unsupported columns for the total distance between cap and sill in simple bents, or for the height of stories in multiple-story construction. The caps and sills are assumed to have a width of 12''. It facilitates the application of bracing to have the columns of the same width and vary the other dimension as required. Unfortunately the experimental work of the U. S. Govern- ment on timber testing has not yet progressed far enough to establish unquestionably a general relation between the strength of lono- columns and the crushing: streno^th of short blocks. The § 157. TRESTLES. 183 following formula has been suggested, but it cannot be consid- ered as established : f = -F X »^r. ■ ^ ^ — . — -.^ in which J ^ '^ 700 + 15c + c'' f = allowable working stress per sq. in. for long columns; ji^== " '^ " " " " " short blocks; I I = length of column in inches ; d = least cross-sectional dimension in inches. Enough work has been done to give great reliability to the two following formulae for white pine and yellow pine, quoted from Johnson's '' Materials of Construction," p. 684 : 1 fiy Working load per sq. in. =^ = 1000 — ^[jj ^ long-leaf pine; «< " '' " " =p = 600 — ^(y-j , white pine; in which I = length of column in inches, and h = least cross-sectional dimension in inches. The frequent practice is to use 12'' X 12'' posts for all tres- tles. If we substitute in the above formula ^ = 20' = 240" and h =. 12", we have p = 1000 - i(\V-)' = ^^0 lbs. 900 X 1-11: = 129600 lbs., the loorking load for each post. This is more than the total load on one trestle bent and il- lustrates the usual great waste of timber. Making the post 8" X 12" and calculating similarly, we have p = T75, and the working load per column is 775 X 06 = 74400 l])s. As considerable must be allowed for " weathering," which destroys the strength of the outer layers of the wood, and also for the dynamic effect of the live load, 8" X 12" may not be too great, 184 BAILROAD CONSTRUCTION, § 158. but it is certainly a safe dimension. 12'' X 6" would possibly prove amply safe in practice. One method of allowing for weathering is to disregard the outer half-inch on all sides of the post, i.e., to calculate the strength of a post one inch smaller in each dimension than the post actually employed. On this basis an 8'' X 12'' X 20' post, computed as a 7" X 11' post, would have a safe columnar strength of 706 lbs. per square inch. With an area of 77 square inches, this gives a working load of 54362 lbs. for each post^ or 217148 lbs. for the four posts. Consider- ing that 74200 lbs. is the maximum load on one cap (14 feet span), the great excess of strength is apparent. 158. Design of caps and sills. The stresses in caps and sills are very indefinite, except as to crushing across the grain. As the stringers are placed almost directly over the inner posts, and as the sills are supported just under the posts, the transverse stresses are almost insignificant. In the above case four posts have an area of 4 X 12" X 8" = 384 sq. in. The total load, 74200 lbs., will then give a pressure of 193 pounds per square inch, which is within the allowable limit. This one feature might require the use of 8" X 12" posts rather than 6" X 12" posts, for the smaller posts, although probably strong enough as posts, would produce an objectionably high pressure. 159. Bracing. Although some idea of the stresses in the bracing could be found from certain assumptions as to wind- pressure, etc. , yet it would probably not be found wise to de- crease, for the sake of economy, the dimensions which practice has shown to be sufiicient for the work. The economy that would be possible would be too insignificant to justify any risk. Therefore the usual dimensions, given in §§ 139 and 140, should be employed. CHAPTEE Y. TUNNELS. SURVEYING. 160. Surface surveys. As tunnels are always dug from each end and frequently from one or more intermediate shafts, it is necessary that an accurate surface survey should be made between the two ends. As the natural surface in a locality where a tunnel is necessary is almost invariably very steep and rough, it requires the employment of unusually refined methods of work to avoid inaccuracies. It is usual to run a line on the surface that will be at every point vertically over the center line of the tunnel. Tunnels are generally made straight unless curves are absolutely necessary, as curves add greatly to the cost. Fig. 85 represents roughly a longitudinal section of the ^ ^^-IQOOD' H* 7000 '>t--""CUWJ \ TyJO- : ^000^ j -5000— ^ Fig. 85 —Sketch of Section of the Hoosac Tunnel. Hoosac Tunnel. Permanent stations were located at ^i, B^ C^ />, E^ and F^ and stone houses were built at A^ B, C^ and 7>. These were located with ordinary field transits at first, and then all the points were placed as nearly as possible in one vertical plane by repeated trials and minute corrections, using a verv large specially constructed transit. The stations 7> and F weva necessary because E and A were invisible from (7 niid /?. 185 186 RAILROAD CONSTRUCTION. § 160. The alignment at A and ^ having been determined with great accuracy, the true ahgnment was easily carried into the tunnel. The relative elevations of A and E were determined with great accuracy. Steep slopes render necessary many settings of the level per unit of horizontal distance and require that the work be unusually accurate to obtain even fair accuracy per unit of distance. The levels are usually re-run many times until the probable error is a very small quantity. The exact horizontal distance between the two ends of the tunnel must also be known, especially if the tunnel is on a grade. The usual steep slopes and rough topography likewise render accurate horizontal measurements very difficult. Fre- quently when the slope is steep the measurement is best obtained by measuring along the slope and allowing for grade. This may be very accurately done by employing two tripods (level or transit tripods serve the purpose very well), setting them up slightly less than one tape-length apart and measuring iDetween horizontal needles set in wooden blocks inserted in the top of each tripod. The elevation of each needle is also observed. The true horizontal distance between two successive positions of the needles then equals the square root of the difference of the squares of the inclined distance and the differ- ence of elevation. Such measurements will probably be more accurate than those made by attempting to hold the tape horizontal and plumbing down with plumb-bobs, because (1) it is practically difficult to hold both ends of the tape truly horizontal; (2) on steep slopes it is impossible to hold the down- Mil end of a 100-foot tape (or even a 25-foot length) on a level with the other end, and the great increase in the number of applications of the unit of measurement very greatly increases the probable error of the whole measurement ; (3) the vibrations of a plumb-bob introduce a large probability of error in trans- ferring the measurement from the elevated end of the tape to the ground, and the increased number of such applications of the unit of measurement still further increases the probable error. ^ 161. TUNNELS. 187 c 161. Surveying down a shaft. If a shaft is sunk, as at S^ Fig. 85, and it is desired to dig out the tunnel in both directions from the foot of the shaft so as to meet tlie headings from the outside, it is necessary to know, when at the bottom of the shaft, the elevation, alignment, and horizontal distance from each end of the tunnel. The elevation is generally carried down a shaft by means of a steel tape. This method involves the least number of ai)pli- cations of the unit of measurement and greatly increases the accuracy of the final result. The horizontal distance from each end may be easily trans- ferred down the shaft by means of a plumb-bob, using sonVe of the precautions described in the next paragraph. 1 To transfer the alignment from the surface to the bottom of a shaft requires the highest skill because the shaft is always small, and to produce a line perhaps several thousand feet long in a direction given by two points 6 or 8 feet apart requires that the two points must be determined with extreme accuracy.. The eminently successful method adopted in the Hoosac Tunnel will be briefly described : Two beams were securely fastened across the top of the shaft (1030 feet deep), tlie beams being: plaged transversely to tlie direction of the tunnel and as far apart as possible and yet allow plumb-lines, hung from the intersection of each beam with the tunnel center line, to swing freely at the bottom of the shaft. These intersections of the beams with the center line were determined by averaging the results of a large number of careful observations for alio-nment. Two fine parallel wires, spaced about J^" apart, were then stretched between the beams so that the center line of the tunnel bisected at all points the space between the wires. Plumb-bobs, weighing 15 pounds, were suspended by fine wires beside each cross-beam, the wires passing between the two parallel alignment wures and bisecting the space. The plumb- bobs were allowed to swing in pails of water at the. bottom. Drafts of air up the shaft required the construction of boxes surrounding the wires. Even these precautions did not suffice 188 RAILROAD CONSTRUCTION. § 162. to absolutely prevent vibration of the wire at the bottom through a very small arc. The mean point of these vibrations in each case was then located on a rigid cross-beam suitably placed at the bottom of the shaft and at about the level of the roof of the tunnel. Short plumb-lines were then suspended from these points whenever desired ; a transit was set (by trial) so that its line of collimation passed through both plumb lines and tlie line at the bottom could thus be prolonged. 162. Underground surveys. Survey marks are frequently placed on the timbering, but they are apt to prove unreliable on account of the shifting of the timbering due to settlement of the surrounding material. They should never be placed at the bottom of the tunnel on account of the danger of being disturbed or covered up. Frequently holes are drilled in the roof and filled with wooden plugs in which a hook is screwed exactly on line. Although this is probably the safest method, even these plugs are not always undisturbed, as the material, unless very hard, will often settle slightly as the excavation proceeds. When a tunnel is perfectly straight and not too long, alignment-points may be given as frequently as desired from permanent stations located outside the tunnel where they are not liable to disturbance. This has been accomplished by running the alignment through the upper part of the cross-section, at one side of the center, where it is out of the way of the piles of masonry material, debris, etc., which are so apt to choke up the lower part of the cross-section. The position of this line relative to the cross-section being fixed, the alignment of any required point of the cross-section is readily found by means of a light frame or template with a fixed target located where this line would intersect the frame FjG. 86. when properly placed. A level -bubble on tlie frame will assist in setting the frame in its proper position. In all tunnel surveying the cross-wires must be illuminated § 163. TUNNELS. 189 by a lantern, and the object sighted at must also be illuminated. A powerful dark-lantern with the opening covered with (jroxind glass has been found useful. This may be used to illuminate a plumb-bob string or a very fine rod, or to place behind a brass plate having a narrow slit in it, the axis of the slit and plate being coincident with the plumb-bob string by which it is hung. On account of the interference to the surveying caused by the work of construction and also by the smoke and dust in the air resulting from the blasting, it is generally necessary to make the surveys at times when construction is tenq^orarily sus- pended. 163. Accuracy of tunnel surveying. Apart from the very natural desire to do surveying which shall check well, there is an important financial side to accurate tunnel surveying. If the survev lines do not meet as desired when the headino^s come together, it may be found necessary, if the error is of appreciable size, to introduce a slight curve, perhaps even a reversed curve, into the alignment, and it is even conceivable that the tunnel section would need to be enlarged somewhat to allow for these curves. The cost of these changes and the perpetual annoyance due to an enforced and undesirable alteration of the original design will justify a considerable increase in the expenses of the survey. Considering that the cost of surveys is usually but a small fraction of the total cost of the work, an increase of 10 or even 20^ in the cost of the surveys will mean an insignificant addition to the total cost and frequently, if not generally, it will result in a saving of many times the increased cost. The accuracy actually attained in two noted American tunnels is given as follows : The Musconetcong tunnel is about 5000 feet long, bored through a mountain -100 feet high. The error of alignment at the meeting of the headings was O'.Oi, error of levels O'.Olo, error of distance 0'.52. The Hoosac tunnel is over 25,000 feet long. The heading from the east end met the heading from the central shaft at a point 11271 feet from the east end and 15G3 feet from the shaft. The error in align- ment was y\ of an inch, that of levels " a few hundredths," 190 RAILROAD CONSTRUCTION. % 164. error of distance ' ' trifling. ' ' Tlie alignment, corrected at the shaft, was carried on through and met the heading from the west end at a point 10138 feet from the west end and 2056 feet from the shaft. Here the error of alignment was -^^" and that of levels 0.134 ft. DESIGN. 164. Cross-sections, l^early all tunnels have cross-sections peculiar to themselves — all varying at least in the details. The general form of a great many tunnels is that of a rectangle sur- mounted by a semi-circle or semi-ellipse. In very soft material an inverted arch is necessary along the bottom. In such cases the sides will generally be arched instead of vertical. The sides are frequently battered. With very long tunnels, several forms of cross-section will often be used in the same tunnel, owing to differences in the material encountered. In solid rock, which w^ill not disintegrate upon exposure, no lining is required, and the cross-section will be the irregular section left by the blasting, the only requirement being that no rock shall be left within the required cross-sectional figure. Farther on, in the same tunnel, when passing through some very soft treacherous material, it may be necessary to put in a full arch lining — top, sides, and bot- tom — which will be nearly circular in cross-section. For an illustration of this see Figs. 87 and 88. The width of tunnels varies as greatly as the designs. Single- track tunnels generally have a width of 15 to 16 feet. Occa- sionally they have been built 14 feet wide, and even less, and also up to 18 feet, especially when on curves. 24 to 26 feet is the most common width for double track. Many double- track tunnels are only 22 feet wide, and some are 28 feet wide. The heights are generally 19 feet for single track and 20 to 22 feet for double track. The variations from these figures are con- siderable. The lower limits depend on the cross-section of the rolling stock, with an indefinite allowance for clearance and ven- tilation. Cross-sections which coincide too closely with what is 164. TUNNELS. 191 Fig. 87.— Housac Tunnel. Section through Solid Rock. Fig. 88.— Hoosac Tunnel Section through Soft Ground, 192 RAILROAD CONSTRUCTION. 165. absolutely required for clearance are objectionable, because any slight settlement of the lining which would otherwise be harm- less would then become troublesome and even dangerous. Figs. 87, 88, and 89 ^ show some typical cross-sections. Fig. 89. — St. Cloud Tunnel. 165. Grade. A grade of at least 0.2^ is needed for drainage. If the tunnel is at the summit of two grades, the tunnel grade should be practically level, with an allowance for drainage, the actual summit being perhaps in the center so as to drain both ways. When the tunnel forms part of a long ascending grade, it is advisable to reduce the grade through the tunnel unless the tunnel is very short. The additional atmospheric resistance and the decreased adhesion of the driver wheels on the damp rails in a tunnel will cause an engine to work very hard and still more rapidly vitiate the atmosphere until the accumulation of poison- ous gases becomes a source of actual danger to the engineer and fireman of the locomotive and of extreme discomfort to the passengers. If the nominal ruling grade of the road were maintained through a tunnel, the maximum resistance would be * Drinker's "Tunneling. PLATE 11. TuN^'El.-TIMBERI^G— English System (6*). TUNNEL-TIMBERTXG— ENGTJSn SYSTEM (6). {To face 'page 192.) PLATE III. ■///■//■/r.'Mi'i)j'''',''i:/M'/'ir'//.'ii//.i//,'///f,/j'/'//'-^^i ', /, • .w ;^y:'y/<>-y//fr/M/',r,"'^////'y, ///i-'^y^v/fif^y^ TUJS2HiiL-TIMJ3EKlIS'G— ENGLISH SYSTEM {c). TuNNEL-TTMBERiKG— English System {d), {To face page 192.) § 167. TUNNELS. 193 found in the tunnel. This would probably cause trains to stall there, which would be objectionable and perhaps dangerous. 166. Lining. It is a characteristic of many kinds of rock and of all earthy material that, although they may be self-sus- taining when first exposed to the atmosphere, they rapidly dis- integrate and require that the top and perhaps the sides and even the bottom shall be lined to prevent caving in. In this country, when timber is cheap, it is occasionally framed as an arch and used as the permanent lining, but masonry is always to be preferred.- Frequently the cross-section is made extra large so that a masonry lining may subsequently be placed inside the wooden lining and thus postpone a large expense until the road is better able to pay for the work. In very soft unstable material, like quicksand, an arch of cut stone voussoirs may be necessary to withstand the pressure. A good quality of brick is occasionally used for lining, as they are easily handled and make good masonry if the pressure is not excessive. Only the best of cement mortar should be used, economy in this feature being the worst of folly. Of course the excavation must include the outside line of the lining. Any excavation which is made out- side of this line (by the fall of earth or loose rock or by exces- sive blasting) must be refilled with stone well packed in. Occa- sionally it is necessary to fill these spaces with concrete. Of course it is not necessary that the lining be uniform throughout the tunnel. 167. Shafts. Shafts are variously made with square, rectan- gular, elliptical, and circular cross-sections. The rectangular cross-section, with the longer axis parallel with the tunnel, is most usually employed. Generally the shaft is directly over the center of the tunnel, but that always implies a complicated con- nection between the linings of the tunnel and shaft, provided such linings are necessary. It is easier to sink a shaft near to one side of the tunnel and make an opening through the nearly vertical side of the tunnel. Such a method was employed in the Church Hill Tunnel, illustrated in Fig. 90.'' Fig. 91 f shows * Drinker's "Tunneling." f Rziha, " Lebrbuch der Gesammteu Tunnelbaukunsi." 194 RAILROAD CONSTRUCTION, 167. a cross-section for a large main sliaft. Many shafts have been built with the idea of being left open permanently for ventila- tion and have therefore been elaborately Uned with masonry. Fig. 90.— Connection with Shaft, Church Hill Tunnel. Fig. 91. — Cross-section, Large Main Shaft. The general consensus of opinion now appears to be that shafts are worse than useless for ventilation ; that the quick passage of a train through the tunnel is the most effective ventilator ; and that shafts only tend to produce cross-currents and are ineffective to clear the air. In consequence, many of these elaborately lined shafts have been permanently closed, and the more recent PLATE IV TuNNEL-TiMBEKiNG— French System (a). TuNN^:L-TIMREHI^•G-FKE^XII System (&). (To face page 194.) PLATE y. Tunnel timbeking— Belgian Sy&tem (a). Mi Tunnel-timbeking— Belglvn System (6). {To face 'page 194.) § 169. TUNNELS. 195 practice is to close up a shaft as soon as the tunnel is completed. Shafts always form drainage-wells for the material they pass through, and sometimes to such an extent that it is a serious matter to dispose of the water that collects at the bottom, requiring the construction of large and expensive drains. 168. Drains. A tunnel will almost invariably strike veins of water which will promptly begin to drain into the tunnel and not only cause considerable trouble and expense during construc- tion, but necessitate the provision of permanent drains for its perpetual disposal. These drains nmst frequently be so large as to appreciably increase the required cross- section of the tunnel. Generally a small open gutter on each side will suffice for this purpose, but in double-track tunnels a large covered drain is often built between the tracks. It is sometimes necessary to thoroughly grout the outside of the lining so that water will not force its way through the masonry and perhaps injure it, but may freely drain down the sides and pass through openings in the side walls near their base into the gutters. CONSTRUCTION. 169. Headings. The methods of all tunnel excavation de- pend on the general principle that all earthy material, except the softest of liquid mud and quicksand, will be self-sustaining over a greater or less area and for a greater or less time after excavation is made, and the work consists in excavating some material and immediately propping up the exposed surface by timbering and poling-boards. The excavation of the cross-sec- tion begins with cutting out a ''heading," which is a small horizontal drift whose breast is constantly kept 15 feet or more in advance of the full cross-sectional excavation. In solid self-sustaining rock, which will not decompose upon exposure to air, it becomes simply a matter of excavating the rock with the least possible expenditure of time and energy. In soft ground the heading must be heavily timbered, and as the heading is gradually enlarged the timbering must be gradually extended 196 RAILROAD CONSTRUCTION, 170. and perhaps replaced, according to some regular system, so that when the full cross-section has been excavated it is supported by such timbering as is intended for it. The heading is some- times made on the center line near the top ; with other plans, on the center line near the bottom ; and sometimes two simultaneous headings are run in the two low^er corners. Headings near the bottom serve the purpose of draining the material above it and facilitating the excava- tion. The simplest case of heading timber- ing is that shown in Fig. 92, in which cross- timbers are placed at intervals just under the roof, set in notches cut in the side walls and supporting poling-boards which sustain what- ever pressure may come on them. Cross-timbers near the bottom support a flooring on which vehicles for transporting material may be run and under which the drainage may freely escape. As the necessity for timbering becomes greater, side timbers and even bottom timbers must be added, these timbers supporting poHng-boards, and even the breast of the heading must be pro- tected by boards suitably braced, as shown in Fig. 93. The Fig. 93. Fk;} 93 — TiMBEEixa FOK Tunnel Heading. supporting timbers are framed into collars in such a manner that added pressure only increases their rigidity. 170. Enlargement. Enlargement is accomplished by remov- ing the poling-boards, one at a time, excavating a greater or less PLATE VI. -■' '7^^ ;~^^ ■'////'"/ j'^^>-4' •: " •^^ -.^./ / Tunnel timeeiung— German System (a). ''':■'::: -^:';:>,'r.,Xir\ •^v/'/'^ii', ■^- >ci v^ ^^^;^^: ., ::-^^:^-.;^J^.i•'^- ;'^^"■ ''^-'^^^Ifv^ :^?^^ TuNNEL-TiMBET? TNG— German Sys'iem (6). {To face page 196.) PLATE VII. pr-T^^i^ -^/^" Tunnel-timbering — German System [c). JmiW^r^''^^ •^' Tunnel-timbering— German System {d) {To face page 1% ) §171. TUNNELS. 197 amount of material, and immediately supporting the exposed material with poling-boards suitably braced. (See Figs. 93 and 94.) This work being systematically done, space is therel)y Fig. 94. obtained in which the framing for the full cross- section may be gradually introduced. The framing is constructed with a cross- section so large that the masonry lining may be constructed within it. 171. Distinctive features of various methods of construction. There are six general systems, known as the English, German, Belgian, French, Austrian, and American. They are so named from the origin of the methods, although their use is not confined to the countries named. Fig. 95 shows by numbers (1 to 5) the order of the excavation within the cross-sections. The Eriir- lish, Austrian, and American systems are alike in excavating the entire cross-section before beginning the construction of the masonry lining. The German method leaves a solid core (5) until practically the whole of the lining is complete. This has the disadvantage of extremely cramped quarters for work, poor ventilation, etc. The Belgian and French methods agree in excavating the upper part of the section, building the arch at once, and supporting it temporarily until the side walls are built. The Belgian method then takes out the core (3), removes very short sections of the sides (4), immediately underpinning the arch with short sections of the side walls and thus gradually constructing the whole side wall. The French method digs out the sides (3), supporting the arch temporarily with timbers and 198 RAILROAD CONSTRUCTION. 171. tlien replacing the timbers with masonry ; the core (4) is taken out last. The French method has the same disadvantage as the German — working in a cramped space. The Belgian and French systems have the disadvantage that the arch, supported tempo- rarily on timber, is very apt to be strained and cracked by the slight settlement that so frequently occurs in soft material. The English, Austrian, and American methods differ mainly in the f' ^1 1 1 1 1 1 1 -+- 1 1 1 -+- 1 1 1 fx ^ 4 3 -4-- 4 5 1 5 ENGLISH AUSTRIAN AMERICAN german belgian french Fig. 95. — Order of Working by the Various Systems. design of the timbering. The English support the roof by lines of very heavy longitudinal timbers which are supported at com- paratively wide intervals by a heavy framework occupying the whole cross-section. The Austrian system uses such frequent cross-frames of timber- work that poling-boards will suffice to support the material between the frames. The American sys- tem agrees with the Austrian in using frequent cross-frames supporting poling-boards, but differs from it in that the '' cross- frames " consist simply of arches of 3 to 15 wooden voussoirs, the voussoirs being blocks of 12" X 12'' timber about 2 to 8 feet long and cut with joints normal to the arch. These, arches are put together on a centering which is removed as soon as the arch PLATE VIII. TUNNEL-TIMBEIIING — AUSTRIAN SYSTEM (a). TUNNEL-TlMliEKlNG — AUSTRIAN JSYSTEM (6). TuNNET,-TiMBERiNG— Austrian System (rt). {To face page 198.) PLATE IX. ^ MM^ ^'^ .0 \\' TuNNEL-TiMiifiiiiNa— Austrian System [d). -^-id. down-stream end. down-stream end. three pipes. Fig. 98.— Standard Vitrified-pipe Culvert. Plant System. (1891.) the supposed extra strength is not therefore obtained. In Fig. 98 are shown the standard plans for vitrified- pipe culverts as used on the "Plant system." Tile pipe is much clieaper than iron pipe, but is made in much shorter lengths and requires nmch more work in laying and especially to obtain a uniform grade. 212 RAILROAD CONSTRUCTION. §188. BOX CULVERTS. 188. Wooden box culverts. This form serves the purpose of a cheap temporary construction which allows the use of a bal- lasted roadbed. As in all temporary constructions, the area should be made considerably larger than the calculated area (§§ 179-182), not only for safety but also in order that, if the smaller area is demonstrated to be sufficiently large, the per- manent construction (probably pipe) may be placed inside with- out disturbing the embankment. All designs agree in using heavy timbers (12'' X 12", 10'' X 12", or 8" X 12") for the side walls, cross-timbers for the roof, every fifth or sixth timber being notched down so as to take up the thrust of the side walls, and planks for the flooring. Fig. 99 shows some of the standard designs as used by the C, M. tfe St. P. Ry. Fig. 99.— Standard Timber Box Culvert. C, M. & St. P. Ry. (Feb. 1889. > 189. Stone box culverts. In localities where a good quality of stone is cheap, stone box culverts are the cheapest form of permanent construction for culverts of medium capacity, but their use is decreasing owing to the frequent difficulty in obtain- ing really suitable stone within a reasonable distance of the culvert. The clear span of the cover-stones varies from 2 to 4 feet. The required thickness of the cover-stones is sometimes calculated by the theory of transverse strains on the basis of cer- tain assumptions of loading — as a function of the height of the embankment and the unit strength of the stone used. Such a method is simply another illustration of a class of calculations § 190. CULVERTS AND MINOR BRIDGES. 213 wliicli look very precise and beautiful, but which are worse than useless (because misleading) on account of the hopeless uncertainty as to the true value of certain quantities which must be used in the computations. In the first place the true value of the unit tensile strength of stone is such an uncertain and variable quantity that calculations based on any assumed value for it are of small reliability. In the second place the weight of the prism of earth lying directly above the stone, plus an allowance for live load, is by no means a measure of the load on the stone nor of the forces that. tend to fracture it. All earthwork will tend to form an arch above any cavity and thus relieve an imcertain and probably variable proportion of the pressure that might other- wise exist. The higher the embankment the less the propor- tionate loading, until at some uncertain height an increase in height will not increase the load on the cover-stones. The effect of frost is likewise large, but uncertain and not computable. The usual practice is therefore to make the thickness such as experi- ence has shown to be safe with a good quality of stone, i.e., about 10 or 12 inches for 2 feet span and up to 16 or 18 inches for 4 feet span. The side walls should be carried down deep enough to prevent their being undermined by scour or heaved by frost. The use of cement mortar is also an important feature of first-class work, especially when there is a rapid scouring cur- rent or a liability that the culvert will run under a head. In Fig. 100 are shown standard plans for single and double stone box culverts as used on the IS^orfolk and Western R.R. 190. Old-rail culverts. It sometimes happens (although very rarely) that it is necessary to bring the grade line within 3 or 4 feet of the bottom of a stream and yet allosv an area of 10 or 12 square feet. A single large pipe of sufficient area could not be used in this case. The use of several smaller pipes side by side would be both expensive and inefficient. For similar reasons neither wooden nor stone box culverts could be used. In sucli cases, as well as in many others where the head-room is not so limited, the plan illustrated in Fig. 101 is a very satisfactory solution of the problem. The old rails, having a length of 8 or 214 RAILROAD CONSTRUCTION. § 190. 191. CULVERTS AND MINOR BRIDGES. 215 9 feet, are laid close togetlier across a G-foot opeiiing. Some- times the rails are held together by long bolts passing through the webs of the rails. In the plan shown the rails are confined [TnTTTT TTTTTTr TTr.TTTT,TT;TT TTTrTTTTTTTrTr 7?1 ^fe Fig. 101.— Standard Old-rail Culvert. N. & W. R.R. (1895.) by low end walls on each abutment. This plan requires only 15 inches between the base of the rail and the top of the culvert channel. It also gives a continuous ballasted roadbed. ARCH CULVERTS. 191. Influence of design on flow. The variations in the design of arch culverts have a very marked influence on the cost and efficiency. To combine the least cost with the great- est efficiency, due weight should be given to the following elements : (a) the amount of masonry, (h) the simplicity of the constructive work, {c) the design of the wing walls, {d ) the design of the junction of the wing walls with the barrel LI (a) Fio. 102. — Types of Culverts. and faces of the arch, and {e) the safety and permanency of the construction. These elements are more or less antagonistic to each other, and the defects of most designs are due to a lack of proper proportion in the design of these opposing interests. The simplest construction (satisfying elements 1> and e) is the straight 216 RAILROAD CONSTRUCTION. § 192. barrel arch between two parallel vertical head walls, as sketched in Fig. 102, a. From a hydraulic standpoint the design is poor, as the water eddies around the corners, causing a great resistance which decreases the flow. Fig. 102, 5, shows a much better de- sign in many respects, but much depends on the details of the design as indicated in elements {h) and (c/). As a general thing a good hydraulic design requires complicated and expensive masonry construction, i.e., elements (b) and {d) are opposed. Design 102, (?, is sometimes inapplicable because the water is liable to work in behind the masonry during floods and perhaps cause scour. This design uses less masonry than {a) or (h). 192. Example of arch culvert design. In Plate XY is shown the design for an 8-foot arch culvert according to the standard of the ISTorfolk and Western R.R. Note that the plan uses the flaring wing walls (Fig. 102, h) on the up-stream side (thus protecting the abutments from scour) and straight wing walls (similar to Fig. 102, c) on the down-stream end. This economizes masonry and also simplifies the constructive work. Note also the simplicity of the junction of the wing walls with the barrel of the arch, there being no re-entrant angles below the springing line of the arch. The design here shown is but one of a set of designs for arches varying in span from 6' to 30'. MINOR OPENINGS. 193. Cattle-guards, (a) Pit guards. Cattle-guards will be considered under the head of minor openings, since the old- fashioned plan of pit guards, which are even now defended and preferred by some railroad men, requires a break in the con- tinuity of the roadbed. A pit about three feet deep, five feet long, and as wide as the width of the roadbed, is w^alled up with stone (sometimes with wood), and the rails are supported on heavy timbers laid longitudinally with the rails. The break in the continuity of the roadbed produces a disturbance in the elastic wave running through the I'ails, the effect of which is noticeable at high velocities. The greatest objection, however, lies in the --41^2-- > X § 193. CULVERTS AND MINOR BRIDGES. 217 dangerous consequences of a derailment or a failure of the tim- bers owing to unobserved decay or destruction by fire — caused perhaps by sparks and cinders from passing locomotives. The very insignificance of the structure often leads to careless in- '■■! .t i:; o" 1 12 X Vl\ l.i 8' m^. 12 10- -Cil2' Fig. 103.— Pit Cattle-guahds. P. R.R. spection. But if a single pair of wheels gets off the rails and drops into the pit, a costly wreck is inevitable. The (once) standard design for such a structure on the Pennsylvania E.R. is shown in Fig. 103. (b) Surface cattle-guards. These are fastened on top of the ties; the continuity of the roadbed is absolutely unbroken and thus is avoided much of the danger of a bad wreck owing to a possible derailment. The device consists essentially of overlay- ing the ties (both inside and outside the rails) with a surface on which cattle w411 not walk. The multitudinous designs for such a surface are variously effective in this respect. An objection, which is often urged indiscriminately against all such designs, is the liability that a brake-chain which may happen to be drag- ging may catch in the rough bars which are used. The bars are sometimes "home-made," of wood, as shown in Fig. 104. Iron, or steel bars are made as shown in Fig. 105. The general construction is the same as for the wooden bars. The 218 MAILROAD CONSTRUCTION. 194. metal bars have far greater dnrabilitv, and it is claimed that they are more effective in discouraging cattle from attempting to cross. Fig. 104.— Cattle-guard with Wooden Slats. Fig. 105.— Merrill- Stevexs Steel Cattle-guard. 194. Cattle-passes. Frequently when a railroad crosses a farm on an embankment, cutting the farm into two parts, the railroad company is obliged to agree to make a passageway through the embankment sufficient for the passage of cattle and perhaps even farm-was^ons. If the embankment is hi^h enousrh so that a stone arch is practicable, the initial cost is the only great objection to such a construction ; but if an open wooden structure is necessary, all the objections against the old-fashioned cattle-guarda apply witli equal force here. The avoidance of a grade crossing which would otherwise be necessary is one of the PLATE XVL iH BOLT EVERY THIRD TIE. -N,, O- STANDARD I-BRIDGES-14-FT. SPAN. NORFOLK AND WESTERN R.R. (1891.) TYPES OF PLATE GIRDER BRIDGES. C. M. & St.P. RY. (Dec. 1895.) TYPE '•£" GIRDER 25 FEET AND UNDER, (To face page 219.) § 195. CULVERTS AND MINOR BHTDGE8. 219 • great compensations for the expense of the construction and maintenance of these structures. The construction is sometimes made by placing two pile trestle bents about 6 to 8 feet apart, supporting the rails by stringers in the usual way, the special feature of this construction being that the embankments are filled in behind the trestle bents, and the thrust of the embank- ments is mutually taken up through the stringers, which are notched at the ends or otherwise constructed so that they may take up such a thrust. The designs for old-rail culverts and arch culverts are also utilized for cattle-passes when suitable and convenient, as well as the designs illustrated in the following section. 195. Standard stringer and I-beam bridges. The advantages of standard designs apply even to the covering of short spans with wooden stringers or with I beams — especially since the methods do not require much vertical space between the rails and the upper side of the clear opening, a feature which is often of prime importance. These designs are chiefly used for culverts or cattle-passes and for crossing over highways — pro- viding such a narrow opening would be tolerated. The plans all imply stone abutments, or at least abutments of sufficient stability to withstand all thrust of the embankments. Some of the designs are illustrated in Plate XVI. The preparation of these standard desii^ns should be attacked bv the same oreneral methods as already illustrated in § 156. When computing the required transverse strength, due allowance should be made for lateral bracing, which should be amply provided for. Xote particularly the methods of bracing illustrated in Plate XYI. The designs calling for iron (or steel) stringers may be classed as permanent constructions, which are cheap, sate, easily in- spected and maintained and therefore a desirable method of construction. CHAPTER YII. BALLAST. 196. Purpose and requirements. " The object of the ballast is to transfer the applied load over a large surface ; to hold the timber work in place horizontally ; to carry off the rain-water from the superstructure and to prevent freezing up in winter ; to afford means of keeping the ties truly up to the grade line ; and to give elasticity to the roadbed." This extremely con- densed statement is a description of an ideally perfect ballast. The value of any given kind of ballast is proportional to the extent to which it fulfills these requirements. The ideally per- fect ballast is not necessarily the most economical ballast for all roads. Light traffic generally justifies something cheaper, but a very common error is to use a very cheap ballast when a small additional expenditure would procure a much better ballast which would be much more economical in the long run. 197. Materials. The materials most commonly employed are gravel and broken stone. Burnt clay, cinders, shells, and small coal are occasionally used as ballast when they are especially cheap and convenient or when better kinds are especially expen- sive. Although it is hardly correct to speak of the natural soil as ballast, yet many miles of cheap railways are "ballasted" with the natural soil, which is then called " mud ballast." Mud ballast. When the natural soil is gravelly so that rain will drain through it quickly, it will make a fair roadbed for light traffic, but for heavy traffic, and for the greater part of the length of most roads, the natural soil is a very poor material for ballast ; for, no matter how suitable the soil might be along 220 § 197. BALLAST. 221 limited sections of the road, it would practically never liappen that the soil would be uniformly good throughout the whole length of the road. Considering that a heavy rain will in one day spoil the results of weeks of patient " surfacing" with mud ballast, it is seldom economical to use "mud" if there is a gravel-bed or other source of ballast anywhere on the line of the road. Cinders. Tlie advantages consist in the excellent facilities for drainage, ease of handling, and cheapness — after the road is in operation. One disadvantage is excessive dust in dry weather. Cinders are considered preferable to gravel in yards. Slag. When slag is readily obtainable it furnishes an ex- cellent ballast, free from dust and perfect in drainage qualities. Some kinds of slag are objectionable on account of their delete- rious chemical effect on the ties and spikes — especially on metallic ties. Shells, small coal, etc. These comparatively inferior kinds of ballast are used for light traffic when they are especially cheap and convenient. They are extremely dusty in dry weather, break up into very fine dust, and are but little better than mud. Gravel. This is the most common form of ballast which may be called good ballast. In 1885, the Roadmasters Associa- tion of America voted in favor of gravel ballast as against rock ballast. Although not so stated, this action was perhaps due to a conviction of its real economy for the average railroad of this country, which may be called a "light traffic" road. Gravel should preferably be screened over a screen having a \" mesh, so as to screen out all dirt and the finest stones. Generally a railroad will be able to find at some point along its line a "gravel-pit" affording a suitable supply. This may be dug out with a steam-shovel, screened if necessary, and sent out over the line by the train-load at a comparatively small cost. Rock or broken stone. Ruck ballast is generally specified to be such as will pass through a \\" (or 2'') ring. Although pref- erably broken by hand, machine-broken stone is much cheaper. It is most easily handled with forks. This also has the effect of 222 RAILROAD CONSTRUCTION. § 198. screening out tlie dirt and fine chips which would interfere with eliectual drainage. Rock ballast is more expensive in first cost, and also more troublesome to handle, than any other kind, but under heavy trafiic will keep in surface better and will require less work for maintenance after the ties have become thoroughly bedded. For roads with very light traffic, running few trains, at comparatively low velocities, the advantages of rock ballast over other kinds are not so pronounced. For such roads rock ballast is an expensive luxury. The amount of trafi^ic which will justify the use of rock ballast will depend on the cost of obtaining ballast of the various kinds. 198. Cross-sections. A depth of 12'' under the tie is gener- ally required on the best roads, but for light trafiic this is some- times reduced to Q" and even less. The width is generally 1 to 2 feet less than the wddth of the roadbed proper — excluding ditches. If the ballast has an average width of 10 feet (12 feet at bottom and 8 feet at top) and an average depth of 15 inches (including that placed between the ties), it will require 2144 cubic yards per mile of track. The P. R.R. estimates 2500 cubic yards of gravel and 2800 cubic yards of stone ballast per mile of single track. On account of the requirements of drain- age the best form of cross-section depends on the kind of ballast used. Mud ballast. Since the great objection to mud ballast lies in its liability to become soft by soaking up the rain that falls, it becomes necessary that it should be drained as quickly and readily as its nature will permit. Fig. 106 shows a typical Fig. OP.— '• Mud " Ballast. cross-section for mud ballast. It should be crowned 2" above the top of the tie at the center, thence sloped so as to leave a slight clearance under the rail between the ties, thence sloping down to the bottom of the tie at each end and continuing to §199. BALLAST. 223 slope down to the ditcli (in cut), wliicli should be 18" or 20" be- low the bottom of the tie. Gravel, cinders, slag, etc. The subgrade is crowned 6" or 8" in the center, as shown in Fig. 107. The ballast is crowned Fig. 107.— Gravel Ballast. to the top of the tie in the center, but is sloped down to the bottom of the tie at each end. This is necessary (and more especially so with mud ballast) to prevent a possible accumula- tion and settlement of water at the ends of the tie, which would readily soak into the end fibers and produce decay. Broken stone. Stone ballast is shouldered out beyond the ends of the ties so as to aiford greater lateral binding. The space betAveen the ties is filled up level with the tops. The Fig. 108.— Broken Stone Bali^ast. perfect drainage of stone ballast permits this to be done w^ithout any danger of causing decay of the ties by the accumulation and retention of water. 199. Methods of laying ballast. The cheapest method of laying ballast on new roads is to lay ties and rails directly on the prepared subgrade and run a construction train over the track to distribute the ballast. Then the track is lifted up until sufficient ballast is worked under the ties and the track is prop- erly surfaced. This method, although cheap, is apt to injure the rails by causing bends and kinks, due to the passage of loaded construction trains when the ties are very unevenly and roughly supported, and the method is therefore condemned and prohibited in some specifications. The best method is to draw 224 RAILROAD CONSTRUCTION, § 200. ill carts (or on a contractor's temporary track) the ballast that is required under the level of the hottoin of the ties. Spread this ballast carefully to the required surface. Then lay the ties and rails, which will then have a very fair surface and uniform sup- port. A construction train can then be run on the rails and distribute sufficient additional ballast to pack around and between the ties and make the required cross-section. The necessity for constructing some lines at an absolute minimum of cost and of opening them for traffic as soon as pos- sible has often led to the policy of starting traffic when there is little or no ballast — perhaps nothing more than a mere tamping of the natural soil under the ties. When this is done ballast may subsequently be drawn where required by the train-load on flat cars and unloaded at a minimum of cost by means of a '' plough." The plough has the same width as the cars and is guided either by a ridge along the center of each car or by short posts set up at the sides of the cars. It is drawn from one end of the train to the other by means of a cable. The cable is sometimes operated by means of a small hoisting-engine carried on a car at one end of the train. Sometimes the locomotive is detached temporarily from the train and is run ahead with the cable attached to it. 200. Cost. The cost of ballast in the trade is quite a variable item for different roads, since it depends (a) on the first cost of the material as it comes to the road, (h) on the distance from the source of supply to the place where it is used, and {c) on the method of handling. The first cost of cinder or slag is frequently insignificant. A gravel-pit may cost nothing except the price of a little additional land beyond the usual limits of the right of way. Broken stone will usually cost $1 or more per cubic yard. If suitable stone is obtainable on the company's land, the cost of blasting and breaking should be somewhat less than this. The cost of loading the ballast on to trains will be small (per cubic yard) if it is handled with steam-shovels — as in the case of gravel taken from a gravehpit. Hand-shovelling will cost more. The cost of hauling will depend on the distance § 200 BALLAST, 225 hauled, and also, to a considerable extent, on the limitations on the operation of the train due to the necessity of keeping out of the way of regular trains. There is often a needless waste in this way. The ''mud train " is considered a pariah and entitled to no rights whatever, regardless of the large daily cost of such a train and of the necessary gang of men. The cost of broken stone ballast m the track is estimated at $1.25 per cubic yard. The cost of gravel ballast is estimated at 60 c. per cubic yard in the track. The cost of placing and tamping gravel ballast is estimated at 20 c. to 24 c. per cubic yard, for cinders 12 c. to 15 c. per cubic yard. The cost of loading gravel on cars, usintr a steam-shovel, is estimated at 6 c. to 10 c. per cubic yard.^ * Report Roadmasters Association, 1885. CHAPTER YIII. TIES, AND OTHER FORMS OF RAIL SUPPORT. 201. Various methods of supporting rails. It is necessary that the rails shall be sufficiently supported and braced, so that the gauge shall be kept constant and that the rails shall not be subjected to excessive transverse stress. It is also preferable that the rail support shall be neither rigid (as if on solid rock) nor too yielding, but shall have a uniform elasticity throughout. These requirements are more or less fulfilled by the following methods. (a) Longitudinals. Supporting the rails throughout their entire length. This method is very seldom used in this country except occasionally on bridges and in terminals when the longitudinals are supported on cross- ties. In § 224: will be described a system of rails, used to some extent in Europe, having such broad bases that they are self-supporting on the ballast and are only connected by tie-rods to maintain the gauge. (b) Cast-iron "bowls" or "pots.'' These are castings resem- bling large inverted bowls or pots, having suitable chairs on top for holding and supporting the rails, and tied together with tie-rods. They will be described more fully later (§ 223). (b) Cross-ties of metal or wood. These will be discussed in the following sections. 202. Economics of ties. The true cost of ties depends on the relative total cost of maintenance for long periods of time. The first cost of the ties delivered to the road is but one item in the 226 § 203. TIES. 227 economics of tlie question. Clieap ties require fre(|uent renew- als, whicli cost for the lahor of each renewal practically the same whether the tie is of oak or hemlock. Clieap ties make a poor roadbed which will require more track labor to keep even in tolerable condition. The roadbed will require to be disturbed so frequently on account of renewals that the ties never get an opportunity to get settled and to form a smooth roadbed for any length of time. Irregularity in width, thickness, or length of ties is especially detrimental in causing the ballast to act and wear unevenly. The life of ties has thus a more or less direct influence on the life of the rails, on the wear of rollinof stock, and on the speed of trains. _/ These last items are not so readily reducible to dollars and cents, but when it can be shown that the total cost, for a long period of time, of several renewals of cheap ties, with all the extra track labor involved, is as great as or greater than that of a few renewals of durable ties, then there is no question as to the real economy. In the following dis- cussions of the mei-its of untreated ties (either cheap or costly), chemically treated ties, or metal ties, the true question is there- fore of the ultimate cost of maintaining any particular kind of ties for an indefinite period, the cost including the flrst cost of the ties, the labor of placing them and maintaining them to surface, and the somewhat uncertain (but not therefore non- existent) effect of frequent renewals on repairs of rolling stock, on possible speed, etc. WOODEN TIES. 203. Choice of wood. This naturally depends, for any partic- ular section of country, on the supply of wood wliicli is most readily available. The woods most commonly used, especially in this country, are oak and pine, oak being the most durable and generally the most expensive. Kedwood is used very ex- tensively in California and proves to be extremely durable, so far as decay is concerned, but it is very soft and is much injured by " rail-cutting." This defect is being partly remedied by the 228 RAILROAD CONSTRUCTION. % 204. use of tie-plates, as will be explained later. Cedar, chestnut, liemlock, and tamarack are frequently used in this country, . In tropical countries very durable ties are frequently obtained from the hard woods peculiar to those countries. According to a re- cent bulletin of tlie U. S. Department of Agriculture the pro- portions of the various kinds used in the United States are about as follows : Oak 60^ Pine 20 Cedar 6 Chestnut 5j Hemlock and Tama- rack 3 Redwood 3 Cypress 2% Various 1 Total 100^ 204. Durability. The durability of ties depends on the cli- mate ; the drainage of the ballast ; the volume, weight, and speed of the traffic ; the curvature, if any ; the use of tie-plates ; the time of year of cutting the timber ; the age of the timber and the degree of its seasoning before placing in the track ; the nature of the soil in which the timber was grown; and, chiefly, on the species of wood employed. The variability in these items will account for the discrepancies in the reports on the life of various woods used for ties. White oak is credited with a life of 5 to 12 years, depending principally on the traffic. Is is both hard and durable, the hardness enabling it to withstand the cutting tendency of the rail-flanges, and the durability enabling it to resist decay. Pine and redwood resist decay very well, but are so soft that they are badly cut by the rail-flanges and do not hold the spikes very well, necessitating frequent respiking. Since the spikes must be driven within certain very limited areas on the face of each tie, it does not require many spike-holes to '^ spike-kill " the tie. On sharp curves, especially with heavy traffic, the vdieel- flange pressure produces a side pressure on the rail tending to overturn it, which tendency is resisted by the spike, aided some- times by rail-braces. Whenever the pressure becomes too great the spike will yield somewhat and will be slightly withdrawn. The resistance is then somewhat less and the spike is soon so loose that it must be redriven in a new hole. If this occurs very §206. TIES. 229 often, the tie may need to be replaced long before any decay has set in. When the traffic is v^ery light, the wood very durable, and the climate favorable ties have been known to last 25 years. 205. Dimensions. The usual dimensions for the best roads (standard gauge) are 8' to 8' <6" long, 6" to 7" thick, and 8" to 10" wide on top and bottom (if they are hewed) or 8" to 9" wide if they are sawed. For cheap roads and light traffic the length is shortened sometimes to 7' and the cross-section also re- duced. On the other hand a very few roads use ties 9' long. Two objections are urged against sawed ties : first, that the grain is torn by the saw, leaving a woolly surface which induces decay ; and secondly, that, since timber is not perfectly straight- grained, some of the fibers are cut obliquely, exposing their ends, which are thus liable to decay. The use of a " planer-saw " ob- viates the first difficulty. Chemical treatment of ties obviates both of these difficulties. Sawed ties are more convenient to handle, are a necessity on bridges and trestles, and it is even, clahned, although against connnonly received opinion, that actual trial has demonstrated that they are more durable than hewed ties. 206. Spacing. The spacing is usually 14 to 16 ties to a 30- foot rail. This number is sometimes reduced to 12 and even 10, and on the other hand occasionally increased to 18 or 20 by employing narrower ties. There is no economy in reducing the number of ties very nuich, since for any required stiffness of track it is more economical to increase the number of supports than to increase the weight of the rail. The decreasinir cost of rails and the increasing cost of ties have materially changed the rela- tion between number of ties and weight of rail to produce a given stiffness at minimum cost, but many roads have found it economical to employ a large number of ties rather than increase the weight of the rail. On the other hand there is a practical limit to the number that may be employed, on account of the necessary space between the ties that is required for proper tamping. This width is ordinarily about twice the width of the tie. At this rate, with light ties 6" wide and with 12^' clear 230 RAILROAD CONSTRUCTION. § 207. space, there would be 20 ties per 30-foot rail, or 3520 per mile. The smaller ties can generally be bought much cheaper (propor- tionately) than the larger sizes, and hence the economy. Track instructions to foremen generally require that the spacing of ties shall 7iot be uniform along the lengtli of any rail. Since the joint is generally the weakest part of the rail structure, the joint requires more support than the center of the rail. Therefore the ties are placed with but 8" or 10'' clear space between them at the joints, this applying to 3 or 4 ties at each joint ; the remaining ties, required for each rail length, are equally spaced along the remaining distance. 207. Specifications. The specifications for ties are apt to include the items of size, kind of wood, and method of con- struction, besides other minor directions about time of cutting, seasoning, delivery, quality of timber, etc. (a) Size. The particular size or sizes required will be some- what as indicated in § 205. (b) Kind of wood. When the kind or kinds of wood are spe- cified, the most suitable kinds that are available in that section of country are usually required. (c) Method of construction. It is generally specified that the ties shall be hewed on two sides; that the two faces thus made shall be parallel planes and that the bark shall be removed. It is sometimes required that the ends shall be sawed off square ; that the timber shall be cut in the w^inter (when the sap is down) ; and that the ties shall be seasoned for six months. These last specifications are not required or lived up to as much as their importance deserves. It is sometimes required that the ties shall be delivered on the right of way, neatly piled in rows, the alter- nate rows at right angles, piled if possible on ground not lower than the rails and at least seven feet away from them, the lower row of ties resting on two ties which are themselves supported so as to be clear of the ground. (d) Q,uality of timber. The usual specifications for sound timber are required, except that they are not so rigid as for a better class of timber work. The ties must be sound, reason- § 208. TIES. 231 ably straight-grained, and not very crooked — one test being that a hne joining the center of one end with the center of the middle shall not pass outside of the other end. Splits or shakes, espe- er ton is the same, adding (say) 10^ to the weight (and cost) adds 21^ to the stiffness and over 15 fo to the strength. As another illustration, using an 80-lb. rail instead of a 75-11). rail adds only i^%% to the cost, but adds about 14^ to the stiffness and neai-ly \\% to the strength. This shows wdiy heavier rails are mure economical and are being adopted even wdien they are not abso- lutely needed on account of heavier rolling stock. The stiffness, strength, and consequent durability are increased in a much greater ratio than the cost. 228. Effect of stiffness on traction. A very important but generally unconsidered feature of a stiff rail is its effect on trac- 248 RAILROAD CONSTRUCTION. § 229. tive force. An extreme illustration of this principle is seen when a vehicle is drawn over a soft sandj road. The constant compression of the sand in front of the wheel has virtually the same effect on traction as drawing the wheel up a grade whose steepness depends on the radius of the wheel and the depth of the rut. On the other hand, if a wheel, made of perfectly elastic material, is rolled over a surface which, while supported with absolute rigidity, is also perfectly elastic, there would be a forward component, caused by the expanding of the compressed metal just behind the center of contact, which would just bal- ance the backward component. If the rail was supported throughout its length by an absolutely rigid support, the high elasticity of the wheel-tires and rails would reduce this form of resistance to an insignificant quantity, but the ballast and even the ties are comparatively inelastic. When a weak rail yields, the ballast is more or less compressed or displaced, and even though the elasticity of the rail brings it back to nearly its former place, the work done in compressing an inelastic material is wholly lost. The effect of this on the fuel account is certainly very considerable and yet is frequently entirely overlooked. It is practically impossible to compute the saving in tractive power, and therefore in cost of fuel, resulting from a given increase in the weight and stiffness of the rail, since the yielding of the rail is so dependent on the spacing of the ties, the tamping, etc. But it is not difficult to perceive in a general way that such an econ- omy is possible and that it should not be neglected in considering the value of stiffness in rails. 229. Length of rails. The standard length of rails with most railroads is 30 feet. In recent years many roads have been try- ing 45-foot and even 60-foot rails. The argument in favor of longer rails is chiefly that of the reduction in track-joints, which are costly to construct and to maintain and are a fruitful source of accidents. Mr. Morrison of the Lehigh Yalley R.R.^ de- clares that, as a result of extensive experience with 45-foot rails Report, Roadrnasters Association, 1895. § 230. RAILS. 249 on that road, he finds that they are much less expensive to handle, and that, being so long, they can be laid around sharp curves without being curved in a machine, as is necessary with the shorter rails. The great objection to longer rails lies in the difficnltv in allowing for the expansion, which will require, in the coldest weather, an opening at the joint of nearly |" for a 60-foot rail. The Pennsylvania K.R. and the Norfolk and Western R.R. each have a considerable mileage laid with GO-foot rails. 230. Expansion of rails. Steel expands at the rate of .0000065 of its length per degree Fahrenheit. The extreme range of tem- perature to which any rail will be subjected will be about 160°, or say from — 20° F. to + 140° F. With the above coefficient and a rail length of 60 feet the expansion would be 0.0624 foot, or about J inch. But it is doubtful Avhether there would ever be such a range of motion even if there were such a range of temperature. Mr. A. Torrey, chief engineer of the Mich. Cent. R.E., experimented with a section over 500 feet long, which, although not a single rail, was made " continuous " by rio-id splicing, and he found that there was no appreciable addi- tional contraction of the rail at any temperature below + ^0 F. The reason is not clear, but i\iQ fact is undeniable. The heavy girder rails, used by the street railroads of the country, are bonded together with perfectly tight rigid joints wliich do not permit expansion. If the rails are laid at a tem- perature of 60° F. and the temperature sinks to 0°, the rails have a tendency to contract .00039 of their length. If this tendency is resisted by the friction of the pavement in which the rails are buried, it only results in a tension amounting to .00039 of the modulus of elasticity, or say 10920 pounds per square inch, assuming 28 000 000 as the modulus of elasticity. This stress is not dangerous and may be permitted. If the tempera- ture rises to 120° F., a tendency to expansion and buckling will take place, which will be resisted as before by the pavement, and a compression of 10920 pounds per square inch will be in- duced, which will likewise be harmless. The range of tempera- 250 RAILROAD CONSTRUCTION. 231 ture of rails which are buried in pavement is much less than when thej are entirely above the ground and will probably never reach the above extremes. Eails supported on ties which are only held in place by ballast must be allowed to expand and con- tract almost freely, as the ballast cannot be depended on to resist the distortion induced by any considerable range of temperature, especially on curves. 231. Rules for allowing for temperature. Track regulations generally require that the track foremen shall use iron {not wooden) shims for placing between the ends of the rails while splicing them. The thickness of these shims should vary with the temperature. Some roads use such approximate rules as the following : " The proper thickness for coldest weather is ^^ of an inch ; during spring and fall use i of an inch, and in the very hottest weather -^^ of an inch should be allowed." This is on the basis of a 30-foot rail. When a more accurate adjustment than this is desired, it may be done by assuming some veiy high temperature (120° to 150° F.) as a maximum, when the joints should be tight; tlien compute in tabular form the spacing for each temperature, varying by 20°, allowing 0".01:68 (almost exactly -f-^") for each 20° change. Such a tabular form would be about as follows (rail length 30 feet) : Temperature 150° 180° 110° 90° 70° 50° 30° 10° - 10° - 30° Rail opening. , . 3 " 6T JL" 3 2 9 " 6¥ 3 " T6 15" 64 9 " 32 21" 64 3" 8 2 7" One practical difficulty in the way of great refinement in this work is the determination of the real temperature of the rail when it is laid. A rail lying in the hot sun has a very nnich Mgher temperature than the air. The temperature of the rail cannot be obtained even by exposing a thermometer directly to the sun, although such a result might be the best that is easily obtainable. On a cloudy or rainy day the rail has practically the same temperature as the air ; therefore on such days there need be no such trouble. § 2'62. KAILS. 251 232. Chemical composition. About OS to 99.5,^ of the com- position of steel rails is iron, but the value of the rail, as a rail, is almost wholly dependent upon the large number of other chemical elements which are, or may be, present in very small amounts. The manager of a steel- rail mill once declared that their aim was to produce rails having in them — Carbon 0.32 to O.±yjyo Silicon 0.04 to 0.06^ Phosphorus 0.09 to 0.105^ Mano-anese 1.00 to 1.50^ &' The analysis of 32 specimens of rails on the Chic, Mil. & St. Paul K.R. showed variations as follows: Carbon 0.211 to 0.52^ Silicon 0.013 to 0.256^ Phosphorus 0.055 to 0.181^ Manganese 0.35 to 1.63^ These quantities have the same general relative proportions as the rail- mill standard given above, the diiferences lying; mainly in the broadening of the limits. Increasing the percent- age of carbon by even a few hundredths of one per cent makes- the rail harder, but likewise more brittle. If a track is well ballasted and not subject to heaving by frost, a harder and more; brittle rail may be used without excessive danger of breakage,, and such a rail will wear much lon^rer than a softer toiiirher rail, although the softer tougher rail may be the better rail for a road having a less perfect roadbed. A small but objectionable percentage of sulphur is some- times found in rails, and very delicate analysis will often show the presence, in very minute quantities, of several other chemical elements. The use of a very small quantity of nickel or aluminum has often been suggested as a means of ])roducing a more durable rail. The added cost and the uncertaintv of 252 RAILROAD CONSTRUCTION. § 233. the amount of advantage to be gained has hitherto prevented the practical nse or manufacture of such rails. 233. Testing. Cliemical and mechanical testing are both necessary for a thorough determination of the value of a rail. The chemical testing has for its main object the determination of those minute quantities of chemical elements which have such a marked influence on the rail for good or bad. The mechanical testing consists of the usual tests for elastic limit, ultimate strength, and elongation at rupture, detennined from pieces cut out of tlie rail, besides a '' drop test." The drop test consists in dropping a weight of 2000 lbs. from a height of 16 to 20 feet on to the center of a rail which is supported on abutm.ents placed three or four feet apart. The number of blows required to produce rupture or to produce a permanent set of specified magnitude gives a measure of the strength and toughness of the rail. 234. Rail wear on tangents. When the wheel loads on a rail are abnormally heavy, and particularly when the rail has but little carbon and is unusually soft, the concentrated pressure on the rail is frequently greater than the elastic limit, and the metal "flows " so that the head, although greatly abraded, will spread somewhat outside of its original lines, as shown in Fig. 115. The rail wear that occurs on tangents is Fig. 115. almost exclusively on top. Statistics show that the rate of rail wear on tangents decreases as the rails are more W'Orn. Tests of a large number of rails on tangents have shown a rail wear averaging nearly one pound per yard per 10 000 000 tons of trafiic. There is about 33 pounds of metal in one yard of the head of an 80 -lb. rail. As an extreme value this may be worn down one-half, thus giving a tonnage of 165 000 000 tons for the life of the rail. Other estimates bring the tonnage down to 125 000 000 tons. Since the locomotive is considered to be responsible for one half (and possibly more) of the damage done to the rail, it is found that the rate of wear on roads with shorter trains is more rapid in proportion to the tonnage, and it § 23o. BAILS. 25-3 is therefore thought that the life of a rail should be expressed in terms of the number of trains. This has been estimated at 300 000 to 500 000 trains. 235. Rail wear on curves. On curves the maximum rail wear occurs on the inner side of the head of the outer rail, irivinir a worn form somewhat as shown in Fig. 116. The dotted line shows the nature and progress of the rail wear on the inner rail of a curve. Since the pressure on the outer rail is somewhat lateral rather than vertical, the " flow '' does not take place to the same extent, if at all, on the outside, and what- ever flow would take place on the inside is Fig. 116. immediately worn off by the wheel-flange. Unlike the wear on tangents, the wear on curves is at a greater rate as the rail becomes more worn. The inside rail on curves wears chiefly on top, the same as on a tangent, except that the wear is much greater owing to the longitudinal slipping of the wheels on the rail, and the lateral slipping that must occur when a rigid four-wheeled truck is guided around a curve. The outside rail is subjected to a greater or less proportion of the longitudinal slipping, likewise to the lateral slipping, and, worst of all, to the grinding action of the flange of the wheel, which grinds off the side of the head. The results of some very elaborate tests, made by Mr. A. M. Wellington, on the Atlantic and Great Western R.R., on the wear of rails, seem to show that the I'ail wear on curves may be expressed by tlie formula: " Total wear of rails on a el degree curve in pounds per yard per 10 000 000 tons duty = 1 -{- O.OScP.''^ "It is not pretended that this fornmla is strictly correct even in theory, but several theoretical consider- ations indicate that it may be nearly so." According to this formula the average rail wear on a 6° curve will be about twice the rail wear on a tangent. While this is approximately true, the various causes modifying the rate of rail wear (length of trains, age and quality of rails, etc.) will result in numerous and 254 RAILUOAD COIslSTRUCTION. § 236. large variations from tlie above formula, which should only be taken as indicating an a^^proximate law. 236. Cost of rails. In 1873 the cost of steel rails was about $120 per ton, and the cost of iron rails about $70 per ton. Although the steel rails were at once recognized as superior to iron rails on account of more uniform wear, they were an expensive luxury. The manufacture of steel rails by the Ues- semer process created a revolution in prices, and they have steadily dropped in price until, during the last few years, steel rails have been manufactured and sold for $22 per ton. At such prices there is no longer any demand for iron rails, since the cost of manufacturing them is substantially the same as that of steel rails, while their durability is unquestionably inferior to that of steel rails. CHAPTEE X. RAIL- FASTENINGS. RAIL-JOINTS. 237. Theoretical requirements for a perfect joint. A perfect rail- joint is one that has the same strength and stifness — no more and ho less — as the rails which it joins, and which will not interfere with the regular and uniform spacing of ties. It should also be reasonably cheap both in first cost and in cost of maintenance. Since the action of heavy loads on an elastic rail is to cause a wave of translation in front of each wheel, any change in the stiffness or elasticity of the rail structure will cause more or less of a shock, which must be taken up and resisted by the joint. The greater the change in stiffness the greater the shock, and the greater the destructive action of the shock. The perfect rail- joint must keep both rail ends trulv in line both laterally and vertically, so that the flange or tread of the wheel need not jump or change its direction of motion sud- denly in passing from one rail to the other. A consideration of all the above requirements will show that only a perfect weldino' of rail-ends would produce a joint of uniform strength and stiff- ness which would give a uniform elastic wave ahead of each wheel. As welding is impracticable for ordinary railroad work (see § 230), some other contrivance is necessary which will approacli this ideal as closely as may be. 238. Efficiency of the ordinary angle-bar. Throughout the middle portion of a rail the rail acts as a continuous girder. If we consider for simplicity that the ties are unyielding, the deflec- tion of such a continuous girder between the ties will be but 255 256 RAILROAD CON'STRUCTIO:sr. § 239. one-fourtli of tlie deflection that would be found if the rail were cut half-way between the ties and an equal concentrated load were divided equally between the two unconnected ends. The maximum stress for the continuous girder would be but one-half of that in the cantilevers. Joining these ends with rail-joints will give the ordinary "suspended" joint. In order to main tain uniform strength and stiffness the angle-bars must supply the deficiency. These theoretical relations are modified to an unknown extent by the unknown and variable yielding of the ties. From some experiments made by the Association of Engineers of Maintenance of Way of the P. R.K.^ the following deduc- tions were made : 1. The capacity of a "suspended " joint is greater than that of a "supported" joint — whether supported on one or three ties. (See § 240.) 2. That (with the particular patterns tested) the angle-bars alone can carry only 53 to 56/^ of a concentrated load placed on a joint. 3. That the capacity of the whole joint (angle-bars and rail) is only 52.4:^ of the strength of the unbroken rail. 4. That the ineffectiveness of the angle-bar is due chiefly to a deficiency in compressive resistance. Although it has been universally recognized that the angle- bar is not a perfect form of joint, its simplicity, cheapness, and reliability have caused its almost universal adoption. Within a very few years other forms (to be described later) have been adopted on trial sections and have been more and more extended, until their present use is very large. The present time (1900) is evidently a transition period, and it is quite probable tliat within a very few years the now common angle-plate will be as unknown in standard practice as the old-fashioned "fish-plate" is at the present time. 239. Eifect of rail gap at joints. It has been found that the jar at a joint is due almost entirely to the deflection of the joint * Roadmasters Association of America — Reports for 1897. § 240. RAIL-FASTENINQS. 257 and scarcely at all to the small gap required for expansion. This gap causes a drop equal to the versed sine of the arc hav- ing a chord equal to the gap and a radius equal to the radius of the wheel. Taking the extreme case (for a 30-foot rail) of a f " gap and a 33" freight-car wheel, the drop is about yijVtt"- ^^^ order to test how much the jarring at a joint is due to a gap be- tween the rails, the experiment was tried of cutting shallow notches in the top of an otherwise solid rail and running a loco- motive and an inspection car over them. The resulting jarring was practically imperceptible and not comparable to the jar pro- duced at joints. Xotwithstanding this fact, many plans have been tried for avoiding this gap. The most of these plans con- sist essentially of some form of compound rail, the sections breaking joints. (Of course the design of the compound rail has also several other objects in view.) In Fig. 117 are shown a Fig. 117.— CoMrouxD Ratl Section^s. few of the very many designs which have been proposed. These designs have invariably been abandoned after triaL Another plan, which has been extensively tried on the Lehigh Yalley U.K., is the use of mitered joints. The advantages gained by their use are as yet doubtful, while the added expense is unques- tionable. The " Eoadmasters Association of America" in 1S95 adopted a resolution recommending mitered joints for double track, l)ut their use does not seem to be growinir. 240. "Supported," "suspended," and "bridge" joints. In a supported joint the ends of the rails are on a tie. If the angle- plates are short, the joint is entirely supported on one tie ; if very long, it may be possible to place three ties under one angle- bar and thus the joint is virtually supported on three ties rather than one. In a suspended joint the ends of the rails are midway between two ties and the joint is supported by the two. There 258 RAILROAD CONSTRUCTION. § 241. have always been advocates of both methods, but suspended joints are more generally used than supported joints. The opponents of three-tie joints claim that either the middle tie will be too strongly tamped, thus making it a supported joint, or that, if the middle tie is weakest, the joint becomes a very long (and therefore weak) suspended joint between the outer joint-ties, or that possibly one of the outer joint-ties gives way, thus breaking the angle-plate at the joint. Another objection which is urged is that unless the bars are very long (say tti inches, as used on the Mich. Cent. E.K.) the ties are too close for proper tamp- ing. The best answer to these objections is the successful use of these joints on several heavy-traffic roads. " Bridge "-joints are similar to suspended joints in that the joint is supported on two ties, but there is the important differ- ence that the bridge- joint supports the rail from underneath and there is no transverse stress in the rail, whereas the supported joint requires the combined transverse strength of both angle- bars and rail. A serious objection to bridge- joints lies in the fact of their considerable thickness between the rail base and the tie. When joints are placed " staggered '* rather than '' oppo- site " (as is now the invariable standard practice), the ties sup- porting a bridge-joint must either be notched down, thus w^eakening the tie and promoting decay at the cut, or else the tie must be laid on a slope and the joint and tlie opposite rail do not get a fair bearing. 241. Failures of rail-joints. It has been observed on double- track roads that the maximum rail wear occurs a few inches be- yond the rail gap at the joint in the direction of the traffic. On single-track roads the maximum rail wear is found a few inches each side of the joint rather than at the extreme ends of the rail, thus showing that the rail end deflects down under the wheel until (with fast trains especially) the wheel actually jumps the space and strikes the rail a few inches beyond the joint, the impact producing excessive wear. This action, which is called the "drop," is apt to cause the first tie beyond the joint to become depi'essed, and unless this tie is carefully watched and main- 242. RAIL-FA STEISINOb. 259 tained at its proper level, the stresses in the aiigle-l)ar may actually become rev^ersed and the bar may break at the tu]). The angle-bars of a suspended joint are normally in coinpression at the top. The mere reversal of the stresses would cause the bars Fig. 118. — Eh-ect of " Wheel Drop " (Exaggerated). to give way with a less stress than if the stress were always the same in kind. A supported joint, and especially a three-tie joint (see § 240), is apt to be broken in the same manner. 242. Standard angle-bars.— An angle-bar must be so made as to closely fit the rails. The great multiplicity in the designs of rails (referred to in Chapter IX) results in nearly as great variety in the detailed dimensions of the angle-bars. The sec- tions here illustrated must be considered only as types of the variable forms necessary for each different shape of rail. The absolutely essential features required for a fit are (1) the angles Fig. 119.— Standard Angle-bar— 80-lb. Rail. M. C. R.R. of the upper and lower surfaces of the bar where they fit against the rail, and (2) the height of the bar. The bolt-holes in the 260 RAILROAD CONSTRUCTION. § 243. bar and rail must also correspond. The holes in the angle-plates are elongated or made oval, so that the track-bolts, which are made of corresponding shape immediately mider the head, will not be turned by jarring or vibration. The holes in the rails are made of larger diameter (by about ^") than the bolts, so as to allow the rail to expand with temperature. 243. Later designs of rail-joints. In Plate XYIII are shown various designs which are competing for adoption. The most j)rominent of these (judging from the discussion in the conven- tion of the Roadmasters Association of America in 1897) are the " Continuous " and the " Weber." Each of them has been very extensively adopted, and where used are universally pre- ferred to angle-plates. [N^early all the later designs embody more or less directly the principle of the bridge- joint, i.e., sup- port the rail from underneath. An experience of several years will be required to demonstrate which form of joint best satis- fies the somewhat opposed requirements of minimum cost (])oth initial and for maintenance) and minimum wear of rails and rolling stock. TIE-PLATES. 244. Advantages. (a) As already indicated in § 204, the life of a soft-wood tie is very much reduced by "rail-cutting" and "spike-killing," such ties frequently requiring renewal long before any serious decay has set in. It has been practi- cally demonstrated that the "rail-cutting" is not due to the mere pressure of the rail on the tie, even with a maximum load on the rail, but is due to the impact resulting from vibration and to the lono^itudinal workino' of the rail. It has been proved that this rail-cutting is practically prevented by the use of tie-plates. (h) On curves there is a tendency to overturn the outer rail due to the lateral pressure on the side of the head. This produces a concentrated pressure of the outer edge of the base on the tie which produces rail-cutting and also draws the inner spikes. Formerly the only method of guarding PLATE XVIII. WEIR BOLTED STIFF FROG. r7?ff;<«y^yw.'y.^\t SECTION THROUGH C-D. SECTION THROUGH A-B. ELLIOT PLATE RIVETED FROG. SECTION THROUGH PLATE AT POINT. Kail Joints and Frogs. SECTION THROtIGH SPRING-HOUSING. {To face page 260.) § 245. RAIL-FASTENiyGS. 201 as'ainst this was bv the use of " rail -braces," one pattern of which is shown in Fig. 12U. But it has been found that tie- FiG. 120. plates serve the purpose even better, and rail-braces have been abandoned where tie-plates are used, {c) Driving spikes through holes in the plate enables the spikes on each side of the rail to mutually support each other, no matter in which (lateral) direc- tion the rail may tend to move, and this probably accounts in large measure for the added stability obtained by the use of tie- plates, id) The wear in spikes, called ' ' necking, ' ' caused by the vertical vibration of the rail against them, is very greatly reduced, {e) The cost is very small compared with the value of the added life of the tie, the large reduction in the work of track maintenance, and the smoother running on the better track which is obtained. It has been estimated that by the use of tie-plates the life of hard-wood ties is increased from one to three years, and the life of soft-wood ties is increased from three to six years. From the very nature of the case, the value of tie-plates is greater when they are used to protect soft ties. 245. Elements of the design. The earliest forms of tie-plates were llat on the bottom, but it was soon found that they would work loose, allow sand and dirt to o^et between the rail and the plate and also between the plate and the tie, which would cause excessive wear. Such plates are also apt to produce an objec- tionable rattle. Another fault of the earlier designs was the use of plates so thin that they would buckle. The latest designs have flanges or " teetli " formed on the lower surface which penetrate the tie about f" to If". Opinion is still divided on the question of whether these teeth should run with the grain 262 BAILBOAD CONSTRUCTION. § 246. or across the grain. If the flanges run with the grain, they generally extend the whole length of the tie-plate — as in the Wolhaupter design. If the grain is to be cut crosswise, several teeth about 1" wide will be used — as in the Goldie design. WOLHAUPTER Fig. 121. — Tie-plates. It is a very important feature that the spike-holes shou/d be so punched that the spikes will fit closely to the base of tlie rail. Otherwise a lateral motion of the rail will be permitted which will defeat one of the main objects of the use of the plate. Another unsettled detail is the use of "shoulders" on the upi^er surface. On the one hand it is claimed that tlie use of shoulders relieves the spikes of side pressure from the rail and prevents "necking." On the other hand it is claimed that if the plain plate is once properly set with new spikes (at least Avith spikes not already necked) the spikes will not neck appre- ciably, and that, as the shouldered plates cost more, the additional expenditure is unnecessary. The above designs should be studied with reference to the manner in which they fulfill the requirements which have been already stated. As in the case of rail-joints, the best forms of tie-plates are of comparatively recent design, and experience with them is still insufticient to determine beyond all question which designs are the best. 246. Methods of setting. A very important detail in the process of setting the tie-plates on the ties is that the flanges or teeth should penetrate the tie as far as desired when the plates are flrst put in position. It requires considerable force to press the teeth into a tie. In a few cases trackmen have depended on the easy process of waiting for passing trains to force the teeth §247. RAIL-FASTENINGS. 263 down. Until tlie teeth arc down the spikes cannot l)e driven home, and this apparently clieap and easy process resiiks in loose spikes and rails. If the trackmen neglect even temporarily to tighten these spikes, it will become impossible to make them tight ultimately. The })lates are generally pomided into place with a 10- to 16-pound sledge-hanmier. A very good method was adopted once during the construction of a bridge when a pile-driver was at hand. The bridge-ties were placed nnder the pile-hammer. The plates, accurately set to gauge, were then forced in by a blow from the 8000-lb. hammer falling 2 or 3 feet. SPIKES. 247. Requirements. The rails must be held to the ties by a fastening wdiich will not only give sufHcient resistance, but which will retain its capacity for resistance. It must also be cheap and easily applied. The ordinary track-spike fulfills the last requirements, but has comparatively small resisting power, com- pared with screws or bolts. Worse than all, the tendency to^ vertical vibration in the rail produces a series of upward pulls on the spike that soon loosens it. When motion has once beo-nn the capacity for resistance is greatly reduced, and but little more vibration is required to pull the spike out so much that redriving is necessary. Driving the spike to place again in the same hole is of small value except as a very temporary expedient, as its holding power is then very small. Redriving the spikes in new holes very soon " spike-kills " the tie. Many plans have been devised to increase the holding power of spikes, such as making them jagged, twisting the spike, swelling the spike at about the center of its length, etc. But it has been easily demon- strated that the fibers of the wood are gen- ^^<^- ^22. erally so crushed and torn by driving such spikes that their holding power is less than that of the plain spike. 264 RAILROAD CONSTRUCTION. 248. The ordinary spike (see Fig. 122) is made with a square cross-section which is uniform through the middle of its length, the lower If tapering down to a chisel edge, the upper part swelling out to the head. The Goldie spike (see Fig. 123) aims to improve this form by reducing to a minimum the destruction of the fibers. To this end, the sides are made smooth^ the edges are clean-cut, and the point, instead of being chisel-shaped, is ground down to a pyramidal form. Such fiber- cutting as occurs is thus accomplished without much crushing, and the fibers are thus pressed away from the spike and slightly downward. Any tendency to draw the spike will therefore cause Fig. 123. the fibers to press still harder on the spike and thus increase the resistance. 248. Driving. The holding power of a spike depends largely on how it is driven. If the blows are eccentric and irregular in direction, the hole will be somewhat enlarged and the holding power largely decreased. The spikes on each side of the rail in any one tie should not be directly opposite, but should be staggered. Placing them directly opposite will tend ( ^ to split the tie, or at least decrease the holding power of the spikes. The direc- tion of staggering should be reversed in the tAvo pairs of spikes in any one tie ^^^- ^24. Spike-driving. (see Fig. 124). This will tend to prevent any twisting of the tie in the ballast, which would otherwise loosen the rail from the tie. 249. Screws and bolts. The use of these abroad is very ex- tensive, but 'their use in this country has not passed the experi- mental stage. The screws are " wood "-screws (see Fig. 125), having large square heads, which are screwed down with a track- wrench. Holes, having the same diameter as the hase of the screw-threads, should first be bored into the tie, at exactly the right position and at the proper angle with the vertical. §249. RAIL-FASTENINGS. 265 A liirlit wooden frame is soinetiiiies used to "niido the aiicer at the proper angle. Sometimes the large head of the screw bears directly against the base of the rail, as with the ordinary spike. Other designs employ a plate, made to tit the rail on one side, bearing on the tie on the other side, and through which the screw passes. These screws cost much more than spikes and require more work to put in place, but their holding power is much greater and the work of track maintenance is very nmcb less. Screw-bolts, passing entirely through the tie, liavinir the head at the bottom of the tie and the nut on Fig 125. the upper side, are also used abroad. These are quite difficult to replace, requiring that the ballast be dug out beneath the tie, but on the other hand the occasions for replacing such a bolt are comparatively rare, as their durability is very great. The ^ ^1 i F'"^ ' 1 — ^-J ill Fig. 126. use of screws or bolts increases the life of the tie by the avoid- ance of " spike-killing." It is capable of demonstration tliat the reduced cost of maintenance and the resulting improvement in track would much more than repay the added cost of screws and bolts, but it seems impossible to induce railroad directors to authorize a large and immediate additional expenditure to make an annual saving whose value, although unquestionably consider- able, cannot be exactly computed. 266 RAILROAD CONSTRUCTION. 250. 250. "Wooden spikes." Among the regulations for track- lajing given in § 208, mention was made of wooden "spikes," or plugs, wliicli are used to fill up the holes wlien spikes are withdrawn. The value of the policy of filling up these holes is unquestionable, since the expense is insignificant compared with the loss due to the quick and certain decay of the tie if these holes are allowed to fill with water and remain so. But the method of making these plugs is variable. On some roads they are "hand-made'- by the trackmen out of otherwise useless scraps of lumber, the work being done at odd moments. This policy, while apparently cheap, is not necessarily so, for the hand-made plugs are ir- reofular in size and therefore more or less inefticient. gang IS a track they may spend which could be Since the holes It is also quite probable that if required to make their own plugs, time on these very cheap articles more profitably employed otherwise, made by the spikes are larger at the top than they are near the bottom, the plugs should not be of uniform cross section but should be slightly wedge-shaped. The " Goldie tie-plug" (see Fig. 127) has been de- signed to fill these requirements. Being machine- made, they are uniform in size ; they are of a shape which will best fit the hole ; they can be furnished of any desired wood, and at a cost which makes it a wasteful economy to at- tempt to cut them by hand. Fig. 127. TRACK-BOLTS AND NUT-LOCKS. 251. Essential requirements. The track-bolts must have sufticient strength and must be screwed up tight enough to hold the angle-plates against the rail with sufficient force to develop the full transverse strength of the angle-bars. On the other hand the bolts should not be screwed so tight that slipping may not take place when the rail expands or contracts with temperature. It would be impossible to screw the bolts tight enough to prevent § 252. liAIL-FA8TENINGS. 267 slipping chiring the contraction due to a considerable fall of temperature on a straight track, but when the track is curved, or when expansion takes place, it is conceivable that the resist- ance of the ties in the ballast to lateral motion may be less than the resistance at the joint. A test to determine this resistance was made by Mr. A. Torrey, chief engineer of the Mich. Cent. R.R., using 80-lb. rails and ordinary angle-bars, the bolts being screwed up as usual. It required a force of about 31000 to 35000 lbs. to start the joint, which would be equivalent to the stress induced by a change of temperature of about 22°. Bnt if the central angle of any given curve is small, a comparatively small lateral component will be sufficient to resist a compression of even 35000 lbs. in the rails. Therefore there Avill ordinarily be no trouble about having the joints screwed too tight. The vibration caused by the passage of a train reduces the resistance to slipping. This vibration also facilitates an objectionable feature, viz., loosening of the nuts of the track-bolts. The bolt is readily prevented from turning by giving it a form wdiich is not circular innnediately under the head and making corre- sponding holes in the angle-plate. Square holes would answer the purpose, except that the square corners in the holes in the angle-plates would increase the danger of fracture of the plates. Therefore the holes (and also the bolts, under the head) are made of an oval form, or perhaps a square form with rounded corners, avoiding angles in the outline. The nut-locks should be simple and cheap, should have a life at least as long as the bolt, should be effective, and should not lose their effectiveness with age. IFany of the designs that have been tried have been failures in one or more of these particulars, as will ])e described in detail below. 252. Design of track-bolts. In Fig. 128 is shown a common design of track-bolt. In its general form this represents die bolt used on nearly all roads, being used not only with the common angle-plates, but also with many of the im- proved designs of rail- joints. The variations are chiefly a general increase in size to correspond with the increased 268 RAILROAD CONSTRUCTION. 253. Fig. 128 —Track-bolt. weight of rails, besides variations in detail dimensions wliicli are frequently unimportant. The diameter is usually f '' to y ; 1" bolts are sometimes used for the heaviest sections of rails. As to length, the bolts should not ex- tend more than -J" outside of the nut when it is screwed up. If it extends farther than this, it is liable to be broken olf by a possible derail- ment at that point. The lengths used vary from 3i^', which may be used "f^ with 60 lbs. rails, to 5'', which is required with 100-lb. rails. The length required depends somewliat on the type of nut- lock used. 253. Design of nut-locks. The designs for nut-locks may be divided into three classes : {a) those depending entirely on an elastic washer which absorbs the vibration which might other- wise induce turning; (Ij) those which jam the threads of the bolt and nut so that, when screwed up, the frictional resistance is too great to be overcome by vibration ; {c) the ' ' positive ' ' nut-locks — those which mechanically hold the nut from turning. Some of the designs combine these principles to some extent. The ' ' vulcanized fiber ' ' nut-lock is an example of the first class. It consists essentially of a rubber washer which is pro- tected by an iron ring. When first placed this lock is effective, but the rubber soon hardens and loses its elasticity and it is then ineffective and worthless. Another illustration of class {a) is the use of wooden blocks, generally of 1" to '■2" oak, which extend the entire length of the angle- bar, a single piece forming the washer for the four or six bolts of a joint. This form is cheap, but the wood soon shrinks, loses its elasticity, or decays so that it soon becomes worthless, and it requires constant adjust- ment to keep it in even tolerable condition. The " Yerona" nut-lock is another illustration of class {a) which also combines some of the positive elements of class {c). It is made of §258. RAIL- FASTE^^INGS. 269 tempered steel and, as shown in Fig. 129, is warped aiid lias sharp edges or points. The warped form furnishes the element of elastic pressure when the nut is screwed up. The steel beino- harder than the iron of the angle-bar or of the nut, it bites into them, owing to the great pressure that must exist VERONA NATIONAL^ JONES excelsior-- Fig. 129.— Types of Nut-locks. Tvhen the washer is squeezed nearly flat, and thus prevents any hackward movement, although forward movement (or tighten- ing the bolt) is not interfered with. The " National " nut-lock is a type of the second class {h), in which, like the " Harvey " nut-lock, the nut and lock are combined in one piece. With six-bolt ande-bars and 30-foot rails, this means a saving of 2112 pieces on each mile of single track. The " National " nuts are open on one side. The hole is drilled and the thread is cut sliijhtly smaller than the bolt, so that when the nut is sci'ewed 270 BAILROAD CONSTRUCTION. § 253. up it is forced slightly open and therefore presses on the threads of the bolt with such force that vibration cannot jar it loose. Unlike the " j^ational " nut, the *' Harvey" nut is solid, but the form of the thread is progressively varied so that the thread pinches the thread of the bolt and the friction al resistance to turning is too great to be affected by vibration. The ''Jones" nut-lock, belonging to class (. (82) BF= L = {g -/sin F) cot 17^+/ cos F = 2gn — y* sin F cot iF-\-f cos F = "^^gn -/(I + cos F) +/ cos F = ^-gn-f. Since r — ig — (Z —f sec F) cot F^ and r -\-^g = {L — f cos F) cosec i% (83) 264. SWITCHES AND CROSSINGS 281 r = iZ (cot jF-\- cosec i^) — if sec F QOt F — \f cos F cosec i'" ~ ~~ 2 / i^ gill J<^ r = Z?i - i/ cot \F = Ln — fn. Then from (S3) r = 2^71^ — 2f?i (8i) 264. Effect of straight point-rails. The "point switches," now so generally used, have straight switcli-rails. This requires an angle in the aHgnnient rather than turning off by a tangential curve. The angle is, however, very small (between 1° and 2°), and the disadvantages of this angle are small compared with the very great advantages of the device. .^\v- ?-a * o \ MN = /C a -a FM= . Fig. 139. g - h ^+ ig = smi{F+ a)' FiV 2 sin i(F — a) g — k 2 sin i{F + a) sin i{F g - ^- cos a — cos F' -a) (85) 282 BAILBOAD CONSTRUCTION. §265. BF=L=: FM cos i(i^+ a) + DJSr = {g- I') cot i{F-\- a) + D^\ (86) 265. Combined effect of straight frog-rails and straight point- rails. It becomes necessary in this case to find a curve which shall be tangent to both the point-rail and the frog-rail. The curve therefore begins at M, its tangent making an angle of (x (nsiiallj 1° 50') with the main rail, and runs to H. The central 1' / FH=/ VMDN=a: P_^ VHMR=M(F-n:) a Fig. 140. angle of the curve is therefore {F — a). The angle of the chord JIM with the main rails is therefore i{F^a)+a=.UF+a); _ g — f sin F — k ^^ ~ sinU-^+a) *' ^-r 29 - ^ sin^(i^- a) g — f sin F — h 2 sin ^X^ + a) sin ^{F — a) g — f m\ F — h cos a — cos F ^ ' ST = 2r sin ^{F - a). . (87) . (88) §266. SWITCHES AND CROSSINGS. 283 BF = L = lUL cos \{F + ^) +/ cos 7<^ + DN =z{g - f sin F - k) cot i{F + a) + / cos 7^^+ D^\ (SO) It may be more simple, if (;• -|- ^fj) lias already been com- puted, to write Z = 2(/' + i(/) sin i{F- a) cos U^+a) +/cos F+ DJV = (/' + 4-^)(sin F - sin «') + / cos Z' + DJS', . . (90) 266. Comparison of the above methods. Computing values for r and Z by tlie various methods, on the uniform basis of a Is^o. 9 frog, standard gauged' sy\f= 3'. 37, k = 5i"= 0'.479, Dy =15' 0", and « = 1° 50', we may tabulate the compara- tive results : Simple circle Curved frog r. Curved s\vitch-r. § 263. Straitrht frop:-r. Curved switch-r. § 204. Curved frop:-r. Straight switch-r. § 265. Straight frog-r. Straight switch r. r Deg. of curve L 762.75 7" 31' 84.75 702.00 8° 10' 81.37 747.48 7° 40' 74.00 681.16 8° 25' 72.13 This shows that the effect of using straight frog-rails and straight switch-rails is to sharpen the curve and shorten the lead in each case separately, and that the combined effect is still greater. The effect of the straight switch -rails is especially marked in reducing the length of lead, and therefore Eq. 78 to 80, although having the advantage of extreme simplicity, can- not be used for point-switches without material error. The effect of the straio-ht froo^-rail is less, and since it can be mate- rially reduced by bending the free end of the frog- rails, the in- fluence of this feature is frequently ignored, the frog-rails are assumed to be curved and Eq. 85 and 86 are used. (Soe § 276 for a further discussion of this point.) 284 RAILROAD CONSTRUCTION. 267. 267. Dimensions for a turnout from the outer side of a curved track. In this demonstration the switch-rails will be considered as uniformly circular from the switch-points to the frog-point. Fig. 141. In the triangle FCD (Fig. 141) we have {FC+ CD) : (FC- CD) : : tan i{FDC+DFC) : tan i(FDC-DFC) ; but i{_FDC+ DFC) = 90° - \d and \{FDC - DFC) = iF. Also FG+ CD = 2E and FC - CD = g; .-. 2B:g: : cot |^: tan iF : : cot ^F: tan ^(^ ; tan ^6 = ^. (91) Also OF : FC: : sin 6 : sin ; but cp = {F — 6)\ then r -X- -ka = ( M -X- ^a^-. — r-^^- — :^. . . (92) (93) ^7^ = L = 2{R + ^g) sin i^. If the curvature of the main track is very sharp or the frog angle unusually small, i^may be less than 6-^ in which case the center will be on the same side of the main track as C, Eq. 92 will become (by calling r =. — r and changing the signs) (r - \g) = (^ + \g) sin B sin {6-F)- (94) §267. SWITCHES AND CROSSINGS. 285 If we call d the degree of curve corresponding to the radius r, and D the degree of curve corresponding to the radius i?, also d ' the degree of curve of a turnout from a straight track (the frog angle F being the same), it may be shown that d — d' — D (very nearly). To illustrate we will take three cases, a number 6 frog (very blunt), a number 9 frog (very commonly used), and a number 12 frog (unusually sharp). Suppose 2> = 4° 0'; also Z) = 10° 0'; g ^ ■^' Si" = Ir'.TOS. D-. = 4°. Frog: number. " L " for straight track. . d' - D Error. L 6 12° 54' 20" 12' 57' 52" 0' 03' 32" 56.57 56.50 9 i 3 30 27 3 31 04 37 84.85 84.75 12 13 33 13 36 03 112.72 113.00 D = 10° Frog: number. , "i" for straight track. d d'— D Error. L 6 6° 53' 24" 6° 57' 52 ' 0° 04' 28" 56.66 56.50 9 2 27 54 2 28 56 01 02 84.86 84.75 12 5 44 26 5 40 24 01 58 112.91 113.00 A brief study of the above tabular form will show that the error involved in the use of the approximate rule for ordinary curves (-1° or less) and for the usual frogs (about Xo. 9) is really insignificant, and that, even for sharper curves (10° or more), or for very blunt frogs, the error would never cause damage, considering the lower probable speed. In the most unfavorable case noted above the change in radius is about Ifc. On account of the closeness of the approximation the method is frequently used. The remarkable agreement of the computed values of Z with the corresponding values for a straight main track (the lead 286 RAILROAD CONSTRUCTION. 268. rails circular tlirougliout) shows that the error is insignificant in using the more easily computed values. 268. Dimensions for a turnout from the inner side of a curved track. (Lead rails circular throughout.) From Fig. 142 we have DC+FC\DC -FC::i^ni{DFC+FI)C) : i^n^{DFC- FDC)] but k{DFC+ FDC) = 90° - ^6 and ^(DFC - FDC) = \F', ,'. 2i?:^: :cot 1(9: tan ^F : : cot ^F: tan |-6'; .*. tan 16 gn R' (95) OF:FC:\du e',^in{F+6). ir + J,) = (i? - ^,)g^^^^. Z = BF = 2{R - ig) sin id. ■ (96) • (97) As in § 267, it may be readily shown that the degree of the turnout (d) is nearly the sum of the degree of the main track (7>) and the degree (d ') of a turnout from a straight track when the frog angle is the same. The discrepancy in this case is § 260. SWITCHES AND CROSSINGS. 287 somewhat greater tlian in the other, especially when the curva- ture of the main track is sharp. If the frog angle is also laro-e, the curvature of the turnout is excessively sharp. If the fro, = F,-iF, + iF„ = i{F, + iF„) ; §270. SWITCHES A^D CliOSSINQS. 289 i^.i^... = KF,. 9 I-*- m in KFiK, 2 sin i{Fi + iF,,) sin -,;. . . 0^02) KFi = KF,,, cot KFiF„, = \(j cot \{F, + W.:) ; (103) {r. + 4^) = F,F. l-*- rn 2 sin i^ ^! 4 sin iCi-', + ii^„,) sin i(JP, - ^F,,) ^9 COS ^Fm — COS i^i (104) Fig. 144. If three frogs, all different, miist be used, the largest may be selected as F^n ; the radius of the lead rails may be found by an inversion of Eq. 98; F,n may be located in the center of the tracks by Eq. 99 ; then each of the smaller frogs may be located by separate applications of Eq. 102 or 103, the radius being determined by Eq. 104. 270. Two turnouts on the same side. In Fig. 145, let 0, bisect 0,D. Then {r, + \g) = ^{r, -f- ^g) ; also, 0,0, = 0,F, and Fr = F^, vers F - ^ - ^^ -*- ni — . - — : : — . (105) ^F„, = {9\ + i^)sin F„, (106) It may readily be shown that the relative values of F^, Fiy and F,n are almost identical with those given in § 269 ; as may 290 RAILROAD CONSTRUCTION. §271. be apparent when it is considered that the middle switch may be regarded simply as a curved main track, and that, as Fig. 145. developed in § 267, the dimensions of turnouts are nearly the, same whether the main track is straight or slightly curved. 271. Connecting curve from a straight track. The "con- necting curve ' ' is the track lying between the frog and the side track where it becomes parallel to the main track (FS in Fig. 146 or 147). Call d the distance between track centers. The angle FO,R = F (see Fig. 146). Call t' the radius of the connecting curve. Then d - g _ Fm. 146. (^' - ¥j) - vers F (107) FR = (r - ig) sin F. . . (108) If it is considered that the distance FI^ consumes too much track room, it may be shortened by the method indicated in Fig. 151. 272. Connecting curve from a curved track to the outside. When the main track is curved, the required quantities are the radius r of the connecting curve from Fto S, Fig. 147, and its length or central angle. In the triangle CSF OS+CF: CS-OF:: tstn i{OFS-i- CSF) : tmi{CFS -CSF); §273. SWITCHES AND CROSSINGS. 291 but 1{CFS+ CSF) == 90 - ^?/^; and, since the triangle 0,SF is isosceles, i{CFS - CSF) = iF; o-o 27?+6Z : 6? — ^ :: cot ^^ : tan ^i^ : : cot ^F : tan ^ip ; 1 _ ^^(^ - (/ ) Fig. 147. From the triangle OO^F we may derive r — ig : J2 -\- ig : : sin tp : sin (i^ + ^) ; sin i/j Also ir^3.2(r-i^)sini(i^+^). (109) (110) (111) 273. Connecting curve from a curved track to the inside. As above, it may readily be deduced from the triangle CFS (see Fig. 148) that {2B - d) : {d - g) :: cot i^p : tan iF, and finally that 2n(d - a) tan i.p = ^^j^l (112) 292 RAILROAD CONSTRUCTION. Similarly we may derive (as in Eq. 110) 273. Also FS = 2{r - kg) sin 4(i^ - ^). . (113) . (114) Fig. 148. Two other cases are possible, {a) r may increase until it becomes infinite (see Fig. 149), then F ziz tp. In such a case we may write, by substituting in Eq. 112, <2R-d^^n\d-g), (115) Fig. 149. This equation shows the value of i?, which renders this case possible with the given values of n^ c/, and g. (b) ip may be greater than F. As before (see Fig. 150) 2^ — ', - id; Fig. 155. vers ip = sin 0(\0, = d(r, -\- 7\ — id) {R - idTYr'M'^id'^^y ' • • ^^"''^ . 00, . B + id-r, s,n^,^=sm^^-^-p^; . . (126) i^+ 0,0,0; (127) 2{/2 - id + i(/) sin J(-A - ^, - ^J. . (128) 298 RAILROAD CONSTRUCTION. 276. Althougli tlie above method introduces a reversed curve, yet it uses up less track than the first method and permits the use of ordinary frogs rather than those having some special angle which must be made to order. 276. Practical rules for switch-laying. A consideration of the previous sections will show that the formulae are compara- tively simple when the lead rails are assumed as circular ; that they become complicated, even for turnouts from a straight main track, when the effect of straight frog and point rails is allowed for, and that they become hopelessly complicated when alio wins; for this effect on turnouts from a curved main track. It is also shown (§ 267) that the length of the lead is practically - — i r '" MN=fc , FH=/ Vhmr^v:; (F-a) ^ Fig. 140. the same whether the main track is straight or is curved with such curves as are commonly used, and that the degree of curve of the lead rails from a curved main track may be found with close approximation by mere addition or subtraction. From this it may be assumed that, if the length of lead (Z) and the radius of the lead rails (r) are computed from Eq. 87 and 90 for various fros: ans^les, the same leads mav be used for curved main track ; also, that the degree of curve of the lead rails may be found by addition or subtraction, as indicated in § 267, and that the approximations involved will not be of practical detriment. §276. SWITCHES AJSD CROSSINGS. 299 In accordance with this pkm Table III has been computed from Eq. 87, 88, and 90. The leads there given may be used for all main tracks straight or curved. The table gives the degree of curve of the lead rails for straight main track; for a turnout to the inside, add the degree of curve of the main track ; for a turnout to the outside, suhtract it. If the position of the switch-block is definitely determined, then the rails must be cut accordingly ; but when some freedom is allowable (wdiich never need exceed 15 feet and may require but a few inches), one rail-cutting may be avoided. Mark on the rails at B, F, and D ; measure off the length of the switch- rails DN\ offset \(j -f li from N for the point S. The point H may be located (temporarily) by meas- uring along the rail a distance i^7/ (=/) and then swinging out a distance of / -^ ii (n being the frog number). HT — \(j and is measured at right angles to FII. Points for track centers between S and T may be laid off by a transit or by the use of a string and tape. Substituting in Eq. 31 the value of R and of chord (= 8T), w^e may compute x (= dh). Locate the middle point d and the quarter points a" and c" . Then a" a and c"c each equal three-fourths of dl. Theoretically this gives a parabola rather than a circle, but the difference for all practical cases is too small for measurement. Example. Given a main track on a 1° curve ; a turnout to the outside, using a number 9 frog; gauge \' 8i" ; /"* = 8'. 87* h = hi" : DY = 15' 0" and a = 1° 50'. Then for a straight track r would equal 681.16 [d = 8° 25']. For this curved track d will be nearly (8° 25' — 4°) = 4" 25', or r will be 1207.6. Z for the straight track would be 72.20; but since the lead is slightly increased (see ^ 2(w) when the turnout is on the outside of a curve, Z may hei-e l)e called 72.5. Z7/ = f = 3'.37;/-- ii = 3.37--9=0'.375=4".5. 7/, T, and >^ may be located as described above. ST may be measured on the ground, or it may be computed from Eq. 88, giviuir the value «h- FiG. 156. 300 RAILROAD CONSTRUCTION. 217. of 53.80 feet for straiglit track. Since it is slightly more for a turnout to the outside of a curve, it may be called 54.0. Then „ (54. oy 8 X 1297.6 "^ ^'^^^ ^^®*' ^'^^ ^^'' ^^^ ^^" = ^-21 foot. CROSSINGS. 277. Two straight tracks. When two straight tracks cross each other, four frogs are necessary, the angles of two of them being supplementary to the angles of the other. Since such crossings are sometimes operated at high speeds, they should be SECTION ON A-B SrCTION^ON OD Fm. 157.— Crossing, very strongly constructed, and the angles should preferably be 90° or as near that as possible. The frogs will not in o-eneral be "stock" frogs of an even number, especially if the angles are large, but must be made to order with the required angles as measured. In Fig. 157 are shown the details of such a cross- ing. Note the fillers, bolts, and guard-rails. §279. SWITCHES AND CIIOSSINGS. 301 278. One straight and one curved track. Structurally the crossino: is about the same as above, but the froir aiiirles are all unequal. In Fig. 158, 7? is known, and the angle J/, made by the center lines of the tracks at their point of inter- section, is also known. J/ = XCM. SC = n COS M. R cos J/+ \(j COS i^,= 7?- i^ o. .1 1 7^ ^C0SJ/+^^ Similarly cos r „ = Trv~y — ■ R cos M— \(j ^ R cos 2f— ^g cos F,=z — p—z;, — • ^(129) Fig. 158. 279. Two curved tracks. The four frogs are unequal, and the angle of each must be computed. The radii if, and 7?, are :i-- Fig. 159. known ; also the angle M. r, , 7\ , r, , and 7\ are therefore known by adding or subtracting ^g, but the lines are so indi- 302 RAILROAD CONSTRUCTION. § 279. Trn""" "7"^"'^' ^'^^ '^^ '^^^^ ^^^^» = ^- t^e angle MC^C = C[, and the line 0,0, = c. Then and KC, + 6^) = 90° -J- Jf tan iiC - C) = cot iM^^^^' C, and C, then become known and sm 6, In the triangle F, (7. C. , call K« + ^, + r,) = s, ■ then vers i^. = ?^^^=i^XfL^Zi) Similarly vers i^, = ^(■^^ - n)(.?, - ^j vers i^. = ^i^^^ll^k^n) ^ • • (130) ' l' 3 vers i^, = ?(fi_ZLZ?)lfi^^ In the above equations APPENDIX. THE ADJUSTMENTS OP INSTRUMENTS. The accuracy of instrumental work may be vitiated by any one of a large number of inaccuracies in the geometrical rela- tions of the parts of the instruments. Some of these relations are so apt to be altered by ordinary usage of the instrument that the makers have provided adjusting-screws so that the inaccura- cies may be readily corrected. There are other possible defects, which, however, will seldom be found to exist, provided the instrument was properly made and has never been subjected to treatment sufficiently rough to distort it. Such defects, when found, can only be corrected by a competent instrument maker or repairer. A WARNING is necessary to those who would test the accuracy of instruments, and especially to those whose experience in such work is small. Lack of skill in handlintr an instrument will often indicate an apparent error of adjustment when tlie real error is very different or perhaps non-existent. It is always a safe plan wlien testing an adjustment to note the amount of the apparent error; then, beginning anew, make another independ- ent determination of the amount of tlie error. When two or \noYe perfectly independent determinations of such an error are made it will generally be found that they differ by an appreciable amount. The differences may be due in variable measure to careless inaccurate manipulation and to instrumental defects which are wholly independent of the particular test being made. Such careful determinations of the amounts of the errors are generally advisable in view of the next paragraph. 303 304 THE ADJUSTMENTS OF INSTRUMENTS. Do NOT DISTURB THE ADJUSTING -SCREWS ANY MORE THAN NECESSARY. Altliougli metals are apparently rigid, tliey are really elastic and yielding. If some parts of a complicated mechanism, which is held together largely by friction, are sub- jected to greater internal stresses than other parts of the mech- anism, the jarring resulting from handling will frequently cause a slight readjustment in the parts which will tend to more nearly equalize the internal stresses. Such action frequently occurs with the adjusting mechanism of instruments. One screw may be strained more than others. The friction of parts may pre- vent the opposing screw from mimediately taking up an equal stress. Perhaps the adjustment appears perfect under these conditions. Jarring diminishes the friction between the j)arts, and the unequal stresses tend to equalize. A motion takes place which, although microscopically minute, is sufficient to indicate an error of adjustment. A readjustment, made by unskillful hands, may not make the final adjustment any more perfect. The frequent shifting of adjusting-screws wears them badly, and when the screws are worn it is still more difficult to keep them from moving enough to vitiate the adjustments. It is therefore preferable in many cases to refrain from disturbing the adjusting-screws, especially as the accuracy of the work done is not necessarily affected by errors of adjustment, as may be illus- trated : {a) Certain operations are absolutely unaffected by certain -errors of adjustment. {J)) Certain operations are so slightly affected by certain small errors of adjustment that their effect may properly be neglected. (c) Certain errors of adjustment may be readily allowed for and neutralized so that no error results from the use of the un- adjusted instrument. Illustrations of all these cases will be given under their proper heads. AD.JUSTMENTS OF THE TRANSIT. 1. To have the jplate-huhhles m the centei' of the tiibes when the axis is vertical. Clamp the upper plate and, with the lower THE ADJUSTMENTS OF INSTRUMENTS. 305 clamp loose, swing the instruineiit so that the plate-bubbles are parallel to the lines of opposite leveling-screws. Level up until both bubbles are central. Swing the instrument 180°. If the bubbles again settle at the center, the adjustment is perfect. If either bubble does not settle in the center, move the leveling- screws until the bubble is half-icaij back to the center. Then, before touching the adjusting-screws, note carefully the position of the bubbles and observe whether the bubbles always settle at the same place in the tube, no matter to what position the in- strument may be rotated. When the instrument is so leveled, the axis is truly vertical and the discrepancies between this con- stant position of the bubbles and the centers of the tubes measure the errors of adjustment. By means of the adjusting-screws bring each bubble to the center of the tube. If this is done so skillfully that the true level of the instrument is not disturbed, the bubbles should settle in the center for all positions of the instrument. Under unskillful hands, two or more such trials may be necessary. When the plates are not horizontal, the measured angle is greater than the true horizontal angle by the difference between the measured ancle ^nd its projection on a horizontal plane. When this angle of inclination is small, the difference is insignificant. Therefore when the plate-bubbles are very nearly in adjustment, the error of measurement of horizontal angles may be far within the lowest unit of measurement used. A smaJl €rror of adjustment of the plate-bubble J9e7pm(izm?ar to the telescope will affect the horizontal angles by only a small proportion of the error, which will be perhaps imperceptible. Vertical angles will be affected by the same insignificant amount. A small error of adjustment of the plate- bubble iMrallel to the telescope will affect horizontal angles very slightly, but will affect vertical angles by the full amount of the error. All error due to unadjusted plate-bubbles may be avoided by noting in what positions in the tubes the bubbles will remain fixed for all positions of azimuth and then keeping the bubbles adjusted to these positions, for the axis is then truly vertical. It will often save time to work in this way temporarily rather than to stop to make the adjustments. This should especially be done when accurate vertical angles are required. When the bubbles are truly adjusted, they should remain stationary, regardless of whether the telescope is revolved with the upper plate loose and the lower plate clamped or whether the whole instrument is revolved, the plates being clamped together. If there is any appreciable difference, 306 THE ADJUSTMENTS OF INSTRUMENTS. it shows that the two vertical axes or " centers" of the plates are not con- centric. This may be due to cheap and faulty construction or to the exces- sive wear that may be sometimes observed in an old instrument originally well made. In either case it can only be corrected by a maker. 2 . To make the revolving axis of the telescoi:>e jyerpendicular to the vertical axis of the instrument. This is best tested by using a long plumb-line, so placed that the telescope must be ]3ointed upward at an angle of about 45° to sight at the top of the plumb-line and downward about the same amount, if pos- sible, to sight at the lower end. The vertical axis of the transit must be made truly vertical. Sight at the upper part of the line, clamping the horizontal plates. Swing the telescope down and see if the cross- wire a 52 54 5(i 58 (io 240 238 49 85 237 235 234 24 65 08 O T 2 S4 2-43833 ■43494 •43157 •42823 .42492 .42163 2.41837 •41513 .41192 •40873 •40557 •40243 2 ■39931 .39622 • 393 • 5 .39016 •38707 • 3840 7 2.38109 •37813 .37519 .37227 .36937 • 36649 231 .01 226.55 222 . 27 218. 15 2.36363 •35517 . 34688 -33875 214. 18 210.36 206.68 203.13 2.33078 •32296 -31529 ■307/6 99.70 96-38 93-19 90.09 - 30037 - 29316 -28597 • 27896 87. 10 81 .40 76.05 71 .02 66.28 61 .80 57-58 53-58 49-79 46.19 42.77 39-52 36-43 33-47 ^o . 66 27 . 97 25-39 22.93 20.57 18.31 14.06 10.13 06. 50 03^13 00.00 27207 25863 24563 23303 22083 20899 19749 18633 17547 16492 2 . 1 5464 -14464 .13489 .12539 •11613 2. 10709 .09827 .08965 .08124 317 TABLE II.— TANGENTS. EXTERNAL DISTANCES, AND LONG CHORDS FOR M 1° CURVE. 1 A Taiigeut Ext.Dist. LoiigCh'd A Taii:;eiit Ext.Dist. LoiigCh d A Tansreut Ext.Dist. LoiigCh'd V T. i;. xc. T. -K. LC. T. E. LC. 50.00 0.218 I 00 . 00 11° 551.70 26 . 500 1098.3 21° 1061 .9 97.58 2088.3 lo' 58.34 0.297 116.67 10 560. 11 27.313 III4.9 10 1070.6 99-15 2104.7 20 66.67 0.388 133-33 20 568.53 28. 137 II3I-5 20 1079.2 100.75 2121 . 1 30 75.01 0.491 1 50 . 00 30 576.95 28.974 I 1 48 . 1 30 1087.8 102.35 2137.4 40 83.34 0.606 166.66 40 585.36 29.824 I164.7 40 1 096 . 4 103.97 2153.8 50 91.68 0.733 183.33 50 593-79 30.686 I181.2 50 1105. I 105.60 2170.2 2° 100.01 0.873 199.99 12= 602.21 31.561 I 197.8 22° 1113.7 107.24 2186.5 10 108.35 1 .024 216.66 10 610.64 32.447 1214.4 10 1122.4 1 08 . 90 2202.9 20 116.68 1.188 233-32 20 619.07 33-347 . 1231 .0 20 II3I.O 110.57 2219.2 30 125.02 1.364 249.98 30 627.50 34-259 1247.5 30 II39-7 112.25 2235.6 40 133.36 1.552 266.65 40 635-93 35-183 1264. I 40 1148.4 113-95 2251 .9 50 141.70 1.752 283.31 50 644-37 36.120 1280.7 50 1157.0 115.66 2268.3 3° I 50 . 04 1.964 299.97 13° 652.81 37.069 1297.2 23° 1165.7 117.38 2284.6 10 158.38 2.188 316.63 10 661 .25 38.031 I313-8 10 1174.4 119. 12 2301 .0 20 166.72 2.425! 333-29i 20 669.70 39 . 006 1330.3 20 1183.1 120.87 2317.3 30 175.06 2.674 349-95 30 678.15 39-993 1346.9 30 1191.8 122.63 2333-6 40 183.40 2.934 366 . 6 1 40 686.60 40.992 1363-4 40 1 200 . 5 124.41 2349.9 50 191-74 3.207 383-27 50 695.06 42 . 004 1380.0 50 I 209 . 2 126. 20 2366.2 4° 200 . 08 3-492 399-92 ir 703.51 43-029 1396.5 24° 1217.9 128.00 2382.5 10 208.43 3.790 416.58 10 711.97 44.066 1413-I 10 1226.6 129.82 2398.8 20 216.77 4.099 433-24 20 720.44 45.116 1429.6 20 1235-3 131.65 2415.1 30 225.12 4.421 449.89 30 728.90 46.178 1446.2 30 1244.0 133.50 2431.4 40 233-47 4.755 466.54 40 737-37 47.253 1462.7 40 1252.8 135-36 2447.7 50 5° 241 .81 5. 100 483.20 50 745-85 48.341 1479-2 50 1261.5 137-23 2464 . 250.16 5.459 499-85 15° 754-32 49.441 1495-7 25° 1270.2 139.11 2480.2 10 258.51 5.829 516.50 10 762.80 50.554 1512.3 10 1279.0 141 .01 2496-5 20 266.86 6. 211 533-15 20 771.29 51.679 1528.8 20 1287.7 142.93 2512.8 30 275.21 6.606 549.80 30 779-77 52.818 1545-3 30 1296.5 144.85 2529.0 40 283.57 7.013 566.44 40 788.26 53-969 1561.8 40 1305-3 146.79 2545-31 50 6° 291 .92 7.432 583-09 50 796.75 55-132 1578.3 50 1314.0 148.75 2561.5 300.28 7.863 599-73 10° 805.25 56.309 1594.8 20° 1322.8 150.71 2577.8 10 308 . 64 8.307 616.38 10 813.75 57-498 1611.3 10 1331.6 152.69 2594.0 20 316.99 8.762 633.02 20 822.25 58-699 1627.8 20 1340.4 154.69 2610.3 30 325.35 9-230 649 . 66 30 830.76 59-914 1644.3 30 1349-2 156.70 2626.5 40 333.71 9.710 666 . 30 40 839.27 61 . 141 1660.8 40 1358.0 158.72 2642.7 50 342.08 10.202 682.94 50 847-78 62.381 1677.3 50 1366.8 160.76 2658.9 7° 350.44 10.707 699-57 17° 856.30 63-634 1693.8 27° 1375-6 162.81 2675.11 10 358.81 11.224 716.21 10 864.82 64 . 900 1710.3 10 1384-4 . 164.87 2691.3 20 367.17 11-753 732-84 20 873-35 66.178 1726.8 20 1393.2 166.95 2707.5! 30 375.54 12.294 749-47 30 881.88 67.470 1743-2 30 1402.0 1 69 . 04 2723.7 40 383.91 12.847 766. 10 40 890.41 68.774 1759.7 40 1410.9 171.15 2739-9 50 392.28 13-413 782.73 50 898.95 70.091 1776.2 50 1419.7 173-27 2756.1 8° 400 . 66 13.991 799-36 18° 907.49 71.421 1792.6 28° 1428.6 175-41 2772.3 10 409.03 14.582 815.99 10 916.03 72.764 1809.1 10 1437.4 177-55 2788.4 20 417.41 15.184 832.61 20 924.58 74.119 1825.5 -20 1446.3 179.72 2804.6 30 425.79 15-799 849-23 30 933-13 Z5-488 1842.0 30 1455. I 181.89 2820.7 40 434.17 16.426 865.85 40 941.69 76.869 1858.4 40 1464.0 184.08 2836.9' 50 442.55 1 7 . 066 882.47 50 950.25 78.264 1874.9 50 1472.9 186.29 2853.0 0° 450.93 17.717 899 . 09 19° 958.81 79.671 1891.3 29° 1481.8 188.51 2S69.2 10 459-32 18.381 915.70 10 967-38 81 .092 1907.8 10 1490.7 190.74 2885.31 20 467.71 19.058 932.31 20 975.96 82.525 1924.2 20 1499.6 192.99 2901 .4 30 476.10 19.746 948.92 30 984-53 83.972 1 940 . 6 30 1508.5 195.25 2917.6 40 484.49 20.447 965-53 40 993.12 85-431 1957.1 40 1517.4 197-53 2933-7 50 492.88 21 . 161 982 14 50 20° I 00 I .70 86 . 904 1973-5 50 1526.3 199.82 2949.8 10° 501 .28 21.886 998.74 1010.29 88.389 1989.9 30° 1535.3 202. 12 2965.9 10 509.68 22.624 io^5-35 10 1018.89 89.888 2006 . 3 10 1544.2 204.44 2982.0 20 518.08 23-375 1031.95 20 1027.49 91-399 2022.7 20 1553-I 206 . J7 2998. 1 30 526.48 24.138 1048.54 30 1036.09 92.924 2039.1 30 1562. I 209 . 1 2 3014.2 40 534.89 24.913 1065. 14 40 1044.70 94.462 2055.5 40 1571.0 211 .48' 3030.2 50 543.29 25.700 1081.73 50 2V 1053-31 96.013 2071 .9 50 1580.0 213.86 3046.3 11° 551-70 26 . 500 1098.33 1061.93 97-577 2088.3 31° T589.0 216.25 3062.4 1 318 TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS FOR A 1° CURVE. A Taiii^ent Kxt.Dist. LoiitfCh'il A Tan ire lit Kxt.Dist. LoiiirCh (1 A Taiiireiit Kxt.nist. LonirChd T. i'. 1 Lt. T. ^. , Lt. T. A'. J.<. 1 31" 1 1589.0 216.25 3062.4 41" 2142.2 387.38 4013.1 51° 2732.9 618.39 4933 4| lo'i 1598.0 218.66 3078.4 10 2151.7 390.71 4028.7 10 2743.1 622.81 4948 4 20 1 606 . 9 221 .08 3094.5 20 2161 .2 394.06: 4044-3 20 2753.4 627 . 24 4963 4 30 1615.9 223.51 3110.5 30 2170.8 397-43; 4059 -9 30 2763.7 631 .69 4978 4 40 1624.9 225.96 3126.6 40 2180.3 400.82 4075-5 40 2773.9 636.16 4993 4 50 1633.9 228.42 3142.6 50 2189.9 404.22 4091 . I 50 2784.2 640 . 66 5008 j4 4 32" 1643.0 230.90 3158.6 42^ 2199.4 407.64 4106.6 52° 2794.5 645.17 5023 10 1652.0 233.39 3174.6 10 2209.0 411.07 4122. 2 10 2804.9 649.70 S038 4 20 1661 .0 235.90 3 1 90 . 6 20 2218.6 414.52 4137-7 20 2815.2 654.25 S053 4 30 1670.0 238.43 3206.6 30 2228. I 417.99 4153-3 30 2825.6 658.83 5068 40 1679.1 240 . 96 3222.6 40 2237.7 421 .48 4108.8 40 2835.9 663.42 5083 3 50 1688. I 243.52 3238.6 50 2247.3 424.98' 4184.3 50 2846.3 668.03 5098 2 33° 1697.2 246.08 3254.6 43° 2257.0 428. 50 ■ 4199.8 53° 2856.7 672.66 5113 I 10 I 706 . 3 248.66 3270.6 10 2266.6 432.04 42153 10 2867.1 677.32 5128 20 I715.3 251 .26 3286.6 20 2276.2 435-59 4230.8 20 2877.5 681 .99 5142 9 30 1724.4 253.87 3302.5 30 2285.9 439-16; 4246.3 30 2S88.O 686.68 5157 8 40 1733.5 256.50 3318.5 40 2295.6 442.75 4261.8 40 2898.4 69 I . 40 5172 7 50 1742.6 259.14 3334.4 50 2305.2 446.35 4277.3 50 2908.9 696 . 1 3 5187 6 34° 1751.7 261 .80 3350.4 44^ 2314-9 449.98 4292.7 54" 2919.4 700.89 5202 4 10 1760.8 264.47 3366.3 10 2324.6 453.62 4308.2 10 2929.9 705.66 5217 3 20 1770.0 267. 16 3382.2 20 2334.3 457-27 43236 20 2940.4 710.46 5232 I 30 1779. I 269.86 3398.2 30 2344.1 460.95 4339-0 30 2951 .0 715.28 5246 9 40 1788.2 272.58 3414.1 40 2353.8 464.64! 4354-5 40 2961.5 720. II 5261 7 50 1797.4 275.31 3430.0 50 2363.5 468.35 4369-9 50 2972. I 2982.7 724.97 5276 5 35" 1806.6 278.05 3445-9 45° 2373-3 472.08 4385-3 55° 729.85 5291 10 1815.7 280.82 3461.8 10 2383-1 475.82 4400.7 10 2993.3 734.76 5306 I 20 1824.9 283.60 3477.7 20 2392.8 479.59 4416.1 20 3003.9 739.68 5320 9 30 1834. I 286.39 3493-5 30 2402 . 6 483.37 4431-4 30 3014.5 744.62 5335 6 40 1843.3 289.20 3509-4 40 2412.4 487.16 4446 . 8 40 3025.2 749.59 5350 4 50 30° 1852.5 1861.7 292 .02 3525-3 50 4(i° 2422.3 490 . 98 4462 . 2 50 3035-8 754-57 5365 . I 294.86 3541. I 2432 . I 494.82' 4477.5 5(r 3046.5 759.58 5379 8 10 1870.9 297.72 3557.0 10 2441.9 498.67 4492.8 10 3057.2 764.61 5,394 S 20 1880. I 300.59 3572.8 20 2451.8 502.54 4508.2 20 3067 . 9 769.66 5409 2 30 1889.4 303.47 3588.6 30 2461 .7 506.42 4523- 5 30 3078.7 774-73 5423 .9 40 1898.6 306.37 3604.5 40 2471.5 510.33 4538.8 40 3089.4 779.83 5438 .6 50 1907.9 309.29 3620.3 50 2481 .4 514.25 4554-1 50 3100.2 784.94! 5453 3 37° I917.I 312.22 3636. I 17° 2491.3 518.20 4569.4 57" 3110.9 790.08 5467 9 10 1926.4 315-17 3651.9 10 2501 .2 522.16 4584.7 10 3121.7 795-241 5482 S 20 1935.7 318.13 3667.7 20 2511 .2 526.13 4599-9 20 3132.6 800.42 5497 2 30 1945.0 321. II 3683.5 30 2521 . 1 530.13 4615.2 30 3143-4 805.62 5511 8 40 1954.3 324.11 3699.3 40 2531.1 534.15 4630.4 40 3154-2 810.85 5526 -4 50 1963.6 327.12 3715.0 50 2541.0 538.18 46457 50 3165.1 816. 10 5541 38" 1972.9 330.15 3730.8 48° 2551.0 542.23 4660.9 58" 3176.0 821.37 5555 6 10 1982.2 333.19 3746.5 10 2561 .0 546.30 4676. I 10 3186.9 826.66 SS70 2 20 I99I.5 336.25 3762.3 20 2571.0 550.39 4691.3 20 3197.8 831.98 5584 7 30 2000 . 9 339.32 3778.0 30 2581 .0 554.50 4706.5 30 3208.8 S37.31 5599 3 40 2010.2 342.41 3793-8 40 2591.1 558.63 4721.7 40 3219.7 842.67 5613 8 50 2019.6 345.52 3809.5 50 4y° 2601 . I 562.77, 4736.9 50 3230.7 848.06 5628 3 8 39° 1 2029.0 348.64 3825.2 2611 .2 566.94 4752.1 51)° 3241-7 853.46 5642 10 2038.4 351.78 3840.9 10 2621 .2 571.12 4767.3 10 3252.7 858.89, 5657 3 20 2047 . 8 354.94 3856.6 20 2631.3 575.32 4782.4 20 3263.7 £64.34; 5671 8 30 2057.2 358.11 3872.3 30 2641,4 579.54 4797-5 30 3274-8 869.82; 5686 3 40 2066 . 6 361 .29 3888.0 40 2651.5 583.78 4812.7 40 3285.8 875.32' 5700 8 50 2076.0 364.50 3903 - 6 50 2661.6 588.04 592.32 4827.8 50 3296.9 880.84 1 5715 2 7 40 2085.4 367.72 3919-3 50° 2671.8 4842.9 (>0" 3308.0 886.38; 5729 10 2094.9 370.95 3935-0 10 2681.9 596.62 4858.0 10 3319-1 891.95 5744 I 20 2104.3 374.20 3950.6 20 2692. I 600.93 4873-1 20 3330.3 897.54 5758 5 30 2113.8 377.47 3966.3 30 2702.3 605.27 4888.2 30 3341-4 903. 15 1 5772 9 40 2123.3 380.76 3981.9 40 2712. 5 609.62 1 4903 - 2 40 3352.6 908.79; 5787 50 2132.7 384.06 3997 . 5 50 51° 2722.7 6 1 4 . 00 4918.3 ^ 3363-81 914-45, 5801 7 41° : 2142.2 387.38 4013. I 2732.9 _6'8.39i 4933.4 «r 1 3375 -oi 920. 14 1 5816. 319 TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS FOR A 1° CURVE. A 61° ID 20 30 40 50 Tangent T. Ext.Dist. LongCh'd A 71° 10 20 30 40 50 Tangent T. Ext.Dist. LongCird LC. A 1 Tangent Ext.Dist. E. LongCh'd LC. 3375-0 3386.3 3397.5 3408 . 8 3420.1 3431.4 920. 14 925-85 931-58 937.34 943.12 948.92 5816.0 5830.4 5844-7 5859.1 5873-4 5887.7 4086 . 9 4099.5 4II2.I 4124.8 4137-4 4150. I 1 308 . 2 1315-5 1322.9 1330-3 1337-7 1345- I ! 6654.4 6668.0 6681.6 6695 • I 6708 . 6 6722. I 81° 10 20 30 40 50 4893 - 6 4908 . 4922 . 5 4937-0 4951-5 4966 . 1 1805.3 1814.7 1824. I 1833-6 1843. I 1852.6 7442 . 2 7454-9 7467 - 5 7480.2 7492 - 8 7505-4 62° 10 20 30 40 50 68° 10 20 30 40 50 3442.7 3454.1 3465.4 3476.8 3488 . 2 3499-7 954.75 960 . 60 966.48 972.39 978.31 984.27 5902.0 5916.3 5930-5 5944.8 5959-0 5973-3 72= 10 20 30 40 50 4162.8 4175-6 4188.4 4201 .2 4214.0 4226.8 1352.6 1360. I 1367.6 1375-2 1382.8 1390-4 6735.6 6749.1 6762.5 6776.0 6789-4 6802 . 8 82° 10 20 30 40 50 4980.7 4995-4 5010.0 5024.8 5039-5 5054-3 1862.2 1871.8 1881.5 189I .2 1 900 . 9 I9IO.7 7518.0 7530.5 7543-1 7555-6 7568.2 7580.7 3511.1 3522.6 3534-1 3545-6 3557-2 3568.7 990.24 996.24 1002.3 1008.3 IOI4.4 1020.5 5987-5 6001 .7 6015.9 6030 . 6044 . 2 6058.4 73° 10 20 30 40 50 4239-7 4252.6 4265.6 4278.5 4291.5 4304-6 1398.0 1405 . 7 i4i3-5 1421 .2 1429.0 1436.8 6816.3 6829.6 6843 . 6856.4 6869.7 6883.1 83° 10 20 30 40 50 5069 . 2 5084.0 5099-0 5113-9 5128.9 5143-9 1920. 5 1930-4 1940.3 1950-3 I 960 . 2 1970.3 7593-2 7605.6 7618.1 7630.5 7643.0 7655-4 64° 10 20 30 40 50 3580.3 3591-9 3603.5 3615-1 3626.8 3638.5 1026.6 1032.8 1039.0 1045.2 I051.4 1057.7 6072. 5 6086.6 6100.7 6114.8 6128.9 6143.0 74° 10 20 30 40 50 75° 10 20 30 40 50 4317-6 4330-7 4343-8 4356.9 4370.1 4383-3 4396.5 4409.8 4423 . I 4436.4 4449-7 4463 - 1 1444.6 1452.5 1460.4 1468.4 1476.4 1484.4 6896.4 6909.7 6923.0 6936 . 2 6949-5 6962 . 8 84° 10 20 30 40 50 5159.0 5174-1 5189.3 5204.4 5219.7 5234-9 1980.4 1990.5 2000 . 6 2010. 8 2021 . I 2031.4 7667.8 7680.1 7692.5 7704-9 7717.2 7729-5 65° 10 20 30 40 50 66° 10 20 30 40 50 3650.2 3661 .9 3673.7 3685.4 3697 - 2 3709.0 3720.9 673^.7 3744.6 3756.5 3768.5 3780.4 1063.9 1070.2 1076.6 1082.9 1089.3 1095.7 1 102. 2 IIO8.6 III5.I II2I.7 II28.2 1134.8 6157. I 6171 . I 6185.2 6199.2 6213.2 6227.2 1492-4 1500.5 1508.6 1516.7 1524.9 1533-1 6976.0 6989.2 7002 . 4 7015.6 7028.8 7041.9 85° 10 20 30 40 50 5250-3 5265.6 5281.0 '5296.4 5311-9 5327-4 2041.7 2052. I 2062 . 5 2073.0 2083.5 2094 . I 7741-8 7754-1 7766.3 7778.6 7790.8 7803.0 6241 .2 6255.2 6269. I 6283.1 6297.0 6310.9 76° 10 20 30 40 50 4476.5 4489.9 4503 -4 4516.9 4530.4 4544-0 1541.4 1549-7 1558.0 1566.3 1574-7 1583. I 7055 7068 . 2 7081.3 7094-4 7107.5 7120.5 86° 10 20 30 40 50 5343-0 5358-6 5374-2 5389-9 5405-6 5421.4 2 1 04 . 7 2115.3 2126.0 2136.7 2147.5 2158.4 7815.2 7827.4 7839.6 7851.7 7863.8 7876.0 67° 10 20 30 40 50 68° 10 20 30 40 50 6d° 10 20 30 40 50 3792.4 3804.4 3816.4 3828.4 3840.5 3852.6 II4I.4 II48.O 1154.7 II61 .3 II68.I ri74.8 6324.8 6338.7 6352.6 6366.4 6380.3 6394-1 77° 10 20 30 40 50 4557-6 4571.2 4584-8 4598-5 4612.2 4626 . 1591.6 1 600 . 1 1608.6 1617. I 1625.7 1634.4 7133-6 7146.6 7159-6 7172.6 7185.6 7198.6 87° 10 20 30 40 50 5437-2 5453-1 5469.0 5484-9 5500.9 5517-0 2169. 2 2180.2 2191 . I 2202.2 2213.2 2224.3 7888.1 7900.1 7912.2 7924.3 7936.3 7948 . 3 3864.7 3876.8 3889.0 3901 .2 3913.4 3925.6 1181.6 1188.4 1195.2 1202.0 1208.9 1215.8 ■ 1222.7 1229.7 1236.7 1243-7 1250.8 1257.9 6408 . 6421.8 6435-6 6449.4 6463 . 1 6476 . 9 78° 10 20 30 40 50 4639.8 4653-6 4667.4 4681.3 4695 . 2 4709.2 1 643 . 1651.7 1 660 . 5 1669.2 1678. 1 1686.9 7211 .6 7224.5 7237-4 7250.4 7263.3 7276.1 88° 10 20 30 40 50 5533-1 5549-2 5565-4 5581-6 5597-8 5614.2 2235-5 2246.7 2258.0 2269.3 2280.6 2292.0 7960.3 7972.3 7984.2 7996.2 8008.1 8020 . 3937.9 3950.2 3962.5 3974.8 3987.2 3999-5 6490 . 6 6504-4 6518. I 6531.8 6545-5 6559.1 7D° 10 20 30 40 50 4723.2 4737-2 4751-2 4765.3 4779-4 4793-6 1695.8 1704.7 1713-7 1722.7 1731-7 1740.8 7289.0 7301.9 7314.7 7327.5 7340.3 7353-1 89° 10 20 30 40 50 5630.5 5646.9 5663.4 5679-9 5696.4 57130 2303.5 2315.0 2326.6 2338.2 2349-8 2361.5 8031 .9 8043 . 8 8055-7 8067.5 S079.3 8091 .2 70=: 10 20 30 40 50 4011.9} 4024.4 4036.8 4049-3 4061.8 4074.4 1265.0 1272. I 1279-3 1286.5 1293.7 1300.9 6572.8 6586.4 6600 . I 6613.7 6627.3 6640 . 9 80° 10 20 30 40 50 4808 . 7 4822.0 4836.2 4850.5 4864.8 4879.2 1749-9 1759-0 1768.2 1777-4 1796.0 7365-9 7378.7 7391-4 7404-1 7416.8 7429-5 J)0° 10 20 30 40 50 5729-7 5746.3 5763-1 5779-9 5796-7 5813.6 2373.3 2385.1 2397.0 2408 . 9 2420.9 2432.9 8103.0 8114.7 8126.5 8138.2 8150.0 8161.7 71° 4086 . 9 1308.2 6654.4 §1° 1 4893.61 1805.3 7442 . 2 t)i° 5830.5 2444.9 8173-4I 320 TABLE III.— SWITCH LEADS AND DISTANCES. LEAD-RAILS CIRCULAR THROUGHOUT; GAUGE 4' S|". See §262. Frog Number («). 4 4-5 5 5.5 6 6.5 7 7-5 8 S.5 9 9-5 10 10.5 II II. 5 12 Frog Angle (F) 14 12 40 II 15 00 59 10 23 9 31 25 16 20 38 47 10 51 16 37 41 09 10 43 59 21 35 01 32 43 29 27 09 12 18 58 45 46 19 Lead (L) (Eq. 79). 37-67 42 -37 47.08 51.79 56.50 61.21 65.92 70.62 75-33 80.04 84-75 89.46 94.17 98.87 103.58 108.29 113.00 Chord (QT) (Eq- 77)- 37-38 42. 12 46.85 51-58 56.30 61.03 65-75 70.47 75.19 79.90 84.62 89.33 94-05 98.76 103.47 108.19 112.90 Radius of Lead Rails (r, Eq.7S). 150.67 190.69 23542 2S4.S5 339.00 397. S5 461.42 529.69 602.67 680.36 762.75 849-85 941.67 1038. 19 1139.42 1245.36 1356.00 Log r. 2.I780I .2S032 •371S3 .45462 .53020 •59972 . 6640<3 .72402 .78007 -83273 .8S238 .92934 2.97389 3.01627 . 05668 .09529 3.13226 Degree of Curve ((/). 38 46' 30 24 24 32 20 13 16 58 14 26 12 26 10 50 9 31 8 26 7 31 Frog Number / 7.5 8 8.5 9 45 05 32 02 36 14 9 10 10 II II 12 TURNOUTS WITH STRAIGHT POINT-RAILS AND STRAIGHT FROG-RAILS ; GAUGE 4' 8^'. See § 265. Frog Number in). 4 4-5 5 5.5 6 6-5 7 7-5 8 8.5 9 9.5 10 10.5 II II-5 12 Switch PointAngle (a). 40' 40 45 45 50 50 50 50 50 50 50 50 50 50 50 50 50 Length of Switch Point 7-5 7.5 10. 10. o 15.0 15.0 15-0 15 15 15 15-0 15 15 15.0 15.0 15-0 15.0 Length of Straight Frog-rail (7"). 1.50 1.69 1.87 2.06 2.25 2.44 2.62 2. Si 3.00 3.19 3-37 3-56 3- 75 3-94 4.12 4-31 4.50 Lead (L) (Eq. 90). 32.20 34.29 41.85 44-16 56.00 58.84 61.65 64.36 67.04 69.60 72.20 74.70 77.04 79.51 81.82 84.09 86.16 Chord (ST) (Eq. 88). 23.09 25-03 29. ss 32.03 38.66 41.34 43.98 46.50 48. 99 51. 38 53.80 56.11 58.28 60.57 62.69 64.78 66.67 Radius of Lead- rails (r.Eq.87). 125.21 159.25 197-65 240.44 2S8.09 340.19 397-65 460.00 527.91 600 . 94 6S1.16 767.11 85S.14 959-00 1065.52 II80. 16 1299.93 Log 09764 20208 2958(3 38100 45953 53172 59950 66276 72256 77883 83325 8S4S6 93356 98182 02756 07194 11392 Degree of Curve (^). 47' 05' 36 36 29 22 24 00 19 59 16 54 14 27 12 29 10 52 9 33 8 25 7 28 6 41 5 59 5 23 4 51 4 24 Frog Number («) 4 4-5 5 5.5 6 6.5 7 7-5 8 8.5 9 9-5 10 10.5 II II-5 12 TRIGONOMETRICAL FUNCTIONS OF THE FROG ANGLES (F). Frog Number FrogAngle (F). 4 4-5 5 5.5 6 6.5 7 7. 8 S. 9 9- 10 10. II II . 12 5 14 12 II 10 9 8 8 7 7 6 6 6 5 5 5 4 4 15' 40 25 23 31 47 10 37 09 43 21 01 43 27 12 58 46 00' 49 16 20 38 51 16 41 10 59 35 32 29 09 18 45 19 Nat. sin F. Na t. COSi^. .24615 96923 •21951 j 97561 .19802 98020 .1S033 9S360 .16552 ! 98621 -15294 98S23 .14213 98985 •13274 99II5 .12452 99222 .11724 99310 .11077 993S5 .10497 99448 •09975 99501 .09502 99548 .09072 995S8 .08679 99623 •08319 99653 Log sin F. 9.39120 •34145 . 29670 .25606 .21884 •18453 .1526s . 1230I .09522 .06909 .04442 9.02107 8. 9989 I .97781 •95770 .93S4S 8. 92007 321 Log cos F. 1.98642 .98927 •9913I .992S2 •99397 .99486 •99557 .99614 . 9966(3 • 99699 •99732 •99759 .997S3 .99803 .99S20 .99S36 1.9SS49 Log cot F. 10.59522 .647S2 .69461 -73675 .77513 .81033 .8428s •87313 •9013s .92790 .9528C3 .97652 IO.99S92 11.02021 .04050 .05987 II.07S42 Log vers .F. 8.48811 .38721 .29670 .21467 . 13966 •0705s S. 0065 5 7.94691 .8911(3 .83S64 .78915 .74232 .6978S •65560 .6152S .57676 7.539S6 Frog Number 4 4.5 5 5- 6 6. 7 7 ■ 8 8. 9 9- 10 10. II II . 12 5 TABLE IV.— ELEMENTS OF TRANSITION CURVES. 1 "0 d LO t^ Cl LO In CO M ^ CO "^ M CO O O 0\ ^-r\ xj-\ \0 O N CO © 5i O O 0\ On CO VO ro O O CN a\ ON ON On oo tN IN NO 00 n LO CO CO VO CO tN CO lA 6 '^ C> ^ On '^ ON -^ On C) LO t\ On f^ "^ t^ On n ':+ N CNl ^ '^ CO CO CO M Cl '"' ^ "i-O CO CO IN tN 00 LO c^ LO ^ LO t\ CO LO LO C CO IN too ("~» (— <0 <^* -^ "^ — ri vO -^ — LTl O ic -^ On ri o •^ >-< Ln On " ro LO t^ OO q CO CO CO Cl Cl d 1— >^ od d\ d\ d\ 6 6 6 d d >-<' ; >1 LO tN Cl LO r^ Q LO Cl X CO LO CO t^ LO NO '^ CO CO CO Cl ON LO ^ CI '^ d LO CO CO ^N. vo n oo o — r^ _ M ON CI CI n Cl h-i H rj m CO -• o oo -• O — to oo >-n -^ 00 NO ON OO LO IN "^ 1 1 b IN LO CJ IN LO Cl LO d d d d -< cj fo LO JN d -^ LO CO CO H- LO " C l> LO CO 00 CO 1— c w NO 10 Cl CO VO CO -e- ("^ (00 -^ CO On 1- (n LO t^ oo "-» vo On r^ (04 M LO In a 3 1-1 °(N M X UN n;^ _C VO O M vo 00 O O CO M NO ^ « t^ ro m i^ M rj t^ CO OO LO r^ CI LO r-^ LO Cl In > m CO On CO t^ o ^' '^ NO 00 m LO HH CO 1— 1 Cl lt\ O rt vr^ vA ^O "O t^ t^ IN. IN. tN c ^ N^ NO 1— CO Cl NO -^ LO CO • ft -a "5. 3 o (J CO Cl Cl LO Cl LO X On (^ (On (vO ('^ On vo O C^ON0O r^>-r\H-^ O OnOnOOnOnOnoo (Q LO ^ '^ Cl CO w LO r^ ON On t^ oo oo NO NO CO c c s 00 LO vo oo "^ O >-o "^ CO '^ ^ X CO -1 w CO - NO CO ON On IN ft NO 00 CO LO CO On Cl CO VO VO tc CO oo -' CO "^ NO t^ CO 0\ o rt -^ CO Cl ^ -H CO Cl LO Cl o 00 co" On b o o V S o ^ " ^ n "0 IN LO n LO IN CJ LO ■■■■^ CO CO l-l vo hm CO Cl '^ ■e- M X © ., ., On (r^ (-^ (On (« 3^ 2^ On On On oo 00 C C On t^ CO - VO LO LO 00 ^ Cl •^ LO ^ CO LO Cl Cl -4^ t/j s >-t C^ ro -^ "-> vo t^ CO On o p *-Cj ■55 rt ©^ fH ^1 cc rj^ Lt C b» 0(0 a 322 TABLE IV.— ELEMENTS OF TRANSITION CURVES. s as 0\ o\ o\ oo r^rt-o "* n -^ r^ On c~» Tf t^ C\ r> "+ H^ „ ►H -I c^ cs vO^O M ro^ »-nr> >-oo O r^-±DO M ^ \D oo O •- CO i-r^mroooooor-^ri fO -< -c n <0 -t CO (ro ro -+ CO <— CAfOOOO O C) t1- tJ- lA \d o t^ r^ i^ 00* CO CO -e- iO\ (vO i-r> O - r\ NO ur> ro s V ON OS On ON On CO NO -t- >-( r^ ON ON C7n On ON ON ON CA ON CO if) ON ON On ON ON ON ON ON ON UN e ON ON ON ON ON On ON ON ON ON (— (D rj CO o rn fON ro -iNO «-/^r^ O — ^^ -+ NO ON r> i/^ On f") OOOOOO-'-'-rJ i-r»w-iO O Lou-iO O i-nvn — •+rom^— 00«-t O O ^ <^l rO"^r^CN« rO -< M cO'^h^^NO t>^co OnO C3 I Ci OC C ^ CO *1 o^ CI lo On M l^ Q i-*"! O 'o O NO "^ CO r^ « O ro n NO "^i O »'^ n -rt O <■! ro ro ro ri ►- I O CO CO r^ t>x NO "^ ro r> ►- I O CO -t- O — CO r-^ CO i'-^ NO n ■-I -rt ^ ro LO to rn o o - yr\ O - o rv NO NO »-^ "+ Tf c<-) n — r>. O Lo ■<^ CO •- 00 ri — O >-o O -* i/~i ro O u^ CO — r\ NO u-iioT^-'^fOri — •- 8 8 O O 8 - O >-o CO Th r^ CO O " O "-> O O — CO O NO r^ ro O CO V? 8 ro *-<"> O r) rf Tt CO CO N lo O -. ro o 8 « O ir^ O O -r CO O ro n o o - "^ 8 — O CO CI O CO O "^ ro — Cl NO O ci CO CO M CI M O »J~i O >-o ro '^ O - M oo O NO LO Tj- l-O LO O >- CJ CO O Lo O CO -^ O c) CO O C) O '^ ir-i o CO o o - CO c o -- O lr^ O •'N O 'i- CO - O CO r^ NO O ro ci ci ro LO o ""> '^ CO — CO t^ VO ro — >-^ « « O O O u-i ro -rj- O io O ro O - ro -t r\ CO ro — "^ NO CI ro O iJ^i lo *-<-> O >-■ ci Cl CO -^ o lO o CO 8 LO ^ o o D o o i-r\ O -t CO ro C> O -i- 'S 8 uo oo •rt CJ NO ^^ CO r\ ci ►- O O O NH «-l C> CO ""^ «J^ 8 o o NO VO o o o o O o O tr\ ro -1- ri CO Cl -t o - O NO Cl "~> O w-i CO «- ro •+ •-'N '-' r^ ro -f — ro C) — " GO — — cico-t-LO o CO 8 O 8 °o ^ CO iT 8 CO f^ CO VO NO O Cl O vi-i O CO vr-i Tt CO — LO CO Cl — -t ro Cl — O OOO '-'<'» «^'fO-t"~» O >o ro -t r-^ CO O " O vn O - \r\ VO ro "-o O "^ ro -f Cl ro Cl "^ O CO NO oo O H- Cl CO O ►- t^ oo ro — 1-0 -+ OOOO^^''^'^''^-*' ^ ^ ^1 W ^ It C t- X Ct O 323 TABLE IV.— ELEMENTS OF TRANSITION CURVES. *l-n fS (S NO C) -* fs -^ CO CO 55 O OS r^ ^ O ON On On NO On O w-i On 'd- CO On NO ■* r^ CO CO o LO LO XI CO LO s- LO C) LTl CO LTi CN "^ On ^ On CO r^ •-<* Tt- r^ ir% rj- n h. On t^ Tf M M '^ r^ On r) 1? J^ On '"' '"' '"' "0 n U-l ON LO CO \_r\ CO LO CO CO un t>~. ("1 r^ (■xt- (O CO ^ tv. oo' oo' oo' oo' On c ^ H^ ^ On LO O CO NO CO C< CO ON oo 00 NO Lr> \0 be B ^ CO 8 UO CO c^ o l< Oo' OO' CO On d\ On Cfv ds dv V N e CO c^ '-' ^ '"' M "i^ VO t^ "0 NO ^ u CO CO CO LO n '* -0- c CO "d N \-r\ t^ VO M On t^ ^ s (On (NO NO 00 o q q q "• i-c n CO CO '^t \> On NO 00 rv ^ CO l-O CO 10 ON n Lo ON « ^ CO uo LO LO ~ '^ n s ^O o o CO CO O O o CO O O CO O O O O O CO CO « K4 M CO ir\ \o CO sr O l-l CO LO t^ O -+ OO n r^ 00 CO LO ■LTt t— 1 HH ■"" ci n CO CO M W) « CO CO 0» '0 u-v t^ n 10 t^ On f^ '* NO c CO LO CO rf NO CO On © M •-" N CO rt ir\ NO t^ 00 ON O c ^^ 0» H CI W Tj^ 10 t' X a rt 1H 324 TABLE v.— LOGARITHMS OF NUMBERS. N. 1 1" 12 1 :5 4 5 , a 1 7 S ! 1 1*. P. 1 100 lOI I02 00 000 . 432 860 043 087 130 173 216 260 303 775 *i99 389 817 ^241 ^7 ^ • 475 902 518 945 561 987 C04 *o3o 646 689 ^072 *ii4 732 *i57 43 [ 4.5 43 4.3 42 4.2 41 41 103 104 01 283 703 326 745 368 787 410 828 452 870 494 911 53t> 953 578 994 619 ^036 661 .. *o77 2 8.7 3 130 8.6 12.9 8.4 12.6 rA 8.3 13 3 ifi A 1 105 02 119 160 201 243 284 325 3^6 407 448 489 : 4 »74 5 21.7 17.2 21. s 21.0 20.5 106 530 571 612 653 694 735 775 816 857 898 6 26. 1 25.8 25-2 24.6, 107 108 938 03 342 979 382 ^019 422 *o6o 463 * ^100 503 *i4i 543 *i8i 583 *22I 623 *262 663 *302 703 ; 7 304 834.8 (^i3y-i 30.1 34-4 38.7 29.4 33.6 37.8 28.7 32.8 36-» 109 110 III 112 742 04 139 782 822 862 901 941 981 *026 *o6o *ioo 178 218 257 297 33^^ 375 415 454 493 883 ^269 532 922 57? 960 616 999 649 *o38 688 *o76 727 '^115 766 *i54 805 *I92 844 *23I 40 I 4.0 40 4.0 39 3-9 38 3.8I 113 05 308 346 384 423 461 499 538 576 614 652 2 8.1 3 12. i 8.0 12.0 7.8 11.7 7.6 II. 4 114 690 728 765 804 842 880 9^8 956 994 *032 ir, 6 It;. 2 115 06 070 107 145 183 220 258 296 333 371 408 . 5 20.2 20.0 '9-5 19.0 116 446 483 520- 558 595 632 670 707 744 781 • 624.3 24.0 23.4 33.8 117 818 855 893 930 967 *oo4 *04o *o77 *i 14 *i5i • 728.3 832.4 28.0 32.0 27 -3 31.2 36.6 30-4 118 07 188 225 261 298 335 372 408 445 481 518 . 9 36.4 36.0 351 34.2 119 120 121 554 591 627 664 700 737 773 809 *i76 529 845 882 3f 77 76 ■?«; 918 954 990 *026 386 *o62 *098 *i34 *206 564 *242 08 278 314 350 422 457 493 606 122 636 671 707 742 778 813 849 884 920 955 I 3-7 3-7 3.6 ^.i;| 123 124 995 09 342 *026 377 *o6i 412 *o96 447 *i3i 482 *i66 517 *202 552 ■*"237 586 *272 621 *307 656 2 7-5 3 II. 2 7-4 11. 1 7.2 10.8 7.0 10.5 I2S 691 725 760 795 830 864 899 933 968 *002 4 150 5 18.7 14.8 18.5 14.4 18.0 14.0 17-5 126 10 037 071 106 140 174 209 243 277 312 346 6 22.5 22.2 21.6 21.0 127 380 414 448 483 517 551 585 619 ^53 687 7 26.2 8 ^0.0 25.9 2Q.6 25.2 ?a 8 24-5 28.0 128 721 755 789 822 ^56 890 924 958 991 *025 9 33-7 33-3 324 31-5 129 130 131 132 133 II 059 092 126 160 193 227 260 294 327 361 394 427 461 494 528 561 594 627 661 694 727 12 057 385 760 096 418 793 123 450 826 156 483 859 189 515 892 221 548 925 254 580 958 287 613 991 320 645 *024 352 678 34 1 3-4 2 6.9 34 3-4 6.8 33 3-3 6.6 32 3-3 6.4 134 7x5 743 775 807 840 872 904 937 969 *OOI 3 10.3 10.2 9.9 9.6 135 13 033 065 097 130 162 194 226 258 290 322 4 138 5 172 13.6 17.0 13.2 16. s 12.8 16.0 136 354 386 417 449 481 513 545 577 608 640 6 20.7 20.4 19.8 19.2 137 138 672 988 703 *oi9 735 *o5i 767 *o82 798 *ii3 830 *i45 862 *i76 893 *207 925 *239 956 ^270 7 24.1 827.6 931.5 23.8 27.2 30.6 23.1 26.4 29.7 22.4 25.6 28.8 139 110 141 14 301 332 364 395 426 457 767 *o75 488 519 550 582 31 -?! ^0 20 613 644 675 706 736 798 *io6 829 866 891 1 922 952 983 *oi4 *o45 *i37 *i67 ^198 142 15 229 259 290 320 351 381 412 442 473 503 1 3.1 3.1 30 3.9 143 144 533 836 564 866 594 896 624 926 655 956 685 987 715 *oi7 745 *o47 770 *o77 806 *io7 2 6.3 3 9-4 4 12.6 5 «5-7 6.2 9-3 6.0 9.0 5.8 8.7 TT ^ 14s 16 137 165 196 226 256 286 316 346 376 405 12.4 15-5 15.0 14-5 146 435 465 494 524 554 584 613 643 672 702 6 18.9 18.6 18.0 17-4 147 731 761 791 820 849 879 908 938 967 997 7 22.0 8 25.2 21.7 74 R 31. 24.0 20.3 23.3 148 17 026 055 085 114 143 T72 202 231 266 289 928.3 27.9 27.0 a6.i T49 150 318 609 348 377 406 435 464 493 522 551 580 638 667 696 725 753 782 811 840 869 N. 1 1 2 3 4 5 1 6 1 7 8 9 P.P. 325 TABLE v.— LOGARITHMS OF NUMBERS. N. 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 N. O 17 609 897 18 184 469 752 19 °33 312 590 865 20 139 2 412 682 951 21 219 484 748 22 on 271 53T 788 23 045 299 553 804 24 055 304 551 797 25 042 285 527 768 26 007 245 482 717 951 27 184 416 646 875 28 103 330 555 780 29 003 225 446 666 885. 3Q 103 O 638 926 213 497 780 061 340 617 893 167 439 709 978 245 511 774 037 297 557 814 070 667 955 241 526 808 089 368 645 926 194 792 031 269 505 740 974 207 439 669 898 466 736 126 352 578 802 025 248 468 688 907 124 350 603 855 105 353 600 846 091 334 575 816 055 292 529 764 998 230 462 692 921 696 984 270 554 836 117 396 673 948 221 493 763 '012 298 582 864 145 423 700 975 249 520 790 •032 f^o58 149 375 600 825 048 270 490 710 929 146 298 564 827 089 349 608 865 121 375 628 880 129 378 625 871 115 358 599 325 590 853 115 375 634 891 147 401 653 905 154 403 650 895 139 382 840 078 316 552 787 '021 254 485 715 944 171 398 623 847 070 292 512 732 950 168 623 863 102 340 576 811 '044 277 508 738 966 194 426 645 869 092 3M 534 754 972 190 753 '041 327 611 893 173 451 728 ^003 276 6 782 811 547 817 ^'085 352 616 880 141 401 660 917 '070 355 639 921 201 479 755 ^030 303 574 172 426 679 930 179 427 674 920 164 406 647 007 126 599 834 *o68 300 531 761 989 844 '112 378 643 906 167 427 686 942 '098 384 667 949 229 507 783 '057 33^ 8 9 840 601 198 451 704 955 204 452 699 944 188 430 672 911 150 387 623 858 '''091 323 554 784 012 239 465 696 914 137 358 578 798 *oi6 233 6 871 '139 405 669 932 193 453 711 968 *'l27 412 695 977 256 534 8i5 *o85 357 869 *i56 446 724 *oo5 284 562 838 112 385 628 223 477 729 980 229 477 723 968 212 455 696 898 ^165 43 J 695 958 219 479 737 994 249 502 754 '005 254 502 748 993 237 479 935 174 411 646 881 = 114 346 577 806 •035 262 488 713 936 159 380 6o5 820 *o38 254 720 959 197 434 670 904 '137 369 600 829 058 285 510 735 959 181 402 622 841 '059 276 8 ^5 924 ^^192 458 722 984 245 505 763 ^019 P. P. 274 527 779 '030 279 526 773 ''017 261 503 744 983 221 458 693 928 *i6i 392 623 _85l *o8o 307 533 758 981 203 424 644 863 »o8i 298 9 29 28 27 . I 2.9 2.8 2. .2 ■3 5-8 8.7 5.6 8.4 5- 8. ■4 11.6 T1.2 10. •5 .6 M-5 17.4 14.0 16.8 13- 16. •7 20.3 ig.6 18. .8 23.2 22.4 21. •9 26.1 25.2 24. 2§ 26 . I 2-6 .2 5-3 •3 7-9 •4 • S TO. 6 13.2 .6 15-9 ■7 18.5 .8 21.2 •9 23-8 2.6 5-2 7.8 10.4 13.0 15-6 20.8 234 2S 25 24 . I 2.5 2. K .2 5-1 50 •3 7-6 7-5 •4 10.2 TO.O •5 12.7 12.5 .6 15-3 150 ■7 17-8 17-5 .8 20.4 20.0 •9 22.9 22.5 2.4 4.8 7.2 9.6 12.0 14.4 16.8 19. a 21 .6 . I 2-3 .2 •3 4-7 7.0 4 9-4 •5 .6 II. 7 14. 1 •7 .8 •9 16.4 18.8 21. 1 23 23 2-3 4.6 6.9 9.2 11-5 13-8 16. 1 18.4 20.7 22 22 21 2.1 4-3 6.4 8.6 TO. 7 12.9 15.0 17.2 19-3 .1 2.2 2.2 .2 •3 6.7 4.4 6.6 •4 9.0 8.8 •s II. 2 II. .6 13-5 13.2 •7 .8 15-7 18.0 15-4 17.6 •9 20.2 19.8 P. p. 326 TABLE V.- —LOG ARITinL^ ( )F XI 'MHERS. Tn. 1 2" 3 4 5 7 S <) r. p. 1 200 30 103 124 146 168 190 211 233 254 276 298 j 20I 319 341 3^3 384 406 427 449 470 492 513 . I 2 2 2 I 202 535 556 578 599 621 642 664 685 707 728 4-4 4.2 203 749 771 792 813 835 856 878 899 920 941 • 3 6.6 6.3 204 963 984 ♦005 *o27 *o48 ^069 *og6 *II2 ^^^33 *i54 R R R 1 205 31 175 196 217 239 260 281 302 323 344 365 •4 .5 II .0 0.4 10.5 206 386 408 429 450 471 492 513 534 555 576 .6 13.2 12.6 207 597 618 639 660 681 702 722 743 764 785 208 806 827 848 869 890 910 931 952 973 994 •7 .8 154 17.6 14.7 16.8 209 210 32 014 222 035 056 077 097 118 139 160 186 387 201 407 •9 19. s 18.9 242 263 284 304 325 346 3^6 35 -1 211 428 449 469 490 510 531 551 572 592 613 20 ^KJ 212 ^33 654 674 695 715 736 756 776 797 817 .2 4. 1 4.0 213 838 858 878 899 919 940 960 980 1*001 *02I •3 6.1 6.0 214 33 041 061 082 102 122 142 163 183 j 203 223 8.0 ! 10. t 215 244 264 284 304 324 344 365 385 405 425 • 4 •5 .6 0.2 10. 2 216 44! 465 485 505 525 546 566 586 606 626 12.3 12.0 217 646 666 6S6 706 726 746 766 786 806 825 L 218 845 865 885 905 925 945 965 985 *oo4 *024 •V 8 14-3 16.4. 14.0 16.0 219 220 221 34 044 064 084 104 123 M3 163 183 203 222 •9 1S.4 18.0 ; 242 439 262 281 301 1 321 341 366 380 400 I 419 19 19 459 478 498 5^8 537 557 j 576 596 , 615 222 ^35 655 674 694 713 733 752 I 772 791 i 811 . I . 2 1.9 3-9 5-8 1.9 3.8 5-7 223 830 850 869 889 908 928 947 1 9^6 986 *oo5 •3 224 35 025 044 063 083 102 121 141 166 179 199 225 218 237 257 276 29! 3''4 334 353 372 391 •4 7.8 9-7 II. 7 7.6 9-5 II. 4 226 411 430 449 468 487 S<^7 526 545 564 583 • 5 .6 227 602 621 641 660 679 698 717 736 755 774 228 793 812 83? 850 869 88h 907 926 945 9^4 •V .8 •9 13-6 15-6 17-5 13-3 229 230 231 983 *002 "^021 *o4o *o59 *o78 *o97 1*116 1*135 ;*i54 15-2 17. 1 36 173 191 216 229 248 267 286 1 305 323 ' 342 „ 1 361 380 399 417 436 455 474 492 511 530 18 10 232 549 567 586 605 623 642 661 679 698 717 . I 2 1-8 3-7 5-5 1 .8 ! 3-6 ; 5-4 1 233 735 754 773 791 810 828 847 866 884 903 ■ 3 |234 921 940 958 977 996 *oi4 *o33 *o5i *o7o *o88 1 235 37 107 125 143 162 186 199 217 236 254 1 273 •4 7-4 9.2 II .1 7.2 9.0 10.8 I236 291 309 328 346 364 383 401 420 438 456 • 5 .6 237 475 493 511 530 548 5^6 584 603 621 639 238 657 676 694 712 730 749 767 785 803 821 •7 .8 •9 12.(3 14.8 16.6 12.6 1239 240 241 840 858 876 894 912 930 III 948 i 967 985 1*003 14.4 16.2 38 021 039 057 075 093 129 ) 147 165 183 201 219 237 255 273 291 309 327 345 3^3 17 1 242 38i 399 417 435 453 471 489 507 525 543 . I 1-7 3-5 5-2 1.7 ^•4 5-1 '243 566 578 596 614 632 650 667 685 703 721 •3 '244 739 757 774 792 810 828 845 863 881 899 |245 1 246 9^6 39 093 934 III 952 129 970 146 987 164 *oo5 181 *023 199 *046 217 *o58 234 *o76 252 •4 •5 .6 7.0 8.7 10.5 6.8 ; 8.5 1 10.2 1247 269 287 305 322 340 357 375 392 410 427 I248 445 462 480 497 515 532 550 567 585 602 .7 .8 •9 12.2 II. 9 13.6 ! 249 250 620 637 655 672 689 707 724 742 759 776 14 -O 157 794 811 828 846 863 881 898 915 933 950 i N. 1 1 2 3 4 5 : 7 1 8 1 9 P. P. 1 327 TABLE V.- -LOGARITHMS < 3F NUMBERS. 1 N. 250 251 1 ! 2 3 4 5 6 7 1 8 p. p. 39 794 811 828 846 863 881 898 915 933 950 967 984 *002 *oi9 *o36 *o54 *o7i *o88 *io5 *I23 252 40 140 157 174 191 209 226 243 266 277 295 1 253 312 329 346 z^i 380 398 415 432 449 466 if 17 254 483 500 517 534 551 569 586 603 620 637 . I 1-7 1.7 255 654 671 688 705 722 739 756 773 790 807 .3 3-5 5.2 3-4 5.1 1 255 824 841 858 875 892 908 925 942 959 976 257 993 *OIO ^027 *044 *o6i *o77 *094 *iii *I28 ""145 •4 7.0 R ^ 6.8 8.5 10.2 25« 41 162 179 195 212 229 246 263 279 296 3^3 •5 .6 0.7 10.5 259 260 261 32,^ 346 3^3 380 397 413 430 447 464 631 480 647 •7 .8 12.2 14.0 15.7 II. 9 13.6 15. -^ 497 664 514 536 547 564 581 597 614 680 697 714 730 747 764 780 797 8n 262 830 846 863 880 896 913 929 946 962 979 263 995 *OI2 •^023 *o45 *o6i ^078 ^094 *iii *I27 *i44 264 42 160 177 193 209 226 242 259 275 292 308 - 5 ^ 265 324 341 357 373 390 406 423 439 455 472 16 10 266 488 504 521 537 553 569 586 602 613 635 .1 1-6 3-3 4.9 1.6 267 651 667 6^% 700 716 732 748 765 781 797 ■3 4.8 268 813 829 846 862 878 894 910 927 943 959 269 270 271 975 991 ^^007 *o23 *o4o ^056 ^072 *o88 *io4 *I26 •4 •5 .6 6.6 8.2 9.9 6.4 8.0 9.6 43 ^36 152 168 184 200 216 233 249 265 281 297 3^Z 329 345 361 377 393 409 425 441 272 457 473 489 505 520 536 552 568 584 600 • / 8 II-5 13.2 14-8 11 .2 12 8 273 6ig 632 648 664 680 695 711 727 743 759 •9 14.4 274 775 791 8og 822 ^Z^^ 854 870 SS6 901 917 i 275 933 949 965 980 996 *OI2 *028 *043 *oS9 *o7s ) 1 276 44091 105 122 138 154 169 185 201 216 232 \ 277 248 263 279 295 310 326 342 357 373 38Q iS 15 i 278 404 420 435 451 467 482 498 513 529 545 2 1-5 3-1 4-6 6.2 7-7 1-5 3-0 4-5 6.0 7-5 279 280 281 560 576 591 607 622 638 653 669 685 700 .3 .4 i 716 870 731 747 762 778 793 809 824 839 855 *oo9 886 901 917 932 948 963 978 994 282 45 025 040 055 071 085 102 117 132 148 163 .6 9-3 9.0 283 178 194 209 224 240 255 270 286 301 3^6 ^ [ 284 332 347 362 377 393 408 423 438 454 469 .7 .8 10. 8 12.4 10.5 1 12.0 2«5 484 499 515 530 545 560 576 591 606 621 •9 13-9 13-5 286 ^2>6 652 667 682 697 712 727 743 758 773 287 788 803 818 ^ZZ 848 864 879 894 909 924 288 939 954 969 984 999 *oi4 *029 *044 *o59 *o75 ^ 289 290 291 46 090 105 120 135 150 165 180 195 210 225 . I .2 •3 14 1.4 4-3 ^4 1-4 2.8 4.2 240 255 269 284 299 314 329 344 359 374 389 404 419 434 449 464 479 493 508 523 292 538 553 568 583 597 612 627 642 657 672 .4 5.8 7.2 5.6 7.0 293 687 701 716 731 746 761 775 790 805 820 294 834 849 864 879 894 908 923 938 952 967 .6 8.7 8.4 295 982 997 '^OII ^025 ^41 *o55 *o75 *o85 *IOO *ii4 .7 .8 ICt T n R 296 47 129 144 158 173 188 202 217 232 246 261 II. 6 9.0 II. 2 297 275 290 305 319 334 348 3^3 378 392 407 •9 13.0 12.6 298 421 436 451 465 480 494 509 523 538 552 299 300 N. 567 58I 596 610 625 639 654 66s 683 697 712 726 741 755 770 784 799 813 828 842 1 2 3 4 5 1 6 7 8 9 P. P. || 328 TABLE v.— T.OGARITTIMS OF yUMT^FRq. 300 301 1 2 3 4 5 7 8 J) r. i». 1 47 712 726 741 755 770 784 799 813 828 842 856 871 885 900 914 928 943 957 972 986 302 48 000 015 029 044 058 072 087 101 11? 130 3<'3 144 158 173 187 201 216 230 244 259 273 304 287 301 316 33^ 344 358 *^ -^ 0/0 387 401 415 30s 430 444 458 472 487 50i 515 529 543 SS8 « 306 572 586 606 614 629 ^43 657 671 685 699 ^'f *^ 307 714 728 742 756 770 784 798 812 827 841 2 1 .4 2.9 1 .4 2.S 308 855 869 ^^3 897 911 925 939 953 967 982 •3 4-3 4.2 309 310 311 996 'OIO *024 *o38 *052 =^^066 *o8o *o94 *io8 *I22 •4 • 5 .6 5.S 7.2 8.7 5.6 7.0 , 8.4 49 136 i 150 164 178 192 206 220 234 248 262 276 290 304 318 332 346 359 373 387 401 312 415 429 443 457 471 485 499 513 526 540 •7 .8 9.8 , II. 2 1 3^3 554 5^8 582 596 610 624 ^37 65 J 665 679 II. 6 314 693 707 720 734 748 762 776 789 803 817 •9 13.0 12.6 315 831 845 858 872 886 900 913 927 941 955 316 9^8 982 996' *OIO =^023 *o37 *o5i *o65 *078 *092 317 50 106 119 ^33 147 160 174 188 201 215 229 31S 242 256 270 283 297 311 324 33^ 352 365 319 320 321 379 515 392 406 420 433 569 447 466 474 488 soil t5 to 528 542 555 583 596 610 623 1 637 650 664 677 691 704 718 731 745 758 772 , I -0 1-3 2.7 '^ 1 1-3 2.6 322 785 799 812 826 839 853 866 880 893 907 .2 323 920 933 947 960 974 987 *OOI *oi4 *02 7 *04i •3 4.0 3-9 324 51 054 068 081 094 108 121 135 148 161 175 •4 5.4 6.7 5-2 6.5 325 188 201 215 228 242 255 268 282 295 308 326 322 335 348 361 375 388 401 415 428 441 .6 8.1 7.8 327 455 468 481 494 508 521 534 547 561 574 •7 .8 9-4 10.8 r> T 328 587 6o5 614 627 640 653 667 680 693 706 y .1 10.4 329 330 33^ 719 733 746 759 772 904 785 798 812 825 838 •9 12. i II. 7 851 983 864 877 891 917 930 943 956 969 996 *oo9 *022 *o35 *o48 *o6i *o74 *o87 *ioo 332 52 114 127 140 153 166 179 192 205 218 231 244 257 270 283 296 309 322 335 348 361 334 374 387 400 413 426 439 452 465 478 491 335 504 517 530 543 556 569 582 595 608 621 r S 1 '> 33^ 634 647 660 672 685 ^98 711 724 737 750 I 2 I 2 337 763 776 789 801 814 827 840 853 866 879 . 2 2.5 2.4 338 891 904 917 930 943 956 9^8 981 994 *oo7 .3 3.7 3.6 339 53 020 033 045 058 071 084 097 109 122 135 /I R 340 341 148 i65 173 186 199 211 224 237 250 377 262 •4 •5 .6 5 -^ 6.2 7.5 4.0 6.0 7.2 275 288 301 313 326 339 352 364 390 342 402 415 428 440 453 466 478 491 504 516 . 7 S.7 10. 8.4 9.6 343 529 542 554 567 580 592 605 618 630 643 .8 344 656 66s 681 693 706 719 731 744 756 76(3 •9 II. 2 10.8 j 345 782 794 807 819 832 845 857 870 882 895 346 907 920 932 945 958 970 983 995 *oo8 *020 347 54033 045 058 076 083 095 108 120 ^33 14! 348 158 170 183 195 208 220 232 245 257 270 349 350 282 295 307 320 332 344 357 369 3S2 394 407 419 431 444 456 469 481 493 506 5^S N. 1 2 1 3 4 »■> 1 7 1 8 9 P.P. 329 TABLE v.- -LOGARITHMS ( 3F NUMBERS. 1 N. 1 2 j 3 4 5 6 7 8 9 1 P. P. 350 351 54 407 419 431 444. 456 469 481 493 506 518 . I 12 1 . 2 530 543 555 568 580 592 605 617 629 642 \352 654 666 679 691 703 716 728 740 753 765 .2 2.5 353 777 790 802 814 826 839 851 863 876 888 •3 3-7 1354 900 912 925 937 949 961 974 986 998 *oi6 .J. K .0 i355 55 023 035 047 059 071 084 096 108 120 133 .5 6.2 1356 145 157 169 181 194 206 218 230 242 254 .6 7.5 1357 267 279 291 303 315 327 340 352 364 376 .7 .8 8.7 10. 1358 3H 400 412 424 437 449 461 473 485 497 359 360 1361 509 630 750 521 533 545 558 570 696 582 594 606 618 •9 II. 2 12 I 2 642 654 775 666 678 702 714 726 738 762 787 799 811 823 835 847 859 3^2 871 883 895 907 919 931 943 955 966 978 . 2 2.4 \3^3 990 *002 *oi4 ^025 "^038 *o56 ^062 *o74 *o86 ^098 •3 3.6 364 56 no 122 134 146 158 170 181 193 205 217 4.8 6.0 365 229 241 253 265 277 288 306 312 324 33^ •4 .5 366 348 360 372 383 395 407 419 431 443 455 .6 7.2 367 465 478 490 502 514 525 537 549 561 573 8.4 9.6 10.8 II 368 585 596 6o§ 620 632 643 655 667 679 691 •7 .8 369 370 371 702 714 726 738 749 761 879 773 785 796 808 •9 820 832 843 855 867 984 896 1 902 914 925 937 949 961 972 996 *oo7 *oi9 *o3i ^042 372 57 054 066 077 089 lOI 112 124 136 147 159 .2 2.3 I373 171 182 194 206 217 229 240 252 264 275 •3 3-4 374 287 299 310 322 333 345 357 3^S 380 391 4-6 "^ . 7 J375 403 414 426 438 449 461 472 484 495 507 •4 376 519 530 542 553 565 576 588 599 611 622 .6 6.9 377 634 645 657 668 680 691 703 714 726 737 8.0 9.2 10.3 II 378 749 760 772 783 795 806 818 829 841 852 •7 8 379 380 381 864 875 887 898 909 921 932 944 955 967 •9 978 990 *OOI *OI2 *024 *o35 149 *o47 *o58 ^'069 *o8i 58 092 104 115 126 138 161 172 183 195 382 206 217 229 240 252 263 274 286 297 3^S .2 2.2 3^3 320 33^ 342 354 365 376 388 399 410 422 •3 3-3 384 433 444 455 467 478 489 501 jI2 523 535 385 546 557 568 580 591 602 613 625 636 647 •4 •5 .6 4.4 5.5 6.6 386 658 670 681 692 703 715 726 737 748 760 387 771 782 793 804 816 827 838 849 861 872 388 883 894 905. 916 928 939 950 961 972 984 •7 Q 7-7 8.8 9 9 10 389 1390 391 995 59 106 *oo6 *oi7 *028 *o39 *o5o 162 *o62 *o73 184 ^084 *o95 •9 117 128 140 ^51 173 195 206 217 229 240 251 262 273 284 295 3^6 317 392 328 339 35^ 362 373 384 395 406 417 428 . I I.O 1393 439 450 461 472 483 494 505 5I6 527 53S .3 3.1 394 549 560 571 582 593 604 615 626 637 648 395 659 670 681 692 703 714 725 736 747 758 •4 4.2 396 769 786 791 802 813 824 835 846 857 868 •5 .6 5-2 6.3 397 879 890 901 912 923 933 944 955 9^6 ^977 398 988 999 *OIO *02I *032 *o43 *o53 *o64 *o75 *o86 .7 7-3 8.4 9.4 399 400 60 097 206 108 119 130 141 151 162 173 184 195 .8 .9 217 227 238 249 260 271 282 293 3^3 1 2 3 4 5 6 7 8 9 1 P. P. 330 TABLE V.- -LOGARITHMS ( )F NUMHERS. In. 400 401 1 '21314 5 e 7 1 8 I* . V. 60 206 217 1 227 1 238 249 260 271 379 282 293 401 412 314 325 1 zz(^ 347 357 z^l 390 402 422 433 444 455 466 476 487 498 509 519 403 530 541 552 563 573 584 595 606 616 627 II ,404 638 649 659 670 681 692 702 713 724 735 . I .2 I . I 2.2 405 745 756 767 777 788 799 810 826 H^ 842 •3 , 3-3 406 852 863 874 884 895 906 916 927 938 949 407 959 970 981 991 *00 2 *oi3 *023 *o34 *044 *o55 .4 .5 4.4 5-5 408 61 066 076 087 098 I08 119 130 140 151 161 .6 6.6 409 410 411 172 183 193 204 ! 215 225 331 236 342 246 352 257 268 .7 .8 •9 7.7 S.8 9.9 278 ' 289 299 1 310 326 Ti^Z 373 384 1 394 405 416 426 437 447 458 468 479 412 489 500 511 521 532 542 553 563 574 584 413 595 605 616 625 637 647 658 668 679 689 414 700 710 721 731 742 752 763 773 784 794 415 805 8'5 825 %l6 846 857 867 878 888 899 416 909 Q20 930 940 951 961 972 982 993 *oo3 .2 2. 1 [417 62 013 024 034 045 ^y:i 065 076 086 097 107 •3 3-1 418 117 128 ^zl 149 159 169 180 190 200 211 419 420 421 221 325 428 232 242 252 263 3^6 469 273 376 480 283 294 304 314 •4 •5 .6 •7 .8 4.2 5-2 6.3 7-3 8.4 335 345 356 459 387 397 407 418 438 449 490 506 510 521 422 531 541 552 562 572 582 593 603 6^s 624 423 634 644 654 665 675 685 695 706 716 726 .9 9.4 424 736 747 757 767 777 7S8 798 808 818 828 425 839 849 859 869 879 890 900 910 926 931 426 941 951 961 971 981 992 *002 *OI2 *022 *032 427 63 043 053 063 073 083 093 104 114 124 134 10 428 144 154 164 175 185 195 205 215 225 235 . I .2 I.O 2.0 429 430 431 245 347 256 266 276 286 387 296 3O6 3I6 326 336 •3 •4 .5 3-0 4.0 5-0 357 367 1 377 397 407 417 427 437 447 458 468 478 488 498 508 518 528 538 432 548 558 5^8 578 588 598 608 618 628 639 .6 6.0 433 649 659 669 679 689 699 709 719 729 739 434 749 759 769 779 789 799 809 819 829 839 •7 .8 7.0 8.0 435 849 859 869 879 889 899 909 919 928 938 •9 9.0 436 948 958 9^^8 978 988 998 '■^^oog *oi8 *028 ^038 437 64 048 058 068 078 088 09S 107 117 127 137 43^ 147 157 167 177 187 197 207 217 226 236 439 440 441 246 256 266 276 286 296 394 306 315 325 335 .1 .2 .3 9 0.9 J ^•9 2-8 345 444 355 453 365 375 384 404 414 424 434 463 473 483 493 503 512 522 532 442 542 552 562 571 58i 591 601 611 621 636 443 640 650 660 670 679 689 699 709 7I8 728 •4 .5 .6 3-s : 4-7 5.7 444 738 748 758 767 777 787 797 806 8I6 826 445 836 846 855 865 875 885 894 904 914 923 446 933 943 953 962 972 982 992 *OOI *oii *02I .7 .8 6.6 7.6 447 65 031 040 050 060 069 079 089 098 108 118 •9 8.5 1 448 128 137 147 157 166 176 186 195 205 215 449 450 224 234 244 253 263 273 282 292 302 311 321 7^7>^ 1 340 350 ! 360 3^^9 379 389 398 408 i ^^• 1 1 2 1 3 1 4 r> is 7 1 8 ! V . P. 331 TABLE V.- -LOGARITHMS OF NUMBERS. 4 I~x7 450 451 1 1 j 2 3 4 5 6 7 8 P .P. 65 321 33^ 340 350 360 369 379 389 398 408 417 427 437 446 456 466 475 485 494 504 452 514 523 533 542 552 562 571 581 590 600 453 610 619 629 ^38 648 657 667 677 686 696 10 454 705 715 724 734 744 753 763 772 782 791 .2 2.0 |455 801 810 820 830 839 849 858 868 877 887 .3 3-0 456 896 906 9^5 925 934 944 953 963 972 982 457 991 *OOI *oi3 *020 ^029 *o39 *o48 ^058 *o67 *o77 •4 .5 4.0 5.0 45 « 66 085 096 105 115 124 134 143 153 162 172 .6 6.0 459 460 461 181 190 200 209 219 228 238 247 257 266 •7 .8 •9 7.0 8.0 9.0 276 285 294 304 313 323 332 342 351 1 3^0 370 379 389 398 408 417 426 436 445 455 462 464 473 483 492 502 511 520 530 539 548 463 558 567 577 586 595 605 614 623 ^33 642 464 652 661 673 680 689 698 708 7x7 726 736 a 465 745 754 764 773 782 792 801 816 820 829 9 466 838 84S 857 S6e 876 885 894 904 913 922 .1 .2 0.9 1.9 2.8 467 931 941 950 959 969 978 987 996 *oo6 *oi5 .3 468 67 024 034 043 052 061 071 080 089 099 108 469 470 471 117 126 136 228 145 154 163 173 182 191 200 • 4 .5 .6 •7 .8 3.8 4.7 5.7 210 302 219 237 246 256 265 274 283 293 311 320 329 339 348 357 3(^6 376 385 472 394 403 412 422 431 440 449 458 467 477 0.5 7.6 473 486 495 504 513 523 532 541 550 559 5^8 •9 8.5 474 578 587 596 605 614 623 ^33 642 651 660 475 669 ^78 687 697 706 715 724 733 742 75 J 476 760 770 779 788 797 806 815 824 ^33 842 477 852 861 870 879 8SS 897 906 915 924 933 9 478 943 952 961 970 979 988 997 *oo6 *oi5 *024 .1 2 0.9 I 8 479 480 481 68 033 042 051 060 070 166 079 088 097 106 115 •3 .4 . 1; 2.7 3.6 4. ^ 124 ^33 142 151 169 259 178 187 196 205 214 223 232 241 250 268 277 286 295 482 304 3^3 322 331 340 349 358 367 376 385 .6 5.4 483 394 403 412 421 430 439 448 457 4^6 475 484 484 493 502 511 520 529 538 547 556 565 •7 .8 6.3 7 2 485 574 583 592 601 610 61Q 628 637 646 654 •9 8.1 480 663 672 681 690 699 708 717 726 735 744 487 753 762 770 779 788 797 806 815 824 ^33 488 842 851 860 S6s 877 886 895 904 913 922 489 490 491 931 940 948 957 966 975 984 993 *002 *oi5 .1 .2 •3 8 o.S 1-7 2.5 69 019 ! 02g 037 046 055 064 073 081 090 099 108 117 126 134 143 152 161 170 179 187 492 196 205 214 223 232 240 249 258 267 276 493 284 293 302 311 320 328 337 346 355 364 .4 .5 3-4 a. 2 494 372 38^^ 390 399 408 41 6 425 434 443 451 .6 5-1 495 460 469 478 487 495 504 513 522 530 539 496 548 557 565 574 583 592 600 609 618 627 .7 8 5.9 6 A 497 635 644 653 662 670 679 688 697 705 714 •9 7.6 498 723 731 740 749 758 766 775 784 792 801 499 500 N. 810 819 827 836 845 853 862 871 879 888 897 905 914 923 931 946 949 958 966 975 1 2 3 4 5 6 7 8 9 P. P. 332 TABLE V.- -LOGARITHMS OF NUMBERS. 500 1 »> •.i 4 »"> (> 7 S 1) p P. 69 897 984 905 914 923 931 946 *027 949 *o36 958 9^6 975 *o6i 992 *OOI *OIO *oi8 *o44 *o53 502 70 070 079 087 096 105 113 122 131 139 148 9 0.0 5^3 157 165 174 182 191 200 208 217 226 234 . I 504 243 251 260 269 277 286 294 303 312 326 .2 1.8 505 329 337 346 355 3(^3 372 386 389 398 406 •3 2.7 506 415 423 432 441 449 458 4^6 475 483 492 •4 • 5 3.6 4-5 507 501 509 518 526 535 543 552 566 569 578 508 586 595 603 612 626 629 637 646 654 663 .6 5-4 509 510 511 672 6Sd 689 697 706 791 714 723 731 740 748 833 9J8 •7 .8 •9 6.3 7-2 S.I 757 765 774 782 799 884 808 816 825 842 856 859 867 876 893 901 910 512 927 935 944 952 961 969 978 986 995 *oo3 5^3 71 oil 020 028 037 045 054 062 071 079 088 514 096 105 113 121 130 138 147 155 164 172 515 186 189 197 206 214 223 231 239 248 256 8 516 265 273 282 290 298 307 315 324 332 340 . I .2 0.8 I -7 517 349 357 366 374 382 391 399 408 416 424 •3 2.5 5^^ 433 441 449 458 465 475 483 491 500 508 519 520 521 516 600 525 ' 533 542 550 633 558 567 575 583 592 •4 •5 .6 3.4 4.2 5.1 608 617 625 642 656 659 667 675 684 692 700 709 717 725 734 742 750 758 522 767 775 783 792 806 808 817 825 833 842 •7 .8 5-9 6.8 523 850 858 867 875 883 891 900 908 916 925 •9 7-6 524 933 941 949 958 9^6 974 983 991 999 *oo7 ; 525 72 016 024 032 040 049 057 065 074 082 090 ! 526 098 107 115 123 131 140 148 156 164 173 527 181 189 197 206 214 222 230 238 247 255 8 528 263 271 280 288 296 304 312 321 329 337 .1 o.S I 6 529 530 r53i 345 354 362 370 378 386 395 403 411 419 •3 •4 2.4 3-2 4..0 427 436 444 452 466 4^8 55Q 476 558 485 493 501 509 517 526 534 542 566 575 583 532 591 599 607 615 624 632 640 648 658 664 .6 4.8 533 672 68i 689 697 705 713 721 729 738 746 534 754 762 770 778 785 795 803 811 819 827 •7 8 5-6 6.4 7.2 535 835 843 851 859 868 876 884 892 900 908 ■9 53^ 916 924 932 941 949 957 965 973 981 989 537 997 *oo5 *oi3 *02i *o3o *o3S *o46 *o54 *o62 *o7o 538 73 078 085 094 102 1 10 118 126 134 143 151 ' 539 540 541 159 239 3^9 167 175 183 191 199 279 207 287 368 215 223 231 . I _ 2 .3 1 0.7 1-5 2.2 247 255 263 344 271 295 303 311 328 336 352 360 376 384 392 542 400 408 416 424 432 440 448 456 464 472 543 480 488 496 504 512 520 528 536 544 552 •4 3-0 544 560 568 576 584 592 600 608 615 623 631 •5 .0 3-7 4-5 545 639 647 655 663 671 679 687 695 703 711 546 719 727 735 743 75^ 759 767 775 783 791 •7 .8 •9 ^.2 6.0 6.7 547 798 806 814 822 830 838 846 854 862 870 548 878 886 894 902 909 917 925 933 941 ! 949 549 550 1 X. 957 965 973 981 989 997 075 *004 *OI2 *026 1*028 107 74 036 044 052 060 068 083 091 1 099 . 1 ! 2 1 3 1 4 5 C, 7 S P . P. 333 TABLE V.- -LOGARITHMS OF NUMBERS. 650 551 1 1 2 3 4 5 6 7 8 9 P^ P. 74 036 n5 044 052 060 068 075 083 091 099 178 107 186 123 131 139 146 154 1O2 170 55- 194 202 209 217 225 233 241 249 257 264 553 272 280 288 296 304 312 319 327 335 343 554 351 359 S'^G 374 3S2 390 398 406 413 421 55^ 5S(^ 429 437 445 453 466 463 476 484 492 499 507 515 523 531 538 546 554 562 570 577 8 557 585 593 601 609 616 624 62,2 640 648 655 .1 0.8 T 6 55S 663 671 679 687 694 702 710 718 725 733 .3 2.4 559 741 819 896 749 756 764 772 850 780 788 795 803 811 •4 •5 .6 3-2 4.0 4.8 560 826 834 842 857 865 873 881 888 5^1 904 912 919 927 935 942 950 958 966 562 973 981 989 997 '•004 *OI2 *020 *027 *o3,S *o43 5^3 75 051 058 065 074 081 089 097 105 112 120 •7 5-6 fi A 565 566 128 135 143 151 158 166 174 182 189 197 •9 D.4 7.2 205 212 220 228 235 243 251 258 266 274 281 289 297 304 312 320 327 335 343 350 5^7 568 569 670 571 358 366 *T -7 J/3 38i 389 396 404 412 419 427 435 442 450 458 465 473 4S0 488 496 503 511 5^9 525 534 541 549 557 564 572 580 1 ^ 587 595 602 6 10 618 625 633 641 648 656 663 671 679 6-6 694 701 709 717 724 732 !57-^ 739 747 755 762 770 M ^ •? ill 785 7Q2 800 808 . I 0.7 1-5 2.2 573 815 823 830 838 846 853 861 868 876 883 • 3 574 891 899 906 914 921 929 936 944 9Si 959 575 967 974 982 989 997 ^■"004 *OI2 ^019 *027 *o34 •4 3-9 570 76 042 050 057 065 072 oSo 087 095 102 no •5 .6 3-7 4.5 577 117 125 132 140 147 155 162 170 178 185 57S 193 200 208 2^5 223 230 238 245 253 260 •7 .8 •9 5-2 6.0 6.7 579 680 581 268 275 283 290 298 305 380 313 387 320 328 335 343 350 358 365 372 395 470 4c 2 410 417 425 432 440 447 455 462 477 485 582 492 500 507 514 522 529 537 544 552 559 5^3 567 574 582 589 596 604 6ii 619 626 634 ^5H 641 648 656 663 671 678 686 693 700 708 |5^5 715 723 730 738 745 752 760 767 775 782 5S0 790 797 804 812 819 827 534 841 849 856 7 i5«7 864 871 878 SS6 893 901 908 9LS 923 930 . I 0.7 588 589 590 591 937 945 952 960 967 974 982 989 997 '^'004 .3 1.4 2.1 77 on 019 026 ^33 041 114 048 055 129 063 070 078 151 225 .4 •5 .6 2.8 3-5 4.2 085 158 092 100 107 122 195 136 210 144 166 173 181 188 2C3 217 592 232 239 247 254 261 269 276 283 2QI 298 593 305 313 320 327 335 342 349 3S6 364 --* 371 •7 s 4-9 ^ 6 594 378 386 393 400 408 415 422 430 437 444 .9 6.3 |595 451 459 465 473 481 488 495 S^3 510 517 59t> 524 532 539 546 554 561 S^d> 575 583 590 1597 597 604 612 619 626 634 641 648 6SS 667, !598 670 677 684 692 699 706 713 721 728 735 j599 600 N. 742 750 757 764 771 779 786 793 8o5 808 815 822 829 837 844 851 858 S66 873 880 1 2 3 4 5 6 7 8 9 P, p- 334 TABLE V.- -LOGARITHMS ( 3F NUMBERS. N. 1 .ad 1 3 1 4 n a 7 S 1) P . P. 600 6oi 602 603 77 ^15 887 959 78 031 822 829 1 ^37 844 85 i 858 866 873 8S0 894 1 967 039 902 974 046 909 981 053 9^6 988 060 923 995 067 931 *oo3 075 938 *OIO 082 945 *oi7 089 952 *024 096 604 605 606 103 175 247 1 1 1 , 182 254 118 190 261 125 197 269 132 204 276 139 211 283 147 218 290 154 226 297 161 233 304 168 240 311 1 607 608 609 319 390 461 i 3^^ \ 397 469 333 404 476 340 412 483 347 419 490 354 426 568 362 433 504 369 446 511 376 447 518 383 454 526 .1 .2 •3 •4 .5 .6 ■7 8 0.7 1-5 2.2 3.0 3-7 4.5 5.2 6 610 533 604 675 746 540 1 547 554 J6i 632 703 774 575 583 590 597 611 612 613 , 61 1 682 753 618 689 760 625 696 767 639 716 781 646 717 788 654 725 795 661 732 802 668 739 8jo 614 615 616 817 887 958 824 894 965 831 901 972 838 908 979 845 915 986 852 923 993 859 930 *ooo 866 937 *oo7 873 944 *oi4 886 951 *02I .9 6.7 617 618 619 79 028 099 169 035 106 176 042 113 183 049 120 190 260 056 127 197 267 063 134 204 076 141 211 078 148 218 085 155 225 092 162 232 .1 .2 •3 7 0.7 1.4 2.1 620 239 309 379 449 246 253 274 281 288 295 302 621 622 623 316 386 456 323 393 462 330 400 469 337 407 476 344 414 483 351 421 490 358 428 497 365 435 504 372 442 511 624 625 626 518 588 657 595 664 532 602 671 539 609 678 546 6r6 685 553 622 692 560 629 699 567 ^36 706 574 643 713 581 656 720 •4 •5 .6 2.8 3-5 4.2 627 628 629 727 796 865 733 803 872 740 810 879 74? 8i§ S86 754 823 892 761 830 _89i_ 968 768 837 9O6 775 844 913 782 85 f 920 789 858 927 •7 .8 •9 4.9 5-6 6.3 680 934 941 ! 948 954 961 975 982 989 058 126 195 996 065 ^33 202 631 632 ^33 80 003 071 140 010 078 147 016 085 154 023 092 i6i 036 099 168 037 106 174 044 113 iSi 051 120 188 634 635 636 209 277 345 216 284 352 222 291 359 229 298 366 236 304 373 243 311 380 250 3^8 386 257 325 393 263 332 406 270 339 407 6 637 639 414 482 550 618 686 753 821 421 489 557 427 495 563 434 502 570 441 509 577 448 516 584 455 523 591 461 529 597 468 536 604 672 475 543 611 679 746 814 882 .1 .2 •3 .4 •5 .6 •7 8 0.6 1-3 1.9 .6 3-2 3-9 4-5 5-2 5-S 640 625 631 638 645 652 658 665 641 642 643 692 760 S2S 699 767 834 706 774 841 713 780 848 719 787 855 726 794 86: 733 801 868 740 807 875 644 645 646 888 956 81 023 895 962 ; 030 902 969 ^36 909 976 043 915 983 050 922 989 057 929 996 063 936 *oo3 076 942 *OIO 077 949 *oi6 083 •9 647 648 649 090 157 224 097 164 231 104 171 238 no ' 177 244 117 184 251 124 191 258 135 197 264 137 204 27! 144 211 278 345 151 218 284 351 650 291 298 304 311 3^8 324 33^ 338 N. 1 1 2 3 4 5 1 7 8 i 9 1 P. P. 335 TABLE v.— LOGARITHMS OF NUMBERS. N. 1 2 3 4 5 6 7 1 8 9 P.P. 650 |65i 81 291 298 304 311 318 324 33^ 33^ 345 351 ' 358 365 371 378 3^5 391 398 405 411 418 652 425 431 438 444 451 458 464 471 478 484 (>5S 491 498 504 511 518 524 531 538 544 551 654 558 564 571 577 584 591 597 604 611 617 655 624 631 637 644 650 657 664 676 677 684 j 656 696 697 703 710 717 723 730 736 743 750 7 657 756 763 770 776 783 789 796 803 809 816 .1 0.7 1 658 822 829 836 842 849 855 862 869 875 882 1.4 2. 1 659 660 661 888 895 901 908 915 921 928 934 941 948 •4 •5 .6 2.8 3.5 4.2 954 961 967 974 986 987 994 *oo6 *oo7 *oi3 82 020 ^^6. <^33 040 046 053 059 066 072 079 662 086 092 099 105 112 118 125 131 ^3^ 145 663 151 158 164 171 177 184 190 197 203 210 •7 « 4.9 664 217 223 230 236 243 249 256 262 269 275 •9 5'0 6.3 665 282 288 295 302 308 3^5 321 328 334 341 666 347 354 360 367 373 380 386 393 399 406 667 412 419 425 432 438 445 451 458 464 471 668 477 484 490 497 5^3 510 516 523 529 53<^ 1 669 670 ,671 542 549 555 562 568 575 581 588 594 601 9 607 614 620 627 ^33 640 646 653 659 666 672 678 685 691 698 704 711 717 724 730 A 2 '672 737 743 750 756 763 769 775 782 788 795 .2 0-6 1 .3 673 801 808 814 821 827 834 840 846 853 859 •3 1.9 '674 866 872 879 885 892 898 904 911 917 924 2.6 3-2 3.9 675 930 937 943 949 956 962 969 975 982 988 •4 676 994 *OOI *oo7 *oi4 *020 *027 *o33 *o39 *046 *052 .6 677 83 059 065 071 078 084 091 097 103 no 116 678 123 129 136 142 148 155 161 168 174 180 •7 s 4-5 5-2 5-8 679 680 j68i 187 251 193 200 206 212 219 225 231 238 244 •9 257 263 270 276 283 289 295 302 308 314 321 327 334 340 346 353 359 365 372 '682 378 385 391 397 404 410 416 4.23 429 435 I683 442 448 455 461 467 474 480 486 493 499 684 50^ 512 518 524 531 537 543 550 556 562 685 569 575 581 S88 594 6o5 607 613 619 626 A 6S6 632 ^38 645 651 657 664 676 676 683 689 J 6 i687 695 702 708 714 721 727 733 740 746 752 .2 1.2 68S 759 765 771 778 784 790 796 803 809 815 •3 1.8 689 690 691 822 885 828 834 841 847 853 859 866 872 878 .4 • 5 .6 2.4 3-0 3.6 891 897 904 910 916 922 929 935 941 948 954 960 966 973 979 985 992 998 *oo4 692 84 010 017 023 029 035 042 048 054 061 067 •7 .8 4.2 4.8 693 073 079 086 092 098 104 III 117 123 129 694 136 142 148 154 161 167 179 186 192 •9 5.4 695 198 204 211 217 223 229 236 242 248 254 696 261 267 273 279 286 292 298 304 311 317 697 323 329 335 342 348 354 360 367 373 379 698 385 392 398 404 410 416 423 429 435 441 699 700 447 510 454 460 465 472 479 485 491 553 497 5^3 516 522 528 534 541 547 559 i 565 L N. 1 2 3 4 5 6 7 8 9 P. P, 1 336 TABLE V.- -LOGARITHMS OF NUMBERS. 700 701 1 ! 12 ! .'i " 4 5 (; 7 s *) !• . P. 84 510 572 516 522 528 534 541 547 553 559 565 578 584 590 596 603 609 615 621 627 702 (>?>l 640 646 652 658 664 671 677 683 689 ,703 695 701 708 714 720 726 732 739 745 751 704 757 763 769 776 782 788 794 806 806 813 705 819 825 831 837 843 849 856 862 868 874 706 880 886 893 899 905 911 917 923 929 936 6 707 942 948 954 960 9^6 972 979 985 991 997 . I .2 0.6 1 .3 708 85 003 009 015 021 028 034 040 046 052 05 8 •3 1.(3 1709 710 711 064 070 077 083 089 095 156 217 lOI 162 107 163 113 119 •4 •5 .6 2.6 3-2 3.9 126 132 138 144 150 174 181 187 193 199 205 211 223 229 236 242 712 248 254 260 265 272 278 284 290 297 303 ^ 713 309 315 321 327 ZZ2> 339 345 351 357 Z^?> •7 .8 4-5 5-2 1 714 370 376 382 388 394 400 406 412 418 424 •9 5-8 715 430 436 443 449 455 461 467 473 479 485 716 491 497 503 509 515 521 527 533 540 546 717 552 558 564 570 576 582 588 594 600 606 718 612 618 624 635 ^36 642 648 655 661 667 719 720 721 673 679 685 691 697 703 709 715 721 727 6 n f\ 733 739 745 751 757 763 769 775 781 787 793 799 805 8ii 817 823 829 835 841 847 722 853 859 865 872 878 884 890 896 902 908 .2 I .2 1723 914 920 926 932 938 944 95Q 956 962 968 •3 I. 8 724 974 980 986 992 998 *oo4 *OTO *oi6 *022 *028 i725 86034 040 046 052 058 063 069 075 081 087 •4 2.4 3.0 1726 093 099 105 III 117 123 129 135 141 147 .6 3.6 J727 153 159 165 171 177 183 189 195 201 207 728 213 219 225 231 237 243 249 255 261 267 •7 .8 4.2 4.8 1 729 730 731 273 332 391 278 284 290 296 302 3O8 314 320 326 '9 5.4 338 397 344 350 356 362 368 374 380 386 403 409 415 421 427 433 439 445 732 451 457 463 469 475 481 486 492 498 504 733 510 S'^G 522 528 534 540 546 552 558 563 |734 569 575 58I 587 593 599 605 611 617 623 735 623 634 640 646 652 658 664 670 676 682 5 736 688 693 699 705 711 717 723 729 735 741 737 746 752 758 764 770 776 782 788 794 800 . I .2 0.5 I . I 738 805 8ii 817 823 829 835 841 847 852 858 •3 1-6 ,739 740 741 864 923 982 870 929 987 876 882 888 894 899 905 911 917 976 *o34 .4 • 5 .6 2.2 2.7 3-3 935 941 946 952 *OII 958 964 970 993 999 *oo5 *oi7 *023 *028 742 87 040 046 052 058 064 069 075 o8i 087 093 ^ 743 099 104 no 116 122 128 134 140 145 151 •7 .8 3-8 4.4 744 157 163 169 175 180 1 86 192 198 204 210 •9 4.9 745 215 221 227 233 239 245 250 256 262 268 '746 274 279 285 291 297 Z^3 309 314 320. 326 747 332 338 343 349 355 361 367 372 378 384 748 390 396 402 407 413 419 425 431 436 442 749 750 448 454 460 465 471 529 477 483 489 546 494 500 5:s8 : 506 512 517 523 535 541 552 " 1 2 3 4 5 G 7 8 J) P. P. 337 TABLE V.- -LOGARITHMS OF NUMBERS. N. 1 1 2 3 1 4 5 1 6 7 8 9 P. P. 750 1751 87 ^06 ! 512 517 523 529 535 541 546 552 558 1 564 570 575 58i 587 593 598 604 610 616 '752 622 627 ^33 639 645 650 656 662 668 673 ;753 679 685 691 697 702 708 714 720 725 731 !754 737 743 748 754 760 766 771 777 783 789 755 794 800 806 812 817 823 829 835 840 846 75t» 852 858 863 869 875 881 886 892 898 904 6 757 909 915 921 927 932 938 944 949 955 961 .1 2 0.6 I 2 I75S 967 972 978 984 990 995 *ooi "^007 *OI2 *oi8 .3 i.'s 1759 i760 761 88 024 030 035 041 047 1 04 053 058 064 070 075 133 •4 .5 .6 2.4 3.0 3.6 081 087 093 098 no 115 121 178 127 138 144 150 155 161 167 172 184 190 762 195 201 207 212 218 224 229 235 241 247 \7^3 252 258 264 269 275 281 286 292 298 303 ■7 « 4.2 4.8 5-4 1764 309 315 320 3^6 332 337 343 349 355 366 •9 i7t^5 366 372 377 383 389 394 400 406 411 417 766 423 428 434 440 445 451 457 462 468 474 7^7 479 485 491 496 502 508 513 519 525 530 I708 536 542 547 553 558 564 570 575 58i 587 (709 770 1771 592 598 604 609 615 671 621 626 632 638 643 s 649 654 660 666 677 683 688 694 700 705 711 716 722 728 733 739 745 750 756 i772 761 767 773 778 784 790 795 801 806 812 .1 0-5 773 818 823 829 835 846 846 851 857 863 S6s .3 1 . I 1-6 774 874 879 885 891 896 902 907 913 919 924 !775 930 936 941 947 952 958 964 969 975 980 •4 2,2 |77^ 986 992 997 *oo3 *oo8 *oi4 *oi9 *025 *o3i *o36 •5 .6 2.7 3-3 |777 89 042 047 053 059 064 070 075 081 087 092 '778 098 103 109 114 120 126 131 137 142 148 .7 3-8 779 1780 1781 153 159 165 170 176 181 187 193 248 198 204 . •9 4-4 i 4.9 209 215 226 226 231 237 243 254 259 265 276 276 282 287 293 298 304 309 315 1782 320 326 332 337 343 348 354 359 365 370 |7«3 376 38i 387 393 398 404 409 415 420 426 1784 431 437 4J2 448 454 459 465 470 476 481 ^785 487 492 498 503 509 514 520 525 531 536 786 542 548 553 559 564 570 575 581 586 592 5 787 597 603 60s 614 619 625 630 636 641 647 .1 0.5 788 652 658 66s 669 674 680 685 691 696 702 .3 1-5 789 790 791 707 7^3 718 724 729 735 740 746 751 757 •4 •5 .6 1 2.0 2.5 ' 3-0 ! 762 768 773 779 784 790 795 801 806 812 867 817 823 828 834 839 845 856 856 861 792 872 878 883 889 894 900 905 911 916 922 793 927 933 938 943 949 954 960 965 971 976 .7 .8 •9 3-5 794 982 987 993 998 *oo4 *oo9 *o,5 *020 *026 *o3i 4.0 4-5 795 90 036 042 047 053 058 064 069 075 080 086 796 091 097 102 107 113 118 124 129 135 140 797 146 151 156 162 167 173 178 184 189 195 798 200 205 211 216 222 227 233 238 244 249 799 800 254 260 265 271 276 282 287 292 298 3^3 309 314 320 325 330 33^ 341 347 352 358 1 N. 1 2 3 1 4 5 6 7 8 9 P .P. 1 338 TABLE V.- -LOGARITHMS OF NUMBERS. 1800 8oi 1 2 3 4 5 <; 7 S 9 P. P. 90 309 314 320 325 330 336 341 347 352 358 363 3^8 374 379 385 390 396 401 406 412 802 417 423 428 433 439 444 450 455 466 466 803 471 477 482 488 493 498 504 509 515 520 804 525 531 536 542 547 552 558 563 569 574 805 579 585 590 596 601 6og 612 617 622 628 ;8o6 633 639 644 649 655 666 666 671 ^76 682 807 687 692 698 703 709 714 719 725 730 736 808 741 746 752 757 762 768 773 778 784 789 809 810 811 795 800 805 811 816 821 827 832 838 891 843 a 848 854 859 864 870 875 886 886 896 902 907 913 918 923 929 934 939 945 950 b 812 955 961 9^6 971 977 982 987 993 998 *oo3 , I .2 0. 3 I.I «i3 91 009 014 019 025 030 036 041 046 052 057 •3 1-6 814 062 068 073 078 084 089 094 100 105 1 16 815 116 121 126 131 137 142 147 153 158 163 •4 .6 2 . 2 2. 7 816 169 174 179 185 190 195 201 206 211 217 3-3 817 222 227 233 238 243 249 254 259 264 270 818 275 280 286 291 296 302 307 312 318 323 •7 8 3-8 4-4 4.Q 819 820 821 328 381 333 339 344 349 402 355 360 365 418 471 371 376 •9 3H 392 397 408 413 423 429 434 439 445 450 455 461 466 476 482 1822 487 492 497 503 508 513 519 524 529 534 823 540 545 550 556 561 5^6 571 577 582 587 824 592 598 603 608 614 619 624 629 635 640 1825 645 655 656 661 666 671 677 682 687 692 826 698 703 708 714 719 724 729 735 740 745 827 750 756 761 766 771 777 782 787 792 798 828 803 808 813 819 824 829 834 839 84s 8so 829 830 831 855 908 960 860 S66 871 876 881 887 892 897 902 f 913 918 923 928 934 939 944 949 955 965 976 976 981 986 991 996 *002 *oo7 I 0.5 I.O 832 92 012 017 023 028 033 038 043 049 054 059 .2 833 064 069 075 080 085 090 096 lOI 106 III .3 1-5 834 116 122 127 132 137 142 148 153 158 163 835 168 174 179 184 189 194 200 205 210 215 •4 .5 2.5 836 220 226 231 236 241 246 252 257 262 267 .6 30 837 272 277 283 288 293 298 3^3 309 314 319 838 324 329 335 340 345 350 355 366 366 37^ •7 .8 3-5 4.0 839 i840 1841 376 381 386 391 397 402 407 412 417 423 474 •9 4.5 428 433 438 490 443 448 500 454 459 464 515 469 i 479 485 495 505 510 521 526 I842 531 536 541 546 552 557 562 567 572 577 843 583 588 593 598 603 608 613 619 624 629 844 634 639 644 649 655 660 665 670 675 686 1845 685 691 696 701 706 711 716 721 727 732 846 737 742 747 752 757 762 768 773 778 783 '847 788 793 798 803 809 814 819 824 829 834 1848 839 844 850 855 860 865 876 875 886 885 '849 850 891 942 896 947 901 906 911 9^6 921 926 977 931 982 937 988 952 957 962 967 972 1 ^• 1 2 3 4 5 G 7 8 P. P. 339 TABLE V.- -LOGARITHMS OF NUMBERS. 850 851 1 2 3 4 5 1 6 1 7 i 8 9 1 P. P. 1 92 942 947 952 957 962 967 972 977 982 988 993 998 *oo3 *oo8 *oi3 *oi8 *023 *028 *o34 *o39 852 93 044 049 054 059 064 069 074 079 084 090 853 095 100 105 no 115 120 125 130 135 140 854 146 151 156 161 166 171 176 181 186 191 855 196 201 207 212 217 222 227 232 237 242 1 856 247 252 257 262 267 272 278 283 288 293 s 857 298 303 308 313 318 323 328 333 338 343 .1 2 0.5 I I 858 348 354 359 364 369 374 379 384 389 394 .3 1.6 859 860 861 399 404 409 414 419 424 429 434 439 445 •4 •5 .6 2.2 2.7 3.3 450 455 460 465 470 475 480 485 490 495 506 505 510 515 526 525 530 535 540 545 862 550 556 561 566 571 576 581 586 591 596 863 601 606 611 616 621 626 631 636 641 646 •7 3-8 864 . 65J 656 661 66s 671 676 681 686 691 696 •9 4.4 4.9 865 701 706 711 716 721 726 731 736 742 747 866 752 757 762 767 772 777 782 787 792 797 867 802 807 812 817 822 827 832 837 842 847 868 852 857 862 867 872 877 882 887 892 897 869 870 871 902 907 912 917 922 927 932 937 942 947 5 952 94 002 957 962 1 967 972 977 982 987 992 997 007 012 017 022 025 031 ^36 041 046 872 051 056 061 065 071 076 081 086 091 096 .1 2 0.5 I 873 lOI 105 III 116 121 126 131 136 141 146 .3 1.5 874 151 156 161 166 171 176 181 186 191 196 875 201 206 210 215 226 225 236 235 246 245 .4 2.0 876 250 255 260 265 270 275 280 285 290 295 •5 .6 2.5 3.0 877 300 305 310 315 320 324 329 334 339 344 878 349 354 359 364 369 374 379 384 389 394 • 7 3-5 879 880 881 399 404 409 413 418 468 423 428 433 438 443 .0 •9 4.0 4.5 448 ' 4S3 458 i 463 473 478 483 487 492 497 502 507 512 517 522 527 532 537 542 882 547 552 556 56i 566 571 576 58i 586 591 I883 596 601 606 611 615 626 625 636 635 646 ' |884 645 650 655 660 665 670 674 679 684 689 885 694 699 704 709 714 719 724 728 733 738 886 743 748 753 758 763 768 773 777 782 787 4 887 792 797 802 807 812 817 821 826 83I 836 .1 0.4 888 841 846 851 856 861 865 870 875 886 885 .3 U.9 1-3 889 890 891 890 895 900 905 909 914 919 924 929 934 .4 .6 1.8 2.2 2.7 939 944 949 1 953 958 963 968 973 978 983 988 992 997 j*002 *oo7 *OI2 *oi7 *022 *026 031 892 95 036 041 046 051 056 061 065 070 075 o85 893 085 090 095 099 104 109 114 119 124 129 .7 g 3-1 3.6 4.6 894 134 138 143 148 153 158 163 167 172 177 •9 895 182 187 192 197 201 206 211 216 221 226 896 231 235 240 245 250 255 260 264 269 274 897 279 284 289 294 298 3<=>3 308 3^3 318 323 898 327 332 337 342 347 352 ! 356 361 366 371 899 900 376 381 385 390 395 400 ! 405 410 414 419 424 429 434 438 443 448 1 453 458 463 467 1 2 3 4 5 ! 6 7 8 9 P , P. 340 TABLE v.— LOGARITHMS OF NUMHERS. 900 901 1 12 3 4 »"> (i 7 S l\ l». . 95 424 472 429 434 438 443 492 448 453 458 4^J3 467 516 477 482 487 496 501 506 511 902 520 525 530 535 540 544 549 554 559 564 903 569 573 578 5 ^^3 588 593 597 602 607 612 904 617 621 626 63? 636 641 ^M5 650 655 660 905 665 669 674 679 684 689 ^93 ^98 703 708 906 713 717 722 727 732 737 741 746 751 756 907 760 765 770 775 780 784 789 794 799 804 908 808 813 8i8 823 827 832 837 842 847 851 909 910 911 ^56 861 866 870 875 923 880 885 890 894 942 899 904 952 _9°9_ 956 9^3 918 928 933 986 937 947 961 966 971 975 985 990 994 5 1 912 999 *oo4 "^009 *oi4 *oi8 ^023 *028 *^33 *o37 *042 .1 0.5 913 96 047 052 056 061 066 071 075 086 085 090 .3 1-5 914 094 099 104 109 113 1^8 123 128 132 137 915 142 147 151 156 161 166 170 175 180 185 •4 2.0 916 189 194 199 204 208 213 218 222 227 232 •5 .6 2-5 3-0 917 237 241 246 251 256 260 265 270 275 279 918 284 289 293 298 303 308 312 317 322 327 •7 .8 • Q 3-5 919 920 921 331 336 341 345 350 355 360 364 369 374 4.0 .1 . c. 379 3^ 388 393 397 402 407 412 416 421 426 430 435 440 445 449 454 459 463 4^8 922 473 478 482 487 492 496 501 506 511 515 923 520 525 529 534 539 543 548 553 558 562 924 567 572 576 58i 586 590 595 600 605 609 925 614 619 623 628 ^33 637 642 647 651 656 926 661 666 670 675 680 684 689 694 ^^98 703 927 708 712 717 722 726 731 736 741 745 750 928 755 759 764 769 773 778 783 787 792 797 929 930 931 801 806 811 815 820 825 829 834 839 843 ^ ^48 S95 853 857 862 867 871 876 881 885 896 899 904 909 913 918 923 927 932 937 4 932 941 946 951 955 960 965 969 974 979 983 . I 0.4 933 ^88 993 997 ^^002 *oo7 *OII *oi6 *020 *025 ^030 •3 u 9 1.3 934 97 034 039 044 048 053 058 062 067 072 076 935 081 086 090 095 099 104 109 113 118 123 .4 1.8 936 127 132 137 141 146 151 155 160 164 169 •5 .6 2.7 937 i74 178 183 188 192 197 202 206 21 1 215 93« 220 225 229 234 239 243 248 252 257 262 •7 .8 • 9 31 a. 6 939 940 941 265 3^3 271 317 276 280 285 289 33^ 294 340 299 345 303 308 322 328 33"^ 377 349 396 354 359 3^3 368 373 382 386 391 406 942 405 409 414 419 423 428 432 437 442 446 943 451 456 460 465 469 474 479 483 488 492 944 497 502 506 511 515 520 525 529 534 538 945 543 548 552 557 561 566 570 575 580 584 946 589 593 598 603 607 61 2 616 621 626 630 947 635 639 644 649 653 658 662 667 671 676 948 681 685 690 694 699 703 708 713 717 722 949 950 725 772 731 736 740 745 749 754 758 763 768 777 781 786 790 795 800 804 809 813 N. 1 2 3 4 5 6 7 8 P. P. 341 TABLE v.— LOGARITHMS OF NUMBERS ' N. 1 2 3 4 5 6 7 8 9 P.P. 950 97 772 818 777 822 781 786 796 795 800 804 809 ^^3 , 951 827 83 J 836 841 845 850 854 859 952 S63 868 873 877 882 886 891 895 900 904 953 909 914 918 923 927 932 936 941 945 950 954 955 959 964 968 973 977 982 986 991 996 1 955 98 000 005 009 014 oig 023 027 032 036 041 5 956 046 050 055 059 064 068 073 077 082 086 .1 0.5 957 091 095 100 105 109 114 118 123 127 132 .2 I.O 958 136 141 145 150 154 159 163 168 173 177 •3 1-5 959 960 961 182 227 272 186 191 195 200 204 209 213 218 222 •4 •5 .6 2.0 2.5 3-0 231 236 246 245 249 254 259 263 268 277 281 286 296 295 299 304 308 313 962 317 322 326 33^ 335 340 344 349 353 358 ■ 7 3-5 1 963 362 367 371 376 380 385 389 394 398 403 .8 4.0 ! 964 407 412 416 421 425 430 434 439 443 448 •9 4-5 965 452 457 461 466 470 475 479 484 488 493 966 497 502 506 511 515 520 524 529 533 538 967 545 547 551 556 560 565 569 574 578 583 968 587 592 596 601 605 610 614 619 623 628 969 970 971 632 677 722 637 641 646 650 655 659 663 668 672 , 4 681 686 696 695 699 704 708 713 717 726 731 735 740 744 749 753 757 762 .1 0.4 972 766 771 775 780 784 789 793 798 802 807 .2 0.9 973 8ii 815 820 824 829 ^33 838 842 847 85? •3 1-3 974 856 860 865 869 873 878 882 887 891 896 •4 1.8 975 900 905 909 914 918 922 927 931 936 940 •5 .6 2.2 976 945 949 954 958 963 967 971 976 980 985 2.7 977 989 994 998 *oo3 *oo7 *OII *oi6 *020 *025 *029 • 7 3-1 ' 978 99 034 038 043 047 051 056 060 065 069 074 .8 3.6 i 979 , 980 1 981 078 082 087 091 096 100 105 109 113 118 = 9 4.0 122 127 131 136 146 145 149 153 158 162 167 171 176 180 184 189 193 198 202 206 982 211 215 220 224 229 233 237 242 246 251 983 255 260 264 263 273 277 282 286 290 295 984 299 304 308 312 317 321 326 330 335 339 985 343 348 352 357 361 365 370 374 379 383 4 986 387 392 396 401 405 409 414 418 423 427 .1 0.4 i 987 431 436 440 445 449 453 458 462 467 471 .2 0.8 1 988 475 480 484 489 493 497 502 506 511 515 •3 1 .2 989 990 991 519 563 607 524 528 533 537 541 54^ 550 554 559 •4 •5 .6 1.6 2.0 2.4 568 572 576 581 585 590 594 598 603 611 616 626 625 629 ^33 638 642 647 992 651 655 660 664 668 673 677 682 686 696 •7 2.8 993 695 699 703 708 712 717 721 725 730 734 .8 3.2 3.6 994 738 743 747 751 756 760 765 769 773 778 •9 i 995 782 786 791 795 800 804 808 813 817 821 996 826 836 834 839 843 847 852 856 861 865 997 869 874 878 882 887 891 895 900 904 908 998 913 917 922 926 93^ 935 939 943 948 952 999 1000 956 00 000 961 965 969 974 978 982 987 991 995 004 oog 013 017 021 026 030 034 039 1 ^• 1 2 3 4 5 G 7 8 9 P. P. 1 342 TABLE v.— LOGARITHMS OF NUMBERS • • N. 1 2 1 3 1 4 5 1 6 7 8 1) P. P . 1000 OI 000 000 434 043 087 1 136 J 73 607 217 266 304 347 390 477 521 564 651 694 737 781 824 02 867 911 954 997 *o4i *o84 *I27 *i7i ^214 *257 03 001 301 344 387 431 474 517 566 604 647 696 04 733 777 820 863 906 950 993 *036 *o79 *I23 '^5 002 166 209 252 295 339 382 425 468 511 555 c6 59S 641 684 727 770 814 857 900 943 9^'^6 07 003 029 072 115 159 202 245 288 331 374 417 47 43 oS 466 503 545 590 ^33 676 719 762 805 848 .1 4-3 4-3 09 1010 1 1 891 934 977 *026 ^063 493 *io6 536 *i49 579 *I92 622 *235 665 *278 .2 •3 •4 • 5 .6 8.7 13.6 174 21.7 26. 1 8.6 12.9 1 17.2 25.8 I 004 321 751 364 407 450 708 794 837 880 923 966 *oo9 *o5i *094 *i37 12 005 186 223 265 309 352 395 438 481 523 566 13 609 652 695 738 781 824 866 909 952 ! 995 •7 .8 30 -4 34.8 30.1 34-4 14 006 038 081 123 165 209 252 295 337 386 423 •9 39- 1 38.7 15 a66 509 551 i 594 637 680 722 765 808 851 16 893 936 979 i*02 2 ^064 *io7 *i5o *i93 *235 *278 17 007 321 3^3 4O6 449 491 534 577 620 662 705 18 748 790 S33 875 918 961 *oo3 *o46 *o89 131 19 1020 21 008 174 600 009 025 217 642 oog 259 302 344 770 196 387 813 430 472 515 557 685 728 153 855 898 946 : 9S3 III 238 281 323 366 1 408 22 451 493 536 578 621 663 706 748 790 i 833 42 42 23 875 918 966 *oo3 *045 *o88 *i36 *I72 ""215 *257 .2 4-^ 8.5 4.2 8.4 24 010 300 342 385 427 469 512 554 1 596 639 681 ■3 12.7 12.6 25 724 766 808 851 893 935 978 *026 *o62 *io5 ■4 17.0 21 .2 16.8 21 .0 26 on 147 189 232 274 316 359 401 443 486 528 .6 25-5 25.2 27 570 612 655 697 739 782 824 866 908 951 •7 29-7 29.4 28 993 *o35 *o77 *I20 *l62 *204 *246 *288 *33^ *373 .8 •9 340 38.2 33-6 37-8 29 1030 31 012 415 837 457 500 542 584 625 ^048 668 710 753 795 879 921 963 i*oo6 *OQO 511 *I32 174 216 637 013 258 301 343 385 427 469 553 595 32 679 722 764 806 848 890 932 974 *oi6 *o58 33 014 100 142 184 225 263 310 352 394 436 478 34 526 562 604 648 688 730 772 814 856 898 i 35 940 982 *024 *o66 *io8 *i50 *I92 *234 *276 *3i8 1 3'^ 015 360 401 443 485 527 569 611 653 695 737 0S Al 41 ' 37 779 820 862 904 946 988 ^030 *072 *TI3 155 .1 41 4.1 9, 1 3^ 016 197 239 281 323 364 406 448 490 532 573 ■3 12.4 12.3 39 1010 41 615 017 033 450 657 699 741 782 824 866 908 950 991 •4 •5 .6 16.6 20.7 24.9 16.4 20.5 . 24.6 , 075 492 117 534 158 206 242 284 325 742 367 409 826 576 617 659 701 784 42 867 909 95 J 992 *034 *o76 *ii7 *i59 *20I ^242 .8 33-2 2fc.7 ?2.8 43 01 S 284 326 367 409 451 492 534 575 617 659 •y 37-3 36.9 1 44 706 742 783 825 867 908 950 991 *os3 *o74 ! 45 019 116 158 199 241 282 324 365 407 448 490 1 46 531 573 614 656 697 739 786 822 863 905 47 946 988 *029 *o7i *II2 *i54 *i95 *237 *278 *320 48 020 361 402 444 485 527 5^8 610 651 692 734 49 1050 775 021 189 817 236 1 858 899 941 982 396 *024 437 *o65 *i06 ♦148 272 2 3^3 354 478 520 ' 561 N. 3 1 4 r> <> 7 8 5) P. P • 343 TABLE v.— LOGARITHMS OF NUMBERS • i ^. 1 2 3 4 5 ! 6 ! 7 I 8 9 P.P. 1050 51 021 189 602 230 272 313 3^)4 396 437 478 892 520 561 644 685 726 '16?, 809 856 933 974 52 022 015 057 098 139 181 222 263 304 346 387 41 53 423 469 511 552 593 634 676 717 758 799 .1 4.1 8 -i 54 840 882 923 964 *oo5 *o46 *o88 *I29 *i7o *2II •3 12.4 : 55 023 252 293 zzs 376 417 458 499 540 58i 623 •4 16.6 ! 56 664 705 746 787 828 869 910 951 993 *o34 •5 .6 20.7 24.9 ' 57 024 075 116 157 198 239 280 321 362 403 444 5^ 485 526 568 609 650 691 732 773 814 855 .8 33-2 59 1060 61 896 025 306 715 937 978 *oi9 *o6o *IOI ^142 *i83 *224 *265 •9 37.3 347 388 429 469 5TO 551 592 ^3% 674 *o83 1 756 797 ^Z^ 879 920 961 *002 ^042 1 62 026 124 165 205 247 288 329 370 410 451 492 .41 1 63 533 574 615 656 696 737 778 819 860 901 .1 .2 4.1 8.2 64 941 982 *023 *o64 *io5 *i45 *i86 *227 *268 *309 •3 12.3 65 027 349 390 431 472 512 553 594 635 675 7^6 .4 16.4 66 757 798 ^z^^ 879 920 961 *OOI *042 *o83 *I23 .6 24.6 1 67 028 164 205 246 285 327 368 408 449 490 530 •7 28.7 1 68 571 612 652 693 734 774 815 856 896 937 .8 ■ Q 32.8 26. Q 69 1070 1 71 977 029 384 789 *oi8 *o59 *099 ^140 *i8i *22I ^262 *302 *343 J 424 465 505 546 586 627 668 708 749 830 870 911 951 992 *032 *o73 *ii4 *i54 72 030 195 235 276 2>^6 357 397 438 478 519 559 .1 4.0 73 599 040 680 721 761 802 842 883 923 964 .2 .•1 8.1 12. 1 74 031 004 044 085 I2g 166 205 247 287 327 368 75 408 449 489 529 570 610 651 691 73? 772 •4 •5 20.2 76 812 852 893 933 973 *oi4 *o54 *094 ""^ZS *i75 .6 24-3 77 032 215 256 296 3Z6 377 417 457 498 538 578 •7 .8 28.3 ^2.4 ^ 78 619 659 699 739 780 820 866 900 941 981 •9 36.4 1 79 1080 ' 81 033 021 424 825 061 102 142 182 222 263 303 343 383 Ar\ 464 504 544 584 625 665 705 745 785 866 906 946 986 *025 *o66 *io7 147 187 82 034 227 267 307 347 388 428 468 508 548 588 .1 4.0 1 ^2 623 668 708 748 789 829 869 909 949 989 .2 .3 8.0 12.0 ' 84 035 029 069 109 149 189 229 269 309 349 3S9 .4 16.0 1 85 429 470 510 550 590 630 670 710 750 790 •S' 20.0 86 830 870 910 950 990 ^029 ^069 *i09 *i49 *i89 .6 24.0 87 036 229 269 309 349 389 429 469 509 549 589 •7 .8 28.0 32.0 88 629 669 708 748 788 828 868 908 948 988 .9 36.0 89 1090 91 037 028 068 107 147 187 227 267 307 705 347 386 !^0 I 425 825 465 506 546 586 625 665 745 ! 785 864 904 944 984 *023 ^063 *io3 143 183 92 038 222 262 302 342 38J 421 461 501 540 580 .1 3.§ 92, 620 660 699 739 779 819 858 898 938 977 .2 .3 7-9 11-8 94 039 017 057 096 ^ze 176 216 255 295 *^ /I •* 374 •4 15-8 95 414 454 493 533 572 612 652 691 73 i 771 •5 19.7 96 810 850 890 929 969 *oo8 ^048 *o88 *I27 *i67 23-/ 97 040 205 246 286 325 365 404 444 483 523 5^3 •7 .8 27-6 ' 31-6 98 602 642 681 721 766 800 839 879 918 958 •9 35-5 1 99 1100 997 041 392 *o37 *o76 *ii6 *i55 *i95 *234 *274 ^Z^l ""353 432 471 511 550 590 629 669 708 748 N. 1 2 3 4 5 6 7 8 9 P.P. 344 VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. log sin ip = log (p" -\- S. 0° log ' = log sin (p -\- S'. 1 = log tan 04- 7". II log tan (p = log ' + T. log (p '' ( S 4.685 57 T Lo^. Sin. S' T LoJT. Tan. | o 57 — 00 5.31442 42 — 00 t 60 I 57 57 6.46 372 42 42 6.46 372 120 2 sf 57 .76475 42 42 .76475 180 3 Si Si .94 084 42 42 .94 084 240 4 Si Si 7.06 578 7.16 269 42 42 7.06 578 7.16 269 300 5 4.685 Si 57 5.31442 42 360 6 Si Si .24187 42 42 .24188 420 7 57 Si .30 882 42 42 .30882 480 8 Si 57 .36681 42 42 .36681 540 600 9 10 Si Si .41 797 42 42 .41 797 4.685 si Si 7.46 372 5.31442 42 7.46 372 660 II 57 Si .50512 42 42 .50512 720 12 Si Si .54 290 42 42 .54291 780 13 Si 57 • 57 767 42 42 • 57 767 840 14 Si 57 .60985 42 42 .60 985 900 15 4.685 57 58 7.63 981 5.31442 42 7.63 982 960 16 Si 58 .66 784 42 42 .66 785 1020 17 57 58 .6941^ 42 42 .69418 1 1080 18 57 58 .71 899 42 42 .71 900 1 1 140 19 57 4.685 57 58 _ ^.74.248 42 42 .74 248 ' 1200 20 58 7.76475 5.31443 42 7.76476 1260 21 57 58 .78 594 43 42 .78 595 1320 22 57 58 .80614 43 42 .80615 1380 23 57 58 .82 545 43 42 .82 546 1440 24 57 58 •84 393 43 42 •84 394 1500 25 4.685 57 58 7.86 166 5.31443 41 7.86 167 1560 26 57 58 .87 869 43 41 .87871 1620 27 57 58 .89 508 43 41 .89 510 1680 28 57 58 .91 088 43 41 .91 089 1740 29 57 58 .92 612 43 41 .92613 1800 30 4.685 57 58 7.94 084 5.31443 41 7.94 086 i860 31 57 58 •95 508 43 41 .95510 1920 32 57 58 .96 887 43 41 .96 889 1980 33 57 59 .98 223 43 41 .98 225 2040 34 57 59 .99 520 43 41 •99 522 2100 35 4.685 56 59 8.00 778 5.31443 41 8.00781 2160 36 56 59 .02 002 43 41 .02 004 2220 37 56 59 .03 192 43 4J .03 194 2280 38 56 59 .04 350 43 40 •04352 2340 39 56 4.685 56 59 .05 478 43 40 .05481 2400 40 59 8.06 577 5.31443 46 8.06 580 2460 41 56 S9 .07 650 43 40 .07 653 2520 42 56 59 .08 695 43 40 .08 699 2580 43 56 60 .09718 43 40 .09721 2640 44 56 60 .10716 43 40 .10720 2700 45 4.685 56 60 8. 1 1 692 5.31444 40 8. 1 1 696 2760 46 56 60 .12647 44 40 .12 651 2820 47 56 60 .13581 44 40 .13585 2880 48 56 60 .14495 44 39 .14499 ^940 49 56 60 .15390 44 39 •15395 3000 50 4.685 56 60 8.16268 5.31444 39 8.16272 3060 51 56 60 .17 128 44 39 • 17 133 3120 52 56 61 .17971 44 39 • 17 976 3180 53 56 61 .18798 44 39 .18803 3240 54 55 61 .19610 44 39 .19615 3300 55 4.685 Si 61 8.20 407 5-3H44 39 8.20412 3360 56 Si 61 .21 189 44 38 .21 195 3420 57 55 61 .21958 44 38 .21 964 3480 S« 55 61 .22713 44 38 .22719 3540 _ 59_ si 62 .23 45J 44 38 J ^23 462 _ 345 TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES.! log sin if) =z log 0" -\- S. 10 log (p" = log sin -f- 5". 1 log tan cp = log .'j6Ail 17609 6.76475 17609 3.23 524 0.00000 58 3 6.94084 6.94084 3-05915 0.00000 57 4 7.06 578 12494 9691 7918 6695 7.06 578 12494 9691 2.93421 0.00000 56 5 7.16 269 7.16 269 2.83736 0.00 000 55 6 7.24 1 8f 7.24188 791 8 2.75 812 0.00000 54 7 7.30882 7.30882 6094 2.69 11^ 0.00000 53 8 7.36681 5799 7.36681 5799 2.63 3I8 0.00000 52 9 7.41 797 5"5 4575 7.41 797 5"5 4575 4139 2.58 203 0.00000 51 10 7.46 372 7.46372 2.5362^ 0.00 000 50 II 7.50512 3778 7.50512 2.49488 0.00000 49 12 7.54290 7.54291 2.45 709 9.99999 48 13 7.57767 3476 7.57767 3476 2.42 233 9.99999 47 14 7.60985 3218 2996 2803 2633 7.60985 3218 2996 2803 2.39014 9.99999 4.6 15 7.63981 7.63982 2,36018 9.99999 45 i6 7.66784: 7.66 785 2.33215 9.99999 44 17 7.6941^ 7.69418 2633 2.30 582 9.99999 43 i8 7.71 899 7.71 900 2482 2.28 099 9.99999 42 19 7.74248 2348 2227 2119 7.74248 2348 2227 2.25751 9.99999 41 20 7.76475 7.76476 2.23 524 9.99999 40 21 T.^Z 594 I'l^ 595 2.21 405 9.99999 39 22 7.80614 7.80615 2.19384 9.99999 38 23 7.82545 1930 7.82 546 1930 2.17454 9.99999 37 24 7.84393 1843 1772 7-84394 1848 1773 2.15 605 9.99999 36 25 7.86 166 7.86 16^ 2.13832 9.99999 35 26 7.87869 1703 1639 7.87871 1703 1639 2.12 129 9.99999 34 27 7.89 508 7.89510 2.10490 9-99 998 33 28 7.91 088 1579 7.91 089 1579 2.08 916 9-99 998 32 29 7.92 612 1524 7.92613 1524 2.07 386 9-99 998 31 30 7.94084 1472 7.94086 1472 1424 2.05 914 9-99 998 30 3i 7.95 508 7.95510 2.04 490 9.99998 29 32 7.96 887 1379 7.96 8S9 1379 2.03 III 9-99998 28 33 7.98 223 1336 7.98225 1336 2.01 774 9.99998 27 34 7.99 520 1296 7.99 522 1296 2.00478 9.99998 26 1 35 8.00 778 1253 1223 1 190 1158 1128 1099 1072 8.00781 ■1^59 1223 iigo 1158 1.99 219 9.9999? 25 i 36 37 8.02 002 8.03 192 8.02 004: 8.03 191 1.97995 1.96 805 9.9999? 9.99997 24 23 38 8.04 350 8.04 352 1.95 647 9.99997 22 39 8.05 478 8.05481 1123 1099 1072 1.94 519 9.99997 21 40 8.06 577 8.06 580 1-93 419 9-99 997 20 41 8.07 650 8.07653 1.92 347 9.99997 19 42 8.08 696 8.08 699 1. 91 300 9-99 997 18 43 8.09 718 998 976 8.09721 1.90278 9-99 996 17 1 44 8.10716 8.10720 999 976 1.89 279 9-99 996 16 45 8. 1 1 692 8. 1 1 696 1.88303 9-99 996 15 46 8.12647 8.12651 954 1.87 349 9.99996 14 47 8.13 581 934 8.13585 934 1.86 415 9.99996 13 48 8.14495 914 895 877 860 8.14499 914 1.85 506 9.99996 12 49 8.15396 8.15395 895 877 860 1.84605 1.8372? 9-99 995 II 50 8.16268 8.16272 9.99995 10 51 8.17 128 843 8.17 133 843 827 1.82867 9-99 995 9 52 8.17 971 8.17 976 1.82023 9-99 995 8 53 8.18798 811 797 782 768 8.18803 1. 81 igg 9-99 995 7 54 8.19610 8.19615 797 783 763 1.80384 9-99 994 6 55 56 8.20407 8.21 189 8.20412 8.21 195 1.79587 1.78804 9-99 994 9-99 994 5 4 57 8.21 958 8.21 964 1.78036 9.99994 3 58 8.22 713 755 8.22 719 755 1.77 286 9.99994 2 59 8.23455 8.24185 742 730 8.23462 742 730 1.76538 9-99 993 I 60 8.24 192 1.75808 9-99 993 1 Log. Cos. D Log. Cot. Com. D. Log. Tau. Log. Sin. / 89^ 348 TABLE VII. — LOGARITHMIC SIXES, COSINES, TANGENTS, AND COTANGENTS. 1 TiOK. sin. 8.24 18S I 8. 24 903 o 8.25 609 3 8.26 304 4 8.26988 5 8.27 661 6 8.28324 7 8.28977 8 8.29620 9 8.30254 10 8.30879 1 II 8.31495 12 8.32 102 ■ 13 8.32 701 14 8.33292 15 8.33 87S i6 8.34450 17 8.35018 i8 8-35 578 19 8.36 1 31 20 8.36677 21 S.37 2r7 22 8.37750 23 8.38276 24 8.38796 25 8.39310 26 8.39818 27 8.40320 28 8.40816 29 8.41 307 30 8.41 792 31 8.42 271 32 8.42 746 33 843215 34 8.43 680 35 8.44 139 36 8.44 594 37 8.45 044 38 8.45489 39 8.45930 40 8.46 365 41 8.46798 42 8.47 226 43 8.47 650 44 8.48069 45 8.48485 46 8.48 896 47 8.49 304 48 8.49 708 49 8.50 108 1 50 8. 50 504 1 51 8.50897 52 8.51286 ' 53 8.51 672 54 8.52055 1 55 8.52434 56 8.52810 ! 57 8.53183 58 8.53552 59 8-53918 GO 8.542S2 Lot;. Cos. D 718 706 694 684 673 663 653 643 634 625 616 607 599 591 583 575 567 560 553 546 539 533 526 520 514 503 502 496 491 485 479 474 469 464 459 454 450 44S 440 436 432 428 423 419 415 411 407 404 400 396 393 389 386 382 379 375 373 369 366 363 I> Loir, Tan. Com, I). 8.24 192 8.24910 8.25 6I6 8.26 311 8.26 99^ 8.27 669 8.28 332 8,28985 8.29 629 8.30263 8,30888 8.31 504 8.32 112 8.32711 8.33302 8,33885 8,34461 8.35029 8.35 589 8.36 143 8.36 689 8.37 229 8.37 762 8.38289 8.38809 8.39323 8.39831 8.40334 8.40830 8.41 321 8.41 807 8.42 287 8.42 762 8.43231 8.43 696 8.44 156 8.44 611 8.45061 8.45 507 8.45 948 8.46 385 8,46817 8.47 245 8.47 669 8.48089 8.48 505 8.48917 8.49325 8.49729 8,50130 8-50526 8.50 920 8.51 310 8,51 696 8,52079 '875^2"458~ 8.52835 8.53208 8.53578 8.53944 8-54 308 Los:. Cot. 718 706 695 6S4 673 663 653 643 634 625 616 607 599 591 583 575 568 560 553 546 539 533 527 520 514 50S 502 496 491 4S5 480 475 469 464 460 455 450 445 441 437 432 428 424 419 416 412 408 404 400 396 393 390 386 383 379 376 373 370 366 364 Com, l». I-oir. Cut. Loir. Cos. 1,75 808 9-99 993 1.75090 9-99 993 1.74383 9-99 993 1.73 688 9.99992 1.73004 9.99992 1.72 331 9.99992 1. 71 667 9.99992 1. 71 014 9.99992 1.70 371 9.99991 1.69736 9.99991 1.69 III 9.99991 1.68495 9.99990 1.67888 9.99990 1.67288 9.99990 1.66697 9.99990 1.66 1 14 9.99989 1.65 539 9-99989 1,64971 9.99989 1. 64410 9.99989 1.63857 9.99988 1.63 310 9.99988 1.62 771 9.99988 1,62 238 9.99987 1,61 711 9.99 9S7 1. 61 191 9.99987 1,60676 9.99986 1,60 168 9,99986 1.59666 9.99986 1.59 169 9.99986 1.58678 9.99985 1.58 193 9.99985 1. 57713 9-99985 1.57238 9.99984 1,56768 9-99984 1,56304 9.99984 1,55844 9.99983 1-55389 9.99983 1-54 938 9.99982 1-54 493 9.99982 1.54052 9.99982 1. 53615 9.9998! 1-53183 9.99981 1-52754 9.99981 1.52330 9.99980 1. 51 911 9,99980 9-99 979 1. 51 495 1. 5 1 083 9-99 979 1.50675 9-99 979 1.50270 9-99 978 1.49870 9.99978 1-49 473 9.99978 1,49080 9.99977 1,48690 9-99 977 1.48 304 9-99 976 1,47921 9.99976 1,47541 9-99 975 1,47 165 9-99 973 1,46792 9-99 975 1,46422 9-99 974 1,46055 1.45 691 9-99 974 9.99973 Loir, I'jin. Loir. Sin. ss 349 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 2 / Log. Sin. D Log. Tan. Com. D. L og. Cot. Log. Cos. 8.54 282 360 8.543O8 ■^(m 45691 9-99 973 60 I 8.54642 8.54669 45331 9 99 973 59 2 8.54999 357 8.55027 358 J 44 973 9 99972 58 3 8-55 354 354 8.55381 354 J 44618 9 99972 57 , 4 8-55 705 8.56054 351 348 8-55 733 8.56083 352 J 44266 9 99971 56 5 349 J 346 J 43917 9 99971 55 6 8. 56 400 8.56429 43571 9 99971 54 7 8.56743 343 8.56772 343 I 4322^ 9 99970 53 8 8.57083 340 8.57 113 341 I 42886 9 99970 52 : 9 8. 57 421 338 335 8.57452 338 I 42 548 9 99969 51 10 8.57 756 ^•Sll^l 335 .42 212 9 •99969 50 II 8.58089 332 8.58 121 333 J 41879 9 99968 49 12 8.58419 330 8.58451 330 I 41 548 9 99 968 48 13 8.58747 327 8.58779 328 J 41 220 9 99967 47 14 8.59072 325 323 8.59105 8.59428 325 I 40895 9 99967 46 15 8-59 395 323 J 40571 9 99966 45 i i6 8.59715 320 8.59749 40251 9 99966 44 i '7 8.60033 3^3 8.60067 318 I 39932 9 99965 43 i8 8.60349 310 8.60384 316 I .39616 9 99965 42 1 19 8.60662 313 311 8.60698 314 I 39302 9 99 964 41 20 8.60973 8.61 009 6^'- "■ J 38990 9 99964 40 1 "^ 8.61 282 309 8.61 319 309 J 38681 9 99963 39 22 8.61 589 306 8.61 626 307 I 38374 9 99963 38 23 8.61 893 304 8.61 931 3C5 I 38068 9 99962 37 1 24 8.62 196 302 300 8.62 234 303 I 37765 9 99962 ?>^ 1 25 8.62 496 8.62 535 300 37465 9 99.961 35 i 26 8.62795 298 8.62 834 299 I 37 166 9 99961 34 1 V 8.63 091 296 8-63 131 297 I 36869 9 99960 33 28 8-6338! 294 8.63425 294 J 36574 9 99 959 32 29 8.63677 292 290 8.63 718 293 I 36281 9 99 959 31 1 1 30 8.63968 8.64009 291 35990 9 99 958 30 31 8.64256 283 8.64 298 283 J 35702 9 99958 29 32 8.64 543 285 8.64585 287 - 35414 9 99 957 28 1 33 8.64827 284 8.64870 285 I 35 129 9 99 957 27 i 34 8.65 no 282 281 8.65153 283 I 34846 9 99 956 26 ! 35 8.65 391 8.65435 2S1 34565 9 99956 25 36 8.65670 279 8.65715 280 34285 9 9995! 24 37 8.65 94f 277 8.65 993 278 J 34007 9 99 954 23 38 8.66223 275 8.66 269 276 J 33731 9 99 954 22 1 39 8.66497 274 272 8.66 543 274 I 33 456 9 99 953 21 1 40 8.66 769 8.66816 -/- 33184 9 99 953 20 41 8.67 039 26§ 8.67087 271 ^ 32913 9 99952 19 42 8.67 308 8.67 356 32643 9 99952 18 43 8-67575 8.67624 32376 9 99951 17 44 8.67 840 265 264 8.67 890 266 . 32 no 9 99950 16 : 1 45 8.68 104 8.68 154 262 31845 9 99950 15 46 8.68 366 8.68417 31 583 9 99 949 14 47 8.68627 8.68 678 31 321 9 99 948 13 48 8.68 886 259 8.68938 259 J 31 062 9 99948 12 1 49 8.69 144 257 256 8.69 196 258 I 30803 9 99 947 II 50 8.69400 8.69453 256 30547 9 99 947 10 ! 51 8.69654 254 8.69708 255 J 30 292 9 99 946 9 52 8.69907 253 8.69 961 253 J 30038 9 99 945 8 53 8.70159 251 8.70214 29786 9 99 945 7 54 8. 70 409 250 248 8.70464 250 , 29 53? 9 99 944 6 55 8.7065^ 8.70714 249 248 29286 9 99 943 5 1 56 8.70905 247 8.70962 29038 9 99 943 4 ! 57 8.71 150 245 8.71 208 246 , 28 791 9 99942 3 1 58 8.71 395 244 8.71453 245 J 28546 9 99942 2 ; 57 8.71 638 243 241 8.71 697 243 J 28303 9 99941 I i 60 8.71 880 8.71 939 242 28060 9- 99940 ^ 1 Log. Cos. D Log. Cot. Com. D. Lc )g. Tan. Log. Sin. j 1 87' 350 TABLE VII. — LOGARITHMIC SINES, COSIXES, TANGENTS, AND C0TAN(;ENTS 10 II 12 13 14 25 26 27 28 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 oO 51 52 53 54 55 56 57 58 59 60 Log. Sin. 8.71 880 8.72 120 8.72359 8.72597 8.72833 8.73069 8.73 302 8-73 533 ^■73 766 8.73997 8.74 226 8-74453 8.74680 8.74905 8.75 129 8-75 353 8-75 574 8-75 795 8.76015 8.76233 8.76451 8.7666; 8.76883 8.77097 8.77310 8.77 522 8.77 733 8.77943 8.78 152 8.78360 8.78 567 8.78773 8.78978 8.79183 8.79386 8.79588 8.79789 8.79989 8.80189 8.80387 8.80585 8.80782 8.80977 8.81 172 8.81 366 8.81 560 8.81 752 8.81 943 8.82 134 8.82324 8.82 513 8.82 701 8.82888 8.83075 8.83260 ^-83445 8.83629 8.83813 8.83995 8.8417; 8-84 358 Log. Cos. 240 239 237 236 235 233 233 231 230 229 227 226 225 224 223 221 221 219 2I§ 217 215 215 214 213 212 211 210 209 208 207 206 205 204 203 202 201 200 199 198 197 197 195 195 194 193 192 191 191 187 186 185 185 184 183 182 182 181 d. Loy. Tan. 8-71 939 8.72186 8.72420 8.72 659 8.72896 8.73 131 8-73366 8-73 599 8.73831 8.74062 8.74 292 8.74 520 8-74748 8.74974 8.75 199 8.75422 8.7564$ 8.75867 8.76087 8.76 306 8.76524 8.76741 8.76958 8.77 172 8.77386 8.77 599 8.77 811 8.78022 8.78232 8.78441 8.78648 8.7885$ 8.79061 8.79 266 8.79470 8.79673 8.79875 8.80075 8.80275 8.80476 8.80674 8.80871 8.81 068 8.81 264 8.81 459 8.81 653 8.81 846 8.82038 8.82 230 8.82 420 8.82616 8.82 799 8.82987 8.83175 8.83 361 8.83547 8.83732 8.83 916 8.84 100 8.84282 8.84464 Log. Cot. c. d. 241 240 2J8 237 235 235 233 232 231 229 228 227 226 225 223 223 221 220 219 218 217 216 214 214 213 212 210 210 209 207 207 206 204 204 203 202 201 200 199 198 197 197 195 195 194 193 192 191 190 190 187 186 185 185 184 183 182 182 3° Lotf. Cot. 1.28 066 1.27 819 1.27 579 1.27 341 1.27 104 1.26868 1.26633 1 . 26 400 1.26 168 1.25937 1.25 708 1.25479 1.25 252 1.25 026 1.24 801 1.24577 1.24 35-? 1-24133 1-23913 •3 693 I. 1-23475 1.23258 1.23 042 1.22 82; 1.22 613 1.22 400 1.22 188 1. 21 978 1. 2 1 768 1. 21 559 1. 21 351 1. 21 144 1.20 938 1.20734 1.20 530 1.20 327 1.20 125 1. 19923 1. 19 723 1. 19 524 1. 19 326 1. 19 128 1. 18 931 1. 18736 1. 18 541 1. 18 347 1. 18 154 1. 17 961 1. 17 770 1. 17 579 1-17389 1. 17 201 1. 17 012 1. 16825 1. 16 638 1. 16453 1. 16 268 1. 16083 1 . 1 5 900 1.1571; ^•15535 c. (1. Log. Tan. Loe. Cos. 9.99940 9.99940 9-99 939 9-99 938 9.99938 9.99937 9-99 936 9-99 935 9-99 935 9-99 93-+ 9-99 933 9-99 933 9-99932 9-99 931 9-99931 9.99930 9-99929 9.99928 9.99928 9-99927 9-99926 9.99925 9-99925 9-99924 9.99923 9.99922 9.99922 9.99921 9.99926 9.99919 9.99919 9.99918 9.9991; 9-99 916 9.99916 9.99915 9.99914 9-99913 9.99912 9.99912 9.99 911 9.99910 9-99 909 9-99 908 9.99907 9.99907 9.99906 9-99905 9.99904 9.99903 9.99902 9.99902 9.99901 9.99900 9-99899 9.99898" 9.99897 9.99895 9.99896 9-99895 9-99894 Lug. Sin, i 50 49 48 47 _46^ 45 44 43 42 41 40 39 38 37 _3i 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 9 8 7 6 r. r. 330 320 310 6 330 32.0 31 7 38.5 37-3 36.1 8 44.0 42-6 41 3 9 49-5 48.0 46.5 10 55 -o 53-3 S'O 20 110. 106.6 J03-3 30 165.0 160.0 1550 40 220.0 213.3 206. A 50 275.0 266.6 258.3 290 280 270 6 29.0 28.0 27.0 7 8 33-8 38.6 32-6 37-3 360 9 10 43 S 48.3 42.0 46.6 40.5 450 20 96.6 93-3 90.0 30 40 145.0 193-3 140.0 186.^ 135-0 180.0 5'^ 241.6 233 -3 225.0 300 30.0 350 40.0 45.0 50.0 100.0 150.0 200.0 250.0 260 26.0 30-3 34-6 39 o 43-3 86.6 130.0 173-3 216.6 250 240 230 220 6 25.0 24.0 23.0 22.0 7 29 I 28.0 26 § 25 6 8 33 3 32.0 30 6 29 3 9 37 5 36.0 34 5 33 10 41 6 40.0 38 3 36 ft 20 83 3 80.0 76 6 73 3 30 12s 120.0 IIS no 40 166 6 160.0 153 3 146 6 50 208 3 200.0 191 6 X83 3 210 200 190 180 6 21 .0 20.0 19.0 18.0 7 24-5 23-3 22.1 21 .0 8 28.0 26.6 25-3 24.0 9 31-5 30.0 28.5 27.0 10 35-0 33-3 31-6 30.0 20 70.0 66.6 63-3 60.0 30 105.0 100.0 05.0 90.0 40 140.0 1.33-3 126.6 120.0 50 175-0 166.6 158-3 150.0 6 9 0.9 9 0.9 8 0.8 7 0.7 6 0.6 7 8 i.i 1.2 1.0 1.2 0.9 1.0 0.8 0.9 0.7 0.8 9 1.4 1-3 1.2 1 .0 0.9 10 1.6 1-5 1-3 1.1 1.0 20 31 3-0 2-6 2-3 2.0 30 40 50 4-7 6-3 7-9 4 5 6.0 7 5 4.0 6-6 3-5 5-8 30 4.0 5-0 0.5 0.6 o 6 0.7 2-5 3| 4-1 4 4 3 2 I 6 0.4 0.4 0.5 0.2 0.1 7 o.,S 0.4 0.3 0.2 0.1 8 0.6 0.5 0.4 0.2 I 9 °? 0.6 0.4 0.3 0.1 10 0.7 0-^ O..S 0.3 0.1 20 1-5 '■3 I.O 0.6 03 30 2.2 2.0 >-5 1.0 0-5 40 3? ^•6 2.0 '•? 0$ 5 10 II 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Lot?. Sip. I d. 50 51 52 53 54 55 56 57 58 00 8.94029 8.94174 8.94317 8.94460 8.94603 8.94745 8.94887 8.95028 8.95 169 8.95310 8.95450 8.95 589 8.95728 8.95867 8.96 005 8.96 143 8.96280 8.96417 8-96553 8.96689 8.96825 8.96960 8.9709^ 8.97 229 8.97 363 8.97496 8.97 629 8.97 762 8.97894 8.98026 8.Q8 ii;7 8.98288 8.98419 8.98 549 8.98679 8.98 808 8.98937 8.99066 8.99194 8.99322 8.99449 8.99 577 8.99703 8.99830 8.99956 9.00081 9.00 207 9.00 332 9.00456 9.00 580 9.00704 9.00828 9.00951 9.01 073 9.01 196 9.01 318 9.01 440 9.01 561 9.01 682 9.01 803 9.QI 923 Log. Cos. 144 143 143 143 142 142 141 141 140 140 139 139 138 138 138 137 137 136 136 135 135 134 »34 134 133 133 132 132 132 131 131 130 130 130 129 129 12§ I2g 127 127 127 126 126 126 125 125 125 124 124 124 123 123 122 122 122 122 121 121 120 126 Log. Tan. | c. d. | Log. <'<>t. 8.94 195 8.94346 8.94485 8.94629 8.94773 8.94917 8.95059 8.95 202 8.95 344 8.95485 8.95626 8.95767 8.95 90^ 8.9604^ 8.96 i8g 8.96325 8.96464 8.96602 8.96739 8.96876 8.97 013 8.97 149 8.97 285 8.97421 8.97 556 8.97 690 8.97 825 8.97 958 8.98 092 8.98225 8.98357 8.98490 8.98621 8.98753 8.98884 8.9901$ 8.99 145 8. 99275 8. 99404 8. 99 533 8. 99662 8. 99791 8. 99919 9 00046 9 00 174 9 00 306 9 00427 9 00553 9 00679 9 00804 9 00 930 9.01 054 9.01 179 9.01 303 9.01 427 9.01 550 9.01 673 9.01 796 9.01 9I8 9. 02 046 9.02 162 Log. Cot. 145 144 144 144 M3 142 142 142 141 141 141 140 140 139 139 138 138 137 137 137 136 136 135 135 134 134 133 133 133 132 132 131 131 131 131 130 130 129 129 129 I2g 128 127 127 126 126 126 125 125 125 124 124 124 124 123 723 123 122 122 121 "cTJT 1.05 803 1.05659 1.05 515 1.05 370 1.05226 1.05 083 1.04946 1.04 798 1.04 656 1.04 514 1.04373 1.04 232 1.04092 1.03952 j_^03_8_i^ 1.03674 1.03 536 1.03398 1.03 266 1.03 123 1.02 986 1.02 856 1.02 714 1.02 579 1.02 444 1.02 309 1.02 175 1.02 041 1. 01 908 1. 01 775 Lour. Cos. 1. 01 642 1. 01 510 1. 01 378 1. 01 247 1. 01 116 1.00985 1.00855 1. 00 725 1.00595 1.00466 1.00337 1. 00 209 1 . 00 08 1 0.99953 0.99 826 0.99699 0.99 573 0.99 446 0.99321 0.99195 0.99070 0.98945 0.98821 0.98 697 0.98 573 0.98 450 0.98327 0.98 204 0.98 081 0.97959 0.97 838 JiOg. Tan. S4t' 9.99834 9-99833 9.99832 9.99831 9.99830 9.99829 9.99827 9.99826 9-99825 9.99824 9.99823 9.99 822 9.99 821 9.99819 9.99 81 8 9.99817 9-99816 9.99815 9.99814 9-99 Si 3 9.99 81 1 9.99 816 9.99809 9.99 808 9.99807 9.99805 9-99804 9.99803 9.99 802 9.99801 9.99799 9-99 798 9.99797 9.99796 9-99 794 9-99 793 9.99 792 9.99791 9-99789 9-99788 9.99787 9.99786 9.99784 9-99783 9-99782 9.99781 9.99779 9-99 778 9.99777 9.99776 9-99 774 9-99 773 9.99772 9.99776 9.99769 9.99768 9.99766 9.99765 9.99764 9.99763 9-99761 Log. Sin. 00 59 58 57 Ji 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 35 34 33 32 31 I'. I'. 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 1 1 To 9 8 7 6 140 139 138 137 14.0 13.9 13.8 13-7 16. s 16.2 16. 1 16.0, 18.6 18.5 18.4 18.2, 21 .0 20.8 20.7 20.5 23-3 23.1 23.0 22.8 46.6 46.3 46.0 45-6 70.0 69-5 69.0 68.5 93-3 92.$ 92.0 9'-3 116. 6 i'5-8 115. 114.1 '3-5 13-4 13 3 »5 7 15-6 15 5 18.0 17-8 17 7 20.2 20. 1 19 9 22.5 22.3 22 1 45.0 675 90.0 44-6 67.0 89.3 44 66 88 3 § <5 112.5 III. 6 110 8 145 144 143 142 141 6 14-5 14.4 14.3 14.2 14 I 7 16.9 16.8 16.7 16 .S 16.4 8 19-3 19.2 19.0 18.9 18.8 9 21.7 21.6 21.4 21.3 31. I 10 24.1 24.0 23 § 23-6 23-5 20 48-3 48.0 47-6 47-3 47.0 30 72 5 72.0 71 5 71.0 70.5 40 9<> 6 96.0 95-3 94-^ 94.0 50 120.8 120.0 119. 1 118.3 H7-5 136 13.6 o; 15-8 - 18. 1 20.4 22.^ 45-3 68.0 90-6 l"3-3 135 134 133 132 13.2 15-4 17.6 19.8 22.0 44.0 66.0 131 130 129 128 6 13-1 13.0 12.9 12.8 7 153 151 15.0 14 9 8 17-4 173 17.2 17.0 9 19-$ 19.5 193 19.2 10 21. § 21.6 2t-5 21-3 20 43-6 43-3 43 42.6 30 6s.,S 65.0 64.5 64.0 40 87.3 86.^ 86.0 85.3 50 109.1 108.3 107-5 106.6 127 126 125 124 123 12.7 12.6 12.5 12.4 12.3 14.8 14-7 14.6 14.4 14-3 16.9 16.8 16.6 16.5 16.4 19.0 18.9 18 7 18.5 18.4 21.1 21.0 20. § 20.^ 20.5 42.3 42.0 41 6 41-3 41.0 63. s 63.0 62.5 62.0 6t.5 84-6 84.0 83.3 82. 6 82.0 105-8 105.0 104.1 103.3 102.5 122 12.2 14.2 16.2 .8.3 20.3 40.6 61 .0 81.3 lOI.fi 121 12. 1 14.1 i6.i 18.1 20. T 40.3 6a. s 80.^ 100. 8 120 I o.i 0.2 0.2 12.0 •4 16.0 18.0 20 40 60 80 100. o O. 2 0-5 0.7 I O 0.0 0.0 0.1 0.1 0.1 o.i o 3 0.4 0-3 o-S o $ 0-8 P. P. 353 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 10 II 12 14 15 16 18 19 20 21 22 23 24 ^5 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Lo^. Sin. 9.01 923 9.02 043 9,02 163 9.02 282 9.02 40T 9.02 520 9.02638 9.02 756 9.02 874 9.02 992 9.03 109 9.03 22^ 9-03 342 9.03458 9-03 574 9.03689 9.03 805 9.03919 9.04034 9.04 148 9.04 262 9.04376 9.04489 9.04 602 9.04715 9.04 828 9.04 940 9.05 052 9.05 163 9.05275 9.05 386 905496 9.05 607 9.05717 9.05 827 9-05 936 9.06046 9.06155 9.06 264 9.06 372 9.06480 9.06 588 9.06 696 9.06 803 9.06 910 9.07 017 9.07 124 9.07 230 9-07 336 9.07 442 9.07 548 9.07653 9-07 758 9.07 863 9.07 96^ 9.08 072 9.08 176 9.08 279 9.08 383 9.08485 9.08 589 Log. Cos. (1. 120 119 119 119 119 118 118 118 117 117 "6 "6 116 116 115 115 114 114 114 114 "3 113 113 113 112 112 III III no log 109 109 109 io§ 108 108 107 107 107 107 log log 106 106 105 105 105 104 104 104 104 103 103 103 103 (1. Lost. Tan. 9.02 162 9.02 283 9.02 404 9.02 525^ 9.02 645' 9.02 765 9.02885 9.03 004 9.03 123 9.03 242 9.03 361 9-03 479 9-03 597 9.03714 9.03 831 9-03 948 9.04065 9.04 18T 9.04 297 9.04413 9.04 528 9.04643 9.04758 9.04 872 9.04987 9.05 lOI 9.05 214 9.05 32^ 9.05 446 9-05 553 c. d. 9.05 666 9.05778 9.05 890 9.06001 9.06 113 9.06 224 9-o6 335 9.06445 9.06555 9.06 665 9.06775 9.06 884 9.06 994 9.07 102 9.07 211 9.07319 9.07 428 9-07 53^ 9.07 643 9.07 756 9.07857 9.07 964 9.08 071 9.08 177 9.08 283 9.08 389 9.08 494 9.08 600 9.08 705 9.08 810 9.08 914 121 121 120 120 120 119 119 119 119 "8 118 118 117 117 117 "6 116 115 114 114 114 114 113 "3 "3 "3 112 112 112 III III III III no no no 109 109 109 log 109 108 log 107 107 107 107 107 log log 106 105 105 105 105 105 104 Log. Cot. 0.97 838 0.97 7I6 0.97595 0.97475 0.97 354 0.97234 0.97 115 0.96995 0.96876 0.96757 0.96639 0.96 521 0.96403 0.96 285 0.96 168 0.96 051 0-95 935 0.95 818 0.95 702 0.95 587 0.95471 0.95 356 0.95 242 0.95 127 0.95013 0.94899 0.94785 0,94672 0.94559 0.94 446 0.94 334 0.94 222 0.94 no 0-93 998 0.93887 Lost. Cos. 9.99761 9.99760 9-99 759 9-99 757 9.99756 99 754 99 753 99752 99756 99 749 9.99748 9-99 746 9-99 745 9-99 744 9.99742 9.99741 9-99 739 9-99 738 9-99 737 9-99 735 9-99 734 9.99732 9-99731 9-99730 9-99 728 9.99727 9.99725 9.99724 9.99723 9.99721 0.93776 0.93 665 0.93 554 0.93444 0.93334 0.93 225 0.93 115 0.93 006 0.92 897 0.92788 0.92 686 0.92 572 0.92 464 0.92357 0.92 249 0.92 142 0.92035 0.91 929 0.91 822 0.91 716 0.91 611 0.91 505 0.91 400 0.91 295 0.91 190 0.91 085 Log. Cot*- I c. (1. i Log. Tan. 83' 9.99720 9-99 718 9.99717 9-99715 9.99714 9.99712 9-99 71 1 9.99710 9-99 708 9-99707 9.99705 9.99 704 9-99702 9.99701 9.99699 9.99698 9.99696 9.99695 9-99693 9.99692 9.99696 9.99689 9.99687 9.99 686 9.99684 9.99683 9.99 681 9.99679 9.99678 999676 9-99675 Log. Sin. 00 59 58 57 Ji 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 ■^i 30 29 28 27 26 25 24 23 22 21 "20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 p. p. 121 121 120 119 118 6 12. 1 12.1 12.0 II. 9 7 8 14.2 16.2 14.1 16. i 14.0 16.0 139 15-8 9 18.2 18. i 18.0 17-8 10 20.2 20. 1 20.0 19-8 20 30 40 40-5 60.7 81.0 40-3 60.5 80. g 40.0 60.0 80.0 39-6 59-5 79-3 50 101.2 100. § 100. 99.1 II 7 117 116 6 n 7 II. 7 11.6 7 13 7 13-6 13-5 8 15 6 15-6 15-4 9 17 6 J7-5 17.4 10 19 6 19-5 19-3 20 39 I 39 -o .38.6 30 5« 7 5«-.5 58.0 40 7» 3 78.0 77-3 50 97 9 97-5 96. g 13-7 15-7 17.7 19-6 39-3 590 78. g "5 "•5 134 15-3 17.2 19. 1 38.3 57-5 76-6 95-8 114 114 113 112 II 6 11.4 II. 4 "•3 II. 2 II. 7 13-3 13-3 13.2 13.0 12. b 15.2 15.2 15.0 14.9 14. 9 17.2 17. 1 16.9 16.8 16. 10 19. 1 19.0 18. H 18. g 18 20 38.1 38.0 37-6 37-3 37 30 57-2 57-0 st>-s 56.0 55 40 76-3 76.0 75-3 74-6 74 50 95-4 95 -o 94-1 93-3 92 IIO IIO 109 6 n.o n.o 10.9 1 7 12.9 12.8 12 7 8 14.7 14 6 14 5 9 16.6 16. s 16 3 10 18.4 18.3 18 I 20 36.8 36-6 36 3 30 55-2 55-0 54 5 40 73-6 73-3 72 6 1 50 92.1 91-6 90 8 1 10^ 107 106 105 6 10.7 10.7 10.6 10.5 7 12. 5 12.5 12.3 12.2 8 14-3 14.2 14. 1 14.0 9 16. 1 16 15-9 15-7 10 17.9 17-8 17 6 17-5 20 35-8 35-6 35-3 35-0 30 53-7 53-5 53-0 52.5 40 71-6 71-3 70-6 70.0 50 89.6 89.1 88.3 87-5 103 103 2 I 6 10.3 10.3 0.2 0.1 7 12 I 12 0.2 0.2 8 13 8 13 7 0.2 0.2 9 15 5 15 4 0.3 0.2 10 17 2 17 I 0-3 0.2 20 34 5 34 3 0.6 0.5 30 40 50 51 69 86 7 2 51 68 85 5 6 8 I.O 1-3 1-6 0.7 I 1.2 108 10.8 12.6 14.4 16.2 18.0 36.0 540 72.0 90 o 104 10.4 12. I 13-8 15-6 17-3 34-g 52.0 69.3 86-6 I 0.1 0.1 o.i o. I 0.1 0-3 0-5 o.^ P. P. 354 TABLE VII. -LOGARITHMIC SINES. COSINES, TANGENTS, AND COTANGENTS. 10 II 12 14 i5 16 18 19 20 21 24 26 27 28 ^9 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 GO Log. Sin. I d. 9.08 589 9.08 692 9.08 794 9.08 897 9.08 999 9.09 lOI 9.09 202 9.09 303 9.09404 909 50! 9.09 606 9.09 706 9.09 806 9 09 906 9 10006 9. 10 lO^ 9. 10 205 9. 10 303 9. 10 402 9. 10 501 9.10599 9. 10 697 9.10795 9. 10 892 9. 10 990 91 9-1 91 9.1 91 9.1 9.1 91 91 9-1 087 184 281 37f 473 570 665 761 856 952 9. 1 2 047 9. 12 14T 9. 12 236 9.12 330 9.12425 9 12 518 9. 12 612 9. 1 2 706 9.12 799 9. 12 892 12985 13078 13 175 13 263 13 355 9- 1 3 447 9 13 538 9.13636 9.13 721 9-13 813 9- 1 3 903 9- 1 3 994 9.14085 9- 14 17! 9 14265 9 14355 Log. Cos. 99 99 99 98 99 98 98 98 97 97 97 97 96 97 96 96 96 95 96 95 9S 95 94 94 94 94 93 94 93 93 93 93 92 92 92 92 92 91 92 91 91 90 91 90 90 90 90 "dT Log. Tail. I c. d. 9.08 914 9.090I8 9.09 123 9.09 226 9- 09 330 9-09 433 909536 9.09 639 9.09 742 9.09 844 9.09 947 9.10048 9.10 150 9,10 252 9 10353 9.10454 9- 10 555 9. JO 655 9.10756 9.10 856 9. 10 956 9. 1 1 055 9.11 155 9 II 254 9ir 353 9. 1 1 452 9. 1 1 553 9. 1 1 649 9. 1 1 747 9. II 845 11 943 12 040 12 137 12 235 12 331 9.12428 9.12 525 9. 12 621 9. 12 717 9.12 813 9. 1 2 908 9. 1 3 004 9. 1 3 099 9-13 194 9.13289 9-13384 9-13 478 9.13 572 9.13665 9. 1 3 766 9- 13 854 9- 1 3 947 9. 14 041 9-14134 9. 14 227 9.14319 9.14412 9.14504 9-14 596 9-14688 9. 14 786 Lost. Cot. 104 104 103 103 103 103 103 102 102 102 loi 102 lOI 101 lOI lOI 100 100 100 lOO 99 99 99 99 98 98 98 97 97 97 96 97 96 96 96 96 95 95 95 95 95 94 94 94 94 94 93 93 93 93 93 92 92 92 92 92 92 c. d. Lot'. Cot. 0.91 085 0.90 98! 0.90 877 0.90773 0.90 670 0.90 566 0.90463 0.90 360 0.90 258 0.90 155 0.90053 0.89 95T 0.89 849 0.89 748 0.89 647 0.89 546 0.89445 0.89344 0.89 244 0.89 144 0.89 044 0.88 944 0.88845 0.88745 0.8 8646 0.88 548 0.88 449 0.88 351 0.88 253 0.88 155 0.88057 0.87 959 0.87 862 0.87 765 0.87 668 0.87 571 0.87475 0.87 379 0.87 283 0.87 187 0.87 091 0.86 996 o. 86 906 0.86805 0.86 716 0.86616 0.86 521 0.86 427 0.86333 0.86 239 0.86 146 0.86 052 0.85 959 0.85 866 0.85773 0.85 686 0.85 588 0.85495 0.85403 0.85 31T ) 85 219 \AMi. r.-iii. \Mii. C(»S. 9.99675 9-99673 9.99672 9.99676 9.99669 9.99667 9.99665 9.99664 9.99 662 9.99661 9-99~65§^ 9.99658 9.99656 9.99654 9-99653 9-99651 9.99650 9.99648 999646 9-99645 9-99643 9-99641 9.99640 9-99638 9-99637 9-99635 9-99633 9.99632 9.99630 9.99628 9.99627 9.99625 9.99623 9.99622 9.99 620 9.99"6Tf 9.99617 9-99615 9.99613 9.99 6 iT 9.99 610 9-99608 9.99606 9.99605 9-99603 9.99 60T 9.99 600 9-99598 9-99 596 9-99 594 9-99 593 9-99 591 9-99 589 9-99 587 9-99 586 9.99 584 9.99582 9.99586 9-99 579 9 99 577 9 99 5/5 Loir. Sin. (>0 59 58 57 _56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II To 9 8 7 6 20 30 40 50 I'. I'. 104 103 102 lOI 10.4 10.3 10.2 10. 1 12 I 12.0 11.9 11.8 13-8 ^3-7 13-6 13-4" 15 6 I5-4 '5 3 '5-1 173 17.1 17.0 16 I 34-6 34-3 34 33-6 52.0 693 86.6 5'-5 68.6 85-8 510 68.0 85.0 50- 5 ^7-3 84.! 100 100 99 6 10. 10. 9-9 i 7 II. 7 II. g "■5 8 13-4 '3-3 13.2 9 10 151 16.7 15.0 16.6 14-8 16.5 20 33-5 33-3 33-0 30 40 50 50.2 67.0 83-7 50.0 66.6 83.3 49-5 66.0 82.5 98 9.8 II. 4 13.0 14.7 16.3 32-6 49.0 65.3 Si. 6 97 97 96 95 6 9-7 9-7 9.6 9-5 7 II. 4 II. 3 II. 2 11 1 8 130 12.9 12.8 12.6 9 14.6 145 14.4 14 2 10 16.2 16. i 16.0 15 § 20 325 32 -3 32.0 31 6 30 48.7 48.5 48.0 47 5 40 65.0 64.6 64.0 63 3 50 81.2 80.8 80.0 79 I 91 91 90 2 6 9.1 9.1 9.0 0.2 7 10.7 10.6 10.5 0.2 U 12.2 12. 1 12.0 0.2 9 13-7 i3-§ '3-5 0.3 10 15.2 15.1 I5-0 ° 3 20 30.5 30. 3 30.0 0.6 30 45-7 45-5 45.0 I.O 40 50 61 .0 76.2 60.^ 75-8 60.0 75 '■2 »-6 0.5 0.7 94 94 93 92 6 9.4 9-4 9-3 9.2 7 II. 10.9 10 R 10.7 8 12.6 12-3 12 4 12.2 9 14.2 14.1 13 9 13-8 10 '5-7 i5-$ 15 5 '5-3 20 3'-5 31-3 3> 30 6 30 47 2 47.0 46 S 46.0 40 63.0 62. A 62 61. 1 50 78.7 78-3 77 5 76.1 r. I' 8*e° 355 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. AND COTANGENTS. 8° Log. hill d. Log. Tan. 10 II 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9-H355 9.14445 9-14 535 9. 14 624 9-14713 9. 14 802 9.14891 9.14 980 9.15068 915 157 9.15245 9-15 333 9. 15 421 9-15 508 9-15595 9.15683 9.15770 9.15857 9-15 943 9. 16 030 9.16 116 9.16 202 9.16 283 9.16374 9. 1 6 460 9.16545 9.16 630 9.16 716 9.16 801 9.16885 9.16 970 9.17054 9.17 139 9.17 223 9.17307 9.17 391 9 17474 9.17558 9-17641 9.17724 9.17 807 9.17 890 9.17972 9.18055 9.18 137 9. 18 219 9.18 301 9.18383 9.18465 9.18 546 9.18628 9. 1 8 709 9. 1 8 790 9.18 871 9.18 952 9. 19032 9.19113 9.19 193 9 19273 9-19353 9-19433 Log. Cos. 90 89 89 89 89 89 88 88 88 88 87 S? 87 87 87 86 86 86 86 86 86 85 85 85 85 85 84 84 84 84 84 84 84 83 83 83 83 83 83 82 82 82 82 82 82 81 81 81 81 81 80 81 80 86 80 80 80 79 9.14 780 9.14872 9.14963 9.15054 9.15 145 9-15236 9.15327 9-15417 9.15 507 9.15 598 9.15687 9-1577? 9.15867 9-15 956 9.16045 c. d. 16 134 16 223 16 312 16 401 16 489 9 16577 9.16665 9.16753 9. 16 841 9.16928 17015 17 103 17 190 17276 17363 9- 9- 9- 9- 9: 9.17450 9.17 536 9.17 622 9.17708 9 17794 9 17880 9.17965 9. 18 051 9.18 136 9. 18 221 9. 1 8 306 9.18 390 9-18475 9.18559 9. 1 8 644 9- 18728 9. 18812 9- 18896 9- 18979 9- 19063 9- 19 146 9- 19 229 9- 19312 9- 19395 9- 19478 9- 19 566 9- 19643 9- 19725 9- 1980^ 9- 19889 9. 19 971 91 91 91 91 91 96 96 90 90 89 90 89 89 89 89 89 89 88 87 88 87 S7 87 87 86 87 86 86 86 86 85 86 85 85 85 84 84 84 84 84 84 84 83 83 83 83 83 83 82 82 82 82 82 82 82 Log . Cot. I c. d. tiog. Cot. 0.85 219 0.85 128 0.85037 0.84945 0.84854 0.84763 0.84673 0.84582 0.84492 0.84 402 0.84 312 0.84 222 0.84 133 o. 84 043 0-83954 0.83865 0.83776 0.83687 0-83 599 0.83 511 0.83 422 0.83334 0.83 247 0.83 159 0.83 071 0.82 984 0.82 897 0.82 810 0.82 723 0.82636 0.82 550 0.82 464 0.82 377 0.82 291 0.82 206 0.82 120 0.82 034 0.81 949 0.81 864 0.81 779 0.81 694 0.81 609 0.81 525 0.81 446 0.81 356 0.81 272 0.81 188 0.81 104 0.81 026 0.80937 0.80 854 0.80 770 0.80687 o. 80 604 0.80 522 0.80439 0.80357 0.80 274 0.80 192 0.80 II 6 0.80 023 JiOg. Tan. 81 Log. Cos. 9-99 575 9-99 ^7% 9.99 371 9.99 570 9-99 568 9.99 566 9-99564 9-99563 9.99561 9-99 559 9-99 55? 9-99 555 9-99 553 9.99552 9.99550 9-99 548 9-99 546 9-99 544 9-99 542 9.99541 9-99 539 9-99 537 9-99 535 9-99 533 9-99 531 9-99 529 9.99528 9.99526 9-99 524 9.99522 9.99520 9-99 518 9-99 516 9.99514 9.99512 9-99 511 9-99 509 9-99 507 9-99 505 9-99 503 9-99 501 9.99499 9-99 497 9.99495 9-99 493 9.99491 9.99489 9.99487 9.99485 9.99484 9.99482 9-99480 9.99478 9-99476 9-99 474 9.99472 9.99470 9-99468 9.99466 9.99464 9.9946: Log. Sin. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II To" 9 p. p. 91 91 90 89 6 9.1 9-1 9-0 8.9 7 10.7 10.6 10.5 10.4 8 12.2 12.T 12.0 II. 8 9 13.7 13-6 13-5 13-3 10 15.2 1 5. 1 15.0 14-8 20 30.5 30.3 30.0 29.6 30 45-7 45-5 45.0 44-5 40 61.0 60.6 60.0 ^9'?> 50 76.2 75-8 75.0 74-1 88 88 87 6 8.8 8.8 8.7 7 10. s 10.2 10. 1 8 11.8 II.? II. 6 9 13-3 1.3-2 13.0 10 14-7 14-6 14.5 20 29.5 29-3 29.0 30 44-2 44-0 43-5 40 59.0 58-6 58.0 50 73-1 73-3 72.5 85 85 84 6 8.5 8.5 8.4 7 lO.O 9-9 9.8 8 11.4 1 1-3 II. 2 9 12.8 12.? 12.6 10 14.2 14. 1 14.0 20 28.5 28.3 28.0 30 42.? 42.5 42.0 40 57.0 56-6 56.0 50 71.2 70.8 70.0 82 82 81 6 8.2 8.2 8.1 7 9.6 9.5 9.4 8 II. 10.9 10.8 9 12.4 12.3 I2„I 10 13-7 13-6 13-5 20 27.5 27-3 27.0 30 41.2 41.0 40.5 40 55.0 54 6 54.0 50 68.^ 68.3 67.5 86 8.6 10.6 II. 4 12.9 14-3 28.6 43-0 57-3 71.6 83 8.3 9-7 II. o 12.4 13-8 27-6 41.5 55-3 69.1 80 8.0 9-3 10.6 12.0 13-3 26.6 40.0 53-3 66.6 10 20 30 40 50 79 7-9 9-3 10.6 II. 9 13.2 26.5 39-? 53.0 66.2 2 0.2 0.2 0.2 0.3 0-3 0.6 i.o 1-3 1-6 I o. I 0.2 0.2 0.2 0.2 0.5 o.? 1.0 1.2 P.P. i 356 TAP>LE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANCiENTS. 9° 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Lr. Sin. (>0 59 58 57 56 55 54 53 52 51 50 49 48 47 J6 45 44 43 42 41 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 9 8 7 6 I', r 81 81 80 6 8.1 8.1 8.0 7 8 9-5 10.8 9-4 10.8 9 3 10.6 9 12.2 12.1 12. ol 10 20 13.6 27.1 135 27.0 13.3I 26.6 i 30 40.7 40.5 40.0 40 50 54-3 67.9 54.0 67.5 53-3 66.6 79 7-9 9.2 10. § II. 8 I3-J 26.3 39-5 52.6 65-8 6 7 8 9 10 20 30 40 50 78 7-8 9- 1 10.4 II. 8 131 26. T 39-2 52.3 65.4 78 7-8 9.1 10.4 II. 7 13.0 26.0 390 52.0 65.0 77 7-7 9.0 10.2 II. 5 12.8 25-6 38.5 51.3 64. T 76 / 8 9 10 20 30 40 7 8 6 9 10 2 ! 1 1 5 12 7i 25 38 5! 2 51 63 1. 76 7.6 8.8 10. T II. 4 12.6 25-3 38.0 50.6 63- 3 75 7-5; 8.^1 10. o! 1 1.2 12.5 25.0 37-5 74 7-4 8.6 98 1 1. 1 12.3 24-6 37-0 73 73 6 7-3 7-3, 7 8.6 8.5! 8 9.8 9-7 9 II. 10.9 10 12.2 12. T 20 24.5 24-3 30 36.7 36.5 40 49.0 48.6 50 61.2 60.8 50.0 49.3 62.5 61.6 72 7.2 8.4 9.6 10.8 12.0 24.0 36.0 48.0 60.0 n 71 5 2 6 7-1 71 0.2 0.2 7 8. .3 8.3 03 0.2 8 9.5 9.4 0.3 0.2 9 10.7 106 0.4 0.3 10 II. 9 II. 8 0.4 03 20 23-8 23-6 0.8 0.6 30 35.7 35-5 1.2 I.O 40 47-6 47.3 '•6 1-3 50 59.6 59.1 2.1 1-6 r. i*. 80' 357 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 10° Log. Sin. d. 9.23967 9.24 038 9.24 1 10 9.24 181 9.24 252 9-24323 9.24394 9.24465 9.24536 9.24 607 10 II 12 13 14 15 16 17 18 19 9.24677 9.24748 9.24818 9.24888 9-24 958 9.25 028 9.25 098 9.25 167 9.25237 9 25 306 20 21 22 23 24 26 27 28 2Q 9.25 376 9.25443 9.25514 9-25 583 9 25652 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.25 721 9.25790 925858 9.25927 9-25 995 9.26 063 9 26 13T 9 26 199 9.26 267 9-26335 9. 26 402 9.26470 9.26 537 9. 26 605 9.26 672 9.26739 9.26 805 9-26873 9 26 940 9.27 007 9.27073 9.27 140 9-27205 9.27 272 9-27 339 9.27405 9.27471 9-27 536 9.27 602 9.27668 9-27 733 9-27 799 9.27 S64 9-27 929 9.27995 9.28 060 Log. Cos. ' 71 7t 71 71 71 71 71 71 70 70 70 70 76 70 69 70 69 70 69 69 69 69 69 69 ^^^ 6q 6§ 68 68 ^^ 68 68 68 67 ^1 68 67 67 67 (^1 67 ^1 66 67 66 66 66 66 66 66 66 65 66 65 65 65 65 65 65 65 Log. Tan. 9.24632 9.24705 9.24779 9.24853 9.24925 9.25 000 9.25073 9.25 146 9.25 219 9.25 292 9.25 365 9-25437 9.25 510 9.25 582 9.25654 d. 9-25 727 9-25799 9.25871 9.25 943 9.26 014 9. 26 085 9.26 158 9.26 229 9. 26 300 9 26 371 9-26443 9.26514 9.26 584 9.26 655 9.26 726 9.26 795 9.26 867 9.26 93f 9.27 007 9.27 078 9.27 148 9.27 218 9.27 287 9-27 357 9.27427 9.27495 9.27 566 9-27635 9.27704 9-27 773 9.27 842 9.27 91T 9.27 9S0 9. 28 049 9.28 117 9.28 186 9.28254 9.28 322 9.28 390 9 28459 9.28 527 9.28 594 9.28 662 9.28 730 9.28 79f 9.28865 Log. Cot. jc^d^ 73 74 73 73 73 11> 73 7j 7Z 73 72 72 72 72 72 72 72 72 71 72 71 71 7? n 71 71 70 71 70 70 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 69 68 69 68 68 6Z 68 68 68 68 67 6Z ^7 Log. Cot. 0.75 368 0.75294 0.75 220 0.75 147 0.75073 0.75 000 0.74927 0.74854 0.74781 0.74708 0.74635 0.74 562 0.74490 0.74417 0.7434! 0.74273 0.74 201 0.74 129 0.74057 0.73985 0-73913 0.73 842 0.73 771 0.73 699 0.73628 0.73 557 0.73486 0.73415 0.73344 0.73274 0.73 203 0.73 J33 0.73 062 0.72 992 0.72 922 0.72 852 0.72 782 0.72 712 0.72 642 0.72 573 0.72 503 0.72434 0.72 365 0.72 295 0.72 225 0.72 157 0.72 088 0.72 020 0.71 951 0.71 882 0.71 814 0.71 746 0.71 677 0.71 609 0.71 541 0.71 473 0.71 405 0.71 337 0.71 270 0.71 202 0.71 135 c. d. I Log. Tan. Log. Cos. 9-99 335 9-99 333 9-99330 9-99 328 9.99326 9.99324 9.99321 9-99319 9-99317 9-99315 9.99312 9.99316 9.99308 9.99 306 9-99303 9.99301 9.99299 9.99295 9.99294 9.99292 9.99290 9.99287 9.99285 9.99283 9.99 280 9.99278 9-99276 9.99273 9.99271 9.99269 9.99265 9.99264 9.99 262 9.99259 9.99257 9.99255 9.99252 9.99250 9.99248 9.99245 9.99243 9.99 246 9.99238 9.99236 9-99233 9.99231 9.99228 9.99225 9.99224 9.99 221 9-99219 9-99 216 9-99214 9.99 212 9.99209 9.99207 9.99204 9 99 202 9-99 199 9-99 197 9 99 194 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 •34 33 32 31 30 29 28 27 26 25 24 23 22 21 p. p. 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 Log. Sin. 74 6 7.4 7 8.6 8 98 9 II. I 10 12.3 20 24 6 30 37.0 40 49-3 50 61.6 73 7-1 8.6 9.8 II. o 12.2 24.5 36.7 49.0 61.2 73 7.3 " 5 7 9 I 3 5 6 9 10 12 24 36 48 60 72 72 7 I 7 6 7.2 7.2 7-1 7. 7 8.4 8.4 8 3 8. 8 9-6 9.6 9 I 9- 9 10.9 10.8 10 7 10. 10 12. 1 12.0 II 9 II. 20 24.1 24.0 23 8 23- 30 36.2 36.0 35 7 35- 40 48.3 48.0 47 6 47. 50 60.4 60.0 59 6 59- 6 7.6 7-0 6.9 j 7 8.2 8.1 8.1 8 9-4 9-3 9.2 9 10.6 10.5 10.4 10 11.^ 11.6 11.6 20 23.5 23.3 23.T 30 35-2 35-0 34-7 40 47.0 46.5 46.3; 50 58.^ 58.3 57-91 6 7 8 9 10 20 30 40 50 70 7-' 8. 9., 0.1 I. ^ ,5.: 7-< 8.' 68 6.8 8.0 9.1 10.3 II. 4 22.8 34.2 45-6 57.1 68 70 7-' 8. 9- o. i.( 3- >5-' .6., 8., 68 6.S 7-9 9.6 10.2 11-3 22-6 34-0 45-3 56.6 66 e 7 c c c ,c c c 2 69 6.C 8.] 9.: o.z I.( 53.1 14-^ .6.1 \7-l 6^ 6.^ 7-9 9.0 10. 1 11. 2 22.5 33-^ 45.0 c6.2 '6S 6 7 8 6.6 7.1 8.8 6.6 7-7 8.8 6.5 7-6 8.^ 6. 7- 8. 9 10 lO.O 11. 1 9.9 II. 9.8 10.9 9. 10. 20 22.1 22.0 21.8 21. 30 40 50 33-2 44- S 55-4 33-0 44-0 55.0 32-^ 43-6 54.6 32. 43. 54- 6 7 8 9 10 20 30 40 50 ^ 2 3 3 4 4 8 I 2 ^ 1 6 2. I 2 0.2 0,2 0.2 0.3 0.3 0.6 I.O 1-3 1-6 6 5 3 I 69 6.9 8.6 9.2 10.3 II. 5 23.0 34-5 46.0 57-5 67 6.7 7.Z 8.9 10.6 II. I 22.3 33-5 44-6 55-8 65 5 6 6 7 8 6 5 3 I P. P. 79 358 T\HLE VII. — LOGARITHMIC SINES, COSINES. TANGENTS. AND COTANGENTS. 11° 10 1 1 12 14 15 i6 17 19 20 21 -J ^4 26 -7 28 -9 ao 34 J3 36 37 38 39 10 41 42 43 44 -15 46 47 4S 49 .'>0 51 52 34 55 56 S7 58 (>0 Lotf. sill. •I. 28 060 28 125 28 189 28254 28 319 28383 28448 28 512 28576 28641 703 769 832 896 960 29 29 29 29 29 087 156 213 277 29340 29403 29 466 29528 29591 29654 29716 29779 29841 29903 29963 30027 33089 30 151 30213 30275 30336 30398 30459 o ;2o 30582 30643 30704 30765 30 826 30886 30947 31 008 068 129 189 249 309 370 429 489 549 609 669 728 Lost. Cos. 65 64 65 64 64 64 64 64 64 64 64 63 64 63 63 63 63 63 63 63 63 63 62 63 62 62 62 62 62 62 62 62 62 61 62 61 61 61 61 61 61 61 61 61 60 6r 60 65 60 60 60 60 60 59 60 60 59 60 59 59 "d7" Loff. Tiiii. c. (1. 1 Loir. int. Lotr. Cos 28 865 28932 29 000 29 067 29 134 29 201 29 268 29 jj:) 29401 29468 29535 29 601 29 667 29734 29 800 29866" 29932 29998 30064 30 129 30195 30 260 30326 30391 30456 30 522 30587 30652 30717 30781 30846 30 91 1 30975 040 104 168 232 297 361 424 488 552 616 679 743 806 869 933 996 32059 -32 I 22 32185 32 248 32310 32 373 32436 32498 32 566 32 623 32 68g 9-3274^ Loir. Cot. 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 66 66 65 65 65 65 65 65 65 65 65 65 64 65 64 64 64 64 64 64 64 64 63 64 64 63 63 63 63 63 63 63 63 63 63 63 62 63 62 62 62 62 62 62 77T 71 135 71 067 71 000 70933 70 866 70798 70732 70 665 70 598 70531 70465 70398 70332 70 266 70 200 70134 70068 70 002 69936 69 870 69 805 69739 69674 69 608 69 543 69478 69413 69348 69 283 69 218 69 153 69 089 69 024 68960 68896 68831 68767 68703 68639 68575 68 511 68447 68384 68 326 68257 68 193 68 136 ,68067 .68 004 '•67 941 .67 878 67815 67752 67689 67 626 .67 564 .67 501 67 439 •67 377 67 314 67 252 otr. Tan. 9-99 9.99 9-99 9.99 9.99 9.99098 9 99 096 9.99093 9.99091 9.99088 9.99085 9-99083 9.99086 9.99077 9.99075 9.99072 9-99069 9.99067 9.99064 9.99062 9.99059 9.99056 9.99054 9.99051 9-99 04 8 9.99046 999043 9.99046 I,ou'. sin. 1'. 1' 21 20 19 18 17 16 / 8 9 10 20 30 40 50 6g 6 6.6 7 8 7-f 8.8 9 lO.O 10 II. I 20 22.1 30 33-2 40 44-3 50 55-4 64 6 6.4 7 8 7-5 8.6 9 9-7 10 10.7 20 21.5 30 32.2 40 43-0 30 53-7 62 6 6.2 7 8 7-3 8.3 9 94 10 10.4 20 20.8 30 31.2 40 41.6 50 52.1 67 7-9 9.0 10.1 i 11. 2 ! 22. ^ 337 45-0 56.2 66 6.6 7-7 8.8 67 6.7 7.8 8.9 lo.o II. T 22.3 33-5 44.6 55-8 65 6 9.9 II. o 22.0 330 44.0 55.0 64 6 4 6.3 7 4 7-4 8 5 8.4 9 6 9-5 10 6 10.6 21 3 21. 1 32 31-7 42 6 42.3 53 J 52.9 62 6.2 7.2 8.2 9.3 10.3 20.6 31.0 6 7 8 9 10 20 30 40 50 41 51 66 6.6 7.6 8.6 9.1 10. 1 20. T 30.2 40.3 50.4 3 9 10 21 32 43 54 63 6.; 7-' 8.. 9-: o.( I.' I.! 2. ■ 2.( 61 6.1 7.2 8.2 9-2 10.2 20.5 30- 7 41.0 51.2 60 6.0 7.0 8.0 9.0 1 0.0 20.0 30.0 40.0 50.0 2 65 6.5 7 ' 8 9 10 21 32 43 54. 63 6.3 7.3 8.4 9-4 10.5 21.0 31-5 42.0 52.5 61 6.1 7-1 8.T 9.1 10. 1 20.3 30.5 40.6 50-8 59 5-9 9 9 9 9 8 7 6 6 6 7 8 9 10 20 30 40 50 0-3 0.2 0.3 0.3 0.4 0-3 0.4 0.4 0.5 0.4 I.O 0.8 1-5 1.2 2.0 1-6 2.5 2.1 0.2 0.2 0.2 03 0.3 0-6 1.0 1-3 1-6 v. I'. 78' 359 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 12° 10 II 12 13 14 liOic. Sin. 9.31 788 9-31 847 9' 3 1 906 9.31 966 9.32025 9.32084 9-32 143 9.32 202 9.32 260 9-32 319 15 16 17 18 19 20 21 22 23 24 9-32378 9-32 436 9-32495 9-32553 9.32611 25 26 27 28 29 30 31 32 33 34 9.32670 9.32 728 9.32 7^6 9.32844 9.32 902 9.32 960 9-33017 9-3307? 9-33 133 9-33 190 9-33248 9-33 305 9-33 362 9-33419 9-33 476 35 36 37 3S 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 GO 9-33 533 9-33590 9-3364? 9-33704 9-33761 9-3381? 9-33874 933930 9-33 9S7 9- 34 043 9.34099 9.34156 9.34212 9.34268 9-34324 9-34 379 9-34 435 9-34 491 9-34 547 9. 34 602 9.34658 9-34713 9-34768 9.34824 9-34879 9-34 934 9-34989 9-35044 9-35099 9-35 154 9.35 209 Log.jCos, d. 59 59 59 59 59 59 59 58 59 58 58 58 58 58 58 58 58 58 58 58 57 58 57 57 5? 57 57 57 57 57 57 57 57 56 56 56 56 56 56 56 56 56 56 56 Si 56 5? 56 Si 5? 5? 55 55 55 55 55 55 54 55 55 Log. Tail. 9-32747 9,32 809 9.32871 9-32933 9-32995 9-33057 9-33 118 9.33 186 9-33242 9-33303 c. d. 9-33364 9-33426 9-33487 9-33548 9.33609 9.33670 9-33731 9-33792 9-33852 9-33913 9-33 974 9-34034 9-34095 9-34155 9-34215 9-34275 9-34336 9-34396 9-34456 9-34515 9-34 575 9-34635 9-34695 9-34 754 9.34814 9-34873 9-34 933 9. 34 992 9-35051 9-35 I TO 9-35 169 9-35 228 9.35287 9-35 346 9-35405 9-35464 9-35 522 9-35 581 9.35640 9-35698 9-35 756 9-35815 9.35873 9.35931 9-35989 9.3604? 9.36 105 9.36163 9.36 221 9-36278 d. 9-36336 62 62 62 62 61 61 62 61 61 61 6i 61 61 6i 60 61 61 66 61 60 60 66 60 66 60 60 60 60 59 60 60 59 59 59 59 59 59 59 59 59 59 59 59 59 58 58 59 58 58 58 58 58 58 58 58 58 5f 58 5? 58 Log. Cot. 0.67 252 0.67 196 0.67 128 o. 67 065 o. 67 004 0.66 943 0.66 881 0.66 819 0.66758 0.66 695 0.66 635 0.66 574 0.66 513 0.66 452 0.66 396 0.66 330 0.66 269 0.66 208 0.66 147 0.66085 0.66 026 0.65 965 0.65 905 0.65 845 0.65 784 0.65 724 0.65 664 0.65 604 0.65 544 0.65 484 0,65 424 0.65364 0.65 305 0.65 245 0.65 186 0.65 125 0.65 067 0.65 008 0.64948 0.64889 0.64 836 0.64 771 o. 64 7 1 2 0.64 653 0.64 594 0.64536 0.6447? 0.64418 o. 64 360 0.64 302 0.64243 0.64 185 0.64 127 o. 64 068 0.64016 0.63952 0.63894 0.63 837 0.63 779 0.63 721 0.63.663 iO g. Cot, le d. I Log. Tan. Log. Cos. 9.99040 9-99038 9-99035 9.99032 9.99029 9-99027 9.99024 9.99021 9.99019 9.99016 999013 9.99 016 9. 99 008 9.99005 9.99 002 9.98999 9.98997 9.98994 9.98991 9.98988 9.98986 9-98983 9.98 986 9-9897? 9.98975 9.98972 9.98969 9.98965 9-98963 9.98 961 9.98958 9.98955 9.98952 9-98949 9-98947 9.98944 9-98 941 998938 9-98935 9-98933 9.98930 9.98 927 9-98924 9.98 921 9.98 9I8 9-98915 9.98913 9.98 910 9.98907 9.98904 9.98 901 9.98898 9-98895 9.98 892 9.98 890 9.98887 9.98884 9.98881 9.98878 9-98875 9.98 872 Log. Sin. 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 P. P. 62 61 61 6 6.2 6.T 6.1 I 7.2 8.2 7.2 8.2 7-1 8.1 9 10 9.3 10.3 9.2 10.2 9-1 10. 1 20 30 40 50 20.6 31.0 41-3 51-0 20.5 30.7 41.0 51.2 20.3 30.5 40.6 50.8 66 60 59 59 6 6.6 6.0 5 9 5-9 7 7.6 7.0 6.9 6.9; 8 8.6 8.0 7.9 7-8 9 9.1 9.0 8.9 8-8 10 10. 1 10.0 9.9 9-8 20 20.1 20.0 19-8 19-6 30 30.2 30.0 29.7 29.5 40 40.3 40.0 39-6 39.3 50 50.4 1 50.0 49-6 49.1 6 7 8 9 10 20 30 40 50 58 5-8 6.8 9-? 19-5 29.2 39-0 48.? 58 5-8 6.? 7-? ^.7 9-6 19-3 29.0 38.6 48.3 Si 5-? 6.7 7-6 8.6 9.6 19.1 28.? 38.3 47-9 57 5-7 6.6 7.6 8.5 9-5 19.0 28.5 38.0 47.5 5S 56 5S 55 6 7 5-6 6.6 5-6 6.5 5-5 6.5 5-5 6.4 8 9 7.5 8-5 7.4 8.4 7-4 8.3 7-3 8.2 10 20 9-4 18.8 9-3 18.6 9-2 18.5 9-1 18.3 30 28.2 28.0 27.? 27.5 40 50 37.6 47.1 37.3 46.6 37-0 46.2 36.6 45-8 54 3 6 S-4 0.3 7 6. .3 0.3 8 7.2 0.4 9 8.2 0.4 10 9-1 0-5 20 18. 1 I.O 30 27.2 1-5 40 36.3 2.0 50 45-4 2.5 2 0.2 0.3 0.3 0.4 0.4 0.8 1.2 1-6 2.1 p. p 77' 360 TABLE VII. — LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS, 13" 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 iO:;. Sill. (I. 9.35 209 9-35 263 9-35 318 9-35 372 9-35427 9-35481 9-35 536 9-35 590 9-35 644 9-35698 9-35 752 9-35805 9.35 865 9-35914 9.35968 9.36 021 9.36075 9.36 123 9.36 182 9-36235 20 9.36 289 21 9-36342 22 9-36393 23 9-36448 24 9.3650? 9- 36 554 9. 36 607 9. 36 660 9 36713 9-36766 9-36818 9.36 871 9-36923 9-36976 9-37 028 9.37081 9-37 133 9-37 185 9-3723^ 9.37 289 9-37341 9 37 393 9-37 445 9-37 497 9-37 548 9. 37 600 9-37652 9-37703 9.37755 9-37 806 9-37857 9.37909 9.37960 9.3801T 9.38 062 9.38 113 9.38 164 9.38215 9.38 266 9-38317 9-38367 54 5-4 5-4 54 54 54 54 54 54 54 54 54 53 54 53 53 53 53 53 53 53 53 53 53 53 53 53 52 53 52 52 52 52 52 52 52 52 52 52 52 52 51 52 51 52 51 51 51 51 51 51 51 51 51 51 51 50 51 51 50 Ian. <-. (1. ( (»t. 9-36336 936394 9.36451 9.36509 9-36566 9.36623 9. 36 68 1 9-36738 9-36795 9.36852 9.36909 9. 36 965 9.37023 9.37080 9-37 136 9-37 193 9.37 250 9-37 306 9- 37 363 9.37419 Log. Cos. I (1. 9-37 475 9-37 532 9.37 588 9-37644 9-37 700 9-37 756 9.37 812 9-37868 9-37924 9-37 979 9-38035 9.38091 9-38 146 9.38 202 9-38257 9-38313 9-38368 9-38423 9-38478 9-38 533 9-38 589 9- 38 644 9-38698 9-38753 9-38 808 9.38863 9.38 918 9-38972 9.39027 9.39081 9-39 136 9.39 190 9- 39 244 9-39299 9-39 353 9-39407 9-39461 9-39 5i§ 9-39569 9-39623 9-39677 Log. Cot. 57 57 57 57 57 57 57 57 57 57 57 56 57 56 57 56 56 56 56 56 56 56 56 56 56 Si 56 56 5^^ 56 5S Si 55 Si Si 55 55 55 55 -? :>D 55 54 55 55 54 55 54 5^ 54 54 54 54 54 54 54 54 54 54 54 53 0.63 663 0.63 606 0.63 548 0.63491 063433 0-63 376 0.63319 0.63 262 0.63 204 0.63 147 0.63 096 0.63033 0.62 977 0.62 920 0.62 863 0.62 806 0.62 730 0.62 693 0.62 637 0.62 580 l.dU'. Cos. 0.62 524 0.62 468 0.62 412 0.62 356 0.62 299 0.62 243 0.62 188 0.62 132 0.62 076 0.62 020 06 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 c. d. 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 964 909 853 798 742 687 632 576 521 466 411 356 301 246 191 137 082 7 0.60973 0.60 913 0.60 8 64 0.60 809 0.60 755 0.60 701 0.60 647 0.60 592 0-60 538 0.60 484 0.60 430 0-60375 0.60 323 Log. Tan. 9.98 872 9. 98 869 9.98865 9-98863 9.98 860 9.98 858 9.98855 9.98852 9.98849 9.98 840 9-98843 9.98 840 9-98837 9.98834 9.98831 9.98 828 9.98 825 9.98822 9.98 819 9.98815 9.98813 9.98 816 9.98 S07 9.98 804 9.98 80T 9-98798 9.98795 9.98792 9-98789 9.98 786 9-98783 9.98 780 9-98777 9-98774 9.98771 9.98768 9-98765 9.98 762 9.98759 9-98755 9.98752 9.98749 9.98745 9-98743 9.98 746 998737 9-98734 9.98731 9.98728 9-98725 9.98 72T 9-98718 9.98/ 15 9.98 712 9-98 709 9.98 706 9-98 703 9.98 700 9.98695 9.98693 9.98 696 40 39 3^ 37 I'. !'. -3 24 23 22 21 20 19 18 17 _i6 15 14 13 12 1 1 To 9 8 7 Loer. Sin. 6 7 8 9 loj 20 i 30' 401 50 I 57 S-7 6.7 7-6 8.6 9.6 19.T 28.^ 38.3 47-9 57 5-7 6 7 8 9 19 28 38 47 56 5-^ 6. 7- 18 28 37 47 •0 .6 3- 6. •5 •5 7- 8. •4 9- •8 18. 28. •6 .1 37- 46. 55 55 54 6 5-5 5-5 5--+ 7 6.5 6 4 6.3 8 7.4 7 3 7.2 9 8.3 8 2 8.2 10 9.2 9 I 9-1 20 18.5 18 3 18.1 30 27-7 27 5 oy n 40 37.0 36 6 36.3 50 46.2 45 8 45-4 56 6 54 5-4 6.3 7-2 8.1 9.0 18.0 27.0 36.0 45.0 53 53 52 52 1 6 7 5-3 6.2 5-3 6.2 5-2 6.1 5-2 6.6 8 7.1 7.0 7.0 6.9 9 10 8.0 8.9 7-9 8-8 7-9 ^-7 7.8 8.6 20 30 17.8 26. f 17-6 26.5 17.5 26.2 26.0 40 35.6 35-3 35.0 34-6 50 44.6 44.1 43-7 43.3 6 7 8 9 10 20 30 40 50 51 5.1 6.0 6-8 7-7 8.6 17.1 25. f 34-3 42.9 6 7 8 9 10 20 30 40 50 3 0.3 0.4 0.4 0.5 0.6 I.I i.f 2.3 2.9 51 5-1 5-9 6.8 7-6 8-5 17.0 25-5 34.0 42.5 0-3 0.3 0.4 0.4 0.5 i.o 1.5 2.0 2.5 50 5.0 5 6 7 8 16 25 33 42 2 0.2 0.3 0.3 0.4 0.4 0.8 1.2 1-6 2.1 1 . 1 76° 361 TABLE Vli.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 14° _9_ 10 II 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 Ji 55 56 57 58 59 60 Log. Sill. d. 9-3836? 9.38418 9-38468 9.38519 9.38569 9.38 620 9.38670 9.38 726 9.38771 9.38821 9.38871 9.38 921 9.38971 9.39 021 9.39071 9.39 120 9.39176 9.39 220 9.39269 9-39319 9-39368 9.39418 9-3946? 9-39515 9.39566 9.39615 9.39664 9-39713 9.39762 9-39 81 1 9. 39 860 9.39909 9-39 95? 9.40006 9.40055 9.40 103 9.40152 9. 40 200 9.40249 9.40 297 9-40345 9-40394 9.40442 9.40490 9.40 538 9.40 586 9.40634 9.40682 9.40 730 9.4077? 9.40 825 9.40873 9.40 920 9.40968 9.41015 9.41 063 9.41 116 9.41 158 9.41 205 9.41 252 9.41 299 Log. Cos. d. 50 50 50 50 50 50 50 50 50 50 50 50 50 50 49 50 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 48 48 49 48 48 48 48 48 48 48 48 48 48 48 48 48 48 47 48 4? 4? 4? 4? 4? 4? 4? 47 4? 47 Log. Tau. 9-39677 9-39 731 9-39784 9-39838 9.39892 9-39 945 9-39 999 9.40052 9.40 106 9.40159 9. 40 2 1 2 9.40 265 9-40 318 9.40372 9-40425 9.40478 9.40531 9.40 583 9-40636 9.40 689 9.40742 9.40794 9.40847 9.40899 9.40952 9.4 9-4 9-4 9-4 9-4 9-4 9-4 9.4 9-4 9-4 9-4 9.4 9-4 9-4 9.4 9.4 9-4 9.4 9.4 9.4 004 057 109 161 213 266 318 370 422 474 525 57? 629 681 732 784 836 887 938 990 9.42041 9.42 092 9-42 144 9-42 195 9-42 246 9.42 29? 9-42 348 9.42 399 9.42450 9.42 501 9.42 552 9.42 602 9.42653 9-42 704 9.42 754 9.42 805 Loe. Cot. c. d. I Log. Cot. 54 53 54 53 53 53 53 53 53 53 53 53 53 53 53 53 52 53 52 53 52 52 52 52 52 52 52 52 52 52 52 52 52 52 51 52 52 51 51 51 52 51 51 51 51 51 51 51 51 51 51 51 51 50 51 50 51 53 50 50 c. d. 0.60 323 0.60 269 0.60 21 5 0.60 1 61 0.60 108 0.60054 0.60001 o. 59 94? 0.59894 0.59 841 0.5978? 0.59734 O.5968T 0.59 628 0.59575 0.59 522 0.59469 0.59 4I6 0.59363 0-5931 I 0.59258 0.59 205 0.59153 0.59 100 o. 59 048 0.58995 0.58943 0.58891 0.58838 0.58786 0.58734 0.58682 0.58 630 9.58578 0.58 526 0.58474 0.58 422 0.58 370 0.58319 0.58 26? 0.58 216 0.58 164 0.58 112 o. 58 061 O.58CIO 0.57 958 0.5790? 0.57856 0.57805 0.57753 0.57 702 0.57651 0.57 606 0.57549 o. 57 499 0.57448 0.5739? 0.57 346 0.57 296 o 57245 o'S7 195 Log. Tan. Log. Cos. 9.98 696 9.9868? 9.98684 9.98681 9.98678 9.98674 9.98 67T 9.98668 9.98 665 9.98 662 9-98658 9.98655 9.98 652 9.98649 9. 98 646 9.98 642 9.98639 9-98636 9.98633 9-98630 9.98626 9-98623 9.98 620 9.98 617 9.98613 9.98 616 9.98 607 9.98 604 9.98 606 9.98 59? 9.98 594 9.98 591 9.98 58? 9.98 584 9.98 581 9.98 578 9.98 574 9.98571 9.98 568 9.98 564 9.98 561 9.98558 9.98 554 9.98551 9.98 548 9.98 544 9.98 541 9.98 538 9-98 534 9.98 531 9.98 528 9.98 524 9.98 521 9.98518 9.98 514 9.98511 9.98 508 9.98 504 9.98 501 9.98498 9-98494 Log. Sin. d. 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 p. P. 6 7 8 9 10 20 30 40 50 54 53 5.4 5-3 6.3 6.2 7.2 7.1 8.1 8.0 9.0 8.9 18.0 17.8 27.0 26.? 36.0 35-6 45-0 44.6 53 5.3 6.2 7.0 7-9 8.8 17-6 26.5 35-3 44.1 52 52 51 51 6 5.2 5.2 5.1 5. 7 6.1 6.6 6.0 5- 8 7.0 6.9 6.8 6. 9 7.9 7.8 7-7 7- 10 8.? 8.6 8.6 8. 20 17.5 17.3 17.! 17. 30 26.2 26.0 25-? 25. 40 35.0 34.6 34.3 34. 50 43-? 43.3 42.9 42. 56 SO 49 49 6 5.S 5.0 4.9 4- 7 5-9 5-8 5.8 5- 8 6.7 6.6 6.6 6. 9 7.6 7.5 7.4 7- 10 8.4 8.. 3 8.2 8. 20 16.8 16.6 16.5 16. 30 25.2 25.0 24.? 24. 40 33.6 33-3 33.0:32. 50 42.1 41-6 41.2 40. 48 48 Al 47 6 4-8 4.8 4-? 4- 7 5-6 5.6 5-5 5- 8 6.4 6.4 6.3 6. 9 7.3 7.2 7.1 7. 10 8.1 8.0 7.9 7. 20 16. 1 16.0 15.8 15. 30 24.2 24.0 23-? 23. 40 32.3 32.0 31.6 31. 50 40.4 40.0 39.6 39. 6 7 8 9 10 20 30 40 50 3 0.3 0.4 0.4 0.5 0.6 I.I I.? 2.3 2.9 3 0.3 0.3 0.4 0.4 0.5 i.o 1.5 2.0 2.5 p. P. 75* 362 TABLE VII. — L()GARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 15° 10 II 12 14 15 i6 17 i8 19 20 21 22 23 24 ^3 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Lotf. Sill. 9-4 9.4 9.4 9.4 9-4 9.4 9-4 9.4 9-4 9.4 9 4 9-4 9.4 9.4 9.4 299 346 394 441 488 534 581 628 675 721 768 815 861 908 954 9.42 000 9.42 047 9.42093 9.42 139 9.42 185 9.42 232 9.42 278 9.42 324 9.42 369 9.42415 9.42 461 9.42 507 9-42 553 9-42 598 9.42644 9.42 690 9-42735 9.42781 9.42825 9.42 871 9.42917 9.42 962 9.4300^ 9.43052 9.43098 50 51 52 53 54 55 56 57 58 59 GO 9-43 143 9.43 188 943233 9.43278 9-43322 9-43 367 9.43412 9-43 457 9-43 501 9-43 546 9.43 591 943635 9.43680 9-43724 9-43 768 9-43813 9-43857 9-43 901 9-43 945 9-43989 9-44034 Log. Cos. 47 47 47 47 46 47 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 46 45 46 46 46 45 45 46 45 45 45 45 45 45 45 45 45 45 45 45 45 45 44 45 44 45 44 4-+ 45 44 44 44 4-1 44 44 4-1 44 44 44 Lou'. Tnii. c. d. 9.42 805 9.42 856 9.42 906 9-42956 9.43007 9-43057 9 43 107 9-43 157 9.43 208 9.43258 9-43308 943358 9-43408 9-43458 9-43 508 9-43 557 9.43607 9-43657 9-43 706 9-43 756 9-43 806 943855 9-43 905 9-43 954 9-44003 9-44053 9.44 102 9-44 151 9.44 200 9-44249 9.44299 9-44 348 9-44 397 9.44446 9-44 494 9-44 543 9-44 592 9.44641 9.44690 9-44 738 944787 9.44835 9.44884 9-44932 9.44981 9.45029 9.4507^ 9.45 126 9.45 174 9.45 222 9.45 270 9-45 318 9-45 367 9.45415 9:_45J:^ 9.45 515 9-45 558 9-45 606 9.45654 9-45 702 9-45 749 Log. Cot. 51 50 50 50 50 50 50 50 50 50 50 50 50 50 49 50 49 49 50 49 49 49 49 49 49 49 49 49 49 49 49 49 49 48 49 49 48 49 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 47 48 48 ^1 48 47 Loe. Col. 0.57 195 0.57 144 0.57094 0.57043 o. 56 993 o. 56 942 0.56 892 o. 56 842 0.56 792 o. 56 742 0.56 692 o. 56 642 0.56 592 0.56 542 o. 56 492 0.56442 o. 56 392 0.56 343 0.56 293 o. 56 243 0.56 194 0.56 144 0.56095 0.56 04^ 0.55996 0.55947 0.55898 0.55848 0.55799 0.55 750 0.55701 0.55652 0.55 603 0.55 554 0.55 505 0.55 456 0.5540^ 0.55359 0.55310 0.55 261 liOir. Cos. 0.55 213 0.55 164 0.55 116 0.55 067 0.55019 0.54970 0.54 922 0.54874 0.54 825 0.5477? 0.54729 o. 5468T 0.54633 0.54585 0.54537 o. 54 489 0.54441 0.54393 0.54346 o 54 298 0.54256 9-98494 9.98491 9-98487 9.98484 9. 98 481 9-9847? 9.98 474 9-98470 9.98467 9.98464 9. 98 466 9-98457 9-98453 9.98 450 9-98446 9-98443 9-98439 9-98436 9-98433 9.98 429 9.98 426 9.98 422 9.98419 9.98415 9.98 412 9.98 408 9.98405 9.98 401 9-98 398 9-98 394 9.98 391 9-98 387 9.98 384 9.98 386 9.98 377 998373 9-98370 9.98365 9-98363 9-98 359 9.98356 9.98352 9-98348 9-98345 9-98 341 c. (1. I Lotr. Tan. 9-98 338 9-98334 9-98331 9.98 32^ 9.98 324 9.98 320 9-98 3I6 9-98313 9-98309 9-98 306 9.98 302 9.98298 9-98295 9.98 29T 9.98288 9-98 284 Loir. Sin. 40 39 38 37 36 35 34 33 r. I' 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 1 1 10 9 8 7 6 50 5< D 6 5.6 5.0 7 5-9 5 8 8 6.? 6 6 9 7.6 7 5 10 8.4 8 3 20 16.8 16 6 30 25.2 25 40 33-6 33 3 50 42.1 41 6 6 7 81 9! 10, 49 4.9 5-8 6.6 7-4 8.2 20 16.5 30,24.? 4033-0 5041.2 47 4.? 5 " 6 9 10 20 15 3o|23 4031 50:39 49 4-9 5 6 , 5 T 16.3 24.5 32.6 40 " 47 4-7 48 4 ' 5 48 4-8 5-6 6.4 7.2 8.0 16.0 24.0 32.0 40.0 7 7 15 ?|23 631 6|39 46 4-6 '5 23 31 TI38 46 4.6 5-3 6.T 6.9 7-6 15-3 2123.0 030.6 ?!38.3 6 7 8 9 10 20 45 4.5 5-3 6.6 6.8 7.6 15.1 30 22.^ 40' 30. 3 50:37.9 10 20 45 4-5 5-2 6.0 6.? 7.5 15.0 22. 5 44 4-4 5 30.0 29 37.5I37 40 50 0.4 0.3 0.4 0.4 0.5 0.4 0.6 o.S 0.6 0.6 1-3 i.i 2.0 I-? 2-6 2-3 3.3 2.9 9 7 4 8 ^^ 2 22 6*29 M36 3 0.3 0.3 0.4 0.4 0.5 i.o 1-5 2.0 2-5 44 4-4 5 ' 5 6 7 74° 563 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 16° 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 GO hog. Si:i. 9-44034 9-44078 9.44122 9.44 166 9.44209 (1. 9-44253 9.44 297 944341 9-44384 9-44 428 9.44472 9-44515 9-44 559 9.44602 9.44646 9.44689 9-44732 9.44776 9.44819 9.44 862 9-44 905 9-44 948 9.44991 9-45 034 9.45077 9.45 120 9.45 163 9.45 206 9.45 249 9.45 291 9-45 334 9-45 377 9.45419 9.45462 9-45 504 9-45 547 9.45 589 9-45631 9.45 674 9.45716 9-45 758 9.45 800 9-45 842 9.45 885 9.45927 9-45 969 9.46 on 9.46052 9.46094 9-46 136 9.46 178 9.46 220 9.46 261 9-46 303 9-46 345 9.46 385 9.46 428 9.46469 9.46 511 9.46552 9-46 593 Lo?. Cos. 44 44 44 43 44 44 43 43 44 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 42 43 43 42 42 43 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 42 42 41 42 41 41 42 41 41 41 41 41 41 d. Lost. Tan. c. d. 9-45 749 9-45 797 9-45 845 9.45 892 9.45 940 9.45 98^ 9.46035 9.46082 9.46 129 9.46 177 9.46 224 9.46 271 9-46318 9.46 366 9.46413 9. 46 460 9.46 507 9.46 554 9.46 601 9.46647 9.46 694 9.46 741 9.46788 9.46834 9.46881 9.46 928 9.46974 9.47 021 9.47 067 9.47 114 9.47 166 9 47 207 9.47253 9-47 299 9 47 345 9-47 392 9.47 438 9.47 484 9-47 530 9-47 576 9.47 622 9-47 668 947714 9.47 760 9.47 806 9-47851 9.47 897 9-47 943 9.47 989 9.48034 9.48 080 9.48 125 9.48 171 9.48 216 9.48 262 9-48 307 948353 9.48 398 9-48443 9-48488 9-48 534 hog. Cot. 48 47 47 4? 47 47 47 47 47 47 47 47 47 47 47 47 47 47 46 47 47 46 46 47 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 45 46 46 45 46 45 46 45 45 45 45 45 45 45 45 45 45 45 — c. d. Log. Cot. 54256 54202 155 107 060 54 54 54 0.54012 0.53965 0.53917 0.53876 0.53823 0.53776 0.53728 0.53681 0.53634 0.53 587 0.53 0.53 0.53 0.53 0.53 540 493 446 399 352 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.52 0.52 0.52 305 258 212 165 iii 072 025 979 932 886 0.52 0.52 0.52 0.52 0.52 839 793 747 706 654 0.52 0.52 0.52 0.52 0.52 608 562 516 469 423 0.52 0.52 0.52 0.52 0.52 377 33? 286 240 194 0.52 0.52 0.52 0.52 o.5t 0.51 0.51 0.51 0.51 0.51 148 102 057 01 1 96|_ 920 874 829 783 738 0.51 0.51 0.51 0.51 0.51 692 647 602 556 511 0.51 466 hog. Tan. I Loar. Cos. 9.98 284 9.98 286 9.98277 9.98273 9.98 269 9.98 266 9.98 262 9.98258 9.98255 9.98 251 9.98 247 9-98 244 9.98 246 9.98236 9-98233 9.98 229 9.98 22^ 9.98 222 9-98218 9.98 214 9.98 211 9.98 207 9.98203 9.98 200 9.98 9.98 9.98 9.98 9.98 9-98 9.98 9.98 9.98 9.98 9-98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9.98 9-98 9.98 9.98 9.98 96 92 85 81 77 73 70 66 62 58 55 51 47 43 40 36 32 28 24 21 17 13 09 05 02 9.98 098 9.98094 9.98 096 9.98086 9.98 082 9.98079 9.98075 9.98 071 9.98 o6f 9.98063 9-98059 liOir. Sin. 3 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 "(iT 00 59 58 57 5^ 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 V. V. 48 4^ 6 4.8 A- 7 7 5.6 5.5 8 6.4 6.3 9 7.2 7-1 10 8.0 7-9 20 16.0 15-8 30 24.0 23-7 40 32.0 31-6 50 40.0 39.6 46 46 4S 6 4-6 4.6 4-5 7 5-4 5-3 5-3 8 6.2 6.T 6.6 9 7.0 6.9 6.8 10 7-9 7-6 7.6 20 15-5 15-3 15.1 30 23.2 23.0 22.7 40 31.0 30.6 30.3 50 38.^ 38.3 37.9 44 43 43 6 4.4 4.3 4. 7 8 9 5-1 5-8 6.6 5-1 5-8 6.5 5- 5- 6. 10 7-3 7.2 7- 20 14-6 14.5 14. 30 22.0 21. f 21. 40 50 29-3 36.6 29.0 36.2 28. 35- 47 4-7 5-5 6.2 7.6 7-8 15-6 23-5 31-3 39-1 45 4.5 5.2 6.0 7.5 15.0 22.5 30.0 37.5 6 7 8 9 10 20 30 40 50 4 04 0.4 0.5 0.6 0.6 1-3 2.0 2-6 3-3 P. I' 3 0.3 0.4 0.4 0.5 0.6 i.i i-'7 2.3 2.9 42 42 41 41 6 4.2 4.2 4.1 4.1 7 8 9 4.9 5-6 6.4 4-9 5.6 6.3 4.8 5-5 6.2 4.8 5-4 6.T 10 7-1 7.0 6.9 6.8 20 14. 1 14.0 13-8 13-6 30 40 21.2 28.3 21.0 28.0 20.^ 27.6 20.5 27-3 50 35-4 35-0 34-6 34-1 364 TAHLE VII. — LOGARITHMIC SINES, COSINES, TANGENTS, AND CO'lAN(iEXTS. 17" 4 5 6 7 8 9 10 II 12 13 U 15 i6 17 i8 19 20 •21 22 23 24 25 26 28 30 31 32 jj 34 35 36 37 38 39 40 41 42 43 44 55 56 57 58 59 (;o Lotf. Sill. (I. 9-46 593 9.46635 9.46676 9.46717 9-46758 9.46799 9.46 840 9.46881 9.46 922 9- 46 963 9.47004 9.47 04^ 9.47086 9.47 127 9.47 168 9.47 208 9.47 249 9.47 290 9-47 330 9 47 371 9.47 41 1 9.47452 9.47492 9-47 532 9-47 573 9.47613 9-47653 9.47694 9-47 734 9 47 774 9.47814 9.47 854 9.47 894 9-47 934 9-47 974 9.48 014 9.48054 9.48093 948 133 9- 48 173 948213 9.48252 9.48 292 948 331 9.48371 9.48 410 9.48450 9.48 489 9.48 529 9.48 568 9.48 607 9.48646 9,48686 9.48725 9.48 764 9.48 803 9.48 842 9.48 881 9.48 920 9.48959 9.48998 Loe. Cos. 41 4i 41 41 41 41 41 4f •41 41 41 40 41 40 40 41 40 40 40 46 40 40 40 40 40 46 40 43 40 4^ 40 40 40 40 40 39 40 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 29 39 38 Loif. Tan. 9-48 534 9.48 579 9.48 624 9.48 669 9.48714 9.48759 9.48 804 9.48 849 9.48 894 948939 9.48984 9.49028 9-49073 9.49 118 9.49 162 9.49 207 9-49252 9.49296 9-49 341 949385 9-49430 9-49 474 9-49 518 9-49 563 9.49607 9-49651 9.49695 9.49 740 9.49784 9.49 828 9.49872 9.49 916 9.49960 9. 50 004 9. 50 048 9. 50 092 9.50136 9.50179 9.50223 9.50 267 9.50 31 1 9-50354 9.50398 9.50442 9.50485 0. d. iOir. Cot. 9.50529 9-50572 9.50616 9.50659 9.50702 d. 9. 50 746 9.50789 9.50832 9.50876 9.50919 9. 50 962 9.51005 9.51 048 9.51 091 95^ 134 9.51 i7f 45 45 45 45 45 45 44 45 45 45 44 45 44 44 45 44 44 44 4^ 44 44 44 44 44 44 44 44 44 44 44 4T 41- 43 44 44 44 43 44 43 44 43 43 44 43 43 43 43 43 43 43 43 .43 43 43 43 43 43 43 43 43 51 46O 51 421 51 376 51 330 51 285 51 240 51 195 51 151 51 106 51 061 51 016 50971 50926 50 882 5083? 50792 50748 50703 50659 50614 50570 50525 50481 50437 5039 2_ 50 348" 50304 50 260 50 216 50 172 50 128 50083 50039 49996 49_952^ 49 908 49 864 49 826 49 776 49 733 IjOU'. Cos. 49 689 49 645 49602 49558 49 5 U 49 47 1 49427 49384 49340 49297 49254 49 216 49167 49 124 49 081 49038 48994 48 951 48908 48 865 48 822 9.98059 9.98 056 9. 98 052 9.98 048 9.98044 9. 98 040 9.98036 9.98032 9.98:028 9.98024 9.98 02 1 9.98 017 9.98013 9.98 009 9.98005 9.98 001 9-97 997 9-97 993 9.97989 9.97985 9.97981 9.97 977 9-97 973 9.97969 9-97966- 9.97962 9.97958 9-97 954 9.97950 9.97946 9-97 942 9-97938 9-97 934 997930 9.97926 ivOi;. Cot. I ('. ^ 3^ 3^ 38 38 38 38 3^ 38 38 37 38 38 37 38 37 37 38 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 36 37 36 37 36 (I. 9.f,i 177 9.51 226 9.51 263 9-51 306 9-51 349 9.51 392 9-51435 9-51477 9.51 520 9.51 563 9.51 605 9.51 648 9.51 691 9-51 733 9.51 776 9.51 818 9.51 861 9.51 903 9.51 946 9.51 988 9.52 030 9.52073 9.52 115 9.52 157 9 52 199 9.52241 9.52284 9.52 326 9.52 368 9.52410 9.52452 9.52494 9.52536 9.52578 9.52 619 9.52 661 9.52703 9-52745 9.52787 9-52828 9.52870 9.52 912 9 52953 9.52995 9 53036 9.53078 9-53 119 9.53 161 9.53 202 9-53244 9.53285 9-53 326 9- 53 368 9-53409 9-53450 9-53491 9-53 533 9-53 574 9-53615 9.53656 9-53697 C. (I. 43 43 43 43 42 43 42 43 42 42 43 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 42 42 41 42 41 41 42 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 Los?. Cot. 0.48822 0.48 779 0.48 736 0.48 693 0.48 656 0.48 608 0.48 565 0.48 522 0.48 479 0-48437 0.48 394 0.48 351 0.48 309 0.48 265 0.48 224 0.48 181 0.48 139 0.48 096 0.48 054 0.48 012 0.47 969 0.47 927 0.47 885 0.47 842 0.47 806 0-47 758 0.47 716 0.47 674 0,47 632 0.47 590 0.47 548 0.47 506 0.47 464 0.47 422 0.47 386 0-47 338 0.47 296 0.47 255 0.47 213 0.47 1 71 0.47 130 0.47 088 0.47 046 0.47 005 0.46 963 0.46 922 0.46 886 0.46 839 0.4679? 0.46 756 0.46 714 0.46673 0.46 632 0.46 591 0.46 549 Lot?. Cot. I c. d. 0.46 5O8 0.46 467 0.46426 0.46 385 0-46 344 0-46 303 Log^. Tan. Los. Cos. 9.97 826 9-97816 9.97 812 9.97 808 9.97 804 9.97 800 9-97796 9-97792 9.97787 9-97783 9-97 779 9-97 775 9.97771 9.97 767 9.97 763 9-97 758 9-97 754 9-97750 9-97746 9-97742 9-97 73? 9-97 733 9.97729 9.97725 9-97721 9-97 716 9.97712 9-97 708 9.97704 9-97 700 9.97695 9.97691 9-97687 9.97 683 9-97678 9.97674 9.97670 9.97 666 9.97 661 9.9765^ 9-97653 9.97649 9.97644 9.97646 9.97636 9.97632 9-97 62f 9.97623 9.97619 9.97614 9.97 616 9.97 606 9.97 601 9-97 59? 9-97 593 9-97 588 9.97584 9.97580 9.97 575 9-97 571 9-97 567 Log. Sin. 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 "dT GO 59 58 57 56 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 19 18 17 16 p. p. 15 14 13 12 II 10 9 8 7 6 43 42 6 4-3 4.2 7 8 9 5.0 5.? 6.4 4.9 5-6 6.4 10 7-1 7-1 20 14-3 14.1 30 21. s 21.2 40 28.6 28.3 50 35-8 354 42 4.2 4-9 5.6 6.3 7.0 14.0 21.0 28.0 35-0 41 6 4-1 7 4-8 8 5-5 9 6.2 10 6.9 20 13-8 30 20.7 40 27-6 50 34-6 41 4.1 4-8 5-4 6.1 6.8 13-6 20.5 27-3 34-1 39 38 6 3-9 3-8 7 4-5 4-5 8 5.2 5-1 9 58 5.8 10 6.5 6.4 20 13.0 12.8 30 195 19.2 40 26.0- 25-6 50 32.5 32.1 6 7 8 9 10 20 30 40 50 31 3.? 4.4 5.0 5.6 6.2 12.5 18.^ 25.0 31.2 37 3-7 4.3 4.9 5-5 6.1 12.3 18.5 24.6 30.8 38 3-8 4.4 5.6 5-7 6.3 12.6 19.0 25.3 31-6 36 3-6 4.2 4-8 5-5 6.1 12. T 18.2 24-3 30.4 6 7 8 9 10 20 30 40 50 4 0.4 0.5 0.6 0.7 o.? 1-5 2.2 3-0 3-? 4 04 0.4 o.S 0.6 0.6 1-3 2.0 2.6 3-3 P. p. 366 TABLE VII. -LOGARITHMIC SINES. COSINES. TANGENTS, AND COTANGENTS 19 10 II 12 13 1± 15 i6 17 i8 19 20 2; 22 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 4£ 45 46 47 ^8 49 r>o 51 52 53 54 55 56 57 58 59 (>0 Lo:;. Sill. (]. 9- 9- 9' 9. 9: 9- 9 9 9 9 9 9 9 9 9 264 301 337 374 416 447 483 520 556 593 629 665 702 738 774 816 847 883 919 955 51 991 52 027 52 063 52099 52 135 52 170 52206 52242 52278 52314 52349 52385 52421 52456 52492 5252? 52563 52598 52634 52 669 52704 52740 52775 52 810 52846 52881 52 916 52951 52986 53021 53056 53091 53 126 5316T 53 196 53231 53 266 53301 53 335 53370 9-53405 Log. Cos. 37 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 3l 36 35 3l 36 35 35 35 35 JD :>:> 35 35 35 35 35 35 35 35 3:) 35 35 35 35 35 35 35 34 35 35 34 35 34 og. Tail. I c. (I. I Log. Cot. 53697 53738 53 779 53 S20 53861 53902 53 943 53983 54024 54065 54 106 54147 54187 54228 54269 54309 54350 54390 54431 54471 54512 54552 54 593 54633 54673 54714 54 754 54 794 54834 54874 54915 54 955 54 995 55035 55075 55115 55 155 55 195 55235 55275 55315 55 355 55 394 55 434 55 474 55 514 55 553 55 593 55633 55672 55712 55751 55791 55831 55870 55909 55 949 55988 56028 5606^ 56 106 LOK. Cot. 41 41 41 41 41 41 40 41 41 46 41 40 40 41 40 40 40 40 40 40 40 40 40 40 40 40 46 40 40 40 40 40 40 40 40 39 40 40 40 40 40 39 40 39 40 39 40 39 39 39 39 40 39 39 39 39 39 39 39 39 0.46 303 0.46 262 0.46 221 0.46 180 0.46 139 0.46 098 0.46057 0.46 015 0.45975 0.45 934 0.45 894 0.45 853 0.45 812 0.45 772 0.45731 0.45 690 0.45 650 0.45 609 0.45 569 o 45 528 0.45 488 0.45 447 0.45 407 0.45 367 0.45 326 0.45 286 0.45 246 0.45 205 0.45 165 0.45 125 0.45 085 0.45045 0.45 005 0.44965 0.44925 0.44884 0.44845 0.44 805 0.44 765 0.44725 0.44 685 0.44645 0.44 60 5 0.44 565 0.44 526 0.44486 0.44 446 0.44406 0.44 367 0.44327 0,44 288 0.44 248 0.44 208 0.44 169 0.44 129 0.44090 0.44 051 0.44 01 T 0.43 972 0.43 932 0.43893 Loir. Tan. Loir. Cos. 9.97 567 9.97 562 9.97 558 9-97 554 9-97 549 9-97 545 9-97 541 9-97 536 9-97 532 9.97 527 9-97 523 9.97 519 9.97514 9.97 510 9.97 505 9.97 501 9-97 497 9-97492 9.97488 997483 9-97 479 9-97 475 9.97476 9.97466 9-97461 9-97 457 9.97452 9.97448 9-97 443 9-97 439 9-97 434 9-97430 9-97425 9.97421 9-97 416 9-97412 9.97407 9-97403 9-97 398 9-97 394 9-97 389 9-97385 9-97 380 9-97376 9-97371 9-97 367 9-97 362 9-97358 9-97 353 9-97 349 9-97 344 9-97 340 9-97 335 9-97330 9-97 326 9-97321 9-97317 9.97312 9-97 308 9-97303 9 97 298 Loif. Sin. o 49 48 47 ^6^ 45 44 43 42 41 24 21 20 '9 18 17 16 15 14 13 12 1 1 1(7 9 8 7 6 5 4 3 r. I'. 41 40 40 6 41 4.6 4.0 1 7 4.8 4-7 46 8 5-4 5-4 5-3 9 6.1 6.1 6.0 10 6.8 6.f 6.6 20 ■3-6 '3-5 13-3 30 20.5 20.2 20.0 40 27.3 27.0 26.6 50 34-1 33-i^ 33-3 39 6 7 8 9 10 20 30 40 50 3.9 3- 4.6 4- 5-2 5- 5.9 5- 6.6 6. 13-1 13- 19-7 19- 26.3 26. 32.9 32- 39 9 37 36 6 3-7 3-6' 7 4-3 4-2 8 4.9 4.8 9 5-5 5-5 10 6.T 6.1 20 12.3 12. 1 30 18.5 18.2 40 24-6 24-3 50 30.8 30.4' 36 3-6 4.2 4-8 5-4 6.0 12.0 18.0 24.0 ^o.o 35 35 34 / 8 9 10 20 30 40 50 J 5 3-5 3- 4 I 4.1 4- 4 7 4-6 4- 5 3 5.2 5- 5 9 5-8 5- II 8 II. 6 1 1. 17 7 17-5 17- 23 6 23-3 23- 29 6 29.1 28. 6 7 8 9 10 20 30 40 50 5 0.5 0.6 0.6 o.'7 0.8 1-6 2-5 3-3 4.1 4 0.4 0.5 0.6 0.7 o.f 1.5 3-f 4 0.4 0.4 0.5 0.6 0.6 1-3 2.0 2.6 3-3 O 3(>1 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 10 II 12 13 14 15 16 17 18 19 20 21 22 23 -7 1 — -T 25 26 27 28 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 i 57 58 59 60 Lost. Mil 53 405 53440 53 474 53509 53 544 53 578 53613 53647 53682 53716 53750 53785 53819 53854 53888 53922 53 956 53990 54025 54059 54093 54127 5416T 9 54195 9 54229 9.54263 9.54297 9-54331 9-54365 9-54 398 9-54 432 9- 54 466 9- 54 500 9-54 534 9-54567 9.54601 9-54634 9.54668 9.54702 9-54 73? 9.54769 9. 54 802 9-54836 9.54869 9 54 902 9-54936 9.54969 9.55002 9- 55 036 9-55069 9-55 102 9-55 135 9-55 168 9.55 202 9-55 235 9-55 268 9.55301 9-55 334 9-55367 9-55400 9-55 433 Loff. Cos. d. 35 34 34 35 34 34 34 34 34 34 34 3-4 34 34 34 34 34 34 34 34 34 34 34 34 33 34 34 34 33 34 34 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 Los. Tail. 9.56 IO6 9.56 146 9.56 185 9.56 224 9.56263 9-56303 9.56342 9-56381 9.56420 9.56459 9.56498 9-56537 9-56576 9.56615 9.56654 56693 56732 56771 56810 56848 9.56 887 9.56 926 9.56965 9-57003 9-57042 9.57081 9.57 119 9.57 158 9-57 196 9-57235 9-57274 9.57 312 9-57350 9-57389 9-57427 9.57466 9-57 504 9-57 542 9.57581 9.57619 9-57657 9.57696 9-57 734 9-57772 9.57810 9-37848 9-57886 9.57925 9-57963 9.58 001 9.58039 9-58077 9.58 115 9-58153 9.58 196 9.58 228 9.58266 9-58304 9.58342 9-58380 9-58417 Lost. Cot. c, (1. 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 39 38 39 38 39 38 38 39 38 38 38 38 39 38 38- 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 37- 38 38 38 37 38 37 Log. Cot. 0.43 893 0.43 854 0.43815 0.43775 0.43 736 0.43 697 0.43658 0.43619 0.43 580 0.43 540 0.43 501 0.43 462 0.43423 0.43 384 0.43 346 0.43 307 0,43 268 0.43 229 0.43 190 0.43 151 0.43 1 12 0.43074 0.4303 s 0.42 996 0.42 958 0.42 919 0.42 880 0.42 842 0.42 803 0.42 765 0.42 726 0.42 687 0.42 649 0.42 611 0.42 572 0.42 534 0.42 495 0.42457 0.42 419 0.42 380 0.42 342 0.42 304 0.42 266 0.42 227 0.42 189 0.42 151 0.42 113 0.42075 0.42037 0.41 999 0.41 961 0.41 923 0.41 885 0.41 847 0.41 809 0.41 771 0.41 733 0.41 6gl 0.41 658 0.41 620 0.41 582 c. (I. I Loi?. Tan. Loir. Cos. 9.97 298 9.97 294 9.97 289 9-97 285 9.97 280 9-97275 9.97271 9.97 266 9.97 261 9.97257 9.97 252 9.97 248 9-97 243 9-97238 9-97 234 9-97 229 9.97 224 9.97 220 9.97215 9.97 210 9.97 206 9.97 201 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9-97 9.97097 9.97092 59 54 49 44 40 35 30 25 21 16 II 06 02 9.97087 9.97 082 9.97078 9.97073 9.97068 9.97063 9-97058 9.97054 9.97049 9.97044 9-97039 9-97034 9.97029 9-97025 9.97 020 9-97015 Lost. Sin. 4 4 4 5 4 4 4 5 4 4 4 5 4 4 5 4 4 5 4 4 5 4 5 4 4 5 4 5 4 4 5 4 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 5 4 5 5 4 5 5 4 5 5 60 59 58 57 56 55 54 53 52 ^i 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 V_ 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 p. P. 39 3S 6 7 8 3.9 4.6 5-2 3- 4. 5- 9 10 5-9 6.6 5- 6. 20 13-1 13- 30 40 19.7 26.3 19. 26. 50 32.9 32- 6 7 8 9 10 20 30 40 50 38 3-8 4-5 5-1 5-8 6.4 12.8 19.2 25-6 32,1 38 3-8 4.4 5.6 5-7 6.3 12.6 19.0 25-3 31-6 27 3-7 4 5 5 6 12 18 25 31 35 34 6 3-5 3-4 7 4.1 4.0 8 4-6 4.6 9 5-2 5.2 10 5-8 5-7 20 II. 6 11-5 30 17-5 17.2 40 23-3 23.0 50 29.1 28.? 34 3-4 3-9 4-5 5-1 5-6 II-3 17.0 22.6 28.3 6 3-3 3- 7 3-9 3. 8 4-4 4- 9 5-0 4- 10 5.6 5- 20 II. I II. 30 16.7 16. 40 22.3 22. 50 27.9 27- 23 22 4 0.4 0.5 0.6 0.7 0.7 1-5 2.2 3-0 3-7 5 6 0.5 7 0.6, 8 0.6! 9 0.7 10 0.8 20 1-6 30 40 2-5 3-3 50 4.1 p. p. G9 368 LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 2\ 10 II 12 14 15 16 18 19 20 21 22 23 24 26 27 28 29 80 31 32 33 34 J3 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Locr. Sill. 9-55 433 9.55466 9-55 498 9-55 531 9-55 564 9-55 597 9.55630 9.55662 9.5569^ 9.55728 9.55760 9-55 793 9.55826 9-55^58 9.55891 (1. 9-55923 9-55956 9-55 9S8 9.56 020 9.56053 '9. 56085 9.56 118 9.56 150 9.56 182 9.56214 9.56247 9.56279 9.56311 9-56343 9-56375 9- 56 407 9-56439 9.56471 9.56503 9-56533 9.56567 9-56599 9.56631 9.56663 9.56695 9.56727 9-56758 9. 56 790 9.56 822 9.56854 50 51 52. 53 54 55 56 57 58 59 60 9.56885 9.56917 9- 56 949 9 56 980 9.57012 9- 57 043 9.57075 9-57 106 9-57 138 9-57 169 9.57 201 9.57232 9-57263 9.57295 9-57 326 9-57 357 lj(»ff. Cos. 33 32 jj 33 32 33 32 33 32 32 32 33 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 31 32 32 31 32 31 32 3f 31 32 31 31 31 31 31 31 31 31 31 31 31 31 31 Lost. Tan. 0. d. I Loe. Cot. 9.58417 9.58455 9- 58 493 9-58531 9.58568 9.58 606 9.58644 9.58681 9.58719 9-58756 9.58794 9-58831 9.58869 9-58906 9 58 944 9.58 981 9.59019 9.59056 9- 59 093 9-59 131 9.59 168 9-59205 9.59242 9.59 280 9.59317 9-5935 + 9-59391 9.59428 9.59465 9.59502 9.59540 9-59 577 9.59614 9.59651 9.59688 9-59724 9-59761 9-59 798 9-59833 9.59872 9. 59 909 9- 59 946 9.59982 9.60019 9 60056 9.60093 9.60 129 9.60 165 9.60 203 9.60239 d. 9.60 276 9.60 312 9-60349 9.60 386 9.60422 9-60459 9.60495 9.60 531 9.60 568 9.60 604 9. 60 64 1 38 3? 38 37 3f 38 37 3f 37 37 37 3? 37 3? 3? 3l 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 36 36 36 37 36 36 36 36 36 36 36 36 0.41 582 0.41 544 0.41 507 0.41 469 0.41 43? 0.41 394 0.41 356 0.41 318 0.41 281 0.41 243 0.41 206 0.41 163 0.41 131 0.41 093 0.41 056 0.41 oig 0.40 981 0.40 944 0.40 906 0.40 869 0.40 832 0.40 794 0.40757 0.40 720 0.40 683 0.40 646 0.40 608 0.40 571 0.40 534 0.40 497 0.40 460 0.40423 0.40 386 0.40 349 0.40 312 l,nir. ("OS. 0.40 275 0.40 238 0.40 201 0.40 164 0.40 128 0.40 091 0.40 054 0.40017 0.39 980 0.39944 o. 39 907 0.39876 0-39833 0.39797 0.39766 0.39724 0.3968^ 0.39650 0.39614 0.3957? 0.39541 0.39 504 0.39468 0.39432 0.39395 9.97015 9.97 016 9.97005 9.97 000 9.96995 9.96991 9.96 986 9.96 981 9.96976 9-96971 9-96966 9.96 961 9-96956 9-96952 9-96947 9-96942 9-96937 9.96932 9-96927 9.96 922 9.96917 9.96 912 9-96907 9. 96 902 9-9689? 9.96 892 9.9688? 9.96882 9.96877 9.96873 9.96868 9.96 863 9.96858 9.96853 9.96848 9.96843 9.96838 9-96833 9.96828 9.96823 9.96 818 9.96 813 9.96808 9.96 802 9.96797 9-96792 9.96787 9.96 782 9-9677? 9.96772 9-96 767 9.96 762 9-9675? 9.96752 9.96747 o- 39 359 Cot. c. d. : liOtr. 'In 11. 9 96 742 9-96737 9.96732 9.96727 9.96 721 9.96 7 16 4 5 5 5 4 5 5 5 4 5 5 5 4 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 I 5 5 5 5 5 5 5 45 44 43 42 41 40 39 38 37 35 34 33 32 31 I'. I'. 30 29 28 27 26 l,ou'. sin. 25 24 23 22 21 20 '9 18 17 16 38 37 37 6 3-8 3-? 3-7 , 7 4.4 4-4 4 3 > 8 S.o 5.0 4 9 9 S-7 5.6 5 ^ 10 6. .3 6.2 6 I 20 12.6 12.5 12 3 30 19.0 18.? 18 5 40 25-3 25.0 24 6 50 31-6 31.2 30 8 36 36 6 3-6 3-6 7 4.2 4-2 8 4.8 4.8 9 5-5 5-4 10 6.1 6.0 20 12. T 12.0 30 18.2 18.0 40 24-3 24.0 50 30.4 30.0 33 32 32 6 7 8 9 10 20 30 40 50 3-3 3-2 3-2 3 8 3 8 3-? 4 4 4 3 4.2 4 9 4 9 4.8 5 5 5 4 5-3 II 10 8 10.6 16 5 16 2 16.0 22 21 6 21.3 27 5 27 I 26.6 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 31 31 3-1 3-6 4.1 4.6 5-1 10.3 15-5 20.6 25 8 3 T 3 7 4 2 4 5 7 2 10 5 15 ? 21 26 T 0.5 0.6 0.5 0.6 0.? 0.6 0.8 0.7 0.9 0.8 1-8 2.? 3-6 4-6 1-6 2-5 3-3 4.1 4 0.4 0.5 0.6 0.7 0.7 '•5 "^ 2 30 3-? 1'. I" G8^ 3'-^y TABLE VII.-LOGARITHMIC SINES, COSINES. TANGENTS. AND COTANGENT^ 23° Log. Siu. 9-57 35^ 9-57389 9.57420 9-57 451 9-57482 9-57513 9-57 544 9-57576 9.57607 9.57638 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 9.57669 9.57700 9-57731 9.57762 9.57792 9-57823 9-57854 9.5788$ 9-57916 9-57 947 25 26 27 28 29 30 31 32 33 -31 35 36 37 38 39 9-57 977 9-58008 9.58039 9.58070 9.58 TOO 9-58 131 9.58 162 9.58192 9.58223 9-58253 9.58284 9-58314 9-58345 9-5837$ 9.58 406 9.58436 9.58465 9.58497 9.58527 9-58557 40 9.58587 41 9.58618 42 9.58648 43 9-58678 9-58708 50 51 52 53 54 9-58738 9.58769 9.58799 9.58 829 9.58859 9.58889 9.58919 9- 58 949 9-58979 9.59009 9-59038 9- 59 068 9- 59 098 9.59128 9-59158 9-59 188 Log. Cos. 31 31 31 31 31 31 31 31 31 31 31 31 31 30 31 31 31 30 31 30 31 30 31 30 30 31 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 29 30 30 d. Log. Tail. c. <1 9.60 641 9.60677 9.60713 9.60 750 9.60 785 9.60 822 9.60 859 9.60 895 9-60931 9.60 967 9.61 003 9.61 039 9.61 076 9.61 112 9.61 148 9.61 184 9.61 220 9.61 256 9.61 292 9.61 328 9-6i 364 9.61 400 9.61 436 9.61 472 9.61 507 9.61 543 9-61 579 9.61 615 9.61 651 9.61 685 9.61 722 9.61 758 9.61 794 9.61 829 9 61 865 9.61 901 9.61 936 9.61 972 9. 62 007 9-62043 9.62078 9.62 114 9.62 149 9.62 185 9.62 220 9.62 256 9.62 291 9.62 327 9.62 362 9-62 39^ 9-62433 9.62468 9.62 503 9-62 539 9-62 574 9.62 609 9.62 644 9.62 679 9.62 715 9.62 750 9.62 785 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 3! 36 36 3§ 36 35 36 3l 36 35 35 36 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 Log. Cot. 0-39359 0.39322 0.39285 0.39250 0.39213 0.39 177 0.39 141 0.39 105 0.39069 0.39032 0.38995 o. 38 966 0.38924 0.38888 0.38852 0.38816 0.38 780 0.38744 0.38 708 0.38 672 0.38636 0.38 600 0.38 564 0.38 528 0.38492 0-38456 0.38 420 0.38385 0.38349 0.38 313 0.3827^ 0.38 242 0.38 206 0.38 170 0.38 135 0.38099 0.38 063 0.38028 0.37 992 0.37 957 0.37 921 0.37 S86 0.37 850 0.37815 0.37779 0-37 744 0.37 708 0.37673 0.3763^ 0.37 602 0-37 567 0.37 531 0-37 496 0.37461 0.37426 Log. Cot. led. 0.37390 0.37355 0.37326 0.37 28 5 0.37250 Log. Cos. o- 37 215 Log. Tan. 9.96715 9.96 711 9.96705 9.96 701 9.96 696 9.96 691 9.96686 9.96681 9-9667$ 9.96 676 9.96665 9.96660 9.96655 9.96 650 9-96644 9.96639 9.96634 9.96 629 9.96 624 9.96 619 9.96 613 9-96608 9. 96 603 9.96598 9.96593 9-96 587 9.96 582 9.96577 9-96572 9-96 567 9.96 561 9-96556 9.96551 9.96546 9-96 540 9-96 535 9-96 530 9.96525 9.96519 9.96514 9.96 509 9-96 503 9.96498 9-96493 9.96488 9.96482 9-96477 9.96472 9.96465 9.96461 9.96456 9-96450 9-96445 9.96440 9-96434 9-96429 9-96424 9-96418 9.96413 9. 96 408 9.96 402 Log. 8111. 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 d. 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 _3i 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II To 9 8 7 6 5 4 3 2 I ~0~ P. p. 36 6 3-6 7 4.2 8 4.8 9 5-5 10 6.1 20 12. 1 30 18.2 40 24-3 50 30.4 3S 6 3-5 7 4.1 8 4.? 9 5-3 10 5-9 20 11.8 30 17-7 40 23-6 50 29.6 7 8 9 10 20 30 40 50 3i 3-1 3-7 4.2 4-7 5.2 10.5 15-^ 21.0 26.2 30 30 6 3-0 3-0 7 3-5 3.5 8 4-0 4.0 9 4.6 4-5 10 5-1 5.0 20 10. 1 lO.O 30 15.2 15.0 40 20.3 20.0 50 25.4 25.0 36 3-6 4.2 4.8 5-4 6.0 12.0 18.0 24.0 30.0 35 3.5 4-1 4-6 5.2 5-8 II. 6 17.5 23-3 29.1 31 3-1 3-6 4-1 4.6 5-1 10.3 15.5 20.6 25-8 29 2.9 3-4 3-9 4.4 4-9 9-8 14.^ 19-6 24.6 B 5 6 0.5 0-5 7 0.5 0.6 8 0.7 0.6 9 0.8 0.7 10 0.9 0.8 20 1-8 1-6 30 2.7 2.5 40 3-6 3-3 50 4-6 4.1 p. p. 67 370 TAl^LE VII.— LOGARITHMIC SINES, COSINES, 2:v TANGENTS, AND COTANGENTS. 10 II 12 13 14 20 21 22 23 24 25 26 27 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Log. Sill. I d. 15 9-59 16 9-59 17 9-59 18 9.59 19 9.59 9-59 9-59 9-59 9-59 9-59 188 21^ 247 277 306 9-59 9-59 9-59 9-59 9-59 336 366 39^ 425 454 9-59 9-59 9-59 9-59 9-59 484 513 543 572 602 631 661 696 719 749 9-59 9-59 9-59 9-59 9-59 778 807 S37 866 895 9-59 9.59 9-59 9.60 9.60 924 953 982 012 041 9.60 9.60 9.60 9.60 9.60 070 099 128 157 186 9.60 9.60 9.60 9.60 9.60 215 244 273 301 330 9.60 9.60 9.60 9.60 9.60 359 388 417 445 474 9.60 9.60 9.60 9.60 9.60 503 532 560 589 618 9.60 9.60 9.60 9.60 9.60 646 675 703 732 766 9.60 9.60 9.60 9.60 9.60 789 817 846 874 903 9-60931 Log. Cos. 29 30 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 28 29 28 29 28 29 28 28 29 28 28 28 28 28 28 28 28 28 28 28 d. Lou. Tan. 9.62 785 9.62 820 9.62855 9.62 890 9.62 925 9.62 966 9.62995 9.63030 9.63065 9.63 106 963 135 9.63 170 9.63 205 9.63 240 9.63275 9.63310 9-63344 9-63379 9.63414 9-63449 9.63484 9-63 5I8 9-63553 9.63 588 9.63 622 9-63657 9.63 692 9-63726 9.63761 9-63795 9.63830 9.63 864 9.63899 9-63933 9-63 968 9. 64 002 9.64037 9.64071 9.64 106 9.64 140 9.64 174 9.64 209 9.64 243 9.64 277 9.64312 9-64 346 9.64 380 9.64415 9.64449 9-64483 9-6451? 9.64551 9.64585 9.64620 9.64654 c. d. 9.64688 9.64 722 9.64756 9.64790 9.64 824 9.64'85§" Loir. Cot. led. 35 35 35 35 35 35 35 35 35 35 35 35 34 35 35 3I 35 35 34 35 34 34 35 3-+ 34 35 34 34 34 34 34 34 34 34 3^ 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 Log. Cot. 0.37 215 0.37 179 0.37 144 0.37 109 0.37074 0.37039 0.37004 o. 36 969 0.36934 0.36899 o. 36 864 0.36 829 0.36794 0.36 760 0.36725 0.36 690 0.36655 0.36 626 0.36585 0.36 551 0.36 516 0.36481 0.36447 0.36 412 0.36 377 0.36 343 0.36308 0.36273 0.36239 o. 36 204 0.36 170 0.36 135 0.36 lOI o. 36 065 0.36 032 o- 3 5 997 03 5 963 0-35928 0.35894 0-35859 0.35825 0.35791 0.35 756 0.35 722 0.35688 0-35653 0.35619 0.35585 0.35551 0-35 517 0.35482 0.35 448 0.35414 0.35 380 0.35 346 0.35 312 0.35 278 0.35244 0.35 209 0-35 i7l 0.35 141 Loir. Tsui. Locr. Cos. 96 402 96 397 96 392 96386 96381 96375 96370 96365 96359 96354 96 349 96343 96338 96332 96327 96 321 96316 96 31 1 9630I 96 300 96 294 96 289 96 283 96278 96 272 96 267 96 261 96 256 96 251 96245 96 240 96234 96 229 96 223 96 218 96 212 96 205 96 201 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 95 90 84 79 73 68 62 57 51 46 40 34 29 23 18 12 06 96 lOI 96095 96 090 96 084 96078 9-96073 Lost. Sin. «0 59 58 57 56 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 I'. 1' 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 3S 3.5 4.1 4-7 5-3 5-9 II. 8 23-6 29.6 35 3 4 4 5 5 1 1 17 23 29 34 34 3-4 3.4 4.0 4.6 3 4 9 5 5.2 5 5 I 6 II. 5 1 1 3 17.2 17 23-0 28.7 22 28 6 3 6 7 8 9 10 20 30 40 50 30 3-0 3.5 4.0 4-5 5.0 lo.o 15.0 20.0 25.0 29 29 28 6 2.9 2.9 2. 7 3-4 3-4 3- 8 3-9 3-8 3- 9 4.4 4-3 4- 10 4.9 4-8 4- 20 9-8 9-6 9- 30 14-7 14.5 14. 40 19-6 19-3 19. 50 24.6 24.1 23- 6 6 0.6 5 0-5 7 8 0.7 0.8 0.6 9 10 0.9 I.O 0.8 0.9 20 2.0 1-8 30 40 50 3-0 4.0 5.0 2.? 3-6 4.6 5 0.5 0.6 0.6 o.? 0.8 1-6 2.5 3-3 4.1 r. r 66 371 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 24° 29 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Loff. Sin. 9.60931 9.60959 9.60988 d. 772 800 828 856 883 911 938 966 994 9.62 021 9.62 049 9.62075 9.62 104 9.62 13T 9.62 158 9.62 186 9.62 213 9.62 241 9.62 268 9.62 295 9.62323 9.62 350 9.62 yj^ 9.62 404 9.62432 9.62459 9.62486 9.62 513 9.62 540 9.62 56^ 9>62 595 Log. Cos. 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 28 27 28 27 28 2^ 28 2f 28 27 2^ 2^ 27 28 27 27 2? 27 2f 27 2f if 27 27 2^ 27 27 27 27 2^ 27 If 27 27 27 2? Los, 'i'an. C. d. Log. Cot. 9.64858 9.64 892 9.64926 9.64 960 9-64994 9.65 028 9.65 062 9.65 096 9.65 129 9-65 163 9.65 197 9.65 231 9.65 265 9.65 299 9-65 332 9-65 366 9.65 400 9-65433 9.65467 9-65 501 9-65 535 9.65 568 9.65 602 9.65635 9.65 669 9.65703 9-65 736 9-65 770 9.65 803 9-65 837 9.65 870 9-65 904 965 937 9.65971 9. 66 004 9.66037 9.66071 9.66 104 9.66 13^ 9.66 171 9.66 204 9.66 23^ 9.66 271 9. 66 304 96633^ 9.66370 9.66404 9.66437 9.66 470 9- 66 503 9.66 9.66 9.66 536 570 603 9.66 636 9.66669 9.66 702 9.66735 9.66768 9.66 801 9.66 834 9^86f Log. Cot. c. d. 34 34 33 34 34 34 34 33 34 34 33 34 34 33 34 33 33 34 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 0.35 141 0.35 107 0-35073 0.35 040 0.35 006 0.34972 0.34938 0.34904 0,34 '^']0 0.34836 o. 34 802 0.34769 0.34735 0.34701 0.3466^ 0.34633 o. 34 600 0.34566 0.34532 0.34499 0.34465 0.34431 0.34398 0.34364 0.34331 0.34297 0.34263 0.34230 0.34196 0.34 163 0.34 129 0.34096 o. 34 062 0.34029 0.33996 0.33 962 0.33929 0-3389? 0.33862 0.33829 0.33795 0.33762 0.33729 0.33696 0.33 662 Log. Cos. 0.33629 0.33 596 0.33 563 0.33529 o. 33 496 0.33463 0.33430 0.33397 0.33364 0.33331 0.33 298 0.33265 0.33232 0.33 198 0.33 i6g 0.33 J 32 Log. Tan. 9.96073 9. 96 067 9.96062 9.96056 9.96050 9.96045 9-96039 9.96033 9.96 028 9.96 022 9-96016 9.96 01 1 9.96005 9-95 999 9.95994 9.95988 9.95982 9-95 977 9.95971 9.95965 9-95 959 9-95 954 9-95 948 9.95942 9-95 937 9-95931 9-95925 9.95919 9.95914 9.95908 9-95 902 9.95 896 9.95891 9.95885 9-95 879 9-95 873 9.95867 9.95 862 9-95 856 9.95850 9.95 844 9-95838 9-95833 9-95 827 9-95821 d. 9.95815 9.95809 9.95 804 9.95798 9.95 792 9-95786 9.95786 9-95 774 9-95 768 9-95 763 9-95 757 9-95 751 9-95 745 9-95 739 9-95 733 9-95 72^ Log. Sin. 6 6 5 6 6 I 6 6 6 I 6 6 6 I 6 6 "dT GO 59 58 57 56 55 54 53 52 51 oO 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 P. p. 34 33 33 6 3.4 3-3 3- 7 3-9 3 9 3- 8 4-5 4 4 4- 9 5-1 5 4- 10 5-6 5 6 5- 20 II-3 II I II. 30 17.0 16 1 16. 40 22.6 22 3 22. 50 28.3 27.9 27. 28 28 3-2 3-^ 4.2 4-6 9- 14. 18. 23- 6 2.8 7 3-3 8 3.8 9 4-3 10 4.7 20 9.5 30 14.2 40 19.0 50 23-/' 2f 6 2.f 7 3-2 8 3-6 9 4.1 10 4.6 20 9.1 30 13-7 40 18.3 50 22.9 27 2.7 3-1 3.6 4.6 4-5 9.0 13-5 18.0 22.5 6 6 0.6 7 8 0.7 0.8 9 10 0.9 I.O 20 2.0 30 40 50 3-0 4.0 5.0 0.5 0.6 0.1 0.8 0.9 1-8 2.^ 3-6 4.6 P. p. 65 372 TABLE VII. LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 25" I 2 3 4 5 6 7 8 9 10 II 12 13 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 ao 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49_ oO 51 52 53 54 55 56 57 58 59 GO liOir. sill. 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 (]. 9.62 730 9.62757 9.62 784 9.62 81 I 9.62838 9.62864 9.62 89T 9.62 9I8 9.6294^ 9.62 972 9.62999 9.63025 9.63 052 9.63079 9.63 106 9.63 132 9.63159 9.63 186 9.63 212 9.63239 9.63 266 9.63 292 9-63319 9-63 34 5 9.63372 9-63 398 9.63425 9.63451 9.63478 9.63 504 9.63 530 9-63557 9-63 583 9.63609 9 63 636 9.63 662 9.63688 9.63715 9.63 741 9.63767 9-63793 9.63819 9.63 846 9.63 872 9.63898 9.63 924 9.63956 9-63976 9.64002 9.64028 9.64054 9.64086 9.64 106 9.64 132 9-64 158 9.64 184 Log. Cos. J d. 27 27 27 27 27 27 27 27 27 26 27 27 27 26 27 ^6 27 26 27 26 27 26 26 26 27 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 Log. Tan. I c d. I Lot?. Cot 9.66867 9. 66 900 9-66933 9.66 966 9.66999 9.67032 9.67 065 9-6709? 9-67 130 9-67 163 9.67 196 9.67 229 9.67 262 967294 9.67 327 9.67 360 9-67393 9-67425 9-67458 9.67 491 9.67 523 9-67 556 9.67 589 9.67 621 9.67654 9.67 687 9.67719 9.67752 9.67784 9.67 817 9.67849 9.67 882 9.67 914 9-67947 9.67979 9.68 012 9.68 044 9.68 077 9.68 109 9.68 14T 9.68 174 9.68 205 9-68 238 9.68 271 9-68 303 9-68 335 9.68 368 9. 68 400 9.68432 9.68 464 9-68497 9.68 529 9.68 561 9-68 593 9.68625 9.6865^ 9.68 690 9.68 722 9-68754 9.68786 9.68818 32 33 33 33 33 33 32 33 33 33 32 33 32 33 32 33 32 33 32 32 33 32 32 33 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 Log. Cot. I c. (1. 0.33 132 0.33 100 0.33067 0-33034 0.33001 0.32 968 0-32935 0.32 902 0.32 869 0.32836 0.32 803 0.32 771 0.32738 0.32 705 0.32 672 0.32 640 0.32 607 0.32 574 0.32 54T 0.32 509 0.32476 0.32443 0.32 41 I 0.32 378 0.32 345 0.32 313 0.32 286 0.32 248 0.32 215 0.32 183 0.32 150 0.32 118 0.32 085 0.32053 0.32 026 0.3 03 0.3 0.3 0.3 0-3 0.3 0.3 3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0-3 0.3 0.3 988 955 923 891 858 826 793 761 729 6% 66if 632 600 56? 535 503 471 439 406 374 342 310 278 246 214 T82" Log. Tan. Locr. Cos. (1. 9.95 727 9.95721 9.95716 9.95710 9.95704 9.95698 9.95692 9.95 686 9.95 686 9.95674 9.95668 9.95 662 9-95656 9.95656 9-95644 9-95 638 9.95632 9.95627 9.95 621 9.95615 9.95609 9.95 603 9-95 597 9-95 591 9-95 585 9-95 579 9-95 573 9.95 567 9.95 561 9-95 555 9-95 549 9-95 543 9-95 537 9-95 530 9-95 524 9 95 518 9.95512 9-95 506 9 95 506 9-95 494 9-95488 9.95482 9.95476 9.95 470 9.95464 9-95458 9.95452 9-95 445 9-95 439 9-95 433 9.95427 9.95421 9.95415 9-95409 9-95403 9-95 397 9-95 390 9.95 384 9-95 378 9-95 372 9-95 366 Log. Sin. I" (iO 59 58 57 56 30 29 23 27 26 25 24 23 22 21 20 ^9 18 17 _i6 IS 14 13 12 II 9 8 7 6 p. I*. 27 6 7 8 9 10 20 30 40 50 18 22. ^ 33 32 32 6 3-3 3-2 3-2 7 3-8 3-8 3-^ 8 4.4 4-3 4.2 9 4-9 4.9 4-8 10 5-5 5-4 5-3 20 II. 10.8 10.6 30 16.5 16.2 16.0 40 22.0 21.6 21.3 50 27.5 27.1 26.6 2S 26 25 6 2.6 2.6 2. 7 8 3 3 I 3 3 4 3- 3- 9 4 3 9 3- 10 20 4 8 4 8 4 8 3 6 4- 8. 30 13 2 13 12. 40 17 6 17 3 17. 50 22 I 21 6 21. 6 7 8 9 10 20 30 40 50 8 6 5 0.5 0.6 0.7 0.8 0.9 1-8 2.f 3-6 4-6 0.6 0.6 0.7 0.7 0.8 0.8 I.O 0.9 1. 1 1.0 2.1 2.0 3-2 3-0 4-3 4.0 5-4 5.0 i P. 1'. G4' 373 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS 26° 10 II 12 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 GO Lo^. Sill. d. 64 184 64 210 64236 64 262 64 287 64313 64339 64365 64391 644I6 64442 64468 64493 64519 64545 64576 64596 64622 6464^ 64673 64698 64724 64749 64775 64 800 64826 64851 64876 64902 64927 64952 64978 65 003 65028 65054 65079 65 104 65 129 65 155 65 180 65 205 65 230 6525.^ 65 286 65305 65331 65356 65381 65 406 65431 65456 65481 65 506 65530 65555 65 586 65605 65 630 65655 65 680 9^65704 26 26 26 25 26 26 25 26 25 26 25 25 26 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 24 25 25 25 24 25 25 24 Log. Cos. i d. Loff. Tan. c. d 69615 69647 69678 69 716 69742 68818 68850 68882 68 914: 68946 68978 69016 69042 69074 69 106 69138 69 170 69 202 69234 69 265 69 297 69329 69 361 69393 69425 69456 69488 69 520 69552 69583 69773 69805 69837 69868 69 900 69931 69 963 69994 70026 70058 70089 70 121 70152 70183 70215 70246 70278 70309 70341 70372 70403 70435 70466 70497 70529 70 566 70591 70623 70654 70685 70716 Log. Cot. 32 32 32 32 32 32 32 31 32 32 32 32 32 31 32 32 31 32 32 31 32 31 32 31 32 31 31 32 31 31 32 31 31 31 31 31 31 32 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 TTd. Log. Cot. 0.31 182 0.31 150 0.31 11^ 0.31085 0.31053 0.31 021 0.30989 0.3095? 0.30 926 0.30894 0.30 862 0.30830 0.30798 0.30 766 0.30734 o. 30 702 0.30 676 0.30639 o. 30 607 0.30575 0.30543 0.30 511 0.30480 o. 30 448 0.30 4I6 0.30384 0.30353 0.30321 0.30 289 0.30 258 0.30226 0.30194 0.30163 0.30 1 31 0.30 100 0.30068 0.30037 0.30005 0.29973 0.29 942 0.29 910 0.29 879 o. 29 847 0.29 8 16 0,29 785 0.29753 0.29 722 o. 29 696 0.29 659 0.29 628 0.29596 0.29 565 0.29533 0.29 502 0.29471 Log. Cos. 0.29439 O.29408 0.29377 0.29346 0.29314 0.29 283 Log. Tan. 95366 95360 95 353 95 34? 95341 95 335 95329 95323 95316 95310 95304 95298 95292 95285 95279 95273 95267 95 266 95254 95248 95242 95235 95229 95223 95 217 95 210 95204 95 95 95 95 95 95 95 95 95 95 95 95 95 98 91 85 79 73 66 60 54 47 41 35 28 22 16 09 03 95097 95090 95084 95078 95071 95065 95058 95052 95046 95039 95033 95026 95 020 95014 9500? 95 001 94 994 9-94988 Log. Sin. d. 00 59 58 57 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 p. p. 32 6 3-2 7 3.8 8 4-3 9 4.9 10 5.4 20 10.8 30 16.2 40 21.6 50 27.1 32 3-2 3-7 4.2 4.8 5-3 10.6 16.0 21.3 26. s 6 7 8 9 10 20 30 40 50 31 3-1 3-7 4.2 4-7 5.2 10.5 I5-? 21.0 26.2 31 3-1 3 4 4 5 10 15 20 25 26 25 6 2.6 2.5 7 3-0 3-0 8 3-4 3-4 9 3-9 3-8 10 4.3 4.2 20 8.6 8.5 30 13.0 i2.^ 40 17-3 17.0 50 21.6 21.2 24 § 6 2.4 0.6 7 2-8 0.7 8 3-2 0.8 9 37 i.o 10 4.1 I.I 20 8.T 2.1 30 12.2 3-2 40 16.3 4-3 50 20.4 5-4 25 2.5 2.9 3-3 3-? 4.1 8.3 12.5 16.6 20.8 6 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 P. P. ■ o 374 TABLE VII. LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. Lo;?. Sin. 9.65 704 9.65729 965754 9.65779 9-65 80 3 9.65828 9.65853 9.65 878 9.65 902 9.65927 9.65 95T 9-65 976 9.66001 9.6602^ 9.66 050 9.66074 9. 66 099 9.66 123 9.66 1 48 9.66 172 9 66 197 9.66 221 9.66 246 9.66 276 9.66 294 9.66 319 9-66343 9.66 367 9.66 392 9.66416 9. 66 440 9.66 465 9.66489 9.66513 9.66 537 9.66 561 9.66 586 9.66 610 9.66634 Q 66 658 9.66682 9-66705 9.66 730 9.66754 9.66778 9.66 802 9.66 825 9.66 850 9.66874 9.66898 9.66 922 9.66946 9.66 976 9-66994 9 67 018 (1. 9.67 042 9.67 066 9.67089 9.67 1 13 9-67 137 9.67 161 Log. Cos. I (1. 25 24 25 24 25 24 25 24 24 24 25 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 23 24 24 24 23 24 23 24 liOff. Tan. c. d. 1 Lot?, lot. 70716 70748 70779 70810 70 841 70872 70903 70935 70 966 70997 028 059 090 121 152 183 214 24^ 276 307 338 369 400 431 462 493 524 555 586 617 647 678 709 740 771 80 f 832 863 894 925 955 986 72 017 72047 72 078 72 109 72 139 72 170 72 201 72231 72 262 72 2G2 72323 72354 72384 72415 7244? 72476 72 5O6 72 537 9-72 567 Lot,'. Cot. 31 31 31 31 31 31 31 o' ^I 31 31 31 30 31 31 31 30 31 31 30 31 30 31 31 33 31 33 30 31 30 31 30 30 30 31 30 30 30 30 31 30 30 30 30 30 30 30 0.29 283 0.29 252 0.29 221 0.29 190 0.29 158 0.29 127 o. 29 096 0.29065 0.29034 o. 29 003 0.28 972 0.28 946 0.28 909 0.28 878 0.28847 0.28 815 0.2878^ 0.28754 0.28 723 0.28 692 0.28 661 0.28 636 0.28 599 0.28 568 0.28537 0.28 505 0.28 476 0.28 445 0.28 414 0.28383 0.28 352 0.28 321 0.28 290 0.28 260 0.28 229 0.28 198 0.28 167 0.28 136 0.28 106 0.28 075 o. 28 044 0.28 014 0.27983 0.27 952 0.27 921 0.27 891 0.27 860 0.27 830 0.27799 0.27 768 0.27 738 0.27707 0.27 677 0.27 646 0.27 615 0.27 585 0.27 554 0.27 524 0.27493 0.27 463 0-27432 c. (1. I Log. Tun. Lot;. Co.s. 94988 94981 94 975 94969 94962 94956 94 949 94 943 94 936 94930 94923 94917 94910 94904 9489? 94891 94884 94878 94871 94865 94858 94852 94845 94839 94832 94825 94819 94812 94806 94 799 94 793 94786 94 779 94 773 94766 94760 94 753 94 746 94740 94733 94727 94720 94713 94707 94706 94693 94687 94680 94674 94667 94 660 94654 94647 94 646 94633 94627 94 620 94613 94 607 94 600 9-94 593 Loir. Sin. 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 ~^ 24 23 21 20 19 18 17 16 15 14 13 12 1 1 To 9 8 7 6 r. i\ 3 I 3 I 6 3-1 3-1 7 3 7 3 6 8 4 4 I 9 4 7 4 6 10 5 -7 5 I 20 10 5 10 3 30 15 1 15 5 40 21 20 6 50 26 2 25 8 25 6 2. 7 2. 8 J- 9 3- 10 4- 20 8. 30 12. 40 16. 50 20. 30 3-3 4.6 4-6 5-1 10. r 15.2 20.3 25.4 24 24 2j 6 2.4 2.4 2. 7 8 9 2 3 8 2 7 2.8 3-2 3-6 2. 3- 10 20 4 8 I I 8.0 3- 7- 30 12 2 12.0 11. 40 16 3 16.0 15- 50 20 4 20.0 19. 6 7 8 9 10 20 3^ 40 50 I'. !• 0.7 0.6 0.6 0.8 0.9 0.7 o-S 0.7 o.S I.O 1.0 0.9 i.T 1. 1 1.0 2.3 3-5 4.6 2.1 3-2 4-3 2.0 3-0 4.0 5-8 5-4 5.0 63 375 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 28" 15 16 17 18 19 20 21 22 23 I ^ 25 26 27 ! 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 ! 51 52 53 54 55 56 57 58 59 60 Los. Siu. 9.67 161 9.67 184 9.67 208 9.67 232 9.67 256 9.67279 9.67 303 9.67 327 9.67356 9-67 374 9.67 397 9.67421 9.67445 9.67468 9.67492 9.67515 9-67 539 9.67 562 9.67 586 9.67 609 967633 9.67 656 9.67679 9.67703 9.67 726 9.67 750 9.67773 9-67 796 9.67 819 9.67 843 9.67 866 9.67889 9.67913 9.67936 9.67959 9.67 982 9.68005 9.68 029 9.68 052 9.68075 9.68098 9.68 12T 9.68 144 9.68 167 9.68 190 9.68 213 9.68236 9 68 259 9.68282 9.68305 9.68328 9.68351 9.68374 9.68 397 9.68 420 9.68443 9.68466 9.68488 9.68 51T 9-68 534 9-68 557 Log. Cos. d. 23 24 23 24 23 23 24 23 23 23 24 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 22 23 23 23 22 23 23 22 "dT Loff. Tail. c. d 9.7256; 9.72 598 9.72628 9.72659 9.72 689 9.72719 9.72750 9.72 780 9.72 811 9.72841 9.72 871 9.72 902 9.72932 9.72 962 9.72993 9.73023 9-73053 9.73084 9-73 114 9-73 144 9-73 ^74 9.73205 9-73235 9.73265 9-73295 9.73 325 973356 9-73386 9.73416 9-73 446 9-73 476 9-73 506 9-73 536 9-73567 9-73 597 9.73627 9-73657 9-73687 9-73717 9-73 747 9-73777 9.73807 9-73837 9.73867 9-73897 9.73927 9.73957 9-73987 9.74017 9.74047 9- 74076 9.74 106 9-74 136 9.74 166 9.74 196 9.74 226 9-74256 9.74 286 9-74315 9-74 34g 9-74 375 Log. Cot. 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 29 30 30 30 29 30 29 Log. Cot. 0.27432 0.27 402 0.27371 0.27341 0.27 311 0.27 286 0.27 250 0.27 219 0.27 189 0.27 159 0.27 128 0.27098 0.27067 0.2703; 0.27 007 0.26976 0.26946 0.26 916 0.26886 0.26855 0.26 825 0.26 795 0.26 765 0.26734 o. 26 704 0.26674 o. 26 644 0.26 614 0.26 584 0.26553 0.26 523 0.26493 0.26463 0.26433 0.26 403 0.26373 0.26343 0.26313 0.26 283 0.26 253 0,26 223 0.26 193 0.26 163 0.26 133 0.26 103 0.26 073 o. 26 043 0.26 013 0.25 983 0.25953 0.25923 0.25893 0.25 863 0.25 833 0.25 804 0.25774 0.25744 0.25 714 0.25 684 0-25654 0.25 625 c. d. i Log. Tan. Los. Cos. 9.94 593 9.94587 9.94580 9.94 573 9.94 566 9.94560 9 94 553 9-94 546 9-94 539 9-94 533 9.94 526 9.94519 9.94512 9.94 506 9-94 499 9.94492 9-94485 9-94 478 9-94472 9-94465 9-94458 9-94451 9-94 444 9-94 437 9-94 431 9.94424 9.94417 9.94416 9-94403 9-94 396 994390 9-94383 9-94376 9-94369 9.94 362 9-94 355 9-94 348 9-94 341 9-94 335 9.94328 9.94321 994314 9-94307 9.94306 9-94295 9.94286 9.94279 9.94272 9.94265 9-94258 9.94251 9-94 245 9.94238 9.94231 9.94224 9.94217 9.94 210 9-94203 9.94 196 9-94 189 9.94 182 Log. Siu. 45 44 43 42 41 40 39 38 37 36 35 34 33 3^- 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 P. P. 30 30 29 6 7 8 9 10 20 30 40 50 24 2.4 2.8 3-2 3-6 4.0 8.0 12.0 16.0 20.0 6 3.0 3-0 -7 7 3-5 3-5 3- 8 4.0 4.0 3- 9 4-6 4-5 4- 10 5-1 5.0 4- 20 10. 1 lo.o 9- 30 15.2 15.0 14. 40 20.3 20.0 19. 50 25.4 25.0 24. 23 23 6 2-3 2.3 7 2.7 2.7 8 3-1 3-0 9 3-5 3-4 10 3-9 3-8 20 7-8 7-6 30 II. 7 II. 5 40 15-6 15-3 50 19.6 19. 1 22 2.2 2.6 3-0 3-4 3-? 7.5 II. 2 15.0 18.7 6 7 8 9 10 20 30 40 50 7 0.7 0.8 0.9 1.6 i-i 2.3 3-5 4-6 5-8 0.6 0.7 0.8 i.o I.I 2.t 3-2 4.3 5-4 P. P. Gl 376 TABLE VII. LOGARITHMIC SINES, COSINES, TANGENTS, AND CO TANGENTS, 2\y I 2 4 5 6 7 8 9 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 -3 26 27 28 29 ;io 3^ 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 liOc:. Sin. 9.68557 9.68 580 9.68 602 9.68625 9.68648 9.68671 9.68 693 9-68716 9.68739 9.68 76T d. 9.68 784 9.68807 9.68 829 9.68852 9.68 874 9.68897 9.68 920 9.68 942 9.68 965 9.68 987 9.69 010 9.69 032 9.69055 9.69077 9.69099 9.69 122 9.69 144 9.69 167 9.69 189 9.69 211 9.69 234 9.69 256 9.69278 9.69 301 9.69323 9 69 34^ 9.69367 9.69390 9.69412 9-69434 9-69 456 9.69478 9.69 506 9.69523 9.69 545 9-69 567 9-69 589 9.69 61T 9.69633 9.69655 9.69677 9.69699 9.69 721 9-69743 9.69765 9-69787 9.69 809 9.69 831 9.69853 9-69875 9-69897 Log. Cos. 23 22 -3 22 23 23 23 22 22 22 22 22 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 21 22 22 "dT Los. Tan. led. 9-74 375 9.74405 9-74 435 9.74464 9-74 49-1 9.74524 9.74554 9-74583 9.74613 9- 74 643 9.74672 9.74702 974732 9.74761 9.74791 9.74821 9-74850 9. 74 880 9.74909 9-74 939 9-74969 9-74 998 9.75 028 9.75057 9.75087 9-75 116 9.75 146 9-75 17^ 9.75205 9-75 234 9.75 264 975293 975323 9-75352 9.75382 9.75 411 9.75441 9.75470 9-75 499 9.75 529 9-75 558 9.75 588 9.75617 9.75 646 9.75676 9.75705 9-75 734 9-75764 9-75 793 9.75 822 9.75851 9.75881 9.75910 9-75 939 9.75968 9-75998 9.76027 9.76056 9.76085 976 115 9.76 144 30 30 29 30 29 30 29 29 30 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 Log. Cot. I c. d. Lo«?. Cot. l-(u:. Cos. d. 7 7 7 7 7 7 7 7 ? 7 7 7 7 7 7 7 1 7 7 7 7 7 7 7 7 1 7 7 7 1 7 7 1 7 0.25625 9.94 182 ()0 0.25 595 9.94175 59 0.25 565 9.94 168 58 0-25 53^ 9.94 161 57 0.25505 9.94154 56 0.25476 9.94 147 55 0.25 446 9.94 140 54 0.25 416 9-94133 53 0.25387 9.94126 52 0.25357 0.25 327 9-94 118 51 50 9.94111 0.25 297 9.94 104 49 0.25 268 9-9409? 48 0.25 238 9-94090 47 0.25 208 9.94083 9- 94076 46 45 0.25 179 0.25 149 9.94069 44 0.25 120 9.94062 43 0.25 090 9.94055 42 0. 2 5 060 9.94048 41 0.25 031 9-94041 40 0.25 001 9-94034 39 0.24972 9-94026 3a 0.24 942 9.94019 37 0.24913 9.94012 36 35 0.24883 9.94005 0.24854 9.93998 34 0.24 824 9.93991 33 0.24795 9-93984 32 0.24 765 9-93 977 3» 0.24736 9.93969 ao 0.24 706 9.93 962 29 0.24677 9-93 955 28 0. 24 64^ 9-93948 27 0.24 618 9-93941 / 7 26 25 0.24588 9-93 934 0.24559 9.93926 7 7 1 7 7 1 7 7 7 1 7 1 7 7 1 7 1 1 7 1 7 1 24 0.24529 9-93919 23 0.24 500 9.93912 22 0.24471 9-93905 21 0.24441 9.93898 20 0.24 412 9.93891 J9 0.24383 9-93883 18 0.24353 9-93876 17 0.24324 9.93 869 9.93 862 i« 0.24 295 15 0.24 265 9-93854 14 0.24 236 9-9384? 13 0.24 207 9.93840 12 0.2417^ 9-93833 11 10 9 0.24 148 0.24 119 9.93 826 9-93 818 0. 24 090 9-93811 8 0. 24 066 9.93 804 7 0.24031 0.24002 9-93 796 6 993789 5 0.23973 9.93782 4 0.23943 9-93 775 3 0.23914 9.93767 2 0.23885 9-93766 7 1 d. I 0.23 856 9-93 753 Loff. Tun. !.(!g. sill. / r. I* 7 8 9 10 20 30 40 50 30 3.0 3.5 4.0 4.5 5.0 lo.o 15.0 20.0 25.0 29 2.9 3-4 3-9 4 4 4.9 9.8 14.? 19.6 24.6 23 6 2-3 7 2 7 8 3 9 3 4 10 3 8 20 7 6 30 1 1 5 40 15 3 50 19 I 22 22 6 2.2 2.2 7 2.6 2.5 8 3.0 2.9 9 3-4 3-3 10 3-7 3-6 20 7.5 7.3 30 II. 2 II. 40 15.0 14-6 50 18.7 18.3 29 2.9 3 3 4 4 9 14 19 24 21 2.T 2.5 2-8 3.2 3.6 7-1 10.? 14-3 17.9 6 7 8 9 10 20 30 40 50 1 0.1 0.9 i.o 1. 1 1.2 2.5 3-? 5.0 6.2 7 0.7 0.8 0.9 1.6 i.i 2-3 3-5 4-6 5.8 p. P. GO 377 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 10 1 1 12 13 14 15 16 17 19 -:> 26 27 28 29 ao 31 32 33 34 35 36 37 38 39 40 42 43 44 55 56 57 58 59 <>0 30 20 9 21 9 22 9 23 9 24 9 Lost. sin. 69897 69919 69 940 69 962 69 984 70006 70028 70050 70071 70093 70 115 70137 70158 70 180 70 202 70 223 70245 70 267 70288 70310 70331 70353 70375 70396 70418 70439 70461 70482 70504 70525 70547 70568 70590 70 61 1 70 632 70654 70675 70696 70718 70739 70 760 70782 70803 70 824 70846 70867 70888 70 909 70930 70 952 70973 70994 7101I 71036 71 05^ 71 078 71099 71 121 71 142 71 163 71 184 hog. Cos. d. 22 21 22 22 21 22 22 21 22 21 22 21 21 22 21 21 22 21 21 21 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 "(IT Loir. Tan. 76 144 76 173 y6 202 76 231 76 266 76 289 76319 76348 76377 76406 76435 76464 76493 76 522 76551 76580 76 609 76638 76667 76696 76725 76754 76783 76812 76 841 76 870 76899 76 928 76957 76986 77015 77043 77 072 77 lOI 77 130 77 159 77 188 77217 77245 77274 77303 77332 77361 77389 77418 77 447 77476 77 504 77 533 77 562 77 591 77619 77648 77677 77705 77 734 77763 77791 77 820 77849 77 87^ Log. Cot. 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 29 28 29 29 29 28 29 29 28 29 29 28 29 28 29 28 29 28 29 28 29 28 28 29- 28 28 29 28 28 29 28 liOg. Cot. 0.23 856 0.23 827 0.23797 0.23768 0.23739 0.23 710 0.23 681 0.23 652 0.23 623 0.23 594 0.23 565 0.23 535 0.23 506 0.23477 0.23 448 0.23419 0.23 396 0.23 361 0.23332 0.23303 0.23 274 0.23245 0.23 216 0.23 18^ 0.23 158 0.23 129 0.23 lOI 0.23 072 0.23043 0.23 014 0.22 985 0.22 956 0.22 92^ 0.22 898 0.22 869 0.22 841 0.22 812 0.22 783 0.22 754 0.22 725 0.22 696 0.22 668 0.22 639 0.22 616 0.22 581 0.22553 0.22 524 0.22 495 0.22 466 0.22 438 0.22 409 0.22 386 0.22 352 0.22 323 0.22 294 0.22 266 0.22 237 0.22 208 0.22 180 0.22 151 0.22 122 Log. Tail. Los:. Cos. 9-93 753 93746 93 738 93731 93724 93716 93709 93 702 93694 93687 93680 93 672 93665 95658 93656 93643 93635 93628 93621 93613 93606 93 599 93591 93584 93 576 93569 93562 93 554 93 547 93 539 93532 93524 93517 93509 93502 93 495 93487 93480 93472 93465 93 457 93450 93442 93 435 93427 93420 93412 93405 93 39? 93390 93382 93 374 93367 93 359 93352 93 344 93 337 93329 93321 93314 9-93 306 Log. Sill. d. 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 p. p. 22 21 6 2.2 2.T 7 2.5 2.5 8 2.9 2.8 9 3-3 3-2 10 3-6 3-6 20 7-3 7.1 30 II. 10.7 40 14-6 14.3 50 18.3 17.9 21 2.1 2.4 2.8 3-1 3-5 7.0 10.5 14.0 i7'S 8 1 6 0.8 O.J 7 8 0.9 1.6 0.9 I.O 9 10 1.2 I-.3 I.I 1.2 20 2-6 2.5 30 4.0 Z-1 40 50 5.3 6.6 5.0 6.2 7 0.7 0.8 0.9 1.6 I.I 2.3 3-5 4--6 5.8 29 29 28 6 2.9 2.9 2.8 7 3-4 3.4 3-3 8 3-9 3-8 3-8 9 4.4 4.3 4-3 10 4.9 4-8 A-1 20 9-8 9-6 9-5 30 14-7 14.5 14.2 40 19-6 19-3 19.0 50 24.6 24.1 23.^ p. p. 59 378 TABLE VII. — LOGARITHMIC SINES. COSINES, TANGENTS, AND COTANGENTS. «> i o O 1 5 6 7 8 9 10 II 12 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 oO 51 52 53 54 55 56 57 58 -59_ 00 liOsr. Sill. d. 9-7 9-7 9-7 9-7 9.7 9.7 9-7 9.7 9-7 9-7 9.7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9.7 9-7 9-7 9-7 9-7 9.7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 9-7 184 205 226 247 268 289 310 331 351 372 393 414 435 456 477 498 518 539 560 581 601 622 643 664 684 705 726 746 767 788 808 829 849 870 891 911 932 952 973 993 9.72 014 9.72034 9.72055 9.72075 9.72 096 9.72 116 9-72 136 9.72 157 9.72 177 9.72 198 9.72 218 9.72 238 9.72259 9.72279 9.72299 9.72319 9.72340 9.72360 9.72380 9.72 400 9.72 421 21 21 21 21 21 21 21 20 21 21 21 20 21 21 21 20 21 20 21 26 21 20. 21 26 21 20 26 21 20 26 20 26 21 20 20 20 26 26 20 20 20 26 20 20 20 20 26 26 26 20 26 20 20 26 20 26 20 20 20 20 I,otr. Tail. c. d. -otr. Cot. 9.77^77 9.77906 9-77 934 9-77963 9.77992 9.78 020 9.78049 9.78077 9.78 106 9-78 134 9.78 163 9.78 19T 9.78 220 9.78 248 9-78277 Log. Cos. i iU 9.78305 9-78334 9.78362 9.78391 9.78419 9.78448 9-78476 9.78505 9-78533 9.78561 9.78590 9-78618 9.78647 9-78675 9.78703 9.78732 9.78 760 9.78788 9.78817 9.78845 9.78873 9.78 902 9.78930 9-78958 9.78987 9.79015 9-79043 9.79071 9.79 100 9.79 128 9-79156 9-79 184 9.79213 9.79241 9.79269 9.79297 9-79325 9-79 354 9.79382 9.79410 9-79 438 9.79465 9.79494 9-79522 9-79551 9.79 579 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 O. 22 I 22 0.22 094 0.22 065 0.22 037 0.22 008 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0,2 0.2 02 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 979 951 922 894 865 837 808 780 751 723 694 666 637 609 586 552 523 495 467 438 410 381 353 325 296 268 239 211 183 154 126 098 070 041 013 0.20 985 0.20 956 0.20 928 o. 20 900 0.20 872 0.20 843 o. 20 8 1 5 0.20 787 0.20 759 0.20 731 0.20 702 0.20 674 o. 20 646 0.20618 0.20 590 0.20 561 0.20533 0.20 505 0.20477 o. 20 449 0.20 421 l.oir. Cos liO^. C<»t. ' c. «1. I, otr. Tiin. 93 306 93 299 93291 93 284 93276 93268 93 261 93253 93245 93238 93230 93 223 93215 93207 93 200 93 192 93 184 93 ^77 93 169 93 161 93 153 93 146 93 138 93 ^'30 93 123 93 115 93 107 93 100 93 092 93084 93076 93069 93 061 93053 93045 93038 93030 93 022 93014 93 006 92 999 92991 92983 92975 9296? 92 960 92952 92944 92936 92928 92 920 92913 92905 92 897 92889 92 881 92873 92865 92858 92 850 92 842 9 9 9 9 9_ LoJT. sill. 0 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 -:> 24 23 22 21 10 19 18 17 16 15 14 13 12 1 1 9 8 7 6 r. 28 21 6 2.8 2.1 7 3-2 3.2 8 3-? 3-6 9 4.2 4.1 10 4-6 4.6 20 9-3 9.1 30 14.0 13-7 40 18.6 18.3 50 23-3 22.9 27 2.7 3-1 3-6 4.6 4-5 9.0 13-5 18.0 22.5 19 19 6 1.9 1.9 7 2-3 2.2 8 2.6 2.5 9 2.9 2-8 10 3-2 3-1 20 6.5 6.3 30 9-? 9.5 40 13.0 12.6 50 li5.2 15.8 18 1-8 2.1 2.4 2.8 3-1 6.T 9.2 12.3 15.4 6 7 8 9 10 20 30 40 SO 8 0.8 i.o I.I 1.3 1-4 2-8 4-2 5-6 7-1 8 0.8 0.9 1.6 1.2 1-3 2.6 4.0 5-3 6.6 r. I'. 5(j 3*1 iAiiLl. V 11. -LOGARITHMIC SiNES, COSINES, TANGENTS, AND COTANGENTS 34 24 Log. Sill. 9-74756 9-74 775 9 74 793 9.74812 9.74831 9.74849 9.74868 9-74887 9.7490? 9.74924 9-74 943 9.74961 9.74980 9-74 998 9.75017 9-75036 9-75054 9.75073 9.75091 9-75 no 9-75 128 9-75 147 9.75 165 9.75 184 9.75 202 9.75 221 9-75239 9-75 257 9.75276 9-75 294 9-75313 9-75331 9-75 349 975368 9-75386 9-75404 9-75423 9-75441 9-75 459 9-75478 9-75496 9-75 54 9-75 532 9-75551 9.75569 9-75587 9.75605 9.75623 9-75642 9.75 660 9-75678 9-75695 9-75714 9-75 732 9-75750 9-75769 975787 9.75805 9-75823 9-75841 975859 liOf?. Cos. (1. 19 18 19 18 18 19 18 19 18 18 18 18 18 19 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 IS Log. Tan, c. d. I Log. Cot. 9.82898 9.82 926 9-82953 9.82 980 9.83007 9-83035 9.83 062 9.83 089 9-83 116 9-83 143 9.83 171 9.83198 9.83 225 9-83252 9-83279 9-83307 9-83334 9-83361 9.83388 9.83415 9.83442 9.83469 9-83496 9-83524 9-83551 9.83578 9.83 605 9-83632 9-83659 9-83686 9-83713 9.83740 9.83767 9-83794 9.83821 9-83848 9.83875 9.83 902 9-83929 9.83957 9.83984 9.84 01 1 9-84038 9.84065 9.84091 9.84 118 9-84 Hi 9.84 172 9.84199 9-84225 9.84253 9.84 286 9-84 307 9-84334 9-84361 9-84388 9.84415 9-84442 9.84469 9.84496 27 27 27 27 27 27 27 27 27 2f 27 27 27 27 27 27 27 27 2f 27 27 27 27 27 27 27 27 2? 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 27 27 27 27 27 26 27 27 27 27 27 26 Log. Cot. I 0. d. 9.84 522 0.17 lOI 0.17074 0.17 047 0.17 019 o. 16 992 0.16 965 0.16938 0.16 916 0.16883 0.16 855 0.16 829 0.16 802 0.16774 0.16 747 o. 16 720 o. 16 693 0.16 666 o. 16 639 0.16 612 0.16 584 0.16557 0.16 536 0.16 503 0.16 476 o. 1 6 449 0.16422 o. 16 395 0.16368 0.16 346 o. 16 313 0.16 285 o, 16 259 o. 16 232 0.16 205 0-16 178 0,16 151 0.16 124 o. 16 097 o. 16076 o. 16 043 O.I60I6 o. 1 5 989 o. 1 5 962 0-15935 o. 1 5 908 0.15 881 0.15854 0.15 827 o. 1 5 800 0.15773 0.1 5 746 0.15719 o. 1 5 692 0.15 665 0-1563 9 0.15 612 0.15 585 0.15 558 0.15 531 0.15504 Log. Cos. 9.91 857 9.91 849 9.91 846 9.91 832 9.91 823 9.91 814 9.91 806 9.91 79^ 9.91 789 9.91 786 9.91 772 9.91 763 9-91 755 9.91 746 9-91 737 9.91 729 9.91 720 9.91 712 9.91 703 9.91 694 9.91 686 9.91 677 9.91 668 9.91 660 9.91 651 9.91 642 9.91 634 9.91 625 9.91 616 9.91 608 9.91 599 9.91 590 9.91 582 9-91 573 9.91 564 9.91 556 9-91 547 9 91 538 9.91 529 9.91 521 9.91 512 9.91 503 9.91 495 9.91 486 9.91 477 9.91468 9.91 460 9-91451 9.91 442 9.91433 9.91424 9.91 416 9.91407 9.91 398 991 389 9.91 386 9.91 372 9-91 363 9-91 354 9-9t 345 0.15 47f 9-91336 Log. Tan. | Log. Sin. P. P. 6 7 8 9 10 20 30 40 50 2f 2.7 3-2 3-6 4.1 4.6 9.1 13-? 18.3 22.9 27 2.7 3-1 3-6 4.6 4-5 9.0 13-5 18.0 22.5 19 18 6 1-9 1.8 7 2.2 2.1 8 2.5 2.4 9 2.8 2.8 10 3-1 .3.1 20 6.3 6.1 30 9-5 9.2 40 12.6 12.3 50 15-8 15.4 I 9 10 20 30 40 50 9 0.9 1.6 1.2 1-3 1-5 3-0 4-5 6.0 7.5 26 2-6 3-1 3-S 4.0 4.4 8.8 13.2 17-6 22.1 18 1.8 2.1 2.4 2.7 30 6.0 9.0 12.0 15.0 8 0.8 o I 3 4 P. P. ai> 382 TABLE VII.-LOGARITHMIC SINES. COSINES. TANGENTS. AND COTANGENTS ;}5 Lop. Sill. 9-75^59 9-75877 9.75895 9-75913 9-75 931 9-75 949 9.75967 9.75985 9.76003 9.76021 9.76039 9.76057 9.76075 9.76092 g.'jd no 19 9.76 128 9.76146 9.76 164 9.76 182 9. 'j6 200 9.76 21^ 9.76235 9.76253 9.76 271 9.76 289 9-76306 9.76324 9.76342 9.76 360 9-7637? 9-76395 9.76413 9.76431 9-76443 9.76466 9-76484 9.76 501 9.76519 9-76536 976554 9.76572 9765S9 9.76607 9 76 624 9.76 642 9.76 660 9.76677 9.76695 9.76 712 9.76730 9-76747 9.76765 9.76 782 9.76 800 9.76 8if 9-76835 9.76 852 9.76 869 9.76887 9.76904 9.76 922 18 18 18 18 18 18 18 18 18 18 18 18 If 18 18 iS If 18 18 If 18 18 If 18 If 18 If 18 If 18 If 18 If If 18 If If If 18 If If If 17 18 17 If If If If If If If If If If 17 If If If If fiOe. Tiin. r. d. 9 84522 9-84549 9-84576 9. 84 603 9.84630 9.84657 9.84684 9.84 71 1 984 73f 9.84764 9.8479! 9.84818 9.84845 9.84871 9.84898 9.84925 9.84952 9.84979 9.85 005 9-85032 9.85059 9.85 086 9-85 113 985 139 9.85 166 9.85 193 9.85 220 9.85246 9.85273 9.85 300 9.85327 985353 9.85 380 9.85407 9-85433 9.85 466 9-85487 9-85513 9.85 540 9-85567 594 9.85 9.85 620 9.85647 9-85673 9.85 700 9.85727 9-85753 9.85780 9.85 807 9.85833 9.85 860 9.85887 9.85913 9.85940 9.85966 Log. Cos. I d. 985993 9.86 020 9.86046 9.86073 9. 86 099 9.86 126 27 27 27 26 27 27 27 26 27 27 26 27 26 27 27 26 27 26 27 27 26 27 26 27 26 27 26 27 26 27 26 26 27 26 27 26 26 27 2S 27 26 26 26 27 2S 26 27 26 2S 26 27 26 26 26 26 27 26 26 26 26 Lotr. Cot. o-i5 47f 0.15456 0.15423 0-15 396 o. 1 5 370 0-15343 O.I53I6 o. 15 289 o. 15 262 0.15235 0.15 208 0.15 182 0.15 155 0.15 128 0.15 loT o. 1 5 074 o. 1 5 048 o 1 5 02 1 o. 14 994 o. 14 967 Loir. Cos. 9-9' 336 9.91 327 9.91 318 9.91 310 9.91 301 9.91 292 9.91 283 9.91 274 9.91 265 9-91 256 9-91 24f 9.91 239 9.91 230 9.91 221 9.91 212 9.91 203 9.91 194 o. 1 4 940 0.14 914 0.14887 0.14 866 0.14833 9.91 9-91 9.91 185 176 167 0.14807 0.14 780 0-14753 0.14726 o. 14700 o. 14 673 0.14646 o. 14620 0.14593 0.14566 9.91 9.91 9.91 9.91 9.91 158 149 146 131 122 9.91 113 9.91 104 9-91 095 9.91086 9.91 077 0.14539 0.14513 0.14486 0.14459 0-14 433 o. 1 4 406 0.14379 o- 14 353 0.14326 o. 14 299 9.91 068 9.91 059 9.91 056 9.91 041 9.91 032 9.91 023 9.91 014 9.91 005 9.90996 9.90987 0.14 273 o. 14 246 0.14 219 0.14 193 0.14 166 0.14 140 0.14 113 o. 14 086 o. 1 4 060 0.14033 9.90978 990969 9. 90 960 9.90951 9.90942 990933 9.90923 9.90914 9.90905 9-90896 Log. Cot. I c. (1. o. 14007 o. 1 3 980 0.13953 0.13927 o. 13 906 Log. Tan. 9.90887 9.90878 9. 90 869 9. 90 860 9.90 856 9.90 841 9-90832 9-90823 9.90814 9.90 805 9.90796 9 9 8 9 9 8 9 9 9 9 8 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 Log. Sill. 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 24 23 22 21 "20 19 18 17 16 15 14 13 12 1 1 To 9 8 7 _6 5 4 1*. i' 6 7 8 9 10 20 30 40 50 27 2.7 3-1 3-6 4.6 4-5 9-0 13 5 18.0 22.5 28 6 7 8 9 10 20 30 40 50 18 1.8 2.1 2.4 2.7 30 6.0 9-0 12.0 15.0 If i-f 2.6 2-3 2.6 2.9 5.8 ir.6 14.6 17 1-7 2.0 2.2 2.5 2-8 5.6 8.5 II-3 14. 1 6 7 8 9 10 20 30 9 0.9 1. 1 1.2 1.4 1.6 3-1 4.f 40 6.3 9 0.9 1.6 1.2 1-3 1-5 3-0 4-5 8 0.8 i.o I.I 13 1-4 2.8 4.2 6.0 5.6 5oj7-9l7.5l7.i \\ V 54 383 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 10 II 12 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 9.76 922 9.76939 9-76950 9.76974 9.76991 Log. Sill. I (I. 17 17 If If 17 if 17 If If 17 If 9.77 008 9.77 026 9-77043 9.77 066 9.77078 9.77095 9.77 112 9.77 130 9.77 147 9.77 164 50 51 52 53 54 55 56 57 58 59 60 9.77 181 9-77 198 9.77 216 9-77233 9.77250 9.77 267 9-77284 9.77302 9-77319 9-77336 9-77 353 9-77370 9-77387 9.77404 9.77421 9-77 439 9-77456 9-77 473 9.77490 9-77 507 9-77 524 9-77 541 9-77558 9-77 575 9-77 592 9-77 609 9.77 626 9-77643 9.77 660 9-77 ^77 9-77693 977710 9-77 72f 9-77 744 9.77761 9-77778 9-77 795 9.77812 9.77828 9-77845 9.77 862 9-77879 9.77896 9-77913 9.77929 9-77 946 Log. Cos. 17 17 If 17 If 17 17 If 17 17 If 17 17 17 17 If 17 17 17 17 Log. Tan. 9.86 126 9.86 152 9.86 179 9.86 206 9.86 232 17 17 17 17 17 17 17 17 16 17 17 17 17 16 17 17 16 17 17 16 17 17 16 17 9.86 259 9.86285 9.86 312 9-86338 9-86365 9.86 391 9.86418 9.86444 9.86471 9-86497 (1. 86524 .86550 .86577 .86603 86630 9 .86656 9.86683 9.86709 9.86 736 9.86762 9-86788 9.86815 9.86841 9.86868 9.86894 9.86 921 9.86 94f 9-86973 9. 87 000 9 -87026 9.87053 9.87079 9.87 105 9.87 132 Q-87 158 1.87 185 9 II 9.872 9.87237 9.87 264 9.87 290 :56 c. d. 9-87 3I6 987343 9.87369 9-87395 9.87422 "9^448 9.87474 9.87501 9-87 52f 9-87553 9.87 580 9.87 606 9.87 632 9-87659 9.87685 9-87 711 Log. Cot. Log. Cot. 26 26 27 26 26 2§ 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 2§ 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 2S 26 26 26 26 26 26 2§ 26 26 26 26 cTd^ 0.13874 0.1384^ 0.13 821 0.13794 o.i3 76f 0.13 741 O.I37I4 0.13688 0.13 661 0.13635 9-90796 9.90786 9-90 77f 9.90 768 9.90759 o. 1 3 608 0.13 582 0.13555 0.13529 o. 1 3 502 0.13476 0.13449 0.13423 0.13396 0.13 370 0.13343 O.I33I7 0.13 290 0.13 264 0.13237 9.90750 9-90740 990731 9.90 722 9.90713 9.90703 9-90694 9.90685 9.90676 9.90666 9.9065^ 9. 90 648 9.90639 9.90 629 9. 90 620 0.13 211 0.13 185 0.13 158 0.13 132 0.13 105 0.13079 0.13 052 0.13026 o. 1 3 000 0.12973 o. 1 2 947 O. I 2 920 0.12 894 0.12868 0.12 841 O. 12 815 0.12 789 O. 12 762 0.12 736 O. I 2 709 0.12 683 0.12 657 O. 1 2 636 0. 1 2 604 0.12 578 9.9061 1 9. 90 602 9.90592 9.90583 9-90 574 9.90 564 9-90 555 9.90 546 990 536 9.90 527 9.90 518 9-90 508 9-90499 9-90490 9.90 486 9.90471 9.90 461 9.90452 9-90443 9-90433 0.12 551 0.12 525 0.12 499 0.12 472 0.1 2 446 0.12 420 0.12 393 o. 12 36f 0.12 341 0.12315 9.90424 9.90414 9-90405 9.90396 9-90386 9.90377 9-90 36f 9.90358 9-90348 9-90339 9-90330 9-90 320 9.90 311 9-90301 9.90 292 o. 12 288 Log. Tan. 9.90 282 9-90 273 9.90 263 9.90254 9.90244 990235 Log. Sin. Log. Cos. I d. 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 d. 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 p. P. 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 27 28 2.7 2.6 3-1 3-1 3-6 3-5 4.0 4.0 4-5 4-4 9.0 8.8 13-5 13-2 18.0 17-6 22. ^ 22.1 26 2.6 3-0 3-4 3-9 4-3 8.6 13.0 17.3 21. § If 17 I.f 1.7 2.0 2.0 2-3 2.2 2.6 2.5 2.9 2.8 5-8 5-6 8.7 8.5 II. 6 II. J 14.6 14.1 18 1.6 1-9 2.2 2.5 2.f 5-5 8.2 II. o i3-f 6 7 8 9 10 20 30 40 50 9 0.9 I.I 1.2 1.4 1.6 3-1 4-7 6.3 7-9 9 0.9 i.o 1.2 1-3 1-5 3-0 4-5 6.0 7.5 P. P. 53° 3^4 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. ' ! Loer. sill. 9 10 1 1 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 J3 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 m 9-77 946 9.77963 9.77980 9-77 996 9.78013 9.78030 9.78046 9 78063 9.78080 9.78097 9.78 113 9.78 130 9.78 147 9.78 163 9.78 180 9.78 196 9.78213 9.78 230 9.78246 9.78263 9.78 279 9.78 296 9.78 312 9.78329 9.78346 9.78 362 9-78379 9-7839? 9.78412 9.78428 9.78444 9.78461 9-78 47f 9.78494 9.78510 9.78 527 9-78 543 978559 9.78576 9-78 592 9.78 609 9.78625 9.78 641 9.78658 9.78674 9.78 696 9.78707 9.78723 9-78739 9-78755 9.78772 9-78788 9.78 804 9.78821 9.78837 9.78853 9.78869 9.78885 9.78 902 9.78918 9- 78 934 liOtr. (;os. I liOer. Tan. r. «l. 9-87711 9-8773? 9.87764 9.87796 9-87 816 9-87843 9.87869 9.87895 9.87 921 9-87948 9.87974 9.88 000 9.88 026 9.88053 9.88 079 9.88 105 988 I3T 9.88157 9.88 184 9.88 210 9.88236 9.88262 9.88288 9.88315 9.88341 9.88 367 9.88393 9.88 419 9-88445 9.88472 9.88498 9.88 524 9.88556 9-88 576 9.88602 9.88629 9.88655 9.88681 9.88707 988733 9.88759 9 88785 9.88 81T 9.88838 9.8 8864 9. 88 890 9.88916 9.88942 9.88968 9-88 994 9.89 026 9.89046 9.89 072 9.89098 9.89 124 9.89 156 9-89 ^11 9.89203 9.89 229 9-89255 9.89 281 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 2S 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 l.op. Cot. o. IjOu. (!()t. c. d. I Loir. Tan 2288 2 262 2 236 2 209 2183 2 157 2 131 2 104 2 078 2 052 2 026 999 973 947 921 895 868 842 816 790 763 IZl 711 685 659 371 345 319 293 266 240 214 188 162 136 1 10 084 058 032 005 0979 0953 092^ o 90T 0875 o 849 o 823 0797 0771 0745 07 '9 Dtf. Con. I d. 90235 90 225 90 216 90 206 90196 90 187 90177 90 168 90158 90 149 90 139 90 130 90 120 90 116 90 lOI 90091 90082 90072 90062 90053 90 043 90033 90024 90014 90004 89995 89985 89975 89966 89 956 89946 89937 89927 8991? 89908 89898 89888 89878 89869 89859 89 849 89839 89 830 89820 89816 89791 89781 89771 89761 89751 89742 89 732 89722 89712 89702 89692 89683 89673 89663 989653 9 9 9 10 9 9 9 9 9 9 9 10 9 9 9 9 lO 9 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 10 9 10 9 10 9 10 9 10 9 10 9 10 10 9 10 9 10 10 9 10 10 9 10 10 10 Lot;. Sill. <>0 59 58 57 55 54 53 52 51 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 "20" 19 18 17 16 r. \\ 2§ 26 6 7 8 9 10 20 30 40 50 2-6 2 3-1 3 3-5 3- 4.0 3- 4.4 4- 8.8 8. 13.2 13- 17-6 17- 22.1 21. 17 18 6 1-7 1-6 7 2.0 1.9 8 2.2 2.2 9 2-5 2-5 10 2-8 2-y 20 5-6 5-5 30 8.5 8.2 40 11-3 II. 50 14.1 13.7 16 1.6 1-8 2.1 2.4 2-6 5-3 8.0 10.6 13-3 10 6 I.O 7 I.I 8 1-3 9 1-5 10 1-6 20 3-3 30 5.0 40 6 6 50 8-3 9 0.9 1. 1 1.2 1.4 1.6 3-1 4-? 6.3 7.9 5:^ 385 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 88^ 10 II 12 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 60 Lo^. Sin. 9-78934 9.78 950 9.78965 9.78 982 9.78 999 9.79015 9.79031 9.79047 9.79063 9.79079 9.79095 9.79 III 9.79 12^ 9-79 143 9.79 159 9.79175 9.79 191 9.7920^ 9.79223 9.79239 9-79255 9.79271 9.79287 9-79303 9 79319 9-79 33^ 979351 979367 979383 9-79 399 9.79415 9-79431 9-79 446 9.79462 9-79 478 9-79 494 9.79510 9.79526 9-79541 9-79 557 9-79 573 9-79589 9.79605 9.79 620 9-79635 9.79652 9.79668 9.79683 9 79 699 979715 9.79730 9-79 746 9.79762 9.79777 9-79 793 9.79809 979824 9.79840 9.79856 9.79871 9-79887 Log.J^os^ d. Loff. Tan. 9.89 281 9.89307 9-89333 9-89359 9.89385 9.89 411 9-89437 9.89463 9.89489 9.89515 9-89 541 9.89 567 9-89 593 9.89 619 9.89645 9.89 671 9.89697 9.89723 9-89749 9.89775 9.89 801 9.89827 9.89853 9.89879 9.89905 9.89931 9-89957 9.89 982 9.90008 9.90034 9.90066 9.90085 9. 90 112 990 138 9.90 164 9.90 190 9.90 216 9.90 242 9 90 268 9.90294 9.90319 9-9c» 345 9.90371 9.9039; 9.90423 9.90449 990475 9.90 501 9.90 525 9:90552 990578 9. 90 604 9.90630 9.90656 9.90 682 9.9070; 9-90733 9.90759 9.90785 9.90 81 1 9.90837 Log. Cot. c. d. 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 26 26 2$ 26 26 26 26 26 25 26 26 26 25 26 26 26 25 26 26 26 2? 26 26 2? 26 26 26 25 26 c. d. Log. Cot. 0719 0693 0667 0641 0615 0589 0563 0537 o 511 0485 0459 0433 0407 0381 0355 0329 0303 o 277 o 251 o 225 o 199 o 173 0147 O 121 0095 o 069 0043 o 01; 0.09 991 0.09 965 0.09939 0.09913 0.09 887 0.09 861 0.09 836 0.09 810 0.09 784 0.09 758 0.09 732 0.09 706 0.09 686 0.09 654 0.09 623 0.09 602 0.09 577 0.09551 0.09 525 0.09 499 o 09 473 0.09 44; 0.09 421 0.09 395 0.09 370 0.09 344 0.09 318 0.09 292 0.09 265 0.09 246 0.09 214 0.09 189 0.09 163 Log. Tan. Los. Cos. 9.89653 9.89643 9-89633 9.89623 9.89613 9.89 604 9-89594 9-89584 9.89574 9.89564 9.89554 9-89544 9-89534 9.89524 9.89514 9.89 504 9-89494 9.89484 9.89474 9.89464 9-89454 9-89444 9-89434 9.89424 9.89414 9.89404 9.89394 9-89384 989374 9.89364 9-89354 9.89344 9-89334 9.89324 9.89314 9.89304 9.89294 9.89 284 9-89274 9.89 264 9.89253 9-89243 9-89233 9.89223 9.89213 9.89 203 9.89193 9.89 182 9.89 172 9.89 162 9.89152 9.89 142 9.89132 9.89 I2T 9 89 1 1 1 9.89 lOI 9.89091 9.89081 9.89076 9.89066 9.89 056 Log. Sin. d. GO 59 58 57 56 55 54 53 52 51 50 49 48 47 45 44 43 42 41 40 39 38 V 36 35 34 33 32 31 30 29 28 27 26 15 14 13 12 II 10 9 8 7 6 p. P. 26 2S ; 6 2.6 2.5 s 7 3-0 3.0 8 3-4 3-4 9 3-9 3-8 10 4-3 4.2 ; 20 8.6 8-5 ; 30 13.0 12.; i 40 17.3 17.0 50 21.6 21.2 1 16 16 6 1-6 1.6 7 1.9 1-8 8 2.2 2. 1 9 2-5 2.4 10 2.7 2-6 20 5-5 5-3 30 8.2 8.0 40 II. 10.6 50 13-y 13-3 15 1.8 2.6 2-3 26 5-1 1-1 i,o-3 12.9 10 10 6 1.6 I.O 7 1.2 I.I 8 1-4 I-.3 9 1.6 i.S 10 1.7 1-6 20 3-5 3-3 30 5-2 5.0 40 7-0 6.5 50 8.; 8.3 9 0.9 I.I 1.2 1.4 1.6 3-1 ^•1 6.3 7.9 p. p 51° 386 TABLE VII. — LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 10 1 1 12 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 80 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 43 50 51 52 53 54 Lojf. Sill. I (1. Loi?. Tan. 55 56 57 58 59 60 9.79887 9.79903 9-79918 9-79 934 9-79 949 9.79965 9.79980 9.79996 9.8001T 9.80027 9.80042 9.80058 9.80073 9.80089 9.80 104 9.80 120 9.80135 9. 80 1 5 1 9.80 165 9 80 182 9.80 197 9. 80 2 1 3 9.80228 9.80243 9.80 259 9.80 274 9.80 289 9.80305 9.80 320 9-80335 980351 9.80366 9.80381 9.80397 9 80412 9 80427 9.80443 9.80458 9 80473 9 8048Q 9.80 504 9.80 519 9.80534 9-80549 g.8o 564 9.80 580 9.80595 9. 80 610 9. 80 62 1 9. 80 646 9.80655 9.80671 9.80686 9.80 701 9-80 716 9-80731 9.80746 9.80 761 9-80776 9.80791 ). 80 806 16 15 15 15 15 15 IS 15 iS 15 15 15 il il 15 IS iS iS 15 15 15 15 15 15 15 iS 15 15 15 15 15 15 15 15 15 15 15 15 15 IS 15 15 IS 15 IS 15 15 15 IS 15 15 IS 15 15 15 15 15 15 jiOg. Cos . i < 1. 90837 90863 90 8S3 90914 90940 90 966 90 992 01^ 043 069 095 121 i4o 172 198 224 250 27S 301 327 353 378 404: 430 456 481 507 533 559 584 616 636 662 68^ 713 739 765 790 816 842 867 893 919 945 970 996 92 022 92 04f 92073 92 099 92 124 92 150 92 176 92 201 92 22^ 92253 92 278 92304 92 330 9235S 92 381 26 2S 26 25 26 26 2S 26 26 2S 26 2S 26 2S 26 26 2S 26 2S 26 2S 26 2S 26 2S 26 2S 26 2S 26 2S 26 2S 26 2S 26 2S 26 25 2S 26 2S 26 2S 2S 26 2S 26 2S 2S 26 2S 2S 26 2S 2S 26 2S 2S 26 L(«r. Cot- 0.09 163 0.09 137 0.09 1 1 T 0.09085 0.09 060 0.09034 0.09008 0.08 982 0.08956 0.08 930 0.08 905 0.08 879 0.08853 0.08827 0.08 802 0.08 776 0.08 750 0.08 724 0.08 693 0.08 673 0.08 647 0.08 621 0.08 595 0.08 570 0.08 544 0.08 518 0.08 492 0.08 467 0.08 441 0.08 415 0.08 389 0.08 364 0.08 338 0.08 312 0.08 286 0.08 261 0.08 235 0.08 209 0.08 183 0.08 158 0.08 132 0.08 106 0.08 081 0.08 055 0.08 029 0.08 004 0.07 978 0.07 952 0.07 926 0.07 901 0.07 875 0.07 849 0.07 824 0.07 798 0.07 772 0.07 747 0.07 721 0.07 695 0.07 670 0.07 644 o. 07 6 1 8 liOp. Cot. 1 c. «1. I liO^. Tan. Lour. Cos. (I. 89056 89040 89030 89019 89009 88 999 88989 88978 88968 88958 88947 88937 88927 88917 88906 88 896 88 886 88875 88865 88855 88844 88834 88823 88813 88803 88792 88782 88772 88761 88751 88746 88730 88720 88 709 88699 88 688 88678 88667 88657 88646 88636 88625 88615 88604 88594 88 583 88573 88562 88 552 88 541 88531 88 526 88 510 88 499 88489 9-88 478 9.88467 9.88457 9.88446 9-88 436 9.88425 liOi?. Sill. 10 10 10 10 16 10 16 16 10 i5 10 10 10 16 16 10 16 16 10 i5 16 16 10 16 16 16 10 16 16 10 10 10 16 16 16 16 16 i5 16 10 16 16 10 10 16 i5 16 16 10 i5 10 10 10 i5 10 I r 16 16 10 10 50 49 48 47 j^ 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 19 18 17 16 ~^ 14 13 12 1 1 lo 9 8 7 6 1'. r. 26 2S 7 8 9 10 20 30 40 50 2.6 2.S 3-0 3-0 3-4 3-9 3-4 3-8 4-3 8.6 4-2 8-5 13.0 12.5^ 17.3 21.5 17.0 21.2 6 7 8 9 10 20 30 40 SO 16 1.6 1-8 2.1 2.4 2-6 5-3 8.0 10.6 13-3 IS i-S 1.8 2.6 2.3 2.6 5-1 7-f 10.3 12.9 15 1-5 I-? 2.0 2.2 2-5 5-0 7.5 1 0.0 12.5 II 10 I 6 I.I 1.6 7 1-3 1.2 8 1.4 1.4 9 1-6 1.6 10 1-8 1-7 20 3.6 3-5 3- 30 5-5 5-2 5- 40 7-. 3 7-0 6. 50 9.1 8.? 8. .0 .1 ■3 .5 •6 ■3 .0 -6 •3 p. I'. 50 337 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 40° 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Loe. Sill. 50 51 52 53 54 55 56 57 58 59 (JO 9.80805 9.80822 9.80837 9.80852 9.80867 9.80882 9.80897 9.80 912 9.80927 9.80942 9.80957 9,80972 9.80987 ooT 016 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 ao 9.8 31 9.8 32 9.8 33 9.8 34 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9^ 9.8 031 046 061 076 091 106 121 136 150 165 186 195 210 225 239 254 269 284 299 313 328 343 358 372 387 402 416 431 446 460 475 490 504 519 534 548 563 578 592 607 621 636 650 665 680 694 Log. Cos. (1. 15 15 15 15 15 15 15 15 15 15 15 14 15 15 15 15 15 14 15 15 15 14 15 15 14 15 15 14 15 14 15 15 15 14 15 14 14 15 14 15 14 14 15 14 14 15 14 1$ 14 15 14 14^ 14^ 14 15 14 Lop. Tan. 9.92 38T 9.92407 9.92432 9-92 458 9.92484 9.92 509 9-92 535 9.92 561 9.92 585 9.92 612 9.92638 9.92663 9.92 689 9.92714 9.92 746 9.92 766 9.92 791 9.92 817 9.92 842 9.92 868 9.92894 9.92919 9.92945 9.92971 9.92996 C. (1. 9.93022 9-9304? 993073 9-93 098 9.93 124 9.93 150 9-93 175 9.93 201 9.93225 9.93 252 9-93 278 9-93 303 993329 9-93 354 9-93380 9-93405 9-93 431 9-93 456 9.93 482 9-93 508 9-93 533 9-93 559 9-93 584 9.93610 9-93635 9.93661 9-93685 9.93712 9-93 73? 9-93763 9-93788 9.93814 9.93 840 9.93 865 9-93891 9-93 9^6 Log. Cot. 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 25 26 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 25 26 25 25 25 cTdT Log. Cot. 0.07 618 0.07 593 0.07 567 0.07 54T 0.07 516 0.07 490 0.07 465 0.07 439 0.07 413 0.07 388 Log. Cos. 9.88425 9.88415 9.88404 9-88393 9-88383 (1. 9.88372 9.88 36T 9.88351 9.88346 9.88 329 0.07 362 0.07 336 0.07 311 0.07 285 0.07 259 0.07 234 0.07 208 0.07 183 0.07 15^ 0.07 1 31 0.07 106 0.07 086 0.07 055 0.07 029 0.07 003 0.06 978 0.06 952 0.06 927 0.06 901 0.06 875 0.06 850 0.06 824 0.06 799 0.06 773 0.06 748 0.06 722 0.06 695 0.06 671 0.06 645 0.06 620 0.06 594 0.06 569 0.06 543 0.06 518 0.06 492 0.06 465 0.06 441 0.06415 0.06 390 0.06 364 9.88319 9-88308 9.8829^ 9.88 287 988275 9.88265 9.88255 9.88 244 9-88233 9.88223 9.88 212 9.88 201 9.88 196 9.88 180 9.88 169 158 147 137 126 115 9.88 104 9. 88 094 9.88083 9.88072 9.88 061 9.88050 9.88039 9.88029 9.88018 9.88007 9.87995 9.87985 9-87974 9.87963 9-87953 0.06 339 0.06 313 0.06 288 0.06 262 0.06 237 0.06 21 T 0.06 186 0.06 160 0.06 134 0.06 109 0.06083 Log. Tan. 9.87942 9.87931 9.87 920 9.87909 9.87898 9.8788? 9-87876 9.87865 9-87854 9.87844 9-87833 9.87 822 9.87 811 9.87 800 987789 9-87778 liOg. Sin. 10 II 16 16 16 II 10 16 II id 16 1 1 16 16 1 1 16 id II 16 1 1 16 II 16 II 16 II 16 II 16 II id II II 16 II II 16 II II 16 II II II 16 II II II 16 II II II II II 16 II II II II II II GO 59 58 57 56 p. p 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 12 31 30 29 28 27 26 25 24 23 22 21 To" 19 18 17 16 15 14 13 12 1 1 10 9 8 7 6 26 2$ 6 7 8 9 10 20 30 40 50 2.6 2.5 30 3-0 3-4 3-4 3-9 4-3 8.6 3-8 4.2 8.5 13.0 12.^ 17-3 21.6 17.0 21.2 15 S5 6 1-5 1-5 7 1.8 I.? 8 2.0 2.0 9 2.3 2.2 10 2.6 2.5 20 5-1 5.0 30 7.? 7.5 40 10.3 lO.O 50 12.9 12.5 14 1.4 1-7 1.9 2.2 2.4 4-8 7.2 9-6 12. 1 II 10 6 I.I I.O 7 1-3 1.2 8 1.4 1.4 9 1-6 1.6 10 1-8 1-7 20 36 3.5 30 5-5 5-2 40 7-3 7-0 50 9.1 8.? P. p. 49 388 TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, AND COTANGENTS. 11 5 6 7 8 9 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 Lotr. sill. 25 26 27 28 29 ao 31 32 33 34 35 36 37 38 39 40 41 \^ 43 44 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 694 709 723 738 752 767 781 796 810 824 839 853 868 882 897 911 925 940 954 q6o 983 997 9.82 012 9.82 025 9.82 040 9.82055 9.82 069 9.82 083 9.82 098 9 82 112 9.82 126 9.82 146 982 155 9.82 169 9.82183 9.82 197 9.82 212 9.82 226 9.82 246 9.82 2U 9.82 269 9.82283 9.82 297 9-82311 9.8232^ 45 46 47 48 49 :>o 51 52 53 54 55 56 57 58 59 m 9-82339 9.82354 9.82368 9.82 382 9-82 396 9.82 410 9.82424 9.82438 9.82452 9.82467 9.82 481 9.82495 9.82 509 9.82 523 9-82537 9-82 551 Lot?. Cos. »K liOur. Tan. c. «l 9939I6 9 93942 9.93967 9-93 993 9.94018 9.94044 9.94069 9.94095 9.94 120 9.94146 9-94171 9.94 197 9.94 222 9.94248 9-94273 9-94299 9.94324 9.94350 9-94 375 9.94400 9.94426 9.94451 9-94 477 9.94502 9-94528 9-94 553 9-94 579 9.94604 9.94630 9-94655 9.94681 9-94 706 9-94732 9-94 757 9-94782 9. 94 808 9-94833 9-94859 9.94884 9.94910 9-94 935 9.94961 9.94986 9.95011 9-95037 9.95 062 9.95 o83 9-95 113 9-95 139 9.95 164 9.95 189 9-95 215 9.95 240 9.95 266 9-95291 9-95 316 9-95 342 9-95367 9-95 393 9-95 418 9-95 443 liOtr. ("ot. c. (I. 25 25 25 25 25 25 25 25 25 25 25 25 25 25 2S 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 2^ 25 25 2? 25 25 25 25 25 25 lA.ir. Cot. Lou'. Cos. (1. 0.06 083 9-87778 ,, <><) 0.06 058 9 87 767 ; ; 59 0.06 032 9 87756 58 0.06 007 9 87745 57 0.05 981 9 87734 56 55 0.05 956 9 87723 0.05 930 9 87712 54 0.05 905 9 87701 53 0.05 879 9 87 690 52 0.05 854 9 87679 51 oO 0.05 828 9 87668 0.05 803 9 87657 49 0.05 777 9 87645 48 0.05752 9 87634 47 0.05 726 9 87623 46 0.05 701 9 87612 45 0.05 67I 9 87 601 44 0.05 650 9 87 590 43 0.05 625 9 87579 42 0.05 599 9 87 568 41 0.05 574 9 87557 40 0.05 548 9 87546 39 0.05 523 9 87535 1 38 0.05 497 9 87523 37 0.05 472 9 87512 ^ '■ 36 0.05 446 9 87 501 35 0.05 421 9 87490 34 0.05 395 9 87479 33 0.05 370 9 87468 32 0.05 344 9 87457 31 0.05 319 9 87445 30 0.05 293 9 87434 ^ 29 0.05 268 9 87423 28 0.05 243 9 87412 27 0.05 217 9 87 401 j^ 26 0.05 192 9 87389 25 0.05 166 9 87 378 ^ 24 0.05 141 9 87367 23 0.05 III 9 87356 22 0.05 090 9 87345 21 0.05 064 9 87333 20 0.05 039 9 87322 •^ ^ 19 0.05 014 9 8731^ 18 0.04 988 9 87300 17 0.04 963 9 .87 2(88 j| 16 0.04 937 9 .87277 15 0,04 912 9 87266 ^ 14 0.04 886 9 87254 13 0.04 861 9 87243 ^ 12 0.04 836 9 87232 II 10 0.04 816 9 87 221 0.04785 9 .87 209 ^ 9 0.04759 9 87 198 8 0.04734 9 87187 7 0.04 708 9 87175 6 5 0.04683 9 87164 0.04658 9 87153 ^ 4 0.04632 9 87 141 ^ 3 0.04607 9 87 130 ^ T 0.04 581 9 87 118 I 0.04 556 9 87 107 \a)S. Tan. 1 our. Sin. (1. t V. V 2S 25 6 7 8 9 10 20 30 40 50 2 5 --> 3 2. 3 4 3- 3 8 3- 4 2 4- 8 5 8. 12 1 12. 17 16. 21 2 20. 14 6 1.4 7 1.7 8 1-9 9 2.2 10 2.4 20 4-8 30 7-2 40 9.6 50 12. 1 14 1-4 1.6 1-8 2.1 2-3 4-6 7-0 9-3 II. 6 II I 6 I.I 7 1-3 8 1-5 9 1-7 10 1.9 20 3-8 3- 30 5-7 5- 40 7-6 7- 50 9-6 9- I'. V 48 3»9 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 42" 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 (io Loff. Siu. 82551 82565 82 579 82593 82 607 82621 82635 82649 82663 82 677 82691 82 705 82 719 82733 82746 82 766 82774 82788 82802 82816 82830 82844 82858 8287T 82885 82899 82 913 82 927 82 940 82954 82968 82982 82 996 83009 83023 83037 83051 83 064 83078 83092 83 106 83 !I9 83 133 83 147 83 166 83 174 83 188 83 201 83215 83229 83 242 83256 83 269 83283 83297 ,10 83 83324 83337 83351 83365 83 378 Log. Cos. (1. Lofr. Tan. c. <1. I Loer. Cot. 9-95 443 9.95 469 9.95494 9.95 520 9-95 54^ 9-95 571 9.95 596 9.95 621 9.95647 9.95672 9.95697 9.95723 9-95 748 9-95 774 9-95 799 9.95 824 9-95850 9-95873 9.95901 9.95 926 95951 95 977 96 002 96027 96053 9.96078 9.96 104 9.96 129 9.96154 9.96 180 9.96 205 9.96230 9.96 256 9.96 281 996306 9-96332 9-96357 9-96383 9. 96 408 9-96433 9-96459 9.96484 9-96 509 9-96535 9.96 560 9-96 58S 9.96 611 9.96636 9.96 661 9.96687 9.96 712 9.96737 9.96763 9-96788 9-96813 9.96839 9.96 864 9.96889 9.96915 9.96 940 9-9696! Lot,'. Cot. c. d 25 25 2l A 25 25 2! 25 2S 25 25 25 25 25 25 25 25 2^ 25 25 25 25 25 2? 2S 25 25 2? 2? 25 25 25 25 25 25 25 25 25 2! 25 25 25 25 25 25 25 25 25 25 25 25 25 2? 25 25 25 25 25 25 25 0.04 556 0.04531 0.04 505 0.04 480 0.04454 0.04429 0.04404 0.04 378 0.04353 0.04327 0.04 302 0.04 277 0.04 251 0.04 226 0.04 206 0.04175 0.04 I 50 0.04 124 o. 04 099 0.04074 0.04048 0.04 023 0.03 997 0.03 972 0.03 947 Loe. Cos. 0.03 921 0.03 896 0.03 871 0.03 84! 0.03 820 0.03 795 0.03 769 0.03 744 0.03 718 0.03693 0.03 668 0.03 642 0.03 617 0.03 592 0.03 565 0.03 541 0.03 516 0.03496 0.03465 0.03 440 0.03414 0.03 389 0.03 364 0-03 338 0.03313 0.03 28^ 0.03 262 0.03 237 0.03 21 T 0.03 186 0.03 161 0.03 135 0.03 1 16 0.03 085 0.03059 0-03034 Log. Tan. 9.87 107 9.87 096 9.87 084 9-87073 9.87 062 9.87 056 9.87039 9.87 027 9.87 016 9.87 004 9-86993 9.86982 9.86 976 9.86959 9-86947 9-86936 9.86 924 9-86913 9.86 90T 9.86890 9-86 878 9.86867 9.86855 9.86 844 9.86832 9.86821 9.86809 9.86798 9.86786 9-867 74 9.86 763 9.86751 9.86 740 9.86728 986 716 9.86 705 9.86 693 9.86682 9.86 676 9-86658 9.86647 9.86 63I 9.86 623 9.86612 9. 86 606 86588 86577 86565 86553 86542 9.86 530 9-86 518 9.86 507 9.86495 9.86483 9.86471 9. 86 460 9.86448 9.86436 9.86424 9.86412 Los. Sin. 00 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 .5 4 3 2 I p. P. 25 6 7 8 9 10 20 30 40 50 2.S 2. 3-0 2. 3-4 3- 3-8 3- 4.2 4- 8.5 8. 12.7 12. 17.0 16. 21.2 20. 25 5 9 3 7 I 3 5 6 14 6 1.4 7 1-6 8 1.8 9 . 2.1 10 2-3 20 4-6 30 7.0 40 9-3 50 II. 6 13 1-3 1.6 1.8 2.0 2.2 4-5 6.7 9.0 II. 2 12 II I] 6 1.2 I.I 7 1.4 1.3 8 1.6 1.5 9 1.8 1.7 10 2.0 1.9 20 4.0 3-8 3- 30 6.0 5-7 5- 40 8.0 7-6 7. 50 lO.O 9.6 9- p. p. 47' 390 TABLE VIL— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 43° 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Lofj. Sin. (1. 83378 83392 83405 83419 83432 83446 83459 83473 83486 83500 83513 83527 83540 83554 83567 83585 83594 83607 83621 83634 83647 83661 83674 83688 83701 83714 83728 83741 83754 83768 83781 83794 83808 83821 83834 83847 83861 83874 83887 83900 83914 83927 83940 83953 83967 83980 83993 84005 84019 84033 84046 84059 84072 8408I 84098 84 III 84 124 84138 84 151 84 164 84177 Log. Cos. (I. Loff. Tnii. c. (1 9.96965 9.96991 9.97 016 9-97041 9.97067 9.97092 9.97 117 9-97 143 9.97 168 9-97 193 9.97219 9.97 244 9.97 269 9.97 295 9.97320 9-97 345 9-97 370 9.97396 9-97421 9-97 446 9.97472 9-97 497 9-97 522 9 97 548 9-97 573 9-97 598 9.97624 9.97649 9-97 674 9-97699 9.97725 9.97 750 9-97 775 9.97 801 9.97 826 9-97 851 9-97 877 9-97902 9.9792^ 9.97952 9-97 978 9.98 003 9.98028 9-98054 9.98079 9.98 104 9.98 129 9.98155 9.98 186 9.98 205 9-98231 9.98 256 9.98 281 9-98 306 9-98 332 9-98 357 9.98 382 9.98 408 998433 9-984 58 9-98483 Lop. Cot. 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 Lop. Cot. 0.03034 0.03 009 0.02 984 0.02 958 0.02933 0.02 908 0.02 882 0.02 857 0.02 832 0.02 806 0.02 781 0.02 756 0.02 736 0.02 705 0.02 680 0.02 654 0.02 629 0,02 604 0.02 578 0.02 553 0.02 528 0.02 502 0.02 47^ 0.02 452 0.02 427 0.02 401 0.02 376 0.02 351 0.02 325 0.02 306 0.02 275 0.02 249 0.02 224 0.02 199 0.02 174 0.02 148 0.02 123 0.02 098 0.02 072 0.02 04^ 0.02 022 0.0 1 996 o.oi 971 0.0 1 946 0.01 921 0.01 895 0.01 876 0.01 845 0.01 819 0.01 794 0.01 769 0.01 744 0.01 718 0.01 693 0.01 668 0.01 642 0.01 61^ 0.01 592 0.01 567 0.01 541 o-oi 5 16 Lor. Tan. Loe. Cos. 9.86354 9.86342 9.86330 9-86318 9- 8630 6 9.86 294 9.86282 9.86 271 9.86 259 9.86 247 86412 86 401 86389 8637? 86365 86235 86 223 86 21 1 86 199 86187 86 176 86164 86 152 86 140 86128 86 116 86 104 86092 86080 86068 86056 ^ 86044 9.86032 9.86 020 - 86 008 85996 85984 85972 85 960 85948 85936 85924 85 912 85 900 85887 85875 85863 85851 85839 85827 85815 85803 85791 85 778 85766 85754 85742 85730 85718 85705 985693 Lot;. Sin. (io 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 I'. V. 2S 25 6 2-5 2.5 7 3-0 2.9 8 3-4 3.3 9 3-8 3.^ 10 4.2 4.1 20 8.5 8.3 30 12. f 12.5 40 17.0 16.6 50 21.2 20.8 13 13 6 1-3 1.3 7 1.6 1.5 8 1.8 i.^ 9 2.0 1.9 10 2.2 2.1 20 4-5 4.3 30 6.7 6.5 40 9.0 8.S 50 II. 2 I0.8 12 12 II 6 1.2 1.2 I.I 7 1-4 1.4 1-3 8 1-6 1.6 1-5 9 1-9 1.8 1-7 10 2.1 2.0 1-9 20 4.T 4.0 3-8 30 6.2 6.0 S-7 40 8.3 8.0 7-6 50 10.4 lO.Q 9.6 1'. F. 46 391 TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS. 44° 10 II 12 14 15 i6 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Log. Sin. 9.84177 9.84 190 9.84 203 9.84 216 9.84229 9.84 242 9.84255 9.84268 9.84281 9.84294 9.84307 9.84320 9-84333 9-84 346 9-84359 9.84372 9-84385 9-84398 9.84411 9.84424 9-84437 9.84450 9.84463 9.84476 9.84489 9.84502 9.84514 9.84 52f 9.84540 9-84553 9.84 566 9-84579 9-84592 9. 84 604 9.8461^ 9. 84 630 9.84643 9.84656 9.84669 9.84681 d. 9.84694 9.84707 9.84720 9.84732 9.84745 9.84758 9-84771 9-84783 9-84796 9.84809 9.84822 9-84834 9.84847 9.84860 9.84872 9.84885 9.84898 9.84 916 9.84923 9-84936 9-84948 Lo g. Cos. 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 12 13 13 13 13 13 12 13 13 13 12 13 13 12 13 13 12 13 13 12 13 12 13 12 13 12 13 12 13 12 13 12 12 13 12 12 13 12 12 13 12 Log. Tan. 9.98483 9.98 509 9-98 534 9-98 559 9.98 585 9.98 610 9.98635 9. 98 666 9.98686 9.98 711 9-98736 9.98 762 9-98787 9.98 812 9.98837 9.98863 9.98888 9.98913 9-98 938 9.98964 9.98989 9.99014 9.99040 9.99065 9.99096 9-99115 9.99 141 9.99 166 9.99 1 91 9-99 216 9.99242 9.99267 9.99292 9.99318 9-99 343 9-99368 9-99 393 9.99419 9-99 444 9.99469 9.99494 9-99 520 9-99 545 9.99570 9-99 59? 9.99 621 9.99646 9.99671 9.99697 9.99722 9-99 74^ 9-99772 9.99798 9-99823 9-99848 c. d. 9-99873 9-99899 9.99924 9-99 949 9-99 974 0.00000 2S 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 2§ 25 25 25 25 25 25 25 25 25 25 Log. Cot. o.oi 516 o.oi 491 O.OI 465 O.OI 446 O.OI 415 O.OI 390 O.OI 364 O.OI 339 O.OI 314 O.OI 289 O.OI 263 O.OI 238 O.OI 213 O.OI 18^ O.OI 162 O.OI 137 O.OI 112 O.OI 086 O.OI 061 O.OI 036 O.OI 010 0.00 985 0.00 960 0.00 935 0.00 909 0.00 884 0.00 859 0.00 834 0.00 808 0.00 783 0.00758 0.00733 0.00 70^ 0.00 682 0.00 657 0.00 631 0.00 606 0.00 581 0.00 556 0.00 536 Log. Co t, led. 0.00 505 0.00 480 0.00455 0.00429 0.00404 0.00 379 0.00353 0.00 328 0.00 303 0.00 278 0.00 252 0.00 227 0.00 202 0.00 177 0.00 151 0.00 126 0.00 lOI 0.00076 0.00056 0.00025 o. 00 000 Lo g. Ta n. Log. Cos. 9.85693 9.85681 9.85 669 9.85657 9.85644 9.85 632 9.85 620 9.85608 9-85 595 9-85 583 9.85571 9.85559 9-85 546 9-85 534 9.85 522 9-85 509 9-8549^ 9.85485 9-85472 9.85 466 9.85448 9-85435 9.85423 9.85411 9-85 398 9.85 386 9-85374 9.85361 9-85349 9-85336 9.85 324 9.85312 9-85299 9.85 287 9-85 274 9,85 262 9-85 249 9.85 237 9.85 224 9.85 212 d. 9.85 199 9-85 187 9.85 174 9.85 162 9.85 149 9-85 137 9.85 124 9.85 112 9.85099 9.85087 9.85074 9.85 062 9.85049 9.85037 9.85 024 9.85 Oil 9.84999 9.84986 9.84974 9.84961 9-84948 Log. aiu. d. GO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 9 8 7 6 P. P. 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 2S 25 2-5 2. 3-0 2. 3-4 3-8 3. 3- 4.2 8.5 4- 8. I2.f 12. 17.0 16. 21.2 20. 12 1.2 1-4 1-6 1-9 2.1 4.1 6.2 8-3 10.4 13 13 6 1-3 1.3 7 1.6 i.S 8 1.8 i-f 9 2.0 1.9 10 2.2 2.1 20 4.5 4.3 30 6-^ 6.5 40 9.0 8.6 50 II. 2 10.8 12 1.2 1.4 1.6 1.8 2.0 4.0 6.0 8.0 lO.O p. p. 45 392 TABLE VIII. LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 0° 1° 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Los. Vers. 2> 50 51 52 53 54 55 56 57 58 i9. 60 CO 2.62642 3.22848 3.58066 3-83054 4.02436 .18272 .31662 .43260 • 53490 4.62642 ,70920 .78478 .85431 .91- ■ 4.97860 5.03466 .08732 •13696 •18393 5.22848 . 27086 •31126 • 34987 .38684 5.42230 •45636 .48915 .52073 .55121 5 . 58066 .60914 .63672 . 66344 •68937 5^7i455 .73902 .76282 •78598 .80854 5-83053 •85198 .87291 •89335 .91332 5.93284 •95193 .97061 5.98890 6 . 00680 6.02435 .04155 .05842 •07496 .09120 6. 10714 . 12279 .13816 .15327 .16811 6.18271 Log. Vers. Loe. Exsec. 60206 352I8 24987 19382 15836 13389 I I 598 10230 915T 8278 7558 6953 6437 5992 5605 5266 4964 4696 4455 4238 4046 3861 3697 3545 3406 3278 3158 3048 2944 2848 2757 2672 2593 2518 2447 2379 23'6 2256 2199 2145 2093 2044 1996 1952 1909 1868 1829 1790 1755 1720 1654 1623 1594 1565 1537 1511 1484 1460 J) — 00 2,62642 3.22848 3.58066 3-83054 4^02436 , 18272 ,31662 ,43260 •53491 4.62642 .70921 .78478 •85431 .91868 4.97861 5^03466 .08732 •13697 •18393 5.22849 .27087 .31127 • 34988 .38685 ■5.42231 •45638 .48916 •52075 •55123 . 58068 .60916 •63674 •66346 . 68940 5^71457 .73904 .76284 .78601 .8085^ 5-83056 .85201 .87295 .89338 -91335 5-93288 •95197 .97065 5.98894 6.00685 6 . 02440 .04160 .05847 .07501 .09125 n 6. 10719 . 12284 . 13822 .15333 .16818 6.18278 Los. Exsec. Log. Vers. J» 60206 35218 2498? 19382 15836 13389 1 1 598 10236 9151 8279 7557 6952 6437 5993 5605 5266 4964 4696 4456 4238 4046 386T 3697 3545 3407 3278 31591 3048 2945 2848 2758 2672 2593 251? 2447 2380 2316 2256 2199 2145 2093 2043 1997 1952 1909 1829 I79I 1755 1720 1687 1654 1623 1594 1565 1537 I5II 1485 1460 7> 6,18271 .19707 .211 19 .22509 .23877 6.25223 .26549 .27856 .29142 .30416 6. 31666 ,32892 .34107 •35305 .36487 6.37653 • 38803 . 39938 ,41059 .42165 43258 .44337 .45403 .46455 . 47496 6.48524 .49539 .50544 •51536 .52518 6.53488 . 54448 .55397 •56336 .57265 6.58184 • 59093 • 59993 .60884 .61766 6.62639 .63503 •64359 .65206 .66045 6.66876 ,67700 .68515 .69323 .70124 6.70917 .71703 .72482 .73254 .74019 6.7477^ .75529 .76275 .77014 .777 aJ 6.78474 Loir. Vers. Log. Exsec. 1435 I412 1389 1368 1346 1326 1306 1286 1268 1250 1232 1214 1 1 98 1182 1 166 1 1 50 1135 1121 1 106 1093 I078 1066 1052 1046 1028 1015 1004 992 981 970 960 949 939 929 919 909 900 891 882 872 864 855 84^ 839 831 823 815 806 793 786 779 772 765 758 752 745 739 733 726 /> 6.18278 .19714 .21126 .225I6 .23884 6.2523T .2655^ .27864 ,29151 •30419 6.31669 ,32901 -34II6 •35315 • 36497 6.37663 .38814 • 39949 .41076 .42177 6.43270 •44349 •45415 •46468 •47509 48537 49553 ■50557 ■51550 ■52532 6^53503 • 54463 •55413 •56352 •57281 . 58201 .59116 .60011 . 60902 .61784 6.62657 ,63522 .64378 .65226 .66065 6.66897 ,67726 ■68536 •69345 .70145 6.70939 .71725 .72505 .7327^ . 74043 6.74802 •75554 , 76306 . 77040 .7777% 6.78506 I, Off. Kxst'C 2) 1436 I412 1390 1368 1347 1326 I3O6 1287 I26g 1250 1232 I215 II98 1182 1 166 1151 1135 1121 1106 1093 1079 1066 1053 1046 1028 1016 1004 993 982 976 960 950 939 929 919 909 906 891 882 873 864 856 848 839 83I 823 816 808 806 794 786 779 772 765 759 752 746 739 733 727 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 4S 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 394 TABLE VIII.- -LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 2° :r 5 6 7 8 9 10 II 12 14 15 i6 17 i8 19 20 21 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 (>0 Lou:. A'ers. /> 78474 79195 79909 8061 8 81322 82019 ,82711 ,83398 , 84079 •84755 85425 86091 8675t ,87407 ,88057 88703 89344 89980 90612 91239 91862 ,92480 93093 93703 94308 94909 95506 , 96099 ,96688 ,97272 •97853 •98430 • 99004 •99573 •00139 .00701 .01259 .01814 .02366 .02914 03458 .03999 .04537 .05071 .05603 . 06 1 30 .06655 .07177 .07695 .0821 1 •08723 .09232 •09739 . 10242 • 10743 1 1240 11735 1 222^ 12716 13203 13687 7_ IjOsj. »rs. 721 71-+ 709 703 697 692 686 681 676 670 665 666 655 656 646 641 636 631 627 622 618 613 609 605 601 597 592 589 58-+ 581 577 573 569 565 562 558 555 551 548 544 541 537 534 531 527 525 521 518 515 512 509 506 503 506 497 495 492 489 486 484 IjO!?. Kxsec T> 6.78500 .79221 • 79937 . 80645 .81350 6.82048 .82740 .83427 .84109 •84785 6.85457 .86123 .86783 •87439 . 88096 6.88737 •89378 .90015 .90647 .91275 6.91898 .92516 •93131 •93741 • 94346 6 . 94948 •95545 .96139 .96728 •973J3 6.97895 .98472 •99046 6.99616 7.00182 7.00745 .01304 .01860 .02412 .02966 7.03505 .04047 •04585 .05126 .05652 7 . 06 1 86 . 06706 .07228 ■0774? .08263 7.08776 .09286 •09793 . 10297 • I0798 1 1297 1 1792 12285 12775 13262 7> \ Loif. Vers. /> 7 • 1 3746 721 715 709 703 698 692 687 682 676 671 666 666 656 651 646 641 636 632 628 623 61 8 614 610 605 601 597 593 589 585 581 577 574 570 566 563 559 555 552 548 545 541 538 535 531 528 525 r '>2 519 516 513 509 507 503 501 498 495 493 490 487 484 Kxser. /> i 7 13687 14168 14646 15122 15595 16066 16534 17000 17463 17923 18382 18837 19291 19742 201 91 20637 21081 21523 21963 22406 22836 23269 23700 24129 24555 24980 25402 25823 2624T 26658 27072 27485 27895 28304 287 II 29116 29518 29919 30319 307 1 6 31112 31505 3189^ 32288 32676 33063 33448 33831 34213 34593 34971 35348 35723 36097 36468 36839 37207 37574 37940 38304 38667 481 478 475 473 470 468 466 463 466 458 455 453 451 448 446 444 442 440 437 435 433 431 429 426 424 422 426 418 4>6 414 412 416 409 406 405 402 401 399 397 395 393 392 390 388 386 385 383 382 380 378 377 375 373 371 370 368 367 366 364 ^62 l,(n:. K\s<'c liOer. Vers. 7> 13746 14228 14707 15183 15657 1 61 29 16598 17064 17528 17989 18448 18905 19359 1 98 II 20260 2070^ 21 152 21595 22035 22473 22909 23343 23775 24204 24632 25057 25486 25902 26321 26738 27153 27567 27978 28387 28795 29200 29604 30006 30406 30804 3 1 201 31595 31988 32379 32768 33 '56 33542 33926 34309 34689 35069 35446 35822 36196 36569_ 36940 373'0 37678 38044 38409 It 38773 481 479 476 474 471 469 466 464 46 1 459 456 454 452 449 447 445 442 446 438 436 434 431 429 427 425 423 421 419 41^ 415 413 411 409 407 405 404 402 400 398 396 394 393 391 389 388 385 38-; 382 386 379 377 376 374 37' 569 368 366 365 363 liOe. Kxscr. /> 395 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 4° 5° 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Lof?. Vers. 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 7.38667 . 39028 •39387 .39745 .40102 7.40457 .40810 .41163 .41513 .41863 D Loff. Exsec. 2> .4221 I •4255^ .42903 .43246 .43589 7.43930 .44270 .44608 .44946 .45281 7.45616 .4594-9 .46281 . 466 1 2 .46941 7.47270 .47597 .47922 .48247 .48570 7.48892 .49213 .49533 .49852 .50169 7. 5048 S . 50800 .51114 .51427 . .51739 7 52050 52359 5266^ .52975 .53281 7.53586 . 53890 .54193 . 54495 . 54796 60 7.55096 .55395 .55692 .55989 .5628^ 7.56580 .56873 .57166 •57458 . 57749 7.58039 361 359 358 356 355 353 352 350 349 348 346 345 343 342 341 339 338 337 335 334 332 330 329 328 327 325 324 323 322 321 320 318 3^7 316 315 314 313 311 311 309 308 30^ 306 305 304 303 302 300 300 299 29? 297 295 295 293 293 292 290 290 Lo:r. Vers. 7-38773 •39134 .39495 .39854 . 402 1 1 7.4056; .40922 .41275 .41627 .41977 7.42326 .42673 .43019 .43364 .43708 7.44050 .44390 .44730 .45068 .4540=; 7.457401 .46075 , .4640^ i .46739 i .47070 i 7.47399 .47727 .48054 .48379 .48703 7.49026 .49348 . 49669 .49989 .5030; 7 . 50624 . 50941 .51256 .51569 .51882 7.52194 .52504 .52814 .53122 .53429 7.53735 .54041 . 54345 . 54648 .54950 7 n .55251 .55550 .55849 .5614^ . 56444 7.56740 •57035 •57329 .57621 •57913 7.58204 361 366 359 35^ 356 354 353 352 350 349 34^ 346 345 343 342 340 339 338 337 335 334 332 332 330 329 328 327 325 324 323 322 321 319 318 317 316 315 313 313 311 316 309 308 307 306 305 304 303 302 301 299 299 298 296 296 295 294 292 292 291 Log. Vers. 7 Los;. Kxsec 7> 7 58039 58328 58615 58902 59188 59473 59758 60041 60323 60604 60885 61 164 61443 61721 61998 62274 62549 62823 63096 63369 63641 63911 6418T 64451 64719 64986 65253 65519 65784 66048 6631 1 66574 66836 67097 6735^ 67617 67875 68133 68396 68647 D Loff. Exsec. I 2> 68902 6915; 6941 1 69665 6991; 70169 70421 70671 70921 71 170 714I8 71666 71913 72159 72404 72649 72893 73137 73379 73621 73863 Lo!.'. Vers. 289 287 287 286 285 284 283 282 281 286 279 279 27? 277 276 275 274 273 272 272 276 270 269 268 26^ 266 266 265 264 263 263 26T 261 266 259 258 258 257 256 255 255 254 253 252 252 251 250 250 249 248 24^ 247 246 245 245 244 243 242 242 241 7. 58204 58494 58783 59071 59358 59645 59930 ,60214 . 60498 .60786 7. 7> 61062 61342 61622 6190T 62179 .62456 .62733 , 63008 .63282 .63556 63829 ,64101 ,64372 . 64643 .64912 ,65181 .65449 .65716 .65982 .6624^ .66512 .66776 .67039 .67301 ,67562 ,67823 ,68083 ,68342 ,68601 ,68858 ,69115 ,69371 ,69627 .69881 .70135 70388 ,70641 70893 ,71144 71394 71644 ,71892 ,72141 •72388 .72635 ,72881 .73126 .73371 .73615 •73859 7-74iot Lour. Kxsec 290 289 288 287 286 285 284 283 282 281 286 280 279 278 277 276 275 274 274 273 272 271 276 269 269 268 267 266 265 264 264 263 262 261 261 260 259 258 25; 257 256 255 254 254 253 252 252 251 250 250 248 248 24? 246 246 245 245 244 243 242 ]> I 2 3 4 5 6 7 8 _9L 10 II 12 13 14 p. P. 15 ]6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6 360 36.0 35.0 7 8 42.0 48.0 40.8 46.6 9 10 54-0 60.0 51.5 58., S 20 120.0 116. 6 30 180.0 175-0 40 240.0 233.3 50 300.0 291.6 GO 330 33^o 38.5 44.0 49.5 55-0 IIO.O 165.0 220.0 275.0 270 27.0 31-5 36.0 40-5 45.0 go.o 135-0 180.0 225.0 320 32.0 37-3 42-6 48.0 53.3 106.6 160.0 213.3 266.6 !6 30.3 34-6 39-0 43.3 8^.-6 130.0 173.3 216.6 340 34-0 39-6 45-3 51.0 56.6 "3-3 170.0 226.6 283.3 310 31.0 36.! 41-3 46-5 51-6 103.3 155-0 206.6 258.3 300 290 280 6 30.0 29.0 7 35.0 33-8 8 40.0 38-6 9 45 -o 43.5 10 50.0 48.3 20 100.0 96.6 30 150.0 145.0 40 200.0 193-3 50 250.0 241.6 28. 32-6 37-3 42.0 46.6 93-3 140. 1&6.6 233 .3 260 250 240 230 6 24.0 23.0 7 28 .0 26.8 8 32.0 30.6 9 36.0 34.5 10 40.0 38.3 20 80.0 76-6 30 120.0 115.0 40 160.0 153-3 50 200.0 191-6 210 200 6 21.0 20.0 7 8 24-5 28.0 23.3 26.6 9 31-5 30.0 10 20 35.0 70.0 33-3 66.6 30 105.0 100.0 40 50 140.0 175.0 133-3 166.6 25- 29. 33-3 37-5 41-6 83-3 125.0 166.6 208.3 220 22.0 25-6 29-3 33-0 36.6 73-3 IIO.O 146.6 ^83.3 190 19. 25 28, 31 63 95 • 126 I', t*. 396 TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. G 7° 10 1 1 12 14 15 i6 1 8 19 20 21 2 2 23 ii_ 23 26 27 28 29 30 3i 32 33 34 35 36 37 i 38 39 40 41 42 43 44 45 46 47 4B 49 50 54 56 57 5S 59 (>0 Loir. Vers. I J> 7 74104 74344 74583 74822 75060 75297 75534 75770 76006 76246 76475 76708 76941 77173 77405 77636 77867 78097 78326 78554 78783 79010 79237 79463 79689 I 79914! 80F38 S0362 805S6 80808 81031 81252 81473 81694 81914 82133 82352 82570 82788 83005 83222 83438 83653 83868 84083 84297 84516 84723 84933 85147 85359 85570 85780 85990 86199 86408 86616 86824 87031 87238" Lou'. Vers. 241 240 239 239 238 237 236 236 235 234 234 233 233 231 236 230 229 2 ''7 227 226 22 S 225 224 22 1 223 221 221 226 220 219 219 218 217 217 217 216 215 215 214 214 213 213 212 212 211 21 I 210 210 209 209 208 208 207 206 It Locr. Kxsec.I I> 74101 74343 74585 74826 75066 75305 75544 75782 76019 76256 76492 76728 76963 77197 77431 77664 77S97 78128 78360 78596 78826 79050 79279 79507 79735 79962 80188 80414 80639 80864 81088 81312 8i53d 81758 81980 82201 82422 82642 82862 83081 83300 83518 83735 83952 84169 84385 84606 84815 85030 85243 85457 85670 85882 86094 86305 86516 86726 86936 87146 87354 87563 liOir. Kxser. 242 24? 241 240 239 239 238 237 237 236 235 235 234 233 233 232 231 231 230 230 229 229 228 228 227 226 226 225 225 224 224 223 222 222 221 -> -> [ 226 I 219 I 219 ' 219 218 217 217 3,6 216 215 215 214 I 213 I :: i 213 ' -'3 212 21T 211 21 1 210 210 209 208 208 7> Locr. Vers. J> Lok- Kxse( 7.87238 87444 87650 87855 88060 88264 88468 88672 88875 8907 f 89279 89481 89682 89882 90082 90282 9048 T 90680 90878 91076 91273 91476 91667 91863 9205^ 92253 92448 92642 92836 9 1029 Q-3 2 22 93415 93607 93799 93990 94181 94371 94561 94751 94940 95129 95317 95505 95693 95880 96066 96253 96439 96624 96809 96994 97178 97362 97546 97729 97912 98094 98276 98458 98639 7 .08820 lAta. Vt'rs. 200 205 205 204 204 204 203 203 202 202 20 f 201 206 200 99 99 98 98 97 97 97 96 96 95 95 95 94 94 93 93 92 92 91 91 90 96 90 89 89 89 88 87 88 87 86 86 86 85 85 84 84 84 83 83 83 82 82 82 81 81 87563 87771 87978 88185 88391 88597 88803 89008 8921 2 894 1 6 89620 89823 90025 90228 90429 90636 90831 91032 91231 91431 It 91630 91828 92027 92224 9242T 926 1 8 92815 93016 93206 93401 93596 93790 93984 94177 94370 94562 94754 94946 9513? 95328 95519 95709 95898 96088 96276 96465 96653 96841 97028 97215 97401 9758? 97773 97958 98143 9832? 98512 98695 98879 99062 90244 /> I.Ki;. Ixsic. 208 207 207 206 206 205 205 204 204 203 203 202 202 201 201 201 2C6 99 99 99 98 98 97 97 97 96 95 95 95 95 94 94 93 93 92 92 92 91 91 96 90 89 89 88 88 88 88 87 87 86 86 85 85 84 84 84 83 83 83 82 5 6 7 8 9 10 1 1 12 13 14 16 17 18 19 20 21 22 23 24 r. I* J5 36 57 38 39 40 41 42 43 44 45 46 47 48 _49 50 51 52 53 54 55 56 57 58 i"^ (iO 180 9 9 6 iS o.y 0. 7 21 .0 I .1 8 24.0 I .2 9 27 1.4 10 30.0 1 6 20 00.0 3-1 30 QO.O 4-7 4- 40 I30.0 6.3 6. 50 150.0 7 9 7 30 40 5" 8 8 0-8 0.8 1 I 9 I I I I 3 I 2 I 4 I 3 2 § 2 6 4 2 4 5 fi ,s 3 1 7 1 6 6 1 7 6 6 0.7 °f> 1 7 8 7 8 9 8 9 I 1 10 I I 1 1 20 2 3 2 I 3" 3 5 3 '2 40 4 6 4 3 50 5 8 5 4 6 7 8 9 10 20 30 40 50 o 7 09 0.6 0.7 0.8 0.9 1 .0 2.0 3-0 4.0 5-0 4 3 6 4 0-3 0. 7 4 4 8 5 4 0. 9 6 5 0. 10 6 6 0. 20 I 3 I I I . 3>3 2 I 7 I 4° 2 6 2 3 2. 50 3 3 2 9 2. 2 2 6 0.2 0.2 7 8 3 3 0.2 2 9 4 0.3 0. 10 20 4 0-3 0-6 30 40 I I 2 6 I.O '•3 0. I. 50 2 I 1-6 1 . I'. ]' ?> 5 4 6 t--5 0.5 0.4 7 °-6 6 5 8 7 $ b 9 0.8 7 7 10 0.9 § 7 20 !•§ I 6 I 5 30 2.7 2 5 2 2 40 3-6 3 ? 3 50 4.t 4 I 3 7 397 TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 8° 9" 10 II 12 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 ao 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Log. Vers. 98820 99000 99186 99360 99539 •99718 ,99897 . 0007 5 ,00253 ,00431 . 00608 .00784 . 0096 r ,01137 ,01313 T> Lost. Exsec. 50 51 52 53 54 55 56 57 58 59 .01488 .01663 ,01838 .02012 .02186 •02359 •02533 ,02706 .02878 ,030^6 ,03222 •03394 ■03565 •03736 03906 .04076 ,04246 .04416 •04585 •04754 ,04922 ,05090 ■05258 .05426 •05593 m ,05760 •05926 ■06093 .06259 .06424 ,06589 ,06754 .06919 • 07083 .07247 .07411 .07575 .07738 .07906 .08063 .08225 ,08387 .08549 .08710 ,08871 0Q03 1 Ijosr. Vers. 86 80 79 79 79 78 78 77 78 77 76 76 76 76 75 75 75 74 74 73 73 73 72 72 72 71 71 71 70 70 70 69 69 69 68 68 68 67 67 (>7 66 66 66 65 65 65 65 64 64 64 63 63 62 62 62 61 62 61 61 66 7 99244 99427 99609 99796 99971 .00152 ■00332 ,00512 , 00692 ,00871 .01050 .01229 ,01407 ■01585 ,01763 .01940 .021 17 .02293 ,02469 ,0264.!; z> ,02820 ,02995 ,03176 .03345 03519 ,03692 ,03866 I ,04039 1 ,04212 ■ 04384 I 1) «i 81 86 86 80 80 79 79 78 7^ 78 77 77 77 76 76 75 75 75 75 74 74 73 73 73 73 72 72 71 71 71 70 70 70 70 69 69 69 68 68 68 67 67 67 66 66 66 65 65 65 64 64 64 63 64 63 63 62 8.09569 •045 56 ,04728 I .04899 j .05076 I .05241 i ,05411 ,05581 ,05751 ,05921 . 06090 ,06259 ,06427 ■06595 .06763 .06931 ,07098 .07265 .07431 .07598 .07764 .07929 ,08095 ,08260 ,08424 ,08589 .08753 ,08917 , 0908 I ,09244 . 09407 Loff. Vers. 8. 0903 T .09192 .09352 ,09512 ,09671 09836 09989 0148 0306 0464 0622 0779 0936 1093 1250 1406 1562 1718 1873 2029 2184 2338 2492 2647 2806 2954 3107 3266 34^3 3565 T> Los. Kxsec. 3717 3869 4021 4172 4323 4474 4625 4775 4925 5075 5225 5374 5523 5672 5826 59^8 6116 6264 6412 6559 6706 6852 6999 7145 729T 7437 7582 7728 7873 8017 8162 66 60 eo 59 59 59 58 58 58 57 57 57 57 56 56 56 55 55 55 55 54 54 54 53 53 53 53 52 52 52 52 51 51 51 51 50 50 50 49 50 49 49 49 48 48 48 48 47 47 47 46 46 46 46 45 45 45 45 44 44 liOjr. Vers. I> 8 8 09569 09732 09894 0056 0217 0378 0539 0700 0866 1026 1 186 1340 1499 1658 1816 1975 2133 2291 2448 2605 D 2762 2919 3075 3232 338? 3543 3698 3854 4008 4163 4317 4471 4625 4778 4932 5085 523^ 5390 5542 5694 5846 5997 6148 6299 6450 6606 6750 6906 7050 7199 7349 7497 7646 7795 7943 8091 8238 8386 8533 8686 8827 62 62 62 61 61 61 66 66 60 60 59 59 59 58 58 58 58 57 57 57 57 56 56 55 56 55 55 54 54 54 54 53 53 53 53 52 52 52 52 52 51 51 51 50 56 50 50 49 49 49 48 49 48 48 48 4? 47 a1 47 46 p. P 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49_ 50 51 52 53 54 55 56 57 58 59 (JO liOff. Kxseo. /> I 180 170 6 18.0 17.0 7 21.0 19-8 8 24.0 22-6 9 27.0 2.S-5 10 30.0 28.3 20 60.0 5^-6 30 90.0 8=i.o 40 120.0 i'3-3 50 150.0 141.6 20 30 40 50 30 40 50 40 5" 40 50 40 50 160 16 .0 18.6 21.3 24.0 26.6 53- 80.0 ic6.6 133- 150 140 14.0 21.0 23-3 46.6 70.0 93-3 116. 6 15 17 5 20 22 5 25 50 75 100 .0 125 .0 0.9 0.9 o 1.1 1.0 I 1.2 1.2 1 1.4 1.3 I 1 .6 1.5 I 3-1 3-0 2 4-7 4-5 4 6.3 6.0 5 7-9 7-5 7 7 o 7 0.9 0.6 s C 05 0.6 0. 7 0. 0.8 c.g 1-8 I . 2.7 3-6 4.6 3- 4 P. p 398 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 10 11 6 7 8 9 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 26 27 29 80 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Lojf. Vers. 1 J> Lop. Kxsec. /> 8. 18162 .18306 .18456 .18594 .1S738 8 .18881 . 19024 .19167 .19309 .1945^ 8.19594 .19736 .19878 .20019 .20166 8.20301 . 20442 .20582 .20723 .20861 8. 21003 .21142 .21282 .21421 . 2 [ 560 8.2169^ .21837 .2197^ .221 13 .22251 8.22389 .22526 .22663 .22800 .22937 8.23073 .23209 .23346 .23481 .23617 8.23752 .23888 .24023 .24158 . 24292 50 51 52 53 54 55 56 57 58 59 8.24425 .24561 .24695 .24828 . 24962 25095 25228 25361 ' 25494 ,25627 (iO 8.25759 .25891 .26023 .26155 .26285 8.2641^ Lotf. Vers. I J> 144 144 144 143 U3 143 142 142 142 142 142 142 141 141 141 146 146 146 140 140 139 139 139 139 138 '38 138 138 137 138 137 137 136 137 136 136 136 13^ 136 135 135 135 135 134 134 134 134 133 133 133 133 133 132 133 132 132 132 132 131 131 8. 18827 .18973 .19120 . 1 9266 .19411 8.19557 .19702 .19847 .19992 .20137 8.20281 .20425 . 20569 .20713 .20857 8.21 000 .21143 .21286 .21428 .21571 8.21713 .21855 .21995 .22138 . 22279 8.22420 .22561 .22701 .22842 .22982 8.23122 .23262 .23401 .23540 .23679 8.23818 .23957 .24095 .24234 .24372 8.24509 . 24647 .24784 .24922 .25059 8.25195 •25332 .25468 .25604 .25746 8.25876 .26012 .2614^ .26282 .26417 8.26552 . 26685 .26821 .26955 .27089 8. 27223 iOp. Kxsec. 146 146 146 145 145 '45 145 145 144 144 144 144 144 143 143 143 143 142 142 142 142 141 141 141 141 146 146 146 140 140 140 139 139 139 139 138 138 138 138 137 138 137 137 137 136 136 136 136 136 136 135 135 135 135 134 I 134 ' 134 ] 134 ,134 ; 134 It Loff. Vers. Jt 8.26417 •26548 .26679 .26816 .26941 8.27071 .27201 •27331 .27461 .27596 8.27719 .27849 .27977 .28105 .28235 8.28363 .28491 j .28619 • 28747 .28S75 8.29002 .29129 ■29^56 .29383 .29510 8.29635 ,29763 .29889 .30015 .30146 8, 30266 3039' 305 1 6 30642 30765 . 3089 1 .31015 .31140 .31264 .31388 8.3151T .3>635 .31758 .31882 .32005 8.32128 .32256 •32373 •32495 .32617 8.32739 .32861 •32983 •33104 •33225 8.33347 • 33468 •33588 . 33709 •33829 8.33930 liOir. Vers. 131 131 131 136 130 130 130 130 129 129 129 128 I 29 '28 128 128 128 128 127 '2^ 127 127 127 126 126 126 126 126 125 125 125 125 125 124 124 124 124 124 124 123 124 '23 123 123 123 122 122 I --) ^ 122 122 122 I2T I2T 121 121 121 126 126 126 126 /> Kxser. n 8.27223 ^ •27356! .27490 .27623 .27756 8.27889 .28021 .28153 .28286 .28418 8.28550 .28681 .28813 . 28944 .29075 8 . 29206 •29336 • 29467 •29597 . 29727 8 .29857 • 29987 .30117 .30245 .30375 8 . 30504 •30633 .30762 .30896 .31019 8 • 3"47 .31275 .31402 •31530 .31657 8.32418 •3254-+ .32676 •32796 .32922 8.31785' .31912 .32039 .32165 .32292 8.33047 •33173 •33298 •33423 •33547 8.33675 •33797 •33921 .34045 .34169 8.34293 .34417 •34540 . 34663 .34785 33 33 33 33 j- 32 30 29 29 29 29 29 28 28 28 28 28 27 27 27 27 27 27 26 26 26 26 26 26 25 25 25 25 25 24 25 24 24 24 23 24 24 3 _4 5 6 7 8 9 10 1 1 I 2 13 i4 15 16 17 18 19 r. 1' 8 . 3490') l.nir. Kxsnr. J> 'JO 21 25 26 27 28 29 '60 3' 32 33 1^ 35 36 37 38 39 40 41 42 43 44 45 40 47 48 49 :>o 5' 52 53 55 56 57 58 59 <>0 6 13 7 S 15.1 J7.5 9 19.5 10 21.^ 20 30 40 50 43-3 65.0 86.^ 108.3 40 50 40 50 20 30 40 50 03 ©•3 0.4 0,4 0.5 i.o 2.0 2-5 120 12 .0 14 .0 16.0 18.0 20.0 40.0 60 o 80.0 100. o 3 0-3 0.4 0.4 0.5 0.6 1 .1 2 0.2 0.3 0.3 0.4 0.4 o.§ 1.2 2 0. 0.2 0. 2 0. 3 0. 3 0. 6 0. I 0. I 3 I I 6 I 03 0.4 I*. V. 399 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 13° 13° 10 II 12 13 14 15 i6 i8 19 20 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Los. Vers. 33950 34070 34190 34309 34429 34549 34668 34787 34906 35025 35143 35262 35386 35498 35616 35734 35852 35969 36086 36204 36321 3643? 36554 36671 36787 36903 37019 37135 37251 37366 37482 3759^ 37712 3782^ 37942 38057 38171 382S6 38400 38514 38628 38741 38855 38969 39082 39195 39308 39421 39534 39646 39758 39871 39983 40095 40207 D 40318 40430 40541 40652 40764 8.40875 Lour. Vers, i I) 20 20 19 20 9 9 9 9 9 Lojf. Exsec. 2> 8 34909 35032 35155 35277 35399 35522 35644 35765 35887 36009 36130 36251 36372 36493 36614 36734 36855 36975 37095 37215 37931 38050 38169 38287 38406 38524 38642 38760 38878 38995 39113 39230 39347 39464 3958T 39698 39814 39931 4004^ 40163 40279 40395 405 II 40626 40742 40857 40972 4108^ 41202 41317 37335 37454 37574 37693 37812 41431 41546 41666 41774 41888 42002 jOfj. Kxsec. 23 22 22 22 22 22 21 22 21 21 21 21 26 21 26 26 20 20 20 20 9 9 9 9 9 8 9 8 8 8 J 1 7 7 7 7 6 6 6 6 6 6 6 5 5 5 5 5 5 5 4 4 4 4 4 4 4 IF Log. Vers. | It |Loif. Exsec. 40875 40985 41096 412O6 41317 41427 4153^ 4164^ 41757 41867 41976 42086 42195 42304 42413 42522 42636 42739 4284^ 42956 43064 43172 43280 43388 43495 43603 43710 43817 43924 44031 44138 44245 44351 44458 44564 44670 44776 44882 44988 45093 45 '99 45304 45409 45514 45619 45724 45829 45934 46038 46142 46247 46351 46455 46558 46662 46766 46869 46972 47076 47179 47282 Lotf. Vers. 10 16 16 16 16 10 10 09 10 09 09 09 09 09 09 08 09 08 08 08 08 08 08 07 07 07 07 07 07 06 07 06 06 06 06 06 05 06 05 05 05 05 05 05 05 04 05 04 04 04 04 04 03 04 03 03 03 03 03 03 "77" 42002 42 II 6 42229 42343 42456 42569 42682 42795 42908 43021 43133 43246 43358 43470 43582 43694 43805 43917 44028 44'39 44251 44362 44473 44583 44694 44804 44915 45025 45135 45245 45355 45465 45574 45684 45793 45902 4601 T 46126 46229 46338 46446 46555 46663 46771 46879 46987 47095 47203 47316 47417 47525 47632 47739 47846 47953 48060 48166 48273 48379 48485 8.48591 Losj. Exsec. D 13 13 13 13 13 13 '3 13 12 12 12 12 12 12 12 i" I I I I I I 16 16 16 16 10 16 09 10 10 09 09 09 09 09 09 08 09 08 08 08 08 08 08 07 08 07 07 07 07 07 07 06 07 05 06 06 06 06 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 55 56 57 58 59 (50 P. P 120 119 6 12,0 II. 9 7 8 9 14.0 16.0 18.0 139 15.8 17-8 10 20.0 19-8 20 30 40 40.0 60.0 80.0 39-6 59-5 79-3 50 100. 99.1 117 II. 7 13-6 15.6 175 19-5 39 o 58.5 78.0 97.5 13-5 15-4 17.4 "9-3 38.6 58 o 77-3 96.6 118 116 115 II-5 13.4 ^5-3 17.2 iQ.i 38.3 57-5 76.6 95.8 II 4 113 11: 6 11.4 "•3 II. 7 13 3 13.2 13- 8 TS 2 15.0 14- 9 17 I 16.3 16. 10 19 18.8 18. 20 38 37-6 37 30 57 5'j-5 5b. 40 76 75.3 74- 50 95 94.1 93- III IIO 6 11 .1 11. 7 12.9 12.8 8 14.8 14-6 9 16.6 16.5 10 18.5 18.3 20 37-0 36-6 30 55-5 55-0 40 74.0 73-3 SO 92-5 91-6 109 14-5 16.3 18.1 36.3 54-5 72-6 90.8 6 108 10.8 107 10.7 7 12.6 12-5 8 9 14.4 16.2 14.2 16.0 10 20 18.0 36.0 17-8 35-6 30 54 53-5 40 50 72.0 90.0 89.1 106 10.6 105 104 6 10.5 10.4 1 7 12.2 12 I 8 14.0 13 8 9 '5-7 13 fi 10 17-5 17 3 20 35 -o 34 6 33 52.5 52 40 70.0 tq 3 50 87.5 86 6 0.0 0.0 0.0 O.I O.I O. I 0.2 0-3 0.4 P. l* 400 TABLE VIII. —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 14° 15" 10 1 1 12 J3 U 15 i6 i8 19 20 21 22 23 24 25 26 27 28 29 80 31 32 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 Loir. V«>rs. I) 8.47282 47384 47487 47590 47692 8 47795 47897 47999 48 1 01 48203 48304 48406 48507 48609 48716 4881 1 48912 49013 491 14 49215 49315 494 Id 49516 49616 49716 49816 49916 50015 501 1 5 50215 50314 50413 50512 5061 1 50710 50809 50908 51006 51105 51203 5 1 301 51399 51497 51595 51693 51791 51888 519.S6 52083 52180 52277 52374 52471 52568 52665 52761 52858 52954 53050 53146 53242 02 03 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01 CO 01 06 00 00 00 00 00 00 99 100 99 99 99 99 99 99 98 99 98 98 98 98 98 98 98 97 98 97 97 97 97 97 97 97 96 97 96 96 96 96 96 96 Loir. KxN«M'. /> Loir. Vers. /> 8 48591 48697 48803 48909 49014 49120 49225 49331 49436 4954 » 49646 49750 49855 49960 50064 50168 50273 50377 50481 50585 50688 50792 50896 50999 51 102 51205 51309 51412 51514 51617 51720 51822 51925 52027 52129 52231 52333 52435 52537 52638 52740 5284T 52943 53044 53'45 53246 53347 53448 53548 53649 53749 53850 53950 54050 54150 54250 54350 54449 54549 54649 54748 Kxser. 06 06 05 05 05 05 05 05 05 05 04 05 04 04 04 04 04 04 04 03 04 03 03 03 03 03 03 02 03 02 02 02 02 02 02 02 02 oT 01 01 01 of 01 01 01 01 01 00 06 06 06 00 00 00 00 00 99 100 99 99 /> Loir. Vers. 8 8 53242 5^ ^ ^ o jjj8 53434 53530 53625 53721 53816 53911 54007 54102 /> 54197 54291 54386 54481 54575 54670 54764 54858 54952 55046 55140 55234 55328 55421 55515 55608 55701 55795 55888 55981 56074 56166 56259 56352 56444 56536 56629 56721 56813 56905 56997 57089 57186 57272 57363 57455 57546 57637 57728 57819 57910 58001 58092 58182 58273 58363 58453 58544 58634 58724 ;88i4 96 95 96 95 95 95 95 95 95 95 94 95 94 94 94 94 94 94 94 94 93 94 93 93 93 93 93 93 93 93 92 92 93 92 92 92 92 92 92 92 92 91 91 91 91 91 91 91 91 91 96 91 90 96 90 90 96 90 90 90 Lojr. Vers. /> /> 8 Lmr 55736 55834 55933 56031 56129 56226 56324 56422 56519 56617 56714 56812 56909 57006 57103 57200 57296 57393 57490 57586 57682 57779 57875 57971 c;8o67 58163 58259 58354 58450 58546 58641 58736 58832 58927 59022 59117 592 1 T 59306 59401 5949^ 59590 59684 59779 59873 59967 60061 60)55 60249 60342 60436 60 5 30 99 99 99 99 99 99 98 98 98 98 98 98 98 98 97 98 97 97 97 97 97 97 97 97 97 96 97 96 96 96 96 96 96 96 95 90 95 96 95 95 95 95 95 95 95 94 95 94 94 94 94 94 94 94 94 94 94 93 94 93 10 1 1 I 2 13 14 15 16 17 18 19 20 21 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 i9 50 51 52 53 54 55 56 57 58 V. V 103 10.3 68.6 102 10 lOI II 9 11.8 13 6 13.4 15 3 15.: 17 .6.fi 34 33-6 5» 50-5 68 67. H 85 84.1 100 99 6 1 0.0 9.9 7 II. 6 "•5 8 '3-3 13.2 9 10 15-0 16.6 14.8 16.5 20 33-3 33-0 30 40 50.0 66.6 4Q-5 66.0 50 83.3 82.5 98 9.S 1 1. 4 13.0 14.7 ,6.5 3«.6 49.0 65.3 8:. 6 6 9 . 9 7 7 96 9.6 95 9-5 7 II 3 II. 2 11. 1 8 12 9 12.8 12.6 9 10 14 16 i 14.4 16.0 14.2 »5-8 20 30 40 32 48 64 3 5 32.0 48.0 64.0 3«-6 47 -5 633 50 80 8 80.0 79.1 94 93 b 9.4 9-i 1 7 10.9 10.8 8 12.5 12.4 9 14. 1 «3-9 10 >5-0 15-5 20 31-3 31.0 30 47.0 46.5 40 62.6 62.0 50 78.3 77.5 92 Q.2 10.7 12.2 13.8 J5 3 30-6 46.0 61.3 76.6 91 90 6 9-1 9.0 7 10.6 10.5 8 12. I 13.0 9 '3-6 »3o 10 J5-I '5 20 30-3 30.0 ^0 45-5 45.0 40 60.^ 60.0 50 75-8 75.0 0.11 06 0.0 O. I O.I o.i o.a o 3 0.4 ^!^s('(• /> I 401 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 16 11° 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 29 80 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Loa:. Vers. 50 51 52 53 54 55 56 57 58 59 (io 8.58814 58904 58993 59083 59173 z> 8 59262 59351 59441 59530 59619 59708 59797 59886 59974 60063 60152 60246 60328 60417 60505 60593 60681 60769 60857 60944 61032 61119 61207 61294 6138T 61469 61556 61643 61730 61816 61903 61990 62076 62163 62249 62336 62422 62508 62594 62680 62766 62852 62937 63023 63108 63194 63279 63364 63449 63534 63619 63704 63789 63874 63959 8 . 64043 Lost. Veix. 7> 90 89 90 89 89 89 89 89 89 89 88 89 8S 88 87 Sf 87 87 87 87 87 87 86 87 86 86 86 86 86 86 86 86 86 85 85 85 85 85 85 85 85 84 85 84 Loff. Kxsec 8.60530 .60623 .607 16 .60810 . 60903 8.60996 .61089 .61182 .61275 .61368 8.61466 •61553 .61645 .61738 .61830 8.61922 .62014 .621O6 .62198 .62296 8.62382 .62474 .62565 .62657 .62748 n i. 62840 .62931 .63022 .63113 .63204 8.63295 .63386 .63477 .63567 .63658 8.63748 .63839 .63929 .64019 .64109 8.64199 .64289 •64379 . 64469 .64559 8 . 64649 •64738 .64828 .6491^ .65006 8.65096 .65185 .65274 •65363 .65452 8.65541 .65629 .65718 .65807 •65895 8.65984 93 93 93 93 93 93 93 92 93 92 92 92 92 92 92 92 92 92 92 91 92 91 91 91 9^ 91 91 91 91 96 91 91 96 90 90 96 90 90 90 90 90 90 90 89 90 89 89 89 89 89 89 89 89 89 89 88 88 89 88 88 Kxsec. I J) Log 8. Vers. 64043 64128 64212 64296 64381 ,64465 ,64549 64633 ,64717 ,64801 ,64884 , 64968 ,65052 .65135 .652I8 ,65302 .65385 .65468 ,65551 •65634 ,65717 ,65806 ,65883 ,65965 , 66048 .66131 .66213 .66295 .66378 , 66460 .66542 .66624 ,66706 ,66788 ,66870 ,66951 67033 ,67115 •67196 ,67277 67359 .67446 .67521 .67602 •67683 ,67764 ,67845 67926 , 68007 ,6808^ ,68168 .68248 .68329 . 68409 ,68489 8.68569 .68650 .68730 .68810 .68889 8.68969 Loir. AVrs. n 84 84 84 84 84 84 84 84 84 83 83 84 83 83 83 83 83 83 83 83 83 82 82 83 82 82 81 81 82 81 81 81 81 81 81 81 81 81 86 81 86 86 86 86 86 80 80 86 80 80 79 80 "tT" Lost. Kxsec. 8. 65984 66072 66166 66248 66336 ,66425 ,66512 . 66606 .66688 .66776 8. 66863 66951 67039 67126 67213 8. 01 67j 67388 67475 67562 67649 ,67736 ,67822 ,67909 .67996 ,68082 ,68169 ,68255 ,68341 ,68428 .68514 . 68600 .68686 .68772 ,68858 , 68944 .69029 .69115 .69201 ,69286 .69372 69457 69542 ,6962^ ,69712 ,69798 69883 6996^ 70052 ,7013^ ,70222 70306 70391 70475 ,70560 70644 8.70728 .70813 . 70897 .70981 .71065 8.71149 iOjr. Kxspc. i» 88 88 88 8? 88 88 87 8? 88 S? 87 8? 8f 87 87 87 87 87 86 87 86 86 86 86 86 86 86 86 86 85 86 86 85 86 85 85 85 85 85 85 85 85 85 84 85 85 84 84 84 84 84 84 84 84 84 84 84 84 "77" p. P. 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 CO 93 92 6 9-3 9.2 7 10. 8 10.7 8 12.4 t2.2 9 13-9 13-8 10 15-5 15.3 20 31.0 30.6 ^0 46.5 46.0 40 62.0 61.3 50 n-s 76.6 90 89 6 9.0 8.9 7 10.5 10.4 8 12.0 II. 8 9 13^5 13-3 10 15.0 14-8 20 30.0, 29^6 30 450 44-5 40 60.0 59-3 50 75-0 74.1 87 86 6 8.7 8.6 7 10. 1 10. 8 II. 6 II. 4 9 13.0 12.9 10 14-5 14.3 20 29.0 28.6 30 43-5 43^o 40 58.0 57-3 50 T^'l 71-6 84 83 6 8.4 8.3 7 9.8 9-7 8 II. 2 1 1.O 9 12.6 12.4' 10 14.0 13.8 20 28.0 27.6 30 42.0 41.5 40 56.0 55.3 50 70.0 69.1 81 80 6 8.1 8.0 7 8 9.4 10.8 9-3 10.6 9 12.1 12.0 10 20 13-5 27 133 26.6 3<3 40 50 40-5 54-0 67.5 40.0 53-3 66.6 7 8 0.0 0.0 9 10 I 0.1 20 0.1 30 40 50 0.2 0-3 0.4 91 9- 10. 1 12. 13-1 15- Z°' 45- 60. 75-8 88 II. 7 13.2 14-6 29-3 44.0 58.6 73- 85 8.S 9.9 "•I 12.7 14. 28.3 42.5 56.6 70.8 82 8.2 9-5 10.9 12.3 13-6 27. 41. 54-6 68.3 79 7-9 9- 10.5 II . 26.3 39-5 52 6 65^8 P. P 402 Table viil— logarithmic versed sines and external secants. 18' 19 10 1 1 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 30 31 32 33 34 35 36 37 38 39 10 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Loff. Vers, i 2> 8.68969 .69049 .69129 .6920^ .69288 8.69367 .69446 .69526 ,69605 .69684 8.69763 .69842 .69921 . 70000 . 70079 8.70157 .70236 .70314 .70393 .70471 8,70550 .70628 .707O6 .70784 . 70862 8 . 70946 .71018 .71096 .71174 .71251 71329 71406 ,71484 ,71561 ,71639 71716 .71793 .71876 ■71947 .72024 8.7210T .72178 .72255 .72331 .72408 8.72485 .72561 ,72637 .72714 ,72796 GO 8.72866 .72942 .73018 •73094 ,73176 8.73246 .73322 •73398 •73473 •73549 8.73625 Log. Vers 79 80 79 79 79 79 79 79 79 79 79 79 7^ 79 78 78 78 78 78 7^ 78 78 78 78 78 78 77 78 77 71 7l 77 77 77 77 77 77 77 77 77 7l 77 76 77 7l 76 76 76 76 76 76 76 76 76 76 76 7% 71 76 7% Log. Kxsec. .71149 .71232 .713I6 , 7 1 400 ,71484 J* 8.71567 ,71651 .71734 ,71817 . 7 1 90 1 8,71984 .72067 .72150 .72233 .72316 8.72399 .72481 .72564 .72647 .72729 8.72812 .72894 .72977 •73059 .73141 8.73223 .73306 .73388 .73470 .73551 8.73633 •73715 .73797 .73878 .73960 74041 .74123 . 74204 ,74286 .74367 8.74448 .74529 .74616 .74691 .74772 8.74853 .74934 .75014 .75095 .75175 8.75256 •75336 •75417 .7549? •75577 8.75658 •75738 .75818 .75898 •75978 8.76058 J> IliOSf. Kxsec 83 84 83 84 83 83 83 83 83 83 83 83 83 83 83 82 83 82 82 82 82 82 82 82 82 82 81 82 82 81 81 81 81 81 81 81 81 81 81 81 81 86 81 81 86 86 86 86 86 86 86 80 86 80 80 80 80 80 Log. \ /> 8 7> 73^23 73700 73775 73851 73926 74001 74076 74151 74226 7430T 74376 74451 74526 74606 74675 74749 74824 74898 74973 75047 75121 75195 75269 75343 75417 75491 75565 75639 75712 75786 75860 75933 76006 76080 76153 76226 76300 76373 76446 76519 76592 76664 7673? 76810 76883 76955 77028 77106 77^72> 772AS 771>^7 77390 77462 77534 77606 77678 77750 77822 77893 77965 78037 75 75 75 75 75 75 75 75 75 75 74 75 74 7-1 74 71 74 7l 74 74 74 74 74 74 74 73 73 73 73 74 73 74 73 73 73 73 73 73 73 73 72 73 72 73 72 72 72 72 72 72 72 72 72 72 72 72 72 71 72 71 Log. KxKec. Log. Vers. -?> 76058 76137 76217 76297 76376 76456 76536 76615 76694 76774 76853 76932 7701T 77096 77169 48 .2? 77 77?>'^ 77406 77485 77563 77642 77799 77^77 77956 It 78034 78112 78191 78269 78347 78425 78503 78581 78659 7873S 78814 78892 78969 79047 79124 79202 79279 79357 79434 79511 79588 79665 79742 79819 79896 79973 80050 80126 80203 80280 80356 80433 80509 80586 S0662 8.80738 79 80 79 79 80 79 79 79 79 79 79 79 79 79 79 79 78 79 78 78 78 79 78 78 78 78 78 78 78 78 78 78 78 71 78 7l 7l 71 7l 7l 7l 7l 77 77 7l 77 77 77 77 76 77 76 77 76 76 76 76 76 76 76 liOir. Kxser.l i> 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 32 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 P. 1*. 84 83 6 84 8.j 7 9 8 9-7 8 II 2 11. 9 12 6 12.4 10 >4 13-8 20 28 27 6 30 42 4^ 5 40 56 55-3 50 70 69.1 81 80 6 8.1 8.0 7 8 9.4 10.8 9^3 10.6 9 12 I 12.0 10 '3.5 i3^3 20 27.0 26.6 30 40.5 40.0 40 50 54.0 67.5 53 • 3 66.6 78 77 6 7.8 7.7 7 9.1 9.0 8 10.4 10.2 9 II. 7 "•5 10 13.0 12. § 20 26.0 25-6 30 39.0 38.5 40 52 .0 5'-3 50 65.0 64.1 75 74 6 7.5 7^4 7 8.7 8.6 8 10.0 9-8 9 II. 2 II .1 10 12.5 12 3 20 25.0 24-6 30 37.5 37.0 40 50.0 49.3 50 62.5 61.6 72 71 7.2 7.« 8.4 8.3 9.6 9.4 10.8 JO. 6 12.0 "8 24.0 23^ 36.0 3S^5 48.0 *7-3 60.0 59.1 82 8.2 j 9-5 : 10. y ' 12.3 ' »3.6 27^3 41 .0 54.6 68.3 79 7.9 9.2 10 5 !'.§ 13.1 26.3 39-5 52.6 65-3 76 7.6 II. 4 ia.§ 25.3 38.0 63.3 73 7^3 8.5 9-7 10.9 12. 24. 36. 48.6 60.8 0.3 0.4 r. i'. 403 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 20° 21° Log. Vers. 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 8.78037 .78108 .78180 .78251 .78323 8.78394 .78466 .78537 . 78608 .78679 8.78753 .78821 .78892 .78963 •79034 8, 79105 79175 79246 79317 79387 jy Log. Exsec. X> 79458 79528 79598 79669 79739 79809 ,79879 •79949 .80019 . 80089 80159 .80229 . 80299 • 80369 ■ 80438 80508 8057^ , 80647 .807 16 . 80786 .80855 . 80924 .80993 .81063 .81132 (>0 81201 .81270 ■81339 . 8 1 407 •81476 81545 .81614 ,81682 .81751 ,81819 8.81888 •81956 .82025 .82093 .82161 8.82229 Log. Vers. 71 71 71 71 71 71 71 71 71 71 71 71 71 70 71 70 71 70 70 70 70 70 70 70 70 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 69 69 69 69 69 69 69 68 69 68 69 68 68 68 68 68 68 68 68 68 8 .80738 .80814 .80891 . 80967 •81043 8.81119 .81195 .81271 .81346 .81422 8.81498 .81573 .81649 .81725 .81806 8.81876 .81951 .82025 .82102 .82177 82252 82327 82402 8247^ 82552 .82627 .82702 .82776 .82851 .82926 . 83006 .83075 .83149 •83224 .83298 •83373 •83447 .83521 .83595 .83670 .83744 .83818 .83892 .83966 • 84039 8.84113 .84187 .84261 .84334 . 84408 .84481 .84555 .84628 . 84702 •84775 8.84848 .84922 .84995 .85068 .8514^ 8.85214 Log. Kxsec. 76 76 76 76 76 76 76 75 76 75 75 76 75 75 75 75 75 75 75 75 75 75 75 74 75 75 74 75 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 73 74 73 74 73 73 73 73 73 73 73 73 73 73 73 73 73 Log. Vers. 82229 .8229^ ,82366 .82434 .82502 82569 8263^ 82705 82773 82841 829O8 ,82976 . 83043 ,83111 83178 83246 83313 83386 8344^ 83515 ,83582 ,83649 ,83716 .83783 83850 839I6 .83983 .84050 .84117 .84183 D Loe. Exsec. 84250 843I6 .84383 . 84449 .84515 84582 84648 ,84714 , 84786 ,84846 ,84912 84978 .85044 .85116 .85176 85242 85308 85373 85439 85505 85570 85626 85701 85766 85832 8.85897 .85962 .8602^ . 86092 .86158 8.86223 J) \ Lost. Vers. 68 68 68 68 6? 68 68 6? 68 67 6J 6? 6^ 6? 6J 67 6J 67 6J 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 65 66 65 65 66 65 65 65 65 65 65 65 65 65 65 65 8. 85214 8528^ 85366 85433 85506 85579 85651 85724 85797 85869 ,85942 86014 86087 ,86159 ,86231 86304 86376 86448 86526 86592 86664 86736 86808 ,86886 86952 87024 87095 ,8716^ .87239 ,87316 87382 87453 87525 87596 87668 87739 ,87816 .87881 ■87953 . 88024 z> 88095 88166 88237 ,88308 .88378 88449 88526 ,88591 ,88661 ,88732 88803 ,88873 88944 89014 ,89085 8.89155 .89225 .89295 .89366 .89436 8.89506 t) Log. Exsec. 73 73 72 73 73 72 73 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 71 72 71 72 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 71 76 71 71 76 70 71 76 76 70 76 76 70 76 70 76 70 70 n 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 r. p. 50 51 52 53 55 56 57 58 i9^ 60 76 75 74 6 7.6 7.5 7 «-8 8.7 « 10. 1 10.0 9 11.4 11.2 10 12 6 12.5 20 25-3 25.0 30 38.0 37.5 40 50.6 50.0 50 63.3 62.5 20 30 40 50 7.4 12.3 24.6 37-0 49-3 61.5 73 72 6 7 8 7.3 8.5 9.7 7.2 8.4 9.6 9 10.9 10.8 10 12.1 12.0 20 30 40 50 24-3 36.5 48.6 60. § 24.0 36.0 48.0 60.0 6 7 70 7.0 8.1 69 6.9 8.5 8 9.3 9.2 9 10.5 10.3 10 II. 6 II. 5 20 233 23.0 30 40 50 35-0 4C.6 58.3 34.5 46.0 57-5 71 7.1 8.3 9.4 10. 6 II. § 23.6 35-5 47.3 59- 68 6.8 7-9 9.0 IO-2 ".§ 22.6 34-0 45-3 56-6 6 67 6.7 66 6.6 6« 6. 7 8 7.8 8.9 7.7 8.8 7- 8. 9 10. 9.9 9. 10 II .1 II. 10 20 22.3 22.0 21. 30 33-5 33.0 32 40 44.6 44.0 43 • 50 55.8 55.0 54- P. p 404 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 22° 2:r 10 1 1 12 15 i6 17 i8 19 20 21 22 23 24 -5 26 27 28 29 30 31 32 34 J3 36 37 38 39 40 41 42 43 44 45 46 47 4S 49 50 51 52 53 54 Loj?. Vers. ! J* 8.S6223 .86287 .86352 .86417 .86482 8.86547 .86612 .86676 .86741 .86805 8.86870 .86934 . 86999 .87063; .87127 I 8.87192 .87256 .87326 .87384 • 87448 8.87512 •87576 .87640 .87704 .87768 8.87832 .87895 .87959 .88023 . 88085 8.88150 .88213 .88277 .88340 . 88404 8.88467 .88536 .88593 .88656 .88720 8.88783 .88846 .88909 .88971 • 89034 8 . 89097 .89160 •89223 .89285 •89348 8.8941 1 •89473 .89536 .89598 . 89666 55 56 57 58 59 (>0 8.89723 •89785 •89847 .89910 •89972 8 • 90Q34 IjOj;. Vers. I 7> 64 65 65 65 64 65 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 63 64 63 64 63 63 63 63 63 63 63 65 63 63 63 63 63 63 63 62 63 63 62 63 62 62 63 62 62 62 62 62 62 62 62 62 62 Kxsec I> 8 89506 89576 89646 897 16 89786 89856 89926 89995 90065 90135 90205 I 90274 I 90344 90413 90483 I 90552 90622 90691 90766 90830 90899 90968 91037 91106 91175 91244 91313 91382 91451 91520 91588 91657 91726 91794 91863 91932 92006 92063 92137 92205 92274 92342 92416 92478 92546 92615 92683 92751 92819 92887 92955 93022 93096 93158 93226 93361 93429 93496 93564 8.93631 iOS. Kxsec. 70 70 70 69 70 70 69 70 69 70 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 68 69 69 68 69 68 68 68 69 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 67 68 67 68 67 68 67 67 67 67 "77" Lotf. Vers. J> 8 . 90034 90096 901 58 90220 90282 90344 90406 90467 90529 90591 90652 90714 90776 90837 90899 90966 91021 91083 91144 91205 91267 91328 91389 91450 91511 91572 91633 91694 91755 91815 91876 91937 91997 92058 921 19 92179 92240 92306 92361 92421 92487 92542 92602 92662 92722 92782 92842 92902 92962 93022 93082 93142 93202 93261 93321 93381 93440 93506 93560 93619 8.93679 lidsr. Vers. 62 62 62 62 62 62 61 62 61 61 62 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 6r 61 66 61 66 66 61 66 66 66 66 66 60 66 66 60 60 66 60 60 60 60 60 60 59 60 59 60 59 59 60 59 59 59 /> Loir. Kxsec. 7> 8 8 93631 93699 93766 93833 93901 93968 94035 94102 94170 94237 94304 94371 94438 94505 94572 94638 94705 94772 94839 94905 94972 95039 95105 95172 95238 95305 95371 9543? 95504 95570 95636 95703 95769 95835 95901 95967 96033 96099 96165 96231 96297 96362 96423 96494 96560 96625 96691 96757 96822 96888 96953 97013 97084 97149 97214 97280 97345 97410 97475 97540 ()76o''> jOir. Kxsec. 67 67 67 67 67 67 67 67 67 67 67 67 67 67 66 67 66 67 66 66 67 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 65 66 65 66 65 65 66 65 65 65 65 65 65 65 65 65 65 65 65 65 /> 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 i9 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 sQ CO 70 69 6 70 0.9 7 8 I 8.0 8 9 3 9.2 9 10 5 10.3 10 II f> 11. 5 20 23 3 23.0 .1° 35 34-5 40 46 fi 46.0 50 5B 3 57-5 68 6.8 7.9 9.0 10.2 It. 3 22.6 34.0 45-3 56.0 67 66 6 6.7 0.6 7 7.8 7.7 8 8.9 8.8 9 10.0 9.9 10 II. I II. 20 22 3 22.0 30 33.5 33.0 40 44-6 44.0 50 55. 8 55.0 65 6.5 7.6 9.7 10. § 21.6 32.5 43-3 54.1 64 63 62 6 6.4 6.3 6. 7 7.4 7-3 7- 8 8.5 8.4 8. 9 9.6 9.4 9- 10 10 6 10.5 10 ■20 = 1-3 21.0 20. 30 32.0 31.5 3» 40 42.6 42.0 41. 50 53.3 52.5 5'. 6 [ 60 6 6.1 6.0 7 8 7 8 I I 7.0 8.0 9 9 I 9.0 10 10 I 10. 30 20 3 20.0 30 30 5 30.0 40 40 6 40.0 50 50 8 50.0 59 5-9 6.9 H 9.§ »9 6 29.5 39.3 49. i 5 6 0.0 7 8 9 I 10 t 20 I 30 2 40 3 50 4 I'. I". 405 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 24° 25° 10 1 1 12 15 i6 17 i8 19 20 21 22 23 24 25 26 27 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Los;. Vers. 8 00 93679 93738 93797 93857 93916 D iLoff. Exsec 93975 94034 94094 94153 94212 94271 94330 94389 94448 94506 94565 94624 94683 94742 94800 94859 94917 94976 95034 95093 95151 95210 95268 95326 95384 95443 95501 95559 95617 9567? 95733 95791 95849 95907 95965 96023 96086 96138 96196 96253 9631 1 96368 96426 96483 96541 96598 96656 96713 96776 9682^ 96885 96942 96999 97056 97113 97170 59 59 59 59 59 59 59 59 59 59 59 59 59 58 59 59 58 59 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 57 58 58 58 Sf SJ 58 57 5f 57 57 5f 57 57 57 57 57 57 5? 57 57 Si 57 57 8.97606 .97671 .97736 .97801 •97865 >. 97930 •97995 . 98060 .98125 .98190 I) 8.98254 .98319 .98383 .98513 8.98577 .98642 .98706 .98776 •98835 8.98899 .98963 .99028 .99092 •99156 .99220 .99284 ■99348 .99412 •99476 ;. 99540 .99604 .99668 .99732 .99796 8.99860 .99923 8.99987 9.00051 .001 14 9.OJ178 . 00242 .00305 ,00369 .00432 9-00495 .00559 .00622 .00686 .00749 Log. Vers. 1 I) Log. Kxsec. J> 9 . 008 I 2 .00875 .00938 .01002 .01065 9.01 128 .01191 .01254 .01317 .01380 9.01443 65 65 65 64 65 65 64 65 65 64 64 64 65 64 64 I 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 63 64 63 64 63 63 64 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 Log. Vers. 8.97176 9722^ 97284 97341 97398 8 I) Log. Exsec. 2> 97455 97511 97568 97625 9768T 97738 97795 97851 97908 97964 98026 98077 98133 98190 98246 98302 98358 98414 98470 98527 98583 98639 98695 98750 98806 98802 989 1 8 98974 99030 99085 99141 99197 99252 99308 99363 99419 99474 99529 99585 99646 99695 99751 99806 99861 999 1 6 99971 00025 00081 00136 00 1 91 00246 00301 00356 0041 1 00466 9.00520 Lost. V«»rs. 57 56 57 57 57 56 57 56 56 56 57 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 55 56 56 56 55 56 55 55 56 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 54 55 54 01443 01505 01568 01631 01694 OI756 01819 01882 01944 02007 02070 02132 02195 02257 02319 02382 02444 02506 02569 02631 02693 02755 0281^ 02880 02942 03004 03066 03128 03190 03252 03313 03375 03437 03499 03561 03622 03684 03746 03807 03869 03930 03992 04053 041 1 5 04176 04238 04299 04366 0442 T 04483 04544 04605 04666 04727 04788 04850 0491 1 04972 05033 £i°93 05154 62 63 62 63 62 63 62 62 63 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 61 62 62 61 62 61 61 62 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 6t 61 61 61 61 66 61 /> Loir. Exsec. I /> 5 6 7 8 9 10 II 12 13 14 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 i». p. 65 64 63 6 6.5 6.4 7 7.6 7-4 8 8-6 8^5 9 9-7 9.6 10 IO-8 10.6 2? 21-6 21.3 30 32-5 32.0 40 43-3 42.6 50 54-1 53 3 40 50 56 55 6 5-6 5^5 7 6^5 6.4 8 7-4 7^3 9 8.4 8.2 10 9-3 9.1 20 i«.6 18.3 30 28.0 27^5 40 37-3 36^6 50 46.6 45-8 8.4 9.4 10.5 21 .0 3i^5 42.0 52.5 62 61 60 6. 7.0 8.0 9^ 10.0 20.0 30.0 40.0 50- 59 58 6 5^9 5.8 7 6.9 6.7 8 7-8 7-7 9 8-8 8.7 lO 9^8 9-6 20 19-6 19-3 30 29^5 29.0 40 39-3 38.6 50 49.1 48.3 57 5-7 6.6 7.6 8.5 9-5 19 28 38 47-5 54 5.4 6. 7- 8. 9- 18. 97. 36.0 45- P. P. 406 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. ^G 27 liOK. VlMV /> 10 1 1 12 13 14 15 16 18 10 '20 21 22 ^3 ^4 -3 26 27 28 29 ;{o 31 32 33 34 35 56 37 38 30 4tl 41 42 43 44 45 46 47 48 49 9.00520 .00575 .00630 .00684 .00739 9.00794 .00848 .00903 .00957 .0101 T 9.0 .0 .0 .0 .0 9.0 .0 .0 .0 .0 9.0 .0 .0 .0 .0 9.0 .0 .0 066 120 174 229 28^, 337 391 44 3 499 t)05 662 715 769 823 877 93' 985 .02038 .02092 9.02146 .02199 .02253 .02307 .02360 9.02414 .02467 .02521 .02574 .02627 9.02681 .02734 .02787 .02840 .02894 9.02947 .03000 .03053 . 03 1 o5 .03159 51 - '^ 54 35 56 57 58 59 00 9.03212 .03265 .03318 •03371 •03423 9 -03476 .03529 .03582 •03634 •03^87 9-03740 liOa:. Vers. I /> 55 54 5-+ 54 55 54 54 54 54 54 5-i 54 54 34 54 54 5 + 54 54 54 54 53 54 54 54 53 54 53 54 53 53 54 53 53 53 - -> 53 53 53 53 53 53 53 53 53 53 53 53 53 53 52 53 53 52 52 53 52 liOir. Kxsec /> 9.03154 •O521S •05276 .05337 •05398 9^05458 .05519 .05580 .05640 .05701 9.05762 .05822 .05883 •05943 . 06004 9 . 06064 . 06 1 24 .06185 .0624^ .06305 9.06366 .06426 .06485 .06546 . 06605 9 . 06667 .06727 .06787 .06847 .06907 9.06967 .07027 .07087 •07146 .07205 9.07265 .07326 .07386 • 074-L^ .07505 9.07565 .07624 .07684 .07743 .07803 9.07863 .07922 .07981 . 0804 I . 08 1 00 9 . 08 1 60 .08219 •08278 •08338 •08397 9.08455 .0851I .08574 .08634 ■ 08693 9.08752 MH. Kxspc. 61 61 66 61 60 61 66 66 66 61 66 66 66 66 66 60 60 ,60 60 66 60 66 60 60 60 60 60 60 60 60 60 60 59 60 60 59 60 59 60 59 59 59 59 60 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 I am:. V«ms. /> 9.03740 .03792 .03845 .03898 .03950 ,04002 04055 .04107 ,04160 ,0421 2 9.04264 .04317 .04369 .O442T •04473 9.0452^ .04577 .04630 .04682 .04734 9.04786 •04837 .04889 .04941 •04993 9.05045 .05097 •05148 .05206 .05252 9^05303 •0535^ .05407 •05458 .05510 905561 .05613 .05664 .05715 .05767 9-05818 .05869 .05921 .05972 .06023 9.06074 . 06 1 2 5 .06175 .06227 .06279 9.06330 .06386 .06431 .06482 -06533 9.06584 .06635 .06686 .06735 .0678^ 9.06838 liOif. Vers. 52 52 53 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 51 52 52 52 51 52 51 52 52 51 52 51 51 51 51 51 5> 51 51 5J 51 51 5' 51 51 51 51 51 51 5' 50 51 51 51 51 50 5' 50 5' 50 /> 1.0 l. \^t'(■. /> 08752 I 0881 I 08870 08929 08988 09047 09106 09164 09223 09282 09341 09400 094 5 8 09517 09576 09634 09693 09752 09816 09869 09927 09986 0044 0102 0161 0219 0278 0336 039-+ 0452 051 1 0569 0627 0685 0743 080T 0859 0917 097^ 033 091 149 207 265 323 386 438 496 554 61T 669 727 784 842 899 95^ 2015 2072 2129 2 1 87 2244 K\s«>r. 59 59 59 59 59 59 58 59 59 58 59 58 59 58 58 58 59 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 57 58 58 57 58 58 57 57 58 57 57 5/ 57 58 3/ 57 57 57 57 "77" 4 5 6 7 8 9 10 1 1 12 J 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ^_ ;jo 31 32 jj 34 35 36 37 38 39 40 41 42 43 45 46 47 48 50 5' 52 53 54 55 56 57 58 59 r. 1' 6 6 6 I 60 6.0 1 7 8 7 8 7.0 8.0 9 10 9 lO 9.0 1 10. 20 20 .3 20.0 30 no 50 30 40 50 5 8 30 40.0 50.0 59 5-9 6.9 7-8 8.8 9-8 '9-§ 29.5 39.3 49.1 58 57 6 5-8 7 6.7 8 7-7 9 8-7 10 9-6 20 19.3 30 29.0 40 3S.6 50 48.3 5-7 6.6 7.6 8.5 9^5 19.0 28.5 38.0 47^5 55 .54 6 5-5 5^4 7 6.4 6.3 8 7-3 7.2 9 8.2 8.1 10 9^1 9.0 20 18.3 18.0 30 27.5 27.0 40 36.^ 36.0 50 45^8 45.0 53 52 6 5^3 5.2 7 6.2 6.0 8 7.0 6.9 9 7-9 7.8 10 8^8 8-6 20 '7-6 17.3 30 26.5 26.0 40 35^3 34. 6 50 44.1 43.3 51 6 5.1 0.6 7 5.9 0.5 8 6.8 0.0 9 7.6 o.x 10 8.5 O.T ao 17.0 O.I 30 25.8 0.2 40 34.0 03 50 42.=; 0.4 I'. p 407 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 28° 29° Log. Vers. I) 9.06838 I .06888 2 .06939 3 . 06996 4 . 07040 5 9.07091 6 .07141 7 .07192 8 .07242 9 .07293 10 9 -07 343 II .97393 12 .07444 13 .07494 14 ■07544 15 9.07594 i6 .07644 17 .07695 i8 .07745 19 .0779^ 20 9.07845 21 .07895 22 .07945 23 .07995 24 .08045 25 9.08095 26 .08145 27 .08195 28 .08244 29 .08294 ^ 30 9-Oii344 31 .08394 32 .08443 33 .08493 34 •08543 35 9.08592 36 .08642 37 .08691 3« .08741 39 .08790 40 9.08840 41 .08889 42 .08939 43 .08988 44 .090^^7 45 9.09087 46 .09136 47 .09185 48 .09234 49 .09284 50 9.09333 51 .09382 52 .09431 53 . 09480 54 .09529 55 9-09578 56 .0962^ 57 •09676 5^ .09725 59 .09774 60 9.09823 liOSj. Vers. 50 51 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 49 50 49 50 49 49 50 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 48 49 49 Loar. Kxsec. ! Z> 7> 2244 2302 2359 24 1 6 2474 2531 2588 2645 2703 2760 2817 2874 293T 2988 3045 3102 3159 3216 3273 33"^o 33^7 3444 3500 3557 3614 3671 3727 3784 3841 3897 3954 401 1 4067 4124 4180 42J7 4293 4350 4406 4462 45^9 457? 4631 4688 4744 4800 4856 4913 4969 5025 5081 513? 5193 5249 5305 5361 541^ 5473 5529 5585 5641 Si 57 Si 57 Si 57 57 57 57 57 57 57 57 57 57 57 56 57 57 57 56 57 56 57 56 57 56 56 57 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 55 56 Loff. Vers. 9.09823 09872 09926 09969 0018 f>Off. Exsec. D Lost. Vers. 0067 01 ll 0164 0213 O26T 0310 0358 0407 0455 0504 0552 0601 0649 069^ 0746 0794 0842 0896 0939 0987 035 083 131 179 227 -D Los. Exsec. 323 371 419 467 515 562 616 658 706 754 801 849 897 944 992 2039 2087 2134 2182 2229 2277 2324 2371 2419 2466 2513 2566 2608 2655 2702 49 48 49 48 49 48 48 49 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 Al 48 48 47 48 48 47 48 47 Al 48 47 Al Al 47 47 47 Al Al 47 47 47 47 Al 47 Al 47 47 n J) 5641 5697 5752 5808 5864 5920 5975 6031 6087 6142 6198 6254 6309 6365 6426 6476 6531 6587 6642 6698 6753 6808 6864 6919 6974 7029 7085 7140 7195 7256 7305 7361 7416 7471 7526 7581 7636 7691 7746 78or 7856 7916 7965 8026 8075 8130 8185 8239 8294 8349 8403 8458 8513 856^ 8622 8676 8731 8786 8846 8894 8949 56 Si 56 55 56 Si 56 Si 55 56 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 54 55 55 54 55 55 54 55 54 54 55 54 54 54 54 54 55 54 54 54 TiOii. Exseo. Jt 20 21 22 23 24 25 26 27 28 29 p. p. 30 31 32 33 34 35 36 37 38 39 50 51 52 53 54 55 56 57 58 59 (JO 5l 57 58 54 5.4 6.3 7.2 8.1 9.0 18.0 27.0 36.0 45.0 51 50 54 6 5.4 7 6.3 8 7.2 9 8.2 10 9.1 20 18. £ 30 27.2 40 36.3 50 45-4 5-7 6.7 5-7 6.6 6.0 7-6 8.6 7.6 8.5 7-5 8.5 9.6 9-5 9.4 10. 1 19.0 18.8 28.7 =8.5 28.2 38.3 38.0 37-6 47.9 47-5 47.1 56 55 6 5-6 5-5 1 7 6.5 6 5 8 7-4 7 4 9 8.4 8 3 10 9.3 9 2 20 18.6 18 5 30 28.0 27 7 40 37-3 37 50 46.6 46 2 9.1 27. 36. 45- 5-1 5-0 5- 5-9 6.8 5-9 6.7 5- 6. 7.6 8.5 17.0 7.6 8.4 16.8 7- 8. 16 25.5 25.2 25- 340 42.5 33-6 42.1 33 • 41. 49 49 Al 6 4.9 4.9 4 7 8 5.8 6.6 5-7 6.5 5- 6. 9 10 7-4 8.2 7-3 8i 7- 8 20 16.5 16.3 16. 30 24.7 24-5 24. 40 33.0 32-6 32- 50 41.2 40.3 40. 48 4f 6 4.8 4-7 7 5-6 5 5 8 6.4 6-3 9 7.2 7.1 10 8.0 7-9 20 16.0 15-8 30 24.0 23.7 40 32.0 31-6 50 40.0 39-6 7- 15- 23. 31- 39- P. P 40a TABLF VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 30° 31° Loc Vers. I) 2702 2749 2796 2843 2896 2937 2984 3031 3078 3172 3219 3266 3313 33^9 3406 3453 3500 3546 3593 3(^39 3686 3733 3779 3826 3872 3919 3965 401 1 4058 4104 4151 4197 4243 428g 433(3 4382 4428 4474 45^0 45% 4612 4658 4704 4750 4796 4842 4888 4934 4980 5026 5071 5117 5163 5209 5254 5306 5346 5391 5437 1483 Los;. Vers. 47 47 47 47 47 47 47 47 47 47 46 47 47 46 47 46 47 46 46 46 47 46 4i, 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 45 46 45 46 46 45 45 46 45 45 45 46 iOJT. Kxsec. J> 9 ■ ' ^949 19003 19058 191 12 19167 n 19221 19275 19329 19384 19438 19492 19546 1 960 1 19655 19709 19763 1 9817 1 987 1 19925 1997^^ 20033 20087 20141 20195 20249 20^0 j^j 20357 204 r I 20465 20518 20572 20626 20680 20733 20787 20841 20894 20948 21002 210; = 21 109 21162 I 21216 21269 21323 21376} 21430 1 21483; 21537 21 596 21643 21697 21750 21803 21857 21910 21963 22015 22070 22123 22176 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 53 54 54 54 53 54 53 54 53 54 53 53 54 53 52 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 Loar. Vei! n I. I<0K. Kxsec. J> I Loii. Vers Q 54^^; 3 5528 5574 5619 5665 5710 5755 5801 5846 589T 5937 5982 6027 6073 6118 6163 6208 6253 6298 6343 6388 6434 6479 6523 6568 6613 6658 6703 6748 6793 6838 6882 692^ 6972 7017 7061 7106 7151 7195 7240 7284 7329 7373 7418 7462 7507 7551 7596 7640 7684 7729 7773 7817 786T 7906 7950 7994 8038 8082 8i26 8170 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 44 45 45 45 45 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 /> K\' /> 22170 2222Q 22282 22335 22388 22441 22494 22547 22606 22653 227O6 22759 22812 22865 22918 22971 23024 23076 23129 23182 23235 23287 23340 23393 23446 23498 I 23551 23603 , 23656 i 23709 23761 23814 23866 23919 23971 24024 24076 24128 24181 24233 24285 24338 24396 24442 24495 24547 24599 24651 24704 24756 24808 24860 24912 24964 25016 25068 25126 25172 25224 2 527(3 25328 L Lofr. Exsee 30 31 32 33 34 35 36 37 38 39 40 ! 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 (io 8170 8214 8258 8302 8346 8390 8434 8478 8522 8566 8610 8654 8697 874t 878s 8829 8872 8916 8959 9003 9047 9090 9134 9177 9221 9264 9308 9351 9395 9438 9481 9525 9568 961 T 9654 9698 9741 9784 9827 9870 9914 9957 20000 20043 20086 20129 20172 20215 20258 2030[ 20343 20385 20429 20472 20515 20558 20600 20643 20686 20728 9.20771 Log. Vers. 44 44 44 44 44 44 44 44 43 44 44 43 44 43 44 43 43 43 44 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 42 43 43 43 42 43 42 43 43 42 43 2> 25328 25386 25432 25484 25536 25588 25640 25692 25743 25793 25847 25899 25950 26002 26054 26105 26157 26209 26260 26312 J) Log. V«^rs. 1} 26364 26415 26467 265I8 26570 26621 26673 26724 26776 26827 26878 26930 2698T 27032 27084 27135 27185 27238 27289 27340 27391 27443 27494 27545 27596 27647 27698 27749 27800 27852 27903 27954 28005 28056 28107 28157 28208 28259 28316 2836T 28412 Log. Exsec. 52 52 52 51 52 52 52 51 52 51 52 51 52 51 51 52 51 51 51 52 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 50 51 51 51 51 50 7> 9.20771 20814 20855 20899 20942 20984 027 069 112 154 196 239 281 324 -.66 408 451 493 535 577 620 662 704 746 788 836 872 914 956 998 22040 22082 22124 22165 22208 22250 22292 22334 22376 2241^ 22459 22501 22543 22584 22626 22668 22709 22751 22792 22834 22875 22917 22959 23006 23042 23083 23124 23166 2320^ 23248 9.23290 Log. Vers. 42 42 42 43 42 42 42 42 42 42 42 42 42 42 42 42 42 42 ZI2 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 41 42 42 41 42 41 42 41 41 42 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 ILoi 9 Exsec. I) 28412 28463 28514 28564 28615 28665 28717 28768 28818 28869 28920 28976 29021 29072 29122 29173 29223 29274 29324 29375 29426 29476 29527 29577 29627 29678 29728 29779 29829 29879 29930 29986 30036 30081 30131 3018T 3023T 30282 30332 30382 I) 30432 30482 30533 30583 30633 30683 30733 30783 30833 30883 30933 30983 31033 31083 3fT33 31183 31233 31283 31333 31383 31432 51 51 50 51 51 50 51 50 50 51 50 50 51 56 51 50 50 50 51 50 50 56 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 49 50 50 50 50 49 15 16 17 18 19 20 21 22, 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 il 55 56 57 58 59 'Log. Exseo.l 7> GO p. p. 52 51 6 5-2 5-1 7 6.0 6.0 8 6.9 6-8 9 7.8 7-7 10 8.6 8.6 20 17-3 17. 1 30 26.0 25-7 40 34.6 34-3 50 43.3 42.9 6 50 5-0 50 5-0 7 8 5-9 6.7 5-8 6.6 9 10 20 7.6 8.4 16.8 7-5 16.6 30 25.2 25.0 40 33-6 33-3 50 42.1 41-6 6 44 4.4 43 4-3 7 8 9 5-1 5-8 6.6 51 5-8 6.5 10 7-3 7.2 20 M.6 14-5 30 22.0 21.7 40 50 29-3 36.6 29.0 36.2 6 42 4.2 42 4.2 7 8 9 4.9 5-6 6.4 4.9 5-6 6.3 10 7-1 7.0 20 14.1 14.0 30 40 21 .2 28.3 21.0 28.0 50 35-4 35-0 6 41 4.1 7 8 4.S 5-4 9 10 6.1 6.8 20 J3-6 30 20.5 40 27-3 50 34-1 P. p. 410 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 34° 85° Vers. 7> 23290 23331 23372 23414 23455 23496 23537 23579 23620 23661 23702 23743 237^4 2382^ 23866 23907 23948 23989 24036 24071 24112 24153 24194 24235 24275 243 1 6 24357 24398 24438 24479 24520 24561 2460 T 24642 24682 24723 24764 24804 24845 24885 24926 24966 25007 25047 25087 25128 25168 25209 25249 25289 25329 25370 25410 25450 25496 25531 25571 25611 25651 2569T 9-25731 Lost. Vers. 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 40 41 41 41 46 41 46 41 46 41 40 41 46 46 46 41 46 46 40 40 40 40 46 40 40 46 46 40 40 40 40 46 40 46 40 46 40 40 40 40 40 "77" li<);r. Kxsec n 432 482 532 582 6^2 681 731 781 83' 886 930 980 32029 32079 32129 32178 32228 32277 32327 32377 32426 32476 32525 32575 32624 32673 32723 32772 32822 32871 32920 32970 33019 33069 33118 33167 33216 33266 33315 33364 33413 33463 33512 33561 33616 33659 33708 33758 33807 33856 33905 33954 34003 34052 34101 34150 34199 34248 34297 34346 34395 50 50 49 50 49 50 49 50 49 50 49 49 49 50 49 49 49 49 50 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 Lost. Kxspo. /> Lour. Vers. 25731 25771 2581I 25851 25891 25931 25971 2601 T 26051 26091 261 31 26171 26216 26256 26296 26330 26370 26409 26449 26489 26528 2656^ 26608 2664^ 26687 26725 26765 26806 26845 26885 26924 26964 27003 27042 27082 27121 27161 27200 27239 27278 27318 27357 27396 2743S 27475 27514 27553 27592 27631 27676 27709 27749 27788 27827 27866 27905 27944 27982 28021 28066 28099 Lost. Vers. n 40 40 40 40 40 40 40 39 40 40 40 39 40 40 39 40 39 40 39 39 40 39 39 39 39 40 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 38 39 39 39 IT \A»i. K.Vht'C. It 34395 34444 34492 3454' 34590 34639 34688 34737 34785 34834 34883 34932 34986 35029 35078 35127 35175 35224 35273 35321 35370 35419 35467 35516 35564 35613 35661 35710 35758 35807 35855 35904 35952 36001 36049 36098 36146 36194 36243 36291 36340 36388 36436 36484 36533 3658T 36629 36678 36726 36774 36822 36876 36919 36967 37015 37063 37111 37159 3720^ 37255 37303 I,nir. Kxser. 49 48 49 49 49 48 49 48 49 49 48 48 49 48 49 48 49 48 48 48 49 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 77" 10 20 21 22 23 24 25 26 27 28 2g 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 :>o 5' 52 53 54 55 56 57 58 r. I'. 6 50 5.0 49 4.9 7 8 6 6 5-8 6.6 9 10 7-5 8.3 7-4 8 2 20 16.6 .6.5 30 25.0 247 40 33-3 33-0 50 41 6 41.2 40 50 20 30 40 50 40 50 20 30 40 50 48 4.§ 5-6 6.4 7-3 8.1 16. 1 24.2 3* -3 40.4 41 4.1 4-S 5-5 6.2 6.9 20.7 27-6 34.6 39 3-9 4.6 5-2 .S-9 6.6 13.1 19.7 26.3 32.9 6 7 8 9 10 20 30 40 5'3 49 4.9 5-7 6.5 7-1 8. 16.3 24-5 32.^ 40.8 48 4.8 5.6 6.4 7.2 8.0 j6.o 24 o 32.0 40.0 41 4.1 4.8 5-4 e.i H 13-6 20. T 27.3 34-1 40 40 4 4- 4-7 4- 5 6 4 I 5- 6. 6 7 6. 13 5 '3- 20 2 20. 27 26. 33 7 33 39 3-9 4-5 5-2 5-8 13.0 19.5 26.0 32.5 38 3-8 4-5 51 S.8 6.4 12. g 19.2 25-6 32-1 I'. I'. 411 TABLE VIII.-LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 36° 37° '_ j Log. Vers. I J> JLog. Exseo 10 II 12 H 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9 . 28099 .28138 .28177 .28216 .28255 9.28293 .28332 .28371 .28410 . 28448 9.28487 .28526 .28564 .28603 .28642 9.28680 .28719 .2875; .28796 .28835 9-28873 .28912 .28950 .28988 .29027 9.2906^ .29104 .29142 .29180 .29219 9.29257 .29295 •29334 .29372 .29410 9-29448 • 29487 .29525 .29563 . 29601 9.29639 .29677 .2971^ •29754 .29792 9.29830 .29868 . 29906 .29944 .29982 9.30020 .30057 •30095 .30133 .30171 9 . 30209 . 30247 .30285 .30322 ■ 30360 9 -30398 Log. Vers. 39 38 39 39 38 39 38 39 38 39 38 38 39 38 38 38 38 38 39 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 3f 38 38 38 38 37 38 3? 38 38 IT' 9-37303 •37352 • 37400 .37448 • 37496 37544 37592 37640 37687 37735 n 37783 37831 37879 37927 37975 38023 38071 38119 38166 38214 38262 38310 3835^ 38405 38453 38501 38548 38596 38644 38692 38739 38787 38834 38882 38930 38977 39025 39072 39120 39168 39215 39263 39310 39358 39405 39453 39506 39548 39595 39642 39690 39737 39785 39832 39879 39927 39974 40021 40069 401 16 40163 48 48 48 48 48 48 48 47 48 48 48 48 48 47 48 48 48 4? 48 4^ 48 4f 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 48 47 47 48 47 47 47 47 47 4? 47 47 4f 47 47 47 4? Al 47 4l 4l 47 41 Loff. Kxsec.l /> Log. Vers. I U Log. Exsec.l 2> 9-30398 • 30436 • 30474 .30511 • 30549 9-30587 . 30624 . 30662 . 30700 -3073? 9-30775 .30812 .30850 . 30887 -30925 9.30962 .31000 -31037 •31075 .31112 9.31150 .31187 .31224 .31262 .31299 9-31336 -31374 .31411 •31448 •31485 9.31523 .31560 •31597 •31634 .31671 9-317O8 •31746 .31783 ,31820 -31857 9.31894 •31931 .31968 .32005 .32042 9-32079 .32116 .32153 .32190 -22227 9.32263 .32300 •32337 •32374 .32411 9-32447 .32484 .32521 •32558 .32594 9-32631 Loe. Vers. 37 38 37 37 38 3l 3l 38 37 31 37 3l 37 37 3l 37 37 3l 3l 3l 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 36 37 37 36 37 36 37 36 37 36 37 "77" 9.40163 .40216 .40258 • 40305 .40352 • 40399 ,40447 ,40494 ,40541 40588 9-40635 .40682 •40730 •40777 . 40824 9.40871 •40918 .40965 9-4 •4 •4 .4 •4 9.4 •4 •4 -4 •4 9-4 •4 •4 •4 •4 9-4 .4 .4 •4 •4 012 059 106 153 206 24^ 294 341 388 435 482 529 576 623 670 717 763 816 857 904 951 998 9.42044 .42091 .42138 .42185 -42231 9-42278 -42325 .42372 .42418 .42465 9.42512 -42558 .42605 .42652 .42698 9-42745 .42792 .42838 .42885 -42931 9-42978 Loar. Kxsec. /> 47 41 47 4l 47 47 47 47 47 47 47 47 47 47 47 4/ 47 47 47 47 47 47 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 47 46 47 46 47 46 47 46 47 46 47 46 46 47 46 46 46 47 46 46 46 46 10 20 21 22 23 24 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 20 50 6 7 8 9 10 20 30 40 50 6 7 8 9 10 20 30 40 50 40 50 P. P. 48 4.8 1 5 6 6 4 7 8 3 1 16 1 24 2 32 3 40 4 48 4^ 6 7 8 4-7 5-5 6.3 9 7-1 10 7-9 20 15. § 30 23-7 40 31-6 50 39-6 16. 24. 32. 40. 47 4-7 5-5 6.; 7-0 7- 15.6 23-5 31-3 39- 20 30 40 50 46 4-6 5-4 6.2 7.0 7-7 15.5 23.2 31.0 38.7 39 38 3-9 3- 4-5 4. 5.2 5- 5-8 5; . 6.5 6. 13.0 12. 19^5 19. 26.0 25- 32.5 32 • 38 Zl 3.8 4.4 3 4 50 5 5-7 6.3 5- 6. 12.6 12. 19.0 18. 25-3 25- 31-6 31 37 36 3-7 3 4-3 4- 4-9 4- 5-5 5- 6.1 6. 12.3 12. 18.5 18. 24-6 24. 30-8 30- P. p 412 TABLE VIII,— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 10 II 12 14 15 i6 i8 19 20 21 22 ^3 24 25 26 27 28 29 80 33 34 33 36 37 38 39 40 41 42 43 44_ 45 46 47 48 49 50 51 52 53 54 56 57 58 59 GO :58' :5i> Lost. Vers. /> |li(»sr. Kxsec /> || liOir. Vers, /> liOC. Kxsec 9.32631 32668 32704 32741 32778 32814 32851 32888 32924 32961 32997 33034 33070 33107 33143 33180 33216 33252 33289 33325 33361 33398 33434 33470 33507 33543 33579 33615 33652 33688 33724 33766 33796 33833 33869 33905 33941 33977 34013 34049 34085 34121 34157 34193 34229 34265 34301 34337 34373 34408 34444 34480 345 '6 34552 34587 34623 34659 34695 34736 34766 9 34802 Log. Vers. 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 35 36 36 3l 36 3d 36 3l 36 35 36 35 /> 42978 43024 43071 43118 43164 432 1 1 4325^ 43304 43356 43396 43443 43489 43536 43582 43629 43675 43721 43768 43814 43861 43907 43953 43999 44046 44092 44138 44185 44231 44277 44323 44370 44416 44462 44508 44554 44601 44647 44693 44739 44785 44831 44877 44924 44970 45016 45062 45108 45154 45200 45246 45292 45338 45384 45430 45476 45522 45568 45614 45660 45706 9-45752 iOC. Kxsec. 46 47 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 "77" 9.34802 3483? 34873 34909 34944 34980 35016 35051 35087 35122 35158 35193 35229 35264 35300 35335 35376 35406 35441 35477 35512 35547 35583 35618 35653 35689 35724 35759 35794 35829 35865 35900 35935 35976 36005 36046 36076 361 II 36146 36181 36216 36251 36286 36321 36356 3639' 36426 36461 36495 36536 36565 36606 36635 36670 36705 36739 36774 36809 36844 36878 9 • 369 ' 3 3i) 36 35 35 35 36 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 34 35 35 35 34 35 35 34 35 34 35 34 35 Vers. /> It 9 45752 4579^ 45843 45889 45935 45981 46027 46073 461 18 46164 46216 46256 46302 4634^ 46393 46439 46485 46536 46576 46622 46668 46713 46759 46805 46856 46896 46942 4698? 47033 47078 47124 47170 47215 47261 47306 47352 47398 47443 47489 47534 47580 47625 47671 477 '6 47762 47807 47852 47898 47943 47989 48034 48080 48125 48176 48216 48261 48306 48352 4839? 48442 48488 l,llir. K\K4T. 45 46 46 46 45 46 46 45 46 46 45 46 45 46 45 46 45 46 45 46 45 45 46 45 45 46 45 45 45 46 45 45 45 45 46 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45' /> 5 6 7 8 9 10 1 1 12 13 ii_ 15 16 17 18 19 •20 21 22 23 24 25 26 27 40 41 42 43 44 45 46 47 48 49 50 5' 52 53 J4 55 56 57 58 4;o V. V 20 30 40 50 20 30 40 50 20 30 40 50 40 50 47 4-7 5vS 6.2 7.0 7-8 15-6 23 -5 39- 1 46 4.6 5-1 6.1 6.9 7-6 15-3 23.0 30-6 38.3 46 4-6 5-4 6.2 7.0 7-7 15-5 23.2 31.0 38.7 4S 4-5 5-3 6.0 6.8 7.6 J51 22.7 30-3 37-9 9 10 20 30 40 50 45 4-5 5-2 6.0 6.7 7-5 150 22.5 30.0 37-5 37 3-7 4-3 4-9 5-5 6.1 12.3 18.5 24-6 30-8 36 3-6 4.2 4-8 5-5 6.1 12.1 18.2 24.3 30-4 36 35 3-6 3-5 42 4.1 4.8 4-7 5-4 5-3 e.o 5-0 J2.0 "•§ 18.0 17-7 24.0 23 6 30.0 29.6 35 34 6 3 5 3-4 7 4.1 4.0 8 4-6 4.6 9 5-2 5-2 10 5-§ 5-7 20 n-6 1 1. 5 30 '7-5 17.2 40 23.3 23.0 50 29.1 28.7 V. V 413 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 40° 41° Log. Vers. | J> |Log. Exsec. Z> \ Log. Vers. 10 II 12 13 15 i6 17 i8 19 20 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 ^8 59 GO 36913 36948 36982 37017 37052 37086 3712T 37156 37196 37225 37259 37294 37328 37363 37397 37432 37466 37501 37535 37570 37604 37639 37673 377of 37742 37776 37816 37845 37879 37913 37947 37982 38016 38056 38084 38118 38153 38187 38221 38255 38289 38323 38357 38391 38425 38459 38493 38527 3856T 38595 38629 38663 38697 38731 38765 38799 38833 38866 38906 38934 9.38968 Loff. Vers. 34 34 35 34 34 35 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 33 34 34 34 33 34 33 34 /> 48488 48533 48578 48624 48669 48714 48759 48805 48850 48895 48946 48986 49031 49076 4912T 49166 492 1 T 49257 49302 49347 49392 49437 49482 49527 49572 49618 49663 49708 49753 49798 49843 49888 49933 49978 50023 50068 50113 50158 50203 50248 50293 50338 50383 50427 50472 50517 50562 50607 50652 50697 50742 50787 5083T 50876 5092T 50966 5101 1 51055 51 106 51145 Q . 5 II 90 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 44 45 45 45 45 44 45 45 45 44 45 45 44 45 44 45 45 44 Log. Hxsec 9.38968 39002 39035 39069 39103 39137 39176 39204 39238 39271 39305 39339 39372 39406 39439 39473 39507 39540 39574 3960^ 39641 39674 39708 39741 39774 39808 39841 39875 39908 39941 39975 40008 40041 40075 40108 40 1 41 40175 40208 4024T 40274 40307 40341 40374 40407 40446 40473 40506 40540 40573 40606 40639 40672 40705 40738 40771 40804 40837 40870 40903 40936 40969 U Loar. Exsec 34 33 34 33 34 33 33 34 33 33 34 33 33 33 33 34 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 9 /> LO!.'. Vers. /> l.n 190 235 279 324 369 n 414 458 503 548 592 63? 682 726 771 816 866 905 950 994 52039 52084 52128 52173 52217 52262 52306 52351 52396 52446 52485 52529 52574 52618 52663 52707 52752 52796 52841 52885 52930 52974 53018 53063 53107 53152 53 '96 53240 53285 53329 53374 53418 53462 53507 53551 53595 53640 53684 53728 53773 53817 53861 45 44 45 44 45 44 45 44 44 45 44 44 45 44 44 45 44 44 44 45 44 44 44 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 Exs"-.. /> P. P. 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 '60 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 ii_ 55 56 57 58 59 ({0 2D 30 40 50 40 40 50 44 4.4 5-2 5-9 6.7 7-4 14.8 22.2 29-6 37-1 35 7 4 8 4 9 5 10 5 20 II 30 17 40 23 50 29 34 20 40 50 4S 45 4-5 4-5 5 • 1 5-2 6.0 6.0 6.8 6.7 7.6 7-5 15. 1 15.0 22.7 22.5 303 30.0 37-9 37-5 44 4.4 5-8 6.6 7-3 14-6 22 .0 29-3 36 6 34 33 3-3 3-9 4.4 50 5.6 t6. 27 33 27 p. P. 414 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 4*e" 4;r Log. Vers. | 1> 9 40969 001 034 067 106 133 166 199 231 264 297 330 362 395 428 461 493 525 559 591 624 657 689 722 754 7^7 819 852 885 917 950 982 42014 42047 42079 421 12 42144 42177 42209 4224T 42274 42306 42338 42371 42403 42435 42467 42500 42532 42564 42596 42629 42661 42693 42725 42757 42789 42822 42854 42886 9-4^9'8 Loc. Vers. 32 33 jj 33 32 33 33 32 33 32 33 32 33 32 33 32 33 32 32 32 33 32 32 32 32 32 32 33 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 7> liOj;. KxHec. /> 53861 53906 53950 53994 54038 54083 54'27 54171 54215 54259 54304 54348 54392 54436 54480 54525 54569 546-13 54657 5470T 54745 54790 54834 54878 54922 54966 55016 55054 55098 55142 55186 55230 55275 55319 55363 55407 55451 55495 55539 55583 55627 55671 55715 55759 55803 55847 55890 55934 55978 56022 56065 561 16 56154 56198 56242 56286 56330 56374 5641^ 5646T 56505 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 44 44 44 43 44 44 44 44 43 44 43 Los:. Kxseo. J> Lotr. Vers. 9 42918 42950 42982 43014 43046 J> 43078 43116 43142 43174 43206 43238 43270 43302 43334 43365 4339? 43429 4346T 43493 43525 43557 43588 43626 43652 43684 43715 43747 43779 43816 43842 43874 43906 43937 43969 44006 44032 44064 44095 44127 44' 58 44190 44221 44253 44284 443 '6 44347 44379 44416 44442 44473 44504 44536 44567 44599 44630 4466 T 44693 44724 44755 44787 448 1 8 L(»e. Vers. 32 32 32 32 31 32 32 32 31 32 32 32 31 32 32 31 32 31 32 31 32 31 y 32 31 32 31 31 31 32 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 3' 31 31 Loir. K.vNi'c' /> W 56505 5^>549 56593 56637 56686 56724 56768 56812 56856 56899 56943 56987 57031 57075 57'J8 57162 57206 57250 57293 57337 57381 57424 57468 57512 57556 57599 57643 57687 57730 57774 57818 5786T 57905 57949 57992 58036 58079 58123 58167 58216 58254 5829? 58341 58385 58428 58472 58515 58559 58602 58646 58689 58733 58776 58826 58S64 5890? 58951 58994 59037 59081 L'iiii y> Ldir. Kxser. 43 44 44 43 44 44 43 44 43 44 44 43 44 43 44 43 44 43 44 43 43 44 43 44 43 43 44 43 43 44 43 43 44 43 43 43 44 43 43 43 43 44 43 43 43 43 43 43 43 43 43 43 44 43 43 43 43 43 43 43 /> 10 1 I 12 13 _LL '5 16 17 18 19 20 21 -J 24 25 26 27 28 30 31 32 33 34 35 36 37 38 39 40 41 42 43 45 46 47 48 50 31 5^ 53 54 55 5^^ 57 58 59 <;o 1'. I*. 40 50 40 50 20 30 40 50 40 50 44 4-4 1 5 2 5 9 6 7 7 4 14 iJ 22 29 6 37 I 33 4.4 4.9 5-5 11 .0 16.5 27-5 32 3-2 3-7 4.2 4.8 5-3 10.6 16.0 21 3 26.6 7 8 9 10 30 30 40 50 44 4.4 5-i 5-8 6.6 7-3 14-6 22 o 29-3 36.6 43 4 3 5 I 5 8 6 5 7 2 J4 5 21 7 29 36 2 43 28.6 35-8 32 3 4- 4- 5- 10. 16. 27.1 31 31 3-7 4.7 4-7 5-2 10.5 »5-7 21 .0 26.2 31 3-» 3-6 4« 4-6 51 10.3 »5 5 20.^ 25-8 '. I*. 415 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 44° 45° Log. Vers. 9.44818 I .44849 2 .44880 3 .44912 4 .44943 5 9-44974 6 .45.005 7 •45036 8 .45068 9 .45099 10 9.45130 II .45161 12 .45192 13 .45223 14 .45254 15 9.45285 i6 .45316 17 .45348 i8 .45379 19 .45410 20 9.45441 21 .45472 22 •45503 23 •45534 24 .45565 25 9.45595 26 .4562^ 27 •4565? 28 .45688 29 .45719 30 9.45750 31 .45781 32 .45812 33 .45843 34 .45873 35 9.45904 36 .4593! 37 •45966 3« •45997 39 .46027 40 9.46058 41 . 46089 42 .46120 43 .46150 44 .46181 45 9.46212 46 .46242 47 .46273 48 .46304 49 .46334 50 9.46365 51 .46396 52 .46426 53 .46457 54 .4648^ 55 9-46518 56 .46549 57 .46579 58 .46610 59 . 46646 GO 9.46671 Z> Log. Ex sec. J> Log. Vers. 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 30 31 31 31 31 31 30 31 31 30 31 31 30 31 30 31 30 31 30 31 30 30 31 30 30 31 30 30 30 30 31 30 30 30 30 30 IT I. 59124 .59168 .59211 .59255 • 59298 59342 59385 59429 59472 59515 59559 59602 59646 59689 59732 59776 59819 59863 59906 59949 9-59993 . 60036 .60079 .60123 .60166 , 60209 ,60253 ,60296 .60339 .60383 , 60426 , 60469 ,60512 ,60556 60599 , 60642 ,60685 ,60729 ,60772 ,60815 ,60858 , 60902 60945 ,60988 ,61031 .61075 ,61118 .61 161 ,61204 .61247 ,61291 .61334 61377 ,61426 .61463 9- 61506 61550 •61593 .61636 .61679 9.61722 Log. Kxsec. I I) 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 Log. Vers. 9 . 4667 1 4670T 46732 46762 46793 46823 46853 46884 46914 46945 46975 47005 47036 47066 47096 47127 4715^ 4718^ 47218 47248 47278 47308 47339 47369 47399 47429 47459 47490 47520 47550 n 47586 47616 47646 47676 47706 47731 47761 47791 47821 47851 47881 4791 1 47941 47971 48001 48031 48061 48096 48126 48156 48186 48216 48240 48270 48300 48329 48359 48389 48419 48449 48478 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 36 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 29 30 30 29 30 30 29 30 29 Log. Vers. I J> Log. Exsec. 9.61722 .61765 .61808 .61852 .61895 61938 ,61981 .62024 .6206^ .621 16 62153 .62196 .62239 .62282 ,62326 .62369 .62412 .62455 .62498 ,62541 9- 62584 62627 62670 62713 62756 9- 62799 62842 62885 62928 62971 63014 ,63057 ,63100 63143 63186 ,63229 .63272 •63315 63358 63401 9- 63443 63486 ,63529 63572 63615 •63658 •63701 •63744 ,63787 ,63830 ,63873 ■63915 •63958 ,64001 64044 9.64087 . 64 1 30 •64173 .64216 •64258 9-64301^ Log. Exsec. J /> J) 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 42 43 43 43 43 43 42 43 43 43 43 42 43 43 43 42 43 43 43 42 43 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 p. P. 20 40 50 40 50 40 50 43 4-3 5-1 5.8 6.5 7.2 14-5 29. 36. 6 7 8 9 10 20 30 40 50 31 3^i 3-7 4.2 4.7 5-2 10.5 15-7 21 .0 26.2 30 30 3-5 4.0 4.6 5-1 10. 1 15.5 20.3 25.4 20 30 40 50 43 4-3 5-0 5-7 6.4 7-1 14-3 21-5 28.6 35-8 42 4.2 4.9 5-6 6.4 7-1 141 21 .2 28.3 35-4 31 3-1 3.6 4.1 4-6 5-1 25-8 30 30 3-5 4.0 4-5 5-0 10. o 15.0 20.0 25.0 29 2.9 3-4 3-9 4.4 4.9 9-8 14.7 ^9-6 24.6 P. P. 416 TABLE V^III.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 40° 47° _9_ 10 II 12 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Log. Vers. I 1> Los- Kxsef 50 51 52 53 S±_ 55 56 57 58 59 (»0 48478 4S508 48538 48568 4859? 48627 48657 48686 487 1 6 48746 4877^ 48805 4S835 48864 48894 48923 48953 48983 49012 49042 49071 49101 49130 49160 49189 49219 49248 49278 49307 49336 49366 49395 49425 49454 49483 49513 49542 49571 49601 49630 49659 49689 49718 49747 49776 49806 49835 49864 49893 49922 ;oi D5 49952 . 49981 50010 50039 I 50068 I 50097 50126 50185 50214 9.50243 liOff. Vers. 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 J> 9 64301 64344 64387 64430 64473 64515 64558 ,64601 64644 64687 64729 ,64772 64815 64858 6490 1 • 64943 .64986 ,65029 ,65072 65114 9- 65157 65200 65243 65285 65328 65371 .65414 .65456 .65499 65542 65585 ,65627 ,65670 .65713 6575? 9- 65798 65841 65884 65926 65969 ,66012 .66054 . 66097 . 66 1 40 .66182 v> ,66225 .66268 .66310 •66353 ,66396 66438 ,66481 66523 .66566 , 66609 ,66651 , 66694 ,66737 66779 66822 9.66864 43 42 43 43 42 43 43 42 43 42 43 43 42 43 42 43 42 43 42 43 42 43 42 43 42 43 42 43 42 43 42 43 42 42 43 42 43 42 42 43 42 42 43 42 42 43 42 42 43 42 42 42 43 42 42 42 43 42 42 42 liOij. Vers. josr. Kxser. /> 9.50243 50272 50301 50330 50359 50388 50417 50446 50475 50504 50533 50562 50591 50619 50648 50677 50706 50735 50764 50793 50821 50850 50879 50908 50937 50905 50994 023 0:;2 9-5 Lost. 080 109 138 167 195 224 253 281 310 338 367 /> 396 424 453 481 510 539 567 596 624 653 681 710 738 767 795 823 852 886 909 937 90 5 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 28 29 29 28 29 28 29 28 29 28 29 28 29 28 28 29 28 28 28 29 28 28 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 /> l.dir. Kxscc /> 9- 66864 66907 66950 66992 67035 67077 67120 67162 67205 67248 ,67296 67333 •67375 ,67418 , 67466 ,67503 67546 .67588 ,67631 ,67673 67716 67758 67801 67843 67886 ,67928 ,67971 ,68013 68056 68098 9- 68I4I 68183 68226 68268 683! I 68353 68396 68438 68481 68523 ,68566 ,68608 .68651 ,68693 .68735 ,68778 ,68826 ,68863 ,68905 ,68948 68996 69033 69075 69II7 69160 69202 69245 69287 69330 69372 60414 42 43 42 42 42 42 42 43 42 42 42 42 42 42 42 43 42 42 42 ,1 -7 T- 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 9- /> _9 16 1 1 12 13 14 15 16 17 18 19 33 34 J5 36 37 3 39 8 40 41 42 43 44 45 46 47 48 49_ 'lO 5' 52 53 54 55 56 57 58 59 <>0 V. v. 20 40 50 6 7 8 9 10 20 30 40 50 43 42 6 4 3 4-'-s 7 5-0 4.9 9 5-7 6.4 5-6 6.4 10 7-1 7' 20 »4-3 14.1 30 40 21.5 28.6 21.2 28.3 50 35-8 35-4 20 30 40 50 30 29 2.9 3 4 3-8 4-3 6 7 8 9 lo 20 30 40 50 42 3 3 5 4 4 5 5 10 15 20 25 29 29 4 9 4 9 28 28 2.8 3-2 3-7 4-a 4- 9 >4 18. 23. I'. V 417 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 48° 49° 10 II 12 Los. Vers . J> 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 3f 32 33 34 35 36 37 38 39 40 41 42 43 ±L 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.51965 51994 52022 52050 52079 52107 52135 52164 52192 52226 52249 52277 52305 52333 52362 52390 52418 52446 52474 52503 52531 52559 52587 52615 52643 52671 52699 5272^ 52756 52784 52812 52840 52896 52924 52952 52980 53008 53036 53064 53092 53120 5314? 53175 53203 53231 53259 53287 53315 53343 53370 53398 53426 53454 53482 53509 5353? 53565 53593 53620 9.53648 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 2? 28 28 28 27 28 28 28 2f 28 28 2? 28 2f 28 2? 28 27 28 Log. Vers.' J> Log. Exsec. 9.69414 69457 69499 69542 69584 69625 69669 697 1 1 69753 69796 69838 69881 69923 69965 70008 70050 70092 70135 70177 70220 2> i Log. Vers. 70262 70304 70347 70389 70431 70474 70516 70558 70601 70643 70685 70728 70770 70812 70854 70897 70939 70981 024 066 I08 151 193 235 278 320 362 404 447 489 531 573 616 658 706 743 785 82^ 869 912 9-71954 Log. Exspc. 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 ^2 42 42 42 42 42 T> 53648 53676 53704 53731 53759 53787 53814 53842 53870 5389? 53925 53952 53986 54008 54035 54063 54096 54118 54'45 54173 1> Los. Exsec. 54200 54228 54255 54283 543'6 54338 54365 54393 54426 54448 54475 54502 54530 54557 54585 54612 54639 54667 54694 54721 54748 54776 54803 54836 54858 54885 54912 54939 54967 54994 55021 55048 55075 55103 55130 55157 55184 5521T 55238 55265 55292 27 28 27 27 28 27 28 2? 27 28 27 27 2? 27 27 27 27 2? 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 2? 27 27 27 27 27 27 2l 27 27 27 27 2? 27 2? 27 27 27 2? 27 27 27 71954 71996 72038 72081 72123 72165 7220^ 72250 72292 72334 72376 72419 72461 72503 72545 72587 72630 72672 72714 72756 72799 72841 72883 72925 7296^ 73010 73052 73094 73136 73"78 73221 73263 73305 7334? 73389 73431 73474 73516 73558 73606 73642 73685 73727 73769 73811 73853 73895 73938 73980 74022 74064 74106 74 '48 74191 74233 74275 7431? 74359 7440T 74444 74486 2) Vers. D Lour. Exseo 418 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 It 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 (JO p. P. 20 40 50 6 7 8 9 10 20 30 40 50 42 4.2 4.9 5- 6. 7- 14- 28.3 35-4 3-3 4-3 4-7 9-5 14.2 19.0 23-7 9 4 10 4 20 9 30 13 40 18 50 22 21 2.7 3-2 3-6 P. P. 42 4.2 4.9 5.6 6.3 7.0 14.0 28.0 35-0 28 28 2.8 3-2 3-7 4.2 4-6 9-3 14.0 18.6 23 3 27 2,7 3-1 3.6 4.0 4.5 9.0 13-5 18.0 22. <; TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 50° 51" 10 1 1 12 14 15 i6 17 i8 19 20 21 22 23 24 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Los. Vers. I 7> Lotf. Kxseo 9.55292 .55319 .55347 .55374 .55401 55428 55455 55482 55509 555^.6 9.55563 .55590 .55617 .55644 .55671 9.55698 .55725 .55751 •55778 .5580^ 9.55832 .55859 .55886 •55913 .55940 9.55966 .55993 . 56020 • 56047 . 56074 9. 56101 .56127 .56154 .56181 . 56208 9.56234 . 56261 .56288 •56315 .56341 9.56368 . 56395 .56421 . 56448 .56475 9.56501 .56528 •56554 .56581 . 56608 9.56634 .56661 . 56687 .56714 .56741 9.56767 • 56794 .56826 .56847 •56873 9 . 56900 Lotf. Vers. 27 2f 27 27 27 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 26 27 27 26 27 27 26 27 27 26 27 26 27 26 27 26 27 26 26 27 26 26 27 26 26 26 27 26 26 26 26 26 27 26 26 26 26 26 26 J) J> ■ 74486 .74528 74570 ,74612 74654 74696 74739 74781 74823 7486^ , 74907 ■ 74949 .74991 .75033 75076 ,75118 , 75 1 60 ,75202 75244 75286 75328 75370 75413 .75455 75497 75539 .75581 ,75623 ,7566^ .7570^ 75750 75792 .75834 .75876 ,75918 .75966 , 76002 .76044 .76086 ,76128 76171 ,76213 ,76255 ,76297 .76339 7638T ,76423 .76465 ,76507 .76549 ,76592 .76634 , 76676 .76718 .76760 9.76802 .76844 .76886 .76928 .76976 9.77012 liOC Kxsec I I> 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 Lotf. Vers. I 7> 9. 56900 56926 56953 56979 57005 57032 57058 57085 5711T 57138 57164 57196 57217 57243 57269 57296 57322 57348 57375 57401 57427 57454 57480 57506 57532 57559 57585 57611 5763^ 57664 57690 57716 57742 57768 57794 57821 57847 57873 57899 57925 57951 57977 58003 58029 58055 58082 58108 58134 58160 58186 58212 58238 58264 58290 58316 58342 58367 58393 58419 5^445 58471 Lotr. Vers. 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 /> iOi;. Hxsec.l /> 9 77012 77055 77097 77139 77181 7722 '^ 77265 7730^ 77349 77391 77433 77475 77517 77560 77602 77644 77686 77728 77770 77812 77854 77896 77938 77986 78022 78064 78107 78149 78191 78233 78275 78317 78359 7840T 78443 78485 78527 78569 7861 1 78653 78696 78738 78780 78822 78864 78906 78948 78996 79032 79074 79H6 79'58 79206 79242 79285 79327 79369 794 II 79453 79495 79537 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 Lotf. Kxsec. /> 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 35 36 37 38 39 40 41 42 43 44 45 46 47 48 i2_ 50 5« 52 53 54 55 56 57 58 (iO V. V 30 40 50 40 50 30 40 50 42 42 13.2 17-6 22.1 28.0 35.0 27 27 2.7 2. 3.2 3- 3-6 3 4.1 4.6 4- 4. g.i 9 13-7 18.3 »3- 18. 22.9 22. 2g 26 .6 3.0 3-4 3 9 4-3 17. 2S 7 3.0 8 3.4 9 3.8 10 4.2 20 8.5 30 12.7 40 17.0 50 21..: i». r 419 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 52° 53° 10 II 12 13 14 15 i6 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Log. Vers. 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 58471 5S49? 58523 58549 58575 58601 58626 58652 58678 58704 58730 58755 58781 5880^ 58833 58859 58884 58916 58936 58962 58987 59013 59039 59064 59096 59116 5914T 5916? 59193 592 1 8 59244 59270 59295 59321 59346 59372 5939f 59423 59449 59474 59500 59525 59551 59576 59602 5962^ 59653 59678 59704 59729 59754 59780 59805 59831 59856 5988T 59907 59932 59958 59983 9 . 60008 Loj;. Vers. D Log. Exsec. ! D 26 25 26 26 26 25 26 26 25 26 25 26 26 25 26 25 26 25 26 25 25 26 25 26 25 25 26 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 D 79537 79579 7962T 79663 79705 7974^ 79789 79831 79874 79916 79958 80000 80042 80084 80126 80168 80216 80252 80294 80336 80378 80426 80463 80505 80547 80589 80631 80673 80715 80757 80799 8084T 80883 80925 80968 010 052 094 136 178 220 262 304 346 388 430 473 515 557 599 641 683 725 76^ 809 851 894 936 978 82020 82062 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 Log. Kxsec.i 1) Log. Vers. 9 . 600O8 60034 60059 60084 60IIO 60135 60166 60185 6021 I 60236 60261 60285 60312 60337 60362 6038^ 60412 60438 60463 60488 60513 60538 60563 60589 60614 60639 60664 60689 60714 60739 60764 60789 60814 60839 60864 60889 60914 60939 60964 60989 014 039 064 089 114 139 164 189 214 239 264 289 313 338 363 388 413 438 462 48^ 512 n 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 24 25 25 25 24 25 25 25 2% 25 24 25 25 I) Log. Exsec. D 82062 82104 82146 82188 82236 82272 82315 82357 82399 82441 82483 82525 8256^ 82609 8265T 82694 82736 82778 82820 82862 82904 82946 82988 83031 83073 83115 83157 83199 8324T 83283 83325 83368 83410 83452 83494 83536 83578 83626 83663 83705 83747 83789 83831 83873 83916 83958 84000 84042 84084 84126 84168 842 II 84253 84295 8433? 84379 84422 84464 84506 84548 84596 Log. Exsec. 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 I> 10 II 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 40 41 42 43 44 p. P 20 40 50 20 30 40 50 42 4.2 4.9 5-6 6.4 7-1 M-i 21.2 28.3 35-4 7 3 8 3 9 3 10 4 20 8 30 13 40 17 50 21 26 25 16 20.8 42 4.2 4.9 5 6 6.3 7.0 14.0 21 .0 28.0 35-0 2% 2-5 3-0 3-4 3-8 4-2 8-5 12.7 17.0 21.2 24 2.4 2 8 3-2 3-7 4.1 16. P. P. 420 TABLE VIII, —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 54 r>i> 5 6 7 8 9 10 II 12 13 U 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44_ 45 46 47 48 49 L()!r. Vers. It 50 51 52 53 54 55 56 57 58 59 9.6 .6 .6 .6 .6 9.0 .6 .6 .6 .6 9.6 .6 .6 .6 .6 9.6 .6 .6 .6 .6 12 3 537 562 586 61T 636 661 685 716 735 760 784 809 834 858 883 908 932 957 982 9.62005 .62031 .62055 .62086 .62105 9.62129 .62154 .62178 .62203 .62227 9.62252 .62275 .62301 .62325 .62350 9.62374 .62399 .62423 .62448 .62472 9.62497 .62521 .62546 .62576 .62594 (JO 9.62619 .62643 .62668 .62692 .627 1 6 9.62741 .62765 .62789 .62814 .62838 9.62862 .62887 6291T .62935 .62960 9.62984 24 25 24 25 24 25 24 25 24 25 24 24 25 24 25 24 24 24 25 24 24 24 25 24 24 24- 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 Ii08?. Kxsec. 1> 84596 84632 84675 ,84717 84759 8480T 84843 84886 84928 84970 85012 85054 85097 85139 8;i8i 9.85223 .85265 .85308 •85350 .85392 85434 85476 ,85519 ,85561 ,85603 85645 85688 85730 ,85775 ,85814 9- 85857 85899 85941 85983 86026 Lost. Vers. /> ,86068 86116 ,86152 .86195 .86237 86279 8632T 86364 86406 86448 86496 86533 86575 ,86617 86659 86702 86744 86786 .86829 ,86871 86913 86956 86998 87046 87082 9.87125 Los;. Kxspr. 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 Lotf. \i'\<. It I. 9.62984 •630O8 .63032 .63057 .63081 9.63105 .63129 •63154 .63178 .63202 9.63226 .63256 .63274 .63299 ■63323 9-63347 •63371 •63393 •63419 •63443 9.63468 .63492 .63516 •63540 •63564 9.63588 .63612 •63636 . 63666 •63684 9-63708 •63732 .63756 •63786 •63804 9.63828 .63852 .63876 .63900 .63924 7> 9.63948 .63972 .63996 .64019 • 64043 9.64067 .64091 .64115 .64139 .64163 9.64187 .64216 .64234 .64258 .64282 9.64306 .64330 .64353 .64377 . 6440 1 9.64425 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 23 24 24 24 24 23 24 24 23 24 24 23 24 24 -J 24 23 24 I,Off. Vers. /> I,oc. Kxser Kxser n CS7125 87167 87209 87252 87294 87336 87379 87421 87463 87506 87548 87596 87633 87675 87717 87760 87802 87844 87887 87929 87971 88014 88056 88099 88141 88183 88226 88268 S8316 88353 88395 88438 88486 88522 88565 88607 88650 88692 88734 88777 88819 88862 88904 88947 88989 8903 T 89074 89116 89159 8920T 89244 89286 89329 8937T 89414 89456 89499 89541 89583 89626 89668 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 _4_ 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 •20 21 ->2 23 24 -5 26 27 28 29 30 32 1 '^ 34 35 36 37 3 39 8 40 41 42 43 44 i> 45 46 47 48 49 :>() 5' 52 53 54 55 56 57 58 59 <>0 I'. I'. 20 30 40 50 40 20 30 40 50 42 42 4.9 5-6 6.4 7 » 14. 1 21 .2 28.3 35.4 24 16 42 4 a 4.9 56 6.3 7.0 14.0 21 .0 28.0 35.0 25 24 2 5 2 2 9 2. 3 3 3. 3 7 3. 4 I 4 8 3 8 12 S 12 16 6 16. 20 8 20 23 3.» 3-5 3-9 7-8 II. 7 15-6 19.6 r. I' 421 TABLE VIIL— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 56° 57 10 II 12 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 Log. Vers. 9.64425 64448 64472 64496 64520 64543 6456^ 64591 64614 64638 64662 6468I 64709 64733 64756 64786 64804 6482? 64851 64875 64898 64922 64945 64969 64992 D Lost. Exsec. D 650I6 65040 65063 65087 651 16 65134 65157 65181 65204 65228 65251 65275 65298 6532T 65345 65368 65392 65415 65439 65462 65485 65509 65532 65556 65579 65602 65626 65649 65672 65696 65719 65742 65765 65789 65812 65835 Log. Vers. 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 24 23 23 23 23 23 24 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 89668 897 1 1 89753 89796 89838 89881 89923 89966 90OO8 90051 90094 90136 90179 90221 90264 903O6 90349 90391 90434 90476 90519 90561 90604 90647 90689 90732 90774 90817 90860 90902 90945 90987 030 073 158 200 243 286 328 371 414 456 499 541 584 627 669 712 755 79? 846 883 926 968 92011 92054 92096 92139 92182 92224 42 42 42 42 42 42 42 42 42 43 42 42 42 42 42 42 42 42 42 42 42 43 42 42 42 42 42 43 42 42 42 42 43 42 42 42 42 43 42 42 43 42 42 42 43 42 42 43 42 42 43 42 43 42 42 43 42 43 42 42 D Log. Exsec. !> Log. Vers. 9.65835 .65859 .65882 .65905 .65928 9.65952 .65975 .65998 .66021 . 66044 9.66068 .66091 .66114 •66i3f . 66 I 66 9.66183 .66207 .66230 •66253 .66276 9.66299 .66322 .66345 .66368 .6639I 9.66415 .66438 . 6646 I .66484 .66507 D Log. Exsec. Z> 9.66530 •66553 .66576 •66599 .66622 9.66645 .66668 .66691 .66714 ■66737 9.66760 .66783 .66805 .66828 .66851 9.66874 .6689^ . 66926 .66943 .66966 9.66989 .67012 .67034 .6705; .67086 9.67103 .67126 .67149 .67171 .67194 9.67217 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 22 23 23 23 22 23 23 22 23 23 22 23 22 Loe. Vers. I) 92224 92267 92310 92353 92395 92438 92481 92524 92566 92609 92052 92695 9273? 92786 92823 92866 92909 92951 92994 93037 93080 93123 93165 932O8 9315J 93294 93337 93380 93422 93465 93508 93551 93594 93637 93680 93722 93765 93808 93851 93894 93937 93980 94023 94066 94109 941 51 94194 94237 94286 94323 94306 94409 94452 94495 94538 9458T 94624 9466^ 94716 94753 94796 Log. Exsec. T> 43 42 43 42 43 42 43 42 43 42 43 42 43 42 43 43 42 43 42 43 43 42 43 43 42 43 43 42 43 43 42 43 43 43 42 43 43 43 42 43 43 43 43 43 42 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 5 6 7 8 _9_ 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 p. P. 40 50 20 40 50 40 50 43 4 3 5 S 7 6 4 7 I M 3 21 5 28 6 35 8 24 2-4 2.8 3.2 3-6 4 o 16. 42 4.2 4.9 5-6 6.4 7-1 14. 1 21 .2 28.3 35-4 23 3.1 3-5 3.9 7-§ 11.7 19.6 23 2 2.3 2. 2.7 2. 3.0 3 3-4 3- 3-8 3. 7^6 7- II-5 II. 15-3 T.S- 19.1 18. P. p. 422 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 58° 59" Lo:;. Vers. 1* 9 10 1 1 12 15 i6 i8 19 20 21 22 23 24 -5 26 27 28 -9_ 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 5S 59 00 9.67217 .67240 .67263 .67285 •67308 9.67331 .67354 •67376 .67399 .67422 9.67445 .67467 .67490 .67513 •67535* 9.67671 .67694 .67717 •67739 .67762 9.67784 .67807 •67830 .67852 •67875 9.67897 .67920 .67942 .67965 .67987 9.68010 .68032 .68055 .68077 .68 1 00 9.67558 .67581 .67603 .67626 .67649 9.68122 .68145 .68167 . 68 1 90 .68212 9.68235 .68257 .68280 .68302 •68324 9.68347 •68369 .68392 .68414 •68436 9.68459 .68481 •68503 .68526 •68548 9-68571 Loir. Vers. 1 /> Loir. Kxsec. /> -^3 -3 22 23 22 23 22 23 22 23 22 22 22 22 23 22 22 22 22 22 22 22 -IT 22 9^94796 94839 94882 94925 94968 95011 95054 95097 95140 95183 95226 95269 95313 95356 95_399 95442 95485 95528 95571 95614 95657 95700 95744 95787 95830 95873 959'6 95959 96002 96046 96089 96132 96175 962 1 8 9626T 96305 96348 96391 96434 96478 96521 96564 96607 96656 96694 96737 96786 96824 96867 969 1 6 96953 96997 97040 97083 97127 97170 97213 97257 97300 97343 9^97387 IjOjf. Kxser. 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 ~~Tr \A)\i. \('IS. It 68571 .68593 ,68615 ,68637 ,68660 ,68682 ,68704 ,68727 ,68749 ,68771 68793 ,68816 ,68838 ,68866 ,68882 68905 ,68927 , 68949 ,68971 ,68993 ,69016 .69038 . 69060 .69082 .69104 69126 69149 691 7 1 ,69193 ,6921 5 69237 .69259 ,69281 69303 ,69325 69347 ,69369 ,69392 .69414 ,69436 ,69458 ,69480 .69502 ,69524 ,69546 ,69568 69590 ,69612 ,69634 ,69656 69678 69700 ,69721 69743 69765 9.6978^ . 69809 .69831 .69853 •69875 9 . 69897 liOsr. V<'rs. 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2 2 O -> 22 22 22 22 22 22 22 22 22 22 21 22 2 2 2 2 ^2 21 l-o:: 9 10 K\MM' /> 97387 97430 97473 97517 97566 97603 97647 97696 97734 97777 97826 97864 97907 97951 97994 98038 9808 T 98125 98168 982 1 T 98255 98298 983421 983851 98429 98472 98516 98559 98603 98647 98696 98734 98777 98821 98864 98908 98952 98995 99039 99082 99126 99170 99213 99257 99300 99344 99388 99431 99475 995 '9 99562 99606 99650 99694 99737 99781 99825! 99868! 99912 99956 00000 I KxsHr. 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 44 43 43 43 43 43 43 43 44 43 43 43 43 44 43 43 43 44 43 43 44 43 43 44 43 44 43 44 43 43 44 43 44 "TT {) 4 5 6 7 8 _9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24_ 25 26 27 28 30 31 32 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 :a) 51 52 53 5-+_ 55 56 57 58 59 (>0 v. V. 20 30 40 so 40 50 30 40 50 44 4.4 4- 5-1 5 5.8 6.6 5 6 7.3 7 M.6 14. 22. 21 29.3 36.6 29. 36 43 43 6 4 3 7 5.0 8 5 7 9 6 4 10 7-1- 20 14 3 30 21-5 40 28.6 50 35. 8 23 2.3 2.7 3^o 19. 22 2.2 2.6 15.0 18.7 22 21 32 3-6 7-1 10.7 14.3 17.9 423 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 60° 61° 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 26 27 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Log. Vers, 9.69897 .69919 . 69946 . 69962 .69984 9 -70005 . 70028 . 70050 .70072 • 70093 9-70115 .70137 .70159 .70181 .70202 D 9.70224 . 70246 .70268 .70289 .70311 9-70333 •70355 .70376 . 70398 .70420 9.70441 .70463 .70485 .70507 •70528 9.70550 .70572 • 70593 .70615 • 70636 9.70658 . 70680 .70701 .70723 .70745 9.70765 .70788 . 70809 .70831 .70852 9.70874 .70896 .7091^ .70939 , 70966 9.70982 .71003 .71025 .71045 .71068 9.71089 .71III .71132 .71154 .71175 9.71197 Log. Vers. 22 21 22 22 22 21 22 22 21 22 21 22 22- 21 22 21 22 21 22 21 22 21 22 21 21 22 21 22 21 21 22 21 21 21 22 21 21 21 22 21 21 21 21 21 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 Log. F.xsec. 10.00000 . 00044 .0008^ .00131 . 00 1 7 5 I o . 002 1 9 .00262 .00305 .00356 .00394 10.00438 .00482 .00525 .00569 . 006 1 3 D Log. Vers. D 10.00657 .00701 .00745 .00789 .00833 10.00875 . 00926 . 00964 .oioog .01052 10.01 095 .01 146 .oi 184 .OI228 .01272 IO.OI315 .01366 .01404 •OI448 .01492 10.01535 .01586 .01624 .01668 .01712 10.01755 .01806 .01844 .01889 .01933 10.01977 .02021 .02065 .02109 .02153 10.02197 .02242 .02286 .02330 .02374 1 0.024 1 8 .02463 .02507 .02551 .02595 10.02639 J> Log. Exsec. 44 43 44 43 44 . 43 44 44 43 44 44 43 44 44 44 43 44 44 44 43 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 197 218 239 261 282 304 325 346 368 389 411 432 453 475 496 51? 539 566 581 603 624 645 667 688 709 730 752 773 794 815 ^1>7 858 879 906 922 71943 71964 71985 72005 72028 72049 72070 7209T 721 12 72133 72154 72176 72197 72218 72239 72266 72281 72302 72323 72344 72365 72386 72408 72429 72450 72471 I) I Loff. Vers. 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 Log. Exsec. 10.02639 .02684 .02728 .02772 .02815 10.02861 .02905 .02949 .02994 .03038 10.03082 .03127 .03171 .03215 .03260 10.03304 •03348 .03393 •0343^ .03481 10.03526 .03576 .03615 .03659 .03704 I0.03748 -03793 .03837 .03881 .03926 n 10.03970 . 040 1 5 .04059 .04104 .04149 10.04193 .04238 .04282 .04327 -04371 10.04416 . 0446 I .04505 .04550 -04594 10.04639 . 04684 .04728 .04773 .04818 10.04862 .0490^ .04952 .04995 .05041 10.05086 .05131 .05175 ,05226 .05265 10.05310 I) 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 45 44 44 44 44 44 44 45 44 44 44 45 44 44 44 45 44 45 44 44 45 44 45 44 45 44 45 10 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Log. Exsec. 1> GO 40 50 40 50 p. P. 45 4-5 5-2 6.0 6.7 7-5 20 T5.0 30 22.5 40 30.0 50 37-5 44 4.4 5-i 5-8 6.6 7-3 14-6 29-3 36.6 44 4.4 5-2 5.9 6.7 7-4 M-8 22.2 29-6 37.1 43 4-3 5-1 5-8 6.5 7.2 14.5 21.7 :>g.o 36.2 22 21 2.2 2. 2-5 2. 2.9 2 3-3 3- ,3-6 3 . ^7-3 7- II. 10. 14.6 14. 18.3 17- ^.6 I 7 3 9 21 20 30 40 50 P. P. 424 TABLE VIII.-L0GARITHM;C VERSED SINES AND EXTERNAL SECANTS 63° 63° 10 II 12 13 14 Lop. Vers. I 7> 9.72471 .72492 •72513 •72534 .72555 9.72576 .72597 .72618 .72639 .72660 15 16 17 18 19 9.72681 .72701 .72722 .72743 .72764 9.72785 .72806 .72S2J .72848 .72869 9.72890 .7291 I .72931 .72952 .72973 9.72994 .73015 •73036 .73057 •73077 9.73098 •73119 .73140 .73161 .73181 9.73202 .73223 •73244 •73265 •73285 9-73306 •73327 .73348 .73368 •73389 9-734IO .73430 •73451 •73472 •73493 9-73513 •73534 •73555 .73575 •73596 ,73617 .7363^ 73(>S^ 73679 73699 9.73720 Loer. Vers. 21 21 21 21 21 21 21 21 21 2r 20 21 21 21 21 21 21 20 21 21 21 20 21 21 21 20 21 21 20 21 21 20 21 26 21 21 20 21 20 21 26 21 20 21 20 20 21 20 21 20 20 21 20 26 21 20 26 21 20 26 /> Loir. Kxsec' J> 10.05310 •05354 •05399 •05444 .05489 10.05534 .05579 •05623 .03668 .05713 I0.05758 .05803 .05848 .05893 .05938 10.05983 .06028 .06072 . 06 1 1 ^ .06162 10.06207J .0625^' ^^ 44 45 45 44 45 45 44 45 45 45 44 45 45 45 45 45 44 45 45 .06297 .06342 .0638^ 10.06432 .0647^ .06522 .06568 .06613 10.06658 .06703 .06748 •06793 .06838 10.06883 .06928 .06974 .07019 . 07064 10,07109 .07154 .07200 .07245 .07290 10.0733$ •07380 .07426 .07471 .07516 10.07562 .07607 .07652 .0769^ •07743 10.07788 .07834 .07879 •07924 .07970 10.08015 iOir. Kxs«'c.l 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 — Loj?. Vers. I 1* 9.73720 .73740 .73761 •73782 .73802 9.73823 .73843 .73864 •7?>^H .73905 9.73926 .73946 .73967 .73987 . 74008 ).74028 . 74049 . 74069 . 74090 .74110 >.74i3i .74151 .74172 .74192 .74213 9.74233 .74254 .74274 .74294 .74315 9-74335 •74356 •75376 •74396 .74417 9-74437 .74458 .74478 . 74498 •74519 9^74539 •74559 .74580 . 74606 .74626 9.74641 .74661 .74681 .74702 .74722 9.74742 •74762 •74783 •74803 •74823 9.74844 .74864 •748S4 .74904 •74924 20 I 26 21 I 26 , 26 I ~° 26 26 21 20 26 26 26 26 26 26 26 26 26 26 26 26 26 26 20 26 20 26 26 26 26 26 20 26 26 26 20 26 26 20 20 26 26 20 26 26 20 26 20 26 20 26 26 20 26 20 26 20 20 26 Lop. Kxsec. 10.0801 5 . 0806 I . 08 1 06 .08151 .08197 /> 10.08242 .0S288 •08333 ■08379 ,08424 10.08470 •08515 ,08561 .08605 .08652 45 45 45 45 45 45 45 10.08697 .08743 ,08789 .08834 .08880 10.08926 .08971 .09017 ,09062 -09I08 10.09154 . 09200 .09245 ,09291 -09337 10.09382 .09428 •09474 ,09520 .09566 I0.096IT •0965? • 09703 • 09749 .09795 10.09841 .09886! .09932! ■0997 8: . 10024 10. 10070 .10116 . 10162', . I0208| , 10254 10. 10300 .10346 .10392 .10438 . 10484 10. 10530 •I0576 . 10622 . 10668 ,10714 9.74945 ' 10. 10766 Loi.'.Vors. /> ILocr. Kxsfr.l 425 45 45 45 45 45 45 45 45 46 45 45 45 46 45 45 45 46 45 46 45 45 46 45 46 46 45 46 45 46 46 45 46 46 45 46 46 46 46 46 46 45 46 46 46 46 46 46 46 46 46 46 46 46 /> 5 6 7 8 _9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 I'. I'. 25 26 27 28 29 30 31 32 33 34 J3 36 37 38 39 52 55 56 57 58 59 (>0 20 40 5^ 30 40 50 20 30 40 50 48 46 5-4 6 2 7 o 7-7 «5-S 23.2 31.0 38.7 6 f 6.9 7 h «5-3 23.0 30 6 38 3 45 4-5 5-3 6 o 6 8 7 6 15 I 22.7 30-3 37-9 45 4-.S 5 2 6.0 6.7 7-5 150 22 5 30.0 37 5 30 40 50 44 4-4 5 2 5 9 6 7 7 4 '4. a 22.2 29-6 37 1 21 2 4 2 8 31 3 10. ij, '7 20 20 6 2,0 7 8 '•3 2-6 9 3^o 10 30 3-3 6-6 30 10.0 40 •3 3 50 16.6 I', r. TABLE VIII. — LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 64 65 10 II 12 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 51 52 53 54 Log. Vers. 55 56 57 58 59 60 9-74945 74965 74985 75005 75026 75046 75066 75086 75106 75126 75147 75167 75187 75207 7522^ n Los 7524? 7526^ 75287 75308 75328 75348 75368 75388 75408 75428 75448 75468 75488 75508 75528 75548 75568 75588 75608 75628 75648 75668 75688 75708 75728 75748 75768 75788 75808 75828 75848 75868 75888 75908 75928 75947 7596^ 75987 76007 76027 76047 76067 76087 76 log 76126 76146 Log. Vers. 20 26 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 26 20 20 20 20 20 26 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 19 20 20 20 20 20 19 20 20 20 19 20 20 20 19 20 20 Z> 10 10 10 10 10 10 10 10 10 10 10 10 10 Log Exsec. 0766 0807 0853 0899 0945 0991 i03f 1084 1 1 30 IJ76 1222 1269 1315 1 361 1407 1454 1506 1546 1593 1639 1685 1732 1778 1825 187T JD I Log. Vers. 191^ 1964 2010 2057 2103 2150 2196 2243 2289 2336 2383 2429 2476 2522 2569 2616 2662 2709 2756 2802 2849 2896 2942 2989 3036 3083 3130 3170 3223 3270 3317 3364 341 1 345^ 3504 3551 Exsec. 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 4g 46 46 46 46 46 46 46 46 46 46 46 47 46 46 46 46 47 46 46 47 46 46 47 46 47 46 47 47 46 47 46 47 47 47 46 47 47 9.76146 .76166 .76186 .76206 .76225 9.76245 .76265 .76285 • 76304 .76324 9.76344 . 76364 .76384 .76403 .76423 9.76443 .76463 .76482 .76502 .76522 9.76541 .76561 .76581 . 76606 .76626 9 . 76640 .76659 .76679 .76699 .76718 X) 9.76738 .76758 .7677^ .76797 .76817 9.76836 .76856 .76875 .76895 .76915 9.76934 .76954 .76973 . 76993 .77012 9.77032 .77052 .77071 .77091 .77116 9.77130 .77149 .77169 .77188 .77208 9.7722^ .77247 .77266 .77286 •77305 9.77325 Log. Vers. 19 20 20 19 20 19 20 19 20 20 19 20 19 20 19 20 19 19 20 19 20 19 19 20 19 19 20 19 19 20 19 19 19 20 19 19 19 20 19 19 19 19 19 19 20 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 Log 10 10 10 10 10 10 10 10 10 10 10 10 10 Log Exsec. 3551 3598 3645 3692 3739 3786 3833 3886 392? 3974 D 4021 4068 4115 4162 4210 4257 4304 4355 4398 4445 4493 4540 4587 4634 4682 4729 4776 4823 4871 491 8 4965 5013 5066 5108 5155 5202 5250 529? 5345 5392 5440 548^ 5535 5582 5630 5678 5725 5773 5826 5868 5916 5963 601 1 6059 6106 6154 6202 6250 6298 6345 6393 47 47 47 47 47 47 47 47 47 47 47 47 47 Al 47 47 47 4f 47 Al A7 47 4? Al A7 Al 47 4? Al 47 Al Al Al 47 Al Al Al Al Al Al Al Al Al Al 48 Al Al Al 48 Al Al 48 47 Al 48 Al 48 48 Al 48 IT 20 p. p. 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 20 40 50 40 50 20 30 40 50 48 4.8 5-6 6.4 7.2 8.0 16.0 24.0 32.0 40.0 47 4-7 5.5 6.2 7.0 7.8 15-6 23.5 3i§ 39.1 20 47 4-7 5-5 6.3 7.1 7-9 15. a 23 -7 31-6 39-6 46 4.6 5.4 6.2 7.0 7.7 15-5 23.2 31.0 38.7 46 6 4.6 7 8 5-3 6,1 9 6.9 10 7-6 20 15-3 30 23.0 40 50 30-6 38.3 20 2.0 2-3 2-6 3.0 3.3 6.6 10. o '3-3 16.6 19 6 1 9 7 8 2 2 3 6 9 2 9 10 3 2 20 6 5 30 9 7 40 50 13 16 2 P. p. 426 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. G6 07° Los. Vers. 10 II 12 15 i6 17 i8 19 20 21 22 23 24 -3 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 77325 77344 77363 77383 77402 77422 77441 77461 77480 774Q9 775'9 77538 7755? 77 S9G 77616 77635 77654 77674 77693 77712 77732 77751 77770 77790 77809 77828 77847 77867 77886 77905 77925 77944 77963 77982 78002 78021 78040 78059 78078 78098 78117 78136 78155 78174 78194 78213 78232 78251 78276 78289 78309 78328 78347 78366 78385 78404 78423 78442 78462 9-78481 Loe. Vers, J> LoK 10 10 10 10 10 10 10 10 10 10 10 1) I -OK 10 10 xsec. 7> 6393 644 T 6489 6537 6585 6633 6680 6728 6776 6824 6872 6926 6968 7016 7064 71 12 7 1 661 7209* 72571 7305' 7353 7401 7449 7498, 7546I 7594 7642 7696 7739 778? 7835 7884 7932 7986 8029 807? 8126 8174 8222 8271 8319 8368 8416 8465 8514 8562 861 1 86:^9 8708 8757 8805 8854 8903 895T 9006 9049 9098 9146 91951 92441 9293! 48 47 48 48 48 4? 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 49 48 48 48 48 48 49 48 48 49 48 49 48 49 48 49 49 48 liOjf. Vers. 9.78481 78500 78519 78538 78557 78576 78595 78614 78633 78652 78671 78696 78709 78728 7874? 78766 78785 78804 78823 78842 78861 78886 78899 789I8 78937 78956 78975 78994 79013 79032 79051 79069 79088 79107 79126 79145 79164 79183 79202 79226 79239 79258 79277 79296 79315 79333 79352 79371 79390 79409 7942? 79446 79465 79484 79503 79521 79540 79559 79578 79596 0.79615 7> \ liOi:. Vers. /> \avj: 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 8 9 9 9 9 8 9 9 8 9 9 8 9 9 8 9 9 8 9 8 9 9 8 9 8 9 8 9 8 9 TT 10 10 10 10 10 10 10 10 10 10 10 10 10 K\scc •9293 19342 1 939 1 19439 19488 19537 19586 19635 19684 19733 19782 1983T 19886 19929 19979 20028 20077 20126 20175 20224 20273 20323 20372 2042! 20476 20520 20569 206 18 20668 2071^ 20767 20816 20865 20915 20964 014 063 113 162 212 262 3" 361 416 466 510 560 609 659 709 759 808 858 908 958 22008 22058 22108 221581 22208 22258I Kxser. /> 49 49 48 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 50 49 49 49 49 50 49 50 49 50 49 50 49 50 50 49 50 50 50 50 50 50 It 10 I 2 3 4 5 6 7 8 40 41 42 43 44 45 46 47 48 ±9 oO 51 52 53 55 ^6 57 58 59 <;o 20 30 40 50 6 7 8 q 10 20 30 40 50 20 30 40 50 I'. 1* 49 4.6 5-6 6.4 7.2 8.0 16 o 24 o 32 o 40.0 19 6 1.9 7 2-3 8 2.6 9 2.9 10 3.2 20 6.5 30 9-7 40 13.0 SO 16.2 48 4.9 4- 5-7 6 5 5- 6. 7-3 8 i 7- 8 .6.3 16. 24-5 24 32 6 3« 40-8 40. 48 4^ 4-7 5-5 6.3 71 7-9 '5-8 23.7 3»-6 39-6 19 1.9 3.2 2-5 2 8 3-» 6.3 9-5 12 6 '5-8 IB 6 1 § 7 2 I 8 3 4 9 3 8 10 3 X 20 6 I 30 9 3 40 13 J 50 15 4 50 49 5-0 5-8 6.6 4.9 5-8 6 6 7-'; 8.3 7-4 8.2 .6.6 16.5 25.0 24 7 33 3 33-0 4'. 6 41.2 I', r. 427 TABLE VIIL— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 68° 09° 10 II 12 14 15 i6 17 i8 19 20 21 23 24 ^3 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Losj. Ters.' I> Loe. Exsec; D 9.7961S •79634 •79653 .79671 .79690 9.79709 .7972? •79746 .79765 •797S3 9.79802 .79821 •79839 .79858 .79877 9.79893 .79914 ■79933 •79951 .79970 9^79988 . 80007 . 80026 .80044 .8006^ 9.80081 .80106 .80119 .8013^ .80156 9.80174 .80193 ,80211 .80230 . 80248 9.80267 .80286 . 80304 •80323 .80341 9.80360 •80378 .80397 .80415 ■ 80434 9.80452 . 80470 . 80489 .8050^ .80526 9.80544 .80563 .80587 .80600 .80618 9.80636 .80655 •80673 . 80692 .80710 9.80728 Log. Vers. 18 19 18 18 19 18 19 18 18 19 18 18 19 18 18 18 19 18 18 18 19 18 18 18 18 19 18 18 18 18 18 18 18 18 19 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 J8 18 18 18 18 10 .222581 .223081 .223581 .224081 .22458 10 225081 22558! 22608 22658 227O8' 10. 22759 22809 22859 22909 22960 10. 23010 23066 231 15 23161 23211 10 23262 23312 23362 23413 23463 10. 23514 23564 23615 23666 23716 10 23767 23817 23868 23919; 239691 10. 24020 24071 24122 24172 24223 10. 24274 24325 24376 24427 24478 10. 24529 24580 24631 24682 24733 10. 24784 24835 24886 24937 24988 10. 25039 25096 25142 25193 25244 10.25295 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 51 50 50 51 50 51 50 51 51 50 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 Ters.! D 80728 80747 80765 80783 80802 80826 80839 80857 80875 80S94 80912 80936 80949 80967 80985 003 022 046 058 077 095 113 131 150 168 186 204 223 241 259 27^ 295 314 332 350 368 386 405 423 441 459 477 495 513 532 550 568 586 604 622 646 658 6/6 695 713 72)^ 749 767 785 803 8182T Log. Exseo.; jf> 10 25295 25347 25398 25449 25501 10. 25552 25604 25655 25707 25758 10 25810 25861 25913 25964 26016 10 2606^ 26 II 9 2617 1 26222 26274 10 26326 26378 26429 26481 26533 10 26585 26637 26689 26741 26793 7> 'Log. Kxsec i D ' Log. Vers.' 7) 428 10 26845 26897 26949 27001 27053 10 ,27105 ,2715^ , 27209 27261 27314 10 27366 274I8 27476 27523 2757? 10 2262^ 27680 27732 27785 2783? 10. 27890 27942 27995 2804^ 28100 10.28152 .28205 .28258 .28316 .28363 10.28416 51 51 51 51 51 51 51 51 51 51 51 51 51 51 51 52 51 51 52 51 52 51 52 52 52 51 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 5- 53 52 52 53 52 Log. Exsec. 1 /> 10 1 1 12 13 14. 15 16 17 18 19 20 21 22 23 24 p. P 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 40 50 20 30 40 50 20 30 40 50 20 30 40 50 53 52 5.3 6.2 7.0 7.9 8.8 17-6 26.5 35-3 44.1 52 5.2 6.0 6. a 7.8 8.6 17^3 26.0 34-6 43 3 51 51 5-9 6.8 7-6 8.5 17.0 25.5 34-0 42.5 5.2 '3.1 7.0 7.0 8.7 17-5 26.2 SS-o 43'7 51 5-1 6.0 6-8 .7-7 8.6 25-7 34^3 42.9 50 5-0 5^0 6.7 7.6 8.4 16.8 25.2 33- 6 42.1 20 30 40 50 50 5-0 16.6 25.0 33.3 41-6 19 I 1.9 1. 2.2 2. 2-5 2-§ 2. 2. 3-1 6.3 3^ 6. 9-5 12.6 0. 12. ^5-1 ^5- 18 7 2. 8 2. 9 2. 10 3- 20 6. 30 9^ 40 12. 50 15- P. p. TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. o 71 i6 18 19 20 21 22 23 24 -5 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 :)5 56 hi 58 59 (>0 Lojf. Vers. ' I> 9.8182I .81839 .81857 .81875 .81893 9 . 8 1 9 1 1 .81929 .81947 .8196I .81983 9.82001 .S2019 .82037 .82055 .82073 9.82091 .82109 .82127 .82145 .82163 9.82449 .82467 .82485 .82503 .82526 9-82538 •82556 •82574 •82592 .82609 9.82627 •82645 .82663 .82681 .82698 9.82716 •82734 .82752 .82769 .82787 9.82805 .82S23 .82840 .82858 .82876 9.82181 .82199 .82217 •82235 9.8.^276 .82288 .82306 ■82324 .82342 9.82360 .82378 • 82396 .82413 .82431 9. 82894 liOs;. Vers. liOtf. K.xsec It 10. 28416 . 28469^ .28521 .28574 .28627 28680 28733' 28786 28839' 28892 10 10 28945 28998: 2905 I \ 29104 29157 10. 29210 29263 293161 29370 29423 10. 29476 29529 29583 29636 29689 10. 29743 29796 29850 29903 29957 10. 30010 30064 30117 3o>7i, ^0225i 10 30278| 30332 30386 30440 30493 10 30547 30601 30655 30709 30763 10. 30817 .30871' •30925 • 30979 •31033 10 31087 31141, 31195 31249 313031 10 31358; 31412 31466 31521 3'575i 10. 31629 53 52 53 53 52 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 54 53 53 54 r ■-> DJ 54 53 54 53 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 Lot. Vers. /> /> |Loc. Kxsec. /> 82894 8291 I 82929 82947 82964 82982 83000 83017 83035 83053 83076 83088 83106 83123 83141 83159 83176 83194 83211 83229 83247 83264 83282 83299 83317 83335 83352 83370 83405 83422 83440 83458 83475 83493 83510 83528 83545 83563 83586 83598 83615 83633 83656 83668 83685 83703 83720 83737 83755 83772 83790 83807 83825 83842 83859 83877 83894 83912 83929 83946 I.OL'. K\M-C.I 7> 10 10 10 lioe. Vers. 7> 10 10 10 10 10 10 10 10 10 10 31629 31684 31738 3 ' 793 31847 31902 3'956 3201 1 32066 32126 32175 32230 32284 32339 3239-1 32449 32504 32558 32613 ^2668 32723 32778 32833 32888 32944 32999 33054 33' 09 33164 33220 33275 33330 33385 33441 33496 33552 3360^: 33663 33718 33774 33829 33885 33941 339961 34052! 34108 34164; 342201 34275J 34331 1 34387 34443 34499 34555 34611 3466^ 34723 34780, 34836I 348q2 34948! 54 54 54 54 54 54 54 55 h\ 54 55 54 55 hi 55 55 54 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 55 56 55 55 56 55 56 56 55 56 56 56 56 56 56 56 56 56 56 56 56 5 6 7 8 _9_ 10 1 1 1 2 '3 14 15 16 17 18 19 r. r 21 22 23 24 26 27 28 29 30 34 j3 36 37 38 39 40 41 42 43 ii_ 45 46 47 48 49 oO 51 52 53 54 ^\^»M• /> 55 56 57 58 59_ (;o 20 40 50 40 50 20 30 40 50 40 50 56 56 5 6 6.6 7-5 8.5 9.4 18.5 28.2 37-6 47-» 5.5 6.5 7-4 8.3 9.2 T8.5 27.7 370 46.2 54 53 5-3 6.2 7-i 8 o 8.9 26.7 35-6 44.6 5.6 6.5 7^4 8.4 9 3 i3.6 28.0 37-3 5S 55 5-5 6.4 7-3 8.2 9.1 18.3 27.5 36$ 45-8 54 4 5 4 5- 6.3 6. 7 2 7 8 2 8. 9 r 9 18 I 18. 27 2 ?!• 36 3 3^'- 45 4 45- 53 5-3 6.2 7.6 7-2 8.| 17-6 26.5 35-1 44 I 55 6 5-2 7 O.i 8 7.0 7-9 10 8.7 20 '7-5 30 a6.2 40 35-0 50 43-7 18 17 17 2.4 2-7 3-0 6.0 9.0 5-8 8.7 ii.fi 14 .6 '•7 I', r 42Q TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 10 II 12 13 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Loff. Vers. 9-83946 83964 83981 83999 84016 84033 84051 84063 84085 84103 84126 8413^ 84155 84172 84189 84207 84224 8424T 84259 84276 84293 84316 84328 84345 84362 84380 84397 84414 84431 84449 D Los. Kxsec 84466 84483 84506 8451^ 84535 84552 84569 84586 84603 84626 84638 84655 84672 84689 84706 84724 84741 84758 84775 84792 84809 84826 84844 84861 84878 84895 84912 84929 84946 84963 9.84986 Log. Vers.! J> 10 10 10 10 10 10 10 10 10 10 10 10 10 Log 34948 35005 35061 35ii? 35174 35230 35286 35343 35399 35456 35513 35569 35626 35683 35739 35796 35853 35910 35967 36023 36086 36137 36194 3625T 36308 36366 36423 36480 36537 36594 36652 36709 36766 36824 36881 36938 36996 37054 3711T 37169 D 37226 37284 37342 37399 3745? 37515 37573 37631 37689 37747 37805 37863 37921 37979 38037 38095 38153 38212 38276 38328 38387 56 56 56 56 56 56 56 56 57 56 56 56 57 56 57 56 57 57 56 57 57 57 57 57 SJ 57 57 Si 57 57 57 Si Si Si 57 57 58 Si Si si si 58 57 58 58 Si 58 58 58 58 58 58 58 58 58 58 58 58 58 58 Log. Vers. 84986 8499? 85014 85031 85049 85066 85083 85100 85117 85134 85151 85168 85185 85202 85219 85236 85253 85270 85287 85304 85321 85338 85355 85372 85389 85405 85422 85439 85456 85473 85496 85507 85524 85541 85558 85575 85592 85608 85625 85642 85659 85676 85693 85710 85726 85743 85766 85777 85794 85811 8582? 85844 85861 85878 8^895 8591 I 85928 85945 85962 85979 85995 £> Log l^lxsec! 7> i Lo:r. Vers.' 7> 10 10 10 10 10 10 10 10 10 10 10 10 10 liOi Exsec 38387 38445 38504 38562 38621 38679 38738 38796 38855 389'4 38973 39031 39096 39149 39208 39267 39326 39385 39444 39503 39562 3962T 39681 39740 39799 39859 39918 3997? 40037 40096 40136 40216 40275 40335 40395 40454 40514 40574 40634 40694 40754 40814 40874 40934 40994 41054 41114 41 '74 41235 41295 j> 41355 41416 41476 41537 41597 41658 41719 41779 41840 4 1 90 1 41962 58 58 58 58 58 58 58 59 58 59 58 59 59 58 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 60 59 59 60 59 60 59 60 60 60 60 60 60 60 66 60 60 66 66 60 66 60 66 66 66 61 66 66 61 61 Kxsec. /> 15 16 17 18 19 20 21 22 23 24 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 P. P. 40 50 40 53 40 50 6 7 8 9 10 20 30 40 50 20 30 40 50 6 I 6.1 1 7 8 I I 9 10 1 i 20 3 30 5 40 50 6 § 60 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 59 5-9 6.9 5- 6. 7-8 8-8 7- 8. 9-8 9- J9-6 39- 29-5 29. 39-3 49.1 39 48. 58 5-8 6.7 7-7 8.7 9-6 19-3 29.0 38.6 48.3 5-7 6.6 7.6 8.5 9-5 IQ.O 28.5 38.0 47-5 65 6.0 7.0 8.0 9.1 10. T 20.1 30.2 40- 3 50.4 59 5-9 6.9 7 9 8.9 9,9 i9-§ 29.7 39-6 49.6 58 57 5-7 6.7 7-6 8.6 9.6 19. 1 28.7 38.3 47-9 57 56 17 17 6 1-7 1-7 7 2 2.0 8 2 3 2.2 9 2 6 2.5 10 2 Q 2-8 20 S X 5 6 30 8 7 8.S iO II 6 1^-3 50 14 6 14.1 5-6 6.6 7-5 8.5 9-4 38.8 28.2 37-6 471 16 1-6 1.9 137 P. P 430 TABLE VIII. — LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 7 4 7 a 10 II 12 14 15 i6 17 i8 19 20 21 2 2 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44^ 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 00 Lost. Vers. 9.85995 .86012 .86029 . 86046 . 86062 9.86079 . 86096 .86113 .86129 .86146 7> Log 9.86163 .86179 .86196 .86213 .86230 9.86246 .86263 .86280 .86296 .86313 9.86330 • 86346 .86363 .86380 ■86396 9.86413 . 86430 • 86446 . 86463 • 86479 86496 86513 86529 86546 86s62 •86579 .86596 .86612 .86629 .86645 9.86662 .86678 .86695 .86712 •86728 9.86745 .86761 .86778 .86794 .86811 9.86827 . 86844 . 86866 .86877 .86893 9.86910 .86926 . 86943 .86959 . 86976 9.86992 Lost. Vers. 17 16 17 '6 17 16 17 16 17 1(5 16 17 16 17 16 16 17 16 16 17 '6 16 '7 16 16 17 16 16 16 17 16 16 16 16 17 16 16 16 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 16 '6 16 16 16 16 16 10 /> Lo:r 10 10 10 10 10 10 10 10 10 10 10 10 Exsecj 1) 41962 42022 42083 42144 42205 42266 4232? 42388 42450 42511 42572 42633 42695 42756 42817 42879 42940 43002 43063 43' 25 43 '87 43249 43310 43372 43434 43496 43558 43620 43682 43744 43806 43868 43931 43993 44055 44118 44180 44242 44305 44368 44430 44493 44556 446 1 8 4468 T 44744 44807 44870 44933 44996 45059 45122 45185 45248 45312 45375 45439 45502 45565 45629 45693 Kxspc. 60 61 61 61 61 61 61 61 61 61 61 6! 61 61 61 61 61 61 62 61 62 61 62 61 62 62 62 62 62 62 62 62 62 62 62 62 62 62 63 62 62 63 62 63 62 63 63 63 63 63 63 63 63 63 63 63 63 63 63 64 Lojf. Vers. /> 9.86992 . 87009 .87025 .87042 .870 58 9.87074 .87091 .87107 .87124 .87146 9.87157 .87173 .87189 .87206 .87222 9.87239 .87255 .8727! .87288 • 87304 9.87326 .87337 •87353 .87370 .87386 9.87402 .87419 .87435 .87451 .87468 9.87484 .87506 .875I6 •87533 .87549 9.87565 .87582 .87598 .87614 .87631 9.87647 .87653 .87679 .87696 .87712 9.87728 .87744 .87761 .^7777 .87793 9.87809 .87825 .87842 .87858 .87874 9.87896 . 87906 .87923 •87939 .87955 9.87971 K\s»'o. 10 10 •45^93 •45756 .45820 .45884 .45947 .4601 1 .46075 •46139 .46203 .4626^ 10 •46331 • 46395; . 46460 .46524' .46588 10 ,46652 ,46717 ,46781 46846 46916 10 •46975 • 47040 .47104 ,47169 .47234 10 .47299 •47364 .47429 ■47494 ■47559 10 .47624 ,47689 .47754 ,47820 47885 10 .47950 .48016, .48081 .48147 •48213 10 •48278 .48344 .48410 .48476 ,48542 10 .48607 .48674 ■ 48740; .48806, .48872 10 .48938 . 49004 ,49071 .4913?, ,49204! 10 ,49270 .49337 49403 49476 49537 10.49604 7> i liO tf. V p r s . I J) IliO g . Kx» 6 6 6 6 6 6 6 6 6 6 6 $ 6 6 6 "TT liOP 10 10 10 10 10 10 10 10 10 10 10 10 10 Kxsec. D 49604 49670 4973? 49804 49871 49939; 50006' 50073 50146 50208; 50275 50342 50410 5047? 50545 50613 5068 1 j 50748; 508 1 6 50^ 50952 51026 51088 51157 51225 51293 51361 51430 51498 51567 51636 51704 51773 51842 51911 51980 52049 52118 52187 52256 52325 52394 52464 52533 52603 52672 52742 52812 52881 52951 53021 53091 5316T 53231 53301 53372 53442 53512 53583 53653 53724 ' Kxsec. 66 67 67 67 6? 67 67 67 6f 67 6f 6f 67 68 6f 68 6? 68 68 68 68 68 68 68 68 68 68 68 68 69 68 68 69 69 69 69 6q 69 69 69 69 69 69 69 69 70 69 69 70 70 70 70 70 70 76 70 76 70 76 70 "TT Los. Vers. 88933 88949 88964 88986 88996 89012 89028 89044 89060 89075 89091 8910^ 89123 89139 89155 89176 89186 89202 89218 89234 89249 89265 89281 89297 893 • 2 89328 89344 89360 89376 89391 89407 89423 89438 89454 89470 89486 89501 8951^ 89533 89548 89564 89580 89596 896 11 89627 89643 89658 89674 89690 89705 89721 89737 89752 89768 89783 89799 89815 89836 89846 89862 9.8987? liOsr. Vers. U Lo;r. Exsec. JJ 10 10 10 10 10 10 10 10 10 10 10 10 10 T) ihuii 53724 53794 53865 53936 54007 54078 54149 54220 54291 54362 54433 54505 54576 5464? 54719 5479' 54862 54934 55006 55078 55150 55222 55294 55366 55438 5551^ 55583 55655 55728 55801 55873 55946 56019 56092 56165 56238 563 1 T 56384 5645? 5653' 56004 56678 56751 56825 56899 56973 57047 57126 57195 57269 57343 5741? 57491 57566 57646 57715 57790 57864 57939 58014 58089 Kxsec. T> 70 71 76 71 71 71 71 71 71 7T 71 71 71 72 71 71 72 71 72 72 72 72 72 72 72 72 72 73 72 72 73 72 73 73 73 73 73 73 73 73 73 73 74 73 74 74 73 74 74 74 74 74 74 74 75 74 74 75 75 75 I 25 26 27 28 ao 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 P. P. 75 74 72 t 7.-S 7-4 7. 7 8.7 8 6 8. 8 10. 9 8 9 9 II. 2 II I 10. 10 12.5 12 3 12. 20 25.0 24 6 24. 30 37.5 37 3b. 40 50.0 49 3 48. 50 62.5 61 6 60. 20 30 40 50 66 6.6 1 7 7 8 8 9 9 1 1 22 33 44 55 16 13 16 T.6 2.4 •^•6 5-3 8.0 10. 6 J3-3 72 71 6 7.2 7.1 7 8.4 8.3 8 9.6 9.4 9 10.8 10. 6 10 12 .0 ".8 20 24.0 23-6 30 36.0 35.5 40 48 47.3 50 60.0 59-1 70 7.0 8.2 9.4 10.6 11.7 23.3 35-2 47.0 58.7 69 68 67 6 6.Q 6.8 7 8,5 7-9 8 9.2 9.0 9 10.3 10 2 10 "5 ".3 20 2:!.0 22-6 30 34. s 34 40 46.0 45-3 50 57.5 50-6 6.- 7.8 8.9 100 II. i 2P.3 33-5 44-6 55.8 0.0 0.0 0.0 o. 1 O. I o. i 0.2 0.3 0.4 15 P. P. 432 TABLE Vlll. — LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 78' ly c 1) Lo?. Vers. | 7> 9.8987^ 15 I? 10.58089 I .89893 .58164 2 • 89908 •58239 3 .89924 •58315 4 • 89939 16 , 3 • 58390 S 9.89955 10.58465 6 .89971 , 2 .58541 7 .89986 I = •586I6 8 .90002 . 58692 9 .90017 t6 .58768 10 9.90033 10.58844; 1 1 • 90048 .58920 12 . 90064 .58995 13 . 90080 I 2 .59072 U .90095 I? .59148 10.59224 15 9.901 1 1 i6 .90125 I = .59300 17 .90142 ' 3 I = •59377; i8 •90157 T » •59453 19 .90173 •59530 20 9.90188 10.59606 21 . 90204 T £ •59683 22 .90219 • 59760; 23 •90235 ^ 5 1 = •59837, 24 .90250 T » • 59914; 10.59991' 25 9.90266 26 .90281 1 ' 5 T P .60068 27 .902971 I5 , 3 .60145 28 .90312 j '5 .60223 29 .90328 ' .60306 30 9 •90343 10.60378 31 •903591 ' 5 T r .60455, 32 •90374 •60533 33 .90389 . 606 1 1 1 34 •90405 Id , p .60688 35 9 . 90426 10.60765 36 .90436 I3 T = . 60844 37 .90451 I 3 . P .60923 3^ .90467 I 5 , p . 6 I 00 I 39 .90482 I3 15 .61079 40 9.90497 10.61158 41 •90513 r P .61236 42 •90528 r = .61315 43 •90544 1 = •61393 44 •90559 ' 3 15 .61472 45 9.90574 10.61551 46 .90590 .61630 47 .9060 5 , P .61709 48 . 9062 1 '5 1 p .61788 49 oO •90636 15 , p .6186^ 9.90651 10.61947 51 . 90667 '3 .62026 52 .90682 I 3 .62105 53 . 90697 T = .62185 54 . 907 1 3 T P .62265 55 9.90-28 10.62345 56 • 90744 15 15 .62424 57 •90759 .62504 5^ .90774 .62585 59 . 90790 .62665 GO 9.90805 10.62745 ' liOtf. Vers. 1 /> liOe. Kxsec. Loff. Kxsec. J> 75 75 75 71 75 75 75 76 75 76 76 75 75 76 76 76 76 76 76 76 77 76 77 77 77 77 77 71 77 77 77 77 78 77 78 78 78 78 78 78 78 78 78 79 78 79 79 79 79 79 79 79 80 79 80 79 80 86 80 80 /> Loir. Vers. 90805 90826 90835 90851 90865 7> 90881 90897 90912 90927 90943 90958 90973 90988 91004 9IOI9 91034 91049 9 1 06 5 91080 91095 91 1 10 91 126 91141 91156 91171 91 187 91202 91217 91232 91247 91263 91278 91293 91308 91323 91338 91354 91369 91384 9' 399 91414 91429 9 '445 91460 91475 91490 91505 91520 91535 91556 91565 91581 91596 9161 1 91626 91641 91656 91671 91685 91701 9 • 7 ' r ) liOjr. Kxser. 10 62745 .62825 ,62906 ,62985 ,6306^ 10 ,63148 ,63229 .63310 .63391 ,63472 10 63553 63634 63716 6379? 63879 10 63961 64043 ,64125 ,64207 ,64289 1) 10 .64371 64453 •64536 ,64618; ,64701! 10 10 64784 64867 64950 65033 ,651 16 ^5199 65283 65366 ,65450 65534 10 6561^ ,657oT .65785 ,65870 •65954 10 66038 J ,66123 ,6620^1 .66292 ,66377 10 10 10 ,66462 .66547 ,66632 .66717 .66803 "66888 ,66974 .67059 ,67145 .6723T 673"? .67403 67490 .67576 • 67663 10,^7749 I IjOtr. Vers. /> l, 83 83 83 83 84 83 84 84 84 84 84 84 84 84 85 85 85 85 85 85 85 85 85 86 86 86 86 86 86 86 86 /> 5 6 7 8 _9_ 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ^9 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 50 5' 52 53 54 55 56 57 58 59 r»o 1'. I' 86 85 84 6 8.6 8.5 7 10. 9 8 11.4 11.3 9 12. g 12.7 10 14.3 14. 1 20 28.6 28.3 ^0 43.0 42.1; 40 57-3 56.6 50 7»^6 70.8 8. 9.8 II .2 12.6 14.0 28.0 42.0 56.0 70.0 83 82 81 6 8.3 8.2 8. 7 9-7 9-5 9- 8 11 .0 10. 2 10. 9 12.4 12.3 12. 10 20 i3-§ 27-6 X3-6 27-3 '3- 27. 30 41. s 41.0 40 40 50 55-3 69.1 54 § 68.3 54 • 67 80 79 6 80 7-9 7 9-3 9.2 8 10.6 10.5 9 12.0 "•§ 10 133 13-1 20 26.6 26.3 30 40.0 39-5 40 53-3 52-6 50 66.6 65-8 78 78 9.1 10.4 11.7 13.0 26.0 39-0 52.0 65.0 77 76 75 6 7^7 7.6 7 9.0 8.8 8 10.2 10. 1 9 "•5 II. 4 10 «« § 12. $ 20 25 6 253 30 38.5 38.0 40 5'-3 50. § 50 64.1 63.3 7-5 8.7 10. o 1 1 .2 12.5 25.0 37-5 50.0 62.5 20 30 40 50 0.3 0.4 16 15 6 1.6 i'5 7 8 3.1 1.8 2.6 9 2-4 23 10 2.5 2.6 20 5-3 5^» 30 8.0 7-7 40 10$ 10.3 50 133 12.9 15 '•5 '•7 2.0 2.3 2-5 7-5 to.o 12.5 I', r TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 80^ 81 10 1 1 12 14 15 i6 17 i8 19 20 21 22 23 24 25 26 27 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Log. Vers 7I6 731 746 761 776 791 807 822 837 852 867 882 897 912 927 942 957 972 987 92002 92016 9203! 92046 92061 92076 92091 92106 92121 92136 92151 92166 921 81 92196 9221 r 92226 92240 92255 92276 92285 92306 92315 92330 92345 92360 92374 92389 92404 92419 92434 92449 92463 92478 92493 92508 92523 92538 92552 9256^ 92582 92597 9.92612 Log. Vers. X) Loff. Exsec. 10 67749 67836 67923 .68010 . 68097 10 ,68184 .6827 68359 .68447 .68534 10 ,68622 68716 ,68798 ,68886 68975 10 69063 ,69152 69246 69329 694 1 8 10 69507 69596 69686 69775 69865 10 69955 , 70044 ■70134 ,70224 .70315 10 70405 70495 70586 70677 70768 10 70859 70950 7104T 71133 71224 10 ,71316 ,71408 7 1 500 71592 ,71684 10 71776 71869 7 1 96 1 72054 72147 10 ,72240 72333 72427 ,72526 72614 10 ,7270^ 7280T 72895 72990 73084 10.73178 /> Log. Kxsec. I) 86 87 87 ^7 8^ 8? 87 8? 87 88 88 89 89 89 89 89 89 89 90 89 90 90 96 90 90 91 96 91 91 91 91 91 91 91 92 92 92 92 92 92 92 93 92 93 93 93 93 93 93 94 94 94 94 94 Log. VerS' I) 92612 92626 92641 92656 92671 92686 92706 92715 92730 92745 92759 92774 92789 92804 92818 92833 92848 92862 9287^ 92892 92907 9292T 92936 92951 92965 92986 92995 93009 93024 9.3039 93053 93068 93083 9309^ 93II2 93127 9314I 93156 9317I 93185 93200 93214 93229 93244 93258 J> Log. Exsec. 93273 93287 93302 93317 93331 93346 93366 93375 93389 93404 93419 93433 93448 93462 93477 9.93491 10 10 10 10 10 10 10 10 10 10 10 10 10 Loe. V«*rs.l 7> |Loi 73178 73273 73368 73463 73558 73653 73748 73844 73940 74035 74131 7422^ 74324 74426 74517 74613 74716 7480^ 74905 75002 ij 75099 7S^9l 75295 75393 7549' 75589 75688 75786 75885 75984 76083 76182 76282 76382 76481 76581 76681 76782 76882 76983 77083 77184 77286 77387 77488 77590 77692 77794 77896 77998 78101 78203 78306 78409 78513 78616 78720 78823 78927 79031 79136 Exsec. 95 94 95 95 95 95 95 96 95 96 96 96 96 96 96 97 97 97 97 97 98 9l 98 98 98 98 98 99 99 99 99 99 100 99 00 00 06 06 06 06 01 01 01 01 01 02 02 02 02 02 02 03 03 03 03 04 03 04 04 04 /> 10 I 2 3 4 20 21 22 23 24 25 26 27 28 29 30 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 «0 p. r 20 40 50 6 7 8 9 10 20 30 40 50 40 50 20 30 40 50 20 30 40 50 90 9 TO 5 12 13 5 15 30 45 60 75 0-5 0.6 o.g 0.7 o§ 1-6 2.5 3-3 41 IS 40 50 7 0. 8 Q 0. I I I 1. 2 3 2. 3 4 5 6 3- 4- 5 8 5- 80 8.0 9-3 13-3 26.6 40.0 53 3 66.6 0.8 0.9 1 .0 1.2 1-3 2.6 4.0 5-3 6.6 IS I S 1. I 8 I. 2 a. 2 3 2. 2 6 2. 5 7 10 I 7 .3 5- 7- 10. 12 9 12. 14 1.4 1-7 1.9 2.2 2.4 4-8 7.2 9-6 12. 1 434 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 83° 8:r 10 II 12 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L(>S. Vers. I 7> 9-93491 93506 'J3- 93535 93549 93564 93578 93593 93607 93622 93636 93651 93665 93680 93694 93709 93723 93738 93752 93767 93781 93796 93816 93824 93839 93^53 93868 93882 93897 9391 1 93925 93940 93954 93969 93983 93997 94012 94025 94041 940 s 5 94069 94084 94098 941 12 94127 94141 94155 94170 94184 94198 94213 94227 94241 94256 94270 94284 94299 94313 9432? 94341 9 94356 liOe. Vers. I 7> 14 u 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 u 14 u 14 14 14 14 14 14 u 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 Log. Ex sec. I D \ IO.79F36J .792401 •79345 •79450 •79555 10.79666 .79766 .79871 •79977 . 80083 10.80189 . 80296 . 80402 .80509 .806 16 10.80723 .50831 •80938 .81046 .81154 10,81202 .813711 .81479 .8i:;88 .81697 io.8i8o6 .81916 .82025 .82135 .82245 10.82356' .82466 .82577 .82688 .82799 10.82916 .83022 •83133' .83245 • 833vS' 10.83470 .83583 .83695 .83809 .83922 10.84035 .84149 •84263 . 84492 10.84607 .84721 .84837 .84952 .85068 10.85183 .85299 .85416 •85532 . 85649 10.85766 104 105 104 105 105 105 105 106 106 106 106 106 107 107 107 I of \oJ 108 108 108 log log 109 109 109 109 109 1 10 no no 1 16 116 1 1 1 III III 1 11 1 11 I 12 I 12 112 I 12 I 12 113 113 113 114 114 114 114 U5 114 115 116 116 116 116 117 117 Log. Vers. I /> L(»;r. Kxscc. Kxspr.l 7> 9 -943 56 94370 94384 94398 94413 94427 94441 94456 94470 94484 94498 94512 94527 94541 94555 945<^9 94584 94598 94612 94626 94646 , 94655 j 94669 94683 94697 I 947 1 1 94726 94740 94754 94768 94782 94796 94816 94825 94839 94853 94867 9488T 94895 94909 94923 94938 94952 94966 94980 94994 95008 95022 95036 , 95050 95064 95078 95093 95107 95121 95135 95M9 95163 95177 95191 995205 liOR. Vers. 7> 10.85766 .85884 . 86001 .861 19 .86237 /> 10 86355 ,86474 ,86592 ,86711 ,86831 10 .86956 .87076 .87196 ,87316 .87431 10 87552 87673 87794 .8;;9i6 ,88038 10. 88160 88282 88405 88528 8865T 10 88775 ,88898 ,89022 .89147 ,89271 10 89396 ,89521 ,89647 .89773 .89899 117 iif 117 118 118 118 118 119 119 119 120 120 120 126 121 121 121 ! 121 122 122 122 122 123 123 124 123 124 124 124 125 125 125 126 126 10 .90025 , 90 1 5 2 ,90279 , 90406 90533 10 ,90661 .90789 ,90917. ,91046 91175 10 91304 .91434 .91564; .91694, 91825, 10 91956 ,92087 ,92218 .92350 ,92482; 10 92614 92747 92886 93014' 9314 71 93281I 126 126 127 127 127 128 127 128 129 129 129 1 130 129 136 130 131 131 3 3 13" 131 132 132 133 133 133 133 134 4 5~ 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 21 23 24 10 l.fijr. Kxsec' /> 25 26 27 28 30 31 32 33 34 I', r. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 :>o 51 52 53 54 55 56 57 58 59 3-3 15.0 16.^ 33.3 50.0 66.6 83.3 6 II .0 1 7 '2.^1 8 M.6 16. s TO ,8.3 20 36.6 30 55.0 40 73 3 50 91-6 6 3 0.3 7 8 0.3 0.4 q 0.4 10 0.5 20 1 .0 30 1.5 40 2.0 50 2-5 I 6 0.. 1 7 I 8 I 9 I 10 I 20 3 30 5 40 ^ 50 8 14 6 1.4 7 ».7 8 » 9 9 2.2 10 a. 4 20 4.? 30 7.2 40 9-6 50 12. 1 0.1 O.I o.i o.a 0.3 0.4 14 1.4 15 » 8 a. I 2-3 4 6 7.0 9.3 11.6 r. r. 435 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 84° 85° 10 II 12 15 i6 17 i8 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Lost. Vers. 9.95205 95219 95233 9524f 9526T 95275 95289 95303 9531^ 95331 95345 95359 95373 95387 95401 95415 95429 95443 95457 95471 2> Log. Exsec. 95485 95499 95513 95527 95540 95554 95568 95582 95596 95610 95624 95638 95652 95666 95680 95693 95707 95721 95735 95749 95763 95777 95791 95804 95818 95832 95846 95860 95874 95888 95901 95915 95929 95943 95957 95970 9598^ 95998 96012 96026 9.96039 Log. Vers. n 10 93281 93416 •93551 93686 93821 10 93957 .94093 ,94229 94366 94503 10 ,94641 94778 .94917 .95055 95194 10, 95333 95473 95613 95753 95894 JD 10 ,96035 .96176 .963I8 , 9646 1 , 96603 10 96746 ,96889 97033 97177 97322 10 .97467 ,97612 .97758 ,97904 ,98056 10 .9819? .98345 ,98492 , 98646 .98789 10 98938 .9908^ .9923^ .9938^ •99538 10 10 1 1 99689 ,99841 99993 ,00145 ,00293 II 00451 00605 00759 00914 01069 II 01225 01381 0153^ 01694 01852 1 1 .02010 Log. Kxsen. 34 35 35 35 35 36 36 37 37 Zl 3l 38 38 39 39 39 40 40 40 41 41 42 42 42 43 43 44 44 44 45 45 45 46 46 47 4^ 4? 48 49 49 49 50 50 51 51 51 52 52 53 53 54 54 55 55 55 56 56 57 S7 58 7> Log. Vers. 9.96039 96053 96067 96081 96095 96 log 96122 96136 96150 96163 9617^ 9619I 96205 962 1 8 96232 96246 96259 96273 96287 96301 96314 96328 96342 96355 96369 96383 96397 96416 96424 96438 1> Log 96451 96465 96479 96492 96506 96519 96533 96547 96566 96574 96588 9660T 96615 96629 96642 96656 96669 96683 96697 96716 96724 9673^ 96751 96764 96778 96792 96805 96819 96832 96846 9.96859 I I 1 1 II 1 1 I I I I II II II 1 1 II II I I Lotf. Vers. 7> |L<»er 436 Exsec. 02010 02163 0232^ 02487 02646 02807 02968 03129 03291 03453 036 1 6 03780 03944 04108 04273 04438 04604 04771 04938 05106 1) 05274 05443 05612 05782 05952 06123 06295 06467 06640 06813 06987 07 161 07336 07512 07688 07865 08043 08221 08400 08579 08759 08940 09121 09303 09486 09669 09853 10038I 10223I 10409 10595 10783 1097 1 1 1 160 1 1 349 1 1 539 1 1 736 11922 12114 1230^ 12501 Kxs«>«' 58 59 59 59 66 61 61 61 62 63 63 64 64 65 65 66 67 67 67 68 69 69 69 70 71 71 72 73 73 74 74 75 76 76 77 71 78 79 79 80 86 81 82 82 83 84 85 85 86 86 87 88 89 89 90 91 91 92 93 93 5 6 7 8 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 (>0 J) p. p. 6 190 19.0 7 8 22.1 25-3 9 28.5 10 20 63-3 30 95 -o 40 50 126.6 158.3 170 17.0 19-8 22.6 25-5 28.3 56.6 85.0 "3-3 141-6 180 18.0 21.0 24.0 27.0 30.0 60.0 00. o 120.0 150.0 160 16.0 18. 6 21.3 24 .0 26.6 53-3 80.0 106.6 ^33-3 150 140 15.0 17-5 20.0 22.5 25.0 50,0 75-0 lOO.O I2S.O 14.0 16.3 18.6 21 .0 23 -3 46.5 70.0 93-3 116.6 130 9 b 6 13.0 0.9 0. 7 15-1 1 0. 8 17-3 I 2 I . 9 19-5 I 3 1. 10 21.6 I 5 I. 20 43-3 3 2. .30 65.0 4 5 4- 40 86.6 60 5- 5« 108.3 7 5 6. 6 7 0.7 6 0.6 7 8 0.8 0.9 0.7 0.8 9 I.O 0.9 10 I.I 1 .0 2D 2.3 2.0 30 3-5 3-0 40 4-6 4.0 50 5-8 5-0 5 05 0.6 0.6 0.7 0-8 1-6 2-5 3-3 4.1 14 14 I 6 1.4 1.4 I. 7 1-7 1-6 I. 8 1.9 1-8 I. 9 2.2 2.1 2. 10 2.4 2-3 2. 20 4-8 4 6 4- 30 7.2 7.0 6. 40 9-6 9-3 9- 50 12. 1 11.6 II. r. »' TABLE VIII. -LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 80° S7'^ 10 II 12 14 15 i6 I? i8 19 20 21 22 23 24 ^5 26 27 29 30 31 32 33 J±. 35 36 37 38 39 40 41 42 43 44 45 46 47 4B 49 50 51 52 53 54 55 56 57 58 5?_ GO Loar. Vers. J) 9.96859 .96873 .96887 .96900 .96914 ,96927 ,96941 96954 96968 96981 9.96995 .970O8 . 97022 •9703? • 97049 .97062 .97076 .97089 97103 97116 ■ 97 1 30 ■97143 ■97157 ,97170 97183 ■97^97 .97216 ,97224 ,9723^ 97251 ,97264 ,97277 ,97291 97304 97318 997331 •97345 •97358 •97371 •97385 9^97398 .97412 •97425 •97438 •97452 9.9746S •97478 .97492 •97505 •97519 9-97532 •97545 •97559 •97572 •97585 9^97599 .97612 .97625 •97639 .97652 9.9766: 13 14 13 13 13 J3 f3 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 J3 13 13 liOp. Kxsec. 7> I I . 12501 . 12696 . I2891 .13087 .13284 I I . 1^482 . I 3680 •13879 .14079 . 14286 '95 195 '96 '96 198 198 199 200 201 1 1 14482 14684 14887 15092 15297 1 1 15502 15709 1-5917 16125 '6334 1 1 16544 16755 16967 17186 17394 1 1 . 1 7609 . 17824 . 1 804 1 .18259 .1847^ II . 18697 .1891^ •i9'38 • '9361 .19584 II . 19809 . 20034 .20261 . 20489I .20717, 11.20947, .211781 .21410I .21643' .21877: I I .221 I2j .22349 .22586 .22825 .230651 ' I • 233O6 •23548 . 23792' -240371 .24283! 11.24530 •24778 •25028 .25279 •255311 11.25785 201 202 203 204 205 205 206 208 208 209 210 21 1 212 213 214 214 215 216 218 218 219 226 221 222 223 224 225 227 227 228 230 236 232 233 234 235 236 237 239 239 24? 242 243 245 246 247 248 250 251 252 254 Lojf. Vers. I It 9.97665 .97679 .97692 •97705 •977I8 9.97732 •97745 •97758 .97772 ■97785 9^97798 .97811 .97825 •97838 •97851 9.97864 •97878 .97891 • 97904 -97917 9-97931 •97944 •97957 .97976 .97984 9-97997 .98016 .98023 • 98036 .98050 9.98063 .98076 . 98089 .98102 .98116 9.98129 .98142 •98155 .98168 .98181 9.98195 .98208 .98221 •98234 •9824? 9.98266 •98273 .98287 .98300 •98313 9.98326 •98339 .98352 .98365 •98378 9.98392 • 98405 .98418 .98431 •984-14 f). 984^7 Loir. Vers. | I> \\,nK. Kxser. /> k Loir. Vers. '3 13 '3 13 '3 13 13 13 '3 '3 '3 13 13 13 13 '3 13 13 '3 13 13 13 13 13 13 13 13 '3 13 13 13 13 13 13 13 13 13 13 13 '3 '3 13 13 13 13 M 13 13 '3 '3 !-(»::. K\> /> "•25785 _ .26046 "^5 .26297 "^0 •26554 ^'^ .26814 1 1 27074 27336 27599 27864 28131, 1 1 •28398 .28668 .28938 .2921 1 •29485 1 1 .29766 • 3003^ .30316 •30596 •30878 1 1 ^,1 162 •3'447 •31734' .32023 •32313 II .32606 . 32900 •33196 •33494 •33793 1 1 . 34095 ■34398 • 34704 .35011 •35321 ^ ''•356321 ^I^ -5/ 259 266 262 263 265 266 267 269 276 272 274 275 277 278 279 282 2S3 285 287 288 296 292 294 296 298 299 301 303 305 307 309 10 1 1 12 '3 14 '5 16 17 18 '9 I'. I'. 35946 .36261 •36579 • 36899 1 1 .37221 •37546 .37872 .38201 •38532 1 1 . 38866 .3920I •39540 •39886 .40224 1 1 .40569 . 409 1 8 .41269 .41622 •41979 7> 11.42338 .42699 .43064 •4343' .43802 "-44175 3^3 3'5 318 320 -7 2 '> 324 326 328 33' 333 335 338 340 343 345 348 351 353 356 359 361 36.1 37 5 20 21 22 23 3_ 25 26 27 28 29 ;iO 31 32 33 ii 35 36 37 3^ 39 j> 40 41 42 43 J4 45 46 47 48 49 50 51 52 53 55 56 57 58 59 (;o 6 250 25.0 7 8 29.1 33-3 9 37-5 10 20 83.3 30 40 50 125.0 166.^ 208.3 230 23.0 26. § 30 6 34-5 38^3 76.6 115. o '53^3 191. 6 210 6 21 .0 7 24^5 8 28.0 9 3i^5 10 35^o 20 70.0 20 105.0 40 140.0 50 »75-o 240 24 .0 28.0 32 o 36.0 40.0 80.0 120.0 160.0 200.0 220 22.0 25-6 29.3 33^o 36.^ 73-3 1 10. o '83.3 200 20.0 2^-3 26.6 30.0 33-3 66.6 100.0 133.3 166.6 0.1 0.1 0-3 0.5 o.^ 14 13 6 1.4 '•3 7 '•6 i.ti 8 1-8 1.8 9 2.1 2.0 10 2-3 3.2 20 4.6 4..S 30 7.0 6.7 40 9.3 9.0 50 ".6 II. 2 0.1 o. t 13 '•3 '•7 1.5 2. 1 4-3 6..S 8.6 10. 8 r. r. 190 4 3 6 ig.o 0.4 o^3 7 8 9 22. 1 25^3 28.5 0.4 0-5 0.6 0-3 0.4 0.4 10 20 3'-6 633 °'6 ^•3 0-5 I.O 30 40 50 95-0 126.6 15S.3 2.0 3.3 '•5 2.0 = •5 1 437 TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 88° 89" ' Lo ?. Vers. J> Loi:. Exsec. 1 J> Loj J. Vers. I) Loff. Exsec. 2> / P. P. 9 9845^ I^ " 44175] 376 9 44551! :^7a 99235 11.75050 742 755 768 781 I 98476 13 99248 1 Z .75792 I 2 3 4 98483 98496 .98509 13 13 13 44931 ^g 45313 ^86 45699 ^., 99261 99274 99287 13 13 13 .7654^ .77316 .78097 2 3 4 5 9 .98522 \l '' 46088 ^^^ 9 99299 12 11.78892 795 809 825 840 856 872 896 908 927 S 6 .98535 13 46486 ^92 •46876 395 99312 13 .79702 6 7 •98548 13 99325 13 .80527 7 8 .98562 ^3 •47275 ii^ 99338 1 z .81367 8 9 •98575 13 •47677 -^°; 99351 13 .82223 9 10 9 .98588 ;^ '■ 48083 4-^ 9 99363 12 11.83095 10 II .98601 ij> 48493 jj^ 99376 13 .83986 II 12 98614 13 •48906 4 3 99389 13 . 84894 12 13 .98627 ^j •49323 l^Q • 99402 .85821 13 14 98640 13 49743 "^^ •99415 13 13 12 .86768 947 967 989 14 15 16 15 9 i6 98653 98666 \l ■' 50168 ;|^| 9 50597 f:^ 99428 99446 11.87735 .88724 17 98679 13 51029 2:^: 99453 13 •89735 17 i8 19 98692 98705 13 13 5H66 436 51906 ^^^ . 99466 •99479 1 2 13 . 90769 .91829 1 1034 1059 1085 1112 1 146 1171 1203 1236 18 19 20 9 21 98718 98731 \i " 52351 445 g 53713 % 54176 4 3 •99491 •99504 12 13 I 1 .92914 .94026 20 21 o -7 98744 13 13 13 •99517 i^ •95167 22 23 24 98757 98770 •99530 •99543 13 12 •96338 •97541 23 24 25 25 9 98783 M " 54643!^? 9 99555 11.98777 26 98796 13 •99568 ^3 I 2 . 00048 ' \ 26 27 98809 13 5|5^l 485 56076 4 56563 "^^l 99581 .01358 1309 27 28 29 98822 98835 13 99594 99606 13 12 13 T .02707 • 04098 1349 1 391 1436 1485 28 29 30 30 9 98848 =3 ": 57334 roi 99619 12.05535 31 98861 99632 .07020 31 32 98874 13 58058 504 58567; 1^:1 59082 513 99645 3 •08557 1537 32 33 34 98887 98900 13 99657 99670 13 . IOI49 . II80I 1592 1652 1716 33 34 35 9 98913 ;^ II ^ -^ ?20 59602 J 9 60129 527 60662 533 99683 I 2 I2.I3517 35 36 98925 99695 • 15302 :^^f 36 37 98938 13 99708 13 .17163 37 38 39 98951 98964 I J 13 61202 539 61747. 545 99721 99734 13 12 13 12 .19106 .21139 1943 2033 2I3I 2246 2361 38 39 10 41 42 10 9 41 42 98977 98996 99003 13 62300 ^52 Q 99746 99759 99772 12.23271 .25511 .27872 6 7 I I I 3 • 3 .6 13 ^•3 1.5 43 44 990 1 6 99029 13 12 S5g: . 99784 99797 13 .3036^ •33013 2495 2645 2815 43 44 9 lO 2 2 2 1-7 2.1 45 9 99042 '5 II 65167 soo 99810 12 12.35828 45 30 4 6 •7 4-3 6.5 46 99055 ij 65762 99823 3 •3883? 3009 46 40 9 .0 8.6 47 99068 13 66366 ^Vl f 978 ^4 67598 . . 99835 1 2 . 42068 3231 3489 3791 4152 4588 512^ 5812 6707 47 50 1 1 . J 1 10. § 48 49 99081 99093 13 12 99848 99861 13 12 12 13 12 12 •4555^ •49349 48 49 50 51 52 53 54 1 50 9 51 52 53 54 99106 991 19 99132 99H5 99158 13 13 12 68227l^;8 9 68865I ^38 695ii'6t^ 70168 ^56 70834: f' 99873 99886 99899 9991 1 99924 12.53501 . 58089 .63217 .69029 •75736 6 7 8 9 ] [2 1.2 1.4 1-6 1.9 55 9 56 57 58 59 99171 99184 99197 99209 99222 13 12 13 1 1 7^509J68| 9 728921 96 736001 / / 74319' II 99937 99949 99962 99974 99987 13 12 12 12 13 12 12.83667 •93371 13.05877 •234991 •53615 7931 9704 I25O6 I762T 301 16 55 56 57 58 59 00 10 20 30 40 50 I 2. 1 4.1 6.2 8.3 0.4 (>0 9 99235 ^ II 75050 10 00000 Infinity 1 Lo u. Vers. D Log. Kxsec' T> Lo! :. Vers. j> Loar. Exsec. /> ' 1 P. P. 1 438 TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. lO 20 30 40 50 1 10 20 30 40 50 2 10 20 30 40 50 3 10 20 30 40 50 4 10 20 30 40 50 5 10 20 30 40 50 e 10 20 30 40 7 10 20 30 40 50 8 10 20 30 40 50 9 10 20 30 40 50 10 Sin. 0.0000 0.0029 0.0058 o.ooSf o.ci 15 0.0I4I 0.0174 0.0203 0.0232 0.0262 0.0291 0.0320 0.0349 0.0378 0.0407 0.0436 0.0465 0.0494 0.0523 0.0552 0.0581 0.0610 0.0639 0.0663 0.0697 0.0726 0.0755 0.0784 0.0813 0.0842 0.0871 o. 0900 0.0929 0.0958 0.0987 o. 1016 0.1045 0.1074 O.I 103 O.I 1 32 0.II6I o. 1 190 0.I2I8 o. 1 247 O.I276 0.1305 0.1334 0.1363 0.I39I 0.1420 0.1449 0.1478 o. 1 507 0.1535 0.1564 o. 1 593 0.1622 0.1656 0.1679 0.1708 0.1736 Cos. 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 2§ 29 d. Tan. J). 0000 0.0029 0.0058 0.0087 o.oi 16 0.014.5 0.0174 0.0203 0.0233 0.0262 0.0291 0.0320 d. 0.0349 0.0378 o. 040^ 0.0436 0.0466 0.0495 0.0524 0-0553 0.0582 0.061 1 0.0641 0.0670 0.0699 0.0728 0.0758 0.0787 0.0816 0.0845 0.0875 o. 0904 0.0933 0.0963 0.0992 o. 102T 0.I05I o. 1 080 o. mo 0.1139 o. 1169 0.1 198 0.1228 0.1257 0.1287 0.1316 0.1346 0.1376 o.i4og 0.1435 0.1465 0.1494 o. 1524 0.1554 0.1584 0.1613 0.1643 0.1673 0.1703 0.1733 0.1763 Cot. Cot. 00 343.773 171.885 114.588 85. 9398 68.7501 ;57_^89§ 49.1039 42.9641 38.1884 34-367^ 31.2416 28.6362 26.4316 24. 54 if 22.903^ 21.4704 20.2055 19.0811 18.0750 17. 1693 16.3498 15.6048 14.9244 14.3005 13.726^ 13.1969 12.7062 12.2505 11.8261 11.4300 11.0594 10.71 19 10.3854 10.0786 9.788T 9.5143 9-2553 9.0098 8.7769 8.5555 8.3449 8.1443 7-9530 7.7703 7-595? 7.4287 7.268^ 7- 1 153 6.9682 6.8269 6.6911 6.5605 _6^3j+8 ^,3i3'7. 6. 1976 6.0844 5-9757 5.8708 5-7693 d. 5-6713 i860 1398 7756 8217 1261 6053 2046 Tan. 6380 4333 2648 1244 0061 9056 8'95 7450 6804 6237 5739 5298 4907 4557 4243 3961 3706 3475 3265 3073 2899 2738 2590 2454 2329 2213 2106 2006 1913 1826 1746 1670 1599 1534 1471 1413 1358 1306 1257 1211 1 1 67 1126 1087 1049 1014 986 <1. Cos. 1. 000 1. 0000 I.GOOO 0.9999 0-9999 0.9999 ^9998 0.9998 0.999? 0-9996 0.9996 0.9995 0-9994 0.9993 0.9991 0.9996 0.9989 0.9988 o.998g 0.9984 0.9983 0.9981 0.5979 0.9977 0997g 0.9973 0.9971 0.9969 0.9967 0.9964 0.9962 0.9959 0-9956 0.9954 0.9951 0.9948 0.9945 0.9942 0.9939 0.9935 0.9932 0.9929 0992g 0.9922 0.9918 0.9914 0.9916 ^.9906 0.9902 0.9898 0.9894 0.9890 0.9886 0.9881 J°^9l77: 0.9872 0.9867 0.9863 0.9858 o.98y_ 0.9848 Sin. d. yo 50 40 30 20 10 so 50 40 30 20 10 88 50 40 30 20 10 87 50 40 30 20 10 86 50 40 30 20 10 85 50 40 30 20 10 84 50 40 30 20 10 83 50 40 30 20 10 82 50 40 30 4 5 20 6 10 7 81 8 Q 50 40 30 20 10 80 r. I' 30 29 29 3-0 6.0 9.0 12.0 15.0 18.0 21 .0 24.0 27.0 14.7 17.7 20-6 23.6 26.5 2.9 5-8 8.7 11.6 M-5 17.4 20.3 23.2 26.1 28 5 4 4 2-8 05 0.4 5-7 8.5 I.O 1-5 0.9 1.3 11.4 2.0 1.8 14.2 2.5 2.2 17. 1 3.0 2.7 19.9 3-5 31 22.8 4.0 3-t 25.6 4-5 4.0 0.4 1.6 3.2 3-6 3322 4 5 62 72 82 9I3 3 0.3 70.6 0.9 1.2 1.5 2.1 2.4 I 2.7 1.5 1.7 2.0 1-4 1.6 1.8 0.4 0.5 0.6 .ojo.7 .20.8 .310.9 0.0 0.1 I X O 0.1 0.3 0.4 0.6 0.7 0.9 0.2 0.3 0.3 0.4 0.4 P. P. 80-90 439 TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 10-20° / Sin. d. 28 Tan. d. Cot. d. Cos. d: p. p. 10 10 0.1736 0.1763 30 5-6713 949 0.9848 5 80 50 0.1765 0.1793 5-5764 0.9843 20 0.1793 ^8 0.1823 i^ 5.4845 919 890 862 8^A 0.9838 b 40 30 40 0.1822 O.1851 29 28 0.1853 0.1883 30 30 30 30 • 30 5-3955 5-3093 0.9832 0.9827 5 5 30 20 33 32 31 50 11 10 0.1879 28 28 O.1913 5-2256 811 787 764 742 0.9822 5 6 10 79 50 I 2 3 3-3 3 6.6 6 9-9 9 .2 3.1 .4 6.2 .6 9.3 0.1908 0.1944 0.1974 5-I44S 0.9816 01 936 5.0658 0.9816 20 0.1965 ^8 28 0. 2004 3^ 30 4.9894 0.9805 5 6 40 4 5 13.2 12 16.5 16 .8 12.4 •0 15.5 30 0.1993 2R 0.2034 30 4.9151 721 0.9799 5 30 6 19.8 19 .2 18.6 40 50 12 10 20 0.2022 0.2050 28 28 28 28 0.2065 0.2095 30 30 .30 30 4.8430 4-7728 7o£ 682 664 646 620 0.9793 0.978^ 6 6 6 6 6 20 10 78 50 40 7 8 9 23.1 22 26.4 25 29.7 28 .4 21-7 .6 24.8 .827.9 0.2079 0.2125 47046 0.9781 0.9775 0.9769 0.2I0f 0.2136 0,2156 0.2185 4.6382 4-5736 30 40 0. 2 1 64 0.2193 28 0.2217 0.224^ 30 4.5107 4.4494 613 0.9763 0.97 56 6 30 20 1 36 30 29 1 3.0 3.0I 2.9 ( 50 13 10 20 0.2221 28 28 28 28 0.2278 3^ 30 31 30 4-3897 597 582 568 553 0.9750 6 6 6 6 10 77 50 40 2 3 4 5 6 6.1 6 9.1 g 12.2 12 15-2 15 18.3 18 .0 5.8 ! .0 8.7 .0 II. 6 .0 14.5 .0 17.4 2249 O.23O8 43315 09743 0.2278 0.2306 0.2339 0.2370 4-2747 4.2193 0-9737 0.9736 30 40 0.2334 0.2362 28 28 0.2401 O.243T 3i 30 4-1653 4.1125 540 527 515 0.9723 0.9717 y 6 30 20 7 8 21.3 21 24-4 24 .0 20.3 .0 23.2 50 0.2391 23 0. 2462 31 4.0616 0.Q710 10 9 27.4 27 .0 26.1 U 10 20 0.2419 28 28 0.2493 31 30 31 4.0108 491 480 469 0.9703 7 7 7 76 50 40 20 28 2*7 0.2447 0.2475 0.2524 0.2555 3-9616 3-9136 0. 9696 0.9688 30 0.2504 ^8 0.2586 31 3.8667 0.9681 7 30 I 2.§ 2 .8 2.7 40 0.2532 28 0.2617 31 3.8208 458 0.9674 7 20 2 5.2 5 .6 5.4 50 15 0.2560 28 28 0. 2648 3i 31 3-7759 449 439 42Q 0. 9665 7 7 8 10 75 3 4 8.5 8 II. 4 II .4 8.1 .2 10.8 0.2588 0.2679 3.7326 0.9659 10 0.2615 ?8 0.2716 31 3.6891 420 0.9651 7 50 6 17.1 16 .8^16. 2 20 30 0.2644 0.2672 28 "8 0.2742 0.2773 31 3.6470 3.6059 411 403 0.9644 0.9636 7 8 40 30 7 8 19.9 19 22.8 22 .618.9 .4 21.6 40 0.2700 28 0.2804 3-565? ■^94 0.9628 8 20 9 25-525 .2 24.3 50 16 10 0.2723 28 28 ?8 0.2836 31 31 31 3.5261 387 379 371 0.9626 8 8 8 10 74 50 10 8 0.2758 0.286^ 3-4874 0.9612 0.2784 0.2899 3-4495 0. 9604 20 0.2812 0. 2930 3-4123 0-9596 8 8 40 3 [ i.o 0. 90.8 30 40 0.2840 0.2868 ■'■1 28 0.2962 0.2994 32 3-3759 3-3402 357 0.9588 0.9580 30 20 > 2.0 I. J 3.02. 8 1.6 72.4 50 17 10 0*2896 28 27 28 0.3025 3i 32 31 3-3052 350 343 337 0.9571 8 8 8 10 78 50 - M-o 3- ; 504- 3 6.0 5. 6 3.2 5 4-0 44.8 2923 0.3057 0.3089 32708 09563 0.2951 3.2371 0.9554 20 0.2979 28 0.31 21 32 3.2040 331 0.9546 8 40 r 7.0 6. i 8.0 7. 3 5-6 2 6.4 30 0. 3007 2y 0.3153 32 3-1716 324 0.9537 y 30 c ) 9.0 8. I 7.2 40 0.3035 28 0.3185 32 3-1397 319 0.9528 8 20 50 0.3062 2y 0.3217 32 3.1084 313 0.9519 y 10 18 0.3090 27 0.3249 32 3.0777 307 0.9516 y 72 A »v A v 10 20 0.31 18 0.3145 27 0.3281 0.3313 32 32 3-0475 3-OI78 302 296 0.950T 0.9492 9 9 50 40 1 c 2 I / / .70.7 < •5 1-4 D.60.5 1 1.2 1.0 30 0.3173 27 0.3346 32 2.9887 0.9483 9 30 32 .22.1 1.8 1.5 40 0.3200 •I] 0.3378 32 2.9606 286 0.9474 y 20 4 3 .0 2.8 : 2.4 2.0 50 19 10 3228 27 27 27 27 0.3411 32 32 32 32 2.9319 277 272 267 0.9464 y 9 9 Q 10 71 50 53 64 7 5 86 -73-5 , .5 4-2 : .2 4.9 t. -0 5.6 < 3-02.5 ?.6 3.0 ^23-5 I..8 4.0 ' 0.3255 03443 2.9042 0.9455 03283 0.3476 2.8770 0.9445 20 0.3310 27 0-3508 2.8502 263 0.9436 5 40 96 •76.3 . 5.4 4-5 1 30 0-3338 0-354I 2.8239 0.9425 30 40 0-3365 27 0-3574 33 2.7980 0.9415 20 50 0.3393 ■^■7 27 0.3607 ii 32 2.7725 254 250 0.9407 y 10 10 70 I20 0.3420 0.3639 2-7475' 0.9397 Cos. 1 d. Cot. d. TaB. I d. Sin. 1 d. ' P. p. 1 70°-80' 440 TABLE IX.— NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 20 lo 20 I 30 I 40 30 21 10 20 30 40 50 22 10 20 30 40 ! 50 23 j 20 I 30 ' 40 24 JO I 20 I 30 40 50 25 10 20 30 40 50 26 10 20 30 40 50 27 10 20 30 40 50 28 10 20 30 40 50 29 10 20 30 40 50 30 Sin. d. o 3420 0-3447 0.3475 0.3502 0.3529 0-3556 035 83 o. 36 1 I 0.3638 0.3665 0.3692 0.3719 03746 0.3773 0.3800 0.3827 0-3853 0.3880 o 3907 0.3934 0.3961 0.398^ 0.4014 o. 404 1 0.406^ 0.4094 0.4120 0.4147 0.4173 0.4200 o 4226 0.4252 0.4279 0.4305 0.4331 0.4357 04383 0.4410 0.4436 0.4462 0.4488 0.4514 04540 0.4566 0.4591 0.4617 0.4643 0.4669 0.4694 0.4720 0.4746 0.4771 0.4797 0.4822 0.4848 0.4873 0.4899 0.4924 0.4949 0.4975 0.5000 Cos. Tan. 3 639 0.3672 0.3705 0.3739 0.3772 0.3805 03838 0.3872 0.3905 0.3939 0.3972 o. 4005 0.4040 0.4074 0.4108 0.4142 0.4176 0.4216 0.4244 0.4279 0.4313 0.4348 0.4383 o.44if 04452 0.4487 0.4522 0.455? 0.4592 0.462^ 0.4663 0.4698 0.4734 0.4770 0.4805 0.484T 0487^ 0.4913 0.4949 0.4986 0.5022 0.5058 0.5095 0.5132 0.5169 0.5205 0.5242 0.5280 05317 0.5354 0.5392 0.5429 0.546? 0.5505 0-5543 0.5581 0.5619 0.565? o. 5696 0.5735 05773 Cot. 33 33 33 33 33 33 33 33 33 34 34 33 34 34 34 34 34 34 34 34 35 34 35 34 35 35 35 35 35 35 35 36 35 36 36 36 36 36 36 36 37 36 37 36 37 37 37 37 37 37 38 37 38 38 38 38 38 39 38 (1. Cot. 2-7475 2.7228 2.6985 2.6746 2.6511 2.6279 2.6051^ "2:7826 2.5604 2.5386 2.517T 2.4959 24751 2.4545 2.4342 2.4142 2-3945 2.3750 2.3558 2.3369 2.3182 2.2998 2.2815 2.263^ 2.2466 2.2285 2.21 13 2.1943 2.1775 2. 1609 2. 1445 2. 1283 2.II23 2.0965 2.0809 2.0655 2.0503 2.0352 2.0204 2.0057 .9911 .9768 9626 .9486 •9347 .9210 .9074 .8940 r88oy .8676 .8546 .841? .8296 ^8165 .8040 -791? ■119% .7675 .7555 .743? 7320 Tan. 247 245 239 235 232 238 225 221 218 215 212 208 206 203 200 197 194 192 189 187 184 182 179 177 175 172 I70 168 166 164 162 159 158 156 ^54 152 »5o M8 147 145 143 142 140 139 137 136 134 132 131 130 I2§ 127 125 124 123 122 126 119 118 117 Cos. 3-1 2.0 2.2 T .2 1-3 25 25 24 24 23 23 2.5 2.5 2.4 2.4 2-3 5-1 S-c 4-9 4.!> 4-7 7-6 7-5 7 3 7-2 70 10. 2 12.7 10.0 12.5 9 £ 12.2 9.6 12.0 9.4 It. 7 15.3 150 14 7 14.4 14.1 ^7-8 17-5 J7.1 16.8 16.4 20.4 22.9 20.0 22.5 19. t 22.6 10.2 21.6 :8.8 21 .1 2-3 4.6 6.9 9.2 lis 13-8 4 20.7 22 22 21 21 20 20 4.4 6.6 13.21 4-3 6.4- 4 2 6.3 8.6 8.4 [0.710.5 12.9 12.6 41 6.1 e.o 8.0 10. o 12.31 12 .0 15.4I15.0 14.7 14-3 M- 17.6117.21 16.8 I6.4J16 19.8119.3 18.9ll8.4il8 19 19 18 3-9 5 8 7.8 9 7 13-6 '3 i5-6|i5 i7-5i'7 P. P. 3-7 5-5 7-4 g.2 12.9 14.8 16.6 446 TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. G0°-70" 7()°-8()" Vers. GO lo 20 30 40 50 Gl 10 20 30 40 50 G2 10 20 30 40 50 G3 10 20 30 40 50 64 10 20 30 40 50 G5 10 20 30 40 50 G6 10 20 30 40 50 67 10 20 30 40 50 68 10 20 30 40 50 69 10 20 30 40 50 70 5000 5025 5050 5076 5101 5126 5152 5177 5203 5228 5254 5279 530g 5331 5356 53^2 5408 5434 5460 5486 5512 5538 5564 5590 5616 5642 5668 5695 5721 574^ 5774 5800 5826 5853 5879 5906 5932 5959 5986 6012 6039 6066 6092 61 19 6i46 6173 6200 6227 6254 6281 6308 6335 6362 6389 6416 6443 6476 6498 6525 6552_ 6580 Vers. 25 25 25 25 25 25 25 25 2S 2S 25 26 25 25 26 26 2l 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 27 26 26 27 26 27 27 26 27 27 27 27 27 27 27 27 2^ 27 27 27 27 2f d. Kxsec. ,0000 .0101 .0204 .030^ •0413 .0519 062§ .073S .0846 .0957 . 1076 ,1184 1300 I4I8 i53§ 1657 1902 2027 2153 2281 2411 2543 2676 28ii 2948 3087 3228 3371 351? 3662 3810 3961 4114 4269 4426 4586 4747 4912 5078 5247 5419 5593 5770 5949 613T 6316 6504 6694 6888 7085 .7285 ,7488 7694 7904 811^ 8334 8554 .8778 9006 9238 Kxsec. 01 02 03 05 06 07 09 16 II 13 14 16 18 26 21 23 25 26 28 30 31 33 35 37 39 40 43 44 46 48 51 52 55 Si 59 61 64 66 69 71 74 77 79 82 85 88 96 94 96 200 203 206 210 213 216 226 224 22^ 232 (I. 70 10 20 30 40 50 71 10 20 30 40 50 72 10 20 30 40 50 73 10 20 30 40 50 74 10 20 30 40 50 75 10 20 30 40 50 76 10 20 30 40 50 77 10 20 30 40 50 78 10 20 30 40 50 79 10 20 30 40 50 80 Vers. 6580^ 6607 6634 6662 6689 67j7_ ^744. 6772 6799 6827 6854 6882 6910 7104 7132 7160 7187 7215 7243 6271 7299 732^ 7355 7383 21 12 7440 7468 7496 7524 7552 7581 7609 763? 7665 7694 7722 775o_ 7779 7807 7835 7864 7892 7921 7949 7978 8005 8035 8063 8092 8126 8149 817^ 8206 8235 8263 Vers. 27 2^ 27 2? 27 27 2^ 2? 27 27 27 28 27 27 28 27 28 28 2? 28 28 2^ 28 28 28 28 28 28 28 28 28 28 28 2g 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 29 28 28 (1. Kxsec. 9238 •9473 •9713 •995^ .0205 • 0458 097 1 .1244 .1515 .1792 .2073 .2366 7265^ .2951 • 3255 • 3565 .3881 2.4203 •4531 .4867 .5209 •5558 J9i,5 .6279 .6651 .7031 .7420 .7816 . 8222 8637 .9061 .9495 .9939 .0394 .0859 1335 1824 2324 2836 3362 3901 4454 5021 5604 6202 6816 7448 809f 8765 9451 OI58 0885 j[636 2408 3205 4026 4874 5749 6653 758? Kxsec. 235 240 244 248 253 257 262 26^ 276 276 281 287 292 298 304 310 316 322 328 33^ 342 349 356 364 372 380 388 396 406 414 424 434 444 454 465 476 488 500 512 525 539 553 56? 582 598 614 631 649 667 686 707 728 749 772 796 821 847 ^7l 904 934 (I. I'. 1* 9 0.9 8 7 2.7 3-6 5-4 6.3 7.2 0.8 1.6 2.4 0.7 1.4 2. 1 3.2 2.8 4-0 3.5 4.8 4.2 5-6 6.4 7.2 4.9 5.6 1.8 2.4 3-0 3.6 4.2 4.8 •3 5-4 5 4 ).S 0.4 : .0 -5 1.2 2.0 1.6 2.5 2.0 302. 4 3.52.8 40 3.2 4.513.6 3 2 19 8 7 0-3 0.2 0.1 0.9 0-8 0. 0.6 0.4 0.2 1.9 1.7 I. 0.9 0.6 0.3 2-8 2.5 2. 1 .2 0.8 0.4 3.8 3-4 3- 1-5 I.O 0.5 4-7 4.2 3- i.fc 1.2 0.6 5-7 5-1 4- 2.1 2.4 1.4 1.6 0.7 0.8 6.6 7.6 5-9 6.8 5- 6. 2.7 1.8 0.9 8-5 7-6 6. 6 5 4 3 2 1 0.6 0.5 0.4 0.3 0.2 oi i.3|i.i 0.9 0.7 0.5 o 3 i.9|i.6 !•§ 10 0-7 0.4 2.6|2.2 1.8 1.4 1.00.6 3.2 2.7 2.21.7 '•2 0.7 3.9 3.3 2.7|2.1 l.S 0.9 4.5 3§ 3-1 2.4 1.7 1.0 5.2 4.4 3.62.^ 2.0 1.2 5-8 4'94-ol3-i 2.2 r.3 29 28 28 2f 2.9 5.8 8.7 II. 6 14.5 17.4 20.3 23.2 26. 1 2-8 5-7 8.5 II. 4 14.2 17.1 19.9 22.8 25-6 2.8 5.6 8.4 14.0 16.8 19.6 22.4 252 27 2g 26 2^ 2.7 2-6 2.6 2-5 5^4 .S • 3 5-2 51 8.1 7-9 7.8 7-6 10.8 10.6 10.4 10.2 »J 5 13-2 13.0 12.7 16.2 '5-9 15.6 »5-3 18.9 18.5 18 2 17-8 21.6 21.2 20.8 30.4 24.3 23-8 23.4 22.9 447 TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 80°-85" 85^-90° so lO 20 40 50 81 10 20 30 40 50 S2 10 20 30 40 50 83 10 20 3« 40 50 84 10 20 30 40 50 85 Vers. 8263 8292 8321 8349 8378 8407 8435 8464 8493 8522 8550 8579 8608 8637 8666 8694 8723 8752 8781 8810 8839 8868 8897 8926 8954 8983 9012 9041 9070 9099 9128 Vers. <]. 29 28 28 29 28 29 28 29 28 29 29 28 29 28 29 29 29 28 29 29 29 29 28 29 29 29 29 29 29 d. Exsec. (1. 4 7587 4-8554 4.9553 5.0588 5 . 1 666 5.2772 5-3924 5.5121 5-6363 5-7654 5.8998 6.0396 6.1853 6.3372 6.4957 6.6613 6.8344 7.0I56 7-2055 7 • 4046 7.6138 7.8336 8.0651 8.309T 8.5667 8.8391 9.1275 9-4334 9-7585 10. 1045 10-4737 Exsec. 966 999 035 072 III 152 196 242 291 343 398 456 519 585 656 731 812 898 991 2091 2198 2315 2440 2576 2723 2884 3059 3250 3466 3691 85 10 20 30 40 50 86 10 20 30 40 50 87 10 20 30 40 50 88 10 20 30 40 50 89 10 20 30 40 50 90 Vers. 9123 9157 9186 9215 9244 9273 9302 9331 9366 9389 94I8 9447 ?47i 9505 9534 9564 9593 9622 9651 9680 9709 9738 9767 9796 982S 9854 9883 9912 9942 9971 0000 Vers. (1. 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 d. Exsec. 0-4737 0.8683 I .2912 1-7455 2.2347 2.7631 d. 3-3356 3-9579 4.6368 5-3804 6. 1984 7. 1026 8.1073 9.2303 20.4937 21 .9256 23. 5621 25-4505 27.6537 30.2576 33.3823 37.2015 41-9757 48, 1 146 56.2987 67-7573 84.9456 113-5930 170.8883 342..7752 00 Exsec. .3946 .4229 .4542 .4892 .5284 •5725 .6223 -6789 -7436 .8180 .9041 I . 0047 I . 1230 I .2634 I. 4319 I .6365 1.8884 2.2032 2.6039 3.1247 3.8192 4.7741 6. 1383 8.1846 d. P. P. 29 29 28 2.9 5.9 8.8 11.8 14.7 17.7 20-6 23.6 26.5 2.9 5.8 8.7 II. 6 M-5 17.4 20.3 23.2 26.1 5-7 8-5 11.4 14.2 17.1 19.9 22.8 25.6 448 TABLE XL— USEFUL TRIGONOMETRICAL FORMUL.€. lo II sin a = I _ tan a / \ cosec a |/i + tan""^^ ~ — cos 2a Vi -\- cot'"^ a cos a tan a = Vi — cos"^ Ricketts and Russell's Notes on Inorganic Clieniistry (Non- metallic) Oblong 8vo, morocco, 75 Ruddimau's Incompatibilities in Prescriptions 8vo, 2 00 Scliimpfs Volumetric Analysis 12mo, 2 50 Spencer's Sugar Manufacturer's Handbook 16mo, morocco, 2 00- Handbook for Chemists of Beet Sugar Houses. 16mo, morocco, 3 00 Stockbridge's Rocks and Soils 8vo, 2 50 * Tillman's Descriptive General Chemistry 8vo, 3 00 Van Deventer's Physical Chemistry for Beginners. (Boltwood.) 12mo, 1 50 Wells's Inorganic Qualitative Analysis 12mo, 1 50 " Laboratory Guide in Qualitative Chemical Analysis. 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